Finite Group Databasebeta

← #10780334

Subgroups — #10780334

Order 768.

606 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2969C211yesmaximal, characteristic, complemented
2#83154C211yesmaximal, characteristic, complemented
3#83150C211yesmaximal, characteristic
4#78528C211yesmaximal, characteristic, complemented
5#70966C211yesmaximal, characteristic, complemented
6#70961C211yesmaximal, characteristic, complemented
7#70946C211yesmaximal, characteristic, complemented
8#70941C211yesmaximal, characteristic, complemented
9#70875C211yesmaximal, characteristic, complemented
10#70870C211yesmaximal, characteristic, complemented
11#70854C211yesmaximal, characteristic, complemented
12#70849C211yesmaximal, characteristic, complemented
13#70778C211yesmaximal, characteristic, complemented
14#70775C211yesmaximal, characteristic, complemented
15#69851C211yesmaximal, characteristic, complemented
16#19383—31nomaximal, Sylow
17C3×#529C2211yescharacteristic
18#5477C2211yescharacteristic
19C2×#869C2211yescharacteristic
20C3×#446C2211yescharacteristic
21C3×C8×Q8C2211yescharacteristic
22C3×#436—21no—
23C3×#436C2211yescharacteristic
24C3×#435—21no—
25C3×C8×D4C2211yescharacteristic
26C3×#433—21no—
27C3×#431C2211yescharacteristic
28C3×#405—21no—
29C3×C4×M16C2211yescharacteristic
30#5133—21no—
31#5133—21no—
32#5069—21no—
33#5066—21no—
34#5060—21no—
35#5057—21no—
36#5050—21no—
37#5049—21no—
38#5040—21no—
39#5039—21no—
40#5031C2211yescharacteristic
41#5031C2211yescharacteristic
42#5027C2211yescharacteristic
43#5020C2211yescharacteristic
44#5019C2211yescharacteristic
45#5019C2211yescharacteristic
46#5015C2211yescharacteristic
47#5007C2211yescharacteristic
48#4926C2211yescharacteristic
49#4924C2211yescharacteristic
50#4714—21no—
51#4714—21no—
52#4694C2211yescharacteristic
53#4694C2211yescharacteristic
54#4681C2211yescharacteristic
55#4681C2211yescharacteristic
56#4486C2211yescharacteristic
57#4486C2211yescharacteristic
58#4486C2211yescharacteristic
59#4486C2211yescharacteristic
60#4485C2211yescharacteristic
61#4485C2211yescharacteristic
62#4482C2211yescharacteristic
63#4482C2211yescharacteristic
64#4458C2211yescharacteristic
65#4458C2211yescharacteristic
66#4457C2211yescharacteristic
67#4457C2211yescharacteristic
68#3251—31no—
69#3218—31no—
70#3141—31no—
71#2969D311yescharacteristic, complemented
72#1702—31no—
73#1701—31no—
74#1700—31no—
75#1699—31no—
76#1682—31no—
77#1681—31no—
78#1680—31no—
79#1679—31no—
80#1569—31no—
81#1553—31no—
82#1546—31no—
83C2×#235—21no—
84C22×Dic6—21no—
85C2×C3×M16—21no—
86C2×C3×M16D411yescharacteristic
87C2×C3×M16D411yescharacteristic
88C22×C3×C8—21no—
89C3×#126—21no—
90C3×#125D411yescharacteristic
91C3×#124D411yescharacteristic
92C3×#123—21no—
93C3×C22⋊Q8D411yescharacteristic
94C3×#121D411yescharacteristic
95C3×C4×Q8C2311yescharacteristic
96C3×C4×D4—21no—
97C3×C4×D4C2311yescharacteristic
98C3×#117—21no—
99C2×C3×C42C2311yescharacteristic
100C2×#216—21no—
101C2×#216—21no—
102C2×#215—21no—
103C2×#214—21no—
104C2×#214—21no—
105C2×#213—21no—
106#930—21no—
107#930—21no—
108C2×C4⋊Dic3—21no—
109C2×C4⋊Dic3—21no—
110C2×#208—21no—
111#875—21no—
112#874C2311yescharacteristic
113C4×D12—21no—
114C4×D12—21no—
115#869—21no—
116#869—21no—
117#869—21no—
118#869C2311yescharacteristic
119#869C2311yescharacteristic
120#869C2311yescharacteristic
121C3×C4⋊C8—21no—
122C3×C4⋊C8—21no—
123C3×C4⋊C8C2311yescharacteristic
124C3×C4⋊C8C2311yescharacteristic
125C3×C22⋊C8—21no—
126C3×C22⋊C8—21no—
127C3×C22⋊C8—21no—
128C3×#97—21no—
129C3×#97D411yescharacteristic
130C3×#97D411yescharacteristic
131C3×C4×C8—21no—
132C3×C4×C8—21no—
133C3×C4×C8C2311yescharacteristic
134C3×C4×C8C2311yescharacteristic
135#835—21no—
136#832—21no—
137#830—21no—
138#821—21no—
139#821—21no—
140#821—21no—
141#821—21no—
142#816—21no—
143#816—21no—
144#816—21no—
145#816—21no—
146#810—21no—
147#810—21no—
148#810—21no—
149#810—21no—
150#809—21no—
151#809—21no—
152#808—21no—
153#807—21no—
154C4⋊(C3⋊C8)—21no—
155C4⋊(C3⋊C8)C2311yescharacteristic
156C4⋊(C3⋊C8)C2311yescharacteristic
157C4⋊(C3⋊C8)C2311yescharacteristic
158C4⋊(C3⋊C8)C2311yescharacteristic
159#803—21no—
160#532—31no—
161C2×C4⋊Q8—31no—
162#529C2×D311yescharacteristic
163#497—61no—
164#491—61no—
165#489—61no—
166#489—61no—
167#487—31no—
168#487—31no—
169#486—31no—
170#486—31no—
171#478—61no—
172#475—31no—
173#474—31no—
174#469—61no—
175#468—61no—
176#463—61no—
177#462—31no—
178#462—31no—
179#461—31no—
180#461—31no—
181#460—31no—
182#459—31no—
183#451—61no—
184#448—61no—
185#446C2×D311yescharacteristic
186C8×Q8C2×D311yescharacteristic
187#436—21no—
188#436C2×D311yescharacteristic
189#435—21no—
190C8×D4C2×D311yescharacteristic
191#433—21no—
192#432—31no—
193#431C2×D311yescharacteristic
194#423—31no—
195#417—61no—
196#417—61no—
197#405—21no—
198C4×M16C2×D311yescharacteristic
199#335—31no—
200#335—31no—
201#334—31no—
202#334—31no—
203#332—31no—
204#332—31no—
205#332—31no—
206#332—31no—
207#332—31no—
208#332—31no—
209#331—31no—
210#331—31no—
211C2×C3×Q8C2×D411yescharacteristic
212C2×C3×D4—21no—
213C2×C3×D4C2×D411yescharacteristic
214C22×C3×C4—21no—
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4#4111yescharacteristic
218C2×#67—41no—
219C22×Dic3—41no—
220#235—42no—
221#235—42no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×C4×D3—41no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—41no—
230C2×Dic6—81no—
231C2×Dic6—81no—
232C3×M16—41no—
233C3×M16—41no—
234C3×M16—42no—
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×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4C2×D411yescharacteristic
247C3×C4⋊C4C2×D411yescharacteristic
248C3×C22⋊C4—21no—
249C3×C22⋊C4—21no—
250C3×C22⋊C4—21no—
251C3×C22⋊C4—21no—
252C3×C22⋊C4—21no—
253C3×C42—21no—
254C3×C42C2×D411yescharacteristic
255C3×C42C2×D411yescharacteristic
256C3×C42#4111yescharacteristic
257C3×C42C2411yescharacteristic
258#216—81no—
259#216—81no—
260#215—81no—
261#214—81no—
262#214—81no—
263#213—81no—
264(C2×D3)⋊C4—41no—
265(C2×D3)⋊C4—41no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—41no—
273C4⋊Dic3—41no—
274#208—81no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×#41—61no—
280C22×Q8—61no—
281C2×Q16—61no—
282C2×Q16—61no—
283C2×SD16—61no—
284C2×SD16—61no—
285C2×SD16—61no—
286C2×D8—61no—
287C2×M16—21no—
288C2×M16—61no—
289C2×M16#6711yescharacteristic
290C2×M16#6711yescharacteristic
291C22×C8—21no—
292C4⋊Q8—31no—
293C4⋊Q8—31no—
294C4⋊Q8—31no—
295C4⋊Q8—61no—
296C4⋊Q8—61no—
297C4⋊Q8—61no—
298#127—31no—
299#126—21no—
300#125D1211yescharacteristic
301#124—61no—
302#124D1211yescharacteristic
303#123—21no—
304C22⋊Q8D1211yescharacteristic
305#121—61no—
306#121—61no—
307#121D1211yescharacteristic
308C4×Q8C22×D311yescharacteristic
309C4×D4—21no—
310C4×D4—61no—
311C4×D4—61no—
312C4×D4C22×D311yescharacteristic
313#117—21no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4—61no—
316C2×C42C22×D311yescharacteristic
317#107—61no—
318#106—61no—
319C4⋊C8—21no—
320C4⋊C8—21no—
321C4⋊C8—31no—
322C4⋊C8—31no—
323C4⋊C8—31no—
324C4⋊C8—31no—
325C4⋊C8—61no—
326C4⋊C8C22×D311yescharacteristic
327C4⋊C8C22×D311yescharacteristic
328#103—61no—
329#103—61no—
330#103—61no—
331#103—61no—
332#103—61no—
333#103—61no—
334#103—61no—
335#103—61no—
336#103—61no—
337#102—61no—
338#102—61no—
339#102—61no—
340#102—61no—
341#102—61no—
342#102—61no—
343#102—61no—
344C22⋊C8—21no—
345C22⋊C8—21no—
346C22⋊C8—21no—
347C22⋊C8—61no—
348#97—21no—
349#97—61no—
350#97#6711yescharacteristic
351#97#6711yescharacteristic
352C4×C8—21no—
353C4×C8—21no—
354C4×C8C22×D311yescharacteristic
355C4×C8C22×D311yescharacteristic
356C23×C3—21no—
357C23×C3#12111yescharacteristic
358C22×D3—41no—
359C3×Q8—21no—
360C3×Q8—21no—
361C3×D4—21no—
362C3×D4—21no—
363C3×D4—81no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4#12111yescharacteristic
375C2×C3×C4#12111yescharacteristic
376C2×C3×C4#12111yescharacteristic
377C2×C3×C4C22×D411yescharacteristic
378C2×C3×C4C22×D411yescharacteristic
379C2×C3×C4C2×#4111yescharacteristic
380#67—82no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—82no—
385D12—41no—
386D12—41no—
387D12—41no—
388D12—41no—
389D12—81no—
390C4×D3—82no—
391Dic6—41no—
392Dic6—41no—
393Dic6—41no—
394Dic6—41no—
395Dic6—41no—
396Dic6—41no—
397Dic6—82no—
398C3×C8—21no—
399C3×C8—21no—
400C3×C8—21no—
401C3×C8—21no—
402C3×C8—41no—
403C3×C8—41no—
404C3⋊C8—41no—
405C3⋊C8—41no—
406C3⋊C8—41no—
407C3⋊C8—41no—
408#41—122no—
409#41—122no—
410C2×Q8—61no—
411C2×Q8—61no—
412C2×Q8—61no—
413C2×Q8—121no—
414C2×Q8—241no—
415C2×Q8—241no—
416C2×Q8C2×D1211yescharacteristic
417C2×D4—21no—
418C2×D4—61no—
419C2×D4—61no—
420C2×D4—61no—
421C2×D4—121no—
422C2×D4C2×D1211yescharacteristic
423C22×C4—21no—
424C22×C4—121no—
425C22×C4—121no—
426C22×C4C2×D1211yescharacteristic
427C22×C4#23511yescharacteristic
428C22×C4C2×#6711yescharacteristic
429Q16—241no—
430Q16—241no—
431SD16—241no—
432SD16—241no—
433SD16—241no—
434D8—241no—
435M16—41no—
436M16—41no—
437M16—42no—
438M16—241no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—42no—
442C2×C8—61no—
443C2×C8—61no—
444C2×C8—61no—
445C2×C8—61no—
446C2×C8C2×#6711yescharacteristic
447C2×C8C2×#6711yescharacteristic
448C2×C8C2×#6711yescharacteristic
449C2×C8C2×#6711yescharacteristic
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—61no—
457C4⋊C4—61no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4—121no—
461C4⋊C4—121no—
462C4⋊C4C2×D1211yescharacteristic
463C4⋊C4C2×D1211yescharacteristic
464C22⋊C4—21no—
465C22⋊C4—21no—
466C22⋊C4—21no—
467C22⋊C4—21no—
468C22⋊C4—21no—
469C22⋊C4—121no—
470C22⋊C4—121no—
471C42—21no—
472C42C2×D1211yescharacteristic
473C42#23511yescharacteristic
474C42C2×#6711yescharacteristic
475C42C23×D311yescharacteristic
476C22×C3—21no—
477C22×C3—21no—
478C22×C3—21no—
479C22×C3—41no—
480C22×C3—41no—
481C22×C3—41no—
482C22×C3C2×#12111yescharacteristic
483C2×D3—81no—
484C2×D3—81no—
485C2×D3—81no—
486C3×C4—21no—
487C3×C4—21no—
488C3×C4—21no—
489C3×C4—21no—
490C3×C4—41no—
491C3×C4—41no—
492C3×C4—41no—
493C3×C4#57511yescharacteristic
494C3×C4#57511yescharacteristic
495C3×C4#57611yescharacteristic
496C3×C4#57711yescharacteristic
497Dic3—81no—
498Dic3—81no—
499Dic3—81no—
500C23—21no—
501C23—121no—
502C23#93011yescharacteristic
503Q8—21no—
504Q8—21no—
505Q8—121no—
506Q8—121no—
507Q8—121no—
508Q8—121no—
509Q8—121no—
510Q8—121no—
511Q8—242no—
512D4—21no—
513D4—21no—
514D4—81no—
515D4—121no—
516D4—121no—
517D4—121no—
518D4—121no—
519D4—241no—
520D4—242no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—121no—
532C2×C4—121no—
533C2×C4—121no—
534C2×C4—242no—
535C2×C4—242no—
536C2×C4#93011yescharacteristic
537C2×C4#93011yescharacteristic
538C2×C4#93011yescharacteristic
539C2×C4C22×D1211yescharacteristic
540C2×C4C2×#23511yescharacteristic
541C2×C4C22×#6711yescharacteristic
542C8—21no—
543C8—21no—
544C8—21no—
545C8—21no—
546C8—41no—
547C8—41no—
548C8—121no—
549C8—121no—
550C8—121no—
551C8—121no—
552C2×C3—41no—
553C2×C3—81no—
554C2×C3#304011yescharacteristic
555C2×C3#306211yescharacteristic
556C2×C3#306911yescharacteristic
557D3—161no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—241no—
565C22—241no—
566C22—241no—
567C22C2×#93011yescharacteristic
568C4—21no—
569C4—21no—
570C4—21no—
571C4—21no—
572C4—41no—
573C4—41no—
574C4—41no—
575C4—241no—
576C4—241no—
577C4—241no—
578C4#575111yescharacteristic
579C4#575211yescharacteristic
580C4#582211yescharacteristic
581C4#582211yescharacteristic
582C3#1938311yescharacteristic, complemented, Sylow
583C2—41no—
584C2—81no—
585C2—481no—
586C2#8252711yescharacteristic
587C2#8380211yescharacteristic
588C2#8381611yescharacteristic
subgroupG/H
centerC22C2×#930
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#2969C2
socleC22×C3C2×#121
prime radical#83150C2
hypercenter#2969D3
hypocenterC3#19383
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2969

Complements

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

NH ≅ G/Ntype
1. C3×#2969C2semidirect
2. #83154C2semidirect
4. #78528C2semidirect
5. #70966C2semidirect
6. #70961C2semidirect
7. #70946C2semidirect
8. #70941C2semidirect
9. #70875C2semidirect
10. #70870C2semidirect
11. #70854C2semidirect
12. #70849C2semidirect
13. #70778C2semidirect
14. #70775C2semidirect
15. #69851C2semidirect
71. #2969D3semidirect
582. C3#19383semidirect

Maximal subgroups

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

subgroupindex
C3×#29692
#831542
#831502
#785282
#709662
#709612
#709462
#709412
#708752
#708702
#708542
#708492
#707782
#707752
#698512
#193833