Finite Group Databasebeta

← #16849028

Subgroups — #16849028

Order 3968.

1056 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#92495C211yesmaximal, characteristic, complemented
2C2×#11758459C211yesmaximal, characteristic, complemented
3C2×#11758514C211yesmaximal, characteristic, complemented
4C2×C22⋊C4×D31C211yesmaximal, characteristic, complemented
5#14520962C211yesmaximal, characteristic, complemented
6#14520961C211yesmaximal, characteristic, complemented
7#14520958C211yesmaximal, characteristic, complemented
8#14520957C211yesmaximal, characteristic, complemented
9#14520928C211yesmaximal, characteristic, complemented
10#14520927C211yesmaximal, characteristic, complemented
11#14520927C211yesmaximal, characteristic, complemented
12#14520925C211yesmaximal, characteristic, complemented
13#14520925C211yesmaximal, characteristic, complemented
14#14520911C211yesmaximal, characteristic, complemented
15C31×#386C211yesmaximal, characteristic, complemented
16C24×D31C2211yescharacteristic
17C22×D4×C31C2211yescharacteristic
18C23×C4×C31C2211yescharacteristic, complemented
19C22×C22⋊D31C2211yescharacteristic
20C22×C22⋊D31C2211yescharacteristic
21C22×C22⋊D31C2211yescharacteristic
22C22×C22⋊D31C2211yescharacteristic, complemented
23C2×#92495—24no—
24C2×#92495—24no—
25C2×#92495—24no—
26C22×D124C2211yescharacteristic
27C22×D124C2211yescharacteristic
28C22×C4×D31C2211yescharacteristic
29C22×C4×D31C2211yescharacteristic
30C22×C4×D31C2211yescharacteristic, complemented
31C22×Dic62C2211yescharacteristic, complemented
32C2×C22⋊C4×C31C2211yescharacteristic
33C2×C22⋊C4×C31C2211yescharacteristic
34C2×C22⋊C4×C31C2211yescharacteristic, complemented
35C2×C22⋊Dic31C2211yescharacteristic
36C2×C22⋊Dic31C2211yescharacteristic
37C2×C22⋊Dic31C2211yescharacteristic, complemented
38#11758514—24no—
39C2×(C2×D31)⋊C4C2211yescharacteristic
40C2×(C2×D31)⋊C4C2211yescharacteristic
41C2×(C2×D31)⋊C4C2211yescharacteristic
42C2×(C2×D31)⋊C4C2211yescharacteristic
43C2×(C2×D31)⋊C4C2211yescharacteristic
44C2×(C2×D31)⋊C4C2211yescharacteristic
45C2×(C2×D31)⋊C4C2211yescharacteristic, complemented
46C2×(C2×D31)⋊C4C2211yescharacteristic, complemented
47#11758459—24no—
48C22⋊C4×D31—24no—
49C31×#95C2211yescharacteristic, complemented
50C31×#95C2211yescharacteristic, complemented
51C31⋊#95C2211yescharacteristic, complemented
52C31⋊#95C2211yescharacteristic, complemented
53C31⋊#95C2211yescharacteristic, complemented
54C31⋊#95C2211yescharacteristic, complemented
55C31⋊#95C2211yescharacteristic, complemented
56C31⋊#95C2211yescharacteristic, complemented
57C24×C31C2311yescharacteristic
58C24×C31C2311yescharacteristic
59C23×D31—22no—
60C23×D31—22no—
61C23×D31—22no—
62C23×D31—22no—
63C23×D31—22no—
64C23×D31—41no—
65C23×D31—41no—
66C23×D31C2311yescharacteristic
67C23×D31C2311yescharacteristic
68C23×D31C2311yescharacteristic
69C2×D4×C31—22no—
70C2×D4×C31—22no—
71C2×D4×C31—41no—
72C22×C4×C31—22no—
73C22×C4×C31—22no—
74C22×C4×C31—22no—
75C22×C4×C31C2311yescharacteristic
76C22×C4×C31C2311yescharacteristic
77C22×C4×C31C2311yescharacteristic
78C22×C4×C31C2311yescharacteristic
79C22×C4×C31C2311yescharacteristic
80C2×C22⋊D31—22no—
81C2×C22⋊D31—22no—
82C2×C22⋊D31—22no—
83C2×C22⋊D31—22no—
84C2×C22⋊D31—22no—
85C2×C22⋊D31—22no—
86C2×C22⋊D31—22no—
87C2×C22⋊D31—22no—
88C2×C22⋊D31—22no—
89C2×C22⋊D31—41no—
90C2×C22⋊D31—41no—
91C2×C22⋊D31D414yescomplemented
92C22×Dic31C2311yescharacteristic
93C22×Dic31C2311yescharacteristic
94C22×Dic31C2311yescharacteristic
95C22×Dic31C2311yescharacteristic
96C22×Dic31C2311yescharacteristic
97#92495—232no—
98C2×D124—22no—
99C2×D124—22no—
100C2×D124—22no—
101C2×D124—22no—
102C2×D124—41no—
103C2×D124D414yescomplemented
104C2×C4×D31—22no—
105C2×C4×D31—22no—
106C2×C4×D31—22no—
107C2×C4×D31—22no—
108C2×C4×D31—22no—
109C2×C4×D31—22no—
110C2×C4×D31—41no—
111C2×C4×D31—41no—
112C2×C4×D31D414yescomplemented
113C2×Dic62—22no—
114C2×Dic62—22no—
115C2×Dic62—22no—
116C22⋊C4×C31—22no—
117C22⋊C4×C31—22no—
118C22⋊C4×C31—41no—
119C22⋊Dic31—22no—
120C22⋊Dic31—22no—
121C22⋊Dic31—41no—
122(C2×D31)⋊C4—22no—
123(C2×D31)⋊C4—22no—
124(C2×D31)⋊C4—22no—
125(C2×D31)⋊C4—22no—
126(C2×D31)⋊C4—22no—
127(C2×D31)⋊C4—22no—
128(C2×D31)⋊C4—41no—
129(C2×D31)⋊C4—41no—
130C23×C31—21no—
131C23×C31—21no—
132C23×C31—21no—
133C23×C31—21no—
134C23×C31—21no—
135C23×C31—21no—
136C23×C31—21no—
137C23×C31—21no—
138C23×C31—21no—
139C23×C31—21no—
140C23×C31—21no—
141C23×C31—21no—
142C23×C31C2×D412yes—
143C23×C31#4112yescomplemented
144C23×C31C2411yescharacteristic
145C22×D31—21no—
146C22×D31—21no—
147C22×D31—21no—
148C22×D31—21no—
149C22×D31—28no—
150C22×D31—21no—
151C22×D31—21no—
152C22×D31—21no—
153C22×D31—21no—
154C22×D31—21no—
155C22×D31—21no—
156C22×D31—21no—
157C22×D31—21no—
158C22×D31—21no—
159C22×D31—21no—
160C22×D31—21no—
161C22×D31—21no—
162C22×D31—44no—
163C22×D31—44no—
164C22×D31—44no—
165C22×D31—44no—
166C22×D31—44no—
167C22×D31—44no—
168C22×D31C2×D412yes—
169C22×D31C2×D412yes—
170C22×D31C2×D412yes—
171C22×D31C2×D412yes—
172C22×D31#4112yescomplemented
173D4×C31—44no—
174C2×C4×C31—28no—
175C2×C4×C31—21no—
176C2×C4×C31—21no—
177C2×C4×C31—21no—
178C2×C4×C31—21no—
179C2×C4×C31—21no—
180C2×C4×C31—21no—
181C2×C4×C31—21no—
182C2×C4×C31—21no—
183C2×C4×C31—21no—
184C2×C4×C31—21no—
185C2×C4×C31—21no—
186C2×C4×C31—21no—
187C2×C4×C31—21no—
188C2×C4×C31C2×D412yes—
189C2×C4×C31C2×D412yes—
190C22⋊D31—28no—
191C22⋊D31—28no—
192C22⋊D31—44no—
193C22⋊D31—44no—
194C2×Dic31—21no—
195C2×Dic31—21no—
196C2×Dic31—21no—
197C2×Dic31—21no—
198C2×Dic31—21no—
199C2×Dic31—21no—
200C2×Dic31—21no—
201C2×Dic31—21no—
202C2×Dic31—21no—
203C2×Dic31—21no—
204C2×Dic31—21no—
205C2×Dic31—21no—
206C2×Dic31—21no—
207C2×Dic31C2×D412yes—
208C2×Dic31C2×D412yes—
209D124—28no—
210D124—44no—
211C4×D31—28no—
212C4×D31—28no—
213C4×D31—44no—
214Dic62—28no—
215#2404—311nomaximal, Sylow
216C22×C31—22no—
217C22×C31—22no—
218C22×C31—22no—
219C22×C31—22no—
220C22×C31—22no—
221C22×C31—22no—
222C22×C31—41no—
223C22×C31—41no—
224C22×C31—41no—
225C22×C31—41no—
226C22×C31—41no—
227C22×C31—41no—
228C22×C31—41no—
229C22×C31—41no—
230C22×C31C22×D411yescharacteristic
231C22×C31C22×D411yescharacteristic
232C22×C31C22×D411yescharacteristic
233C22×C31C2×#4111yescharacteristic
234C22×C31C2×#4111yescharacteristic
235C22×C31H(5,𝔽2)11yescharacteristic
236C22×C31H(5,𝔽2)11yescharacteristic
237C2×D31—22no—
238C2×D31—22no—
239C2×D31—22no—
240C2×D31—22no—
241C2×D31—22no—
242C2×D31—22no—
243C2×D31—22no—
244C2×D31—22no—
245C2×D31—22no—
246C2×D31—22no—
247C2×D31—22no—
248C2×D31—22no—
249C2×D31—416no—
250C2×D31—41no—
251C2×D31—41no—
252C2×D31—41no—
253C2×D31—41no—
254C2×D31—41no—
255C2×D31—41no—
256C2×D31—41no—
257C2×D31—41no—
258C2×D31#12014yes—
259C4×C31—22no—
260C4×C31—22no—
261C4×C31—41no—
262C4×C31—41no—
263C4×C31—41no—
264Dic31—22no—
265Dic31—22no—
266Dic31—41no—
267Dic31—41no—
268Dic31—41no—
269C22×#41—311no—
270C2×#120—311no—
271C2×#120—311no—
272C22×C22⋊C4—311no—
273#393—311no—
274#393—311no—
275#392—311no—
276#392—311no—
277#386—311no—
278#386—311no—
279#386—311no—
280#386—311no—
281#386—311no—
282#386D3111yescharacteristic, complemented
283#381—311no—
284C2×C31—42no—
285C2×C31—42no—
286C2×C31C2×#12011yescharacteristic
287C2×C31#52511yescharacteristic
288C2×C31#53511yescharacteristic
289C2×C31#54611yescharacteristic
290C2×C31#54611yescharacteristic
291C2×C31#54611yescharacteristic
292C2×C31#54611yescharacteristic
293D31—24no—
294D31—42no—
295D31—42no—
296C25—311no—
297C2×#41—624no—
298C2×#41—624no—
299C2×#41—624no—
300C22×Q8—311no—
301C22×D4—311no—
302C22×D4—311no—
303C22×D4—311no—
304C22×D4—311no—
305C22×D4—311no—
306C22×D4—311no—
307C22×D4C2×D3111yescharacteristic
308C23×C4—311no—
309C23×C4—311no—
310C23×C4—311no—
311C23×C4C2×D3111yescharacteristic, complemented
312#120—624no—
313#120—624no—
314C2×C22⋊C4—311no—
315C2×C22⋊C4—311no—
316C2×C22⋊C4—311no—
317C2×C22⋊C4—311no—
318C2×C22⋊C4—311no—
319C2×C22⋊C4—311no—
320C2×C22⋊C4—311no—
321C2×C22⋊C4—311no—
322C2×C22⋊C4—311no—
323C2×C22⋊C4—311no—
324C2×C22⋊C4—311no—
325C2×C22⋊C4—624no—
326C2×C22⋊C4C2×D3111yescharacteristic
327C2×C22⋊C4C2×D3111yescharacteristic
328C2×C22⋊C4C2×D3111yescharacteristic, complemented
329#95—311no—
330#95—311no—
331#95—311no—
332#95—311no—
333#95—311no—
334#95—311no—
335#95C2×D3111yescharacteristic, complemented
336#95C2×D3111yescharacteristic, complemented
337C31#240411yescharacteristic, complemented, Sylow
338C24—311no—
339C24—311no—
340C24—311no—
341C24—622no—
342C24—622no—
343C24—622no—
344C24—622no—
345C24—622no—
346C24—1241no—
347C24—1241no—
348C24C22×D3111yescharacteristic
349C24C22×D3111yescharacteristic
350#41—6232no—
351C2×Q8—622no—
352C2×Q8—622no—
353C2×Q8—622no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—41no—
357C2×D4—314no—
358C2×D4—314no—
359C2×D4—622no—
360C2×D4—622no—
361C2×D4—622no—
362C2×D4—622no—
363C2×D4—622no—
364C2×D4—622no—
365C2×D4—622no—
366C2×D4—622no—
367C2×D4—622no—
368C2×D4—622no—
369C2×D4—622no—
370C2×D4—622no—
371C2×D4—622no—
372C2×D4—1241no—
373C2×D4—1241no—
374C2×D4—1241no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—311no—
379C22×C4—311no—
380C22×C4—314no—
381C22×C4—311no—
382C22×C4—311no—
383C22×C4—311no—
384C22×C4—622no—
385C22×C4—622no—
386C22×C4—622no—
387C22×C4—622no—
388C22×C4—622no—
389C22×C4—622no—
390C22×C4—1241no—
391C22×C4—1241no—
392C22×C4C22×D3111yescharacteristic
393C22×C4C22×D3111yescharacteristic
394C22×C4C22×D3111yescharacteristic
395C22×C4C22×D3111yescharacteristic
396C22×C4C22×D3111yescharacteristic
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—41no—
400C22⋊C4—622no—
401C22⋊C4—622no—
402C22⋊C4—622no—
403C22⋊C4—622no—
404C22⋊C4—622no—
405C22⋊C4—622no—
406C22⋊C4—622no—
407C22⋊C4—622no—
408C22⋊C4—1241no—
409C22⋊C4—1241no—
410C22⋊C4—1241no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—312no—
424C23—312no—
425C23—312no—
426C23—312no—
427C23—312no—
428C23—628no—
429C23—621no—
430C23—621no—
431C23—621no—
432C23—621no—
433C23—621no—
434C23—621no—
435C23—621no—
436C23—621no—
437C23—621no—
438C23—621no—
439C23—621no—
440C23—621no—
441C23—621no—
442C23—621no—
443C23—621no—
444C23—621no—
445C23—1244no—
446C23—1244no—
447C23—1244no—
448C23—1244no—
449C23—1244no—
450C23—1244no—
451C23#9249512yescomplemented
452C23D4×D3112yes—
453C23C23×D3111yescharacteristic
454Q8—628no—
455D4—44no—
456D4—628no—
457D4—628no—
458D4—628no—
459D4—1244no—
460D4—1244no—
461D4—1244no—
462C2×C4—28no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—312no—
477C2×C4—312no—
478C2×C4—628no—
479C2×C4—628no—
480C2×C4—621no—
481C2×C4—621no—
482C2×C4—621no—
483C2×C4—621no—
484C2×C4—621no—
485C2×C4—621no—
486C2×C4—621no—
487C2×C4—621no—
488C2×C4—621no—
489C2×C4—621no—
490C2×C4—621no—
491C2×C4—621no—
492C2×C4—621no—
493C2×C4—1244no—
494C2×C4D4×D3112yes—
495C2×C4D4×D3112yes—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—314no—
511C22—622no—
512C22—622no—
513C22—622no—
514C22—622no—
515C22—622no—
516C22—622no—
517C22—622no—
518C22—622no—
519C22—622no—
520C22—622no—
521C22—622no—
522C22—622no—
523C22—12416no—
524C22—1241no—
525C22—1241no—
526C22—1241no—
527C22—1241no—
528C22—1241no—
529C22—1241no—
530C22—1241no—
531C22—1241no—
532C22C2×#9249511yescharacteristic
533C22C2×D4×D3111yescharacteristic
534C22C2×D4×D3111yescharacteristic
535C22C2×D4×D3111yescharacteristic
536C22#1175856111yescharacteristic
537C22#41×D3111yescharacteristic
538C22#1175856611yescharacteristic
539C4—22no—
540C4—22no—
541C4—41no—
542C4—41no—
543C4—41no—
544C4—622no—
545C4—622no—
546C4—1241no—
547C4—1241no—
548C4—1241no—
549C2—42no—
550C2—42no—
551C2—624no—
552C2—1242no—
553C2—1242no—
554C2#120×D3111yescharacteristic
555C2#1452155811yescharacteristic
556C2#1452161511yescharacteristic
557C2#1452168911yescharacteristic
558C2#1452169111yescharacteristic
559C2#1452169511yescharacteristic
560C2#1452169611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D31
commutator, socle, cosocleC23×C31C24
FittingC31×#386C2
hypercenter#386D31
hypocenterC31#2404
LCSG ▷ C23×C31 ▷ C31
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C22×#92495C2semidirect
2. C2×#11758459C2semidirect
3. C2×#11758514C2semidirect
4. C2×C22⋊C4×D31C2semidirect
5. #14520962C2semidirect
6. #14520961C2semidirect
7. #14520958C2semidirect
8. #14520957C2semidirect
9. #14520928C2semidirect
10. #14520927C2semidirect
11. #14520927C2semidirect
12. #14520925C2semidirect
13. #14520925C2semidirect
14. #14520911C2semidirect
15. C31×#386C2semidirect
18. C23×C4×C31C22semidirect
22. C22×C22⋊D31C22semidirect
30. C22×C4×D31C22semidirect
31. C22×Dic62C22semidirect
34. C2×C22⋊C4×C31C22semidirect
37. C2×C22⋊Dic31C22semidirect
45. C2×(C2×D31)⋊C4C22semidirect
46. C2×(C2×D31)⋊C4C22semidirect
49. C31×#95C22semidirect
50. C31×#95C22semidirect
51. C31⋊#95C22semidirect
52. C31⋊#95C22semidirect
53. C31⋊#95C22semidirect
54. C31⋊#95C22semidirect
55. C31⋊#95C22semidirect
56. C31⋊#95C22semidirect
91. C2×C22⋊D31D4semidirect
103. C2×D124D4semidirect
112. C2×C4×D31D4semidirect
143. C23×C31#41semidirect
172. C22×D31#41semidirect
282. #386D31semidirect
311. C23×C4C2×D31semidirect
328. C2×C22⋊C4C2×D31semidirect
335. #95C2×D31semidirect
336. #95C2×D31semidirect
337. C31#2404semidirect
451. C23#92495semidirect

Maximal subgroups

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

subgroupindex
C22×#924952
C2×#117584592
C2×#117585142
C2×C22⋊C4×D312
#145209622
#145209612
#145209582
#145209572
#145209282
#145209272
#145209252
#145209112
C31×#3862
#240431