Finite Group Databasebeta

← #27153583

Subgroups — #27153583

Order 12416.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261438C211yesmaximal, characteristic, complemented
2#20261380C211yesmaximal, characteristic, complemented
3#20261092C211yesmaximal, characteristic, complemented
4#20261072C211yesmaximal, characteristic, complemented
5#20261061C211yesmaximal, characteristic, complemented
6#20260972C211yesmaximal, characteristic, complemented
7#20260965C211yesmaximal, characteristic, complemented
8#20260951C211yesmaximal, characteristic, complemented
9#20260881C211yesmaximal, characteristic, complemented
10#20260870C211yesmaximal, characteristic, complemented
11C4×#13095496C211yesmaximal, characteristic, complemented
12#20260800C211yesmaximal, characteristic, complemented
13#20260728C211yesmaximal, characteristic, complemented
14#102×D97C211yesmaximal, characteristic, complemented
15#478×C97C211yesmaximal, characteristic, complemented
16C2×#13095510C2211yescharacteristic
17C2×D4×D97C2211yescharacteristic
18C2×#13095500C2211yescharacteristic, complemented
19C2×D776C2211yescharacteristic
20C2×#13095496C2211yescharacteristic
21C2×C8×D97C2211yescharacteristic
22C2×C8×D97C2211yescharacteristic, complemented
23#15690805—21no—
24#15690798C2211yescharacteristic
25#15690795—21no—
26#15690794—21no—
27#15690790C2211yescharacteristic
28#15690785—21no—
29#15690785C2211yescharacteristic
30#15690785C2211yescharacteristic
31#15690785C2211yescharacteristic, complemented
32#15690784—21no—
33#15690780—21no—
34C4×D388C2211yescharacteristic
35C4×D388C2211yescharacteristic, complemented
36#15690772—21no—
37#15690768C2211yescharacteristic, complemented
38C4⋊C4×D97C2211yescharacteristic
39C2×#13095488—21no—
40#15690743C2211yescharacteristic, complemented
41#15690735C2211yescharacteristic, complemented
42#15690733C2211yescharacteristic, complemented
43#15690733C2211yescharacteristic, complemented
44#15690732C2211yescharacteristic
45#15690731C2211yescharacteristic
46#15690730C2211yescharacteristic
47#15690730C2211yescharacteristic
48#15690730C2211yescharacteristic, complemented
49#15690722C2211yescharacteristic
50#15690722C2211yescharacteristic, complemented
51C4×#11715895C2211yescharacteristic, complemented
52#125×C97C2211yescharacteristic, complemented
53C4×D4×C97C2211yescharacteristic
54#107×C97C2211yescharacteristic, complemented
55#106×C97C2211yescharacteristic, complemented
56C4⋊C8×C97C2211yescharacteristic, complemented
57#102×C97C2211yescharacteristic
58#102×C97C2211yescharacteristic
59C23×D97—21no—
60#13095510—22no—
61#13095510—41no—
62C2×Dic194C2311yescharacteristic
63C2×D388—21no—
64C2×D388—21no—
65C2×D388—21no—
66C2×D388D411yescharacteristic
67C2×D388C2311yescharacteristic
68C2×D388C2311yescharacteristic
69C2×C22⋊D97—21no—
70C2×C22⋊D97—21no—
71D4×D97—22no—
72D4×D97—41no—
73C22×Dic97—21no—
74C2×C4×D97—21no—
75C2×C4×D97—21no—
76C2×C4×D97C2311yescharacteristic
77C2×C4×D97C2311yescharacteristic
78#13095500—41no—
79D776—41no—
80#13095496—22no—
81C2×#11715895C2311yescharacteristic
82C2×#11715895C2311yescharacteristic
83C8×D97—41no—
84C8×D97—41no—
85C4⋊Dic97—21no—
86C4⋊Dic97D411yescharacteristic
87C4⋊Dic97C2311yescharacteristic
88#13095490—21no—
89#13095488—21no—
90#13095488—21no—
91#13095488—21no—
92#13095488—21no—
93#13095488—21no—
94#13095488—21no—
95#13095488—21no—
96#13095488—21no—
97#13095488D411yescharacteristic
98#13095488D411yescharacteristic
99C4×Dic97—21no—
100C2×D4×C97C2311yescharacteristic
101C22×C4×C97—21no—
102C2×C8×C97C2311yescharacteristic
103C2×C8×C97C2311yescharacteristic
104C4⋊C4×C97—21no—
105C4⋊C4×C97—21no—
106C4⋊C4×C97C2311yescharacteristic
107C4⋊C4×C97C2311yescharacteristic
108C4⋊C4×C97C2311yescharacteristic
109C22⋊C4×C97—21no—
110C42×C97C2311yescharacteristic
111C23×C97—21no—
112C22×D97—21no—
113C22×D97—21no—
114C22×D97—42no—
115C22×D97—42no—
116C22×D97—81no—
117C22×D97C2×D411yescharacteristic
118C22×D97C2×D411yescharacteristic
119D4×C97—21no—
120D4×C97#4112yes—
121C2×C4×C97—21no—
122C2×C4×C97—21no—
123C2×C4×C97—21no—
124C2×C4×C97—81no—
125C2×C4×C97C2×D411yescharacteristic
126C2×C4×C97C2×D411yescharacteristic
127C2×C4×C97C2411yescharacteristic
128C22⋊D97—42no—
129C22⋊D97—42no—
130C2×Dic97—21no—
131C2×Dic97—21no—
132C2×Dic97—42no—
133C2×Dic97C2×D411yescharacteristic
134C2×Dic97C2×D411yescharacteristic
135D388—22no—
136D388—21no—
137D388—21no—
138D388—21no—
139D388—21no—
140D388—42no—
141D388—81no—
142D388—81no—
143C4×D97—21no—
144C4×D97—21no—
145C4×D97—21no—
146C4×D97—21no—
147C4×D97—81no—
148C4×D97—81no—
149Dic194—21no—
150Dic194—21no—
151C8×C97—21no—
152C8×C97—21no—
153#11715895—21no—
154#11715895—21no—
155C22×C97—22no—
156C22×C97—41no—
157C22×C97—41no—
158C22×C97C22×D411yescharacteristic
159C2×D97—21no—
160C2×D97—21no—
161C2×D97—21no—
162C2×D97—21no—
163C2×D97—21no—
164C2×D97—41no—
165C2×D97—41no—
166C2×D97—41no—
167C2×D97—41no—
168C2×D97—41no—
169C2×D97—41no—
170C2×D97—84no—
171C2×D97#13512yes—
172C4×C97—21no—
173C4×C97—21no—
174C4×C97—41no—
175C4×C97—41no—
176C4×C97—41no—
177C4×C97C2×#4111yescharacteristic
178C4×C97H(5,𝔽2)11yescharacteristic
179Dic97—21no—
180Dic97—21no—
181Dic97—41no—
182Dic97—41no—
183C2×C97—42no—
184C2×C97#54611yescharacteristic
185C2×C97C2×#13511yescharacteristic
186C2×C97#57611yescharacteristic
187D97—22no—
188D97—41no—
189D97—81no—
190D97—81no—
191#3297—971nomaximal, Sylow
192C97#329711yescharacteristic, complemented, Sylow
193#553—971no—
194#546—971no—
195#488—971no—
196#484—971no—
197#482—971no—
198#478D9711yescharacteristic, complemented
199#466—971no—
200#465—971no—
201#463—971no—
202#449—971no—
203#447—971no—
204C4×D8—971no—
205C8×D4—971no—
206#416—971no—
207C2×#102—971no—
208C2×#41—971no—
209C22×D4—971no—
210C2×SD16—971no—
211C2×D8—971no—
212C2×D8—971no—
213C22×C8—971no—
214C22×C8—971no—
215#126—1941no—
216#125C2×D9711yescharacteristic, complemented
217#124—971no—
218#123—1941no—
219#123—1941no—
220C22⋊Q8—971no—
221#121—971no—
222#121—971no—
223#121—971no—
224#121—1941no—
225#121—1941no—
226#120—1941no—
227C4×D4—971no—
228C4×D4—971no—
229C4×D4—1941no—
230C4×D4C2×D9711yescharacteristic
231#117—971no—
232C2×C4⋊C4—971no—
233C2×C22⋊C4—1941no—
234#107—971no—
235#107C2×D9711yescharacteristic, complemented
236#106C2×D9711yescharacteristic, complemented
237C4⋊C8C2×D9711yescharacteristic, complemented
238#103—971no—
239#103—971no—
240#103—971no—
241#102—971no—
242#102—971no—
243#102—971no—
244#102—971no—
245#102—971no—
246#102C2×D9711yescharacteristic
247#102C2×D9711yescharacteristic
248C22⋊C8—971no—
249C22⋊C8—971no—
250C4×C8—971no—
251C24—1941no—
252#41—3881no—
253#41—1942no—
254C2×Q8—971no—
255C2×D4—971no—
256C2×D4—971no—
257C2×D4—971no—
258C2×D4—3881no—
259C2×D4—1941no—
260C2×D4—1941no—
261C2×D4—1941no—
262C2×D4—1941no—
263C2×D4—1941no—
264C2×D4—1942no—
265C2×D4C22×D9711yescharacteristic
266C22×C4—21no—
267C22×C4—971no—
268C22×C4—971no—
269C22×C4—1941no—
270C22×C4—1941no—
271C22×C4—1941no—
272SD16—3881no—
273D8—3881no—
274D8—1942no—
275C2×C8—971no—
276C2×C8—971no—
277C2×C8—3881no—
278C2×C8—3881no—
279C2×C8C22×D9711yescharacteristic
280C2×C8C22×D9711yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—971no—
284C4⋊C4—971no—
285C4⋊C4—1941no—
286C4⋊C4—1941no—
287C4⋊C4C22×D9711yescharacteristic
288C4⋊C4C22×D9711yescharacteristic
289C4⋊C4C22×D9711yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—971no—
292C22⋊C4—971no—
293C22⋊C4—1941no—
294C22⋊C4—1941no—
295C22⋊C4—1941no—
296C22⋊C4—1941no—
297C22⋊C4—1941no—
298C22⋊C4—1941no—
299C22⋊C4—1941no—
300C22⋊C4—1941no—
301C42—1941no—
302C42C22×D9711yescharacteristic
303C23—21no—
304C23—7761no—
305C23—971no—
306C23—971no—
307C23—3882no—
308C23—3882no—
309C23—1941no—
310C23—1941no—
311Q8—1941no—
312Q8—1941no—
313D4—21no—
314D4—7761no—
315D4—7761no—
316D4—3882no—
317D4—3882no—
318D4—3882no—
319D4—1942no—
320D4—1941no—
321D4—1941no—
322D4—1941no—
323D4—1941no—
324D4#1309551212yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—81no—
329C2×C4—7761no—
330C2×C4—7761no—
331C2×C4—971no—
332C2×C4—971no—
333C2×C4—3882no—
334C2×C4—1941no—
335C2×C4—1941no—
336C2×C4—1941no—
337C2×C4—1941no—
338C2×C4—1941no—
339C2×C4—1941no—
340C2×C4D4×D9711yescharacteristic
341C2×C4D4×D9711yescharacteristic
342C2×C4C23×D9711yescharacteristic
343C8—21no—
344C8—21no—
345C8—1941no—
346C8—1941no—
347C22—22no—
348C22—41no—
349C22—41no—
350C22—7764no—
351C22—972no—
352C22—3881no—
353C22—3881no—
354C22—3881no—
355C22—3881no—
356C22—3881no—
357C22—3881no—
358C22—1941no—
359C22—1941no—
360C22—1941no—
361C22—1941no—
362C22—1941no—
363C22C2×D4×D9711yescharacteristic
364C4—21no—
365C4—21no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—3881no—
370C4—3881no—
371C4—1941no—
372C4—1941no—
373C4C2×#1309551211yescharacteristic
374C4#1569085611yescharacteristic
375C2—42no—
376C2—7761no—
377C2—7761no—
378C2—3881no—
379C2—1942no—
380C2#2026137611yescharacteristic
381C2#135×D9711yescharacteristic
382C2#2026153911yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#478×C97C2
socleC22×C97C22×D4
hypercenter#478D97
hypocenterC97#3297
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #20261438C2semidirect
2. #20261380C2semidirect
3. #20261092C2semidirect
4. #20261072C2semidirect
5. #20261061C2semidirect
6. #20260972C2semidirect
7. #20260965C2semidirect
8. #20260951C2semidirect
9. #20260881C2semidirect
10. #20260870C2semidirect
11. C4×#13095496C2semidirect
12. #20260800C2semidirect
13. #20260728C2semidirect
14. #102×D97C2semidirect
15. #478×C97C2semidirect
18. C2×#13095500C22semidirect
22. C2×C8×D97C22semidirect
31. #15690785C22semidirect
35. C4×D388C22semidirect
37. #15690768C22semidirect
40. #15690743C22semidirect
41. #15690735C22semidirect
42. #15690733C22semidirect
43. #15690733C22semidirect
48. #15690730C22semidirect
50. #15690722C22semidirect
51. C4×#11715895C22semidirect
52. #125×C97C22semidirect
54. #107×C97C22semidirect
55. #106×C97C22semidirect
56. C4⋊C8×C97C22semidirect
192. C97#3297semidirect
198. #478D97semidirect
216. #125C2×D97semidirect
235. #107C2×D97semidirect
236. #106C2×D97semidirect
237. C4⋊C8C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202614382
#202613802
#202610922
#202610722
#202610612
#202609722
#202609652
#202609512
#202608812
#202608702
C4×#130954962
#202608002
#202607282
#102×D972
#478×C972
#329797