Finite Group Databasebeta

← Q8×#908 #11675767

Subgroups — Q8×#908 #11675767

Order 768.

2010 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#235C211yesmaximal, characteristic, complemented
2C2×Q8×D12C211yesmaximal, characteristic, complemented
3C3×Q8×M16C211yesmaximal, characteristic, complemented
4#79977C216yesmaximal, complemented
5#79975C216yesmaximal, complemented
6Q8×#204C212yesmaximal, complemented
7Q8×D24C212yesmaximal, complemented
8#79060C213yesmaximal, complemented
9#79059C213yesmaximal, complemented
10C4×#908C213yesmaximal
11#78506C213yesmaximal, complemented
12Q8×AGL(1,ℤ/8)—31nomaximal, Sylow
13C22×D3×Q8—21no—
14C2×C3×C4×Q8C2211yescharacteristic
15Q8×#67—21no—
16C2×#908C2211yescharacteristic
17#5579C2213yescomplemented
18#5578—23no—
19Q8×D12—21no—
20Q8×D12—23no—
21Q8×D12C2212yes—
22Q8×D12C2216yes—
23Q8×D12C2211yescharacteristic, complemented
24C4×D3×Q8—21no—
25#5567C2213yescomplemented
26Q8×Dic6C2211yescharacteristic, complemented
27#5511—23no—
28C2×#897—23no—
29C4×#235C2213yes—
30C2×C4×D12C2213yes—
31#5470C2213yes—
32C2×#869C2213yes—
33C3×#446C2216yescomplemented
34C3×C8×Q8C2212yescomplemented
35C3×#423C2213yes—
36C3×C4×M16C2213yes—
37#5133C2213yes—
38#5131C2213yes—
39#5121C2213yes—
40#5121C2216yes—
41#4851C2216yescomplemented
42#4850C2216yescomplemented
43#4847C22112yes—
44#4846C2216yes—
45#4846C2216yescomplemented
46#4842C2216yes—
47#4842C2216yescomplemented
48#4711C2216yes—
49#4710C2216yes—
50#4702C22112yescomplemented
51C4×D24C2216yes—
52C4×#204C2216yes—
53#4682C2216yescomplemented
54#4680C2216yescomplemented
55Q8×#41—31no—
56C2×D4×Q8—31no—
57#3386—36no—
58#3384—36no—
59Q8×SD16—32no—
60Q8×D8—32no—
61#3165—33no—
62#3079—33no—
63#3077—33no—
64Q8×M16D311yescharacteristic, complemented
65C4×AGL(1,ℤ/8)—33no—
66C22×C3×Q8D411yescharacteristic
67C2×D3×Q8—26no—
68C2×D3×Q8—22no—
69C2×D3×Q8—21no—
70C2×D3×Q8—48no—
71C2×#235C2311yescharacteristic
72C22×D12C2311yescharacteristic
73C22×C4×D3—23no—
74C22×Dic6—23no—
75C2×C3×M16C2313yes—
76C2×C3×M16C2311yescharacteristic
77C3×C4×Q8D411yescharacteristic
78C3×C4×Q8D413yes—
79C3×C4×Q8C2311yescharacteristic
80C3×C4×Q8C2313yes—
81C2×C3×C4⋊C4D413yes—
82C2×C3×C42C2313yes—
83#946—23no—
84Q8×Dic3—21no—
85#944—23no—
86C4×#67—23no—
87C4×#67—23no—
88C2×(C2×D3)⋊C4—23no—
89#926C2313yes—
90C2×C4⋊Dic3—23no—
91C2×C4⋊Dic3C2313yes—
92#924—23no—
93#924—23no—
94#908Q818yesdirect product factor
95C2×D24C2312yes—
96C2×#204C2312yes—
97#897—26no—
98#897—26no—
99#897—212no—
100#897—23no—
101#896—26no—
102D3×C4⋊C4—23no—
103#889—26no—
104#888—23no—
105#887—23no—
106C4×D12—23no—
107C4×D12—23no—
108C4×D12C2316yes—
109C4×D12C2313yes—
110C4×D12C2316yes—
111C4×D12C2313yes—
112#872—23no—
113C42×D3—23no—
114#870C2316yes—
115#869—23no—
116#869—23no—
117#869C2313yes—
118#869C2316yes—
119#869C2313yes—
120#869C2316yes—
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#821C2316yes—
132#818C2316yes—
133#818C23112yes—
134#817C2316yes—
135#817C23112yes—
136#816C2316yes—
137C23×Q8—61no—
138C2×AGL(1,ℤ/8)—31no—
139Q82—31no—
140#557—33no—
141#554—33no—
142#554—63no—
143D4×Q8—31no—
144D4×Q8—32no—
145D4×Q8—36no—
146D4×Q8—61no—
147D4×Q8—61no—
148D4×Q8—63no—
149#533—33no—
150#533—63no—
151C2×C4⋊Q8—33no—
152C2×C22⋊Q8—63no—
153C4×#41—33no—
154C2×C4×Q8—61no—
155C2×C4×Q8C2×D311yescharacteristic
156C2×C4×D4—33no—
157#501—312no—
158#500—36no—
159#498—36no—
160#478—312no—
161#477—36no—
162#476—36no—
163#476—36no—
164#475—36no—
165#474—36no—
166#474—36no—
167#446C2×D316yescomplemented
168C8×Q8C2×D312yescomplemented
169#442—36no—
170#440—36no—
171C4×SD16—36no—
172C4×D8—36no—
173#428—36no—
174#428—33no—
175#423C2×D313yes—
176#418—33no—
177#417—33no—
178C4×M16C2×D313yes—
179C23×D3—21no—
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—23no—
188D3×Q8—424no—
189D3×Q8—44no—
190D3×Q8—48no—
191#235—22no—
192#235C2×Q814yescomplemented
193C2×D12—21no—
194C2×D12—22no—
195C2×D12C2×Q814yescomplemented
196C2×D12C2412yes—
197C2×D12C2411yescharacteristic
198C2×C4×D3—21no—
199C2×C4×D3—23no—
200C2×C4×D3—26no—
201C2×C4×D3—26no—
202C2×C4×D3—23no—
203C2×C4×D3—412no—
204C2×Dic6—23no—
205C2×Dic6—26no—
206C2×Dic6—26no—
207C2×Dic6—23no—
208C2×Dic6—412no—
209C2×Dic6C2411yescharacteristic
210C3×M16—26no—
211C3×M16C2×Q814yescomplemented
212C2×C3×C8C2416yes—
213C2×C3×C8C2412yes—
214C3×C4⋊C4C2×D413yes—
215C3×C4⋊C4C2×D413yes—
216C3×C4⋊C4C2×D416yes—
217C3×C42C2×D413yes—
218C3×C42C2×D413yes—
219C3×C42C2413yes—
220C3×C42C2413yes—
221#217—23no—
222(C2×D3)⋊C4—23no—
223(C2×D3)⋊C4—23no—
224(C2×D3)⋊C4—26no—
225(C2×D3)⋊C4—26no—
226C4⋊Dic3—23no—
227C4⋊Dic3—23no—
228C4⋊Dic3—23no—
229C4⋊Dic3—26no—
230C4⋊Dic3—26no—
231C4⋊Dic3C2416yes—
232C4⋊Dic3C2413yes—
233C4⋊Dic3C2413yes—
234Dic3⋊C4—26no—
235Dic3⋊C4—26no—
236Dic3⋊C4—23no—
237Dic3⋊C4—23no—
238C4×Dic3—23no—
239C4×Dic3—23no—
240C4×Dic3—23no—
241D24C2×Q818yescomplemented
242#204C2×Q818yescomplemented
243C2×#41—31no—
244C22×Q8—61no—
245C22×Q8—66no—
246C22×Q8—62no—
247C22×Q8—63no—
248C22×Q8—128no—
249C22×Q8D1211yescharacteristic
250C22×D4—31no—
251C23×C4—63no—
252AGL(1,ℤ/8)—38no—
253C2×SD16—32no—
254C2×D8—32no—
255C2×M16C22×D313yes—
256C2×M16C22×D311yescharacteristic
257C4⋊Q8—36no—
258C4⋊Q8—36no—
259C4⋊Q8—33no—
260C4⋊Q8—33no—
261C4⋊Q8—63no—
262C4⋊Q8—63no—
263C4⋊Q8—63no—
264C4⋊Q8—63no—
265#125—36no—
266#125—66no—
267C22⋊Q8—63no—
268C22⋊Q8—63no—
269C22⋊Q8—63no—
270C22⋊Q8—66no—
271C22⋊Q8—612no—
272C22⋊Q8—66no—
273C22⋊Q8—63no—
274C22⋊Q8—66no—
275C4×Q8—33no—
276C4×Q8—33no—
277C4×Q8—61no—
278C4×Q8—63no—
279C4×Q8—63no—
280C4×Q8—63no—
281C4×Q8D1211yescharacteristic
282C4×Q8D1213yes—
283C4×Q8C22×D311yescharacteristic
284C4×Q8C22×D313yes—
285C4×D4—36no—
286C4×D4—36no—
287C4×D4—33no—
288C4×D4—33no—
289C4×D4—63no—
290C4×D4—63no—
291C4×D4—63no—
292C4×D4—63no—
293#117—33no—
294#117—63no—
295C2×C4⋊C4—33no—
296C2×C4⋊C4—63no—
297C2×C4⋊C4—63no—
298C2×C4⋊C4D1213yes—
299C2×C22⋊C4—63no—
300C2×C42—63no—
301C2×C42C22×D313yes—
302#107—36no—
303#107—312no—
304#106—36no—
305#106—312no—
306C4⋊C8C22×D316yes—
307C4⋊C8C22×D316yes—
308#103—36no—
309#102—36no—
310#102—36no—
311#102—36no—
312#97C22×D316yes—
313C4×C8C22×D316yes—
314C23×C3C22×D411yescharacteristic
315C22×D3—24no—
316C22×D3—21no—
317C22×D3—22no—
318C22×D3—42no—
319C22×D3—42no—
320C3×Q8—26no—
321C3×Q8AGL(1,ℤ/8)14yescomplemented
322C2×C3×C4—26no—
323C2×C3×C4—43no—
324C2×C3×C4C22×D413yes—
325C2×C3×C4C22×D413yes—
326C2×C3×C4C22×D413yes—
327C2×C3×C4C22×D411yescharacteristic
328C2×C3×C4C22×D413yes—
329C2×C3×C4C22×Q812yes—
330C2×C3×C42−1+412yes—
331C2×C3×C4C2511yescharacteristic
332#67—24no—
333C2×Dic3—21no—
334C2×Dic3—23no—
335C2×Dic3—23no—
336C2×Dic3—26no—
337C2×Dic3—46no—
338D12—24no—
339D12—22no—
340D12—21no—
341D12C22×Q814yes—
342D12C22×Q812yescomplemented
343C4×D3—24no—
344C4×D3—412no—
345C4×D3—412no—
346C4×D3—46no—
347C4×D3—83no—
348Dic6—21no—
349Dic6—412no—
350Dic6—412no—
351Dic6—46no—
352Dic6—83no—
353Dic6C22×Q812yescomplemented
354C3×C8—26no—
355C3×C8C22×Q814yescomplemented
356C24—61no—
357#41—34no—
358#41—62no—
359C2×Q8—24no—
360C2×Q8—31no—
361C2×Q8—63no—
362C2×Q8—66no—
363C2×Q8—66no—
364C2×Q8—63no—
365C2×Q8—1212no—
366C2×Q8—1224no—
367C2×Q8—128no—
368C2×Q8—124no—
369C2×Q8C2×D1213yes—
370C2×Q8C2×D1211yescharacteristic
371C2×D4—34no—
372C2×D4—31no—
373C2×D4—32no—
374C2×D4—61no—
375C2×D4—61no—
376C2×D4—62no—
377C22×C4—61no—
378C22×C4—63no—
379C22×C4—63no—
380C22×C4—66no—
381C22×C4—66no—
382C22×C4—63no—
383C22×C4—1212no—
384C22×C4C2×D1213yes—
385C22×C4C2×D1213yes—
386C22×C4C23×D311yescharacteristic
387SD16—38no—
388D8—38no—
389M16—26no—
390M16D3×Q814yescomplemented
391C2×C8C23×D316yes—
392C2×C8C23×D312yes—
393C4⋊C4—33no—
394C4⋊C4—33no—
395C4⋊C4—36no—
396C4⋊C4—63no—
397C4⋊C4—66no—
398C4⋊C4—63no—
399C4⋊C4—66no—
400C4⋊C4—66no—
401C4⋊C4—63no—
402C4⋊C4—63no—
403C4⋊C4—66no—
404C4⋊C4—63no—
405C4⋊C4C2×D1213yes—
406C4⋊C4C2×D1216yes—
407C4⋊C4C2×D1213yes—
408C22⋊C4—63no—
409C22⋊C4—66no—
410C22⋊C4—66no—
411C22⋊C4—63no—
412C22⋊C4—63no—
413C42—63no—
414C42—63no—
415C42—63no—
416C42C2×D1213yes—
417C42C2×D1213yes—
418C42C23×D313yes—
419C42C23×D313yes—
420C22×C3—21no—
421C22×C3—21no—
422C22×C3D4×Q812yes—
423C22×C3C23×D411yescharacteristic
424C2×D3—22no—
425C2×D3—24no—
426C2×D3—48no—
427C2×D3—41no—
428C2×D3—42no—
429C2×D3—42no—
430C2×D3—41no—
431C3×C4—23no—
432C3×C4—23no—
433C3×C4—23no—
434C3×C4D4×Q812yes—
435C3×C4C2×AGL(1,ℤ/8)16yes—
436C3×C4C23×Q811yescharacteristic
437C3×C4C2×2−1+411yescharacteristic
438Dic3—22no—
439Dic3—43no—
440Dic3—46no—
441Dic3—43no—
442C23—61no—
443C23—62no—
444C23—64no—
445C23—122no—
446C23—122no—
447C23C22×D1211yescharacteristic
448Q8—26no—
449Q8—32no—
450Q8—61no—
451Q8—1212no—
452Q8—1212no—
453Q8—126no—
454Q8—243no—
455Q8#90814yesdirect product factor
456D4—32no—
457D4—34no—
458D4—64no—
459D4—62no—
460D4—61no—
461D4—64no—
462C2×C4—26no—
463C2×C4—43no—
464C2×C4—61no—
465C2×C4—64no—
466C2×C4—63no—
467C2×C4—63no—
468C2×C4—66no—
469C2×C4—1212no—
470C2×C4—126no—
471C2×C4—1212no—
472C2×C4—126no—
473C2×C4—243no—
474C2×C4C22×D1213yes—
475C2×C4C22×D1213yes—
476C2×C4C22×D1213yes—
477C2×C4C22×D1213yes—
478C2×C4C22×D1211yescharacteristic
479C2×C4C2×D3×Q812yes—
480C2×C4#101012yes—
481C2×C4C24×D311yescharacteristic
482C8—26no—
483C8C2×D3×Q814yescomplemented
484C2×C3—22no—
485C2×C3C2×D4×Q811yescharacteristic
486C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
487C2×C3#358611yescharacteristic
488D3—42no—
489D3—44no—
490C22—21no—
491C22—21no—
492C22—62no—
493C22—64no—
494C22—128no—
495C22—122no—
496C22—121no—
497C22—121no—
498C22—122no—
499C22Q8×D1212yes—
500C22C23×D1211yescharacteristic
501C4—23no—
502C4—23no—
503C4—23no—
504C4—62no—
505C4—126no—
506C4—123no—
507C4—123no—
508C4Q8×D1212yes—
509C4C2×#90816yes—
510C4C22×D3×Q811yescharacteristic
511C4C2×#101011yescharacteristic
512C3Q8×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
513C2—22no—
514C2—124no—
515C2—122no—
516C2C2×Q8×D1211yescharacteristic
517C2C22×#90811yescharacteristic
518C2#8716811yescharacteristic
subgroupG/H
centerC22C23×D12
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×Q8×M16C2
socleC22×C3C23×D4
prime radicalC2×#908C22
hypercenterQ8×M16D3
hypocenterC3Q8×AGL(1,ℤ/8)
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×Q8×D12C2semidirect
3. C3×Q8×M16C2semidirect
4. #79977C2semidirect
5. #79975C2semidirect
6. Q8×#204C2semidirect
7. Q8×D24C2semidirect
8. #79060C2semidirect
9. #79059C2semidirect
11. #78506C2semidirect
17. #5579C22semidirect
23. Q8×D12C22semidirect
25. #5567C22semidirect
26. Q8×Dic6C22semidirect
33. C3×#446C22semidirect
34. C3×C8×Q8C22semidirect
41. #4851C22semidirect
42. #4850C22semidirect
45. #4846C22semidirect
47. #4842C22semidirect
50. #4702C22semidirect
53. #4682C22semidirect
54. #4680C22semidirect
64. Q8×M16D3semidirect
94. #908Q8direct (N × H)
167. #446C2×D3semidirect
168. C8×Q8C2×D3semidirect
192. #235C2×Q8semidirect
195. C2×D12C2×Q8semidirect
211. C3×M16C2×Q8semidirect
241. D24C2×Q8semidirect
242. #204C2×Q8semidirect
321. C3×Q8AGL(1,ℤ/8)semidirect
342. D12C22×Q8semidirect
353. Dic6C22×Q8semidirect
355. C3×C8C22×Q8semidirect
390. M16D3×Q8semidirect
455. Q8#908direct (N × H)
483. C8C2×D3×Q8semidirect
512. C3Q8×AGL(1,ℤ/8)semidirect

Maximal subgroups

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

subgroupindex
Q8×#2352
C2×Q8×D122
C3×Q8×M162
#799772
#799752
Q8×#2042
Q8×D242
#790602
#790592
C4×#9082
#785062
Q8×AGL(1,ℤ/8)3