Finite Group Databasebeta

← #11887543

Subgroups — #11887543

Order 1152.

980 conjugacy classes of subgroups.

The subgroup lattice has 577 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×#634C211yesmaximal, characteristic, complemented
2C3×#5555C211yesmaximal, characteristic
3C3×D4×C3⋊C8C211yesmaximal, characteristic, complemented
4#10591181C212yesmaximal, complemented
5#10591175C212yesmaximal, complemented
6#10591124C212yesmaximal, complemented
7C2×#63567C212yesmaximal, complemented
8#10590365C211yesmaximal, characteristic, complemented
9C4×#3716C211yesmaximal, characteristic, complemented
10#10590307C211yesmaximal, characteristic, complemented
11#10590288C211yesmaximal, characteristic, complemented
12#83531—31nomaximal
13#79447—31nomaximal
14C22×#634C2212yes—
15C2×C3×#237—21no—
16C32×C4×D4C2211yescharacteristic
17D4×C3⋊Dic3—21no—
18#64133C2212yes—
19#64097—21no—
20#64087—22no—
21#64078C2211yescharacteristic
22C4×#634C2211yescharacteristic
23C3×#930C2212yes—
24C2×C3×C4⋊Dic3C2212yes—
25C22×C3×C3⋊C8C2212yes—
26C3×#897—21no—
27C3×#886—22no—
28C3×C4×D12C2211yescharacteristic
29C3×#869C2211yescharacteristic
30C2×#3716C2211yescharacteristic
31C2×#3716C2212yes—
32C3×#830C2212yes—
33C3×C4⋊(C3⋊C8)C2211yescharacteristic
34C3×C4×C3⋊C8C2211yescharacteristic
35#63573C2212yes—
36#63573C2211yescharacteristic
37#63567C2212yes—
38#63567C2212yes—
39#63567C2212yes—
40#63567C2211yescharacteristic
41#63567C2212yes—
42#63563C2211yescharacteristic
43C3×#547—31no—
44#5555D311yescharacteristic
45D4×Dic6—31no—
46D4×Dic6—31no—
47D4×Dic6—61no—
48C3×C8×D4—31no—
49#5114—32no—
50C2×#816—32no—
51#5094—31no—
52#5093—31no—
53#5041—32no—
54#5040—32no—
55#5019—31no—
56C4×#214—31no—
57D4×C3⋊C8D311yescharacteristic, complemented
58#4971—32no—
59C2×#810—32no—
60#4851—31no—
61#4841—31no—
62#4738—32no—
63#4721—32no—
64C4×#204—31no—
65#4681—31no—
66C2×C32×D4D411yescharacteristic
67C22×C32×C4C2312yes—
68C22×C3⋊Dic3—22no—
69C2×#634—24no—
70C2×#634—22no—
71C2×#634—42no—
72C2×#634C2312yes—
73C2×#634C2311yescharacteristic
74C2×C3×#67—22no—
75C22×C3×Dic3—22no—
76C3×#237—44no—
77C2×C3×D12C2311yescharacteristic
78C2×C3×C4×D3—21no—
79C2×C3×Dic6—21no—
80C32×C4⋊C4D411yescharacteristic
81C32×C22⋊C4D412yes—
82C32×C42C2311yescharacteristic
83#3757—22no—
84C4⋊(C3⋊Dic3)—21no—
85C4⋊(C3⋊Dic3)—21no—
86C4⋊(C3⋊Dic3)C2311yescharacteristic
87#3750—21no—
88#3750—22no—
89C4×C3⋊Dic3—21no—
90C3×(C2×D3)⋊C4—21no—
91C3×(C2×D3)⋊C4—22no—
92C3×C4⋊Dic3—21no—
93C3×C4⋊Dic3—22no—
94C3×C4⋊Dic3C2311yescharacteristic
95C3×C4⋊Dic3C2312yes—
96C2×C3×C3⋊C8—42no—
97C2×C3×C3⋊C8C2311yescharacteristic
98C2×C3×C3⋊C8C2312yes—
99C2×C3×C3⋊C8C2311yescharacteristic
100#3716—22no—
101#3716—42no—
102#3298—91noSylow
103C2×C3×#41—61no—
104C2×#237—21no—
105C22×Dic6—32no—
106C22×Dic6—32no—
107C22×Dic6—62no—
108C22×C3×C8—32no—
109C3×C4⋊Q8—31no—
110C3×#123—62no—
111C3×C22⋊Q8—61no—
112C3×#121—32no—
113C3×C4×D4—21no—
114C3×C4×D4—31no—
115C3×C4×D4C2×D311yescharacteristic
116C3×C4×D4C2×D311yescharacteristic
117C2×C3×C4⋊C4—32no—
118D4×Dic3—62no—
119D4×Dic3—61no—
120D4×Dic3—61no—
121C2×#214—31no—
122C2×#214—32no—
123#930C2×D312yes—
124C2×C4⋊Dic3C2×D312yes—
125#924—32no—
126#924—32no—
127#924—62no—
128C22×C3⋊C8C2×D312yes—
129C2×#204—31no—
130C2×#204—32no—
131#897—21no—
132#888—62no—
133#888—61no—
134#888—61no—
135#886—22no—
136#878—64no—
137#878—62no—
138#878—62no—
139C4×D12C2×D311yescharacteristic
140#869—31no—
141#869—31no—
142#869—61no—
143#869C2×D311yescharacteristic
144C4×Dic6—31no—
145C4×Dic6—31no—
146C4×Dic6—61no—
147C3×C4⋊C8—31no—
148C3×C22⋊C8—32no—
149C3×C4×C8—31no—
150#832—31no—
151#830C2×D312yes—
152#821—31no—
153#817—31no—
154#817—32no—
155#816—32no—
156#816—32no—
157#816—31no—
158#816—32no—
159#816—32no—
160#810—31no—
161#810—32no—
162#810—32no—
163#810—32no—
164#810—32no—
165#808—31no—
166#808—32no—
167C4⋊(C3⋊C8)C2×D311yescharacteristic
168C4×C3⋊C8C2×D311yescharacteristic
169C23×C32C2×D412yes—
170C22×C3×D3—21no—
171C32×D4SD1614yescomplemented
172C2×C32×C4—22no—
173C2×C32×C4—22no—
174C2×C32×C4C2×D411yescharacteristic
175C2×C32×C4C2×D412yes—
176C2×C32×C4C2×D411yescharacteristic
177C2×C32×C4C2411yescharacteristic
178C2×C3⋊Dic3—21no—
179C2×C3⋊Dic3—21no—
180C2×C3⋊Dic3—22no—
181C2×C3⋊Dic3—44no—
182#634—21no—
183#634—22no—
184#634—22no—
185#634—44no—
186#634#4112yescomplemented
187C3×#67—44no—
188C2×C3×Dic3—22no—
189C2×C3×Dic3—21no—
190C2×C3×Dic3—44no—
191C3×D12—21no—
192C3×D12—21no—
193C3×C4×D3—42no—
194C3×Dic6—42no—
195C3×C3⋊C8—21no—
196C3×C3⋊C8—22no—
197C3×C3⋊C8—21no—
198D4×Q8—91no—
199#547—31no—
200#487—91no—
201#483—92no—
202#475—91no—
203#465—92no—
204#461—91no—
205#448—92no—
206C4×SD16—91no—
207C8×D4—31no—
208C2×#103—92no—
209C3×#41—124no—
210C2×C3×Q8—61no—
211C2×C3×D4—21no—
212C2×C3×D4—31no—
213C2×C3×D4—62no—
214C2×C3×D4D1211yescharacteristic
215C2×C3×D4#6711yescharacteristic
216C22×C3×C4—22no—
217C22×C3×C4—62no—
218C22×C3×C4—61no—
219C22×C3×C4C22×D312yes—
220C22×C3×C4C22×D312yes—
221C2×#67—22no—
222C22×Dic3—22no—
223C22×Dic3—64no—
224C22×Dic3—62no—
225C22×Dic3—62no—
226#237—44no—
227C2×D12C22×D311yescharacteristic
228C2×C4×D3—21no—
229C2×Dic6—21no—
230C2×Dic6—31no—
231C2×Dic6—31no—
232C2×Dic6—32no—
233C2×Dic6—32no—
234C2×Dic6—61no—
235C2×Dic6—62no—
236C2×Dic6—64no—
237C2×Dic6—64no—
238C2×Dic6—64no—
239C2×Dic6—62no—
240C2×Dic6—62no—
241C2×Dic6—124no—
242C2×Dic6—122no—
243C2×Dic6—122no—
244C2×Dic6—124no—
245C2×C3×C8—31no—
246C2×C3×C8—31no—
247C2×C3×C8—32no—
248C2×C3×C8—122no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—31no—
251C3×C4⋊C4—32no—
252C3×C4⋊C4—62no—
253C3×C4⋊C4—61no—
254C3×C4⋊C4D1211yescharacteristic
255C3×C4⋊C4#6711yescharacteristic
256C3×C22⋊C4—22no—
257C3×C22⋊C4—61no—
258C3×C22⋊C4—62no—
259C3×C22⋊C4D1212yes—
260C3×C22⋊C4#6712yes—
261C3×C42—21no—
262C3×C42C22×D311yescharacteristic
263C3×C42C22×D311yescharacteristic
264#217—64no—
265#217—62no—
266#217—62no—
267#214—62no—
268#214—122no—
269(C2×D3)⋊C4—21no—
270(C2×D3)⋊C4—22no—
271C4⋊Dic3—21no—
272C4⋊Dic3—22no—
273C4⋊Dic3—31no—
274C4⋊Dic3—31no—
275C4⋊Dic3—61no—
276C4⋊Dic3—62no—
277C4⋊Dic3—62no—
278C4⋊Dic3—61no—
279C4⋊Dic3—61no—
280C4⋊Dic3—61no—
281C4⋊Dic3—61no—
282C4⋊Dic3C22×D312yes—
283C4⋊Dic3C22×D311yescharacteristic
284Dic3⋊C4—62no—
285Dic3⋊C4—64no—
286Dic3⋊C4—61no—
287Dic3⋊C4—62no—
288Dic3⋊C4—61no—
289Dic3⋊C4—62no—
290C4×Dic3—62no—
291C4×Dic3—61no—
292C4×Dic3—61no—
293C2×C3⋊C8—42no—
294C2×C3⋊C8C22×D311yescharacteristic
295C2×C3⋊C8C22×D311yescharacteristic
296C2×C3⋊C8C22×D312yes—
297#204—62no—
298#204—122no—
299C22×C32—22no—
300C22×C32—22no—
301C22×C32C2×SD1614yes—
302C22×C32C22×D411yescharacteristic
303C2×C3×D3—41no—
304C2×C3×D3—41no—
305C2×C3×D3—41no—
306C32×C4—22no—
307C32×C4—21no—
308C32×C4C2×SD1612yes—
309C32×C4C2×#4111yescharacteristic
310C32×C4H(5,𝔽2)11yescharacteristic
311C3⋊Dic3—22no—
312C3⋊Dic3—42no—
313C3⋊Dic3—41no—
314C3×Dic3—42no—
315C3×Dic3—41no—
316C2×#41—61no—
317C22×Q8—92no—
318C2×SD16—91no—
319C2×SD16—92no—
320C22×C8—32no—
321C4⋊Q8—31no—
322C4⋊Q8—91no—
323C4⋊Q8—181no—
324#123—62no—
325C22⋊Q8—61no—
326C22⋊Q8—92no—
327C22⋊Q8—182no—
328#121—32no—
329C4×Q8—91no—
330C4×D4—31no—
331C4×D4—181no—
332C4×D4D3211yescharacteristic
333C2×C4⋊C4—32no—
334#106—91no—
335#106—92no—
336C4⋊C8—31no—
337#103—91no—
338#103—92no—
339#103—92no—
340#103—92no—
341#103—92no—
342#102—91no—
343C22⋊C8—32no—
344C4×C8—31no—
345C23×C3—22no—
346C23×C3—61no—
347C23×C3C2×D1212yes—
348C23×C3C2×#6712yes—
349C22×D3—21no—
350C3×Q8—122no—
351C3×D4—24no—
352C3×D4—61no—
353C3×D4—61no—
354C3×D4—124no—
355C3×D4#20414yescomplemented
356C3×D4#21414yescomplemented
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—22no—
365C2×C3×C4—42no—
366C2×C3×C4—42no—
367C2×C3×C4—61no—
368C2×C3×C4—62no—
369C2×C3×C4—124no—
370C2×C3×C4—122no—
371C2×C3×C4C2×D1211yescharacteristic
372C2×C3×C4C2×D1212yes—
373C2×C3×C4C2×D1211yescharacteristic
374C2×C3×C4C2×#6711yescharacteristic
375C2×C3×C4C2×#6712yes—
376C2×C3×C4C2×#6711yescharacteristic
377C2×C3×C4C23×D311yescharacteristic
378C2×C3×C4C23×D311yescharacteristic
379#67—44no—
380C2×Dic3—22no—
381C2×Dic3—21no—
382C2×Dic3—44no—
383C2×Dic3—62no—
384C2×Dic3—61no—
385C2×Dic3—61no—
386C2×Dic3—62no—
387C2×Dic3—64no—
388C2×Dic3—61no—
389C2×Dic3—62no—
390C2×Dic3—61no—
391C2×Dic3—62no—
392C2×Dic3—124no—
393C2×Dic3—124no—
394C2×Dic3—128no—
395D12—21no—
396D12—21no—
397C4×D3—42no—
398Dic6—32no—
399Dic6—32no—
400Dic6—42no—
401Dic6—62no—
402Dic6—62no—
403Dic6—61no—
404Dic6—61no—
405Dic6—62no—
406Dic6—62no—
407Dic6—62no—
408Dic6—62no—
409Dic6—122no—
410Dic6—122no—
411Dic6—124no—
412Dic6—124no—
413Dic6—128no—
414C3×C8—61no—
415C3×C8—62no—
416C3×C8—61no—
417C3⋊C8—21no—
418C3⋊C8—22no—
419C3⋊C8—21no—
420C2×C32—24no—
421C2×C32#54611yescharacteristic
422C2×C32C22×SD1611yescharacteristic
423C2×C32#57811yescharacteristic
424C3×D3—81no—
425#41—124no—
426C2×Q8—61no—
427C2×Q8—91no—
428C2×Q8—92no—
429C2×Q8—184no—
430C2×Q8—182no—
431C2×Q8—362no—
432C2×D4—31no—
433C2×D4—62no—
434C2×D4#62611yescharacteristic
435C22×C4—62no—
436C22×C4—61no—
437C22×C4—182no—
438C22×C4C2×D3212yes—
439SD16—182no—
440SD16—362no—
441C2×C8—31no—
442C2×C8—31no—
443C2×C8—32no—
444C2×C8—122no—
445C4⋊C4—31no—
446C4⋊C4—32no—
447C4⋊C4—62no—
448C4⋊C4—61no—
449C4⋊C4—91no—
450C4⋊C4—181no—
451C4⋊C4—181no—
452C4⋊C4—182no—
453C4⋊C4—181no—
454C4⋊C4#62611yescharacteristic
455C22⋊C4—61no—
456C22⋊C4—62no—
457C22⋊C4—182no—
458C22⋊C4#62612yes—
459C42—181no—
460C42C2×D3211yescharacteristic
461C22×C3—22no—
462C22×C3—22no—
463C22×C3—22no—
464C22×C3—22no—
465C22×C3—21no—
466C22×C3—24no—
467C22×C3—42no—
468C22×C3—42no—
469C22×C3—121no—
470C22×C3—121no—
471C22×C3—121no—
472C22×C3C2×#20414yes—
473C22×C3C2×#21414yes—
474C22×C3C22×D1211yescharacteristic
475C22×C3C22×#6711yescharacteristic
476C2×D3—41no—
477C2×D3—41no—
478C2×D3—41no—
479C3×C4—21no—
480C3×C4—22no—
481C3×C4—22no—
482C3×C4—21no—
483C3×C4—21no—
484C3×C4—21no—
485C3×C4—22no—
486C3×C4—42no—
487C3×C4—41no—
488C3×C4—121no—
489C3×C4—122no—
490C3×C4C2×#20412yes—
491C3×C4C2×#21412yes—
492C3×C4#100411yescharacteristic
493C3×C4D3×#4111yescharacteristic
494C3×C4D3×#4111yescharacteristic
495C3×C4#100911yescharacteristic
496Dic3—42no—
497Dic3—41no—
498Dic3—64no—
499Dic3—62no—
500Dic3—62no—
501Dic3—124no—
502Dic3—122no—
503Dic3—121no—
504Dic3—122no—
505Dic3—122no—
506Dic3—121no—
507C32#329811yescharacteristic, complemented, Sylow
508C23—61no—
509C23C2×#62612yes—
510Q8—92no—
511Q8—122no—
512Q8—181no—
513Q8—182no—
514Q8—182no—
515Q8—364no—
516D4—61no—
517D4—61no—
518D4—124no—
519D4#371614yescomplemented
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—61no—
523C2×C4—62no—
524C2×C4—124no—
525C2×C4—122no—
526C2×C4—181no—
527C2×C4—181no—
528C2×C4—182no—
529C2×C4—364no—
530C2×C4C2×#62611yescharacteristic
531C2×C4C2×#62612yes—
532C2×C4C2×#62611yescharacteristic
533C2×C4C22×D3211yescharacteristic
534C8—61no—
535C8—61no—
536C8—62no—
537C2×C3—24no—
538C2×C3—24no—
539C2×C3—21no—
540C2×C3—21no—
541C2×C3—21no—
542C2×C3—44no—
543C2×C3—241no—
544C2×C3#555411yescharacteristic
545C2×C3C22×#20411yescharacteristic
546C2×C3#575311yescharacteristic
547C2×C3C22×#21411yescharacteristic
548C2×C3#582911yescharacteristic
549C2×C3#583811yescharacteristic
550D3—81no—
551C22—22no—
552C22—22no—
553C22—121no—
554C22—121no—
555C22—121no—
556C22C2×#371614yes—
557C22C22×#62611yescharacteristic
558C4—21no—
559C4—22no—
560C4—121no—
561C4—122no—
562C4—182no—
563C4—361no—
564C4—362no—
565C4C2×#371612yes—
566C4#6431211yescharacteristic
567C4#6431411yescharacteristic
568C3—21no—
569C3#7944711yescharacteristic, complemented
570C3#8353111yescharacteristic, complemented
571C2—24no—
572C2—241no—
573C2C22×#371611yescharacteristic
574C2#1059525411yescharacteristic
575C2#1059545011yescharacteristic
subgroupG/H
centerC22C22×#626
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×C4×D4C22
socleC22×C32C22×D4
prime radicalC3×#5555C2
hypercenterC4×D4D32
hypocenterC32#3298
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. D4×#634C2semidirect
3. C3×D4×C3⋊C8C2semidirect
4. #10591181C2semidirect
5. #10591175C2semidirect
6. #10591124C2semidirect
7. C2×#63567C2semidirect
8. #10590365C2semidirect
9. C4×#3716C2semidirect
10. #10590307C2semidirect
11. #10590288C2semidirect
57. D4×C3⋊C8D3semidirect
171. C32×D4SD16semidirect
186. #634#41semidirect
355. C3×D4#204semidirect
356. C3×D4#214semidirect
507. C32#3298semidirect
519. D4#3716semidirect
569. C3#79447semidirect
570. C3#83531semidirect

Maximal subgroups

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

subgroupindex
D4×#6342
C3×#55552
C3×D4×C3⋊C82
#105911812
#105911752
#105911242
C2×#635672
#105903652
C4×#37162
#105903072
#105902882
#835313
#794473