Finite Group Databasebeta

← #10780306

Subgroups — #10780306

Order 768.

572 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79323C211yesmaximal, characteristic
2#79063C211yesmaximal, characteristic
3#78537C211yesmaximal, characteristic, complemented
4#78511C211yesmaximal, characteristic
5C3×#1569C211yesmaximal, characteristic, complemented
6#70821C211yesmaximal, characteristic, complemented
7#70508C211yesmaximal, characteristic, complemented
8#68595C211yesmaximal, characteristic, complemented
9#68591C211yesmaximal, characteristic, complemented
10#68585C211yesmaximal, characteristic, complemented
11#68581C211yesmaximal, characteristic, complemented
12#68555C211yesmaximal, characteristic, complemented
13#68549C211yesmaximal, characteristic, complemented
14#68545C211yesmaximal, characteristic, complemented
15#68539C211yesmaximal, characteristic, complemented
16#19381—31nomaximal, Sylow
17#5479C2211yescharacteristic
18#5470C2211yescharacteristic
19C2×#869C2211yescharacteristic
20C3×#432C2211yescharacteristic
21C3×#423C2211yescharacteristic
22C3×C4×M16C2211yescharacteristic
23#5137—21no—
24#5134—21no—
25#5128—21no—
26#5127—21no—
27#5120—21no—
28#5120—21no—
29#4925C2211yescharacteristic
30#4851C2211yescharacteristic
31#4851C2211yescharacteristic
32#4850C2211yescharacteristic
33#4849C2211yescharacteristic
34#4846C2211yescharacteristic
35#4846C2211yescharacteristic
36#4845—21no—
37#4842C2211yescharacteristic
38#4841C2211yescharacteristic
39#4829—21no—
40#4735—21no—
41#4722—21no—
42#4714—21no—
43#4709—21no—
44#4708—21no—
45#4703—21no—
46#4703—21no—
47#4702—21no—
48#4688C2211yescharacteristic
49C8×D12C2211yescharacteristic
50#4681C2211yescharacteristic
51#4681C2211yescharacteristic
52#4681C2211yescharacteristic
53#4681C2211yescharacteristic
54#4679C2211yescharacteristic
55C8×Dic6C2211yescharacteristic
56C3×#335C2211yescharacteristic
57C3×#335C2211yescharacteristic
58C3×#334C2211yescharacteristic
59C3×#334C2211yescharacteristic
60#4486C2211yescharacteristic
61#4486C2211yescharacteristic
62#4486C2211yescharacteristic
63#4486C2211yescharacteristic
64#4482C2211yescharacteristic
65#4482C2211yescharacteristic
66#4481C2211yescharacteristic
67#4481C2211yescharacteristic
68#3252—31no—
69#3239—31no—
70#3139—31no—
71#2969—31no—
72#1704—31no—
73#1703—31no—
74#1700—31no—
75#1699—31no—
76#1684—31no—
77#1683—31no—
78#1680—31no—
79#1679—31no—
80#1569D311yescharacteristic, complemented
81#1554—31no—
82#1549—31no—
83C22×Dic6—21no—
84C2×C3×M16—21no—
85C2×C3×M16D411yescharacteristic
86C2×C3×M16D411yescharacteristic
87C2×C3×C42C2311yescharacteristic
88#930D411yescharacteristic
89#929—21no—
90C4×#67—21no—
91#926—21no—
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3—21no—
94#924—21no—
95#924—21no—
96#924D411yescharacteristic
97C2×Dic12—21no—
98C2×#204—21no—
99C2×ΓL(1,𝔽25)—21no—
100C2×D3×C8—21no—
101#876—21no—
102#875D411yescharacteristic
103C4×D12C2311yescharacteristic
104#872—21no—
105#870—21no—
106#870D411yescharacteristic
107#869—21no—
108#869—21no—
109#869—21no—
110#869C2311yescharacteristic
111#869C2311yescharacteristic
112#869C2311yescharacteristic
113#869C2311yescharacteristic
114C4×Dic6—21no—
115C4×Dic6—21no—
116C4×Dic6C2311yescharacteristic
117C3×C4⋊C8—21no—
118C3×C4⋊C8C2311yescharacteristic
119C3×C4⋊C8C2311yescharacteristic
120C3×C4⋊C8C2311yescharacteristic
121C3×C4⋊C8C2311yescharacteristic
122C3×C22⋊C8—21no—
123C3×#97—21no—
124C3×#97D411yescharacteristic
125C3×#97D411yescharacteristic
126C3×C4×C8C2311yescharacteristic
127C3×C4×C8C2311yescharacteristic
128#830—21no—
129#821—21no—
130#821—21no—
131#821—21no—
132(C2×D3)⋊C8—21no—
133(C2×D3)⋊C8—21no—
134#818—21no—
135#818—21no—
136#818—21no—
137#817—21no—
138#817—21no—
139#817—21no—
140#816—21no—
141#816—21no—
142#816—21no—
143#816—21no—
144#816—21no—
145#816—21no—
146#816—21no—
147#816—21no—
148#816—21no—
149#816—21no—
150#816—21no—
151#816—21no—
152#816—21no—
153#815—21no—
154Dic3⋊C8—21no—
155Dic3⋊C8—21no—
156C8×Dic3—21no—
157C4⋊(C3⋊C8)C2311yescharacteristic
158C4⋊(C3⋊C8)C2311yescharacteristic
159C4×C3⋊C8—21no—
160#533—31no—
161C2×C4⋊Q8—31no—
162#529—31no—
163#501—61no—
164#491—61no—
165#491—61no—
166#489—61no—
167#487—31no—
168#487—31no—
169#487—31no—
170#487—31no—
171#484—61no—
172#481—61no—
173#479—61no—
174#477—31no—
175#476—31no—
176#475—31no—
177#475—31no—
178#474—31no—
179#474—31no—
180#470—61no—
181#468—61no—
182#464—61no—
183#462—31no—
184#461—31no—
185#446—31no—
186C8×Q8—31no—
187#436—31no—
188#436—61no—
189#435—61no—
190C8×D4—31no—
191#433—61no—
192#432C2×D311yescharacteristic
193#431—31no—
194#423C2×D311yescharacteristic
195#419—61no—
196#419—61no—
197#405—61no—
198C4×M16C2×D311yescharacteristic
199#335C2×D311yescharacteristic
200#335C2×D311yescharacteristic
201#334C2×D311yescharacteristic
202#334C2×D311yescharacteristic
203#333—31no—
204#333—31no—
205#332—31no—
206#332—31no—
207#332—31no—
208#332—31no—
209#332—31no—
210#332—31no—
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4#4111yescharacteristic
214C2×#67—21no—
215C22×Dic3—41no—
216C2×D12C2×D411yescharacteristic
217C2×C4×D3—21no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6—21no—
221C2×Dic6—21no—
222C2×Dic6—42no—
223C2×Dic6—42no—
224C2×Dic6—41no—
225C2×Dic6C2×D411yescharacteristic
226C3×M16—41no—
227C3×M16—41no—
228C3×M16—81no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8C2×D411yescharacteristic
234C2×C3×C8C2×D411yescharacteristic
235C2×C3×C8C2×D411yescharacteristic
236C2×C3×C8C2×D411yescharacteristic
237C3×C42C2×D411yescharacteristic
238C3×C42C2×D411yescharacteristic
239C3×C42#4111yescharacteristic
240C3×C42C2411yescharacteristic
241#217—21no—
242#217—41no—
243(C2×D3)⋊C4—21no—
244(C2×D3)⋊C4—21no—
245(C2×D3)⋊C4—21no—
246(C2×D3)⋊C4—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—41no—
256C4⋊Dic3—41no—
257C4⋊Dic3C2×D411yescharacteristic
258C4⋊Dic3C2×D411yescharacteristic
259Dic3⋊C4—21no—
260Dic3⋊C4—21no—
261Dic3⋊C4—21no—
262Dic3⋊C4—21no—
263Dic3⋊C4—41no—
264Dic3⋊C4—41no—
265C4×Dic3—21no—
266C4×Dic3—41no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269Dic12—81no—
270#204—81no—
271ΓL(1,𝔽25)—42no—
272D3×C8—42no—
273C22×Q8—61no—
274C2×Q16—61no—
275C2×SD16—61no—
276C2×M16—21no—
277C2×M16—61no—
278C2×M16D1211yescharacteristic
279C2×M16D1211yescharacteristic
280C22×C8—61no—
281C4⋊Q8—31no—
282C4⋊Q8—31no—
283C4⋊Q8—31no—
284C4⋊Q8—31no—
285C4⋊Q8—61no—
286C4⋊Q8—61no—
287C4⋊Q8—61no—
288#126—61no—
289#125—31no—
290#125—61no—
291#124—31no—
292#123—61no—
293C22⋊Q8—31no—
294C22⋊Q8—61no—
295C22⋊Q8—61no—
296#121—31no—
297C4×Q8—31no—
298C4×Q8—61no—
299C4×Q8—61no—
300C4×D4—31no—
301C4×D4—61no—
302#117—61no—
303#117—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C42C22×D311yescharacteristic
307#107—61no—
308#107—61no—
309#107—61no—
310#106—61no—
311#106—61no—
312#106—61no—
313C4⋊C8—21no—
314C4⋊C8—31no—
315C4⋊C8—31no—
316C4⋊C8—61no—
317C4⋊C8—61no—
318C4⋊C8C22×D311yescharacteristic
319C4⋊C8C22×D311yescharacteristic
320C4⋊C8C22×D311yescharacteristic
321C4⋊C8C22×D311yescharacteristic
322#103—61no—
323#103—61no—
324#103—61no—
325#103—61no—
326#103—61no—
327#103—61no—
328#103—61no—
329#103—61no—
330#103—61no—
331#103—61no—
332#103—61no—
333#103—61no—
334#103—61no—
335#102—61no—
336#102—61no—
337#102—61no—
338C22⋊C8—21no—
339C22⋊C8—61no—
340C22⋊C8—61no—
341C22⋊C8—61no—
342#97—21no—
343#97—61no—
344#97D1211yescharacteristic
345#97D1211yescharacteristic
346C4×C8—61no—
347C4×C8—61no—
348C4×C8C22×D311yescharacteristic
349C4×C8C22×D311yescharacteristic
350C23×C3#12111yescharacteristic
351C22×D3—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—41no—
357C2×C3×C4#12111yescharacteristic
358C2×C3×C4#12111yescharacteristic
359C2×C3×C4#12111yescharacteristic
360C2×C3×C4C22×D411yescharacteristic
361C2×C3×C4C22×D411yescharacteristic
362C2×C3×C4C2×#4111yescharacteristic
363#67—81no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—82no—
372D12—21no—
373D12—21no—
374C4×D3—41no—
375C4×D3—41no—
376Dic6—21no—
377Dic6—21no—
378Dic6—41no—
379Dic6—41no—
380Dic6—41no—
381Dic6—41no—
382Dic6—41no—
383Dic6—41no—
384Dic6—82no—
385Dic6—81no—
386C3×C8—21no—
387C3×C8—21no—
388C3×C8—21no—
389C3×C8—21no—
390C3×C8—41no—
391C3×C8—41no—
392C3×C8—41no—
393C3×C8—41no—
394C3⋊C8—41no—
395C3⋊C8—41no—
396C2×Q8—31no—
397C2×Q8—61no—
398C2×Q8—61no—
399C2×Q8—61no—
400C2×Q8—61no—
401C2×Q8—122no—
402C2×Q8—122no—
403C2×Q8—121no—
404C2×D4—31no—
405C2×D4—61no—
406C22×C4—61no—
407C22×C4—121no—
408C22×C4C2×D1211yescharacteristic
409C22×C4D3×D411yescharacteristic
410C22×C4#23911yescharacteristic
411Q16—241no—
412SD16—241no—
413M16—41no—
414M16—41no—
415M16—81no—
416M16—122no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—61no—
422C2×C8—61no—
423C2×C8—122no—
424C2×C8C2×D1211yescharacteristic
425C2×C8C2×D1211yescharacteristic
426C2×C8C2×D1211yescharacteristic
427C2×C8C2×D1211yescharacteristic
428C4⋊C4—31no—
429C4⋊C4—31no—
430C4⋊C4—61no—
431C4⋊C4—61no—
432C4⋊C4—61no—
433C4⋊C4—61no—
434C4⋊C4—61no—
435C4⋊C4—61no—
436C4⋊C4—61no—
437C4⋊C4—61no—
438C4⋊C4—61no—
439C4⋊C4—61no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—121no—
443C4⋊C4—121no—
444C4⋊C4—121no—
445C4⋊C4—121no—
446C22⋊C4—61no—
447C22⋊C4—61no—
448C22⋊C4—61no—
449C22⋊C4—61no—
450C22⋊C4—61no—
451C22⋊C4—121no—
452C42—61no—
453C42—121no—
454C42C2×D1211yescharacteristic
455C42D3×D411yescharacteristic
456C42#23911yescharacteristic
457C42C23×D311yescharacteristic
458C22×C3—21no—
459C22×C3—21no—
460C22×C3—21no—
461C22×C3C2×#12111yescharacteristic
462C2×D3—41no—
463C2×D3—41no—
464C2×D3—41no—
465C3×C4—21no—
466C3×C4—21no—
467C3×C4—21no—
468C3×C4—21no—
469C3×C4#57511yescharacteristic
470C3×C4#57511yescharacteristic
471C3×C4#57711yescharacteristic
472C3×C4#57811yescharacteristic
473Dic3—41no—
474Dic3—41no—
475Dic3—41no—
476Dic3—81no—
477Dic3—81no—
478Dic3—81no—
479Dic3—81no—
480C23—61no—
481C23#89511yescharacteristic
482Q8—61no—
483Q8—61no—
484Q8—121no—
485Q8—121no—
486Q8—121no—
487Q8—121no—
488Q8—121no—
489Q8—121no—
490Q8—241no—
491Q8—242no—
492D4—61no—
493D4—61no—
494D4—241no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—41no—
500C2×C4—61no—
501C2×C4—61no—
502C2×C4—61no—
503C2×C4—121no—
504C2×C4—121no—
505C2×C4—121no—
506C2×C4—121no—
507C2×C4—121no—
508C2×C4—121no—
509C2×C4—242no—
510C2×C4#89511yescharacteristic
511C2×C4#89511yescharacteristic
512C2×C4#89511yescharacteristic
513C2×C4C22×D1211yescharacteristic
514C2×C4C2×D3×D411yescharacteristic
515C2×C4C2×#23911yescharacteristic
516C8—21no—
517C8—21no—
518C8—21no—
519C8—21no—
520C8—41no—
521C8—41no—
522C8—41no—
523C8—41no—
524C8—121no—
525C8—121no—
526C2×C3—41no—
527C2×C3#304011yescharacteristic
528C2×C3#306311yescharacteristic
529C2×C3#306911yescharacteristic
530D3—81no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—121no—
535C22—121no—
536C22—121no—
537C22C2×#89511yescharacteristic
538C4—21no—
539C4—21no—
540C4—21no—
541C4—21no—
542C4—121no—
543C4—121no—
544C4—121no—
545C4—241no—
546C4—241no—
547C4—241no—
548C4—241no—
549C4#574811yescharacteristic
550C4#574811yescharacteristic
551C4#577511yescharacteristic
552C4#577911yescharacteristic
553C3#1938111yescharacteristic, complemented, Sylow
554C2—41no—
555C2—241no—
556C2#7905811yescharacteristic
557C2#7921411yescharacteristic
558C2#7926211yescharacteristic
subgroupG/H
centerC22C2×#895
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1569C2
socleC22×C3C2×#121
prime radical#930D4
hypercenter#1569D3
hypocenterC3#19381
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1569

Complements

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

NH ≅ G/Ntype
3. #78537C2semidirect
5. C3×#1569C2semidirect
6. #70821C2semidirect
7. #70508C2semidirect
8. #68595C2semidirect
9. #68591C2semidirect
10. #68585C2semidirect
11. #68581C2semidirect
12. #68555C2semidirect
13. #68549C2semidirect
14. #68545C2semidirect
15. #68539C2semidirect
80. #1569D3semidirect
553. C3#19381semidirect

Maximal subgroups

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

subgroupindex
#793232
#790632
#785372
#785112
C3×#15692
#708212
#705082
#685952
#685912
#685852
#685812
#685552
#685492
#685452
#685392
#193813