Finite Group Databasebeta

← Q8×#3799 #11903763

Subgroups — Q8×#3799 #11903763

Order 1152.

2276 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C3×C4×D3×Q8C211yesmaximal, characteristic, complemented
3C3×Q8×Dic6C211yesmaximal, characteristic, complemented
4Q8×#626C212yesmaximal, complemented
5Q8×#624C212yesmaximal, complemented
6#10594537C216yesmaximal, complemented
7#10594397C216yesmaximal, complemented
8C4×#3799C213yesmaximal
9#10593958C213yesmaximal, complemented
10#10593860C213yesmaximal, complemented
11#10593859C213yesmaximal, complemented
12Q8×#239—31nomaximal
13Q8×#235—31nomaximal
14C2×Q8×C3⋊D3C2212yes—
15C2×C3×D3×Q8C2211yescharacteristic
16C2×#3799C2211yescharacteristic
17C32×C4×Q8C2211yescharacteristic
18#64106C2216yes—
19C4×C4⋊(C3⋊D3)C2213yes—
20#64078C2213yes—
21C3×Q8×Dic3C2211yescharacteristic
22C3×Q8×Dic3C2212yes—
23C3×D3×C4⋊C4C2213yes—
24C3×#888C2216yes—
25C3×#887C2213yes—
26C3×C42×D3C2213yes—
27C3×#869C2213yes—
28C3×C4×Dic6C2213yes—
29C3×C4×Dic6C2213yes—
30#63904C2216yes—
31C4×#626C2216yes—
32#63900C2216yes—
33#63894C2213yes—
34#63888C22112yes—
35C4×#624C2216yes—
36#63871C2216yes—
37#63869C2216yes—
38#63866C2216yes—
39#63857C2216yes—
40#63856C2216yes—
41#63845C22112yes—
42#63844C2216yes—
43#63844C2216yes—
44Dic3×Dic6C2216yes—
45Dic3×Dic6C2213yes—
46#63841C2213yes—
47#63836C2216yes—
48C3×Q82—31no—
49C2×C3×C4×Q8—31no—
50Q8×#67—32no—
51#5730—36no—
52#5730—33no—
53#5728—36no—
54#5728—33no—
55#5579—33no—
56#5578—36no—
57Q8×D12—32no—
58Q8×D12—31no—
59Q8×D12—31no—
60Q8×D12—61no—
61C4×#239—33no—
62C4×D3×Q8—32no—
63C4×D3×Q8—32no—
64C4×D3×Q8D311yescharacteristic, complemented
65#5570—33no—
66#5567—33no—
67Q8×Dic6D311yescharacteristic, complemented
68#5511—36no—
69C4×#235—33no—
70#5470—33no—
71C2×C32×Q8C2311yescharacteristic
72Q8×C3⋊D3—28no—
73C2×C4⋊(C3⋊D3)C2311yescharacteristic
74C2×C4×C3⋊D3C2316yes—
75C2×#634C2316yes—
76C3×D3×Q8—24no—
77C2×C3×C4×D3C2313yes—
78C2×C3×C4×D3C2313yes—
79C2×C3×C4×D3C2311yescharacteristic
80C2×C3×Dic6C2313yes—
81C2×C3×Dic6C2316yes—
82C2×C3×Dic6C2311yescharacteristic
83C2×#626C2312yes—
84C2×#624C2316yes—
85C2×#624C2312yes—
86#3799Q818yesdirect product factor
87C32×C4⋊C4C2313yes—
88C32×C42C2313yes—
89#3752C2316yes—
90C4⋊(C3⋊Dic3)C2313yes—
91C4⋊(C3⋊Dic3)C2316yes—
92C3×C4⋊Dic3C2316yes—
93C3×C4⋊Dic3C2313yes—
94C3×C4⋊Dic3C2313yes—
95C3×Dic3⋊C4C2316yes—
96C3×Dic3⋊C4C2316yes—
97C3×C4×Dic3C2313yes—
98C3×C4×Dic3C2313yes—
99C3×C4×Dic3C2316yes—
100C3×C4×Dic3C2313yes—
101#3724C23112yes—
102Dic3⋊Dic3C2316yes—
103Dic3⋊Dic3C23112yes—
104#3722C2316yes—
105(C2×D3)⋊Dic3C2316yes—
106Dic32C2316yes—
107Dic32C2316yes—
108Q8×#41—91noSylow
109C22×C3×Q8—31no—
110C2×#239—31no—
111C2×D3×Q8—32no—
112C2×D3×Q8—32no—
113C2×D3×Q8—62no—
114C2×D3×Q8C2×D311yescharacteristic
115C2×#235—31no—
116C3×C4⋊Q8—36no—
117C3×C4⋊Q8—33no—
118C3×C4×Q8—21no—
119C3×C4×Q8—32no—
120C3×C4×Q8—31no—
121C3×C4×Q8—33no—
122C3×C4×Q8—33no—
123C3×C4×Q8—33no—
124C3×C4×Q8C2×D311yescharacteristic
125C3×C4×Q8C2×D311yescharacteristic
126C2×C3×C4⋊C4—33no—
127C2×C3×C42—33no—
128#946—36no—
129#946—36no—
130#946—312no—
131Q8×Dic3—36no—
132Q8×Dic3—33no—
133Q8×Dic3C2×D312yes—
134Q8×Dic3C2×D311yescharacteristic
135#944—36no—
136C4×#67—36no—
137C4×#67—36no—
138#926—33no—
139#924—36no—
140#924—36no—
141#897—36no—
142#897—36no—
143#897—36no—
144#897—36no—
145#897—66no—
146#896—312no—
147#893—36no—
148#893—33no—
149#892—33no—
150#892—36no—
151D3×C4⋊C4—36no—
152D3×C4⋊C4—36no—
153D3×C4⋊C4C2×D313yes—
154#890—312no—
155#890—36no—
156#889—312no—
157#888C2×D316yes—
158#887—36no—
159#887—36no—
160#887C2×D313yes—
161C4×D12—36no—
162C4×D12—33no—
163C4×D12—33no—
164C4×D12—33no—
165C4×D12—63no—
166#872—36no—
167C42×D3—36no—
168C42×D3—36no—
169C42×D3C2×D313yes—
170#870—36no—
171#869—36no—
172#869—33no—
173#869—33no—
174#869—63no—
175#869C2×D313yes—
176C4×Dic6—36no—
177C4×Dic6—33no—
178C4×Dic6—36no—
179C4×Dic6—36no—
180C4×Dic6C2×D313yes—
181C4×Dic6C2×D313yes—
182C22×C3⋊D3C2412yes—
183C22×C3×D3C2411yescharacteristic
184C32×Q8#4114yescomplemented
185C2×C32×C4C2413yes—
186C2×C32×C4C2411yescharacteristic
187C2×C32×C4C2413yes—
188C2×C3⋊Dic3C2416yes—
189C4⋊(C3⋊D3)C2×Q814yescomplemented
190C4×C3⋊D3—212no—
191#634—212no—
192C2×C3×Dic3C2413yes—
193C2×C3×Dic3C2416yes—
194C2×C3×Dic3C2413yes—
195C2×C3×Dic3C2411yescharacteristic
196C2×C3×Dic3C2412yes—
197C3×C4×D3—26no—
198C3×C4×D3—43no—
199C3×C4×D3C2×Q814yescomplemented
200C3×Dic6—26no—
201C3×Dic6—46no—
202C3×Dic6C2×Q814yescomplemented
203#626C2×Q818yescomplemented
204#624—46no—
205#624C2×Q818yescomplemented
206Q82—31no—
207#557—93no—
208#554—96no—
209#554—93no—
210D4×Q8—92no—
211D4×Q8—91no—
212#533—93no—
213#533—96no—
214C4×#41—93no—
215C2×C4×Q8—31no—
216C2×C4×Q8—92no—
217C2×C3×Q8—21no—
218C2×C3×Q8—31no—
219C2×C3×Q8—33no—
220C2×C3×Q8—36no—
221C2×C3×Q8—64no—
222C2×C3×Q8C22×D311yescharacteristic
223C2×C3×Q8C22×D311yescharacteristic
224C22×C3×C4—31no—
225C22×C3×C4—33no—
226C22×C3×C4—33no—
227C2×#67—32no—
228#239—38no—
229D3×Q8—24no—
230D3×Q8—68no—
231D3×Q8—68no—
232D3×Q8—128no—
233#235—38no—
234C2×D12—32no—
235C2×D12—31no—
236C2×D12—31no—
237C2×D12—61no—
238C2×C4×D3—32no—
239C2×C4×D3—32no—
240C2×C4×D3—36no—
241C2×C4×D3—36no—
242C2×C4×D3—36no—
243C2×C4×D3—36no—
244C2×C4×D3—66no—
245C2×C4×D3C22×D313yes—
246C2×C4×D3C22×D313yes—
247C2×C4×D3C22×D311yescharacteristic
248C2×Dic6—36no—
249C2×Dic6—36no—
250C2×Dic6—66no—
251C2×Dic6C22×D313yes—
252C2×Dic6C22×D316yes—
253C2×Dic6C22×D311yescharacteristic
254C3×C4⋊C4—23no—
255C3×C4⋊C4—33no—
256C3×C4⋊C4—36no—
257C3×C4⋊C4—36no—
258C3×C4⋊C4—36no—
259C3×C4⋊C4—33no—
260C3×C4⋊C4C22×D313yes—
261C3×C4⋊C4C22×D313yes—
262C3×C42—23no—
263C3×C42—33no—
264C3×C42—33no—
265C3×C42—36no—
266C3×C42—33no—
267C3×C42C22×D313yes—
268C3×C42C22×D313yes—
269#217—36no—
270(C2×D3)⋊C4—36no—
271(C2×D3)⋊C4—36no—
272(C2×D3)⋊C4—36no—
273(C2×D3)⋊C4—36no—
274(C2×D3)⋊C4—36no—
275(C2×D3)⋊C4—66no—
276C4⋊Dic3—33no—
277C4⋊Dic3—36no—
278C4⋊Dic3—33no—
279C4⋊Dic3—36no—
280C4⋊Dic3—36no—
281C4⋊Dic3—312no—
282C4⋊Dic3—63no—
283C4⋊Dic3—66no—
284C4⋊Dic3C22×D313yes—
285C4⋊Dic3C22×D316yes—
286C4⋊Dic3C22×D313yes—
287Dic3⋊C4—312no—
288Dic3⋊C4—312no—
289Dic3⋊C4—36no—
290Dic3⋊C4—312no—
291Dic3⋊C4C22×D316yes—
292Dic3⋊C4C22×D316yes—
293C4×Dic3—36no—
294C4×Dic3—36no—
295C4×Dic3—36no—
296C4×Dic3—36no—
297C4×Dic3C22×D313yes—
298C4×Dic3C22×D316yes—
299C4×Dic3C22×D313yes—
300C4×Dic3C22×D313yes—
301C22×C32C2511yescharacteristic
302C2×C3⋊D3—22no—
303C2×C3⋊D3—22no—
304C2×C3⋊D3C22×Q814yes—
305C2×C3×D3—21no—
306C2×C3×D3—21no—
307C2×C3×D3C22×Q812yes—
308C32×C4—23no—
309C32×C4C22×Q812yes—
310C32×C4C2×#4116yes—
311C3⋊Dic3—26no—
312C3×Dic3—23no—
313C3×Dic3—23no—
314C3×Dic3—26no—
315C3×Dic3C22×Q812yes—
316C3×Dic3C22×Q814yes—
317C2×#41—91no—
318C22×Q8—31no—
319C22×Q8—92no—
320C4⋊Q8—36no—
321C4⋊Q8—33no—
322C4⋊Q8—96no—
323C4⋊Q8—93no—
324#125—96no—
325#125—912no—
326C22⋊Q8—96no—
327C22⋊Q8—96no—
328C22⋊Q8—912no—
329C22⋊Q8—96no—
330C22⋊Q8—96no—
331C4×Q8—31no—
332C4×Q8—32no—
333C4×Q8—33no—
334C4×Q8—33no—
335C4×Q8—33no—
336C4×Q8—96no—
337C4×Q8—96no—
338C4×Q8—93no—
339C4×Q8—96no—
340C4×Q8D3211yescharacteristic
341C4×D4—93no—
342C4×D4—96no—
343C4×D4—96no—
344C4×D4—93no—
345#117—96no—
346#117—93no—
347C2×C4⋊C4—33no—
348C2×C4⋊C4—96no—
349C2×C42—33no—
350C2×C42—96no—
351C23×C3—31no—
352C22×D3—32no—
353C22×D3—32no—
354C22×D3—62no—
355C22×D3C23×D311yescharacteristic
356C3×Q8—24no—
357C3×Q8—34no—
358C3×Q8—66no—
359C3×Q8—126no—
360C3×Q8#23514yescomplemented
361C3×Q8#23914yescomplemented
362C2×C3×C4—21no—
363C2×C3×C4—23no—
364C2×C3×C4—23no—
365C2×C3×C4—31no—
366C2×C3×C4—32no—
367C2×C3×C4—34no—
368C2×C3×C4—33no—
369C2×C3×C4—36no—
370C2×C3×C4—33no—
371C2×C3×C4—66no—
372C2×C3×C4—123no—
373C2×C3×C4C23×D313yes—
374C2×C3×C4C23×D313yes—
375C2×C3×C4C23×D313yes—
376C2×C3×C4C23×D311yescharacteristic
377C2×C3×C4C23×D311yescharacteristic
378C2×C3×C4C23×D313yes—
379#67—38no—
380C2×Dic3—36no—
381C2×Dic3—36no—
382C2×Dic3—66no—
383C2×Dic3C23×D313yes—
384C2×Dic3C23×D316yes—
385C2×Dic3C23×D313yes—
386C2×Dic3C23×D311yescharacteristic
387C2×Dic3C23×D312yes—
388D12—38no—
389D12—34no—
390D12—34no—
391D12—64no—
392C4×D3—26no—
393C4×D3—38no—
394C4×D3—38no—
395C4×D3—43no—
396C4×D3—612no—
397C4×D3—612no—
398C4×D3—1212no—
399C4×D3—126no—
400C4×D3—126no—
401C4×D3D3×Q814yescomplemented
402Dic6—26no—
403Dic6—46no—
404Dic6—612no—
405Dic6—612no—
406Dic6—1212no—
407Dic6D3×Q814yescomplemented
408C2×C32C23×Q811yescharacteristic
409C2×C32C22×#4111yescharacteristic
410C2×C32#58511yescharacteristic
411C3⋊D3—24no—
412C3×D3—22no—
413#41—98no—
414C2×Q8—31no—
415C2×Q8—33no—
416C2×Q8—36no—
417C2×Q8—64no—
418C2×Q8—96no—
419C2×Q8—188no—
420C2×Q8C2×D3211yescharacteristic
421C2×D4—92no—
422C2×D4—91no—
423C22×C4—31no—
424C22×C4—33no—
425C22×C4—33no—
426C22×C4—92no—
427C22×C4—96no—
428C22×C4—96no—
429C4⋊C4—33no—
430C4⋊C4—36no—
431C4⋊C4—36no—
432C4⋊C4—36no—
433C4⋊C4—33no—
434C4⋊C4—912no—
435C4⋊C4—96no—
436C4⋊C4—912no—
437C4⋊C4—93no—
438C4⋊C4—96no—
439C4⋊C4C2×D3213yes—
440C22⋊C4—96no—
441C22⋊C4—96no—
442C22⋊C4—96no—
443C42—33no—
444C42—33no—
445C42—33no—
446C42—36no—
447C42—96no—
448C42—96no—
449C42C2×D3213yes—
450C22×C3—21no—
451C22×C3—32no—
452C22×C3—61no—
453C22×C3—61no—
454C22×C3C24×D311yescharacteristic
455C22×C3C24×D311yescharacteristic
456C2×D3—21no—
457C2×D3—21no—
458C2×D3—34no—
459C2×D3—34no—
460C2×D3—64no—
461C2×D3—62no—
462C2×D3—62no—
463C2×D3—62no—
464C2×D3—62no—
465C2×D3—122no—
466C2×D3—122no—
467C2×D3C2×D3×Q812yes—
468C3×C4—23no—
469C3×C4—23no—
470C3×C4—26no—
471C3×C4—22no—
472C3×C4—32no—
473C3×C4—34no—
474C3×C4—43no—
475C3×C4—63no—
476C3×C4—66no—
477C3×C4—63no—
478C3×C4C2×#23516yes—
479C3×C4C2×D3×Q812yes—
480C3×C4C2×D3×Q812yes—
481C3×C4C2×#23916yes—
482Dic3—23no—
483Dic3—23no—
484Dic3—26no—
485Dic3—66no—
486Dic3—66no—
487Dic3—126no—
488Dic3C2×D3×Q814yes—
489Dic3C2×D3×Q812yes—
490C32Q8×#4111yescharacteristic, complemented, Sylow
491C23—31no—
492C23—92no—
493Q8—34no—
494Q8—66no—
495Q8—126no—
496Q8—1812no—
497Q8#379914yesdirect product factor
498D4—98no—
499D4—94no—
500C2×C4—31no—
501C2×C4—32no—
502C2×C4—34no—
503C2×C4—33no—
504C2×C4—33no—
505C2×C4—36no—
506C2×C4—66no—
507C2×C4—98no—
508C2×C4—96no—
509C2×C4—123no—
510C2×C4—1812no—
511C2×C4—366no—
512C2×C4C22×D3213yes—
513C2×C4C22×D3213yes—
514C2×C4C22×D3211yescharacteristic
515C2×C3—21no—
516C2×C3—21no—
517C2×C3—21no—
518C2×C3—62no—
519C2×C3C22×#23511yescharacteristic
520C2×C3C22×D3×Q811yescharacteristic
521C2×C3C22×D3×Q811yescharacteristic
522C2×C3C22×#23911yescharacteristic
523C2×C3#596411yescharacteristic
524C2×C3#596611yescharacteristic
525D3—22no—
526D3—64no—
527D3—64no—
528D3—124no—
529C22—32no—
530C22—61no—
531C22—61no—
532C22—94no—
533C22—182no—
534C22—182no—
535C22C23×D3211yescharacteristic
536C4—23no—
537C4—32no—
538C4—34no—
539C4—63no—
540C4—63no—
541C4—66no—
542C4—186no—
543C4C2×#379916yes—
544C4D32×Q812yes—
545C3—21no—
546C3Q8×#23511yescharacteristic, complemented
547C3Q8×#23911yescharacteristic, complemented
548C2—62no—
549C2—184no—
550C2C22×#379911yescharacteristic
551C2C2×D32×Q811yescharacteristic
552C2#1059669611yescharacteristic
subgroupG/H
center, FrattiniC22C23×D32
commutator, socle, cosocleC22×C32C25
FittingC32×C4×Q8C22
prime radicalC2×#3799C22
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. Q8×C4⋊(C3⋊D3)C2semidirect
2. C3×C4×D3×Q8C2semidirect
3. C3×Q8×Dic6C2semidirect
4. Q8×#626C2semidirect
5. Q8×#624C2semidirect
6. #10594537C2semidirect
7. #10594397C2semidirect
9. #10593958C2semidirect
10. #10593860C2semidirect
11. #10593859C2semidirect
64. C4×D3×Q8D3semidirect
67. Q8×Dic6D3semidirect
86. #3799Q8direct (N × H)
184. C32×Q8#41semidirect
189. C4⋊(C3⋊D3)C2×Q8semidirect
199. C3×C4×D3C2×Q8semidirect
202. C3×Dic6C2×Q8semidirect
203. #626C2×Q8semidirect
205. #624C2×Q8semidirect
360. C3×Q8#235semidirect
361. C3×Q8#239semidirect
401. C4×D3D3×Q8semidirect
407. Dic6D3×Q8semidirect
490. C32Q8×#41semidirect
497. Q8#3799direct (N × H)
546. C3Q8×#235semidirect
547. C3Q8×#239semidirect

Maximal subgroups

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

subgroupindex
Q8×C4⋊(C3⋊D3)2
C3×C4×D3×Q82
C3×Q8×Dic62
Q8×#6262
Q8×#6242
#105945372
#105943972
C4×#37992
#105939582
#105938602
#105938592
Q8×#2393
Q8×#2353