Finite Group Databasebeta

← #10778230

Subgroups — #10778230

Order 768.

592 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3185C211yesmaximal, characteristic, complemented
2#83352C211yesmaximal, characteristic
3#83349C211yesmaximal, characteristic
4#78512C211yesmaximal, characteristic
5#71832C211yesmaximal, characteristic, complemented
6#71831C211yesmaximal, characteristic, complemented
7#71348C211yesmaximal, characteristic, complemented
8#71345C211yesmaximal, characteristic, complemented
9#71308C211yesmaximal, characteristic, complemented
10#71306C211yesmaximal, characteristic, complemented
11#71241C211yesmaximal, characteristic, complemented
12#71240C211yesmaximal, characteristic, complemented
13#71235C211yesmaximal, characteristic, complemented
14#71234C211yesmaximal, characteristic, complemented
15#69850C211yesmaximal, characteristic, complemented
16#19185—31nomaximal, Sylow
17C3×#533C2211yescharacteristic
18C3×#533C2211yescharacteristic
19C4×#235C2211yescharacteristic
20C3×#498C2211yescharacteristic
21C3×#494C2211yescharacteristic
22C3×#487C2211yescharacteristic
23C3×#487C2211yescharacteristic
24C3×#470—21no—
25C3×#470—21no—
26C3×#441—21no—
27C3×#441—21no—
28C3×C4×M16C2211yescharacteristic
29#5128—21no—
30#5128—21no—
31#5094C2211yescharacteristic
32#5094C2211yescharacteristic
33#5091C2211yescharacteristic
34#5090C2211yescharacteristic
35#5087C2211yescharacteristic
36#5087C2211yescharacteristic
37#5084C2211yescharacteristic
38#5083C2211yescharacteristic
39#5066—21no—
40#5063—21no—
41#5047—21no—
42#5045—21no—
43C4×#216—21no—
44C4×#215—21no—
45#5002—21no—
46#4998—21no—
47#4926C2211yescharacteristic
48#4925C2211yescharacteristic
49C2×C4⋊(C3⋊C8)C2211yescharacteristic
50#4707—21no—
51#4705—21no—
52#4688C2211yescharacteristic
53#4688C2211yescharacteristic
54C8×Dic6C2211yescharacteristic
55C8×Dic6C2211yescharacteristic
56#4547C2211yescharacteristic
57#4547C2211yescharacteristic
58#4547C2211yescharacteristic
59#4547C2211yescharacteristic
60#4547C2211yescharacteristic
61#4547C2211yescharacteristic
62#4546C2211yescharacteristic
63#4546C2211yescharacteristic
64#4458C2211yescharacteristic
65#4458C2211yescharacteristic
66#4457C2211yescharacteristic
67#4457C2211yescharacteristic
68#3185D311yescharacteristic, complemented
69#3107—31no—
70#3076—31no—
71#2971—31no—
72#1703—31no—
73#1699—31no—
74#1691—31no—
75#1689—31no—
76#1684—31no—
77#1680—31no—
78#1672—31no—
79#1670—31no—
80#1565—31no—
81#1554—31no—
82#1552—31no—
83C2×#235D411yescharacteristic
84C2×C3×Q16—21no—
85C2×C3×Q16—21no—
86C2×C3×M16D411yescharacteristic
87C2×C3×M16D411yescharacteristic
88C3×C4⋊Q8C2311yescharacteristic
89C3×C4⋊Q8C2311yescharacteristic
90C3×C4⋊Q8C2311yescharacteristic
91C3×C4⋊Q8C2311yescharacteristic
92C3×#125—21no—
93C3×#125—21no—
94C3×C22⋊Q8—21no—
95C3×C22⋊Q8—21no—
96C3×C22⋊Q8—21no—
97C3×C22⋊Q8—21no—
98C3×C4×Q8—21no—
99C3×C4×Q8—21no—
100C3×C4×Q8—21no—
101C3×C4×Q8—21no—
102C3×#117—21no—
103C3×#117—21no—
104C2×C3×C42C2311yescharacteristic
105C2×#216—21no—
106C2×#215—21no—
107C4×#67—21no—
108C4×#67—21no—
109#926D411yescharacteristic
110C22×C3⋊C8—21no—
111C2×ΓL(1,𝔽25)—21no—
112C2×ΓL(1,𝔽25)—21no—
113C4×D12D411yescharacteristic
114C4×D12C2311yescharacteristic
115#872—21no—
116C42×D3—21no—
117C4×Dic6D411yescharacteristic
118C4×Dic6C2311yescharacteristic
119C3×#106—21no—
120C3×#106—21no—
121C3×#103—21no—
122C3×#103—21no—
123C3×#103—21no—
124C3×#103—21no—
125C3×#97D411yescharacteristic
126C3×#97D411yescharacteristic
127C3×C4×C8C2311yescharacteristic
128C3×C4×C8C2311yescharacteristic
129#835—21no—
130#835—21no—
131#830—21no—
132#830—21no—
133(C2×D3)⋊C8—21no—
134(C2×D3)⋊C8—21no—
135Dic3⋊C8—21no—
136Dic3⋊C8—21no—
137C8×Dic3—21no—
138C8×Dic3—21no—
139#810—21no—
140#810—21no—
141#810—21no—
142#809—21no—
143#809—21no—
144#809—21no—
145#808—21no—
146#808—21no—
147#808—21no—
148#807—21no—
149#807—21no—
150#807—21no—
151C4⋊(C3⋊C8)—21no—
152C4⋊(C3⋊C8)C2311yescharacteristic
153C4⋊(C3⋊C8)C2311yescharacteristic
154C4⋊(C3⋊C8)C2311yescharacteristic
155C4⋊(C3⋊C8)C2311yescharacteristic
156C4⋊(C3⋊C8)C2311yescharacteristic
157C4⋊(C3⋊C8)C2311yescharacteristic
158#803—21no—
159C4×C3⋊C8—21no—
160#533C2×D311yescharacteristic
161#533C2×D311yescharacteristic
162C4×#41—31no—
163#498C2×D311yescharacteristic
164#494C2×D311yescharacteristic
165#487C2×D311yescharacteristic
166#487C2×D311yescharacteristic
167#485—61no—
168#479—61no—
169#478—61no—
170#477—31no—
171#476—31no—
172#475—31no—
173#475—31no—
174#474—31no—
175#474—31no—
176#470—21no—
177#470—21no—
178#462—31no—
179#461—31no—
180#452—61no—
181C8×Q8—31no—
182C8×Q8—31no—
183#441—21no—
184#441—21no—
185C4×Q16—61no—
186C4×SD16—61no—
187#436—31no—
188#436—31no—
189#436—61no—
190#436—61no—
191#432—31no—
192#432—61no—
193#431—31no—
194#427—61no—
195C2×C4⋊C8—31no—
196#416—61no—
197C4×M16—61no—
198C4×M16C2×D311yescharacteristic
199#335—31no—
200#335—31no—
201#334—31no—
202#334—31no—
203#333—31no—
204#333—31no—
205#332—31no—
206#332—31no—
207#332—31no—
208#332—31no—
209#332—31no—
210#332—31no—
211C2×C3×Q8—21no—
212C2×C3×Q8—21no—
213C2×C3×Q8—21no—
214C2×C3×Q8—21no—
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4#4111yescharacteristic
218C2×#67—21no—
219#235—22no—
220#235—41no—
221C2×D12C2×D411yescharacteristic
222C2×C4×D3—21no—
223C2×C4×D3—21no—
224C2×C4×D3—21no—
225C2×Dic6C2×D411yescharacteristic
226C3×Q16—42no—
227C3×Q16—42no—
228C3×M16—22no—
229C3×M16—41no—
230C2×C3×C8C2×D411yescharacteristic
231C2×C3×C8C2×D411yescharacteristic
232C2×C3×C8C2×D411yescharacteristic
233C2×C3×C8C2×D411yescharacteristic
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—41no—
243C3×C4⋊C4—41no—
244C3×C4⋊C4—41no—
245C3×C4⋊C4—41no—
246C3×C22⋊C4—41no—
247C3×C22⋊C4—41no—
248C3×C42—41no—
249C3×C42—41no—
250C3×C42C2×D411yescharacteristic
251C3×C42C2×D411yescharacteristic
252C3×C42#4111yescharacteristic
253C3×C42C2411yescharacteristic
254#217—21no—
255#216—42no—
256#215—42no—
257(C2×D3)⋊C4—21no—
258(C2×D3)⋊C4—21no—
259C4⋊Dic3C2×D411yescharacteristic
260C4⋊Dic3C2×D411yescharacteristic
261Dic3⋊C4—21no—
262Dic3⋊C4—21no—
263C4×Dic3—21no—
264C4×Dic3—21no—
265C4×Dic3—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—42no—
273ΓL(1,𝔽25)—42no—
274ΓL(1,𝔽25)—42no—
275C2×#41—31no—
276C2×Q16—21no—
277C2×Q16—21no—
278C2×Q16—61no—
279C2×SD16—61no—
280C2×M16—61no—
281C2×M16—61no—
282C2×M16#6711yescharacteristic
283C2×M16#6711yescharacteristic
284C22×C8—61no—
285C4⋊Q8C22×D311yescharacteristic
286C4⋊Q8C22×D311yescharacteristic
287C4⋊Q8C22×D311yescharacteristic
288C4⋊Q8C22×D311yescharacteristic
289#125—21no—
290#125—21no—
291C22⋊Q8—21no—
292C22⋊Q8—21no—
293C22⋊Q8—21no—
294C22⋊Q8—21no—
295C4×Q8—21no—
296C4×Q8—21no—
297C4×Q8—21no—
298C4×Q8—21no—
299C4×Q8—31no—
300C4×Q8—31no—
301C4×D4—31no—
302C4×D4—31no—
303C4×D4—61no—
304C4×D4—61no—
305#117—21no—
306#117—21no—
307#117—31no—
308#117—61no—
309C2×C42—61no—
310C2×C42C22×D311yescharacteristic
311#107—61no—
312#107—61no—
313#107—61no—
314#106—21no—
315#106—21no—
316#106—61no—
317#106—61no—
318#106—61no—
319C4⋊C8—31no—
320C4⋊C8—31no—
321C4⋊C8—31no—
322C4⋊C8—31no—
323C4⋊C8—31no—
324C4⋊C8—31no—
325C4⋊C8—61no—
326C4⋊C8—61no—
327C4⋊C8—61no—
328#103—21no—
329#103—21no—
330#103—21no—
331#103—21no—
332#103—61no—
333#103—61no—
334#103—61no—
335#103—61no—
336#103—61no—
337#102—61no—
338#102—61no—
339#102—61no—
340C22⋊C8—61no—
341C22⋊C8—61no—
342C22⋊C8—61no—
343C22⋊C8—61no—
344#97—61no—
345#97#6711yescharacteristic
346#97#6711yescharacteristic
347C4×C8—61no—
348C4×C8—61no—
349C4×C8—61no—
350C4×C8C22×D311yescharacteristic
351C4×C8C22×D311yescharacteristic
352C23×C3#12111yescharacteristic
353C22×D3—21no—
354C3×Q8—41no—
355C3×Q8—41no—
356C3×Q8—41no—
357C3×Q8—41no—
358C3×Q8—41no—
359C3×Q8—41no—
360C3×Q8—81no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4#12111yescharacteristic
371C2×C3×C4#12111yescharacteristic
372C2×C3×C4#12111yescharacteristic
373C2×C3×C4C22×D411yescharacteristic
374C2×C3×C4C22×D411yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376#67—42no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—21no—
380D12—21no—
381D12—21no—
382C4×D3—42no—
383C4×D3—41no—
384C4×D3—41no—
385C4×D3—81no—
386Dic6—21no—
387Dic6#13712yes—
388C3×C8—21no—
389C3×C8—21no—
390C3×C8—21no—
391C3×C8#13512yes—
392C3⋊C8—41no—
393C3⋊C8—41no—
394C3⋊C8—41no—
395C3⋊C8—41no—
396C3⋊C8—41no—
397C3⋊C8—41no—
398#41—62no—
399#41—121no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—31no—
405C2×D4—31no—
406C2×D4—61no—
407C22×C4—61no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4D3×D411yescharacteristic
411C22×C4#23711yescharacteristic
412C22×C4C2×#6711yescharacteristic
413Q16—42no—
414Q16—42no—
415Q16—122no—
416SD16—122no—
417M16—22no—
418M16—41no—
419M16—122no—
420M16—122no—
421C2×C8—61no—
422C2×C8—61no—
423C2×C8—61no—
424C2×C8—61no—
425C2×C8—61no—
426C2×C8—61no—
427C2×C8—122no—
428C2×C8C2×#6711yescharacteristic
429C2×C8C2×#6711yescharacteristic
430C2×C8C2×#6711yescharacteristic
431C2×C8C2×#6711yescharacteristic
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—31no—
441C4⋊C4—31no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C22⋊C4—41no—
449C22⋊C4—41no—
450C22⋊C4—61no—
451C22⋊C4—61no—
452C22⋊C4—61no—
453C42—41no—
454C42—41no—
455C42—61no—
456C42—61no—
457C42—61no—
458C42D3×D411yescharacteristic
459C42#23711yescharacteristic
460C42C2×#6711yescharacteristic
461C42C23×D311yescharacteristic
462C22×C3—21no—
463C22×C3—21no—
464C22×C3—21no—
465C22×C3C2×#12111yescharacteristic
466C2×D3—41no—
467C2×D3—41no—
468C2×D3—41no—
469C3×C4—21no—
470C3×C4—21no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—81no—
474C3×C4—81no—
475C3×C4—81no—
476C3×C4—81no—
477C3×C4C2×#13511yescharacteristic
478C3×C4C2×#13711yescharacteristic
479C3×C4#57511yescharacteristic
480C3×C4#57511yescharacteristic
481Dic3—22no—
482Dic3—41no—
483Dic3—41no—
484C23—61no—
485C23#93811yescharacteristic
486Q8—32no—
487Q8—41no—
488Q8—41no—
489Q8—41no—
490Q8—41no—
491Q8—41no—
492Q8—41no—
493Q8—61no—
494Q8—81no—
495D4—61no—
496D4—61no—
497D4—122no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—61no—
508C2×C4—61no—
509C2×C4—61no—
510C2×C4—122no—
511C2×C4—121no—
512C2×C4—121no—
513C2×C4—241no—
514C2×C4#93811yescharacteristic
515C2×C4#93811yescharacteristic
516C2×C4#93811yescharacteristic
517C2×C4C2×D3×D411yescharacteristic
518C2×C4C2×#23711yescharacteristic
519C2×C4C22×#6711yescharacteristic
520C8—21no—
521C8—21no—
522C8—21no—
523C8—121no—
524C8—121no—
525C8—121no—
526C8—121no—
527C8—121no—
528C8—121no—
529C8#95012yes—
530C2×C3—41no—
531C2×C3#303511yescharacteristic
532C2×C3#305611yescharacteristic
533C2×C3#306911yescharacteristic
534D3—81no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—121no—
539C22—121no—
540C22—121no—
541C22C2×#93811yescharacteristic
542C4—21no—
543C4—21no—
544C4—21no—
545C4—21no—
546C4—62no—
547C4—81no—
548C4—81no—
549C4—81no—
550C4—81no—
551C4—121no—
552C4—121no—
553C4D3×#13711yescharacteristic
554C4#577711yescharacteristic
555C4C2×#95011yescharacteristic
556C4#582211yescharacteristic
557C3#1918511yescharacteristic, complemented, Sylow
558C2—41no—
559C2—241no—
560C2#8329611yescharacteristic
561C2#8398511yescharacteristic
562C2#8399811yescharacteristic
subgroupG/H
centerC22C2×#938
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#3185C2
socleC22×C3C2×#121
prime radicalC2×#235D4
hypercenter#3185D3
hypocenterC3#19185
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3185

Complements

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

NH ≅ G/Ntype
1. C3×#3185C2semidirect
5. #71832C2semidirect
6. #71831C2semidirect
7. #71348C2semidirect
8. #71345C2semidirect
9. #71308C2semidirect
10. #71306C2semidirect
11. #71241C2semidirect
12. #71240C2semidirect
13. #71235C2semidirect
14. #71234C2semidirect
15. #69850C2semidirect
68. #3185D3semidirect
557. C3#19185semidirect

Maximal subgroups

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

subgroupindex
C3×#31852
#833522
#833492
#785122
#718322
#718312
#713482
#713452
#713082
#713062
#712412
#712402
#712352
#712342
#698502
#191853