Finite Group Databasebeta

← #27152119

Subgroups — #27152119

Order 12416.

400 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 117 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#20261291C211yesmaximal, characteristic, complemented
2#124×D97C211yesmaximal, characteristic
3#20261096C212yesmaximal, complemented
4#20261093C212yesmaximal, complemented
5#20261092C211yesmaximal, characteristic, complemented
6#20261090C211yesmaximal, characteristic, complemented
7#20261089C211yesmaximal, characteristic, complemented
8#20260742C212yesmaximal, complemented
9#20260738C212yesmaximal
10C4×#13095494C211yesmaximal, characteristic, complemented
11#488×C97C211yesmaximal, characteristic, complemented
12C2×Q8×D97C2211yescharacteristic
13C2×D4×D97C2211yescharacteristic
14C2×#13095494C2212yes—
15#15690809C2212yescomplemented
16#15690802C2211yescharacteristic, complemented
17#15690799C2211yescharacteristic
18#15690798C2211yescharacteristic
19#15690797C2211yescharacteristic
20#15690796—22no—
21#15690788—22no—
22#15690775—22no—
23#15690768C2212yes—
24C22⋊C4×D97—22no—
25C42×D97C2211yescharacteristic
26#15690735C2212yescomplemented
27#15690734C2212yescomplemented
28#15690733C2212yescomplemented
29#15690732C2212yes—
30#15690731C2212yes—
31#15690730C2212yes—
32#15690720C2212yescomplemented
33C4×#11715895C2211yescharacteristic, complemented
34#125×C97C2211yescharacteristic, complemented
35#124×C97C2211yescharacteristic
36#103×C97C2212yescomplemented
37#102×C97C2212yes—
38C4×C8×C97C2211yescharacteristic, complemented
39C23×D97—21no—
40C2×Dic194—22no—
41C2×Dic194—21no—
42C2×Dic194C2311yescharacteristic
43Q8×D97—24no—
44C2×D388C2311yescharacteristic
45C2×C22⋊D97—21no—
46D4×D97—24no—
47C2×C4×D97—21no—
48C2×C4×D97D412yes—
49C2×C4×D97C2311yescharacteristic
50#13095494—24no—
51C2×#11715895C2312yes—
52C4⋊Dic97C2312yes—
53#13095490—22no—
54#13095490—22no—
55C22⋊Dic97—22no—
56#13095488—22no—
57#13095488—22no—
58#13095488—22no—
59C4×Dic97—22no—
60C4×Dic97D412yes—
61C4×Dic97C2311yescharacteristic
62C2×Q8×C97C2311yescharacteristic
63C2×D4×C97C2311yescharacteristic
64C2×C8×C97C2312yes—
65C4⋊C4×C97—22no—
66C4⋊C4×C97C2312yes—
67C22⋊C4×C97—22no—
68C42×C97C2311yescharacteristic
69C23×C97—21no—
70C22×D97—21no—
71C22×D97—44no—
72C22×D97—42no—
73C22×D97C2×D411yescharacteristic
74Q8×C97—22no—
75D4×C97—22no—
76C2×C4×C97—21no—
77C2×C4×C97—22no—
78C2×C4×C97C2×D412yes—
79C2×C4×C97C2411yescharacteristic
80C22⋊D97—42no—
81C2×Dic97—22no—
82C2×Dic97—21no—
83C2×Dic97C2×D412yes—
84C2×Dic97C2×D411yescharacteristic
85D388—22no—
86C4×D97—42no—
87C4×D97—42no—
88C4×D97#4114yes—
89Dic194—22no—
90Dic194—44no—
91Dic194—42no—
92C8×C97—22no—
93#11715895—22no—
94C22×C97—41no—
95C22×C97—42no—
96C22×C97C22×D411yescharacteristic
97C2×D97—22no—
98C2×D97—41no—
99C2×D97—42no—
100C2×D97—84no—
101C2×D97#12412yes—
102C4×C97—22no—
103C4×C97—41no—
104C4×C97—42no—
105C4×C97C2×#4112yes—
106Dic97—22no—
107Dic97—41no—
108Dic97—42no—
109Dic97#12412yes—
110C2×C97—81no—
111C2×C97C2×#12411yescharacteristic
112C2×C97#57512yes—
113D97—22no—
114D97—81no—
115#3140—971nomaximal, Sylow
116C97#314011yescharacteristic, complemented, Sylow
117#533—971no—
118C2×#124—971no—
119#489—972no—
120#489—972no—
121#488—971no—
122#488—971no—
123#488—971no—
124#488D9711yescharacteristic, complemented
125#419—972no—
126#418—972no—
127C4×M16—971no—
128C22×Q8—971no—
129C22×D4—971no—
130C2×M16—972no—
131C4⋊Q8—972no—
132#125—971no—
133#125C2×D9711yescharacteristic, complemented
134#124—971no—
135#124—971no—
136#124—971no—
137#124—1942no—
138#124C2×D9711yescharacteristic
139C22⋊Q8—1942no—
140C4×Q8—1942no—
141#117—972no—
142C2×C22⋊C4—1942no—
143C2×C42—971no—
144#103—972no—
145#103—972no—
146#103—972no—
147#103C2×D9712yescomplemented
148#102—972no—
149#102—972no—
150#102—972no—
151#102C2×D9712yes—
152#97—972no—
153C4×C8—971no—
154C4×C8C2×D9711yescharacteristic, complemented
155C24—1941no—
156C2×Q8—971no—
157C2×Q8—1942no—
158C2×Q8—1941no—
159C2×Q8—1944no—
160C2×Q8C22×D9711yescharacteristic
161C2×D4—971no—
162C2×D4—1944no—
163C2×D4—1941no—
164C2×D4C22×D9711yescharacteristic
165C22×C4—971no—
166C22×C4—972no—
167C22×C4—1941no—
168M16—1944no—
169C2×C8—972no—
170C2×C8C22×D9712yes—
171C4⋊C4—22no—
172C4⋊C4—972no—
173C4⋊C4—1942no—
174C4⋊C4—1942no—
175C4⋊C4C22×D9712yes—
176C22⋊C4—22no—
177C22⋊C4—1942no—
178C22⋊C4—1942no—
179C22⋊C4—1942no—
180C22⋊C4—1942no—
181C42—972no—
182C42—971no—
183C42—1942no—
184C42C22×D9711yescharacteristic
185C23—21no—
186C23—971no—
187C23—3882no—
188C23—3884no—
189C23—1941no—
190Q8—22no—
191Q8—3882no—
192Q8—3884no—
193Q8—1942no—
194D4—22no—
195D4—3882no—
196D4—1942no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—971no—
200C2×C4—974no—
201C2×C4—972no—
202C2×C4—3882no—
203C2×C4—3882no—
204C2×C4—1942no—
205C2×C4—1941no—
206C2×C4D4×D9712yes—
207C2×C4C23×D9711yescharacteristic
208C8—22no—
209C8—1942no—
210C22—41no—
211C22—42no—
212C22—7764no—
213C22—972no—
214C22—3881no—
215C22—3882no—
216C22—1942no—
217C22C2×D4×D9711yescharacteristic
218C4—22no—
219C4—41no—
220C4—42no—
221C4—972no—
222C4—3881no—
223C4—3882no—
224C4—1942no—
225C4#41×D9712yes—
226C2—81no—
227C2—7761no—
228C2—1942no—
229C2#124×D9711yescharacteristic
230C2#2026153212yes—
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#488×C97C2
socleC22×C97C22×D4
prime radicalC2×D4×D97C22
hypercenter#488D97
hypocenterC97#3140
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #20261291C2semidirect
3. #20261096C2semidirect
4. #20261093C2semidirect
5. #20261092C2semidirect
6. #20261090C2semidirect
7. #20261089C2semidirect
8. #20260742C2semidirect
10. C4×#13095494C2semidirect
11. #488×C97C2semidirect
15. #15690809C22semidirect
16. #15690802C22semidirect
26. #15690735C22semidirect
27. #15690734C22semidirect
28. #15690733C22semidirect
32. #15690720C22semidirect
33. C4×#11715895C22semidirect
34. #125×C97C22semidirect
36. #103×C97C22semidirect
38. C4×C8×C97C22semidirect
116. C97#3140semidirect
124. #488D97semidirect
133. #125C2×D97semidirect
147. #103C2×D97semidirect
154. C4×C8C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202612912
#124×D972
#202610962
#202610932
#202610922
#202610902
#202610892
#202607422
#202607382
C4×#130954942
#488×C972
#314097