Finite Group Databasebeta

← #11904255

Subgroups — #11904255

Order 1152.

2510 conjugacy classes of subgroups.

The subgroup lattice has 561 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#634C211yesmaximal, characteristic, complemented
2C3×#5577C211yesmaximal, characteristic, complemented
3C3×C4×#239C211yesmaximal, characteristic, complemented
4#10595412C212yesmaximal, complemented
5Dic3×#239C212yesmaximal, complemented
6#10594486C216yesmaximal, complemented
7#10594367C216yesmaximal, complemented
8#10593988C213yesmaximal, complemented
9#10593906C213yesmaximal, complemented
10#10593897C213yesmaximal, complemented
11C4×#3795C213yesmaximal, complemented
12#86768—31nomaximal
13#85845—31nomaximal
14C2×C3×#239C2212yes—
15C2×C3×#239C2211yescharacteristic
16C2×#3795C2213yes—
17C2×#3795C2216yes—
18C32×C4×Q8C2211yescharacteristic
19Q8×C3⋊Dic3C2212yes—
20#64097C2216yes—
21#64078C2213yes—
22C4×#634C2213yes—
23C3×Q8×Dic3C2211yescharacteristic
24C3×#895C2216yes—
25C3×#893C2213yes—
26C3×#892C2213yes—
27C3×#874C2213yes—
28C3×C4×D12C2213yes—
29C3×C4×D12C2213yes—
30C3×C42×D3C2213yes—
31#63914C2216yes—
32#63913C22112yes—
33#63908C2216yes—
34#63908C2216yes—
35#63908C2216yes—
36C4×#625C2216yes—
37#63898C2216yes—
38Dic3×D12C2216yes—
39Dic3×D12C2213yes—
40#63885C22112yes—
41#63879C2216yes—
42#63877C2216yes—
43C4×D3×Dic3C2216yes—
44#63860C2216yes—
45#63844C2211yescharacteristic
46#63841C2216yes—
47#63841C2213yes—
48C3×#550—31no—
49C3×C4×#41—31no—
50#5834—32no—
51#5834—32no—
52Dic3×#41—32no—
53#5718—33no—
54#5716—33no—
55#5616—36no—
56#5596—36no—
57#5577D311yescharacteristic, complemented
58C4×#239—32no—
59C4×#239D311yescharacteristic, complemented
60Q8×Dic6—31no—
61Q8×Dic6—31no—
62Q8×Dic6—61no—
63#5563—33no—
64#5563—33no—
65#5553—33no—
66#5548—36no—
67C4×#237—33no—
68#5532—36no—
69#5477—33no—
70C4×#235—33no—
71C2×C32×Q8C2311yescharacteristic
72C2×#634C2316yes—
73C2×#634C2311yescharacteristic
74C3×#239—24no—
75C3×#239—28no—
76C2×C3×D12C2313yes—
77C2×C3×D12C2316yes—
78C2×C3×D12C2313yes—
79C2×C3×D12C2316yes—
80C2×C3×C4×D3C2313yes—
81C2×C3×C4×D3C2313yes—
82C2×C3×C4×D3C2316yes—
83C2×C3×C4×D3C2313yes—
84C2×#625C2316yes—
85C2×#625C23112yes—
86C2×D3×Dic3C2316yes—
87C2×D3×Dic3C2316yes—
88#3795—212no—
89#3795—412no—
90C32×C4⋊C4C2313yes—
91C32×C42C2313yes—
92C4⋊(C3⋊Dic3)C2313yes—
93C4⋊(C3⋊Dic3)C2316yes—
94#3750C2316yes—
95C4×C3⋊Dic3C2316yes—
96C3×(C2×D3)⋊C4C2316yes—
97C3×(C2×D3)⋊C4C2316yes—
98C3×C4⋊Dic3C2313yes—
99C3×C4⋊Dic3C2311yescharacteristic
100C3×C4×Dic3C2313yes—
101C3×C4×Dic3C2313yes—
102C3×C4×Dic3C2311yescharacteristic
103#3724C2312yes—
104(C2×D3)⋊Dic3C23112yes—
105(C2×D3)⋊Dic3C2316yes—
106(C2×D3)⋊Dic3C2316yes—
107Dic32C2316yes—
108Dic32C2312yes—
109#3483—91noSylow
110C2×C3×#41—32no—
111C2×C3×#41—31no—
112C2×#239C2×D312yes—
113C2×#239C2×D311yescharacteristic
114C2×#237—33no—
115C2×#237—36no—
116C2×#235—33no—
117C2×#235—36no—
118C3×#127—33no—
119C3×#121—36no—
120C3×C4×Q8—21no—
121C3×C4×Q8—31no—
122C3×C4×Q8C2×D311yescharacteristic
123C3×C4×Q8C2×D311yescharacteristic
124C3×C4×D4—33no—
125C3×C4×D4—33no—
126C3×C4×D4—33no—
127C3×#117—33no—
128C2×C3×C42—33no—
129Q8×Dic3—32no—
130Q8×Dic3—32no—
131Q8×Dic3—62no—
132Q8×Dic3C2×D311yescharacteristic
133#940—36no—
134#940—36no—
135#939—312no—
136#938—36no—
137#938—36no—
138#938—36no—
139#936—36no—
140D4×Dic3—36no—
141D4×Dic3—33no—
142D4×Dic3—36no—
143#930—36no—
144#930—36no—
145#930—36no—
146C4×#67—36no—
147C4×#67—36no—
148C4×#67—36no—
149#926—33no—
150#926—36no—
151C2×C4×Dic3—36no—
152#895C2×D316yes—
153#893—36no—
154#893C2×D313yes—
155#892—36no—
156#892—33no—
157#892C2×D313yes—
158#888—36no—
159#888—36no—
160#888—66no—
161#887—31no—
162#885—312no—
163#885—312no—
164#884—312no—
165#67⋊C4—36no—
166#877—36no—
167#875—36no—
168#874C2×D313yes—
169C4×D12—33no—
170C4×D12—36no—
171C4×D12C2×D313yes—
172C4×D12C2×D313yes—
173#872—36no—
174C42×D3—36no—
175C42×D3C2×D313yes—
176#869—33no—
177#869—33no—
178#869—63no—
179C4×Dic6—33no—
180C4×Dic6—33no—
181C4×Dic6—31no—
182C4×Dic6—63no—
183C22×C3×D3C2416yes—
184C22×C3×D3C2413yes—
185C32×Q8#4114yescomplemented
186C2×C32×C4C2413yes—
187C2×C32×C4C2413yes—
188C2×C32×C4C2411yescharacteristic
189C2×C3⋊Dic3C2416yes—
190C2×C3⋊Dic3C2412yes—
191#634—46no—
192#634#4114yescomplemented
193C2×C3×Dic3C2412yes—
194C2×C3×Dic3C2413yes—
195C2×C3×Dic3C2411yescharacteristic
196C2×C3×Dic3C2411yescharacteristic
197C3×D12—26no—
198C3×D12—26no—
199C3×D12—212no—
200C3×D12—46no—
201C3×C4×D3—26no—
202C3×C4×D3—26no—
203C3×C4×D3—212no—
204C3×C4×D3—26no—
205#625—212no—
206#625—412no—
207D3×Dic3—212no—
208D3×Dic3—212no—
209Q82—91no—
210#555—96no—
211#555—93no—
212#550—31no—
213#550—93no—
214#550—92no—
215#532—93no—
216#532—96no—
217C4×#41—31no—
218C4×#41—93no—
219C4×#41—92no—
220C3×#41—68no—
221C3×#41—64no—
222C2×C3×Q8—21no—
223C2×C3×Q8C22×D311yescharacteristic
224C2×C3×Q8C22×D311yescharacteristic
225C2×C3×D4—36no—
226C2×C3×D4—33no—
227C2×C3×D4—33no—
228C2×C3×D4—36no—
229C22×C3×C4—36no—
230C22×C3×C4—33no—
231C22×C3×C4—33no—
232C22×C3×C4—33no—
233C2×#67—36no—
234C2×#67—36no—
235C2×#67—312no—
236C2×#67—312no—
237C22×Dic3—36no—
238C22×Dic3—36no—
239#239—24no—
240#239—28no—
241#237—612no—
242#237—1212no—
243#235—612no—
244#235—1212no—
245C2×D12C22×D313yes—
246C2×D12C22×D316yes—
247C2×D12C22×D313yes—
248C2×D12C22×D316yes—
249C2×C4×D3—36no—
250C2×C4×D3—36no—
251C2×C4×D3C22×D316yes—
252C2×C4×D3C22×D313yes—
253C2×C4×D3C22×D313yes—
254C2×C4×D3C22×D313yes—
255C2×Dic6—31no—
256C2×Dic6—31no—
257C2×Dic6—36no—
258C2×Dic6—36no—
259C2×Dic6—61no—
260C2×Dic6—66no—
261C3×C4⋊C4—23no—
262C3×C4⋊C4—31no—
263C3×C4⋊C4—33no—
264C3×C4⋊C4C22×D313yes—
265C3×C4⋊C4C22×D313yes—
266C3×C22⋊C4—36no—
267C3×C22⋊C4—36no—
268C3×C42—23no—
269C3×C42—31no—
270C3×C42—33no—
271C3×C42—33no—
272C3×C42C22×D313yes—
273C3×C42C22×D313yes—
274#217—36no—
275#217—36no—
276#217—312no—
277(C2×D3)⋊C4—36no—
278(C2×D3)⋊C4—312no—
279(C2×D3)⋊C4—36no—
280(C2×D3)⋊C4C22×D316yes—
281(C2×D3)⋊C4C22×D316yes—
282C4⋊Dic3—33no—
283C4⋊Dic3—33no—
284C4⋊Dic3—36no—
285C4⋊Dic3—36no—
286C4⋊Dic3—63no—
287C4⋊Dic3—66no—
288C4⋊Dic3C22×D311yescharacteristic
289C4⋊Dic3C22×D313yes—
290Dic3⋊C4—36no—
291Dic3⋊C4—36no—
292Dic3⋊C4—32no—
293Dic3⋊C4—32no—
294Dic3⋊C4—66no—
295C4×Dic3—36no—
296C4×Dic3—36no—
297C4×Dic3—32no—
298C4×Dic3—36no—
299C4×Dic3—36no—
300C4×Dic3—32no—
301C4×Dic3—66no—
302C4×Dic3C22×D311yescharacteristic
303C4×Dic3C22×D313yes—
304C4×Dic3C22×D313yes—
305C22×C32C2511yescharacteristic
306C2×C3×D3—26no—
307C2×C3×D3—23no—
308C2×C3×D3—23no—
309C2×C3×D3—26no—
310C2×C3×D3—26no—
311C2×C3×D3—23no—
312C32×C4—23no—
313C32×C4C2×#4116yes—
314C32×C4C2×#4112yes—
315C3⋊Dic3—26no—
316C3⋊Dic3C2×#4114yes—
317C3×Dic3—23no—
318C3×Dic3—21no—
319C3×Dic3—22no—
320C3×Dic3H(5,𝔽2)12yes—
321C2×#41—31no—
322C2×#41—32no—
323C2×#41—93no—
324C2×#41—96no—
325C4⋊Q8—96no—
326C4⋊Q8—93no—
327#127—33no—
328#127—96no—
329#124—96no—
330#124—912no—
331#121—36no—
332#121—912no—
333#121—96no—
334#121—96no—
335#121—96no—
336C4×Q8—31no—
337C4×Q8—91no—
338C4×Q8—92no—
339C4×Q8—93no—
340C4×Q8D3211yescharacteristic
341C4×D4—33no—
342C4×D4—33no—
343C4×D4—33no—
344C4×D4—93no—
345C4×D4—96no—
346C4×D4—96no—
347C4×D4—96no—
348C4×D4—96no—
349#117—33no—
350#117—96no—
351#117—96no—
352#117—93no—
353C2×C42—33no—
354C2×C42—96no—
355C23×C3—36no—
356C23×C3—33no—
357C22×D3C23×D316yes—
358C22×D3C23×D313yes—
359C3×Q8—24no—
360C3×Q8#23514yescomplemented
361C3×Q8#23714yescomplemented
362C3×D4—612no—
363C3×D4—66no—
364C3×D4—66no—
365C3×D4—126no—
366C2×C3×C4—23no—
367C2×C3×C4—21no—
368C2×C3×C4—23no—
369C2×C3×C4—31no—
370C2×C3×C4—32no—
371C2×C3×C4—33no—
372C2×C3×C4—31no—
373C2×C3×C4—66no—
374C2×C3×C4—612no—
375C2×C3×C4—66no—
376C2×C3×C4—66no—
377C2×C3×C4C23×D313yes—
378C2×C3×C4C23×D313yes—
379C2×C3×C4C23×D313yes—
380C2×C3×C4C23×D313yes—
381C2×C3×C4C23×D311yescharacteristic
382C2×C3×C4C23×D311yescharacteristic
383#67—612no—
384#67—612no—
385#67—1212no—
386#67—1212no—
387C2×Dic3—32no—
388C2×Dic3—32no—
389C2×Dic3—36no—
390C2×Dic3—36no—
391C2×Dic3—62no—
392C2×Dic3—66no—
393C2×Dic3—612no—
394C2×Dic3—612no—
395C2×Dic3C23×D312yes—
396C2×Dic3C23×D313yes—
397C2×Dic3C23×D311yescharacteristic
398C2×Dic3C23×D311yescharacteristic
399D12—26no—
400D12—212no—
401D12—26no—
402D12—46no—
403C4×D3—26no—
404C4×D3—212no—
405C4×D3—26no—
406C4×D3—26no—
407C4×D3—612no—
408C4×D3—612no—
409Dic6—34no—
410Dic6—34no—
411Dic6—64no—
412Dic6—126no—
413Dic6—126no—
414Dic6—246no—
415C2×C32C22×#4111yescharacteristic
416C2×C32C22×#4111yescharacteristic
417C2×C32C2×H(5,𝔽2)11yescharacteristic
418C3×D3—43no—
419C3×D3—46no—
420#41—64no—
421#41—68no—
422#41—1812no—
423#41—3612no—
424C2×Q8—91no—
425C2×Q8—96no—
426C2×Q8C2×D3211yescharacteristic
427C2×D4—33no—
428C2×D4—33no—
429C2×D4—36no—
430C2×D4—36no—
431C2×D4—96no—
432C2×D4—912no—
433C22×C4—33no—
434C22×C4—33no—
435C22×C4—33no—
436C22×C4—36no—
437C22×C4—96no—
438C22×C4—96no—
439C4⋊C4—33no—
440C4⋊C4—31no—
441C4⋊C4—92no—
442C4⋊C4—96no—
443C4⋊C4—96no—
444C4⋊C4—93no—
445C4⋊C4C2×D3213yes—
446C22⋊C4—36no—
447C22⋊C4—36no—
448C22⋊C4—96no—
449C22⋊C4—912no—
450C22⋊C4—96no—
451C42—31no—
452C42—33no—
453C42—33no—
454C42—92no—
455C42—96no—
456C42—96no—
457C42C2×D3213yes—
458C22×C3—21no—
459C22×C3—66no—
460C22×C3—63no—
461C22×C3—66no—
462C22×C3—63no—
463C22×C3—66no—
464C22×C3—63no—
465C22×C3C24×D311yescharacteristic
466C22×C3C24×D311yescharacteristic
467C2×D3—23no—
468C2×D3—26no—
469C2×D3—23no—
470C2×D3—26no—
471C2×D3—23no—
472C2×D3—26no—
473C3×C4—23no—
474C3×C4—23no—
475C3×C4—26no—
476C3×C4—22no—
477C3×C4—32no—
478C3×C4—43no—
479C3×C4—62no—
480C3×C4—61no—
481C3×C4—63no—
482C3×C4C2×#23516yes—
483C3×C4C2×#23716yes—
484C3×C4D3×#4112yes—
485C3×C4D3×#4112yes—
486Dic3—21no—
487Dic3—22no—
488Dic3—23no—
489Dic3—34no—
490Dic3—34no—
491Dic3—64no—
492Dic3—66no—
493Dic3—66no—
494Dic3—126no—
495Dic3#100912yes—
496C32#348311yescharacteristic, complemented, Sylow
497C23—33no—
498C23—36no—
499Q8—94no—
500Q8—366no—
501Q8#379514yescomplemented
502D4—66no—
503D4—612no—
504D4—66no—
505D4—126no—
506D4—1812no—
507D4—3612no—
508C2×C4—31no—
509C2×C4—31no—
510C2×C4—33no—
511C2×C4—32no—
512C2×C4—66no—
513C2×C4—66no—
514C2×C4—612no—
515C2×C4—66no—
516C2×C4—92no—
517C2×C4—96no—
518C2×C4—1812no—
519C2×C4—1812no—
520C2×C4C22×D3213yes—
521C2×C4C22×D3213yes—
522C2×C4C22×D3211yescharacteristic
523C2×C3—21no—
524C2×C3—21no—
525C2×C3—21no—
526C2×C3—126no—
527C2×C3—123no—
528C2×C3C22×#23511yescharacteristic
529C2×C3C22×#23711yescharacteristic
530C2×C3C2×D3×#4111yescharacteristic
531C2×C3C2×D3×#4111yescharacteristic
532C2×C3C2×#100911yescharacteristic
533C2×C3D3×H(5,𝔽2)11yescharacteristic
534D3—43no—
535D3—46no—
536C22—63no—
537C22—66no—
538C22—63no—
539C22—66no—
540C22—63no—
541C22—66no—
542C22C23×D3211yescharacteristic
543C4—23no—
544C4—32no—
545C4—61no—
546C4—62no—
547C4—63no—
548C4—94no—
549C4—186no—
550C4C2×#379516yes—
551C4#6431912yes—
552C3—21no—
553C3#8584511yescharacteristic, complemented
554C3#8676811yescharacteristic, complemented
555C2—123no—
556C2—126no—
557C2C22×#379511yescharacteristic
558C2C2×#6431911yescharacteristic
559C2D3×#100911yescharacteristic
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. Q8×#634C2semidirect
2. C3×#5577C2semidirect
3. C3×C4×#239C2semidirect
4. #10595412C2semidirect
5. Dic3×#239C2semidirect
6. #10594486C2semidirect
7. #10594367C2semidirect
8. #10593988C2semidirect
9. #10593906C2semidirect
10. #10593897C2semidirect
11. C4×#3795C2semidirect
57. #5577D3semidirect
59. C4×#239D3semidirect
185. C32×Q8#41semidirect
192. #634#41semidirect
360. C3×Q8#235semidirect
361. C3×Q8#237semidirect
496. C32#3483semidirect
501. Q8#3795semidirect
553. C3#85845semidirect
554. C3#86768semidirect

Maximal subgroups

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

subgroupindex
Q8×#6342
C3×#55772
C3×C4×#2392
#105954122
Dic3×#2392
#105944862
#105943672
#105939882
#105939062
#105938972
C4×#37952
#867683
#858453