Finite Group Databasebeta

← #11675080

Subgroups — #11675080

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
1C3×Q8×#41C211yesmaximal, characteristic, complemented
2C2×#5577C211yesmaximal, characteristic, complemented
3#83703C212yesmaximal, complemented
4#83690C216yesmaximal, complemented
5#83638C216yesmaximal, complemented
6#83562C212yesmaximal, complemented
7#83350C213yesmaximal, complemented
8#83336C213yesmaximal, complemented
9#82526C213yesmaximal, complemented
10C4×#949C213yesmaximal, complemented
11Q8×#208C211yesmaximal, characteristic, complemented
12#61321—31nomaximal, Sylow
13C22×#239—21no—
14C3×Q82C2211yescharacteristic, complemented
15C3×#557C2213yescomplemented
16C3×#554—23no—
17C3×#554C2213yescomplemented
18C3×D4×Q8—21no—
19C3×D4×Q8C2211yescharacteristic, complemented
20C3×#533—23no—
21C3×#533C2213yes—
22C3×C4×#41C2213yes—
23C2×C3×C4×Q8—21no—
24C2×C3×C4×Q8C2211yescharacteristic
25C2×#949C2213yes—
26#5577—21no—
27#5577—23no—
28#5577C2212yes—
29#5577C2216yes—
30C2×#895—23no—
31C2×#874C2213yes—
32C2×C4×D12C2213yes—
33#5232C2211yescharacteristic
34#5088C2216yescomplemented
35#5088C2216yescomplemented
36#5087C2216yescomplemented
37#5084C2216yescomplemented
38#5067C22112yescomplemented
39#5066C22112yescomplemented
40#5027C2216yescomplemented
41#5027C2216yescomplemented
42#5026C2216yescomplemented
43C4×#215C2216yescomplemented
44#5024C2216yescomplemented
45Q8×C3⋊C8C2212yescomplemented
46#5015C2216yescomplemented
47#5015C2216yescomplemented
48#5014C2216yescomplemented
49C4×#213C2216yescomplemented
50#5001C2213yes—
51#5001C2216yes—
52#4995C2213yes—
53#4924C2213yes—
54C4×#208C2213yes—
55Q8×#41D311yescharacteristic, complemented
56C2×#550—31no—
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×#41C2311yescharacteristic
67C22×C3×Q8—21no—
68C22×C3×Q8D411yescharacteristic
69C2×#239—26no—
70C2×#239—22no—
71C2×#239—48no—
72C22×D12—23no—
73C22×D12—23no—
74C22×D12C2313yes—
75C22×C4×D3—23no—
76C3×C4⋊Q8—23no—
77C3×C4⋊Q8—23no—
78C3×C4⋊Q8C2313yes—
79C3×C4⋊Q8C2313yes—
80C3×#125—26no—
81C3×#125C2316yes—
82C3×C22⋊Q8—23no—
83C3×C22⋊Q8—23no—
84C3×C22⋊Q8—23no—
85C3×C22⋊Q8—23no—
86C3×C22⋊Q8—26no—
87C3×C4×Q8—21no—
88C3×C4×Q8—23no—
89C3×C4×Q8—23no—
90C3×C4×Q8—23no—
91C3×C4×Q8D411yescharacteristic
92C3×C4×Q8D413yes—
93C3×C4×Q8C2313yes—
94C3×C4×Q8C2313yes—
95C3×C4×Q8C2311yescharacteristic
96C3×C4×Q8C2313yes—
97C3×C4×D4—23no—
98C3×C4×D4—23no—
99C3×C4×D4C2313yes—
100C3×C4×D4C2313yes—
101C3×#117—23no—
102C3×#117C2313yes—
103C2×C3×C4⋊C4—23no—
104C2×C3×C4⋊C4D413yes—
105C2×C3×C42—23no—
106C2×C3×C42C2313yes—
107#949—212no—
108C2×#215C2316yes—
109C2×#213C2316yes—
110C2×(C2×D3)⋊C4—23no—
111C2×C4⋊Dic3C2311yescharacteristic
112C2×#208C2311yescharacteristic
113C2×#208C2313yes—
114#895—26no—
115#895—212no—
116#895—26no—
117#874—23no—
118#874—23no—
119#874C2316yes—
120#874C2316yes—
121C4×D12—23no—
122C4×D12—23no—
123C4×D12C2316yes—
124C4×D12C2316yes—
125#835C2312yes—
126#832C2312yes—
127#809C2316yes—
128#809C2316yes—
129#809C23112yes—
130#809C23112yes—
131#808C2316yes—
132#807C2316yes—
133C4⋊(C3⋊C8)C2316yes—
134C4⋊(C3⋊C8)C2316yes—
135#803C2316yes—
136C4×C3⋊C8C2316yes—
137C22×#41—61no—
138C2×AGL(1,ℤ/8)—33no—
139Q82C2×D311yescharacteristic, complemented
140#557C2×D313yescomplemented
141#554—23no—
142#554C2×D313yescomplemented
143#550—32no—
144#550—36no—
145#550—61no—
146#550—63no—
147D4×Q8—21no—
148D4×Q8C2×D311yescharacteristic, complemented
149#533—23no—
150#533C2×D313yes—
151C2×#127—33no—
152C2×#121—63no—
153C4×#41C2×D313yes—
154C2×C4×Q8—21no—
155C2×C4×Q8C2×D311yescharacteristic
156C2×C4×D4—33no—
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—
179C23×D3—23no—
180C3×#41—22no—
181C3×#41#4114yescomplemented
182C2×C3×Q8—24no—
183C2×C3×Q8—23no—
184C2×C3×Q8—23no—
185C2×C3×Q8—44no—
186C2×C3×Q8C2×D413yes—
187C2×C3×Q8C2×D411yescharacteristic
188C2×C3×Q8C2411yescharacteristic
189C2×C3×D4—21no—
190C2×C3×D4C2411yescharacteristic
191C22×C3×C4—21no—
192C22×C3×C4—23no—
193C22×C3×C4—23no—
194C22×C3×C4C2×D413yes—
195C22×C3×C4C2×D413yes—
196C22×C3×C4C2411yescharacteristic
197C22×Dic3—21no—
198#239—424no—
199#239—48no—
200C2×D12—26no—
201C2×D12—26no—
202C2×D12—23no—
203C2×D12—26no—
204C2×D12—26no—
205C2×D12—26no—
206C2×D12—26no—
207C2×D12—412no—
208C2×D12—86no—
209C2×D12C2416yes—
210C2×C4×D3—26no—
211C2×C4×D3—26no—
212C2×C4×D3—412no—
213C3×C4⋊C4—26no—
214C3×C4⋊C4—23no—
215C3×C4⋊C4—23no—
216C3×C4⋊C4—23no—
217C3×C4⋊C4—23no—
218C3×C4⋊C4—26no—
219C3×C4⋊C4C2×D413yes—
220C3×C4⋊C4C2×D416yes—
221C3×C4⋊C4C2×D413yes—
222C3×C4⋊C4C2413yes—
223C3×C4⋊C4C2413yes—
224C3×C22⋊C4—23no—
225C3×C22⋊C4—23no—
226C3×C22⋊C4—23no—
227C3×C42—23no—
228C3×C42—23no—
229C3×C42—23no—
230C3×C42C2×D413yes—
231C3×C42C2×D413yes—
232C3×C42C2413yes—
233C3×C42C2413yes—
234#215—212no—
235#213—212no—
236(C2×D3)⋊C4—26no—
237(C2×D3)⋊C4—26no—
238C4⋊Dic3—21no—
239C4⋊Dic3C2412yes—
240#208—26no—
241#208—22no—
242C2×C3⋊C8C2412yes—
243C2×C3⋊C8C2416yes—
244C2×#41—66no—
245C2×#41—62no—
246C2×#41—128no—
247C2×#41C22×D311yescharacteristic
248C22×Q8—21no—
249C22×Q8#6711yescharacteristic
250C22×D4—33no—
251C22×D4—63no—
252C22×D4—63no—
253C23×C4—63no—
254AGL(1,ℤ/8)—612no—
255C2×SD16—36no—
256C2×D8—36no—
257C2×M16—33no—
258C2×M16—31no—
259C4⋊Q8—23no—
260C4⋊Q8—23no—
261C4⋊Q8C22×D313yes—
262C4⋊Q8C22×D313yes—
263#127—36no—
264#127—36no—
265#127—63no—
266#127—63no—
267#125—26no—
268#125C22×D316yes—
269C22⋊Q8—23no—
270C22⋊Q8—23no—
271C22⋊Q8—23no—
272C22⋊Q8—23no—
273C22⋊Q8—26no—
274#121—66no—
275#121—612no—
276#121—66no—
277C4×Q8—21no—
278C4×Q8—23no—
279C4×Q8—23no—
280C4×Q8—23no—
281C4×Q8#6711yescharacteristic
282C4×Q8#6713yes—
283C4×Q8C22×D313yes—
284C4×Q8C22×D313yes—
285C4×Q8C22×D311yescharacteristic
286C4×Q8C22×D313yes—
287C4×D4—23no—
288C4×D4—23no—
289C4×D4—36no—
290C4×D4—36no—
291C4×D4—63no—
292C4×D4—63no—
293C4×D4C22×D313yes—
294C4×D4C22×D313yes—
295#117—23no—
296#117C22×D313yes—
297C2×C4⋊C4—23no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4#6713yes—
300C2×C22⋊C4—63no—
301C2×C42—23no—
302C2×C42C22×D313yes—
303#107—36no—
304#106—36no—
305C4⋊C8—36no—
306C4⋊C8—36no—
307#103—32no—
308#102—32no—
309#102—36no—
310#102—36no—
311#102—312no—
312#102—312no—
313#97—36no—
314C4×C8—36no—
315C23×C3—21no—
316C23×C3C22×D411yescharacteristic
317C22×D3—26no—
318C22×D3—46no—
319C22×D3—46no—
320C22×D3—46no—
321C3×Q8—26no—
322C3×Q8—21no—
323C3×Q8—46no—
324C3×Q8—83no—
325C3×Q8AGL(1,ℤ/8)14yescomplemented
326C3×Q8C2×#4112yescomplemented
327C3×D4—24no—
328C3×D4—21no—
329C3×D4C2×#4112yescomplemented
330C2×C3×C4—21no—
331C2×C3×C4—24no—
332C2×C3×C4—26no—
333C2×C3×C4—23no—
334C2×C3×C4—23no—
335C2×C3×C4—43no—
336C2×C3×C4—46no—
337C2×C3×C4—83no—
338C2×C3×C4C22×D413yes—
339C2×C3×C4C22×D413yes—
340C2×C3×C4C22×D413yes—
341C2×C3×C4C22×D411yescharacteristic
342C2×C3×C4C22×D413yes—
343C2×C3×C4C2×#4112yes—
344C2×C3×C42−1+412yes—
345C2×C3×C4C2511yescharacteristic
346C2×Dic3—22no—
347C2×Dic3—42no—
348D12—26no—
349D12—26no—
350D12—412no—
351D12—46no—
352D12—412no—
353D12—86no—
354D12—86no—
355C4×D3—412no—
356C4×D3—412no—
357C3⋊C8—26no—
358C3⋊C8—22no—
359C24—63no—
360#41—22no—
361#41—1224no—
362#41—128no—
363#41#23914yescomplemented
364C2×Q8—24no—
365C2×Q8—23no—
366C2×Q8—23no—
367C2×Q8—44no—
368C2×Q8C2×#6713yes—
369C2×Q8C2×#6711yescharacteristic
370C2×Q8C23×D311yescharacteristic
371C2×D4—21no—
372C2×D4—36no—
373C2×D4—66no—
374C2×D4—66no—
375C2×D4—63no—
376C2×D4—66no—
377C2×D4—66no—
378C2×D4—66no—
379C2×D4—66no—
380C2×D4—1212no—
381C2×D4—246no—
382C2×D4C23×D311yescharacteristic
383C22×C4—21no—
384C22×C4—23no—
385C22×C4—23no—
386C22×C4—61no—
387C22×C4—66no—
388C22×C4—66no—
389C22×C4—1212no—
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—26no—
400C4⋊C4—23no—
401C4⋊C4—23no—
402C4⋊C4—23no—
403C4⋊C4—23no—
404C4⋊C4—26no—
405C4⋊C4—32no—
406C4⋊C4—61no—
407C4⋊C4C2×#6713yes—
408C4⋊C4C2×#6716yes—
409C4⋊C4C2×#6713yes—
410C4⋊C4C23×D313yes—
411C4⋊C4C23×D313yes—
412C22⋊C4—23no—
413C22⋊C4—23no—
414C22⋊C4—23no—
415C22⋊C4—66no—
416C22⋊C4—66no—
417C42—23no—
418C42—23no—
419C42—23no—
420C42C2×#6713yes—
421C42C2×#6713yes—
422C42C23×D313yes—
423C42C23×D313yes—
424C22×C3—21no—
425C22×C3—21no—
426C22×C3—22no—
427C22×C3—41no—
428C22×C3—41no—
429C22×C3#54812yes—
430C22×C3C23×D411yescharacteristic
431C2×D3—46no—
432C2×D3—46no—
433C2×D3—46no—
434C2×D3—812no—
435C3×C4—23no—
436C3×C4—23no—
437C3×C4—22no—
438C3×C4—23no—
439C3×C4—43no—
440C3×C4—43no—
441C3×C4#54812yes—
442C3×C4C2×AGL(1,ℤ/8)16yes—
443C3×C4C22×#4111yescharacteristic
444C3×C4C2×2−1+411yescharacteristic
445Dic3—42no—
446C23—21no—
447C23—66no—
448C23—126no—
449C23—126no—
450C23—126no—
451C23C22×#6711yescharacteristic
452Q8—21no—
453Q8—26no—
454Q8—46no—
455Q8—83no—
456Q8#94914yescomplemented
457Q8C2×#23912yescomplemented
458D4—24no—
459D4—21no—
460D4—66no—
461D4—66no—
462D4—1212no—
463D4—126no—
464D4—1212no—
465D4—246no—
466D4—246no—
467D4C2×#23912yescomplemented
468C2×C4—21no—
469C2×C4—24no—
470C2×C4—26no—
471C2×C4—23no—
472C2×C4—23no—
473C2×C4—43no—
474C2×C4—46no—
475C2×C4—62no—
476C2×C4—83no—
477C2×C4—1212no—
478C2×C4—122no—
479C2×C4—1212no—
480C2×C4C2×#23912yes—
481C2×C4#100712yes—
482C2×C4C22×#6713yes—
483C2×C4C22×#6713yes—
484C2×C4C22×#6713yes—
485C2×C4C22×#6711yescharacteristic
486C2×C4C22×#6713yes—
487C2×C4C24×D311yescharacteristic
488C8—66no—
489C8—62no—
490C2×C3—22no—
491C2×C3—42no—
492C2×C3C2×#54811yescharacteristic
493C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
494C2×C3#358511yescharacteristic
495D3—86no—
496C22—21no—
497C22—21no—
498C22—22no—
499C22—41no—
500C22—41no—
501C22—126no—
502C22—126no—
503C22—126no—
504C22—2412no—
505C22#581712yes—
506C22C23×#6711yescharacteristic
507C4—23no—
508C4—23no—
509C4—22no—
510C4—23no—
511C4—43no—
512C4—43no—
513C4—122no—
514C4#581712yes—
515C4C2×#94916yes—
516C4C22×#23911yescharacteristic
517C4C2×#100711yescharacteristic
518C3#6132111yescharacteristic, complemented, Sylow
519C2—22no—
520C2—42no—
521C2—246no—
522C2C2×#581711yescharacteristic
523C2C22×#94911yescharacteristic
524C2#8729811yescharacteristic
subgroupG/H
centerC22C23×#67
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×Q8×#41C2
socleC22×C3C23×D4
hypercenterQ8×#41D3
hypocenterC3#61321
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ Q8×#41

Complements

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

NH ≅ G/Ntype
1. C3×Q8×#41C2semidirect
2. C2×#5577C2semidirect
3. #83703C2semidirect
4. #83690C2semidirect
5. #83638C2semidirect
6. #83562C2semidirect
7. #83350C2semidirect
8. #83336C2semidirect
9. #82526C2semidirect
10. C4×#949C2semidirect
11. Q8×#208C2semidirect
14. C3×Q82C22semidirect
15. C3×#557C22semidirect
17. C3×#554C22semidirect
19. C3×D4×Q8C22semidirect
34. #5088C22semidirect
35. #5088C22semidirect
36. #5087C22semidirect
37. #5084C22semidirect
38. #5067C22semidirect
39. #5066C22semidirect
40. #5027C22semidirect
41. #5027C22semidirect
42. #5026C22semidirect
43. C4×#215C22semidirect
44. #5024C22semidirect
45. Q8×C3⋊C8C22semidirect
46. #5015C22semidirect
47. #5015C22semidirect
48. #5014C22semidirect
49. C4×#213C22semidirect
55. Q8×#41D3semidirect
139. Q82C2×D3semidirect
140. #557C2×D3semidirect
142. #554C2×D3semidirect
148. D4×Q8C2×D3semidirect
181. C3×#41#41semidirect
325. C3×Q8AGL(1,ℤ/8)semidirect
326. C3×Q8C2×#41semidirect
329. C3×D4C2×#41semidirect
363. #41#239semidirect
456. Q8#949semidirect
457. Q8C2×#239semidirect
467. D4C2×#239semidirect
518. C3#61321semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×#412
C2×#55772
#837032
#836902
#836382
#835622
#833502
#833362
#825262
C4×#9492
Q8×#2082
#613213