Finite Group Databasebeta

← C3×#26417 #10645024

Subgroups — C3×#26417 #10645024

Order 768.

738 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2441C211yesmaximal, characteristic
2C3×#2362C211yesmaximal, characteristic
3C3×#2100C211yesmaximal, characteristic, complemented
4C3×#2086C211yesmaximal, characteristic, complemented
5C3×#2068C211yesmaximal, characteristic, complemented
6C3×#2026C211yesmaximal, characteristic, complemented
7C3×#1989C211yesmaximal, characteristic, complemented
8C3×#1919C211yesmaximal, characteristic, complemented
9C3×#1892C211yesmaximal, characteristic
10C3×#1873C211yesmaximal, characteristic
11C3×#1865C211yesmaximal, characteristic, complemented
12C3×#1854C211yesmaximal, characteristic
13C3×#1850C211yesmaximal, characteristic
14C3×#1803C211yesmaximal, characteristic, complemented
15C3×#1735C211yesmaximal, characteristic
16#26417C311yesmaximal, characteristic, direct product factor, Sylow
17C23×C3×C8C2211yescharacteristic
18C2×C3×#125C2211yescharacteristic
19C2×C3×C22⋊Q8C2211yescharacteristic
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×#117C2211yescharacteristic
22C2×C3×#117C2211yescharacteristic
23C2×C3×#107C2211yescharacteristic
24C2×C3×#106C2211yescharacteristic
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#103C2211yescharacteristic
28C2×C3×#103C2211yescharacteristic
29C2×C3×#103C2211yescharacteristic
30C2×C3×#103C2211yescharacteristic
31C2×C3×C22⋊C8C2211yescharacteristic
32C3×#399—21no—
33C3×#396—21no—
34C3×#391C2211yescharacteristic
35C3×#389—21no—
36C3×#389C2211yescharacteristic
37C3×#389C2211yescharacteristic
38C3×#389C2211yescharacteristic
39C3×#388—21no—
40C3×#388—21no—
41C3×#387—21no—
42C3×#387—21no—
43C3×#384C2211yescharacteristic
44C3×#380C2211yescharacteristic
45C3×C4×C4⋊C4—21no—
46C3×C4×C4⋊C4C2211yescharacteristic
47C3×C4×C22⋊C4C2211yescharacteristic
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#2441C2×C311yescharacteristic
61#2362C2×C311yescharacteristic
62#2100C2×C311yescharacteristic, complemented
63#2086C2×C311yescharacteristic, complemented
64#2068C2×C311yescharacteristic, complemented
65#2026C2×C311yescharacteristic, complemented
66#1989C2×C311yescharacteristic, complemented
67#1919C2×C311yescharacteristic, complemented
68#1892C2×C311yescharacteristic
69#1873C2×C311yescharacteristic
70#1865C2×C311yescharacteristic, complemented
71#1854C2×C311yescharacteristic
72#1850C2×C311yescharacteristic
73#1803C2×C311yescharacteristic, complemented
74#1735C2×C311yescharacteristic
75C22×C3×Q8C2311yescharacteristic
76C23×C3×C4C2311yescharacteristic
77C22×C3×C8—22no—
78C22×C3×C8—41no—
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⋊Q8Q814yes—
88C3×C4×Q8—22no—
89C3×C4×Q8—41no—
90C3×#117—22no—
91C3×#117—22no—
92C3×#117—41no—
93C3×#117—41no—
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⋊C4—21no—
102C2×C3×C4⋊C4—21no—
103C2×C3×C4⋊C4C2311yescharacteristic
104C2×C3×C4⋊C4C2311yescharacteristic
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C4⋊C4C2311yescharacteristic
108C2×C3×C4⋊C4C2311yescharacteristic
109C2×C3×C4⋊C4C2311yescharacteristic
110C2×C3×C22⋊C4—21no—
111C2×C3×C22⋊C4—21no—
112C2×C3×C22⋊C4—21no—
113C2×C3×C22⋊C4—21no—
114C2×C3×C22⋊C4D411yescharacteristic
115C2×C3×C22⋊C4D411yescharacteristic
116C2×C3×C42—21no—
117C2×C3×C42—21no—
118C2×C3×C42—21no—
119C2×C3×C42—21no—
120C2×C3×C42C2311yescharacteristic
121C2×C3×C42C2311yescharacteristic
122C3×#107—41no—
123C3×#106—22no—
124C3×C4⋊C8—41no—
125C3×C4⋊C8—41no—
126C3×#103—22no—
127C3×#103—41no—
128C3×#103—41no—
129C3×#103—41no—
130C3×C22⋊C8—22no—
131C3×#95—21no—
132C3×#95—21no—
133C3×#95—21no—
134C3×#95—21no—
135C3×#95—21no—
136C3×#95—21no—
137C3×#95—21no—
138C3×#95D411yescharacteristic
139C3×#95D411yescharacteristic
140C23×C8C22×C311yescharacteristic
141C2×#125C22×C311yescharacteristic
142C2×C22⋊Q8C22×C311yescharacteristic
143C2×C4×Q8C22×C311yescharacteristic
144C2×#117C22×C311yescharacteristic
145C2×#117C22×C311yescharacteristic
146C2×#107C22×C311yescharacteristic
147C2×#106C22×C311yescharacteristic
148C2×C4⋊C8C22×C311yescharacteristic
149C2×C4⋊C8C22×C311yescharacteristic
150C2×#103C22×C311yescharacteristic
151C2×#103C22×C311yescharacteristic
152C2×#103C22×C311yescharacteristic
153C2×#103C22×C311yescharacteristic
154C2×C22⋊C8C22×C311yescharacteristic
155#399—21no—
156#396—21no—
157#391C22×C311yescharacteristic
158#389—21no—
159#389C22×C311yescharacteristic
160#389C22×C311yescharacteristic
161#389C22×C311yescharacteristic
162#388—21no—
163#388—21no—
164#387—21no—
165#387—21no—
166#384C22×C311yescharacteristic
167#380C22×C311yescharacteristic
168C4×C4⋊C4—21no—
169C4×C4⋊C4C22×C311yescharacteristic
170C4×C22⋊C4C22×C311yescharacteristic
171#340C22×C311yescharacteristic
172#340C22×C311yescharacteristic
173#340C22×C311yescharacteristic
174#340C22×C311yescharacteristic
175#340C22×C311yescharacteristic
176#340C22×C311yescharacteristic
177#340C22×C311yescharacteristic
178#340C22×C311yescharacteristic
179#340C22×C311yescharacteristic
180#340C22×C311yescharacteristic
181#336C22×C311yescharacteristic
182#336C22×C311yescharacteristic
183C24×C3C2×D411yescharacteristic
184C2×C3×Q8—21no—
185C2×C3×Q8—21no—
186C2×C3×Q8—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8C2×Q812yes—
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×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×Q812yes—
208C22×C3×C4#4112yes—
209C22×C3×C4C2411yescharacteristic
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—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—44no—
223C3×C4⋊C4—24no—
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—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—42no—
236C3×C4⋊C4—42no—
237C3×C4⋊C4—42no—
238C3×C4⋊C4—42no—
239C3×C4⋊C4—42no—
240C3×C4⋊C4—42no—
241C3×C4⋊C4—81no—
242C3×C4⋊C4—81no—
243C3×C4⋊C4C2×Q812yes—
244C3×C4⋊C4#4112yes—
245C3×C4⋊C4#4112yes—
246C3×C22⋊C4—24no—
247C3×C22⋊C4—22no—
248C3×C22⋊C4—41no—
249C3×C22⋊C4—42no—
250C3×C22⋊C4—42no—
251C3×C22⋊C4—81no—
252C3×C42—21no—
253C3×C42—21no—
254C3×C42—21no—
255C3×C42—21no—
256C3×C42—42no—
257C3×C42—42no—
258C3×C42—42no—
259C3×C42—42no—
260C22×Q8C23×C311yescharacteristic
261C23×C4C23×C311yescharacteristic
262C22×C8—22no—
263C22×C8—41no—
264C22×C8—41no—
265C22×C8C23×C311yescharacteristic
266C22×C8C23×C311yescharacteristic
267C22×C8C23×C311yescharacteristic
268C22×C8C23×C311yescharacteristic
269#125—22no—
270#125—41no—
271C22⋊Q8—22no—
272C22⋊Q8C3×Q814yes—
273C4×Q8—22no—
274C4×Q8—41no—
275#117—22no—
276#117—22no—
277#117—41no—
278#117—41no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4C23×C311yescharacteristic
289C2×C4⋊C4C23×C311yescharacteristic
290C2×C4⋊C4C23×C311yescharacteristic
291C2×C4⋊C4C23×C311yescharacteristic
292C2×C4⋊C4C23×C311yescharacteristic
293C2×C4⋊C4C23×C311yescharacteristic
294C2×C4⋊C4C23×C311yescharacteristic
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4C3×D411yescharacteristic
300C2×C22⋊C4C3×D411yescharacteristic
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42C23×C311yescharacteristic
306C2×C42C23×C311yescharacteristic
307#107—41no—
308#106—22no—
309C4⋊C8—41no—
310C4⋊C8—41no—
311#103—22no—
312#103—41no—
313#103—41no—
314#103—41no—
315C22⋊C8—22no—
316#95—21no—
317#95—21no—
318#95—21no—
319#95—21no—
320#95—21no—
321#95—21no—
322#95—21no—
323#95C3×D411yescharacteristic
324#95C3×D411yescharacteristic
325C23×C3—21no—
326C23×C3—21no—
327C23×C3—21no—
328C23×C3—21no—
329C23×C3—21no—
330C23×C3—21no—
331C23×C3C22⋊Q812yes—
332C23×C3C22×D411yescharacteristic
333C3×Q8—22no—
334C3×Q8—22no—
335C3×Q8—41no—
336C3×Q8—41no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
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—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4C22⋊Q812yes—
379C2×C3×C4#13512yes—
380C2×C3×C4C22×Q811yescharacteristic
381C2×C3×C4C2×#4111yescharacteristic
382C2×C3×C4C2×#4111yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C2×C3×C4H(5,𝔽2)11yescharacteristic
385C2×C3×C4H(5,𝔽2)11yescharacteristic
386C3×C8—41no—
387C3×C8—41no—
388C3×C8—41no—
389C3×C8—41no—
390C24C2×C3×D411yescharacteristic
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8C2×C3×Q812yes—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4C2×C3×D411yescharacteristic
410C22×C4C2×C3×D411yescharacteristic
411C22×C4C2×C3×D411yescharacteristic
412C22×C4C2×C3×D411yescharacteristic
413C22×C4C2×C3×D411yescharacteristic
414C22×C4C2×C3×Q812yes—
415C22×C4C3×#4112yes—
416C22×C4C24×C311yescharacteristic
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—44no—
430C4⋊C4—24no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—81no—
449C4⋊C4—81no—
450C4⋊C4C2×C3×Q812yes—
451C4⋊C4C3×#4112yes—
452C4⋊C4C3×#4112yes—
453C22⋊C4—24no—
454C22⋊C4—22no—
455C22⋊C4—41no—
456C22⋊C4—42no—
457C22⋊C4—42no—
458C22⋊C4—81no—
459C42—21no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—42no—
464C42—42no—
465C42—42no—
466C42—42no—
467C22×C3—22no—
468C22×C3—22no—
469C22×C3—22no—
470C22×C3—41no—
471C22×C3—41no—
472C22×C3—41no—
473C22×C3—41no—
474C22×C3C2×C22⋊Q811yescharacteristic
475C22×C3#54611yescharacteristic
476C22×C3#54611yescharacteristic
477C22×C3C2×#13511yescharacteristic
478C22×C3#57511yescharacteristic
479C22×C3#57711yescharacteristic
480C22×C3#57811yescharacteristic
481C3×C4—22no—
482C3×C4—22no—
483C3×C4—42no—
484C3×C4—42no—
485C3×C4—41no—
486C3×C4—42no—
487C3×C4—41no—
488C3×C4—42no—
489C3×C4—41no—
490C3×C4—81no—
491C3×C4—81no—
492C3×C4—81no—
493C3×C4—81no—
494C3×C4#52911yescharacteristic
495C3×C4#53811yescharacteristic
496C3×C4#55411yescharacteristic
497C3×C4#55411yescharacteristic
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23C3×C22⋊Q812yes—
505C23C22×C3×D411yescharacteristic
506Q8—22no—
507Q8—22no—
508Q8—41no—
509Q8—41no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4C3×C22⋊Q812yes—
552C2×C4C3×#13512yes—
553C2×C4C22×C3×Q811yescharacteristic
554C2×C4C2×C3×#4111yescharacteristic
555C2×C4C2×C3×#4111yescharacteristic
556C2×C4C2×C3×#4111yescharacteristic
557C2×C4C3×H(5,𝔽2)11yescharacteristic
558C2×C4C3×H(5,𝔽2)11yescharacteristic
559C8—41no—
560C8—41no—
561C8—41no—
562C8—41no—
563C2×C3—42no—
564C2×C3#245011yescharacteristic
565C2×C3#307611yescharacteristic
566C2×C3#308211yescharacteristic
567C2×C3#330111yescharacteristic
568C2×C3#330211yescharacteristic
569C2×C3#331511yescharacteristic
570C2×C3#331711yescharacteristic
571C22—22no—
572C22—22no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22C2×C3×C22⋊Q811yescharacteristic
579C22C3×#54611yescharacteristic
580C22C3×#54611yescharacteristic
581C22C2×C3×#13511yescharacteristic
582C22C3×#57511yescharacteristic
583C22C3×#57711yescharacteristic
584C22C3×#57811yescharacteristic
585C4—22no—
586C4—22no—
587C4—42no—
588C4—41no—
589C4—42no—
590C4—41no—
591C4—42no—
592C4—41no—
593C4—42no—
594C4—81no—
595C4—81no—
596C4—81no—
597C4—81no—
598C4C3×#52911yescharacteristic
599C4C3×#53811yescharacteristic
600C4C3×#55411yescharacteristic
601C4C3×#55411yescharacteristic
602C3#2641711yescharacteristic, direct product factor, Sylow
603C2—42no—
604C2C3×#245011yescharacteristic
605C2C3×#307611yescharacteristic
606C2C3×#308211yescharacteristic
607C2C3×#330111yescharacteristic
608C2C3×#330211yescharacteristic
609C2C3×#331511yescharacteristic
610C2C3×#331711yescharacteristic
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×#2100C2semidirect
4. C3×#2086C2semidirect
5. C3×#2068C2semidirect
6. C3×#2026C2semidirect
7. C3×#1989C2semidirect
8. C3×#1919C2semidirect
11. C3×#1865C2semidirect
14. C3×#1803C2semidirect
16. #26417C3direct (N × H)
62. #2100C2×C3semidirect
63. #2086C2×C3semidirect
64. #2068C2×C3semidirect
65. #2026C2×C3semidirect
66. #1989C2×C3semidirect
67. #1919C2×C3semidirect
70. #1865C2×C3semidirect
73. #1803C2×C3semidirect
602. C3#26417direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#24412
C3×#23622
C3×#21002
C3×#20862
C3×#20682
C3×#20262
C3×#19892
C3×#19192
C3×#18922
C3×#18732
C3×#18652
C3×#18542
C3×#18502
C3×#18032
C3×#17352
#264173