Finite Group Databasebeta

← C2×C3×#2739 #10670781

Subgroups — C2×C3×#2739 #10670781

Order 768.

2070 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2739C2116yesmaximal, direct product factor
2C2×C3×#400C211yesmaximal, characteristic, complemented
3C2×C3×#400C211yesmaximal, characteristic, complemented
4C2×C3×#400C211yesmaximal, characteristic, complemented
5C2×C3×#398C211yesmaximal, characteristic, complemented
6C2×C3×#397C211yesmaximal, characteristic
7C2×C3×#396C211yesmaximal, characteristic
8C2×C3×#389C211yesmaximal, characteristic, complemented
9C2×C3×#388C211yesmaximal, characteristic
10C2×C3×#388C211yesmaximal, characteristic
11C2×C3×#387C211yesmaximal, characteristic, complemented
12C2×C3×#387C211yesmaximal, characteristic, complemented
13C2×C3×#387C211yesmaximal, characteristic, complemented
14C2×C3×#385C211yesmaximal, characteristic
15C2×C3×#381C211yesmaximal, characteristic
16C2×C3×#380C211yesmaximal, characteristic
17C2×#2739C311yesmaximal, characteristic, direct product factor, Sylow
18C24×C3×C4C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C22×C3×C4⋊C4C2211yescharacteristic
21C22×C3×C4⋊C4C2211yescharacteristic
22C22×C3×C4⋊C4C2211yescharacteristic
23C22×C3×C4⋊C4C2211yescharacteristic
24C22×C3×C4⋊C4C2211yescharacteristic
25C22×C3×C4⋊C4C2211yescharacteristic
26C22×C3×C4⋊C4C2211yescharacteristic
27C22×C3×C4⋊C4C2211yescharacteristic
28C22×C3×C4⋊C4C2211yescharacteristic
29C22×C3×C22⋊C4C2211yescharacteristic
30C22×C3×C22⋊C4C2211yescharacteristic
31C22×C3×C22⋊C4C2211yescharacteristic
32C22×C3×C22⋊C4C2211yescharacteristic
33C22×C3×C22⋊C4C2211yescharacteristic
34C22×C3×C22⋊C4C2211yescharacteristic
35C22×C3×C42C2211yescharacteristic
36C22×C3×C42C2211yescharacteristic
37C3×#400C2218yescomplemented
38C3×#400C2218yescomplemented
39C3×#400C2218yescomplemented
40C3×#398C2218yescomplemented
41C3×#397C2218yes—
42C3×#396C2218yes—
43C3×#389C2218yescomplemented
44C3×#388C2218yes—
45C3×#388C2218yes—
46C3×#387C2218yescomplemented
47C3×#387C2218yescomplemented
48C3×#387C2218yescomplemented
49C3×#385C2218yes—
50C3×#381C2218yes—
51C3×#380C2218yes—
52C2×C3×#95C2211yescharacteristic
53C2×C3×#95C2211yescharacteristic
54C2×C3×#95C2211yescharacteristic
55C2×C3×#95C2211yescharacteristic
56C2×C3×#95C2211yescharacteristic
57C2×C3×#95C2211yescharacteristic
58C2×C3×#95C2211yescharacteristic
59C2×C3×#95C2211yescharacteristic
60C2×C3×#95C2211yescharacteristic
61C2×C3×#95C2211yescharacteristic
62C2×C3×#95C2211yescharacteristic
63C2×C3×#95C2211yescharacteristic
64C2×C3×#95C2211yescharacteristic
65C2×C3×#95C2211yescharacteristic
66C2×C3×#95C2211yescharacteristic
67C2×C3×#95C2211yescharacteristic
68#2739C2×C3116yesdirect product factor
69C2×#400C2×C311yescharacteristic, complemented
70C2×#400C2×C311yescharacteristic, complemented
71C2×#400C2×C311yescharacteristic, complemented
72C2×#398C2×C311yescharacteristic, complemented
73C2×#397C2×C311yescharacteristic
74C2×#396C2×C311yescharacteristic
75C2×#389C2×C311yescharacteristic, complemented
76C2×#388C2×C311yescharacteristic
77C2×#388C2×C311yescharacteristic
78C2×#387C2×C311yescharacteristic, complemented
79C2×#387C2×C311yescharacteristic, complemented
80C2×#387C2×C311yescharacteristic, complemented
81C2×#385C2×C311yescharacteristic
82C2×#381C2×C311yescharacteristic
83C2×#380C2×C311yescharacteristic
84C25×C3C2311yescharacteristic
85C23×C3×C4—24no—
86C23×C3×C4—24no—
87C23×C3×C4—42no—
88C23×C3×C4C2311yescharacteristic
89C23×C3×C4C2311yescharacteristic
90C23×C3×C4C2311yescharacteristic
91C23×C3×C4C2311yescharacteristic
92C23×C3×C4C2311yescharacteristic
93C23×C3×C4C2311yescharacteristic
94C23×C3×C4C2311yescharacteristic
95C23×C3×C4C2311yescharacteristic
96C23×C3×C4C2311yescharacteristic
97C23×C3×C4C2311yescharacteristic
98C23×C3×C4C2311yescharacteristic
99C23×C3×C4C2311yescharacteristic
100C23×C3×C4C2314yes—
101C23×C3×C4C2311yescharacteristic
102C23×C3×C4C2311yescharacteristic
103C2×C3×C4⋊C4—24no—
104C2×C3×C4⋊C4—24no—
105C2×C3×C4⋊C4—24no—
106C2×C3×C4⋊C4—24no—
107C2×C3×C4⋊C4—42no—
108C2×C3×C4⋊C4—42no—
109C2×C3×C4⋊C4—42no—
110C2×C3×C4⋊C4—42no—
111C2×C3×C4⋊C4—42no—
112C2×C3×C4⋊C4—42no—
113C2×C3×C4⋊C4C2314yes—
114C2×C3×C4⋊C4C2314yes—
115C2×C3×C4⋊C4C2314yes—
116C2×C3×C4⋊C4C2314yes—
117C2×C3×C4⋊C4C2314yes—
118C2×C3×C4⋊C4C2314yes—
119C2×C3×C4⋊C4C2314yes—
120C2×C3×C4⋊C4C2314yes—
121C2×C3×C4⋊C4C2314yes—
122C2×C3×C4⋊C4C2314yes—
123C2×C3×C22⋊C4—24no—
124C2×C3×C22⋊C4—24no—
125C2×C3×C22⋊C4—24no—
126C2×C3×C22⋊C4—42no—
127C2×C3×C22⋊C4—42no—
128C2×C3×C22⋊C4—42no—
129C2×C3×C22⋊C4C2314yes—
130C2×C3×C22⋊C4C2314yes—
131C2×C3×C22⋊C4C2314yes—
132C2×C3×C22⋊C4C2314yes—
133C2×C3×C22⋊C4C2314yes—
134C2×C3×C22⋊C4C2314yes—
135C2×C3×C42—42no—
136C2×C3×C42—42no—
137C2×C3×C42C2314yes—
138C2×C3×C42C2314yes—
139C3×#95C2314yes—
140C3×#95C2314yes—
141C3×#95C2314yes—
142C3×#95C2314yes—
143C3×#95C2314yes—
144C3×#95C2314yes—
145C3×#95C2314yes—
146C3×#95C2314yes—
147C3×#95C2314yes—
148C3×#95C2314yes—
149C3×#95C2314yes—
150C3×#95C2314yes—
151C3×#95C2314yes—
152C3×#95C2314yes—
153C3×#95C2314yes—
154C3×#95C2314yes—
155C24×C4C22×C311yescharacteristic
156C22×C4⋊C4C22×C311yescharacteristic
157C22×C4⋊C4C22×C311yescharacteristic
158C22×C4⋊C4C22×C311yescharacteristic
159C22×C4⋊C4C22×C311yescharacteristic
160C22×C4⋊C4C22×C311yescharacteristic
161C22×C4⋊C4C22×C311yescharacteristic
162C22×C4⋊C4C22×C311yescharacteristic
163C22×C4⋊C4C22×C311yescharacteristic
164C22×C4⋊C4C22×C311yescharacteristic
165C22×C4⋊C4C22×C311yescharacteristic
166C22×C22⋊C4C22×C311yescharacteristic
167C22×C22⋊C4C22×C311yescharacteristic
168C22×C22⋊C4C22×C311yescharacteristic
169C22×C22⋊C4C22×C311yescharacteristic
170C22×C22⋊C4C22×C311yescharacteristic
171C22×C22⋊C4C22×C311yescharacteristic
172C22×C42C22×C311yescharacteristic
173C22×C42C22×C311yescharacteristic
174#400C22×C318yescomplemented
175#400C22×C318yescomplemented
176#400C22×C318yescomplemented
177#398C22×C318yescomplemented
178#397C22×C318yes—
179#396C22×C318yes—
180#389C22×C318yescomplemented
181#388C22×C318yes—
182#388C22×C318yes—
183#387C22×C318yescomplemented
184#387C22×C318yescomplemented
185#387C22×C318yescomplemented
186#385C22×C318yes—
187#381C22×C318yes—
188#380C22×C318yes—
189C2×#95C22×C311yescharacteristic
190C2×#95C22×C311yescharacteristic
191C2×#95C22×C311yescharacteristic
192C2×#95C22×C311yescharacteristic
193C2×#95C22×C311yescharacteristic
194C2×#95C22×C311yescharacteristic
195C2×#95C22×C311yescharacteristic
196C2×#95C22×C311yescharacteristic
197C2×#95C22×C311yescharacteristic
198C2×#95C22×C311yescharacteristic
199C2×#95C22×C311yescharacteristic
200C2×#95C22×C311yescharacteristic
201C2×#95C22×C311yescharacteristic
202C2×#95C22×C311yescharacteristic
203C2×#95C22×C311yescharacteristic
204C2×#95C22×C311yescharacteristic
205C24×C3—22no—
206C24×C3—22no—
207C24×C3—22no—
208C24×C3—22no—
209C24×C3—22no—
210C24×C3—22no—
211C24×C3#4114yes—
212C24×C3C2412yes—
213C24×C3C2411yescharacteristic
214C22×C3×C4—22no—
215C22×C3×C4—22no—
216C22×C3×C4—22no—
217C22×C3×C4—22no—
218C22×C3×C4—22no—
219C22×C3×C4—22no—
220C22×C3×C4—22no—
221C22×C3×C4—22no—
222C22×C3×C4—22no—
223C22×C3×C4—22no—
224C22×C3×C4—22no—
225C22×C3×C4—22no—
226C22×C3×C4—22no—
227C22×C3×C4—22no—
228C22×C3×C4—22no—
229C22×C3×C4—22no—
230C22×C3×C4—28no—
231C22×C3×C4—22no—
232C22×C3×C4—22no—
233C22×C3×C4—22no—
234C22×C3×C4—22no—
235C22×C3×C4—22no—
236C22×C3×C4—22no—
237C22×C3×C4—22no—
238C22×C3×C4—22no—
239C22×C3×C4—22no—
240C22×C3×C4—22no—
241C22×C3×C4—22no—
242C22×C3×C4—22no—
243C22×C3×C4—22no—
244C22×C3×C4—22no—
245C22×C3×C4—22no—
246C22×C3×C4—22no—
247C22×C3×C4—22no—
248C22×C3×C4—22no—
249C22×C3×C4—22no—
250C22×C3×C4—22no—
251C22×C3×C4—22no—
252C22×C3×C4—22no—
253C22×C3×C4—22no—
254C22×C3×C4—22no—
255C22×C3×C4—28no—
256C22×C3×C4—44no—
257C22×C3×C4—416no—
258C22×C3×C4#4114yes—
259C22×C3×C4#4114yes—
260C22×C3×C4C2412yes—
261C22×C3×C4C2412yes—
262C22×C3×C4C2412yes—
263C22×C3×C4C2412yes—
264C22×C3×C4C2412yes—
265C22×C3×C4C2412yes—
266C22×C3×C4C2412yes—
267C22×C3×C4C2412yes—
268C22×C3×C4C2412yes—
269C22×C3×C4C2412yes—
270C22×C3×C4C2412yes—
271C22×C3×C4C2412yes—
272C22×C3×C4C2412yes—
273C22×C3×C4C2412yes—
274C3×C4⋊C4—28no—
275C3×C4⋊C4—28no—
276C3×C4⋊C4—28no—
277C3×C4⋊C4—28no—
278C3×C4⋊C4—44no—
279C3×C4⋊C4—44no—
280C3×C4⋊C4—44no—
281C3×C4⋊C4—44no—
282C3×C4⋊C4—44no—
283C3×C4⋊C4—44no—
284C3×C22⋊C4—28no—
285C3×C22⋊C4—28no—
286C3×C22⋊C4—28no—
287C3×C22⋊C4—44no—
288C3×C22⋊C4—44no—
289C3×C22⋊C4—44no—
290C3×C42—44no—
291C3×C42—44no—
292C25C23×C311yescharacteristic
293C23×C4—24no—
294C23×C4—24no—
295C23×C4—42no—
296C23×C4C23×C311yescharacteristic
297C23×C4C23×C311yescharacteristic
298C23×C4C23×C311yescharacteristic
299C23×C4C23×C311yescharacteristic
300C23×C4C23×C311yescharacteristic
301C23×C4C23×C311yescharacteristic
302C23×C4C23×C311yescharacteristic
303C23×C4C23×C311yescharacteristic
304C23×C4C23×C311yescharacteristic
305C23×C4C23×C311yescharacteristic
306C23×C4C23×C311yescharacteristic
307C23×C4C23×C311yescharacteristic
308C23×C4C23×C314yes—
309C23×C4C23×C311yescharacteristic
310C23×C4C23×C311yescharacteristic
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—42no—
316C2×C4⋊C4—42no—
317C2×C4⋊C4—42no—
318C2×C4⋊C4—42no—
319C2×C4⋊C4—42no—
320C2×C4⋊C4—42no—
321C2×C4⋊C4C23×C314yes—
322C2×C4⋊C4C23×C314yes—
323C2×C4⋊C4C23×C314yes—
324C2×C4⋊C4C23×C314yes—
325C2×C4⋊C4C23×C314yes—
326C2×C4⋊C4C23×C314yes—
327C2×C4⋊C4C23×C314yes—
328C2×C4⋊C4C23×C314yes—
329C2×C4⋊C4C23×C314yes—
330C2×C4⋊C4C23×C314yes—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—42no—
335C2×C22⋊C4—42no—
336C2×C22⋊C4—42no—
337C2×C22⋊C4C23×C314yes—
338C2×C22⋊C4C23×C314yes—
339C2×C22⋊C4C23×C314yes—
340C2×C22⋊C4C23×C314yes—
341C2×C22⋊C4C23×C314yes—
342C2×C22⋊C4C23×C314yes—
343C2×C42—42no—
344C2×C42—42no—
345C2×C42C23×C314yes—
346C2×C42C23×C314yes—
347#95C23×C314yes—
348#95C23×C314yes—
349#95C23×C314yes—
350#95C23×C314yes—
351#95C23×C314yes—
352#95C23×C314yes—
353#95C23×C314yes—
354#95C23×C314yes—
355#95C23×C314yes—
356#95C23×C314yes—
357#95C23×C314yes—
358#95C23×C314yes—
359#95C23×C314yes—
360#95C23×C314yes—
361#95C23×C314yes—
362#95C23×C314yes—
363C23×C3—28no—
364C23×C3—28no—
365C23×C3—28no—
366C23×C3—22no—
367C23×C3—22no—
368C23×C3—22no—
369C23×C3—22no—
370C23×C3—22no—
371C23×C3—22no—
372C23×C3—44no—
373C23×C3—44no—
374C23×C3—44no—
375C23×C3—44no—
376C23×C3C2×#4114yes—
377C23×C3C2×#4112yes—
378C23×C3C2×#4112yes—
379C23×C3C2×#4112yes—
380C23×C3H(5,𝔽2)12yes—
381C23×C3H(5,𝔽2)12yes—
382C23×C32−1+412yes—
383C23×C32−1+412yes—
384C23×C3C2511yescharacteristic
385C2×C3×C4—28no—
386C2×C3×C4—28no—
387C2×C3×C4—22no—
388C2×C3×C4—22no—
389C2×C3×C4—22no—
390C2×C3×C4—22no—
391C2×C3×C4—22no—
392C2×C3×C4—22no—
393C2×C3×C4—22no—
394C2×C3×C4—22no—
395C2×C3×C4—22no—
396C2×C3×C4—22no—
397C2×C3×C4—22no—
398C2×C3×C4—22no—
399C2×C3×C4—22no—
400C2×C3×C4—22no—
401C2×C3×C4—22no—
402C2×C3×C4—22no—
403C2×C3×C4—22no—
404C2×C3×C4—22no—
405C2×C3×C4—22no—
406C2×C3×C4—22no—
407C2×C3×C4—22no—
408C2×C3×C4—22no—
409C2×C3×C4—22no—
410C2×C3×C4—22no—
411C2×C3×C4—22no—
412C2×C3×C4—22no—
413C2×C3×C4—22no—
414C2×C3×C4—22no—
415C2×C3×C4—22no—
416C2×C3×C4—22no—
417C2×C3×C4—22no—
418C2×C3×C4—22no—
419C2×C3×C4—22no—
420C2×C3×C4—22no—
421C2×C3×C4—22no—
422C2×C3×C4—22no—
423C2×C3×C4—22no—
424C2×C3×C4—22no—
425C2×C3×C4—22no—
426C2×C3×C4—22no—
427C2×C3×C4—416no—
428C2×C3×C4—44no—
429C2×C3×C4—44no—
430C2×C3×C4—44no—
431C2×C3×C4—44no—
432C2×C3×C4—44no—
433C2×C3×C4—44no—
434C2×C3×C4—44no—
435C2×C3×C4—44no—
436C2×C3×C4—44no—
437C2×C3×C4—44no—
438C2×C3×C4—44no—
439C2×C3×C4—44no—
440C2×C3×C4C2×#4114yes—
441C2×C3×C4C2×#4114yes—
442C24—22no—
443C24—22no—
444C24—22no—
445C24—22no—
446C24—22no—
447C24—22no—
448C24C3×#4114yes—
449C24C24×C312yes—
450C24C24×C311yescharacteristic
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—28no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—28no—
493C22×C4—44no—
494C22×C4—416no—
495C22×C4C3×#4114yes—
496C22×C4C3×#4114yes—
497C22×C4C24×C312yes—
498C22×C4C24×C312yes—
499C22×C4C24×C312yes—
500C22×C4C24×C312yes—
501C22×C4C24×C312yes—
502C22×C4C24×C312yes—
503C22×C4C24×C312yes—
504C22×C4C24×C312yes—
505C22×C4C24×C312yes—
506C22×C4C24×C312yes—
507C22×C4C24×C312yes—
508C22×C4C24×C312yes—
509C22×C4C24×C312yes—
510C22×C4C24×C312yes—
511C4⋊C4—28no—
512C4⋊C4—28no—
513C4⋊C4—28no—
514C4⋊C4—28no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C22⋊C4—28no—
522C22⋊C4—28no—
523C22⋊C4—28no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C42—44no—
528C42—44no—
529C22×C3—24no—
530C22×C3—24no—
531C22×C3—24no—
532C22×C3—42no—
533C22×C3—42no—
534C22×C3—42no—
535C22×C3—416no—
536C22×C3—42no—
537C22×C3#53914yes—
538C22×C3#54114yes—
539C22×C3#54214yes—
540C22×C3#55114yes—
541C22×C3#55214yes—
542C22×C3#55314yes—
543C22×C3#56214yes—
544C22×C3C22×#4111yescharacteristic
545C22×C3C22×#4111yescharacteristic
546C22×C3C22×#4111yescharacteristic
547C22×C3C2×H(5,𝔽2)11yescharacteristic
548C22×C3C2×H(5,𝔽2)11yescharacteristic
549C22×C3C2×2−1+411yescharacteristic
550C22×C3C2×2−1+411yescharacteristic
551C3×C4—24no—
552C3×C4—24no—
553C3×C4—42no—
554C3×C4—42no—
555C3×C4—42no—
556C3×C4—42no—
557C3×C4—42no—
558C3×C4—42no—
559C3×C4—42no—
560C3×C4—42no—
561C3×C4—42no—
562C3×C4—42no—
563C3×C4—42no—
564C3×C4—42no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—28no—
572C23—28no—
573C23—28no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23C2×C3×#4114yes—
579C23C2×C3×#4112yes—
580C23C2×C3×#4112yes—
581C23C2×C3×#4112yes—
582C23C3×H(5,𝔽2)12yes—
583C23C3×H(5,𝔽2)12yes—
584C23C3×2−1+412yes—
585C23C3×2−1+412yes—
586C23C25×C311yescharacteristic
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—416no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4C2×C3×#4114yes—
643C2×C4C2×C3×#4114yes—
644C2×C3—44no—
645C2×C3#273918yesdirect product factor
646C2×C3C2×#53911yescharacteristic
647C2×C3C2×#54111yescharacteristic
648C2×C3C2×#54211yescharacteristic
649C2×C3C2×#55111yescharacteristic
650C2×C3C2×#55211yescharacteristic
651C2×C3C2×#55311yescharacteristic
652C2×C3C2×#56211yescharacteristic
653C22—24no—
654C22—24no—
655C22—24no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—416no—
661C22C3×#53914yes—
662C22C3×#54114yes—
663C22C3×#54214yes—
664C22C3×#55114yes—
665C22C3×#55214yes—
666C22C3×#55314yes—
667C22C3×#56214yes—
668C22C22×C3×#4111yescharacteristic
669C22C22×C3×#4111yescharacteristic
670C22C22×C3×#4111yescharacteristic
671C22C2×C3×H(5,𝔽2)11yescharacteristic
672C22C2×C3×H(5,𝔽2)11yescharacteristic
673C22C2×C3×2−1+411yescharacteristic
674C22C2×C3×2−1+411yescharacteristic
675C4—24no—
676C4—24no—
677C4—42no—
678C4—42no—
679C4—42no—
680C4—42no—
681C4—42no—
682C4—42no—
683C4—42no—
684C4—42no—
685C4—42no—
686C4—42no—
687C4—42no—
688C4—42no—
689C3C2×#273911yescharacteristic, direct product factor, Sylow
690C2—44no—
691C2C3×#273918yesdirect product factor
692C2C2×C3×#53911yescharacteristic
693C2C2×C3×#54111yescharacteristic
694C2C2×C3×#54211yescharacteristic
695C2C2×C3×#55111yescharacteristic
696C2C2×C3×#55211yescharacteristic
697C2C2×C3×#55311yescharacteristic
698C2C2×C3×#56211yescharacteristic
subgroupG/H
center, socleC24×C3C24
commutator, Frattini, cosocleC23C25×C3
prime radicalC25×C3C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C3×#2739C2direct (N × H)
2. C2×C3×#400C2semidirect
3. C2×C3×#400C2semidirect
4. C2×C3×#400C2semidirect
5. C2×C3×#398C2semidirect
8. C2×C3×#389C2semidirect
11. C2×C3×#387C2semidirect
12. C2×C3×#387C2semidirect
13. C2×C3×#387C2semidirect
17. C2×#2739C3direct (N × H)
37. C3×#400C22semidirect
38. C3×#400C22semidirect
39. C3×#400C22semidirect
40. C3×#398C22semidirect
43. C3×#389C22semidirect
46. C3×#387C22semidirect
47. C3×#387C22semidirect
48. C3×#387C22semidirect
68. #2739C2×C3direct (N × H)
69. C2×#400C2×C3semidirect
70. C2×#400C2×C3semidirect
71. C2×#400C2×C3semidirect
72. C2×#398C2×C3semidirect
75. C2×#389C2×C3semidirect
78. C2×#387C2×C3semidirect
79. C2×#387C2×C3semidirect
80. C2×#387C2×C3semidirect
174. #400C22×C3semidirect
175. #400C22×C3semidirect
176. #400C22×C3semidirect
177. #398C22×C3semidirect
180. #389C22×C3semidirect
183. #387C22×C3semidirect
184. #387C22×C3semidirect
185. #387C22×C3semidirect
645. C2×C3#2739direct (N × H)
689. C3C2×#2739direct (N × H)
691. C2C3×#2739direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C3×#27392
C2×C3×#4002
C2×C3×#3982
C2×C3×#3972
C2×C3×#3962
C2×C3×#3892
C2×C3×#3882
C2×C3×#3872
C2×C3×#3852
C2×C3×#3812
C2×C3×#3802
C2×#27393