Finite Group Databasebeta

← #27151874

Subgroups — #27151874

Order 12416.

450 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 129 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#20261269C211yesmaximal, characteristic, complemented
2#20261065C211yesmaximal, characteristic, complemented
3#20261048C211yesmaximal, characteristic, complemented
4#20261035C212yesmaximal, complemented
5#20260950C212yesmaximal, complemented
6#20260886C211yesmaximal, characteristic, complemented
7#20260869C211yesmaximal, characteristic
8#20260845C212yesmaximal, complemented
9#20260835C212yesmaximal, complemented
10#20260792C211yesmaximal, characteristic, complemented
11C2×C4×D4×C97C211yesmaximal, characteristic, complemented
12C2×#13095499C2212yes—
13C2×#13095496C2212yes—
14#15690804—21no—
15#15690802C2211yescharacteristic, complemented
16#15690798C2211yescharacteristic, complemented
17#15690793—21no—
18#15690789C2211yescharacteristic
19#15690784C2211yescharacteristic
20C4×Dic194C2211yescharacteristic, complemented
21C4×D388C2211yescharacteristic, complemented
22C4×C22⋊D97—21no—
23#15690767—21no—
24#15690743C2212yescomplemented
25#15690741C2212yescomplemented
26C4⋊#11715895C2212yescomplemented
27#15690733C2212yescomplemented
28#15690731C2212yes—
29#15690731C2212yes—
30#15690730C2212yescomplemented
31C22⋊#11715895C2211yescharacteristic
32C22⋊#11715895C2211yescharacteristic
33#15690721C2212yescomplemented
34C22×D4×C97C2211yescharacteristic
35C23×C4×C97—21no—
36C4×D4×C97—21no—
37C4×D4×C97—21no—
38C4×D4×C97C2212yes—
39C4×D4×C97C2212yes—
40C2×C4⋊C4×C97C2211yescharacteristic
41C2×C22⋊C4×C97—21no—
42C2×C42×C97C2211yescharacteristic
43C2×Dic194C2311yescharacteristic
44C2×D388C2311yescharacteristic
45C2×C22⋊D97—21no—
46C2×C4×D97—21no—
47#13095499—24no—
48#13095496—24no—
49C2×#11715895C2312yes—
50C2×#11715895C2312yes—
51C4⋊Dic97C2311yescharacteristic
52C4⋊Dic97C2311yescharacteristic
53#13095490—21no—
54#13095490—21no—
55#13095490—21no—
56#13095490—21no—
57C22⋊Dic97—21no—
58#13095488—21no—
59#13095488—21no—
60#13095488—21no—
61#13095488—21no—
62C4×Dic97—21no—
63C24×C97—21no—
64C2×D4×C97—24no—
65C2×D4×C97—21no—
66C2×D4×C97C2312yes—
67C22×C4×C97—21no—
68C22×C4×C97—22no—
69C22×C4×C97—22no—
70C22×C4×C97—44no—
71C22×C4×C97D411yescharacteristic
72C22×C4×C97D411yescharacteristic
73C22×C4×C97C2311yescharacteristic
74C4⋊C4×C97—21no—
75C4⋊C4×C97C2312yes—
76C22⋊C4×C97—22no—
77C22⋊C4×C97—22no—
78C42×C97D411yescharacteristic
79C42×C97D411yescharacteristic
80C42×C97C2311yescharacteristic
81C42×C97C2311yescharacteristic
82C23×C97—24no—
83C23×C97—22no—
84C23×C97—42no—
85C23×C97—42no—
86C23×C97C2×D411yescharacteristic
87C22×D97—21no—
88D4×C97—24no—
89D4×C97—22no—
90D4×C97#4114yes—
91C2×C4×C97—22no—
92C2×C4×C97—22no—
93C2×C4×C97—22no—
94C2×C4×C97—41no—
95C2×C4×C97—44no—
96C2×C4×C97—44no—
97C2×C4×C97—42no—
98C2×C4×C97C2×D411yescharacteristic
99C2×C4×C97C2×D411yescharacteristic
100C2×C4×C97C2×D411yescharacteristic
101C2×C4×C97C2×D411yescharacteristic
102C2×C4×C97C2×D411yescharacteristic
103C2×C4×C97C2411yescharacteristic
104C22⋊D97—42no—
105C2×Dic97—21no—
106C2×Dic97—21no—
107C2×Dic97—21no—
108D388—22no—
109C4×D97—42no—
110Dic194—22no—
111#11715895—22no—
112#11715895—22no—
113C22×C97—21no—
114C22×C97—21no—
115C22×C97—24no—
116C22×C97—48no—
117C22×C97—42no—
118C22×C97—42no—
119C22×C97AGL(1,ℤ/8)12yes—
120C22×C97C22×D411yescharacteristic
121C2×D97—41no—
122C2×D97—41no—
123C2×D97—41no—
124C4×C97—21no—
125C4×C97—21no—
126C4×C97—21no—
127C4×C97—21no—
128C4×C97—21no—
129C4×C97—42no—
130C4×C97C2×#4112yes—
131Dic97—41no—
132Dic97—41no—
133Dic97—41no—
134C2×C97—22no—
135C2×C97—44no—
136C2×C97#52511yescharacteristic
137C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C97#57511yescharacteristic
139D97—81no—
140#3114—971nomaximal, Sylow
141C97#311411yescharacteristic, complemented, Sylow
142#529—971no—
143C2×C4×D4D9711yescharacteristic, complemented
144#483—971no—
145#480—971no—
146#478—972no—
147#463—972no—
148#450—971no—
149#447—971no—
150#442—972no—
151#440—972no—
152#432—971no—
153C22×D4C2×D9711yescharacteristic
154C23×C4—21no—
155C2×SD16—972no—
156C2×D8—972no—
157#126—1941no—
158#125—971no—
159#124—971no—
160#123—1941no—
161C22⋊Q8—971no—
162#121—971no—
163C4×Q8—971no—
164C4×D4—21no—
165C4×D4—21no—
166C4×D4—971no—
167C4×D4—1941no—
168C4×D4C2×D9712yes—
169C4×D4C2×D9712yes—
170#117—1941no—
171C2×C4⋊C4C2×D9711yescharacteristic
172C2×C22⋊C4—21no—
173C2×C42C2×D9711yescharacteristic
174#107—972no—
175#106—972no—
176C4⋊C8—972no—
177#103—972no—
178#102—972no—
179#102—972no—
180#102—972no—
181C22⋊C8—971no—
182C22⋊C8—971no—
183#97—972no—
184C24—21no—
185C2×Q8—971no—
186C2×D4—24no—
187C2×D4—21no—
188C2×D4—971no—
189C2×D4—1941no—
190C2×D4C22×D9712yes—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—44no—
195C22×C4—1941no—
196C22×C4C22⋊D9711yescharacteristic
197C22×C4C22⋊D9711yescharacteristic
198C22×C4C22×D9711yescharacteristic
199SD16—1944no—
200D8—1944no—
201C2×C8—972no—
202C2×C8—972no—
203C4⋊C4—21no—
204C4⋊C4—971no—
205C4⋊C4—971no—
206C4⋊C4—1941no—
207C4⋊C4—1941no—
208C4⋊C4—1941no—
209C4⋊C4—1941no—
210C4⋊C4C22×D9712yes—
211C22⋊C4—22no—
212C22⋊C4—22no—
213C22⋊C4—1941no—
214C22⋊C4—1941no—
215C22⋊C4—1941no—
216C22⋊C4—1941no—
217C22⋊C4—1941no—
218C42—1941no—
219C42C22⋊D9711yescharacteristic
220C42C22⋊D9711yescharacteristic
221C42C22×D9711yescharacteristic
222C42C22×D9711yescharacteristic
223C23—24no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23—1941no—
228C23C2×C22⋊D9711yescharacteristic
229Q8—1942no—
230D4—24no—
231D4—22no—
232D4—3882no—
233D4—1942no—
234D4#1309551114yes—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—41no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—42no—
242C2×C4—3882no—
243C2×C4—1941no—
244C2×C4—1941no—
245C2×C4—1941no—
246C2×C4C2×C22⋊D9711yescharacteristic
247C2×C4C2×C22⋊D9711yescharacteristic
248C2×C4C2×C22⋊D9711yescharacteristic
249C2×C4C2×C22⋊D9711yescharacteristic
250C2×C4C2×C22⋊D9711yescharacteristic
251C2×C4C23×D9711yescharacteristic
252C8—1942no—
253C8—1942no—
254C22—21no—
255C22—21no—
256C22—24no—
257C22—48no—
258C22—42no—
259C22—42no—
260C22—3881no—
261C22—3881no—
262C22—3881no—
263C22#1569083612yes—
264C22C22×C22⋊D9711yescharacteristic
265C4—21no—
266C4—21no—
267C4—21no—
268C4—21no—
269C4—21no—
270C4—42no—
271C4—3881no—
272C4—3881no—
273C4—3881no—
274C4C2×#1309551112yes—
275C2—22no—
276C2—44no—
277C2—7761no—
278C2#2026124611yescharacteristic
279C2C2×#1569083611yescharacteristic
280C2#2026153311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×C4×D4×C97C2
socleC22×C97C22×D4
prime radical#20260869C2
hypercenterC2×C4×D4D97
hypocenterC97#3114
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #20261269C2semidirect
2. #20261065C2semidirect
3. #20261048C2semidirect
4. #20261035C2semidirect
5. #20260950C2semidirect
6. #20260886C2semidirect
8. #20260845C2semidirect
9. #20260835C2semidirect
10. #20260792C2semidirect
11. C2×C4×D4×C97C2semidirect
15. #15690802C22semidirect
16. #15690798C22semidirect
20. C4×Dic194C22semidirect
21. C4×D388C22semidirect
24. #15690743C22semidirect
25. #15690741C22semidirect
26. C4⋊#11715895C22semidirect
27. #15690733C22semidirect
30. #15690730C22semidirect
33. #15690721C22semidirect
141. C97#3114semidirect
143. C2×C4×D4D97semidirect

Maximal subgroups

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

subgroupindex
#202612692
#202610652
#202610482
#202610352
#202609502
#202608862
#202608692
#202608452
#202608352
#202607922
C2×C4×D4×C972
#311497