Finite Group Databasebeta

← #27153356

Subgroups — #27153356

Order 12416.

556 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690835C212yesmaximal, complemented
2D4×D388C211yesmaximal, characteristic, complemented
3#20261111C211yesmaximal, characteristic, complemented
4#20260986C212yesmaximal, complemented
5#20260935C211yesmaximal, characteristic, complemented
6#20260927C211yesmaximal, characteristic, complemented
7#20260880C212yesmaximal, complemented
8#20260868C212yesmaximal, complemented
9C4×#13095498C211yesmaximal, characteristic, complemented
10#20260813C211yesmaximal, characteristic, complemented
11#550×C97C211yesmaximal, characteristic, complemented
12C22×D388C2212yes—
13C2×D4×D97—21no—
14#15690835—28no—
15C2×#13095498C2211yescharacteristic
16C2×#13095498C2212yescomplemented
17C2×#13095496C2212yes—
18C2×#13095496C2212yescomplemented
19C2×#13095496C2212yescomplemented
20C2×#13095495C2212yescomplemented
21#15690807C2211yescharacteristic
22#15690785—21no—
23#15690784C2212yes—
24#15690780—22no—
25C4×D388C2211yescharacteristic
26#15690741C2211yescharacteristic, complemented
27C4⋊#11715895C2211yescharacteristic, complemented
28#15690734C2211yescharacteristic, complemented
29#15690731C2212yescomplemented
30#15690730C2211yescharacteristic
31#15690730C2212yescomplemented
32C22⋊#11715895C2212yescomplemented
33C4×#11715895C2211yescharacteristic, complemented
34C2×#41×C97C2212yescomplemented
35#127×C97—21no—
36#127×C97C2211yescharacteristic, complemented
37#121×C97—22no—
38#121×C97C2212yescomplemented
39C4×Q8×C97C2211yescharacteristic, complemented
40C4×D4×C97—21no—
41C4×D4×C97C2211yescharacteristic
42C23×D97—22no—
43C2×D388—24no—
44C2×D388—22no—
45C2×D388—21no—
46C2×D388—21no—
47C2×D388—42no—
48C2×D388C2312yes—
49C2×D388C2311yescharacteristic
50C2×C22⋊D97—22no—
51D4×D97—44no—
52C2×C4×D97—21no—
53#13095498—24no—
54#13095498D414yescomplemented
55#13095496—24no—
56#13095496—24no—
57#13095496—42no—
58#13095495—24no—
59C2×#11715895C2312yes—
60C2×#11715895C2311yescharacteristic
61C2×#11715895C2311yescharacteristic
62C4⋊Dic97C2311yescharacteristic
63#13095488—21no—
64#13095488—22no—
65#41×C97—24no—
66#41×C97—42no—
67C2×Q8×C97C2311yescharacteristic
68C2×D4×C97—22no—
69C2×D4×C97—22no—
70C2×D4×C97—21no—
71C2×D4×C97—21no—
72C2×D4×C97D411yescharacteristic
73C2×D4×C97C2312yes—
74C22×C4×C97—22no—
75C22×C4×C97C2312yes—
76C4⋊C4×C97—21no—
77C4⋊C4×C97D411yescharacteristic
78C4⋊C4×C97C2311yescharacteristic
79C22⋊C4×C97—22no—
80C22⋊C4×C97D412yes—
81C42×C97—21no—
82C42×C97C2311yescharacteristic
83C23×C97—22no—
84C23×C97C2×D412yes—
85C22×D97—21no—
86C22×D97—22no—
87C22×D97—44no—
88C22×D97—44no—
89C22×D97—82no—
90Q8×C97—21no—
91Q8×C97C2×D412yescomplemented
92D4×C97—22no—
93D4×C97—22no—
94D4×C97—22no—
95D4×C97—44no—
96D4×C97—42no—
97D4×C97—81no—
98D4×C97—82no—
99C2×C4×C97—21no—
100C2×C4×C97—22no—
101C2×C4×C97—22no—
102C2×C4×C97—21no—
103C2×C4×C97—44no—
104C2×C4×C97C2×D411yescharacteristic
105C2×C4×C97C2×D412yes—
106C2×C4×C97C2×D411yescharacteristic
107C2×C4×C97C2411yescharacteristic
108C22⋊D97—44no—
109C2×Dic97—21no—
110D388—22no—
111D388—21no—
112D388—22no—
113D388—44no—
114D388—42no—
115D388—81no—
116D388C2×D412yes—
117C4×D97—42no—
118#11715895—22no—
119#11715895—21no—
120#11715895C2×D412yescomplemented
121C22×C97—22no—
122C22×C97—22no—
123C22×C97—22no—
124C22×C97—42no—
125C22×C97—42no—
126C22×C97—42no—
127C22×C97C22×D411yescharacteristic
128C2×D97—22no—
129C2×D97—41no—
130C2×D97—41no—
131C2×D97—42no—
132C2×D97—42no—
133C2×D97—42no—
134C2×D97—88no—
135C4×C97—22no—
136C4×C97—22no—
137C4×C97—21no—
138C4×C97—41no—
139C4×C97AGL(1,ℤ/8)12yescomplemented
140C4×C97C22×D411yescharacteristic
141C4×C97H(5,𝔽2)11yescharacteristic
142Dic97—41no—
143C2×C97—42no—
144C2×C97—82no—
145C2×C97D4211yescharacteristic
146C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C97#57611yescharacteristic
148D97—42no—
149D97—82no—
150#3274—971nomaximal, Sylow
151C97#327411yescharacteristic, complemented, Sylow
152C2×AGL(1,ℤ/8)—972no—
153#550D9711yescharacteristic, complemented
154D42—971no—
155#493—971no—
156#469—972no—
157#460—971no—
158#459—971no—
159#449—972no—
160#447—972no—
161C4×SD16—971no—
162#436—971no—
163C2×#41C2×D9712yescomplemented
164C22×D4—972no—
165C22×D4—1941no—
166AGL(1,ℤ/8)—1948no—
167C2×SD16—971no—
168C2×SD16—972no—
169C2×D8—972no—
170C2×D8—972no—
171C2×D8—972no—
172C2×M16—972no—
173#127—21no—
174#127—971no—
175#127C2×D9711yescharacteristic, complemented
176#121—22no—
177#121—972no—
178#121—1941no—
179#121C2×D9712yescomplemented
180#120—1942no—
181C4×Q8C2×D9711yescharacteristic, complemented
182C4×D4—21no—
183C4×D4—971no—
184C4×D4C2×D9711yescharacteristic
185#106—971no—
186C4⋊C8—971no—
187#103—971no—
188#102—972no—
189#102—971no—
190#102—972no—
191C22⋊C8—972no—
192C4×C8—971no—
193C24—1942no—
194#41—24no—
195#41—42no—
196C2×Q8C22×D9711yescharacteristic
197C2×D4—22no—
198C2×D4—22no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—971no—
202C2×D4—972no—
203C2×D4—3882no—
204C2×D4—3884no—
205C2×D4—1944no—
206C2×D4—1942no—
207C2×D4—1941no—
208C2×D4—1942no—
209C2×D4—1941no—
210C2×D4C22⋊D9711yescharacteristic
211C2×D4C22×D9712yes—
212C22×C4—22no—
213C22×C4—1941no—
214C22×C4C22×D9712yes—
215SD16—974no—
216SD16—1944no—
217D8—3882no—
218D8—1944no—
219D8—1944no—
220M16—1944no—
221C2×C8—971no—
222C2×C8—971no—
223C2×C8—972no—
224C4⋊C4—21no—
225C4⋊C4—971no—
226C4⋊C4C22⋊D9711yescharacteristic
227C4⋊C4C22×D9711yescharacteristic
228C22⋊C4—22no—
229C22⋊C4—1941no—
230C22⋊C4—1942no—
231C22⋊C4C22⋊D9712yes—
232C42—21no—
233C42C22×D9711yescharacteristic
234C23—22no—
235C23—7762no—
236C23—3884no—
237C23—3884no—
238C23—1942no—
239C23—1941no—
240C23C2×C22⋊D9712yes—
241Q8—21no—
242Q8D4×D9712yescomplemented
243D4—22no—
244D4—22no—
245D4—22no—
246D4—44no—
247D4—42no—
248D4—81no—
249D4—82no—
250D4—7761no—
251D4—972no—
252D4—3884no—
253D4—3884no—
254D4—3882no—
255D4—1942no—
256D4—1942no—
257D4—1941no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—44no—
263C2×C4—3882no—
264C2×C4—1941no—
265C2×C4C2×C22⋊D9711yescharacteristic
266C2×C4C2×C22⋊D9712yes—
267C2×C4C2×C22⋊D9711yescharacteristic
268C2×C4C23×D9711yescharacteristic
269C8—972no—
270C8—1942no—
271C8—1941no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—7768no—
279C22—3882no—
280C22—3882no—
281C22—3882no—
282C22—3881no—
283C22—3881no—
284C22—1942no—
285C22C22×C22⋊D9711yescharacteristic
286C4—21no—
287C4—22no—
288C4—22no—
289C4—41no—
290C4—3881no—
291C4#1569083512yescomplemented
292C4C2×D4×D9711yescharacteristic
293C4#1569085511yescharacteristic
294C2—42no—
295C2—82no—
296C2—7762no—
297C2—3882no—
298C2D4×C22⋊D9711yescharacteristic
299C2C2×#1569083511yescharacteristic
300C2#2026153811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#550×C97C2
socleC22×C97C22×D4
hypercenter#550D97
hypocenterC97#3274
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#15690835C2semidirect
2. D4×D388C2semidirect
3. #20261111C2semidirect
4. #20260986C2semidirect
5. #20260935C2semidirect
6. #20260927C2semidirect
7. #20260880C2semidirect
8. #20260868C2semidirect
9. C4×#13095498C2semidirect
10. #20260813C2semidirect
11. #550×C97C2semidirect
16. C2×#13095498C22semidirect
18. C2×#13095496C22semidirect
19. C2×#13095496C22semidirect
20. C2×#13095495C22semidirect
26. #15690741C22semidirect
27. C4⋊#11715895C22semidirect
28. #15690734C22semidirect
29. #15690731C22semidirect
31. #15690730C22semidirect
32. C22⋊#11715895C22semidirect
33. C4×#11715895C22semidirect
34. C2×#41×C97C22semidirect
36. #127×C97C22semidirect
38. #121×C97C22semidirect
39. C4×Q8×C97C22semidirect
54. #13095498D4semidirect
91. Q8×C97C2×D4semidirect
120. #11715895C2×D4semidirect
139. C4×C97AGL(1,ℤ/8)semidirect
151. C97#3274semidirect
153. #550D97semidirect
163. C2×#41C2×D97semidirect
175. #127C2×D97semidirect
179. #121C2×D97semidirect
181. C4×Q8C2×D97semidirect
242. Q8D4×D97semidirect
291. C4#15690835semidirect

Maximal subgroups

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

subgroupindex
C2×#156908352
D4×D3882
#202611112
#202609862
#202609352
#202609272
#202608802
#202608682
C4×#130954982
#202608132
#550×C972
#327497