Finite Group Databasebeta

← #27153580

Subgroups — #27153580

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#20261381C211yesmaximal, characteristic, complemented
2#20261092C211yesmaximal, characteristic, complemented
3#20261070C211yesmaximal, characteristic, complemented
4#20261059C211yesmaximal, characteristic, complemented
5#20261032C211yesmaximal, characteristic, complemented
6#20260969C211yesmaximal, characteristic, complemented
7#20260962C211yesmaximal, characteristic, complemented
8#20260950C211yesmaximal, characteristic, complemented
9#20260878C211yesmaximal, characteristic, complemented
10#20260868C211yesmaximal, characteristic, complemented
11C4×#13095496C211yesmaximal, characteristic, complemented
12D4×#11715895C211yesmaximal, characteristic, complemented
13#20260729C211yesmaximal, characteristic, complemented
14C2×#15690730C211yesmaximal, characteristic, complemented
15#553×C97C211yesmaximal, characteristic, complemented
16C2×#13095511C2211yescharacteristic
17C22×D388C2211yescharacteristic
18C2×#13095499C2211yescharacteristic, complemented
19C2×#13095496C2211yescharacteristic
20C2×#13095496C2211yescharacteristic
21C22×#11715895C2211yescharacteristic
22C22×#11715895C2211yescharacteristic, complemented
23#15690798C2211yescharacteristic
24#15690795—21no—
25#15690789C2211yescharacteristic
26#15690784C2211yescharacteristic
27#15690783—21no—
28#15690778—21no—
29C4×D388C2211yescharacteristic
30C22⋊C4×D97—21no—
31#15690743C2211yescharacteristic, complemented
32#15690743C2211yescharacteristic, complemented
33#15690741C2211yescharacteristic, complemented
34C4⋊#11715895C2211yescharacteristic, complemented
35#15690733C2211yescharacteristic, complemented
36#15690733C2211yescharacteristic, complemented
37#15690733C2211yescharacteristic, complemented
38#15690731C2211yescharacteristic, complemented
39#15690730C2211yescharacteristic
40#15690730C2211yescharacteristic
41#15690730C2211yescharacteristic
42#15690730C2211yescharacteristic
43#15690730C2211yescharacteristic
44#15690730C2211yescharacteristic
45C22⋊#11715895C2211yescharacteristic
46C22⋊#11715895C2211yescharacteristic, complemented
47C4×#11715895C2211yescharacteristic, complemented
48#126×C97—21no—
49#125×C97C2211yescharacteristic, complemented
50#123×C97—21no—
51#121×C97—21no—
52#121×C97C2211yescharacteristic
53#121×C97C2211yescharacteristic, complemented
54C4×D4×C97—21no—
55C4×D4×C97C2211yescharacteristic
56C4×D4×C97C2211yescharacteristic, complemented
57#117×C97C2211yescharacteristic, complemented
58C2×C4⋊C4×C97C2211yescharacteristic
59C23×D97—21no—
60#13095511—22no—
61#13095511—41no—
62C2×Dic194C2311yescharacteristic
63C2×D388—22no—
64C2×D388—21no—
65C2×D388—41no—
66C2×D388C2311yescharacteristic
67C2×D388C2311yescharacteristic
68C2×C22⋊D97—21no—
69C2×C22⋊D97—21no—
70C2×C4×D97—21no—
71C2×C4×D97—21no—
72#13095499—41no—
73#13095496—22no—
74#13095496—41no—
75C2×#11715895—41no—
76C2×#11715895—41no—
77C2×#11715895C2311yescharacteristic
78C2×#11715895C2311yescharacteristic
79C2×#11715895C2311yescharacteristic
80C2×#11715895C2311yescharacteristic
81C4⋊Dic97C2311yescharacteristic
82#13095490—21no—
83C22⋊Dic97—21no—
84#13095488—21no—
85#13095488—21no—
86#13095488—21no—
87#13095488—21no—
88C2×D4×C97—21no—
89C2×D4×C97—21no—
90C2×D4×C97D411yescharacteristic
91C2×D4×C97C2311yescharacteristic
92C22×C4×C97—21no—
93C22×C4×C97—21no—
94C22×C4×C97C2311yescharacteristic
95C22×C4×C97C2311yescharacteristic
96C4⋊C4×C97—21no—
97C4⋊C4×C97—21no—
98C4⋊C4×C97—21no—
99C4⋊C4×C97D411yescharacteristic
100C4⋊C4×C97C2311yescharacteristic
101C4⋊C4×C97C2311yescharacteristic
102C4⋊C4×C97C2311yescharacteristic
103C22⋊C4×C97—21no—
104C22⋊C4×C97—21no—
105C22⋊C4×C97—21no—
106C22⋊C4×C97—21no—
107C22⋊C4×C97D411yescharacteristic
108C22⋊C4×C97D411yescharacteristic
109C42×C97—21no—
110C42×C97C2311yescharacteristic
111C23×C97—21no—
112C23×C97C2×D411yescharacteristic
113C23×C97C2×D411yescharacteristic
114C22×D97—21no—
115C22×D97—21no—
116C22×D97—42no—
117C22×D97—42no—
118C22×D97—81no—
119D4×C97—22no—
120D4×C97—21no—
121D4×C97—21no—
122D4×C97—42no—
123D4×C97—81no—
124C2×C4×C97—21no—
125C2×C4×C97—21no—
126C2×C4×C97—21no—
127C2×C4×C97—21no—
128C2×C4×C97—21no—
129C2×C4×C97—21no—
130C2×C4×C97—21no—
131C2×C4×C97—81no—
132C2×C4×C97—81no—
133C2×C4×C97C2×D411yescharacteristic
134C2×C4×C97C2×D411yescharacteristic
135C2×C4×C97C2×D411yescharacteristic
136C2×C4×C97C2×D411yescharacteristic
137C2×C4×C97C2411yescharacteristic
138C22⋊D97—42no—
139C22⋊D97—81no—
140C2×Dic97—21no—
141C2×Dic97—21no—
142D388—21no—
143D388—21no—
144D388—21no—
145D388—42no—
146D388#4112yes—
147C4×D97—42no—
148C4×D97—81no—
149Dic194—21no—
150Dic194—21no—
151#11715895—21no—
152#11715895—21no—
153#11715895—21no—
154#11715895—21no—
155C22×C97—21no—
156C22×C97—21no—
157C22×C97—21no—
158C22×C97—21no—
159C22×C97—21no—
160C22×C97—41no—
161C22×C97—41no—
162C22×C97—41no—
163C22×C97#13512yes—
164C22×C97C22×D411yescharacteristic
165C2×D97—22no—
166C2×D97—41no—
167C2×D97—41no—
168C2×D97—41no—
169C2×D97—41no—
170C2×D97—41no—
171C2×D97—84no—
172C4×C97—21no—
173C4×C97—21no—
174C4×C97—21no—
175C4×C97—21no—
176C4×C97—41no—
177C4×C97—41no—
178C4×C97—41no—
179C4×C97C2×#4111yescharacteristic
180C4×C97H(5,𝔽2)11yescharacteristic
181Dic97—41no—
182Dic97—41no—
183C2×C97—22no—
184C2×C97—41no—
185C2×C97—81no—
186C2×C97#54611yescharacteristic
187C2×C97C2×#13511yescharacteristic
188C2×C97#57611yescharacteristic
189D97—42no—
190D97—81no—
191#3297—971nomaximal, Sylow
192C97#329711yescharacteristic, complemented, Sylow
193#553D9711yescharacteristic, complemented
194#546—971no—
195#488—971no—
196#484—971no—
197#482—971no—
198#478—971no—
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—21no—
216#125C2×D9711yescharacteristic, complemented
217#124—971no—
218#123—21no—
219#123—1941no—
220C22⋊Q8—971no—
221#121—21no—
222#121—971no—
223#121—1941no—
224#121C2×D9711yescharacteristic
225#121C2×D9711yescharacteristic, complemented
226#120—1941no—
227C4×D4—21no—
228C4×D4—971no—
229C4×D4C2×D9711yescharacteristic
230C4×D4C2×D9711yescharacteristic, complemented
231#117C2×D9711yescharacteristic, complemented
232C2×C4⋊C4C2×D9711yescharacteristic
233C2×C22⋊C4—1941no—
234#107—971no—
235#107—971no—
236#106—971no—
237C4⋊C8—971no—
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#102—971no—
247#102—971no—
248C22⋊C8—971no—
249C22⋊C8—971no—
250C4×C8—971no—
251C24—1941no—
252#41—3881no—
253#41—1942no—
254C2×Q8—971no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—971no—
258C2×D4—971no—
259C2×D4—3881no—
260C2×D4—1942no—
261C2×D4—1941no—
262C2×D4—1941no—
263C2×D4—1941no—
264C2×D4C22⋊D9711yescharacteristic
265C2×D4C22×D9711yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—1941no—
269C22×C4—1941no—
270C22×C4C22×D9711yescharacteristic
271C22×C4C22×D9711yescharacteristic
272SD16—3881no—
273D8—3881no—
274D8—1942no—
275C2×C8—971no—
276C2×C8—971no—
277C2×C8—971no—
278C2×C8—971no—
279C2×C8—3881no—
280C2×C8—3881no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—971no—
285C4⋊C4—1941no—
286C4⋊C4C22⋊D9711yescharacteristic
287C4⋊C4C22×D9711yescharacteristic
288C4⋊C4C22×D9711yescharacteristic
289C4⋊C4C22×D9711yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—1941no—
295C22⋊C4—1941no—
296C22⋊C4—1941no—
297C22⋊C4—1941no—
298C22⋊C4—1941no—
299C22⋊C4C22⋊D9711yescharacteristic
300C22⋊C4C22⋊D9711yescharacteristic
301C42—21no—
302C42C22×D9711yescharacteristic
303C23—21no—
304C23—7761no—
305C23—3882no—
306C23—3882no—
307C23—1941no—
308C23—1941no—
309C23C2×C22⋊D9711yescharacteristic
310C23C2×C22⋊D9711yescharacteristic
311Q8—1941no—
312Q8—1941no—
313D4—22no—
314D4—21no—
315D4—21no—
316D4—42no—
317D4—81no—
318D4—7761no—
319D4—972no—
320D4—3882no—
321D4—3882no—
322D4—1941no—
323D4—1941no—
324D4—1941no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—7761no—
335C2×C4—3882no—
336C2×C4—1941no—
337C2×C4—1941no—
338C2×C4C2×C22⋊D9711yescharacteristic
339C2×C4C2×C22⋊D9711yescharacteristic
340C2×C4C2×C22⋊D9711yescharacteristic
341C2×C4C2×C22⋊D9711yescharacteristic
342C2×C4C23×D9711yescharacteristic
343C8—1941no—
344C8—1941no—
345C8—1941no—
346C8—1941no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—7764no—
356C22—3881no—
357C22—3881no—
358C22—3881no—
359C22—3881no—
360C22—3881no—
361C22—1942no—
362C22#1569083012yes—
363C22C22×C22⋊D9711yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—3881no—
372C4—3881no—
373C4#41×D9711yescharacteristic
374C4#1569085611yescharacteristic
375C2—22no—
376C2—41no—
377C2—81no—
378C2—7761no—
379C2—3882no—
380C2#2026137311yescharacteristic
381C2C2×#1569083011yescharacteristic
382C2#2026153811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#553×C97C2
socleC22×C97C22×D4
hypercenter#553D97
hypocenterC97#3297
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. #20261381C2semidirect
2. #20261092C2semidirect
3. #20261070C2semidirect
4. #20261059C2semidirect
5. #20261032C2semidirect
6. #20260969C2semidirect
7. #20260962C2semidirect
8. #20260950C2semidirect
9. #20260878C2semidirect
10. #20260868C2semidirect
11. C4×#13095496C2semidirect
12. D4×#11715895C2semidirect
13. #20260729C2semidirect
14. C2×#15690730C2semidirect
15. #553×C97C2semidirect
18. C2×#13095499C22semidirect
22. C22×#11715895C22semidirect
31. #15690743C22semidirect
32. #15690743C22semidirect
33. #15690741C22semidirect
34. C4⋊#11715895C22semidirect
35. #15690733C22semidirect
36. #15690733C22semidirect
37. #15690733C22semidirect
38. #15690731C22semidirect
46. C22⋊#11715895C22semidirect
47. C4×#11715895C22semidirect
49. #125×C97C22semidirect
53. #121×C97C22semidirect
56. C4×D4×C97C22semidirect
57. #117×C97C22semidirect
192. C97#3297semidirect
193. #553D97semidirect
216. #125C2×D97semidirect
225. #121C2×D97semidirect
230. C4×D4C2×D97semidirect
231. #117C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202613812
#202610922
#202610702
#202610592
#202610322
#202609692
#202609622
#202609502
#202608782
#202608682
C4×#130954962
D4×#117158952
#202607292
C2×#156907302
#553×C972
#329797