Finite Group Databasebeta

← Q8×#3796 #11903764

Subgroups — Q8×#3796 #11903764

Order 1152.

2285 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×Q8×C3⋊D3C211yesmaximal, characteristic, complemented
2C3×Q8×D12C211yesmaximal, characteristic, complemented
3C3×Q8×Dic6C211yesmaximal, characteristic, complemented
4Q8×#626C212yesmaximal, complemented
5D3×Q8×Dic3C212yesmaximal, complemented
6#10594393C216yesmaximal, complemented
7#10594387C216yesmaximal, complemented
8#10593956C213yesmaximal, complemented
9C4×#3796C213yesmaximal
10#10593866C213yesmaximal, complemented
11#10593861C213yesmaximal, complemented
12Q8×#239—31nomaximal
13Q8×#237—31nomaximal
14C2×Q8×C3⋊D3C2211yescharacteristic
15C2×C3×D3×Q8C2212yes—
16C2×#3796C2211yescharacteristic
17C32×C4×Q8C2211yescharacteristic
18Q8×C3⋊Dic3C2211yescharacteristic
19C4⋊C4×C3⋊D3C2213yes—
20#64096C2213yes—
21C42×C3⋊D3C2213yes—
22C4×#634C2213yes—
23C3×Q8×Dic3C2212yes—
24C3×#897C2216yes—
25C3×#888C2216yes—
26C3×C4×D12C2213yes—
27C3×#869C2213yes—
28C3×#869C2213yes—
29C3×C4×Dic6C2213yes—
30C4×#626C2216yes—
31#63898C2213yes—
32D3×C4⋊Dic3C2216yes—
33#63887C22112yes—
34#63881C2216yes—
35#63878C2216yes—
36C4×D3×Dic3C2216yes—
37#63872C2216yes—
38#63871C2216yes—
39#63871C2216yes—
40#63866C2216yes—
41#63857C2216yes—
42#63849C2213yes—
43#63846C22112yes—
44#63843C2216yes—
45#63843C2213yes—
46#63843C2216yes—
47Dic3×Dic6C2216yes—
48C3×Q82—31no—
49C3×D4×Q8—31no—
50Q8×#67—32no—
51C2×Q8×Dic3—32no—
52#5730—36no—
53#5730—33no—
54#5728—36no—
55#5728—33no—
56#5728—33no—
57#5721—33no—
58#5635—36no—
59#5622—36no—
60Q8×D12—32no—
61Q8×D12D311yescharacteristic, complemented
62C4×#239—33no—
63C4×D3×Q8—32no—
64C4×D3×Q8—31no—
65C4×D3×Q8—31no—
66C4×D3×Q8—61no—
67#5570—33no—
68Q8×Dic6D311yescharacteristic, complemented
69#5543—33no—
70C4×#237—33no—
71C2×C32×Q8C2311yescharacteristic
72Q8×C3⋊D3—24no—
73C2×C4×C3⋊D3C2313yes—
74C2×C4×C3⋊D3C2313yes—
75C2×C4×C3⋊D3C2311yescharacteristic
76C2×#634C2313yes—
77C3×D3×Q8—28no—
78C2×C3×D12C2311yescharacteristic
79C2×C3×C4×D3C2316yes—
80C2×C3×Dic6C2316yes—
81C2×C3×Dic6C2316yes—
82C2×C3×Dic6C2311yescharacteristic
83C2×#626C2312yes—
84C2×D3×Dic3C2316yes—
85C2×D3×Dic3C2312yes—
86#3796Q818yesdirect product factor
87C32×C4⋊C4C2313yes—
88C32×C42C2313yes—
89C4⋊(C3⋊Dic3)C2313yes—
90#3750C2316yes—
91C4×C3⋊Dic3C2313yes—
92C4×C3⋊Dic3C2313yes—
93C4×C3⋊Dic3C2313yes—
94C3×(C2×D3)⋊C4C2316yes—
95C3×C4⋊Dic3C2316yes—
96C3×C4⋊Dic3C2316yes—
97C3×C4⋊Dic3C2313yes—
98C3×C4⋊Dic3C2313yes—
99C3×Dic3⋊C4C2316yes—
100C3×C4×Dic3C2316yes—
101Dic3⋊Dic3C23112yes—
102Dic3⋊Dic3C2316yes—
103Dic3⋊Dic3C23112yes—
104#3722C2316yes—
105(C2×D3)⋊Dic3C2316yes—
106Dic32C2316yes—
107Dic32C2316yes—
108Q8×#41—91noSylow
109C22×C3×Q8—32no—
110C2×#239—31no—
111C2×D3×Q8—31no—
112C2×D3×Q8—31no—
113C2×D3×Q8—61no—
114C2×D3×Q8C2×D312yes—
115C2×#237—31no—
116C3×C4⋊Q8—36no—
117C3×C4⋊Q8—33no—
118C3×C4⋊Q8—33no—
119C3×C22⋊Q8—36no—
120C3×C4×Q8—21no—
121C3×C4×Q8—32no—
122C3×C4×Q8—33no—
123C3×C4×Q8C2×D311yescharacteristic
124C3×C4×Q8C2×D311yescharacteristic
125C3×C4×D4—33no—
126#946—312no—
127#946—36no—
128#946—36no—
129Q8×Dic3—31no—
130Q8×Dic3—31no—
131Q8×Dic3—36no—
132Q8×Dic3—36no—
133Q8×Dic3—33no—
134Q8×Dic3—36no—
135Q8×Dic3—61no—
136Q8×Dic3C2×D312yes—
137#944—36no—
138D4×Dic3—33no—
139C4×#67—36no—
140C2×C4⋊Dic3—36no—
141#924—36no—
142C2×C4×Dic3—36no—
143#897—36no—
144#897—36no—
145#897—36no—
146#897C2×D316yes—
147#893—33no—
148#893—36no—
149#892—36no—
150#892—33no—
151#892—33no—
152D3×C4⋊C4—36no—
153D3×C4⋊C4—33no—
154D3×C4⋊C4—33no—
155D3×C4⋊C4—63no—
156#890—36no—
157#890—312no—
158#890—36no—
159#889—312no—
160#888C2×D316yes—
161#887—33no—
162#887—36no—
163#887—36no—
164#887—33no—
165#887—33no—
166#887—63no—
167#67⋊C4—36no—
168#878—312no—
169#877—36no—
170C4×D12—36no—
171C4×D12C2×D313yes—
172C42×D3—36no—
173C42×D3—33no—
174C42×D3—33no—
175C42×D3—63no—
176#869—36no—
177#869C2×D313yes—
178#869C2×D313yes—
179C4×Dic6—36no—
180C4×Dic6—33no—
181C4×Dic6—33no—
182C4×Dic6—63no—
183C4×Dic6C2×D313yes—
184C22×C3⋊D3C2411yescharacteristic
185C22×C3×D3C2412yes—
186C32×Q8#4114yescomplemented
187C2×C32×C4C2413yes—
188C2×C32×C4C2411yescharacteristic
189C2×C32×C4C2413yes—
190C2×C3⋊Dic3C2413yes—
191C2×C3⋊Dic3C2413yes—
192C2×C3⋊Dic3C2411yescharacteristic
193C4×C3⋊D3—26no—
194C4×C3⋊D3—43no—
195C4×C3⋊D3C2×Q814yescomplemented
196#634—26no—
197C2×C3×Dic3C2416yes—
198C2×C3×Dic3C2416yes—
199C2×C3×Dic3C2412yes—
200C3×D12C2×Q814yescomplemented
201C3×C4×D3—212no—
202C3×Dic6—212no—
203C3×Dic6—46no—
204C3×Dic6C2×Q814yescomplemented
205#626C2×Q818yescomplemented
206D3×Dic3—46no—
207D3×Dic3C2×Q818yescomplemented
208Q82—31no—
209#557—93no—
210#554—93no—
211#554—96no—
212D4×Q8—31no—
213D4×Q8—92no—
214#533—96no—
215#533—93no—
216C4×#41—93no—
217C2×C4×Q8—92no—
218C2×C4×Q8—91no—
219C2×C3×Q8—21no—
220C2×C3×Q8—31no—
221C2×C3×Q8—36no—
222C2×C3×Q8—36no—
223C2×C3×Q8—68no—
224C2×C3×Q8C22×D311yescharacteristic
225C2×C3×Q8C22×D311yescharacteristic
226C2×C3×D4—31no—
227C22×C3×C4—36no—
228C2×#67—32no—
229C22×Dic3—32no—
230C22×Dic3—36no—
231#239—38no—
232D3×Q8—28no—
233D3×Q8—68no—
234D3×Q8—64no—
235D3×Q8—64no—
236#237—38no—
237C2×D12—32no—
238C2×D12C22×D311yescharacteristic
239C2×C4×D3—32no—
240C2×C4×D3—31no—
241C2×C4×D3—31no—
242C2×C4×D3—33no—
243C2×C4×D3—33no—
244C2×C4×D3—36no—
245C2×C4×D3—33no—
246C2×C4×D3—33no—
247C2×C4×D3—61no—
248C2×C4×D3—63no—
249C2×C4×D3—63no—
250C2×C4×D3C22×D316yes—
251C2×Dic6—33no—
252C2×Dic6—33no—
253C2×Dic6—63no—
254C2×Dic6C22×D316yes—
255C2×Dic6C22×D316yes—
256C2×Dic6C22×D311yescharacteristic
257C3×C4⋊C4—23no—
258C3×C4⋊C4—36no—
259C3×C4⋊C4—33no—
260C3×C4⋊C4—36no—
261C3×C4⋊C4—36no—
262C3×C4⋊C4—33no—
263C3×C4⋊C4C22×D313yes—
264C3×C4⋊C4C22×D313yes—
265C3×C22⋊C4—36no—
266C3×C42—23no—
267C3×C42—36no—
268C3×C42C22×D313yes—
269C3×C42C22×D313yes—
270#217—36no—
271(C2×D3)⋊C4—36no—
272(C2×D3)⋊C4—36no—
273(C2×D3)⋊C4—36no—
274(C2×D3)⋊C4C22×D316yes—
275C4⋊Dic3—33no—
276C4⋊Dic3—33no—
277C4⋊Dic3—36no—
278C4⋊Dic3—312no—
279C4⋊Dic3—312no—
280C4⋊Dic3—63no—
281C4⋊Dic3C22×D313yes—
282C4⋊Dic3C22×D313yes—
283C4⋊Dic3C22×D316yes—
284C4⋊Dic3C22×D316yes—
285Dic3⋊C4—36no—
286Dic3⋊C4—36no—
287Dic3⋊C4—312no—
288Dic3⋊C4—36no—
289Dic3⋊C4—312no—
290Dic3⋊C4—66no—
291Dic3⋊C4C22×D316yes—
292C4×Dic3—33no—
293C4×Dic3—33no—
294C4×Dic3—33no—
295C4×Dic3—33no—
296C4×Dic3—33no—
297C4×Dic3—33no—
298C4×Dic3—36no—
299C4×Dic3—36no—
300C4×Dic3—36no—
301C4×Dic3—36no—
302C4×Dic3—63no—
303C4×Dic3—63no—
304C4×Dic3—63no—
305C4×Dic3C22×D316yes—
306C22×C32C2511yescharacteristic
307C2×C3⋊D3—21no—
308C2×C3⋊D3—21no—
309C2×C3⋊D3C22×Q812yes—
310C2×C3×D3—22no—
311C2×C3×D3—22no—
312C2×C3×D3C22×Q814yes—
313C32×C4—23no—
314C32×C4C22×Q812yes—
315C32×C4C2×#4116yes—
316C3⋊Dic3—23no—
317C3⋊Dic3—23no—
318C3⋊Dic3C22×Q812yes—
319C3×Dic3—26no—
320C3×Dic3—26no—
321C3×Dic3C22×Q814yes—
322C2×#41—91no—
323C22×Q8—32no—
324C22×Q8—91no—
325C4⋊Q8—36no—
326C4⋊Q8—33no—
327C4⋊Q8—33no—
328C4⋊Q8—96no—
329#125—912no—
330#125—96no—
331C22⋊Q8—36no—
332C22⋊Q8—912no—
333C22⋊Q8—96no—
334C22⋊Q8—96no—
335C22⋊Q8—96no—
336C4×Q8—32no—
337C4×Q8—33no—
338C4×Q8—96no—
339C4×Q8—96no—
340C4×Q8—93no—
341C4×Q8—96no—
342C4×Q8—91no—
343C4×Q8—93no—
344C4×Q8—93no—
345C4×Q8D3211yescharacteristic
346C4×D4—33no—
347C4×D4—96no—
348C4×D4—96no—
349C4×D4—93no—
350#117—96no—
351#117—93no—
352C2×C4⋊C4—96no—
353C2×C4⋊C4—93no—
354C2×C42—96no—
355C2×C42—93no—
356C23×C3—32no—
357C22×D3—31no—
358C22×D3—31no—
359C22×D3—61no—
360C22×D3C23×D312yes—
361C3×Q8—24no—
362C3×Q8—34no—
363C3×Q8—612no—
364C3×Q8—126no—
365C3×Q8#23714yescomplemented
366C3×Q8#23914yescomplemented
367C3×D4—34no—
368C2×C3×C4—21no—
369C2×C3×C4—23no—
370C2×C3×C4—23no—
371C2×C3×C4—32no—
372C2×C3×C4—36no—
373C2×C3×C4—36no—
374C2×C3×C4—612no—
375C2×C3×C4C23×D313yes—
376C2×C3×C4C23×D313yes—
377C2×C3×C4C23×D313yes—
378C2×C3×C4C23×D311yescharacteristic
379C2×C3×C4C23×D311yescharacteristic
380C2×C3×C4C23×D313yes—
381#67—38no—
382C2×Dic3—31no—
383C2×Dic3—31no—
384C2×Dic3—38no—
385C2×Dic3—33no—
386C2×Dic3—33no—
387C2×Dic3—33no—
388C2×Dic3—33no—
389C2×Dic3—61no—
390C2×Dic3—63no—
391C2×Dic3—63no—
392C2×Dic3—126no—
393C2×Dic3C23×D316yes—
394C2×Dic3C23×D316yes—
395C2×Dic3C23×D312yes—
396D12—38no—
397D12D3×Q814yescomplemented
398C4×D3—212no—
399C4×D3—38no—
400C4×D3—34no—
401C4×D3—34no—
402C4×D3—64no—
403C4×D3—612no—
404C4×D3—66no—
405C4×D3—66no—
406C4×D3—123no—
407C4×D3—123no—
408C4×D3—126no—
409C4×D3—126no—
410Dic6—212no—
411Dic6—46no—
412Dic6—612no—
413Dic6—66no—
414Dic6—66no—
415Dic6D3×Q814yescomplemented
416C2×C32C23×Q811yescharacteristic
417C2×C32C22×#4111yescharacteristic
418C2×C32#58511yescharacteristic
419C3⋊D3—22no—
420C3×D3—24no—
421#41—98no—
422C2×Q8—31no—
423C2×Q8—36no—
424C2×Q8—36no—
425C2×Q8—68no—
426C2×Q8—93no—
427C2×Q8—184no—
428C2×Q8C2×D3211yescharacteristic
429C2×D4—31no—
430C2×D4—92no—
431C22×C4—36no—
432C22×C4—92no—
433C22×C4—96no—
434C22×C4—91no—
435C22×C4—93no—
436C22×C4—93no—
437C4⋊C4—36no—
438C4⋊C4—33no—
439C4⋊C4—36no—
440C4⋊C4—33no—
441C4⋊C4—36no—
442C4⋊C4—912no—
443C4⋊C4—96no—
444C4⋊C4—912no—
445C4⋊C4—93no—
446C4⋊C4—96no—
447C4⋊C4C2×D3213yes—
448C22⋊C4—36no—
449C22⋊C4—96no—
450C22⋊C4—96no—
451C42—36no—
452C42—96no—
453C42—96no—
454C42—93no—
455C42—93no—
456C42—93no—
457C42C2×D3213yes—
458C22×C3—21no—
459C22×C3—34no—
460C22×C3—62no—
461C22×C3—62no—
462C22×C3C24×D311yescharacteristic
463C22×C3C24×D311yescharacteristic
464C2×D3—22no—
465C2×D3—22no—
466C2×D3—32no—
467C2×D3—32no—
468C2×D3—62no—
469C2×D3—62no—
470C2×D3—62no—
471C2×D3—61no—
472C2×D3—61no—
473C2×D3—61no—
474C2×D3—61no—
475C2×D3C2×D3×Q814yes—
476C3×C4—23no—
477C3×C4—23no—
478C3×C4—26no—
479C3×C4—22no—
480C3×C4—34no—
481C3×C4—43no—
482C3×C4—66no—
483C3×C4—66no—
484C3×C4C2×#23716yes—
485C3×C4C2×D3×Q812yes—
486C3×C4C2×D3×Q812yes—
487C3×C4C2×#23916yes—
488Dic3—26no—
489Dic3—26no—
490Dic3—32no—
491Dic3—32no—
492Dic3—62no—
493Dic3—66no—
494Dic3—63no—
495Dic3—63no—
496Dic3—63no—
497Dic3—63no—
498Dic3—123no—
499Dic3C2×D3×Q814yes—
500C32Q8×#4111yescharacteristic, complemented, Sylow
501C23—32no—
502C23—91no—
503Q8—34no—
504Q8—612no—
505Q8—126no—
506Q8—186no—
507Q8#379614yesdirect product factor
508D4—34no—
509D4—98no—
510C2×C4—32no—
511C2×C4—36no—
512C2×C4—36no—
513C2×C4—612no—
514C2×C4—98no—
515C2×C4—91no—
516C2×C4—94no—
517C2×C4—93no—
518C2×C4—93no—
519C2×C4—186no—
520C2×C4—363no—
521C2×C4—366no—
522C2×C4C22×D3213yes—
523C2×C4C22×D3213yes—
524C2×C4C22×D3211yescharacteristic
525C2×C3—21no—
526C2×C3—21no—
527C2×C3—21no—
528C2×C3—64no—
529C2×C3C22×#23711yescharacteristic
530C2×C3C22×D3×Q811yescharacteristic
531C2×C3C22×D3×Q811yescharacteristic
532C2×C3C22×#23911yescharacteristic
533C2×C3#596611yescharacteristic
534C2×C3#596811yescharacteristic
535D3—24no—
536D3—64no—
537D3—62no—
538D3—62no—
539C22—34no—
540C22—62no—
541C22—62no—
542C22—92no—
543C22—181no—
544C22—181no—
545C22C23×D3211yescharacteristic
546C4—23no—
547C4—34no—
548C4—66no—
549C4—66no—
550C4—92no—
551C4—183no—
552C4—183no—
553C4C2×#379616yes—
554C4D32×Q812yes—
555C3—21no—
556C3Q8×#23711yescharacteristic, complemented
557C3Q8×#23911yescharacteristic, complemented
558C2—64no—
559C2—182no—
560C2C22×#379611yescharacteristic
561C2C2×D32×Q811yescharacteristic
562C2#1059669811yescharacteristic
subgroupG/H
center, FrattiniC22C23×D32
commutator, socle, cosocleC22×C32C25
FittingC32×C4×Q8C22
prime radicalC2×#3796C22
hypercenterC4×Q8D32
hypocenterC32Q8×#41
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. C4×Q8×C3⋊D3C2semidirect
2. C3×Q8×D12C2semidirect
3. C3×Q8×Dic6C2semidirect
4. Q8×#626C2semidirect
5. D3×Q8×Dic3C2semidirect
6. #10594393C2semidirect
7. #10594387C2semidirect
8. #10593956C2semidirect
10. #10593866C2semidirect
11. #10593861C2semidirect
61. Q8×D12D3semidirect
68. Q8×Dic6D3semidirect
86. #3796Q8direct (N × H)
186. C32×Q8#41semidirect
195. C4×C3⋊D3C2×Q8semidirect
200. C3×D12C2×Q8semidirect
204. C3×Dic6C2×Q8semidirect
205. #626C2×Q8semidirect
207. D3×Dic3C2×Q8semidirect
365. C3×Q8#237semidirect
366. C3×Q8#239semidirect
397. D12D3×Q8semidirect
415. Dic6D3×Q8semidirect
500. C32Q8×#41semidirect
507. Q8#3796direct (N × H)
556. C3Q8×#237semidirect
557. C3Q8×#239semidirect

Maximal subgroups

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

subgroupindex
C4×Q8×C3⋊D32
C3×Q8×D122
C3×Q8×Dic62
Q8×#6262
D3×Q8×Dic32
#105943932
#105943872
#105939562
C4×#37962
#105938662
#105938612
Q8×#2393
Q8×#2373