Finite Group Databasebeta

← #10778208

Subgroups — #10778208

Order 768.

614 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79301C211yesmaximal, characteristic
2#79025C211yesmaximal, characteristic, complemented
3#78533C211yesmaximal, characteristic, complemented
4#78512C211yesmaximal, characteristic, complemented
5C3×#1565C211yesmaximal, characteristic, complemented
6#70781C211yesmaximal, characteristic, complemented
7#70433C211yesmaximal, characteristic, complemented
8#68591C211yesmaximal, characteristic, complemented
9#68589C211yesmaximal, characteristic, complemented
10#68584C211yesmaximal, characteristic, complemented
11#68582C211yesmaximal, characteristic, complemented
12#68548C211yesmaximal, characteristic, complemented
13#68546C211yesmaximal, characteristic, complemented
14#68545C211yesmaximal, characteristic, complemented
15#68543C211yesmaximal, characteristic, complemented
16#19183—31nomaximal, Sylow
17#5477C2211yescharacteristic
18C4×#235C2211yescharacteristic
19#5470C2211yescharacteristic
20C3×#431C2211yescharacteristic
21C2×C3×C4⋊C8C2211yescharacteristic
22C3×C4×M16C2211yescharacteristic
23#5134—21no—
24#5134—21no—
25#5128—21no—
26#5128—21no—
27#5113—21no—
28#5106—21no—
29#4926C2211yescharacteristic
30#4851C2211yescharacteristic
31#4850C2211yescharacteristic
32#4850C2211yescharacteristic
33#4848C2211yescharacteristic
34#4847—21no—
35#4846C2211yescharacteristic
36#4843C2211yescharacteristic
37#4842C2211yescharacteristic
38#4842C2211yescharacteristic
39#4829—21no—
40#4737—21no—
41#4734—21no—
42#4710—21no—
43#4710—21no—
44#4707—21no—
45#4705—21no—
46#4694C2211yescharacteristic
47#4693C2211yescharacteristic
48C4×D24—21no—
49C4×#204—21no—
50#4688C2211yescharacteristic
51#4688C2211yescharacteristic
52#4682C2211yescharacteristic
53#4681C2211yescharacteristic
54C8×Dic6C2211yescharacteristic
55C8×Dic6C2211yescharacteristic
56C3×#335C2211yescharacteristic
57C3×#335C2211yescharacteristic
58C3×#334C2211yescharacteristic
59C3×#334C2211yescharacteristic
60#4486C2211yescharacteristic
61#4486C2211yescharacteristic
62#4485C2211yescharacteristic
63#4485C2211yescharacteristic
64#4481C2211yescharacteristic
65#4481C2211yescharacteristic
66#4481C2211yescharacteristic
67#4481C2211yescharacteristic
68#3182—31no—
69#3106—31no—
70#3076—31no—
71#2971—31no—
72#1703—31no—
73#1697—31no—
74#1695—31no—
75#1685—31no—
76#1684—31no—
77#1678—31no—
78#1676—31no—
79#1666—31no—
80#1565D311yescharacteristic, complemented
81#1554—31no—
82#1551—31no—
83C2×#235—21no—
84C2×#235D411yescharacteristic
85C2×C3×M16D411yescharacteristic
86C2×C3×M16D411yescharacteristic
87C22×C3×C8—21no—
88C2×C3×C42C2311yescharacteristic
89#930—21no—
90#930—21no—
91C4×#67—21no—
92C4×#67—21no—
93#926—21no—
94#926D411yescharacteristic
95#924—21no—
96#924—21no—
97C2×D24—21no—
98C2×#204—21no—
99C2×#204—21no—
100C2×#204—21no—
101C2×ΓL(1,𝔽25)—21no—
102C2×ΓL(1,𝔽25)—21no—
103#875—21no—
104#874C2311yescharacteristic
105C4×D12—21no—
106C4×D12—21no—
107C4×D12D411yescharacteristic
108C4×D12C2311yescharacteristic
109#872—21no—
110C42×D3—21no—
111#870—21no—
112#869C2311yescharacteristic
113#869C2311yescharacteristic
114#869C2311yescharacteristic
115C4×Dic6—21no—
116C4×Dic6—21no—
117C4×Dic6D411yescharacteristic
118C4×Dic6C2311yescharacteristic
119C3×C4⋊C8—21no—
120C3×C4⋊C8C2311yescharacteristic
121C3×C4⋊C8C2311yescharacteristic
122C3×C4⋊C8C2311yescharacteristic
123C3×C4⋊C8C2311yescharacteristic
124C3×C22⋊C8—21no—
125C3×#97D411yescharacteristic
126C3×#97D411yescharacteristic
127C3×C4×C8—21no—
128C3×C4×C8C2311yescharacteristic
129C3×C4×C8C2311yescharacteristic
130#830—21no—
131#821—21no—
132#821—21no—
133#821—21no—
134#821—21no—
135#821—21no—
136#821—21no—
137#821—21no—
138(C2×D3)⋊C8—21no—
139(C2×D3)⋊C8—21no—
140#818—21no—
141#818—21no—
142#818—21no—
143#818—21no—
144#818—21no—
145#817—21no—
146#817—21no—
147#817—21no—
148#816—21no—
149#816—21no—
150#816—21no—
151#816—21no—
152#816—21no—
153Dic3⋊C8—21no—
154Dic3⋊C8—21no—
155C8×Dic3—21no—
156C8×Dic3—21no—
157C4⋊(C3⋊C8)C2311yescharacteristic
158C4⋊(C3⋊C8)C2311yescharacteristic
159#803—21no—
160#533—31no—
161#532—31no—
162C4×#41—31no—
163#500—31no—
164#492—31no—
165#487—31no—
166#486—31no—
167#482—61no—
168#479—61no—
169#478—61no—
170#477—31no—
171#477—31no—
172#476—31no—
173#476—31no—
174#475—31no—
175#474—31no—
176#468—61no—
177#468—61no—
178#460—31no—
179#459—31no—
180#449—61no—
181C8×Q8—31no—
182C8×Q8—31no—
183#440—61no—
184#440—61no—
185C4×SD16—61no—
186C4×D8—61no—
187#436—31no—
188#436—31no—
189#436—61no—
190#436—61no—
191#432—31no—
192#432—61no—
193#431C2×D311yescharacteristic
194#427—61no—
195C2×C4⋊C8C2×D311yescharacteristic
196#416—61no—
197C4×M16—61no—
198C4×M16C2×D311yescharacteristic
199#335C2×D311yescharacteristic
200#335C2×D311yescharacteristic
201#334C2×D311yescharacteristic
202#334C2×D311yescharacteristic
203#333—31no—
204#333—31no—
205#333—31no—
206#333—31no—
207#332—31no—
208#332—31no—
209#331—31no—
210#331—31no—
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4#4111yescharacteristic
214C2×#67—21no—
215C2×#67—41no—
216#235—22no—
217#235—41no—
218#235—82no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12C2×D411yescharacteristic
223C2×C4×D3—21no—
224C2×C4×D3—21no—
225C2×C4×D3—21no—
226C2×C4×D3—41no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6C2×D411yescharacteristic
231C3×M16—22no—
232C3×M16—41no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—42no—
238C2×C3×C8C2×D411yescharacteristic
239C2×C3×C8C2×D411yescharacteristic
240C2×C3×C8C2×D411yescharacteristic
241C2×C3×C8C2×D411yescharacteristic
242C3×C42C2×D411yescharacteristic
243C3×C42C2×D411yescharacteristic
244C3×C42#4111yescharacteristic
245C3×C42C2411yescharacteristic
246#217—21no—
247#217—41no—
248(C2×D3)⋊C4—21no—
249(C2×D3)⋊C4—21no—
250(C2×D3)⋊C4—41no—
251(C2×D3)⋊C4—41no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3C2×D411yescharacteristic
259C4⋊Dic3C2×D411yescharacteristic
260Dic3⋊C4—21no—
261Dic3⋊C4—21no—
262Dic3⋊C4—41no—
263Dic3⋊C4—41no—
264C4×Dic3—21no—
265C4×Dic3—21no—
266C4×Dic3—21no—
267C4×Dic3—41no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270D24—42no—
271#204—42no—
272#204—42no—
273#204—42no—
274ΓL(1,𝔽25)—42no—
275ΓL(1,𝔽25)—42no—
276C2×#41—31no—
277C2×#41—61no—
278C2×SD16—61no—
279C2×SD16—61no—
280C2×SD16—61no—
281C2×D8—61no—
282C2×M16—61no—
283C2×M16—61no—
284C2×M16D1211yescharacteristic
285C2×M16D1211yescharacteristic
286C22×C8—21no—
287C4⋊Q8—31no—
288C4⋊Q8—31no—
289C4⋊Q8—31no—
290#127—31no—
291#125—61no—
292#124—61no—
293C22⋊Q8—61no—
294C22⋊Q8—61no—
295#121—61no—
296#121—61no—
297C4×Q8—31no—
298C4×Q8—31no—
299C4×Q8—61no—
300C4×Q8—61no—
301C4×D4—31no—
302C4×D4—31no—
303C4×D4—61no—
304C4×D4—61no—
305C4×D4—61no—
306C4×D4—61no—
307#117—31no—
308#117—61no—
309#117—61no—
310C2×C42—61no—
311C2×C42C22×D311yescharacteristic
312#107—61no—
313#107—61no—
314#107—61no—
315#107—61no—
316#107—61no—
317#106—61no—
318#106—61no—
319#106—61no—
320C4⋊C8—21no—
321C4⋊C8—31no—
322C4⋊C8—31no—
323C4⋊C8—61no—
324C4⋊C8—61no—
325C4⋊C8C22×D311yescharacteristic
326C4⋊C8C22×D311yescharacteristic
327C4⋊C8C22×D311yescharacteristic
328C4⋊C8C22×D311yescharacteristic
329#103—61no—
330#103—61no—
331#103—61no—
332#103—61no—
333#103—61no—
334#102—61no—
335#102—61no—
336#102—61no—
337#102—61no—
338#102—61no—
339#102—61no—
340#102—61no—
341C22⋊C8—21no—
342C22⋊C8—61no—
343C22⋊C8—61no—
344C22⋊C8—61no—
345#97—61no—
346#97D1211yescharacteristic
347#97D1211yescharacteristic
348C4×C8—21no—
349C4×C8—61no—
350C4×C8—61no—
351C4×C8C22×D311yescharacteristic
352C4×C8C22×D311yescharacteristic
353C23×C3#12111yescharacteristic
354C22×D3—21no—
355C22×D3—41no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—41no—
361C2×C3×C4#12111yescharacteristic
362C2×C3×C4#12111yescharacteristic
363C2×C3×C4#12111yescharacteristic
364C2×C3×C4C22×D411yescharacteristic
365C2×C3×C4C22×D411yescharacteristic
366C2×C3×C4C2×#4111yescharacteristic
367#67—42no—
368#67—82no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375D12—21no—
376D12—21no—
377D12—41no—
378D12—41no—
379D12—41no—
380D12—41no—
381D12—81no—
382C4×D3—42no—
383C4×D3—41no—
384C4×D3—41no—
385C4×D3—81no—
386C4×D3—82no—
387Dic6—21no—
388Dic6—41no—
389Dic6—41no—
390Dic6—81no—
391Dic6—81no—
392Dic6#13712yes—
393C3×C8—21no—
394C3×C8—21no—
395C3×C8—21no—
396C3×C8—41no—
397C3×C8—41no—
398C3×C8—41no—
399C3×C8—41no—
400C3×C8#13512yes—
401C3⋊C8—41no—
402C3⋊C8—41no—
403#41—62no—
404#41—121no—
405#41—242no—
406C2×Q8—31no—
407C2×Q8—61no—
408C2×Q8—61no—
409C2×Q8—61no—
410C2×D4—31no—
411C2×D4—61no—
412C2×D4—61no—
413C2×D4—61no—
414C2×D4—61no—
415C2×D4—121no—
416C22×C4—61no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—121no—
420C22×C4C2×D1211yescharacteristic
421C22×C4D3×D411yescharacteristic
422C22×C4#23911yescharacteristic
423SD16—122no—
424SD16—122no—
425SD16—122no—
426D8—122no—
427M16—22no—
428M16—41no—
429M16—122no—
430M16—122no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—42no—
436C2×C8—61no—
437C2×C8—61no—
438C2×C8C2×D1211yescharacteristic
439C2×C8C2×D1211yescharacteristic
440C2×C8C2×D1211yescharacteristic
441C2×C8C2×D1211yescharacteristic
442C4⋊C4—31no—
443C4⋊C4—31no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4—121no—
453C4⋊C4—121no—
454C22⋊C4—61no—
455C22⋊C4—61no—
456C22⋊C4—61no—
457C22⋊C4—121no—
458C22⋊C4—121no—
459C22⋊C4—121no—
460C42—61no—
461C42—61no—
462C42—61no—
463C42—121no—
464C42C2×D1211yescharacteristic
465C42D3×D411yescharacteristic
466C42#23911yescharacteristic
467C42C23×D311yescharacteristic
468C22×C3—21no—
469C22×C3—21no—
470C22×C3—21no—
471C22×C3C2×#12111yescharacteristic
472C2×D3—41no—
473C2×D3—41no—
474C2×D3—41no—
475C2×D3—81no—
476C2×D3—81no—
477C2×D3—81no—
478C3×C4—21no—
479C3×C4—21no—
480C3×C4—21no—
481C3×C4—21no—
482C3×C4C2×#13511yescharacteristic
483C3×C4C2×#13711yescharacteristic
484C3×C4#57511yescharacteristic
485C3×C4#57511yescharacteristic
486Dic3—22no—
487Dic3—41no—
488Dic3—41no—
489Dic3—81no—
490Dic3—81no—
491Dic3—81no—
492C23—61no—
493C23—121no—
494C23#89511yescharacteristic
495Q8—32no—
496Q8—61no—
497Q8—121no—
498Q8—121no—
499Q8—241no—
500Q8—241no—
501D4—61no—
502D4—61no—
503D4—122no—
504D4—121no—
505D4—121no—
506D4—121no—
507D4—121no—
508D4—241no—
509D4—242no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—41no—
515C2×C4—61no—
516C2×C4—61no—
517C2×C4—61no—
518C2×C4—122no—
519C2×C4—121no—
520C2×C4—121no—
521C2×C4—121no—
522C2×C4—121no—
523C2×C4—121no—
524C2×C4—241no—
525C2×C4—242no—
526C2×C4#89511yescharacteristic
527C2×C4#89511yescharacteristic
528C2×C4#89511yescharacteristic
529C2×C4C22×D1211yescharacteristic
530C2×C4C2×D3×D411yescharacteristic
531C2×C4C2×#23911yescharacteristic
532C8—21no—
533C8—21no—
534C8—21no—
535C8—41no—
536C8—41no—
537C8—41no—
538C8—41no—
539C8—121no—
540C8—121no—
541C8#90412yes—
542C2×C3—41no—
543C2×C3#303511yescharacteristic
544C2×C3#305611yescharacteristic
545C2×C3#306811yescharacteristic
546D3—81no—
547D3—161no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—121no—
552C22—121no—
553C22—121no—
554C22—241no—
555C22—241no—
556C22—241no—
557C22C2×#89511yescharacteristic
558C4—21no—
559C4—21no—
560C4—21no—
561C4—21no—
562C4—62no—
563C4—121no—
564C4—121no—
565C4—241no—
566C4—241no—
567C4—241no—
568C4C2×#90411yescharacteristic
569C4#574811yescharacteristic
570C4#577311yescharacteristic
571C4D3×#13711yescharacteristic
572C3#1918311yescharacteristic, complemented, Sylow
573C2—41no—
574C2—241no—
575C2—481no—
576C2#7902111yescharacteristic
577C2#7921211yescharacteristic
578C2#7926111yescharacteristic
subgroupG/H
centerC22C2×#895
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1565C2
socleC22×C3C2×#121
prime radical#79301C2
hypercenter#1565D3
hypocenterC3#19183
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1565

Complements

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

NH ≅ G/Ntype
2. #79025C2semidirect
3. #78533C2semidirect
4. #78512C2semidirect
5. C3×#1565C2semidirect
6. #70781C2semidirect
7. #70433C2semidirect
8. #68591C2semidirect
9. #68589C2semidirect
10. #68584C2semidirect
11. #68582C2semidirect
12. #68548C2semidirect
13. #68546C2semidirect
14. #68545C2semidirect
15. #68543C2semidirect
80. #1565D3semidirect
572. C3#19183semidirect

Maximal subgroups

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

subgroupindex
#793012
#790252
#785332
#785122
C3×#15652
#707812
#704332
#685912
#685892
#685842
#685822
#685482
#685462
#685452
#685432
#191833