Finite Group Databasebeta

← #10778204

Subgroups — #10778204

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#83349C211yesmaximal, characteristic, complemented
2#83347C211yesmaximal, characteristic, complemented
3#79329C211yesmaximal, characteristic, complemented
4#79301C211yesmaximal, characteristic
5C3×#1554C211yesmaximal, characteristic, complemented
6#70816C211yesmaximal, characteristic, complemented
7#70746C211yesmaximal, characteristic, complemented
8#68510C211yesmaximal, characteristic, complemented
9#68509C211yesmaximal, characteristic, complemented
10#68502C211yesmaximal, characteristic, complemented
11#68501C211yesmaximal, characteristic, complemented
12#68471C211yesmaximal, characteristic, complemented
13#68464C211yesmaximal, characteristic, complemented
14#68455C211yesmaximal, characteristic, complemented
15#68448C211yesmaximal, characteristic, complemented
16#19183—31nomaximal, Sylow
17C3×#533C2211yescharacteristic
18#5477C2211yescharacteristic
19C4×#235C2211yescharacteristic
20C3×#432C2211yescharacteristic
21C3×#431C2211yescharacteristic
22#5094C2211yescharacteristic
23#5091C2211yescharacteristic
24#5088C2211yescharacteristic
25#5087C2211yescharacteristic
26#5085C2211yescharacteristic
27#5084C2211yescharacteristic
28#5079C2211yescharacteristic
29#5078C2211yescharacteristic
30#5066—21no—
31#5063—21no—
32#5049—21no—
33#5049—21no—
34#5026—21no—
35#5026—21no—
36#5002—21no—
37#4998—21no—
38C2×C4⋊(C3⋊C8)C2211yescharacteristic
39C4×#208C2211yescharacteristic
40#4850C2211yescharacteristic
41#4848C2211yescharacteristic
42#4843C2211yescharacteristic
43#4842C2211yescharacteristic
44#4838C2211yescharacteristic
45#4838C2211yescharacteristic
46#4830C2211yescharacteristic
47#4830C2211yescharacteristic
48#4737—21no—
49#4734—21no—
50#4729—21no—
51#4729—21no—
52#4707—21no—
53#4705—21no—
54C4×D24—21no—
55C4×#204—21no—
56C3×#333C2211yescharacteristic
57C3×#333C2211yescharacteristic
58C3×#332C2211yescharacteristic
59C3×#332C2211yescharacteristic
60#4485C2211yescharacteristic
61#4485C2211yescharacteristic
62#4481C2211yescharacteristic
63#4481C2211yescharacteristic
64#4480C2211yescharacteristic
65#4480C2211yescharacteristic
66#4479C2211yescharacteristic
67#4479C2211yescharacteristic
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#1565—31no—
81#1554D311yescharacteristic, complemented
82#1551—31no—
83C2×#235—21no—
84C2×#235D411yescharacteristic
85C3×C4⋊Q8C2311yescharacteristic
86C3×C4⋊Q8C2311yescharacteristic
87C3×#125—21no—
88C3×C22⋊Q8—21no—
89C3×C22⋊Q8—21no—
90C3×C4×Q8—21no—
91C3×C4×Q8—21no—
92C3×#117—21no—
93C2×C3×C42C2311yescharacteristic
94C2×#215—21no—
95C2×#215—21no—
96#930—21no—
97#930—21no—
98C4×#67—21no—
99C4×#67—21no—
100#926D411yescharacteristic
101C2×#208D411yescharacteristic
102C2×#208D411yescharacteristic
103C22×C3⋊C8—21no—
104C2×D24—21no—
105C2×#204—21no—
106C2×ΓL(1,𝔽25)—21no—
107C2×ΓL(1,𝔽25)—21no—
108#875—21no—
109#874C2311yescharacteristic
110C4×D12—21no—
111C4×D12—21no—
112C4×D12D411yescharacteristic
113C4×D12C2311yescharacteristic
114#872—21no—
115C42×D3—21no—
116#869C2311yescharacteristic
117C4×Dic6D411yescharacteristic
118C4×Dic6C2311yescharacteristic
119C3×C4⋊C8C2311yescharacteristic
120C3×C4⋊C8C2311yescharacteristic
121C3×C4⋊C8C2311yescharacteristic
122C3×C4⋊C8C2311yescharacteristic
123C3×C22⋊C8—21no—
124C3×C22⋊C8—21no—
125C3×#97—21no—
126C3×C4×C8—21no—
127#835—21no—
128#835—21no—
129#821—21no—
130#821—21no—
131#821—21no—
132(C2×D3)⋊C8—21no—
133(C2×D3)⋊C8—21no—
134#818—21no—
135#817—21no—
136#816—21no—
137Dic3⋊C8—21no—
138Dic3⋊C8—21no—
139C8×Dic3—21no—
140C8×Dic3—21no—
141#810—21no—
142#810—21no—
143#809—21no—
144#809—21no—
145#809—21no—
146#809—21no—
147#808—21no—
148#808—21no—
149#807—21no—
150#807—21no—
151#807—21no—
152#807—21no—
153C4⋊(C3⋊C8)—21no—
154C4⋊(C3⋊C8)C2311yescharacteristic
155C4⋊(C3⋊C8)C2311yescharacteristic
156#803D411yescharacteristic
157#803D411yescharacteristic
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#533C2×D311yescharacteristic
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—61no—
192#432C2×D311yescharacteristic
193#431C2×D311yescharacteristic
194#427—61no—
195C2×C4⋊C8—31no—
196#416—61no—
197C4×M16—31no—
198C4×M16—61no—
199#335—31no—
200#335—31no—
201#334—31no—
202#334—31no—
203#333—31no—
204#333—31no—
205#333C2×D311yescharacteristic
206#333C2×D311yescharacteristic
207#332C2×D311yescharacteristic
208#332C2×D311yescharacteristic
209#331—31no—
210#331—31no—
211C2×C3×Q8—21no—
212C2×C3×Q8—21no—
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4#4111yescharacteristic
216C2×#67—21no—
217C2×#67—41no—
218#235—22no—
219#235—41no—
220#235—82no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12C2×D411yescharacteristic
225C2×C4×D3—21no—
226C2×C4×D3—21no—
227C2×C4×D3—21no—
228C2×C4×D3—41no—
229C2×Dic6—21no—
230C2×Dic6C2×D411yescharacteristic
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—41no—
240C3×C4⋊C4—41no—
241C3×C22⋊C4—41no—
242C3×C42—41no—
243C3×C42C2×D411yescharacteristic
244C3×C42C2×D411yescharacteristic
245C3×C42#4111yescharacteristic
246C3×C42C2411yescharacteristic
247#217—21no—
248#215—42no—
249#215—42no—
250(C2×D3)⋊C4—21no—
251(C2×D3)⋊C4—21no—
252(C2×D3)⋊C4—41no—
253(C2×D3)⋊C4—41no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3C2×D411yescharacteristic
257C4⋊Dic3C2×D411yescharacteristic
258Dic3⋊C4—21no—
259Dic3⋊C4—21no—
260C4×Dic3—21no—
261C4×Dic3—21no—
262C4×Dic3—21no—
263#208—22no—
264#208—41no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—42no—
268C2×C3⋊C8C2×D411yescharacteristic
269C2×C3⋊C8C2×D411yescharacteristic
270C2×C3⋊C8C2×D411yescharacteristic
271C2×C3⋊C8C2×D411yescharacteristic
272D24—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—31no—
283C2×M16—31no—
284C2×M16—61no—
285C2×M16—61no—
286C22×C8—61no—
287C4⋊Q8—31no—
288C4⋊Q8C22×D311yescharacteristic
289C4⋊Q8C22×D311yescharacteristic
290#127—31no—
291#125—21no—
292#124—61no—
293C22⋊Q8—21no—
294C22⋊Q8—21no—
295#121—61no—
296#121—61no—
297C4×Q8—21no—
298C4×Q8—21no—
299C4×Q8—31no—
300C4×Q8—31no—
301C4×D4—31no—
302C4×D4—31no—
303C4×D4—61no—
304C4×D4—61no—
305C4×D4—61no—
306C4×D4—61no—
307#117—21no—
308#117—31no—
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—31no—
321C4⋊C8—31no—
322C4⋊C8—61no—
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—21no—
343C22⋊C8—61no—
344C22⋊C8—61no—
345#97—21no—
346#97—31no—
347#97—31no—
348C4×C8—21no—
349C4×C8—31no—
350C4×C8—31no—
351C4×C8—61no—
352C4×C8—61no—
353C23×C3#12111yescharacteristic
354C22×D3—21no—
355C22×D3—41no—
356C3×Q8—41no—
357C3×Q8—41no—
358C3×Q8—81no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4#12111yescharacteristic
367C2×C3×C4#12111yescharacteristic
368C2×C3×C4#12111yescharacteristic
369C2×C3×C4C22×D411yescharacteristic
370C2×C3×C4C22×D411yescharacteristic
371C2×C3×C4C2×#4111yescharacteristic
372#67—42no—
373#67—82no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—41no—
378D12—21no—
379D12—21no—
380D12—41no—
381D12—41no—
382D12—41no—
383D12—41no—
384D12—81no—
385C4×D3—42no—
386C4×D3—41no—
387C4×D3—41no—
388C4×D3—81no—
389C4×D3—82no—
390Dic6—21no—
391Dic6—81no—
392Dic6#13712yes—
393C3×C8—41no—
394C3×C8—41no—
395C3×C8—41no—
396C3×C8—41no—
397C3⋊C8—21no—
398C3⋊C8—21no—
399C3⋊C8—21no—
400C3⋊C8—41no—
401C3⋊C8—41no—
402C3⋊C8#13512yes—
403#41—62no—
404#41—121no—
405#41—242no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—31no—
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×C4#23511yescharacteristic
421C22×C4D3×D411yescharacteristic
422C22×C4D3×D411yescharacteristic
423SD16—122no—
424SD16—122no—
425SD16—122no—
426D8—122no—
427M16—62no—
428M16—122no—
429M16—122no—
430M16—121no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—31no—
436C2×C8—31no—
437C2×C8—31no—
438C2×C8—31no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8—122no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—31no—
447C4⋊C4—31no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C22⋊C4—41no—
455C22⋊C4—61no—
456C22⋊C4—61no—
457C22⋊C4—61no—
458C22⋊C4—121no—
459C22⋊C4—121no—
460C42—41no—
461C42—61no—
462C42—61no—
463C42—61no—
464C42#23511yescharacteristic
465C42D3×D411yescharacteristic
466C42D3×D411yescharacteristic
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×C4—81no—
483C3×C4—81no—
484C3×C4C2×#13511yescharacteristic
485C3×C4C2×#13711yescharacteristic
486C3×C4#57511yescharacteristic
487C3×C4#57511yescharacteristic
488Dic3—22no—
489Dic3—41no—
490Dic3—41no—
491Dic3—81no—
492C23—61no—
493C23—121no—
494C23#88411yescharacteristic
495Q8—32no—
496Q8—41no—
497Q8—41no—
498Q8—61no—
499Q8—81no—
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—41no—
516C2×C4—41no—
517C2×C4—61no—
518C2×C4—61no—
519C2×C4—61no—
520C2×C4—122no—
521C2×C4—121no—
522C2×C4—121no—
523C2×C4—121no—
524C2×C4—241no—
525C2×C4—242no—
526C2×C4#88411yescharacteristic
527C2×C4#88411yescharacteristic
528C2×C4#88411yescharacteristic
529C2×C4C2×#23511yescharacteristic
530C2×C4C2×D3×D411yescharacteristic
531C2×C4C2×D3×D411yescharacteristic
532C8—32no—
533C8—41no—
534C8—41no—
535C8—41no—
536C8—41no—
537C8—61no—
538C8—61no—
539C8—61no—
540C8—121no—
541C8—121no—
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×#88411yescharacteristic
558C4—21no—
559C4—21no—
560C4—21no—
561C4—21no—
562C4—62no—
563C4—81no—
564C4—81no—
565C4—121no—
566C4—121no—
567C4—241no—
568C4D3×#13511yescharacteristic
569C4#576811yescharacteristic
570C4D3×#13711yescharacteristic
571C4#577711yescharacteristic
572C3#1918311yescharacteristic, complemented, Sylow
573C2—41no—
574C2—241no—
575C2—481no—
576C2#7872011yescharacteristic
577C2#7886111yescharacteristic
578C2#7896211yescharacteristic
subgroupG/H
centerC22C2×#884
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1554C2
socleC22×C3C2×#121
prime radical#79301C2
hypercenter#1554D3
hypocenterC3#19183
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1554

Complements

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

NH ≅ G/Ntype
1. #83349C2semidirect
2. #83347C2semidirect
3. #79329C2semidirect
5. C3×#1554C2semidirect
6. #70816C2semidirect
7. #70746C2semidirect
8. #68510C2semidirect
9. #68509C2semidirect
10. #68502C2semidirect
11. #68501C2semidirect
12. #68471C2semidirect
13. #68464C2semidirect
14. #68455C2semidirect
15. #68448C2semidirect
81. #1554D3semidirect
572. C3#19183semidirect

Maximal subgroups

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

subgroupindex
#833492
#833472
#793292
#793012
C3×#15542
#708162
#707462
#685102
#685092
#685022
#685012
#684712
#684642
#684552
#684482
#191833