Finite Group Databasebeta

← Q8×#932 #11675765

Subgroups — Q8×#932 #11675765

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
1C2×C3×D4×Q8C211yesmaximal, characteristic, complemented
2Q8×#235C211yesmaximal, characteristic, complemented
3#83564C216yesmaximal, complemented
4#83558C216yesmaximal, complemented
5Q8×#214C212yesmaximal, complemented
6Q8×#213C212yesmaximal, complemented
7#83212C213yesmaximal, complemented
8#83207C213yesmaximal, complemented
9Q8×#208C211yesmaximal, characteristic, complemented
10#82296C213yesmaximal, complemented
11C4×#932C213yesmaximal
12Q8×AGL(1,ℤ/8)—31nomaximal, Sylow
13C23×C3×Q8—21no—
14C3×D4×Q8—21no—
15C3×D4×Q8—23no—
16C3×D4×Q8C2212yes—
17C3×D4×Q8C2216yes—
18C2×C3×C4⋊Q8C2213yes—
19C2×C3×C22⋊Q8—23no—
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×D4C2213yes—
22Q8×#67—21no—
23C2×#932C2211yescharacteristic
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#5094C2216yescomplemented
35#5091C2216yescomplemented
36#5087C2216yescomplemented
37#5084C2216yescomplemented
38#5081C22112yescomplemented
39#5080C2216yescomplemented
40#5078C2216yescomplemented
41#5024C2216yescomplemented
42Q8×C3⋊C8C2212yescomplemented
43#5018C2216yes—
44C4×#214C2216yes—
45#5014C2216yes—
46C4×#213C2216yes—
47#5009C22112yes—
48#5008C2216yes—
49#5007C2216yes—
50#4966C2213yes—
51#4964C2213yes—
52#4964C2216yes—
53#4924C2213yes—
54C4×#208C2213yes—
55Q8×#41—31no—
56C2×D4×Q8D311yescharacteristic, complemented
57#3386—36no—
58#3384—36no—
59Q8×SD16—32no—
60Q8×D8—32no—
61#3165—33no—
62#3079—33no—
63#3077—33no—
64Q8×M16—31no—
65C4×AGL(1,ℤ/8)—33no—
66C22×C3×Q8—26no—
67C22×C3×Q8—22no—
68C22×C3×Q8—23no—
69C22×C3×Q8—48no—
70C22×C3×Q8D411yescharacteristic
71C22×C3×D4C2311yescharacteristic
72C23×C3×C4—23no—
73C2×D3×Q8—21no—
74C2×#235C2311yescharacteristic
75C3×C4⋊Q8—23no—
76C3×C4⋊Q8—23no—
77C3×C4⋊Q8C2316yes—
78C3×C4⋊Q8C2316yes—
79C3×C22⋊Q8—26no—
80C3×C22⋊Q8—26no—
81C3×C22⋊Q8—212no—
82C3×C4×Q8D411yescharacteristic
83C3×C4×Q8D413yes—
84C3×C4×Q8C2311yescharacteristic
85C3×C4×Q8C2313yes—
86C3×C4×D4—23no—
87C3×C4×D4—23no—
88C3×C4×D4C2316yes—
89C3×C4×D4C2316yes—
90C2×C3×C4⋊C4—23no—
91C2×C3×C4⋊C4D413yes—
92C2×C3×C4⋊C4C2313yes—
93C2×C3×C22⋊C4—23no—
94C2×C3×C42C2313yes—
95#946—23no—
96Q8×Dic3—21no—
97#944—23no—
98C2×#214C2312yes—
99#932Q818yesdirect product factor
100C2×#213C2312yes—
101C4×#67—23no—
102C4×#67—23no—
103#926C2313yes—
104#924—23no—
105#924—23no—
106C2×#208C2313yes—
107C2×#208C2311yescharacteristic
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#832C2316yes—
126#832C2316yes—
127#810C2316yes—
128#809C2316yes—
129#808C23112yes—
130#808C2316yes—
131#807C23112yes—
132#807C2316yes—
133C4⋊(C3⋊C8)C2316yes—
134C4⋊(C3⋊C8)C2316yes—
135#803C2316yes—
136C4×C3⋊C8C2316yes—
137C23×Q8—21no—
138C2×AGL(1,ℤ/8)—31no—
139Q82—31no—
140#557—33no—
141#554—33no—
142#554—63no—
143D4×Q8—21no—
144D4×Q8—23no—
145D4×Q8—31no—
146D4×Q8—61no—
147D4×Q8C2×D312yes—
148D4×Q8C2×D316yes—
149#533—33no—
150#533—63no—
151C2×C4⋊Q8C2×D313yes—
152C2×C22⋊Q8—23no—
153C4×#41—33no—
154C2×C4×Q8—61no—
155C2×C4×Q8C2×D311yescharacteristic
156C2×C4×D4C2×D313yes—
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#446—36no—
168C8×Q8—32no—
169#442—36no—
170#440—36no—
171C4×SD16—36no—
172C4×D8—36no—
173#428—36no—
174#428—33no—
175#423—33no—
176#418—33no—
177#417—33no—
178C4×M16—33no—
179C24×C3—21no—
180C2×C3×Q8—24no—
181C2×C3×Q8—26no—
182C2×C3×Q8—26no—
183C2×C3×Q8—412no—
184C2×C3×Q8—424no—
185C2×C3×Q8—48no—
186C2×C3×Q8C2×D413yes—
187C2×C3×Q8C2×D411yescharacteristic
188C2×C3×D4—21no—
189C2×C3×D4—22no—
190C2×C3×D4C2×Q814yescomplemented
191C2×C3×D4C2412yes—
192C22×C3×C4—23no—
193C22×C3×C4—26no—
194C22×C3×C4—26no—
195C22×C3×C4—412no—
196C22×C3×C4C2×D413yes—
197C22×C3×C4C2×D413yes—
198C22×C3×C4C2411yescharacteristic
199C2×#67—21no—
200D3×Q8—44no—
201#235—22no—
202#235C2×Q814yescomplemented
203C2×D12C2411yescharacteristic
204C2×C4×D3—21no—
205C2×C4×D3—23no—
206C2×C4×D3—23no—
207C2×Dic6—23no—
208C2×Dic6—23no—
209C2×Dic6C2411yescharacteristic
210C3×C4⋊C4—26no—
211C3×C4⋊C4—26no—
212C3×C4⋊C4—23no—
213C3×C4⋊C4C2×D413yes—
214C3×C4⋊C4C2×D413yes—
215C3×C4⋊C4C2×D416yes—
216C3×C4⋊C4C2416yes—
217C3×C22⋊C4—26no—
218C3×C22⋊C4—26no—
219C3×C42C2×D413yes—
220C3×C42C2×D413yes—
221C3×C42C2413yes—
222C3×C42C2413yes—
223#217—23no—
224#214C2×Q818yescomplemented
225#213C2×Q818yescomplemented
226(C2×D3)⋊C4—23no—
227(C2×D3)⋊C4—23no—
228C4⋊Dic3—23no—
229C4⋊Dic3—23no—
230C4⋊Dic3C2413yes—
231C4⋊Dic3C2413yes—
232Dic3⋊C4—26no—
233Dic3⋊C4—26no—
234Dic3⋊C4—23no—
235Dic3⋊C4—23no—
236C4×Dic3—23no—
237C4×Dic3—23no—
238C4×Dic3—23no—
239#208—26no—
240#208C2×Q814yescomplemented
241C2×C3⋊C8C2416yes—
242C2×C3⋊C8C2412yes—
243C2×#41—31no—
244C22×Q8—26no—
245C22×Q8—22no—
246C22×Q8—23no—
247C22×Q8—48no—
248C22×Q8—61no—
249C22×Q8#6711yescharacteristic
250C22×D4C22×D311yescharacteristic
251C23×C4—23no—
252AGL(1,ℤ/8)—38no—
253C2×SD16—32no—
254C2×D8—32no—
255C2×M16—31no—
256C2×M16—33no—
257C4⋊Q8—23no—
258C4⋊Q8—23no—
259C4⋊Q8—33no—
260C4⋊Q8—33no—
261C4⋊Q8—63no—
262C4⋊Q8—63no—
263C4⋊Q8C22×D316yes—
264C4⋊Q8C22×D316yes—
265#125—36no—
266#125—66no—
267C22⋊Q8—26no—
268C22⋊Q8—212no—
269C22⋊Q8—26no—
270C22⋊Q8—63no—
271C22⋊Q8—63no—
272C22⋊Q8—63no—
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×Q8#6711yescharacteristic
282C4×Q8#6713yes—
283C4×Q8C22×D311yescharacteristic
284C4×Q8C22×D313yes—
285C4×D4—23no—
286C4×D4—23no—
287C4×D4—33no—
288C4×D4—33no—
289C4×D4—63no—
290C4×D4—63no—
291C4×D4C22×D316yes—
292C4×D4C22×D316yes—
293#117—33no—
294#117—63no—
295C2×C4⋊C4—23no—
296C2×C4⋊C4—63no—
297C2×C4⋊C4#6713yes—
298C2×C4⋊C4C22×D313yes—
299C2×C22⋊C4—23no—
300C2×C42—63no—
301C2×C42C22×D313yes—
302#107—36no—
303#107—312no—
304#106—36no—
305#106—312no—
306C4⋊C8—36no—
307C4⋊C8—36no—
308#103—36no—
309#102—36no—
310#102—36no—
311#102—36no—
312#97—36no—
313C4×C8—36no—
314C23×C3—24no—
315C23×C3—22no—
316C23×C3—42no—
317C23×C3—42no—
318C23×C3C22×D411yescharacteristic
319C22×D3—21no—
320C3×Q8—26no—
321C3×Q8—412no—
322C3×Q8—412no—
323C3×Q8AGL(1,ℤ/8)14yescomplemented
324C3×D4—24no—
325C3×D4—22no—
326C3×D4C22×Q814yes—
327C2×C3×C4—26no—
328C2×C3×C4—26no—
329C2×C3×C4—43no—
330C2×C3×C4—412no—
331C2×C3×C4—46no—
332C2×C3×C4—412no—
333C2×C3×C4C22×D413yes—
334C2×C3×C4C22×D413yes—
335C2×C3×C4C22×D413yes—
336C2×C3×C4C22×D411yescharacteristic
337C2×C3×C4C22×D413yes—
338C2×C3×C4C22×Q812yes—
339C2×C3×C42−1+412yes—
340C2×C3×C4C2511yescharacteristic
341#67—24no—
342C2×Dic3—21no—
343C2×Dic3—23no—
344C2×Dic3—23no—
345D12—21no—
346D12C22×Q812yescomplemented
347C4×D3—24no—
348C4×D3—46no—
349C4×D3—83no—
350Dic6—21no—
351Dic6—46no—
352Dic6—83no—
353Dic6C22×Q812yescomplemented
354C3⋊C8—26no—
355C3⋊C8C22×Q814yescomplemented
356C24—21no—
357#41—34no—
358#41—62no—
359C2×Q8—24no—
360C2×Q8—26no—
361C2×Q8—26no—
362C2×Q8—31no—
363C2×Q8—412no—
364C2×Q8—424no—
365C2×Q8—48no—
366C2×Q8—63no—
367C2×Q8—63no—
368C2×Q8—124no—
369C2×Q8C2×#6713yes—
370C2×Q8C2×#6711yescharacteristic
371C2×D4—22no—
372C2×D4—21no—
373C2×D4—31no—
374C2×D4—61no—
375C2×D4D3×Q814yescomplemented
376C2×D4C23×D312yes—
377C22×C4—23no—
378C22×C4—26no—
379C22×C4—26no—
380C22×C4—412no—
381C22×C4—61no—
382C22×C4—63no—
383C22×C4—63no—
384C22×C4C2×#6713yes—
385C22×C4C2×#6713yes—
386C22×C4C23×D311yescharacteristic
387SD16—38no—
388D8—38no—
389M16—34no—
390M16—66no—
391C2×C8—32no—
392C2×C8—36no—
393C4⋊C4—26no—
394C4⋊C4—26no—
395C4⋊C4—23no—
396C4⋊C4—33no—
397C4⋊C4—33no—
398C4⋊C4—66no—
399C4⋊C4—63no—
400C4⋊C4—63no—
401C4⋊C4—63no—
402C4⋊C4—66no—
403C4⋊C4—63no—
404C4⋊C4C2×#6713yes—
405C4⋊C4C2×#6716yes—
406C4⋊C4C2×#6713yes—
407C4⋊C4C23×D316yes—
408C22⋊C4—26no—
409C22⋊C4—26no—
410C22⋊C4—63no—
411C22⋊C4—63no—
412C22⋊C4—63no—
413C42—63no—
414C42—63no—
415C42—63no—
416C42C2×#6713yes—
417C42C2×#6713yes—
418C42C23×D313yes—
419C42C23×D313yes—
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—24no—
423C22×C3—48no—
424C22×C3—42no—
425C22×C3—42no—
426C22×C3D4×Q812yes—
427C22×C3C23×D411yescharacteristic
428C2×D3—22no—
429C2×D3—41no—
430C2×D3—41no—
431C3×C4—23no—
432C3×C4—23no—
433C3×C4—23no—
434C3×C4—46no—
435C3×C4D4×Q812yes—
436C3×C4C2×AGL(1,ℤ/8)16yes—
437C3×C4C23×Q811yescharacteristic
438C3×C4C2×2−1+411yescharacteristic
439Dic3—22no—
440Dic3—43no—
441Dic3—43no—
442C23—22no—
443C23—24no—
444C23—42no—
445C23—42no—
446C23—61no—
447C23C22×#6711yescharacteristic
448Q8—26no—
449Q8—32no—
450Q8—412no—
451Q8—412no—
452Q8—61no—
453Q8—126no—
454Q8—243no—
455Q8#93214yesdirect product factor
456D4—22no—
457D4—24no—
458D4—32no—
459D4—64no—
460D4—61no—
461D4C2×D3×Q814yes—
462C2×C4—26no—
463C2×C4—26no—
464C2×C4—43no—
465C2×C4—412no—
466C2×C4—46no—
467C2×C4—412no—
468C2×C4—61no—
469C2×C4—64no—
470C2×C4—63no—
471C2×C4—63no—
472C2×C4—126no—
473C2×C4—243no—
474C2×C4C2×D3×Q812yes—
475C2×C4#100712yes—
476C2×C4C22×#6713yes—
477C2×C4C22×#6713yes—
478C2×C4C22×#6713yes—
479C2×C4C22×#6713yes—
480C2×C4C22×#6711yescharacteristic
481C2×C4C24×D311yescharacteristic
482C8—34no—
483C8—66no—
484C2×C3—22no—
485C2×C3—44no—
486C2×C3C2×D4×Q811yescharacteristic
487C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
488C2×C3#358611yescharacteristic
489D3—42no—
490C22—21no—
491C22—21no—
492C22—24no—
493C22—48no—
494C22—42no—
495C22—42no—
496C22—62no—
497C22—121no—
498C22—121no—
499C22Q8×#6712yes—
500C22C23×#6711yescharacteristic
501C4—23no—
502C4—23no—
503C4—23no—
504C4—46no—
505C4—62no—
506C4—123no—
507C4—123no—
508C4C2×#93216yes—
509C4Q8×#6712yes—
510C4C22×D3×Q811yescharacteristic
511C4C2×#100711yescharacteristic
512C3Q8×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
513C2—22no—
514C2—44no—
515C2—122no—
516C2C22×#93211yescharacteristic
517C2C2×Q8×#6711yescharacteristic
518C2#8728711yescharacteristic
subgroupG/H
centerC22C23×#67
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×D4×Q8C2
socleC22×C3C23×D4
prime radicalC2×#932C22
hypercenterC2×D4×Q8D3
hypocenterC3Q8×AGL(1,ℤ/8)
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×D4×Q8C2semidirect
2. Q8×#235C2semidirect
3. #83564C2semidirect
4. #83558C2semidirect
5. Q8×#214C2semidirect
6. Q8×#213C2semidirect
7. #83212C2semidirect
8. #83207C2semidirect
9. Q8×#208C2semidirect
10. #82296C2semidirect
24. #5579C22semidirect
26. Q8×D12C22semidirect
28. #5567C22semidirect
29. Q8×Dic6C22semidirect
34. #5094C22semidirect
35. #5091C22semidirect
36. #5087C22semidirect
37. #5084C22semidirect
38. #5081C22semidirect
39. #5080C22semidirect
40. #5078C22semidirect
41. #5024C22semidirect
42. Q8×C3⋊C8C22semidirect
56. C2×D4×Q8D3semidirect
99. #932Q8direct (N × H)
190. C2×C3×D4C2×Q8semidirect
202. #235C2×Q8semidirect
224. #214C2×Q8semidirect
225. #213C2×Q8semidirect
240. #208C2×Q8semidirect
323. C3×Q8AGL(1,ℤ/8)semidirect
346. D12C22×Q8semidirect
353. Dic6C22×Q8semidirect
355. C3⋊C8C22×Q8semidirect
375. C2×D4D3×Q8semidirect
455. Q8#932direct (N × H)
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
C2×C3×D4×Q82
Q8×#2352
#835642
#835582
Q8×#2142
Q8×#2132
#832122
#832072
Q8×#2082
#822962
C4×#9322
Q8×AGL(1,ℤ/8)3