Finite Group Databasebeta

← #27152121

Subgroups — #27152121

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#20261290C211yesmaximal, characteristic, complemented
2#124×D97C211yesmaximal, characteristic
3#20261097C211yesmaximal, characteristic, complemented
4#20261095C211yesmaximal, characteristic, complemented
5#20261094C211yesmaximal, characteristic, complemented
6#20261091C212yesmaximal, complemented
7#20261088C212yesmaximal, complemented
8#20260742C212yesmaximal, complemented
9#20260738C212yesmaximal
10#20260687C211yesmaximal, characteristic, complemented
11#489×C97C211yesmaximal, characteristic, complemented
12C2×Q8×D97C2211yescharacteristic
13C2×D4×D97C2211yescharacteristic
14C2×#13095494C2212yes—
15#15690810C2211yescharacteristic, complemented
16#15690801C2212yescomplemented
17#15690799C2211yescharacteristic
18#15690798C2211yescharacteristic
19#15690797C2211yescharacteristic
20#15690796—21no—
21#15690796—21no—
22#15690790—21no—
23#15690786—21no—
24Q8×Dic97—21no—
25#15690775—21no—
26#15690768C2212yes—
27C22⋊C4×D97—21no—
28C22⋊C4×D97—21no—
29C42×D97C2211yescharacteristic
30#15690735C2212yescomplemented
31#15690734C2212yescomplemented
32#15690733C2212yescomplemented
33#15690732C2212yes—
34#15690731C2212yes—
35#15690730C2212yes—
36#15690721C2211yescharacteristic, complemented
37C8×Dic97C2212yescomplemented
38C4⋊Q8×C97C2211yescharacteristic, complemented
39#124×C97C2211yescharacteristic
40#103×C97C2212yescomplemented
41#102×C97C2212yes—
42#97×C97C2211yescharacteristic, complemented
43C23×D97—21no—
44C2×Dic194—21no—
45C2×Dic194—21no—
46C2×Dic194C2311yescharacteristic
47Q8×D97—24no—
48C2×D388C2311yescharacteristic
49C2×C22⋊D97—21no—
50D4×D97—24no—
51C2×C4×D97—21no—
52C2×C4×D97D411yescharacteristic
53C2×C4×D97D411yescharacteristic
54C2×C4×D97C2311yescharacteristic
55#13095494—24no—
56C2×#11715895C2312yes—
57C4⋊Dic97—21no—
58C4⋊Dic97—22no—
59C4⋊Dic97C2312yes—
60#13095490—22no—
61C22⋊Dic97—21no—
62C22⋊Dic97—21no—
63#13095488—22no—
64#13095488—21no—
65#13095488—21no—
66#13095488—21no—
67#13095488—21no—
68C4×Dic97—22no—
69C4×Dic97D411yescharacteristic
70C4×Dic97D411yescharacteristic
71C4×Dic97C2311yescharacteristic
72C2×Q8×C97—21no—
73C2×Q8×C97C2311yescharacteristic
74C2×D4×C97C2311yescharacteristic
75C2×C8×C97C2312yes—
76C4⋊C4×C97—21no—
77C4⋊C4×C97C2312yes—
78C22⋊C4×C97—21no—
79C22⋊C4×C97—21no—
80C42×C97C2311yescharacteristic
81C23×C97—21no—
82C22×D97—21no—
83C22×D97—42no—
84C22×D97—42no—
85C22×D97—42no—
86C22×D97C2×D411yescharacteristic
87Q8×C97—22no—
88Q8×C97—42no—
89D4×C97—22no—
90C2×C4×C97—21no—
91C2×C4×C97—22no—
92C2×C4×C97C2×D411yescharacteristic
93C2×C4×C97C2×D411yescharacteristic
94C2×C4×C97C2411yescharacteristic
95C22⋊D97—42no—
96C2×Dic97—22no—
97C2×Dic97—21no—
98C2×Dic97C2×D411yescharacteristic
99C2×Dic97C2×D411yescharacteristic
100C2×Dic97C2×D411yescharacteristic
101D388—22no—
102C4×D97—41no—
103C4×D97—41no—
104C4×D97—42no—
105C4×D97#4114yes—
106Dic194—22no—
107Dic194—42no—
108Dic194—42no—
109C8×C97—22no—
110#11715895—22no—
111C22×C97—41no—
112C22×C97—41no—
113C22×C97—41no—
114C22×C97C22×D411yescharacteristic
115C2×D97—21no—
116C2×D97—21no—
117C2×D97—41no—
118C2×D97—41no—
119C2×D97—41no—
120C2×D97—84no—
121C2×D97#12412yes—
122C4×C97—21no—
123C4×C97—21no—
124C4×C97—41no—
125C4×C97—42no—
126C4×C97C2×#4112yes—
127Dic97—21no—
128Dic97—21no—
129Dic97—41no—
130Dic97—42no—
131Dic97#12412yes—
132C2×C97—81no—
133C2×C97C2×#12411yescharacteristic
134C2×C97#57511yescharacteristic
135C2×C97#57511yescharacteristic
136D97—22no—
137D97—81no—
138#3140—971nomaximal, Sylow
139C97#314011yescharacteristic, complemented, Sylow
140#533—971no—
141C2×#124—971no—
142#489—971no—
143#489—971no—
144#489—971no—
145#489D9711yescharacteristic, complemented
146#488—972no—
147#488—972no—
148#419—972no—
149#418—972no—
150C4×M16—971no—
151C22×Q8—971no—
152C22×D4—971no—
153C2×M16—972no—
154C4⋊Q8—971no—
155C4⋊Q8C2×D9711yescharacteristic, complemented
156#125—972no—
157#124—971no—
158#124—971no—
159#124—971no—
160#124—1941no—
161#124—1941no—
162#124C2×D9711yescharacteristic
163C22⋊Q8—1941no—
164C22⋊Q8—1941no—
165C4×Q8—1941no—
166C4×Q8—1941no—
167#117—972no—
168C2×C22⋊C4—1941no—
169C2×C22⋊C4—1941no—
170C2×C42—971no—
171#103—972no—
172#103—972no—
173#103—972no—
174#103C2×D9712yescomplemented
175#102—972no—
176#102—972no—
177#102—972no—
178#102C2×D9712yes—
179#97—971no—
180#97C2×D9711yescharacteristic, complemented
181C4×C8—972no—
182C24—1941no—
183C2×Q8—21no—
184C2×Q8—971no—
185C2×Q8—1941no—
186C2×Q8—1944no—
187C2×Q8—1941no—
188C2×Q8C22×D9711yescharacteristic
189C2×D4—971no—
190C2×D4—1944no—
191C2×D4—1941no—
192C2×D4C22×D9711yescharacteristic
193C22×C4—971no—
194C22×C4—971no—
195C22×C4—971no—
196C22×C4—1941no—
197M16—1944no—
198C2×C8—972no—
199C2×C8C22×D9712yes—
200C4⋊C4—21no—
201C4⋊C4—972no—
202C4⋊C4—1942no—
203C4⋊C4—1942no—
204C4⋊C4—1941no—
205C4⋊C4C22×D9712yes—
206C22⋊C4—21no—
207C22⋊C4—21no—
208C22⋊C4—1941no—
209C22⋊C4—1941no—
210C22⋊C4—1942no—
211C22⋊C4—1941no—
212C22⋊C4—1941no—
213C22⋊C4—1941no—
214C22⋊C4—1941no—
215C42—971no—
216C42—971no—
217C42—971no—
218C42—1942no—
219C42C22×D9711yescharacteristic
220C23—21no—
221C23—971no—
222C23—3882no—
223C23—3882no—
224C23—3882no—
225C23—1941no—
226Q8—22no—
227Q8—42no—
228Q8—3882no—
229Q8—3882no—
230Q8—1942no—
231D4—22no—
232D4—3882no—
233D4—1942no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—971no—
237C2×C4—974no—
238C2×C4—971no—
239C2×C4—971no—
240C2×C4—3881no—
241C2×C4—3881no—
242C2×C4—3882no—
243C2×C4—1942no—
244C2×C4—1941no—
245C2×C4D4×D9711yescharacteristic
246C2×C4D4×D9711yescharacteristic
247C2×C4C23×D9711yescharacteristic
248C8—22no—
249C8—1942no—
250C22—41no—
251C22—41no—
252C22—41no—
253C22—7764no—
254C22—972no—
255C22—3881no—
256C22—3881no—
257C22—3881no—
258C22—1941no—
259C22—1941no—
260C22C2×D4×D9711yescharacteristic
261C4—21no—
262C4—21no—
263C4—41no—
264C4—42no—
265C4—972no—
266C4—3881no—
267C4—3882no—
268C4—1941no—
269C4—1941no—
270C4#41×D9712yes—
271C2—81no—
272C2—7761no—
273C2—1942no—
274C2#124×D9711yescharacteristic
275C2#2026153111yescharacteristic
276C2#2026153411yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#489×C97C2
socleC22×C97C22×D4
prime radicalC2×D4×D97C22
hypercenter#489D97
hypocenterC97#3140
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #20261290C2semidirect
3. #20261097C2semidirect
4. #20261095C2semidirect
5. #20261094C2semidirect
6. #20261091C2semidirect
7. #20261088C2semidirect
8. #20260742C2semidirect
10. #20260687C2semidirect
11. #489×C97C2semidirect
15. #15690810C22semidirect
16. #15690801C22semidirect
30. #15690735C22semidirect
31. #15690734C22semidirect
32. #15690733C22semidirect
36. #15690721C22semidirect
37. C8×Dic97C22semidirect
38. C4⋊Q8×C97C22semidirect
40. #103×C97C22semidirect
42. #97×C97C22semidirect
139. C97#3140semidirect
145. #489D97semidirect
155. C4⋊Q8C2×D97semidirect
174. #103C2×D97semidirect
180. #97C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202612902
#124×D972
#202610972
#202610952
#202610942
#202610912
#202610882
#202607422
#202607382
#202606872
#489×C972
#314097