Finite Group Databasebeta

← C3×#27700 #10646307

Subgroups — C3×#27700 #10646307

Order 768.

702 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2499C211yesmaximal, characteristic
2C3×#2441C211yesmaximal, characteristic
3C3×#2101C211yesmaximal, characteristic, complemented
4C3×#2099C211yesmaximal, characteristic, complemented
5C3×#2072C211yesmaximal, characteristic, complemented
6C3×#2026C211yesmaximal, characteristic, complemented
7C3×#1991C211yesmaximal, characteristic, complemented
8C3×#1933C211yesmaximal, characteristic, complemented
9C3×#1933C211yesmaximal, characteristic, complemented
10C3×#1892C211yesmaximal, characteristic
11C3×#1873C211yesmaximal, characteristic
12C3×#1834C211yesmaximal, characteristic, complemented
13C3×#1825C211yesmaximal, characteristic
14C3×C8×C22⋊C4C211yesmaximal, characteristic
15C3×#1735C211yesmaximal, characteristic
16#27700C311yesmaximal, characteristic, direct product factor, Sylow
17C23×C3×C8C2211yescharacteristic
18C2×C3×#125C2211yescharacteristic
19C2×C3×C22⋊Q8C2211yescharacteristic
20C2×C3×#117C2211yescharacteristic
21C2×C3×#117C2211yescharacteristic
22C2×C3×#103C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×C22⋊C8C2211yescharacteristic
27C2×C3×C4×C8C2211yescharacteristic
28C2×C3×C4×C8C2211yescharacteristic
29C3×#399—21no—
30C3×#397—21no—
31C3×#397—21no—
32C3×#396—21no—
33C3×#391C2211yescharacteristic
34C3×#391C2211yescharacteristic
35C3×#389—21no—
36C3×#389C2211yescharacteristic
37C3×#389C2211yescharacteristic
38C3×#387—21no—
39C3×#387—21no—
40C3×#387—21no—
41C3×#384C2211yescharacteristic
42C3×#384C2211yescharacteristic
43C3×#382C2211yescharacteristic
44C3×#380C2211yescharacteristic
45C3×C4×C22⋊C4C2211yescharacteristic
46C3×#340C2211yescharacteristic
47C3×#340C2211yescharacteristic
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58C3×#336C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#2499C2×C311yescharacteristic
61#2441C2×C311yescharacteristic
62#2101C2×C311yescharacteristic, complemented
63#2099C2×C311yescharacteristic, complemented
64#2072C2×C311yescharacteristic, complemented
65#2026C2×C311yescharacteristic, complemented
66#1991C2×C311yescharacteristic, complemented
67#1933C2×C311yescharacteristic, complemented
68#1933C2×C311yescharacteristic, complemented
69#1892C2×C311yescharacteristic
70#1873C2×C311yescharacteristic
71#1834C2×C311yescharacteristic, complemented
72#1825C2×C311yescharacteristic
73C8×C22⋊C4C2×C311yescharacteristic
74#1735C2×C311yescharacteristic
75C22×C3×Q8C2311yescharacteristic
76C23×C3×C4C2311yescharacteristic
77C22×C3×C8—22no—
78C22×C3×C8—22no—
79C22×C3×C8—41no—
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C3×#125—22no—
85C3×#125—41no—
86C3×C22⋊Q8—22no—
87C3×C22⋊Q8—22no—
88C3×#117—22no—
89C3×#117—22no—
90C3×#117—22no—
91C3×#117—41no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C4⋊C4C2311yescharacteristic
104C2×C3×C4⋊C4C2311yescharacteristic
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C4⋊C4C2311yescharacteristic
108C2×C3×C22⋊C4—21no—
109C2×C3×C22⋊C4—21no—
110C2×C3×C22⋊C4—21no—
111C2×C3×C22⋊C4—21no—
112C2×C3×C22⋊C4D411yescharacteristic
113C2×C3×C22⋊C4D411yescharacteristic
114C2×C3×C42—21no—
115C2×C3×C42—21no—
116C2×C3×C42C2311yescharacteristic
117C2×C3×C42C2311yescharacteristic
118C3×#103—22no—
119C3×#103—22no—
120C3×#103—41no—
121C3×#103—41no—
122C3×C22⋊C8—22no—
123C3×C4×C8—22no—
124C3×C4×C8—41no—
125C3×#95—21no—
126C3×#95—21no—
127C3×#95—21no—
128C3×#95—21no—
129C3×#95—21no—
130C3×#95—21no—
131C3×#95—21no—
132C3×#95—21no—
133C3×#95—21no—
134C3×#95D411yescharacteristic
135C3×#95D411yescharacteristic
136C23×C8C22×C311yescharacteristic
137C2×#125C22×C311yescharacteristic
138C2×C22⋊Q8C22×C311yescharacteristic
139C2×#117C22×C311yescharacteristic
140C2×#117C22×C311yescharacteristic
141C2×#103C22×C311yescharacteristic
142C2×#103C22×C311yescharacteristic
143C2×#103C22×C311yescharacteristic
144C2×#103C22×C311yescharacteristic
145C2×C22⋊C8C22×C311yescharacteristic
146C2×C4×C8C22×C311yescharacteristic
147C2×C4×C8C22×C311yescharacteristic
148#399—21no—
149#397—21no—
150#397—21no—
151#396—21no—
152#391C22×C311yescharacteristic
153#391C22×C311yescharacteristic
154#389—21no—
155#389C22×C311yescharacteristic
156#389C22×C311yescharacteristic
157#387—21no—
158#387—21no—
159#387—21no—
160#384C22×C311yescharacteristic
161#384C22×C311yescharacteristic
162#382C22×C311yescharacteristic
163#380C22×C311yescharacteristic
164C4×C22⋊C4C22×C311yescharacteristic
165#340C22×C311yescharacteristic
166#340C22×C311yescharacteristic
167#340C22×C311yescharacteristic
168#340C22×C311yescharacteristic
169#340C22×C311yescharacteristic
170#340C22×C311yescharacteristic
171#340C22×C311yescharacteristic
172#340C22×C311yescharacteristic
173#340C22×C311yescharacteristic
174#340C22×C311yescharacteristic
175#340C22×C311yescharacteristic
176#340C22×C311yescharacteristic
177#336C22×C311yescharacteristic
178#336C22×C311yescharacteristic
179C24×C3C2×D411yescharacteristic
180C2×C3×Q8—21no—
181C2×C3×Q8—21no—
182C2×C3×Q8—21no—
183C2×C3×Q8—21no—
184C2×C3×Q8—21no—
185C2×C3×Q8—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4#4112yes—
204C22×C3×C4#4112yes—
205C22×C3×C4C2411yescharacteristic
206C2×C3×C8—21no—
207C2×C3×C8—21no—
208C2×C3×C8—21no—
209C2×C3×C8—21no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C2×C3×C8—21no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8—44no—
218C2×C3×C8#4112yes—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—42no—
233C3×C4⋊C4—42no—
234C3×C4⋊C4—42no—
235C3×C4⋊C4—42no—
236C3×C4⋊C4—42no—
237C3×C4⋊C4—42no—
238C3×C4⋊C4—42no—
239C3×C4⋊C4—81no—
240C3×C4⋊C4—81no—
241C3×C4⋊C4#4112yes—
242C3×C22⋊C4—22no—
243C3×C22⋊C4—22no—
244C3×C22⋊C4—42no—
245C3×C22⋊C4—42no—
246C3×C22⋊C4—42no—
247C3×C22⋊C4—81no—
248C3×C42—21no—
249C3×C42—21no—
250C3×C42—21no—
251C3×C42—21no—
252C3×C42—42no—
253C3×C42—42no—
254C22×Q8C23×C311yescharacteristic
255C23×C4C23×C311yescharacteristic
256C22×C8—22no—
257C22×C8—22no—
258C22×C8—41no—
259C22×C8C23×C311yescharacteristic
260C22×C8C23×C311yescharacteristic
261C22×C8C23×C311yescharacteristic
262C22×C8C23×C311yescharacteristic
263#125—22no—
264#125—41no—
265C22⋊Q8—22no—
266C22⋊Q8—22no—
267#117—22no—
268#117—22no—
269#117—22no—
270#117—41no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4C23×C311yescharacteristic
281C2×C4⋊C4C23×C311yescharacteristic
282C2×C4⋊C4C23×C311yescharacteristic
283C2×C4⋊C4C23×C311yescharacteristic
284C2×C4⋊C4C23×C311yescharacteristic
285C2×C4⋊C4C23×C311yescharacteristic
286C2×C4⋊C4C23×C311yescharacteristic
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4C3×D411yescharacteristic
292C2×C22⋊C4C3×D411yescharacteristic
293C2×C42—21no—
294C2×C42—21no—
295C2×C42C23×C311yescharacteristic
296C2×C42C23×C311yescharacteristic
297#103—22no—
298#103—22no—
299#103—41no—
300#103—41no—
301C22⋊C8—22no—
302C4×C8—22no—
303C4×C8—41no—
304#95—21no—
305#95—21no—
306#95—21no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—21no—
312#95—21no—
313#95C3×D411yescharacteristic
314#95C3×D411yescharacteristic
315C23×C3—21no—
316C23×C3—21no—
317C23×C3—21no—
318C23×C3—21no—
319C23×C3—21no—
320C23×C3—21no—
321C23×C3#12412yes—
322C23×C3C22×D411yescharacteristic
323C3×Q8—41no—
324C3×Q8—41no—
325C3×Q8—41no—
326C3×Q8—41no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—22no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4#12412yes—
368C2×C3×C4#13512yes—
369C2×C3×C4#13512yes—
370C2×C3×C4C2×#4111yescharacteristic
371C2×C3×C4C2×#4111yescharacteristic
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4H(5,𝔽2)11yescharacteristic
375C2×C3×C42−1+411yescharacteristic
376C3×C8—22no—
377C3×C8—41no—
378C3×C8—41no—
379C3×C8—41no—
380C24C2×C3×D411yescharacteristic
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4C2×C3×D411yescharacteristic
400C22×C4C2×C3×D411yescharacteristic
401C22×C4C2×C3×D411yescharacteristic
402C22×C4C2×C3×D411yescharacteristic
403C22×C4C2×C3×D411yescharacteristic
404C22×C4C3×#4112yes—
405C22×C4C3×#4112yes—
406C22×C4C24×C311yescharacteristic
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—44no—
419C2×C8C3×#4112yes—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—81no—
441C4⋊C4—81no—
442C4⋊C4C3×#4112yes—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—42no—
446C22⋊C4—42no—
447C22⋊C4—42no—
448C22⋊C4—81no—
449C42—21no—
450C42—21no—
451C42—21no—
452C42—21no—
453C42—42no—
454C42—42no—
455C22×C3—22no—
456C22×C3—22no—
457C22×C3—22no—
458C22×C3—41no—
459C22×C3—41no—
460C22×C3—41no—
461C22×C3—41no—
462C22×C3C2×#12411yescharacteristic
463C22×C3#54611yescharacteristic
464C22×C3#54711yescharacteristic
465C22×C3C2×#13511yescharacteristic
466C22×C3C2×#13511yescharacteristic
467C22×C3#57711yescharacteristic
468C22×C3#57811yescharacteristic
469C3×C4—22no—
470C3×C4—22no—
471C3×C4—22no—
472C3×C4—41no—
473C3×C4—42no—
474C3×C4—41no—
475C3×C4—81no—
476C3×C4—81no—
477C3×C4—81no—
478C3×C4—81no—
479C3×C4—81no—
480C3×C4—81no—
481C3×C4—81no—
482C3×C4#52911yescharacteristic
483C3×C4#53911yescharacteristic
484C3×C4#55111yescharacteristic
485C3×C4#55211yescharacteristic
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23C3×#12412yes—
493C23C22×C3×D411yescharacteristic
494Q8—41no—
495Q8—41no—
496Q8—41no—
497Q8—41no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4C3×#12412yes—
539C2×C4C3×#13512yes—
540C2×C4C3×#13512yes—
541C2×C4C2×C3×#4111yescharacteristic
542C2×C4C2×C3×#4111yescharacteristic
543C2×C4C2×C3×#4111yescharacteristic
544C2×C4C2×C3×#4111yescharacteristic
545C2×C4C3×H(5,𝔽2)11yescharacteristic
546C2×C4C3×2−1+411yescharacteristic
547C8—22no—
548C8—41no—
549C8—41no—
550C8—41no—
551C2×C3—42no—
552C2×C3#248811yescharacteristic
553C2×C3#313111yescharacteristic
554C2×C3#313711yescharacteristic
555C2×C3#330211yescharacteristic
556C2×C3#330411yescharacteristic
557C2×C3#333611yescharacteristic
558C2×C3#333711yescharacteristic
559C22—22no—
560C22—22no—
561C22—22no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22C2×C3×#12411yescharacteristic
567C22C3×#54611yescharacteristic
568C22C3×#54711yescharacteristic
569C22C2×C3×#13511yescharacteristic
570C22C2×C3×#13511yescharacteristic
571C22C3×#57711yescharacteristic
572C22C3×#57811yescharacteristic
573C4—22no—
574C4—22no—
575C4—22no—
576C4—41no—
577C4—42no—
578C4—41no—
579C4—81no—
580C4—81no—
581C4—81no—
582C4—81no—
583C4—81no—
584C4—81no—
585C4—81no—
586C4C3×#52911yescharacteristic
587C4C3×#53911yescharacteristic
588C4C3×#55111yescharacteristic
589C4C3×#55211yescharacteristic
590C3#2770011yescharacteristic, direct product factor, Sylow
591C2—42no—
592C2C3×#248811yescharacteristic
593C2C3×#313111yescharacteristic
594C2C3×#313711yescharacteristic
595C2C3×#330211yescharacteristic
596C2C3×#330411yescharacteristic
597C2C3×#333611yescharacteristic
598C2C3×#333711yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C4×C22⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
3. C3×#2101C2semidirect
4. C3×#2099C2semidirect
5. C3×#2072C2semidirect
6. C3×#2026C2semidirect
7. C3×#1991C2semidirect
8. C3×#1933C2semidirect
9. C3×#1933C2semidirect
12. C3×#1834C2semidirect
16. #27700C3direct (N × H)
62. #2101C2×C3semidirect
63. #2099C2×C3semidirect
64. #2072C2×C3semidirect
65. #2026C2×C3semidirect
66. #1991C2×C3semidirect
67. #1933C2×C3semidirect
68. #1933C2×C3semidirect
71. #1834C2×C3semidirect
590. C3#27700direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#24992
C3×#24412
C3×#21012
C3×#20992
C3×#20722
C3×#20262
C3×#19912
C3×#19332
C3×#18922
C3×#18732
C3×#18342
C3×#18252
C3×C8×C22⋊C42
C3×#17352
#277003