Finite Group Databasebeta

← #27153550

Subgroups — #27153550

Order 12416.

494 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 139 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
1#20261389C211yesmaximal, characteristic, complemented
2D4×D388C211yesmaximal, characteristic, complemented
3#20261082C211yesmaximal, characteristic, complemented
4#20261049C212yesmaximal, complemented
5#20261016C211yesmaximal, characteristic, complemented
6#20260971C212yesmaximal, complemented
7#20260931C211yesmaximal, characteristic, complemented
8#20260871C212yesmaximal, complemented
9C4×D776C211yesmaximal, characteristic, complemented
10C2×#15690732C212yesmaximal, complemented
11C8×D4×C97C211yesmaximal, characteristic, complemented
12C2×#13095510—21no—
13C22×D388C2212yes—
14C2×D4×D97—21no—
15C2×D776C2211yescharacteristic
16C2×D776C2212yescomplemented
17#15690810C2211yescharacteristic, complemented
18#15690807C2211yescharacteristic
19#15690794—22no—
20#15690790—21no—
21#15690785—21no—
22#15690784C2212yes—
23#15690784C2212yescomplemented
24#15690780—22no—
25C4×D388C2211yescharacteristic
26C4×D388C2211yescharacteristic, complemented
27C2×C4⋊Dic97C2212yescomplemented
28#15690744C2212yescomplemented
29#15690744C2211yescharacteristic, complemented
30#15690732C2212yes—
31#15690732C2211yescharacteristic
32#15690732C2212yescomplemented
33#15690732C2212yescomplemented
34#15690732C2211yescharacteristic, complemented
35#15690732C2212yescomplemented
36C22×C8×C97C2212yescomplemented
37C4×D4×C97C2211yescharacteristic
38C4⋊C8×C97C2211yescharacteristic, complemented
39C22⋊C8×C97C2212yescomplemented
40C4×C8×C97C2211yescharacteristic, complemented
41C23×D97—22no—
42#13095510—44no—
43C2×Dic194—21no—
44C2×D388—24no—
45C2×D388—22no—
46C2×D388—21no—
47C2×D388—21no—
48C2×D388—42no—
49C2×D388C2311yescharacteristic
50C2×D388C2312yes—
51C2×D388C2311yescharacteristic
52C2×C22⋊D97—22no—
53C2×C22⋊D97—22no—
54D4×D97—44no—
55C22×Dic97—22no—
56C2×C4×D97—21no—
57C2×C4×D97—21no—
58D776—22no—
59D776—42no—
60C4⋊Dic97—22no—
61C4⋊Dic97—21no—
62C4⋊Dic97C2311yescharacteristic
63C4⋊Dic97C2312yes—
64C4⋊Dic97C2311yescharacteristic
65#13095488—21no—
66#13095488—21no—
67#13095488—22no—
68#13095488—22no—
69C2×D4×C97D411yescharacteristic
70C22×C4×C97C2312yes—
71C2×C8×C97—42no—
72C2×C8×C97C2311yescharacteristic
73C2×C8×C97C2312yes—
74C2×C8×C97C2311yescharacteristic
75C4⋊C4×C97D411yescharacteristic
76C22⋊C4×C97D412yes—
77C42×C97C2311yescharacteristic
78C23×C97C2×D412yes—
79C22×D97—21no—
80C22×D97—21no—
81C22×D97—22no—
82C22×D97—44no—
83C22×D97—44no—
84C22×D97—82no—
85D4×C97D814yescomplemented
86C2×C4×C97—22no—
87C2×C4×C97—22no—
88C2×C4×C97C2×D411yescharacteristic
89C2×C4×C97C2×D412yes—
90C2×C4×C97C2×D411yescharacteristic
91C2×C4×C97C2411yescharacteristic
92C22⋊D97—44no—
93C22⋊D97—44no—
94C2×Dic97—21no—
95C2×Dic97—21no—
96C2×Dic97—22no—
97C2×Dic97—44no—
98D388—21no—
99D388—22no—
100D388—21no—
101D388—21no—
102D388—22no—
103D388—44no—
104D388—42no—
105D388—81no—
106D388#4112yescomplemented
107C4×D97—42no—
108C4×D97—42no—
109Dic194—42no—
110C8×C97—21no—
111C8×C97—21no—
112C8×C97—22no—
113C22×C97—22no—
114C22×C97—22no—
115C22×C97C2×D814yes—
116C22×C97C22×D411yescharacteristic
117C2×D97—22no—
118C2×D97—41no—
119C2×D97—41no—
120C2×D97—42no—
121C2×D97—41no—
122C2×D97—41no—
123C2×D97—42no—
124C2×D97—42no—
125C2×D97—41no—
126C2×D97—88no—
127C4×C97—22no—
128C4×C97—21no—
129C4×C97C2×D812yes—
130C4×C97C2×#4111yescharacteristic
131C4×C97H(5,𝔽2)11yescharacteristic
132Dic97—42no—
133Dic97—41no—
134Dic97—41no—
135C2×C97—24no—
136C2×C97#54611yescharacteristic
137C2×C97C22×D811yescharacteristic
138C2×C97#57711yescharacteristic
139D97—42no—
140D97—81no—
141D97—82no—
142#3294—971nomaximal, Sylow
143C97#329411yescharacteristic, complemented, Sylow
144#547—971no—
145D42—971no—
146#486—971no—
147#480—972no—
148#474—971no—
149#466—972no—
150#459—971no—
151#447—972no—
152C4×D8—971no—
153C8×D4D9711yescharacteristic, complemented
154C2×#102—972no—
155C2×#41—1941no—
156C22×D4—972no—
157C22×D4—1941no—
158C2×D8—972no—
159C2×D8—971no—
160C22×C8C2×D9712yescomplemented
161C4⋊Q8—971no—
162#127—971no—
163#123—1942no—
164C22⋊Q8—1941no—
165#121—972no—
166#121—972no—
167#121—1941no—
168#120—1942no—
169C4×D4—971no—
170C4×D4—971no—
171C4×D4C2×D9711yescharacteristic
172C2×C4⋊C4—972no—
173#107—971no—
174#107—972no—
175C4⋊C8C2×D9711yescharacteristic, complemented
176#102—971no—
177#102—972no—
178#102—972no—
179#102—972no—
180#102—971no—
181#102—972no—
182C22⋊C8C2×D9712yescomplemented
183C4×C8C2×D9711yescharacteristic, complemented
184C24—1942no—
185#41—3884no—
186C2×Q8—1941no—
187C2×D4—971no—
188C2×D4—972no—
189C2×D4—971no—
190C2×D4—3882no—
191C2×D4—3884no—
192C2×D4—1944no—
193C2×D4—1942no—
194C2×D4—1942no—
195C2×D4—1941no—
196C2×D4—1942no—
197C2×D4—1941no—
198C2×D4D38811yescharacteristic
199C22×C4—1942no—
200C22×C4—1941no—
201C22×C4—1941no—
202C22×C4C22×D9712yes—
203D8—3882no—
204D8—1942no—
205C2×C8—42no—
206C2×C8C22×D9711yescharacteristic
207C2×C8C22×D9711yescharacteristic
208C2×C8C22×D9712yes—
209C4⋊C4—971no—
210C4⋊C4—972no—
211C4⋊C4—971no—
212C4⋊C4—1942no—
213C4⋊C4—1941no—
214C4⋊C4D38811yescharacteristic
215C22⋊C4—1941no—
216C22⋊C4—1941no—
217C22⋊C4—1942no—
218C22⋊C4—1942no—
219C22⋊C4D38812yes—
220C42C22×D9711yescharacteristic
221C23—7762no—
222C23—3884no—
223C23—3884no—
224C23—1941no—
225C23—1942no—
226C23—1941no—
227C23C2×D38812yes—
228Q8—3882no—
229D4—7761no—
230D4—972no—
231D4—3884no—
232D4—3884no—
233D4—3884no—
234D4—3882no—
235D4—1941no—
236D4—1942no—
237D4—1941no—
238D4—1942no—
239D4—1941no—
240D4D77614yescomplemented
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—3884no—
244C2×C4—3882no—
245C2×C4—3882no—
246C2×C4—1941no—
247C2×C4—1941no—
248C2×C4—1942no—
249C2×C4C2×D38811yescharacteristic
250C2×C4C2×D38812yes—
251C2×C4C2×D38811yescharacteristic
252C2×C4C23×D9711yescharacteristic
253C8—21no—
254C8—21no—
255C8—22no—
256C22—22no—
257C22—22no—
258C22—7768no—
259C22—3882no—
260C22—3881no—
261C22—3881no—
262C22—3882no—
263C22—3882no—
264C22—3881no—
265C22—3881no—
266C22—3881no—
267C22—1942no—
268C22C2×D77614yes—
269C22C22×D38811yescharacteristic
270C4—22no—
271C4—21no—
272C4—3881no—
273C4—3881no—
274C4—3882no—
275C4C2×D77612yes—
276C4#41×D9711yescharacteristic
277C4#1569085511yescharacteristic
278C2—24no—
279C2—7762no—
280C2—7761no—
281C2—3882no—
282C2#2026138011yescharacteristic
283C2C22×D77611yescharacteristic
284C2#2026154711yescharacteristic
subgroupG/H
centerC22C22×D388
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC8×D4×C97C2
socleC22×C97C22×D4
hypercenterC8×D4D97
hypocenterC97#3294
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #20261389C2semidirect
2. D4×D388C2semidirect
3. #20261082C2semidirect
4. #20261049C2semidirect
5. #20261016C2semidirect
6. #20260971C2semidirect
7. #20260931C2semidirect
8. #20260871C2semidirect
9. C4×D776C2semidirect
10. C2×#15690732C2semidirect
11. C8×D4×C97C2semidirect
16. C2×D776C22semidirect
17. #15690810C22semidirect
23. #15690784C22semidirect
26. C4×D388C22semidirect
27. C2×C4⋊Dic97C22semidirect
28. #15690744C22semidirect
29. #15690744C22semidirect
32. #15690732C22semidirect
33. #15690732C22semidirect
34. #15690732C22semidirect
35. #15690732C22semidirect
36. C22×C8×C97C22semidirect
38. C4⋊C8×C97C22semidirect
39. C22⋊C8×C97C22semidirect
40. C4×C8×C97C22semidirect
85. D4×C97D8semidirect
106. D388#41semidirect
143. C97#3294semidirect
153. C8×D4D97semidirect
160. C22×C8C2×D97semidirect
175. C4⋊C8C2×D97semidirect
182. C22⋊C8C2×D97semidirect
183. C4×C8C2×D97semidirect
240. D4D776semidirect

Maximal subgroups

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

subgroupindex
#202613892
D4×D3882
#202610822
#202610492
#202610162
#202609712
#202609312
#202608712
C4×D7762
C2×#156907322
C8×D4×C972
#329497