Finite Group Databasebeta

← #11675074

Subgroups — #11675074

Order 768.

2126 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#550C211yesmaximal, characteristic, complemented
2Q8×#235C211yesmaximal, characteristic, complemented
3#83587C216yesmaximal, complemented
4#83586C212yesmaximal, complemented
5#83581C216yesmaximal, complemented
6#83580C212yesmaximal, complemented
7#83165C213yesmaximal, complemented
8#83159C213yesmaximal, complemented
9Q8×#208C211yesmaximal, characteristic, complemented
10#82296C213yesmaximal, complemented
11C4×#932C213yesmaximal, complemented
12#61321—31nomaximal, Sylow
13C22×C3×#41—21no—
14C3×#550—21no—
15C3×#550—23no—
16C3×#550C2212yes—
17C3×#550C2216yes—
18C2×C3×#127C2213yes—
19C2×C3×#121—23no—
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×D4C2213yes—
22Q8×#67—21no—
23C2×#932C2213yes—
24#5579C2213yescomplemented
25#5578—23no—
26Q8×D12C2211yescharacteristic, complemented
27C4×D3×Q8—21no—
28#5567C2213yescomplemented
29Q8×Dic6C2211yescharacteristic, complemented
30#5511—23no—
31C4×#235C2213yes—
32#5470C2213yes—
33#5215C2213yes—
34#5215C2216yes—
35#5075C2216yescomplemented
36#5075C2216yescomplemented
37#5072C2216yescomplemented
38#5072C2216yescomplemented
39#5071C22112yescomplemented
40#5070C2216yescomplemented
41#5070C2216yescomplemented
42#5024C2216yescomplemented
43Q8×C3⋊C8C2212yescomplemented
44#5018C2216yescomplemented
45C4×#214C2216yescomplemented
46#5014C2216yescomplemented
47C4×#213C2216yescomplemented
48#5009C22112yescomplemented
49#5008C2216yescomplemented
50#5007C2216yescomplemented
51#4966C2211yescharacteristic
52#4964C2213yes—
53#4924C2213yes—
54C4×#208C2213yes—
55Q8×#41—31no—
56C2×#550D311yescharacteristic, complemented
57#3368—36no—
58#3366—36no—
59#3362—32no—
60#3358—32no—
61#3138—33no—
62#3077—33no—
63#3038—33no—
64Q8×M16—31no—
65C4×AGL(1,ℤ/8)—33no—
66C2×C3×#41—26no—
67C2×C3×#41—22no—
68C2×C3×#41—48no—
69C22×C3×Q8D411yescharacteristic
70C22×C3×D4—23no—
71C22×C3×D4—23no—
72C22×C3×D4C2313yes—
73C23×C3×C4—23no—
74C2×D3×Q8—21no—
75C2×#235C2311yescharacteristic
76C3×#127—23no—
77C3×#127—23no—
78C3×#127C2316yes—
79C3×#127C2316yes—
80C3×#121—26no—
81C3×#121—26no—
82C3×#121—212no—
83C3×C4×Q8D411yescharacteristic
84C3×C4×Q8D413yes—
85C3×C4×Q8C2311yescharacteristic
86C3×C4×Q8C2313yes—
87C3×C4×D4—23no—
88C3×C4×D4—23no—
89C3×C4×D4C2316yes—
90C3×C4×D4C2316yes—
91C2×C3×C4⋊C4D413yes—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C22⋊C4—23no—
94C2×C3×C42C2313yes—
95#946—23no—
96Q8×Dic3—21no—
97#944—23no—
98C2×#214C2316yes—
99#932—212no—
100C2×#213C2316yes—
101C4×#67—23no—
102C4×#67—23no—
103#926C2313yes—
104#924—23no—
105#924—23no—
106C2×#208C2311yescharacteristic
107C2×#208C2313yes—
108#897—23no—
109#896—26no—
110D3×C4⋊C4—23no—
111#889—26no—
112#888—23no—
113#887—23no—
114C4×D12C2313yes—
115C4×D12C2313yes—
116#872—23no—
117C42×D3—23no—
118#870C2316yes—
119#869C2313yes—
120#869C2313yes—
121C4×Dic6—23no—
122C4×Dic6—23no—
123C4×Dic6C2313yes—
124C4×Dic6C2313yes—
125#832C23112yes—
126#832C23112yes—
127#832C2316yes—
128#832C2316yes—
129#810C2312yes—
130#809C2312yes—
131#808C2316yes—
132#807C2316yes—
133C4⋊(C3⋊C8)C2316yes—
134C4⋊(C3⋊C8)C2316yes—
135#803C2316yes—
136C4×C3⋊C8C2316yes—
137C22×#41—21no—
138C2×AGL(1,ℤ/8)—33no—
139Q82—31no—
140#557—33no—
141#554—33no—
142#554—63no—
143#550—21no—
144#550—23no—
145#550C2×D312yes—
146#550C2×D316yes—
147D4×Q8—31no—
148D4×Q8—61no—
149#533—33no—
150#533—63no—
151C2×#127C2×D313yes—
152C2×#121—23no—
153C4×#41—33no—
154C2×C4×Q8—61no—
155C2×C4×Q8C2×D311yescharacteristic
156C2×C4×D4C2×D313yes—
157#490—312no—
158#486—36no—
159#486—36no—
160#478—312no—
161#476—36no—
162#474—36no—
163#460—36no—
164#460—36no—
165#459—36no—
166#459—36no—
167#446—36no—
168C8×Q8—32no—
169#442—36no—
170#440—36no—
171C4×SD16—36no—
172C4×D8—36no—
173#428—33no—
174#423—33no—
175#418—36no—
176#418—33no—
177#417—31no—
178C4×M16—33no—
179C24×C3—23no—
180C3×#41—424no—
181C3×#41—48no—
182C2×C3×Q8—24no—
183C2×C3×Q8C2×D413yes—
184C2×C3×Q8C2×D411yescharacteristic
185C2×C3×D4—23no—
186C2×C3×D4—26no—
187C2×C3×D4—26no—
188C2×C3×D4—26no—
189C2×C3×D4—26no—
190C2×C3×D4—26no—
191C2×C3×D4—26no—
192C2×C3×D4—412no—
193C2×C3×D4—86no—
194C2×C3×D4C2416yes—
195C22×C3×C4—21no—
196C22×C3×C4—26no—
197C22×C3×C4—26no—
198C22×C3×C4—412no—
199C22×C3×C4C2×D413yes—
200C22×C3×C4C2×D413yes—
201C22×C3×C4C2411yescharacteristic
202C2×#67—21no—
203D3×Q8—44no—
204#235—22no—
205#235#4114yescomplemented
206C2×D12C2411yescharacteristic
207C2×C4×D3—21no—
208C2×C4×D3—23no—
209C2×C4×D3—23no—
210C2×Dic6—23no—
211C2×Dic6—23no—
212C2×Dic6C2411yescharacteristic
213C3×C4⋊C4—21no—
214C3×C4⋊C4C2×D413yes—
215C3×C4⋊C4C2×D413yes—
216C3×C4⋊C4C2×D416yes—
217C3×C4⋊C4C2412yes—
218C3×C22⋊C4—26no—
219C3×C22⋊C4—26no—
220C3×C42C2×D413yes—
221C3×C42C2×D413yes—
222C3×C42C2413yes—
223C3×C42C2413yes—
224#217—23no—
225#214—212no—
226#213—212no—
227(C2×D3)⋊C4—23no—
228(C2×D3)⋊C4—23no—
229C4⋊Dic3—23no—
230C4⋊Dic3—23no—
231C4⋊Dic3C2413yes—
232C4⋊Dic3C2413yes—
233Dic3⋊C4—26no—
234Dic3⋊C4—26no—
235Dic3⋊C4—23no—
236Dic3⋊C4—23no—
237C4×Dic3—23no—
238C4×Dic3—23no—
239C4×Dic3—23no—
240#208—26no—
241#208—22no—
242C2×C3⋊C8C2412yes—
243C2×C3⋊C8C2416yes—
244C2×#41—26no—
245C2×#41—22no—
246C2×#41—31no—
247C2×#41—48no—
248C22×Q8—61no—
249C22×Q8#6711yescharacteristic
250C22×D4—23no—
251C22×D4—23no—
252C22×D4C22×D313yes—
253C23×C4—23no—
254AGL(1,ℤ/8)—612no—
255C2×SD16—36no—
256C2×D8—36no—
257C2×M16—33no—
258C2×M16—31no—
259C4⋊Q8—33no—
260C4⋊Q8—33no—
261C4⋊Q8—63no—
262C4⋊Q8—63no—
263#127—23no—
264#127—23no—
265#127C22×D316yes—
266#127C22×D316yes—
267#125—36no—
268#125—66no—
269C22⋊Q8—63no—
270C22⋊Q8—63no—
271C22⋊Q8—63no—
272C22⋊Q8—63no—
273C22⋊Q8—66no—
274#121—26no—
275#121—212no—
276#121—26no—
277C4×Q8—33no—
278C4×Q8—33no—
279C4×Q8—61no—
280C4×Q8—63no—
281C4×Q8—63no—
282C4×Q8—63no—
283C4×Q8#6711yescharacteristic
284C4×Q8#6713yes—
285C4×Q8C22×D311yescharacteristic
286C4×Q8C22×D313yes—
287C4×D4—23no—
288C4×D4—23no—
289C4×D4—33no—
290C4×D4—33no—
291C4×D4—63no—
292C4×D4—63no—
293C4×D4C22×D316yes—
294C4×D4C22×D316yes—
295#117—33no—
296#117—63no—
297C2×C4⋊C4—63no—
298C2×C4⋊C4#6713yes—
299C2×C4⋊C4C22×D311yescharacteristic
300C2×C22⋊C4—23no—
301C2×C42—63no—
302C2×C42C22×D313yes—
303#107—36no—
304#106—36no—
305C4⋊C8—36no—
306C4⋊C8—36no—
307#103—32no—
308#102—36no—
309#102—36no—
310#102—312no—
311#102—312no—
312#102—32no—
313#97—36no—
314C4×C8—36no—
315C23×C3—26no—
316C23×C3—46no—
317C23×C3—46no—
318C23×C3—46no—
319C23×C3C22×D411yescharacteristic
320C22×D3—21no—
321C3×Q8—26no—
322C3×Q8AGL(1,ℤ/8)14yescomplemented
323C3×D4—26no—
324C3×D4—26no—
325C3×D4—412no—
326C3×D4—46no—
327C3×D4—412no—
328C3×D4—86no—
329C3×D4—86no—
330C2×C3×C4—26no—
331C2×C3×C4—22no—
332C2×C3×C4—43no—
333C2×C3×C4—412no—
334C2×C3×C4—42no—
335C2×C3×C4—412no—
336C2×C3×C4C22×D413yes—
337C2×C3×C4C22×D413yes—
338C2×C3×C4C22×D413yes—
339C2×C3×C4C22×D411yescharacteristic
340C2×C3×C4C22×D413yes—
341C2×C3×C4C2×#4112yes—
342C2×C3×C42−1+412yes—
343C2×C3×C4C2511yescharacteristic
344#67—24no—
345C2×Dic3—21no—
346C2×Dic3—23no—
347C2×Dic3—23no—
348D12—21no—
349D12C2×#4112yescomplemented
350C4×D3—24no—
351C4×D3—46no—
352C4×D3—83no—
353Dic6—21no—
354Dic6—46no—
355Dic6—83no—
356Dic6C2×#4112yescomplemented
357C3⋊C8—26no—
358C3⋊C8—22no—
359C24—23no—
360#41—34no—
361#41—424no—
362#41—48no—
363#41—62no—
364C2×Q8—24no—
365C2×Q8—31no—
366C2×Q8—63no—
367C2×Q8—63no—
368C2×Q8—124no—
369C2×Q8C2×#6713yes—
370C2×Q8C2×#6711yescharacteristic
371C2×D4—26no—
372C2×D4—26no—
373C2×D4—23no—
374C2×D4—26no—
375C2×D4—26no—
376C2×D4—26no—
377C2×D4—26no—
378C2×D4—31no—
379C2×D4—412no—
380C2×D4—61no—
381C2×D4—86no—
382C2×D4C23×D316yes—
383C22×C4—21no—
384C22×C4—26no—
385C22×C4—26no—
386C22×C4—412no—
387C22×C4—61no—
388C22×C4—63no—
389C22×C4—63no—
390C22×C4C2×#6713yes—
391C22×C4C2×#6713yes—
392C22×C4C23×D311yescharacteristic
393SD16—612no—
394D8—612no—
395M16—66no—
396M16—62no—
397C2×C8—32no—
398C2×C8—36no—
399C4⋊C4—21no—
400C4⋊C4—33no—
401C4⋊C4—33no—
402C4⋊C4—66no—
403C4⋊C4—63no—
404C4⋊C4—63no—
405C4⋊C4—63no—
406C4⋊C4—66no—
407C4⋊C4—63no—
408C4⋊C4C2×#6713yes—
409C4⋊C4C2×#6716yes—
410C4⋊C4C2×#6713yes—
411C4⋊C4C23×D312yes—
412C22⋊C4—26no—
413C22⋊C4—26no—
414C22⋊C4—63no—
415C22⋊C4—63no—
416C22⋊C4—63no—
417C42—63no—
418C42—63no—
419C42—63no—
420C42C2×#6713yes—
421C42C2×#6713yes—
422C42C23×D313yes—
423C42C23×D313yes—
424C22×C3—21no—
425C22×C3—21no—
426C22×C3—46no—
427C22×C3—46no—
428C22×C3—46no—
429C22×C3—812no—
430C22×C3#54812yes—
431C22×C3C23×D411yescharacteristic
432C2×D3—22no—
433C2×D3—41no—
434C2×D3—41no—
435C3×C4—23no—
436C3×C4—23no—
437C3×C4—23no—
438C3×C4—42no—
439C3×C4#54812yes—
440C3×C4C2×AGL(1,ℤ/8)16yes—
441C3×C4C22×#4111yescharacteristic
442C3×C4C2×2−1+411yescharacteristic
443Dic3—22no—
444Dic3—43no—
445Dic3—43no—
446C23—26no—
447C23—46no—
448C23—46no—
449C23—46no—
450C23—61no—
451C23C22×#6711yescharacteristic
452Q8—26no—
453Q8—32no—
454Q8—61no—
455Q8—126no—
456Q8—243no—
457Q8#93214yescomplemented
458D4—26no—
459D4—26no—
460D4—32no—
461D4—412no—
462D4—412no—
463D4—46no—
464D4—64no—
465D4—61no—
466D4—86no—
467D4—86no—
468C2×C4—26no—
469C2×C4—22no—
470C2×C4—43no—
471C2×C4—412no—
472C2×C4—42no—
473C2×C4—412no—
474C2×C4—61no—
475C2×C4—64no—
476C2×C4—63no—
477C2×C4—63no—
478C2×C4—126no—
479C2×C4—243no—
480C2×C4D3×#4112yes—
481C2×C4#101012yes—
482C2×C4C22×#6713yes—
483C2×C4C22×#6713yes—
484C2×C4C22×#6713yes—
485C2×C4C22×#6713yes—
486C2×C4C22×#6711yescharacteristic
487C2×C4C24×D311yescharacteristic
488C8—66no—
489C8—62no—
490C2×C3—22no—
491C2×C3—86no—
492C2×C3C2×#54811yescharacteristic
493C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
494C2×C3#358511yescharacteristic
495D3—42no—
496C22—21no—
497C22—21no—
498C22—46no—
499C22—46no—
500C22—46no—
501C22—62no—
502C22—812no—
503C22—121no—
504C22—121no—
505C22#583112yes—
506C22C23×#6711yescharacteristic
507C4—23no—
508C4—23no—
509C4—23no—
510C4—42no—
511C4—62no—
512C4—123no—
513C4—123no—
514C4C2×#93216yes—
515C4#583112yes—
516C4C2×D3×#4111yescharacteristic
517C4C2×#101011yescharacteristic
518C3#6132111yescharacteristic, complemented, Sylow
519C2—22no—
520C2—86no—
521C2—122no—
522C2C22×#93211yescharacteristic
523C2C2×#583111yescharacteristic
524C2#8727711yescharacteristic
subgroupG/H
centerC22C23×#67
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#550C2
socleC22×C3C23×D4
hypercenterC2×#550D3
hypocenterC3#61321
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ C2×#550

Complements

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

NH ≅ G/Ntype
1. C2×C3×#550C2semidirect
2. Q8×#235C2semidirect
3. #83587C2semidirect
4. #83586C2semidirect
5. #83581C2semidirect
6. #83580C2semidirect
7. #83165C2semidirect
8. #83159C2semidirect
9. Q8×#208C2semidirect
10. #82296C2semidirect
11. C4×#932C2semidirect
24. #5579C22semidirect
26. Q8×D12C22semidirect
28. #5567C22semidirect
29. Q8×Dic6C22semidirect
35. #5075C22semidirect
36. #5075C22semidirect
37. #5072C22semidirect
38. #5072C22semidirect
39. #5071C22semidirect
40. #5070C22semidirect
41. #5070C22semidirect
42. #5024C22semidirect
43. Q8×C3⋊C8C22semidirect
44. #5018C22semidirect
45. C4×#214C22semidirect
46. #5014C22semidirect
47. C4×#213C22semidirect
48. #5009C22semidirect
49. #5008C22semidirect
50. #5007C22semidirect
56. C2×#550D3semidirect
205. #235#41semidirect
322. C3×Q8AGL(1,ℤ/8)semidirect
349. D12C2×#41semidirect
356. Dic6C2×#41semidirect
457. Q8#932semidirect
518. C3#61321semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5502
Q8×#2352
#835872
#835862
#835812
#835802
#831652
#831592
Q8×#2082
#822962
C4×#9322
#613213