Finite Group Databasebeta

← #11675081

Subgroups — #11675081

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
1Q8×#235C211yesmaximal, characteristic, complemented
2C2×#5577C211yesmaximal, characteristic, complemented
3C3×Q8×M16C211yesmaximal, characteristic, complemented
4#79980C216yesmaximal, complemented
5#79969C216yesmaximal, complemented
6#79932C212yesmaximal, complemented
7#79924C212yesmaximal, complemented
8#79059C213yesmaximal, complemented
9#79056C213yesmaximal, complemented
10#78526C213yesmaximal, complemented
11C4×#908C213yesmaximal, complemented
12#61321—31nomaximal, Sylow
13C22×#239—21no—
14C2×C3×C4×Q8C2211yescharacteristic
15Q8×#67—21no—
16C2×#908C2213yes—
17#5579C2213yescomplemented
18#5578—23no—
19#5577—21no—
20#5577—23no—
21#5577C2212yes—
22#5577C2216yes—
23Q8×D12C2211yescharacteristic, complemented
24C4×D3×Q8—21no—
25#5567C2213yescomplemented
26Q8×Dic6C2211yescharacteristic, complemented
27#5511—23no—
28C2×#895—23no—
29C2×#874C2213yes—
30C4×#235C2213yes—
31C2×C4×D12C2213yes—
32#5470C2213yes—
33C3×#446C2216yescomplemented
34C3×C8×Q8C2212yescomplemented
35C3×#423C2213yes—
36C3×C4×M16C2213yes—
37#5133C2211yescharacteristic
38#5131C2213yes—
39#5131C2216yes—
40#5121C2213yes—
41#4848C2216yescomplemented
42#4848C2216yescomplemented
43#4847C22112yescomplemented
44#4846C2216yescomplemented
45#4843C2216yescomplemented
46#4843C2216yescomplemented
47#4842C2216yescomplemented
48#4713C22112yescomplemented
49#4711C2216yescomplemented
50#4710C2216yescomplemented
51#4694C2216yescomplemented
52#4694C2216yescomplemented
53C4×D24C2216yescomplemented
54C4×#204C2216yescomplemented
55Q8×#41—31no—
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×M16D311yescharacteristic, complemented
65C4×AGL(1,ℤ/8)—33no—
66C22×C3×Q8D411yescharacteristic
67C2×#239—26no—
68C2×#239—22no—
69C2×#239—48no—
70C2×D3×Q8—21no—
71C2×#235C2311yescharacteristic
72C22×D12—23no—
73C22×D12—23no—
74C22×D12C2313yes—
75C22×C4×D3—23no—
76C2×C3×M16C2313yes—
77C2×C3×M16C2311yescharacteristic
78C3×C4×Q8D411yescharacteristic
79C3×C4×Q8D413yes—
80C3×C4×Q8C2311yescharacteristic
81C3×C4×Q8C2313yes—
82C2×C3×C4⋊C4D413yes—
83C2×C3×C42C2313yes—
84#946—23no—
85Q8×Dic3—21no—
86#944—23no—
87C4×#67—23no—
88C4×#67—23no—
89C2×(C2×D3)⋊C4—23no—
90#926C2313yes—
91C2×C4⋊Dic3C2311yescharacteristic
92#924—23no—
93#924—23no—
94#908—212no—
95C2×D24C2316yes—
96C2×#204C2316yes—
97#897—23no—
98#896—26no—
99#895—26no—
100#895—26no—
101#895—212no—
102D3×C4⋊C4—23no—
103#889—26no—
104#888—23no—
105#887—23no—
106#874—23no—
107#874—23no—
108#874C2316yes—
109#874C2316yes—
110C4×D12—23no—
111C4×D12—23no—
112C4×D12C2316yes—
113C4×D12C2313yes—
114C4×D12C2316yes—
115C4×D12C2313yes—
116#872—23no—
117C42×D3—23no—
118#870C2316yes—
119#869C2313yes—
120#869C2313yes—
121C4×Dic6—23no—
122C4×Dic6—23no—
123C4×Dic6C2313yes—
124C4×Dic6C2313yes—
125C3×C4⋊C8C2316yes—
126C3×C4⋊C8C2316yes—
127C3×#97C2316yes—
128C3×C4×C8C2316yes—
129#821C2316yes—
130#821C2316yes—
131#821C23112yes—
132#821C23112yes—
133#821C2312yes—
134#818C2316yes—
135#817C2316yes—
136#816C2312yes—
137C22×#41—61no—
138C2×AGL(1,ℤ/8)—33no—
139Q82—31no—
140#557—33no—
141#554—33no—
142#554—63no—
143#550—32no—
144#550—36no—
145#550—61no—
146#550—63no—
147D4×Q8—31no—
148D4×Q8—61no—
149#533—33no—
150#533—63no—
151C2×#127—33no—
152C2×#121—63no—
153C4×#41—33no—
154C2×C4×Q8—61no—
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#446C2×D316yescomplemented
168C8×Q8C2×D312yescomplemented
169#442—36no—
170#440—36no—
171C4×SD16—36no—
172C4×D8—36no—
173#428—33no—
174#423C2×D313yes—
175#418—36no—
176#418—33no—
177#417—31no—
178C4×M16C2×D313yes—
179C23×D3—23no—
180C2×C3×Q8—24no—
181C2×C3×Q8C2×D413yes—
182C2×C3×Q8C2×D411yescharacteristic
183C22×C3×C4C2×D413yes—
184C22×C3×C4C2×D413yes—
185C22×C3×C4C2411yescharacteristic
186C2×#67—21no—
187C22×Dic3—21no—
188#239—424no—
189#239—48no—
190D3×Q8—44no—
191#235—22no—
192#235#4114yescomplemented
193C2×D12—26no—
194C2×D12—26no—
195C2×D12—23no—
196C2×D12—26no—
197C2×D12—26no—
198C2×D12—26no—
199C2×D12—26no—
200C2×D12—412no—
201C2×D12—86no—
202C2×D12C2416yes—
203C2×D12C2411yescharacteristic
204C2×C4×D3—21no—
205C2×C4×D3—23no—
206C2×C4×D3—26no—
207C2×C4×D3—26no—
208C2×C4×D3—23no—
209C2×C4×D3—412no—
210C2×Dic6—23no—
211C2×Dic6—23no—
212C2×Dic6C2411yescharacteristic
213C3×M16—26no—
214C3×M16—22no—
215C2×C3×C8C2412yes—
216C2×C3×C8C2416yes—
217C3×C4⋊C4C2×D413yes—
218C3×C4⋊C4C2×D413yes—
219C3×C4⋊C4C2×D416yes—
220C3×C42C2×D413yes—
221C3×C42C2×D413yes—
222C3×C42C2413yes—
223C3×C42C2413yes—
224#217—23no—
225(C2×D3)⋊C4—23no—
226(C2×D3)⋊C4—23no—
227(C2×D3)⋊C4—26no—
228(C2×D3)⋊C4—26no—
229C4⋊Dic3—23no—
230C4⋊Dic3—23no—
231C4⋊Dic3—21no—
232C4⋊Dic3C2412yes—
233C4⋊Dic3C2413yes—
234C4⋊Dic3C2413yes—
235Dic3⋊C4—26no—
236Dic3⋊C4—26no—
237Dic3⋊C4—23no—
238Dic3⋊C4—23no—
239C4×Dic3—23no—
240C4×Dic3—23no—
241C4×Dic3—23no—
242D24—212no—
243#204—212no—
244C2×#41—31no—
245C2×#41—66no—
246C2×#41—62no—
247C2×#41—128no—
248C22×Q8—61no—
249C22×Q8D1211yescharacteristic
250C22×D4—33no—
251C22×D4—63no—
252C22×D4—63no—
253C23×C4—63no—
254AGL(1,ℤ/8)—612no—
255C2×SD16—36no—
256C2×D8—36no—
257C2×M16C22×D313yes—
258C2×M16C22×D311yescharacteristic
259C4⋊Q8—33no—
260C4⋊Q8—33no—
261C4⋊Q8—63no—
262C4⋊Q8—63no—
263#127—36no—
264#127—36no—
265#127—63no—
266#127—63no—
267#125—36no—
268#125—66no—
269C22⋊Q8—63no—
270C22⋊Q8—63no—
271C22⋊Q8—63no—
272C22⋊Q8—63no—
273C22⋊Q8—66no—
274#121—66no—
275#121—612no—
276#121—66no—
277C4×Q8—33no—
278C4×Q8—33no—
279C4×Q8—61no—
280C4×Q8—63no—
281C4×Q8—63no—
282C4×Q8—63no—
283C4×Q8D1211yescharacteristic
284C4×Q8D1213yes—
285C4×Q8C22×D311yescharacteristic
286C4×Q8C22×D313yes—
287C4×D4—36no—
288C4×D4—36no—
289C4×D4—33no—
290C4×D4—33no—
291C4×D4—63no—
292C4×D4—63no—
293C4×D4—63no—
294C4×D4—63no—
295#117—33no—
296#117—63no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—63no—
299C2×C4⋊C4D1213yes—
300C2×C22⋊C4—63no—
301C2×C42—63no—
302C2×C42C22×D313yes—
303#107—36no—
304#106—36no—
305C4⋊C8C22×D316yes—
306C4⋊C8C22×D316yes—
307#103—32no—
308#102—32no—
309#102—36no—
310#102—36no—
311#102—312no—
312#102—312no—
313#97C22×D316yes—
314C4×C8C22×D316yes—
315C23×C3C22×D411yescharacteristic
316C22×D3—21no—
317C22×D3—26no—
318C22×D3—46no—
319C22×D3—46no—
320C22×D3—46no—
321C3×Q8—26no—
322C3×Q8AGL(1,ℤ/8)14yescomplemented
323C2×C3×C4—26no—
324C2×C3×C4—43no—
325C2×C3×C4C22×D413yes—
326C2×C3×C4C22×D413yes—
327C2×C3×C4C22×D413yes—
328C2×C3×C4C22×D411yescharacteristic
329C2×C3×C4C22×D413yes—
330C2×C3×C4C2×#4112yes—
331C2×C3×C42−1+412yes—
332C2×C3×C4C2511yescharacteristic
333#67—24no—
334C2×Dic3—21no—
335C2×Dic3—23no—
336C2×Dic3—23no—
337C2×Dic3—22no—
338C2×Dic3—42no—
339D12—26no—
340D12—21no—
341D12—26no—
342D12—412no—
343D12—46no—
344D12—412no—
345D12—86no—
346D12—86no—
347D12C2×#4112yescomplemented
348C4×D3—24no—
349C4×D3—412no—
350C4×D3—412no—
351C4×D3—46no—
352C4×D3—83no—
353Dic6—21no—
354Dic6—46no—
355Dic6—83no—
356Dic6C2×#4112yescomplemented
357C3×C8—26no—
358C3×C8—22no—
359C24—63no—
360#41—34no—
361#41—62no—
362#41—1224no—
363#41—128no—
364C2×Q8—24no—
365C2×Q8—31no—
366C2×Q8—63no—
367C2×Q8—63no—
368C2×Q8—124no—
369C2×Q8C2×D1213yes—
370C2×Q8C2×D1211yescharacteristic
371C2×D4—31no—
372C2×D4—36no—
373C2×D4—61no—
374C2×D4—63no—
375C2×D4—66no—
376C2×D4—66no—
377C2×D4—66no—
378C2×D4—66no—
379C2×D4—66no—
380C2×D4—66no—
381C2×D4—1212no—
382C2×D4—246no—
383C22×C4—61no—
384C22×C4—61no—
385C22×C4—63no—
386C22×C4—66no—
387C22×C4—66no—
388C22×C4—63no—
389C22×C4—1212no—
390C22×C4C2×D1213yes—
391C22×C4C2×D1213yes—
392C22×C4C23×D311yescharacteristic
393SD16—612no—
394D8—612no—
395M16—22no—
396M16—26no—
397C2×C8C23×D312yes—
398C2×C8C23×D316yes—
399C4⋊C4—33no—
400C4⋊C4—33no—
401C4⋊C4—32no—
402C4⋊C4—61no—
403C4⋊C4—66no—
404C4⋊C4—63no—
405C4⋊C4—63no—
406C4⋊C4—63no—
407C4⋊C4—66no—
408C4⋊C4—63no—
409C4⋊C4C2×D1213yes—
410C4⋊C4C2×D1216yes—
411C4⋊C4C2×D1213yes—
412C22⋊C4—63no—
413C22⋊C4—66no—
414C22⋊C4—66no—
415C22⋊C4—63no—
416C22⋊C4—63no—
417C42—63no—
418C42—63no—
419C42—63no—
420C42C2×D1213yes—
421C42C2×D1213yes—
422C42C23×D313yes—
423C42C23×D313yes—
424C22×C3—21no—
425C22×C3—21no—
426C22×C3#54812yes—
427C22×C3C23×D411yescharacteristic
428C2×D3—22no—
429C2×D3—41no—
430C2×D3—41no—
431C2×D3—46no—
432C2×D3—46no—
433C2×D3—46no—
434C2×D3—812no—
435C3×C4—23no—
436C3×C4—23no—
437C3×C4—23no—
438C3×C4#54812yes—
439C3×C4C2×AGL(1,ℤ/8)16yes—
440C3×C4C22×#4111yescharacteristic
441C3×C4C2×2−1+411yescharacteristic
442Dic3—22no—
443Dic3—43no—
444Dic3—43no—
445Dic3—42no—
446C23—61no—
447C23—66no—
448C23—126no—
449C23—126no—
450C23—126no—
451C23C22×D1211yescharacteristic
452Q8—26no—
453Q8—32no—
454Q8—61no—
455Q8—126no—
456Q8—243no—
457Q8#90814yescomplemented
458D4—32no—
459D4—64no—
460D4—61no—
461D4—66no—
462D4—66no—
463D4—1212no—
464D4—1212no—
465D4—126no—
466D4—246no—
467D4—246no—
468C2×C4—26no—
469C2×C4—43no—
470C2×C4—61no—
471C2×C4—64no—
472C2×C4—63no—
473C2×C4—63no—
474C2×C4—62no—
475C2×C4—1212no—
476C2×C4—122no—
477C2×C4—1212no—
478C2×C4—126no—
479C2×C4—243no—
480C2×C4C22×D1213yes—
481C2×C4C22×D1213yes—
482C2×C4C22×D1213yes—
483C2×C4C22×D1213yes—
484C2×C4C22×D1211yescharacteristic
485C2×C4#100712yes—
486C2×C4D3×#4112yes—
487C2×C4C24×D311yescharacteristic
488C8—22no—
489C8—26no—
490C2×C3—22no—
491C2×C3C2×#54811yescharacteristic
492C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
493C2×C3#358511yescharacteristic
494D3—42no—
495D3—86no—
496C22—21no—
497C22—21no—
498C22—62no—
499C22—121no—
500C22—121no—
501C22—126no—
502C22—126no—
503C22—126no—
504C22—2412no—
505C22#557612yes—
506C22C23×D1211yescharacteristic
507C4—23no—
508C4—23no—
509C4—23no—
510C4—62no—
511C4—122no—
512C4—123no—
513C4—123no—
514C4#557612yes—
515C4C2×#90816yes—
516C4C2×#100711yescharacteristic
517C4C2×D3×#4111yescharacteristic
518C3#6132111yescharacteristic, complemented, Sylow
519C2—22no—
520C2—122no—
521C2—246no—
522C2C2×#557611yescharacteristic
523C2C22×#90811yescharacteristic
524C2#8716711yescharacteristic
subgroupG/H
centerC22C23×D12
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×Q8×M16C2
socleC22×C3C23×D4
hypercenterQ8×M16D3
hypocenterC3#61321
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ Q8×M16

Complements

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

NH ≅ G/Ntype
1. Q8×#235C2semidirect
2. C2×#5577C2semidirect
3. C3×Q8×M16C2semidirect
4. #79980C2semidirect
5. #79969C2semidirect
6. #79932C2semidirect
7. #79924C2semidirect
8. #79059C2semidirect
9. #79056C2semidirect
10. #78526C2semidirect
11. C4×#908C2semidirect
17. #5579C22semidirect
23. Q8×D12C22semidirect
25. #5567C22semidirect
26. Q8×Dic6C22semidirect
33. C3×#446C22semidirect
34. C3×C8×Q8C22semidirect
41. #4848C22semidirect
42. #4848C22semidirect
43. #4847C22semidirect
44. #4846C22semidirect
45. #4843C22semidirect
46. #4843C22semidirect
47. #4842C22semidirect
48. #4713C22semidirect
49. #4711C22semidirect
50. #4710C22semidirect
51. #4694C22semidirect
52. #4694C22semidirect
53. C4×D24C22semidirect
54. C4×#204C22semidirect
64. Q8×M16D3semidirect
167. #446C2×D3semidirect
168. C8×Q8C2×D3semidirect
192. #235#41semidirect
322. C3×Q8AGL(1,ℤ/8)semidirect
347. D12C2×#41semidirect
356. Dic6C2×#41semidirect
457. Q8#908semidirect
518. C3#61321semidirect

Maximal subgroups

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

subgroupindex
Q8×#2352
C2×#55772
C3×Q8×M162
#799802
#799692
#799322
#799242
#790592
#790562
#785262
C4×#9082
#613213