Finite Group Databasebeta

← Q8×#949 #11675760

Subgroups — Q8×#949 #11675760

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
1C3×Q8×#41C211yesmaximal, characteristic, complemented
2C2×Q8×D12C211yesmaximal, characteristic, complemented
3Q8×#215C212yesmaximal, complemented
4#83694C216yesmaximal, complemented
5#83630C216yesmaximal, complemented
6Q8×#213C212yesmaximal, complemented
7#83350C213yesmaximal, complemented
8#83341C213yesmaximal, complemented
9C4×#949C213yesmaximal
10#82508C213yesmaximal, complemented
11Q8×#208C211yesmaximal, characteristic, complemented
12Q8×AGL(1,ℤ/8)—31nomaximal, Sylow
13C22×D3×Q8—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×#949C2211yescharacteristic
26Q8×D12—21no—
27Q8×D12—23no—
28Q8×D12C2212yes—
29Q8×D12C2216yes—
30C2×#897—23no—
31C2×C4×D12C2213yes—
32C2×#869C2213yes—
33#5232C2213yes—
34#5087C2216yes—
35#5084C2216yes—
36#5080C2216yescomplemented
37#5078C2216yescomplemented
38#5066C22112yes—
39#5064C22112yescomplemented
40#5026C2216yes—
41C4×#215C2216yes—
42#5024C2216yescomplemented
43Q8×C3⋊C8C2212yescomplemented
44#5021C2216yescomplemented
45#5020C2216yescomplemented
46#5014C2216yes—
47C4×#213C2216yes—
48#5008C2216yescomplemented
49#5007C2216yescomplemented
50#5001C2213yes—
51#4995C2216yes—
52#4995C2213yes—
53#4924C2213yes—
54C4×#208C2213yes—
55Q8×#41D311yescharacteristic, complemented
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×M16—31no—
65C4×AGL(1,ℤ/8)—33no—
66C2×C3×#41C2311yescharacteristic
67C22×C3×Q8—21no—
68C22×C3×Q8D411yescharacteristic
69C2×D3×Q8—26no—
70C2×D3×Q8—22no—
71C2×D3×Q8—48no—
72C22×D12C2311yescharacteristic
73C22×C4×D3—23no—
74C22×Dic6—23no—
75C3×C4⋊Q8—23no—
76C3×C4⋊Q8—23no—
77C3×C4⋊Q8C2313yes—
78C3×C4⋊Q8C2313yes—
79C3×#125—26no—
80C3×#125C2316yes—
81C3×C22⋊Q8—23no—
82C3×C22⋊Q8—23no—
83C3×C22⋊Q8—23no—
84C3×C22⋊Q8—23no—
85C3×C22⋊Q8—26no—
86C3×C4×Q8—21no—
87C3×C4×Q8—23no—
88C3×C4×Q8—23no—
89C3×C4×Q8—23no—
90C3×C4×Q8D411yescharacteristic
91C3×C4×Q8D413yes—
92C3×C4×Q8C2313yes—
93C3×C4×Q8C2313yes—
94C3×C4×Q8C2311yescharacteristic
95C3×C4×Q8C2313yes—
96C3×C4×D4—23no—
97C3×C4×D4—23no—
98C3×C4×D4C2313yes—
99C3×C4×D4C2313yes—
100C3×#117—23no—
101C3×#117C2313yes—
102C2×C3×C4⋊C4—23no—
103C2×C3×C4⋊C4D413yes—
104C2×C3×C42—23no—
105C2×C3×C42C2313yes—
106#949Q818yesdirect product factor
107C2×#215C2312yes—
108C2×#213C2312yes—
109C2×(C2×D3)⋊C4—23no—
110C2×C4⋊Dic3—23no—
111C2×C4⋊Dic3C2313yes—
112C2×#208C2313yes—
113C2×#208C2311yescharacteristic
114#897—26no—
115#897—212no—
116#897—26no—
117C4×D12—23no—
118C4×D12—23no—
119C4×D12C2316yes—
120C4×D12C2316yes—
121#869—23no—
122#869—23no—
123#869C2316yes—
124#869C2316yes—
125#835C2316yes—
126#832C2316yes—
127#809C2316yes—
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—61no—
138C2×AGL(1,ℤ/8)—31no—
139Q82C2×D311yescharacteristic, complemented
140#557C2×D313yescomplemented
141#554—23no—
142#554C2×D313yescomplemented
143D4×Q8—21no—
144D4×Q8—32no—
145D4×Q8—36no—
146D4×Q8—61no—
147D4×Q8—63no—
148D4×Q8C2×D311yescharacteristic, complemented
149#533—23no—
150#533C2×D313yes—
151C2×C4⋊Q8—33no—
152C2×C22⋊Q8—63no—
153C4×#41C2×D313yes—
154C2×C4×Q8—21no—
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#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—
179C23×D3—21no—
180C3×#41—22no—
181C3×#41C2×Q814yescomplemented
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—23no—
198D3×Q8—424no—
199D3×Q8—48no—
200C2×D12—21no—
201C2×D12—22no—
202C2×D12C2×Q814yescomplemented
203C2×D12C2412yes—
204C2×C4×D3—26no—
205C2×C4×D3—26no—
206C2×C4×D3—412no—
207C2×Dic6—26no—
208C2×Dic6—26no—
209C2×Dic6—412no—
210C3×C4⋊C4—26no—
211C3×C4⋊C4—23no—
212C3×C4⋊C4—23no—
213C3×C4⋊C4—23no—
214C3×C4⋊C4—23no—
215C3×C4⋊C4—26no—
216C3×C4⋊C4C2×D413yes—
217C3×C4⋊C4C2×D416yes—
218C3×C4⋊C4C2×D413yes—
219C3×C4⋊C4C2413yes—
220C3×C4⋊C4C2413yes—
221C3×C22⋊C4—23no—
222C3×C22⋊C4—23no—
223C3×C22⋊C4—23no—
224C3×C42—23no—
225C3×C42—23no—
226C3×C42—23no—
227C3×C42C2×D413yes—
228C3×C42C2×D413yes—
229C3×C42C2413yes—
230C3×C42C2413yes—
231#215C2×Q818yescomplemented
232#213C2×Q818yescomplemented
233(C2×D3)⋊C4—26no—
234(C2×D3)⋊C4—26no—
235C4⋊Dic3—23no—
236C4⋊Dic3—26no—
237C4⋊Dic3—26no—
238C4⋊Dic3C2416yes—
239#208—26no—
240#208C2×Q814yescomplemented
241C2×C3⋊C8C2416yes—
242C2×C3⋊C8C2412yes—
243C2×#41C22×D311yescharacteristic
244C22×Q8—21no—
245C22×Q8—66no—
246C22×Q8—62no—
247C22×Q8—63no—
248C22×Q8—128no—
249C22×Q8#6711yescharacteristic
250C22×D4—31no—
251C23×C4—63no—
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—36no—
260C4⋊Q8—36no—
261C4⋊Q8—63no—
262C4⋊Q8—63no—
263C4⋊Q8C22×D313yes—
264C4⋊Q8C22×D313yes—
265#125—26no—
266#125C22×D316yes—
267C22⋊Q8—23no—
268C22⋊Q8—23no—
269C22⋊Q8—23no—
270C22⋊Q8—23no—
271C22⋊Q8—26no—
272C22⋊Q8—66no—
273C22⋊Q8—612no—
274C22⋊Q8—66no—
275C4×Q8—21no—
276C4×Q8—23no—
277C4×Q8—23no—
278C4×Q8—23no—
279C4×Q8#6711yescharacteristic
280C4×Q8#6713yes—
281C4×Q8C22×D313yes—
282C4×Q8C22×D313yes—
283C4×Q8C22×D311yescharacteristic
284C4×Q8C22×D313yes—
285C4×D4—23no—
286C4×D4—23no—
287C4×D4—36no—
288C4×D4—36no—
289C4×D4—63no—
290C4×D4—63no—
291C4×D4C22×D313yes—
292C4×D4C22×D313yes—
293#117—23no—
294#117C22×D313yes—
295C2×C4⋊C4—23no—
296C2×C4⋊C4—33no—
297C2×C4⋊C4—63no—
298C2×C4⋊C4#6713yes—
299C2×C22⋊C4—63no—
300C2×C42—23no—
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—21no—
315C23×C3C22×D411yescharacteristic
316C22×D3—24no—
317C22×D3—22no—
318C22×D3—42no—
319C22×D3—42no—
320C3×Q8—26no—
321C3×Q8—21no—
322C3×Q8—46no—
323C3×Q8—83no—
324C3×Q8AGL(1,ℤ/8)14yescomplemented
325C3×Q8C22×Q812yescomplemented
326C3×D4—24no—
327C3×D4—21no—
328C3×D4C22×Q812yescomplemented
329C2×C3×C4—21no—
330C2×C3×C4—24no—
331C2×C3×C4—26no—
332C2×C3×C4—23no—
333C2×C3×C4—23no—
334C2×C3×C4—43no—
335C2×C3×C4—46no—
336C2×C3×C4—83no—
337C2×C3×C4C22×D413yes—
338C2×C3×C4C22×D413yes—
339C2×C3×C4C22×D413yes—
340C2×C3×C4C22×D411yescharacteristic
341C2×C3×C4C22×D413yes—
342C2×C3×C4C22×Q812yes—
343C2×C3×C42−1+412yes—
344C2×C3×C4C2511yescharacteristic
345C2×Dic3—26no—
346C2×Dic3—46no—
347D12—24no—
348D12—22no—
349D12C22×Q814yes—
350C4×D3—412no—
351C4×D3—412no—
352Dic6—412no—
353Dic6—412no—
354C3⋊C8—26no—
355C3⋊C8C22×Q814yescomplemented
356C24—61no—
357#41—22no—
358#41D3×Q814yescomplemented
359C2×Q8—24no—
360C2×Q8—23no—
361C2×Q8—23no—
362C2×Q8—44no—
363C2×Q8—66no—
364C2×Q8—66no—
365C2×Q8—1212no—
366C2×Q8—1224no—
367C2×Q8—128no—
368C2×Q8C2×#6713yes—
369C2×Q8C2×#6711yescharacteristic
370C2×Q8C23×D311yescharacteristic
371C2×D4—21no—
372C2×D4—34no—
373C2×D4—32no—
374C2×D4—62no—
375C2×D4—61no—
376C2×D4C23×D311yescharacteristic
377C22×C4—21no—
378C22×C4—23no—
379C22×C4—23no—
380C22×C4—63no—
381C22×C4—66no—
382C22×C4—66no—
383C22×C4—1212no—
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—23no—
395C4⋊C4—23no—
396C4⋊C4—23no—
397C4⋊C4—23no—
398C4⋊C4—26no—
399C4⋊C4—36no—
400C4⋊C4—63no—
401C4⋊C4—66no—
402C4⋊C4—66no—
403C4⋊C4C2×#6713yes—
404C4⋊C4C2×#6716yes—
405C4⋊C4C2×#6713yes—
406C4⋊C4C23×D313yes—
407C4⋊C4C23×D313yes—
408C22⋊C4—23no—
409C22⋊C4—23no—
410C22⋊C4—23no—
411C22⋊C4—66no—
412C22⋊C4—66no—
413C42—23no—
414C42—23no—
415C42—23no—
416C42C2×#6713yes—
417C42C2×#6713yes—
418C42C23×D313yes—
419C42C23×D313yes—
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—22no—
423C22×C3—41no—
424C22×C3—41no—
425C22×C3D4×Q812yes—
426C22×C3C23×D411yescharacteristic
427C2×D3—24no—
428C2×D3—48no—
429C2×D3—42no—
430C2×D3—42no—
431C3×C4—23no—
432C3×C4—23no—
433C3×C4—22no—
434C3×C4—23no—
435C3×C4—43no—
436C3×C4—43no—
437C3×C4D4×Q812yes—
438C3×C4C2×AGL(1,ℤ/8)16yes—
439C3×C4C23×Q811yescharacteristic
440C3×C4C2×2−1+411yescharacteristic
441Dic3—46no—
442C23—21no—
443C23—62no—
444C23—64no—
445C23—122no—
446C23—122no—
447C23C22×#6711yescharacteristic
448Q8—21no—
449Q8—26no—
450Q8—46no—
451Q8—83no—
452Q8—1212no—
453Q8—1212no—
454Q8#94914yesdirect product factor
455Q8C2×D3×Q812yescomplemented
456D4—24no—
457D4—21no—
458D4—34no—
459D4—62no—
460D4—64no—
461D4C2×D3×Q812yescomplemented
462C2×C4—21no—
463C2×C4—24no—
464C2×C4—26no—
465C2×C4—23no—
466C2×C4—23no—
467C2×C4—43no—
468C2×C4—46no—
469C2×C4—66no—
470C2×C4—83no—
471C2×C4—1212no—
472C2×C4—126no—
473C2×C4—1212no—
474C2×C4C2×D3×Q812yes—
475C2×C4#100712yes—
476C2×C4C22×#6713yes—
477C2×C4C22×#6713yes—
478C2×C4C22×#6713yes—
479C2×C4C22×#6711yescharacteristic
480C2×C4C22×#6713yes—
481C2×C4C24×D311yescharacteristic
482C8—34no—
483C8—66no—
484C2×C3—22no—
485C2×C3—42no—
486C2×C3C2×D4×Q811yescharacteristic
487C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
488C2×C3#358611yescharacteristic
489D3—44no—
490C22—21no—
491C22—21no—
492C22—22no—
493C22—41no—
494C22—41no—
495C22—64no—
496C22—128no—
497C22—122no—
498C22—122no—
499C22Q8×#6712yes—
500C22C23×#6711yescharacteristic
501C4—23no—
502C4—23no—
503C4—22no—
504C4—23no—
505C4—43no—
506C4—43no—
507C4—126no—
508C4Q8×#6712yes—
509C4C2×#94916yes—
510C4C22×D3×Q811yescharacteristic
511C4C2×#100711yescharacteristic
512C3Q8×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
513C2—22no—
514C2—42no—
515C2—124no—
516C2C2×Q8×#6711yescharacteristic
517C2C22×#94911yescharacteristic
518C2#8729911yescharacteristic
subgroupG/H
centerC22C23×#67
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×Q8×#41C2
socleC22×C3C23×D4
prime radicalC2×#949C22
hypercenterQ8×#41D3
hypocenterC3Q8×AGL(1,ℤ/8)
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×Q8×D12C2semidirect
3. Q8×#215C2semidirect
4. #83694C2semidirect
5. #83630C2semidirect
6. Q8×#213C2semidirect
7. #83350C2semidirect
8. #83341C2semidirect
10. #82508C2semidirect
11. Q8×#208C2semidirect
14. C3×Q82C22semidirect
15. C3×#557C22semidirect
17. C3×#554C22semidirect
19. C3×D4×Q8C22semidirect
36. #5080C22semidirect
37. #5078C22semidirect
39. #5064C22semidirect
42. #5024C22semidirect
43. Q8×C3⋊C8C22semidirect
44. #5021C22semidirect
45. #5020C22semidirect
48. #5008C22semidirect
49. #5007C22semidirect
55. Q8×#41D3semidirect
106. #949Q8direct (N × H)
139. Q82C2×D3semidirect
140. #557C2×D3semidirect
142. #554C2×D3semidirect
148. D4×Q8C2×D3semidirect
181. C3×#41C2×Q8semidirect
202. C2×D12C2×Q8semidirect
231. #215C2×Q8semidirect
232. #213C2×Q8semidirect
240. #208C2×Q8semidirect
324. C3×Q8AGL(1,ℤ/8)semidirect
325. C3×Q8C22×Q8semidirect
328. C3×D4C22×Q8semidirect
355. C3⋊C8C22×Q8semidirect
358. #41D3×Q8semidirect
454. Q8#949direct (N × H)
455. Q8C2×D3×Q8semidirect
461. D4C2×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
C3×Q8×#412
C2×Q8×D122
Q8×#2152
#836942
#836302
Q8×#2132
#833502
#833412
C4×#9492
#825082
Q8×#2082
Q8×AGL(1,ℤ/8)3