Finite Group Databasebeta

← #11904254

Subgroups — #11904254

Order 1152.

2559 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#3832C211yesmaximal, characteristic, complemented
2C3×#5577C211yesmaximal, characteristic, complemented
3C3×Q8×Dic6C211yesmaximal, characteristic, complemented
4#10595410C212yesmaximal, complemented
5Dic3×#239C212yesmaximal, complemented
6#10594463C216yesmaximal, complemented
7#10594369C216yesmaximal, complemented
8#10593987C213yesmaximal, complemented
9#10593907C213yesmaximal, complemented
10#10593900C213yesmaximal, complemented
11C4×#3796C213yesmaximal, complemented
12#87040—31nomaximal
13#86768—31nomaximal
14C2×#3832C2211yescharacteristic
15C2×C3×#239C2212yes—
16C2×#3796C2213yes—
17C2×#3796C2216yes—
18C32×C4×Q8C2211yescharacteristic
19Q8×C3⋊Dic3C2211yescharacteristic
20#64102C2213yes—
21#64101C2213yes—
22C4×C4⋊(C3⋊D3)C2213yes—
23C42×C3⋊D3C2213yes—
24C3×Q8×Dic3C2212yes—
25C3×#895C2216yes—
26C3×#888C2216yes—
27C3×#874C2213yes—
28C3×C4×D12C2213yes—
29C3×#869C2213yes—
30C3×C4×Dic6C2213yes—
31#63916C2216yes—
32#63916C2216yes—
33#63911C2216yes—
34#63909C2216yes—
35#63907C22112yes—
36C4×#626C2216yes—
37#63898C2216yes—
38#63898C2213yes—
39Dic3×D12C2216yes—
40#63886C22112yes—
41#63881C2216yes—
42#63878C2216yes—
43C4×D3×Dic3C2216yes—
44#63863C2216yes—
45#63849C2213yes—
46#63843C2211yescharacteristic
47#63841C2216yes—
48C3×Q82—31no—
49C3×#550—31no—
50#5834—32no—
51Dic3×#41—32no—
52#5734—36no—
53#5734—33no—
54#5725—36no—
55#5725—33no—
56#5718—33no—
57#5716—33no—
58#5616—36no—
59#5596—36no—
60#5577—32no—
61#5577—33no—
62#5577D311yescharacteristic, complemented
63C4×#239—32no—
64C4×#239—31no—
65C4×#239—31no—
66C4×#239—33no—
67C4×#239—61no—
68Q8×Dic6D311yescharacteristic, complemented
69#5563—33no—
70C4×#237—33no—
71C2×C32×Q8C2311yescharacteristic
72#3832—24no—
73C2×C4⋊(C3⋊D3)C2313yes—
74C2×C4×C3⋊D3C2313yes—
75C2×C4×C3⋊D3C2313yes—
76C2×C4×C3⋊D3C2313yes—
77C3×#239—28no—
78C2×C3×D12C2316yes—
79C2×C3×D12C2313yes—
80C2×C3×D12C2316yes—
81C2×C3×C4×D3C2316yes—
82C2×C3×Dic6C2316yes—
83C2×C3×Dic6C2311yescharacteristic
84C2×#626C2316yes—
85C2×#626C23112yes—
86C2×D3×Dic3C2316yes—
87C2×D3×Dic3C2316yes—
88#3796—212no—
89#3796—412no—
90C32×C4⋊C4C2313yes—
91C32×C42C2313yes—
92#3752C2316yes—
93C4⋊(C3⋊Dic3)C2313yes—
94C4×C3⋊Dic3C2311yescharacteristic
95C4×C3⋊Dic3C2313yes—
96C4×C3⋊Dic3C2313yes—
97C3×(C2×D3)⋊C4C2316yes—
98C3×C4⋊Dic3C2316yes—
99C3×C4⋊Dic3C2311yescharacteristic
100C3×C4⋊Dic3C2313yes—
101C3×Dic3⋊C4C2316yes—
102C3×C4×Dic3C2316yes—
103Dic3⋊Dic3C2312yes—
104#3722C2316yes—
105(C2×D3)⋊Dic3C23112yes—
106(C2×D3)⋊Dic3C2316yes—
107Dic32C2316yes—
108Dic32C2312yes—
109#3483—91noSylow
110C2×C3×#41—32no—
111C2×#239—31no—
112C2×#239—31no—
113C2×#239—33no—
114C2×#239—36no—
115C2×#239—61no—
116C2×#239C2×D312yes—
117C2×#237—33no—
118C2×#237—36no—
119C3×C4⋊Q8—36no—
120C3×C4⋊Q8—33no—
121C3×#127—33no—
122C3×#121—36no—
123C3×C4×Q8—21no—
124C3×C4×Q8—32no—
125C3×C4×Q8—33no—
126C3×C4×Q8C2×D311yescharacteristic
127C3×C4×Q8C2×D311yescharacteristic
128C3×C4×D4—33no—
129#947—36no—
130#947—312no—
131Q8×Dic3—31no—
132Q8×Dic3—31no—
133Q8×Dic3—31no—
134Q8×Dic3—61no—
135Q8×Dic3C2×D312yes—
136#940—36no—
137#939—312no—
138#938—36no—
139#938—36no—
140#936—36no—
141D4×Dic3—33no—
142D4×Dic3—36no—
143D4×Dic3—36no—
144#930—36no—
145C4×#67—36no—
146#926—36no—
147C2×C4×Dic3—36no—
148#895—312no—
149#895—36no—
150#895—36no—
151#895—36no—
152#895C2×D316yes—
153#893—36no—
154#893—33no—
155#893—33no—
156#893—33no—
157#893—36no—
158#893—63no—
159#892—36no—
160#892—33no—
161#892—33no—
162#892—36no—
163#892—33no—
164#892—33no—
165#892—63no—
166#888C2×D316yes—
167#887—31no—
168#885—312no—
169#67⋊C4—36no—
170#877—36no—
171#874—36no—
172#874C2×D313yes—
173C4×D12—36no—
174C4×D12—36no—
175C4×D12—33no—
176C4×D12—33no—
177C4×D12—63no—
178C4×D12C2×D313yes—
179C42×D3—36no—
180C42×D3—33no—
181C42×D3—33no—
182C42×D3—63no—
183#869C2×D313yes—
184C4×Dic6C2×D313yes—
185C22×C3⋊D3C2413yes—
186C22×C3×D3C2416yes—
187C32×Q8#4114yescomplemented
188C2×C32×C4C2413yes—
189C2×C32×C4C2413yes—
190C2×C32×C4C2411yescharacteristic
191C2×C3⋊Dic3C2413yes—
192C2×C3⋊Dic3C2411yescharacteristic
193C2×C3⋊Dic3C2411yescharacteristic
194C4⋊(C3⋊D3)—26no—
195C4×C3⋊D3—26no—
196C4×C3⋊D3—26no—
197C4×C3⋊D3—26no—
198C2×C3×Dic3C2412yes—
199C2×C3×Dic3C2416yes—
200C2×C3×Dic3C2412yes—
201C3×D12—26no—
202C3×D12—212no—
203C3×D12—46no—
204C3×C4×D3—212no—
205C3×Dic6—46no—
206C3×Dic6#4114yescomplemented
207#626—212no—
208#626—412no—
209D3×Dic3—212no—
210D3×Dic3—212no—
211Q82—31no—
212#555—93no—
213#555—96no—
214#550—31no—
215#550—93no—
216#550—92no—
217#532—96no—
218#532—93no—
219C4×#41—93no—
220C4×#41—92no—
221C4×#41—91no—
222C3×#41—68no—
223C2×C3×Q8—21no—
224C2×C3×Q8—31no—
225C2×C3×Q8—36no—
226C2×C3×Q8C22×D311yescharacteristic
227C2×C3×Q8C22×D311yescharacteristic
228C2×C3×D4—36no—
229C2×C3×D4—33no—
230C2×C3×D4—36no—
231C22×C3×C4—36no—
232C2×#67—36no—
233C2×#67—312no—
234C22×Dic3—36no—
235C22×Dic3—36no—
236#239—28no—
237#239—612no—
238#239—68no—
239#239—64no—
240#239—64no—
241#239—1212no—
242#237—612no—
243#237—1212no—
244C2×D12—36no—
245C2×D12—33no—
246C2×D12—33no—
247C2×D12—312no—
248C2×D12—63no—
249C2×D12C22×D316yes—
250C2×D12C22×D313yes—
251C2×D12C22×D316yes—
252C2×C4×D3—36no—
253C2×C4×D3—36no—
254C2×C4×D3—33no—
255C2×C4×D3—33no—
256C2×C4×D3—33no—
257C2×C4×D3—33no—
258C2×C4×D3—33no—
259C2×C4×D3—33no—
260C2×C4×D3—63no—
261C2×C4×D3—63no—
262C2×C4×D3—63no—
263C2×C4×D3C22×D316yes—
264C2×Dic6C22×D316yes—
265C2×Dic6C22×D311yescharacteristic
266C3×C4⋊C4—23no—
267C3×C4⋊C4—31no—
268C3×C4⋊C4—36no—
269C3×C4⋊C4—36no—
270C3×C4⋊C4—33no—
271C3×C4⋊C4C22×D313yes—
272C3×C4⋊C4C22×D313yes—
273C3×C22⋊C4—36no—
274C3×C42—23no—
275C3×C42—36no—
276C3×C42C22×D313yes—
277C3×C42C22×D313yes—
278#217—36no—
279#217—312no—
280(C2×D3)⋊C4—312no—
281(C2×D3)⋊C4—36no—
282(C2×D3)⋊C4—36no—
283(C2×D3)⋊C4—36no—
284(C2×D3)⋊C4—36no—
285(C2×D3)⋊C4—36no—
286(C2×D3)⋊C4—66no—
287(C2×D3)⋊C4C22×D316yes—
288C4⋊Dic3—33no—
289C4⋊Dic3—33no—
290C4⋊Dic3—32no—
291C4⋊Dic3—63no—
292C4⋊Dic3C22×D313yes—
293C4⋊Dic3C22×D316yes—
294C4⋊Dic3C22×D311yescharacteristic
295Dic3⋊C4—32no—
296Dic3⋊C4C22×D316yes—
297C4×Dic3—31no—
298C4×Dic3—33no—
299C4×Dic3—31no—
300C4×Dic3—33no—
301C4×Dic3—33no—
302C4×Dic3—33no—
303C4×Dic3—32no—
304C4×Dic3—36no—
305C4×Dic3—36no—
306C4×Dic3—32no—
307C4×Dic3—61no—
308C4×Dic3—63no—
309C4×Dic3—63no—
310C4×Dic3C22×D316yes—
311C22×C32C2511yescharacteristic
312C2×C3⋊D3—23no—
313C2×C3⋊D3—23no—
314C2×C3⋊D3—23no—
315C2×C3×D3—26no—
316C2×C3×D3—26no—
317C2×C3×D3—26no—
318C32×C4—23no—
319C32×C4C2×#4116yes—
320C32×C4C2×#4112yes—
321C3⋊Dic3—21no—
322C3⋊Dic3—23no—
323C3⋊Dic3H(5,𝔽2)12yes—
324C3×Dic3—22no—
325C3×Dic3—26no—
326C3×Dic3C2×#4114yes—
327C2×#41—32no—
328C2×#41—93no—
329C2×#41—96no—
330C2×#41—91no—
331C4⋊Q8—36no—
332C4⋊Q8—33no—
333#127—33no—
334#127—96no—
335#124—912no—
336#124—96no—
337#121—36no—
338#121—96no—
339#121—96no—
340#121—96no—
341#121—912no—
342C4×Q8—32no—
343C4×Q8—33no—
344C4×Q8—91no—
345C4×Q8—91no—
346C4×Q8D3211yescharacteristic
347C4×D4—33no—
348C4×D4—96no—
349C4×D4—96no—
350C4×D4—96no—
351C4×D4—93no—
352C4×D4—96no—
353C4×D4—93no—
354C4×D4—93no—
355#117—96no—
356#117—93no—
357#117—96no—
358#117—93no—
359C2×C42—96no—
360C2×C42—93no—
361C23×C3—36no—
362C22×D3—33no—
363C22×D3—33no—
364C22×D3—63no—
365C22×D3C23×D316yes—
366C3×Q8—24no—
367C3×Q8—34no—
368C3×Q8—126no—
369C3×Q8#23714yescomplemented
370C3×Q8#23914yescomplemented
371C3×D4—66no—
372C3×D4—612no—
373C3×D4—126no—
374C2×C3×C4—23no—
375C2×C3×C4—21no—
376C2×C3×C4—23no—
377C2×C3×C4—32no—
378C2×C3×C4—32no—
379C2×C3×C4—36no—
380C2×C3×C4—612no—
381C2×C3×C4C23×D313yes—
382C2×C3×C4C23×D313yes—
383C2×C3×C4C23×D313yes—
384C2×C3×C4C23×D313yes—
385C2×C3×C4C23×D311yescharacteristic
386C2×C3×C4C23×D311yescharacteristic
387#67—612no—
388#67—1212no—
389C2×Dic3—31no—
390C2×Dic3—31no—
391C2×Dic3—31no—
392C2×Dic3—31no—
393C2×Dic3—33no—
394C2×Dic3—33no—
395C2×Dic3—61no—
396C2×Dic3—61no—
397C2×Dic3—63no—
398C2×Dic3—612no—
399C2×Dic3—612no—
400C2×Dic3C23×D312yes—
401C2×Dic3C23×D316yes—
402C2×Dic3C23×D312yes—
403D12—212no—
404D12—26no—
405D12—46no—
406D12—612no—
407D12—612no—
408D12—66no—
409D12—66no—
410D12—1212no—
411C4×D3—212no—
412C4×D3—66no—
413C4×D3—66no—
414C4×D3—612no—
415C4×D3—612no—
416C4×D3—612no—
417C4×D3—66no—
418C4×D3—66no—
419C4×D3—66no—
420C4×D3—66no—
421C4×D3—126no—
422C4×D3—126no—
423Dic6—46no—
424Dic6#23914yescomplemented
425C2×C32C22×#4111yescharacteristic
426C2×C32C22×#4111yescharacteristic
427C2×C32C2×H(5,𝔽2)11yescharacteristic
428C3⋊D3—43no—
429C3×D3—46no—
430#41—68no—
431#41—1812no—
432#41—184no—
433#41—3612no—
434C2×Q8—31no—
435C2×Q8—36no—
436C2×Q8C2×D3211yescharacteristic
437C2×D4—33no—
438C2×D4—36no—
439C2×D4—36no—
440C2×D4—96no—
441C2×D4—912no—
442C2×D4—93no—
443C22×C4—36no—
444C22×C4—96no—
445C22×C4—96no—
446C22×C4—93no—
447C22×C4—93no—
448C22×C4—93no—
449C4⋊C4—31no—
450C4⋊C4—36no—
451C4⋊C4—36no—
452C4⋊C4—33no—
453C4⋊C4—92no—
454C4⋊C4—93no—
455C4⋊C4C2×D3213yes—
456C22⋊C4—36no—
457C22⋊C4—96no—
458C22⋊C4—96no—
459C22⋊C4—912no—
460C22⋊C4—96no—
461C42—36no—
462C42—92no—
463C42—96no—
464C42—93no—
465C42—91no—
466C42—93no—
467C42C2×D3213yes—
468C22×C3—21no—
469C22×C3—66no—
470C22×C3—66no—
471C22×C3—66no—
472C22×C3C24×D311yescharacteristic
473C22×C3C24×D311yescharacteristic
474C2×D3—26no—
475C2×D3—26no—
476C2×D3—26no—
477C2×D3—66no—
478C2×D3—63no—
479C2×D3—63no—
480C2×D3—63no—
481C2×D3—63no—
482C2×D3—63no—
483C2×D3—63no—
484C2×D3—123no—
485C2×D3—123no—
486C3×C4—23no—
487C3×C4—23no—
488C3×C4—26no—
489C3×C4—22no—
490C3×C4—34no—
491C3×C4—43no—
492C3×C4—62no—
493C3×C4—66no—
494C3×C4C2×#23716yes—
495C3×C4C2×#23916yes—
496C3×C4C2×#23912yes—
497C3×C4D3×#4112yes—
498Dic3—22no—
499Dic3—26no—
500Dic3—32no—
501Dic3—32no—
502Dic3—62no—
503Dic3—62no—
504Dic3—61no—
505Dic3—61no—
506Dic3—63no—
507Dic3—63no—
508Dic3—123no—
509Dic3C2×#23914yes—
510C32#348311yescharacteristic, complemented, Sylow
511C23—36no—
512C23—93no—
513Q8—34no—
514Q8—126no—
515Q8#379614yescomplemented
516D4—612no—
517D4—66no—
518D4—126no—
519D4—1812no—
520D4—186no—
521D4—3612no—
522C2×C4—32no—
523C2×C4—36no—
524C2×C4—32no—
525C2×C4—612no—
526C2×C4—91no—
527C2×C4—91no—
528C2×C4—93no—
529C2×C4—186no—
530C2×C4—1812no—
531C2×C4—1812no—
532C2×C4—186no—
533C2×C4—186no—
534C2×C4C22×D3213yes—
535C2×C4C22×D3213yes—
536C2×C4C22×D3211yescharacteristic
537C2×C3—21no—
538C2×C3—21no—
539C2×C3—21no—
540C2×C3—126no—
541C2×C3C22×#23711yescharacteristic
542C2×C3C22×#23911yescharacteristic
543C2×C3C22×#23911yescharacteristic
544C2×C3C2×D3×#4111yescharacteristic
545C2×C3D3×H(5,𝔽2)11yescharacteristic
546C2×C3D3×H(5,𝔽2)11yescharacteristic
547D3—46no—
548D3—126no—
549D3—123no—
550D3—123no—
551C22—66no—
552C22—66no—
553C22—66no—
554C22—183no—
555C22—183no—
556C22—183no—
557C22C23×D3211yescharacteristic
558C4—23no—
559C4—34no—
560C4—62no—
561C4—66no—
562C4—92no—
563C4—181no—
564C4—183no—
565C4C2×#379616yes—
566C4D3×#23912yes—
567C3—21no—
568C3#8676811yescharacteristic, complemented
569C3#8704011yescharacteristic, complemented
570C2—126no—
571C2—363no—
572C2C22×#379611yescharacteristic
573C2C2×D3×#23911yescharacteristic
574C2#1059669211yescharacteristic
subgroupG/H
center, FrattiniC22C23×D32
commutator, socle, cosocleC22×C32C25
FittingC32×C4×Q8C22
hypercenterC4×Q8D32
hypocenterC32#3483
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. C4×#3832C2semidirect
2. C3×#5577C2semidirect
3. C3×Q8×Dic6C2semidirect
4. #10595410C2semidirect
5. Dic3×#239C2semidirect
6. #10594463C2semidirect
7. #10594369C2semidirect
8. #10593987C2semidirect
9. #10593907C2semidirect
10. #10593900C2semidirect
11. C4×#3796C2semidirect
62. #5577D3semidirect
68. Q8×Dic6D3semidirect
187. C32×Q8#41semidirect
206. C3×Dic6#41semidirect
369. C3×Q8#237semidirect
370. C3×Q8#239semidirect
424. Dic6#239semidirect
510. C32#3483semidirect
515. Q8#3796semidirect
568. C3#86768semidirect
569. C3#87040semidirect

Maximal subgroups

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

subgroupindex
C4×#38322
C3×#55772
C3×Q8×Dic62
#105954102
Dic3×#2392
#105944632
#105943692
#105939872
#105939072
#105939002
C4×#37962
#870403
#867683