Finite Group Databasebeta

← #38786627

Subgroups — #38786627

Order 25664.

330 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 109 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×#20264367C211yesmaximal, characteristic, complemented
2C22×C4×D401C211yesmaximal, characteristic, complemented
3#27165722C212yesmaximal, complemented
4#27165713C211yesmaximal, characteristic, complemented
5#27165712C211yesmaximal, characteristic, complemented
6#27165711C211yesmaximal, characteristic, complemented
7#27165707C212yesmaximal, complemented
8#27165699C212yesmaximal, complemented
9#27165697C212yesmaximal, complemented
10#27165695C211yesmaximal, characteristic, complemented
11C22⋊Q8×C401C211yesmaximal, characteristic, complemented
12C23×D401C2211yescharacteristic
13#20264367—24no—
14C2×D1604C2212yes—
15C2×D1604C2212yes—
16C2×D1604C2211yescharacteristic, complemented
17C2×C22⋊D401C2212yes—
18C22×Dic401C2211yescharacteristic
19C2×C4×D401—24no—
20C2×C4×D401C2211yescharacteristic
21C2×C4×D401C2211yescharacteristic
22C2×C4×D401C2211yescharacteristic
23C2×C4×D401C2211yescharacteristic
24C2×C4×D401C2212yescomplemented
25C4⋊Dic401C2211yescharacteristic, complemented
26#20264345C2212yescomplemented
27#20264343C2212yes—
28#20264343C2212yes—
29#20264343C2212yescomplemented
30#20264343C2212yescomplemented
31C4×Dic401C2212yescomplemented
32C2×Q8×C401C2211yescharacteristic, complemented
33C22×C4×C401C2211yescharacteristic
34C4⋊C4×C401C2212yescomplemented
35C4⋊C4×C401C2211yescharacteristic, complemented
36C22⋊C4×C401C2212yes—
37C22×D401—22no—
38C22×D401—22no—
39C22×D401—22no—
40C22×D401C2312yes—
41C22×D401C2311yescharacteristic
42C22×D401C2311yescharacteristic
43D1604—24no—
44D1604—24no—
45D1604—22no—
46C22⋊D401—24no—
47C2×Dic401—22no—
48C2×Dic401C2312yes—
49C2×Dic401C2311yescharacteristic
50C2×Dic401C2311yescharacteristic
51C4×D401—22no—
52C4×D401—22no—
53C4×D401—22no—
54C4×D401—24no—
55C4×D401D414yescomplemented
56C23×C401C2311yescharacteristic
57Q8×C401—22no—
58C2×C4×C401—22no—
59C2×C4×C401C2312yes—
60C2×C4×C401C2312yes—
61C2×C4×C401C2311yescharacteristic
62C2×C4×C401C2311yescharacteristic
63C22×C401—21no—
64C22×C401—21no—
65C22×C401#4112yescomplemented
66C22×C401C2411yescharacteristic
67C2×D401—28no—
68C2×D401—21no—
69C2×D401—21no—
70C2×D401—21no—
71C2×D401—21no—
72C2×D401—22no—
73C2×D401—22no—
74C2×D401—22no—
75C2×D401C2×D412yes—
76C2×D401#4112yescomplemented
77C4×C401—21no—
78C4×C401—22no—
79C4×C401—22no—
80C4×C401C2×D412yes—
81Dic401—21no—
82Dic401—22no—
83Dic401C2×D412yes—
84C2×C401—22no—
85C2×C401C22×D411yescharacteristic
86C2×C401C2×#4111yescharacteristic
87C2×C401C2×#4111yescharacteristic
88D401—22no—
89D401—22no—
90D401—42no—
91C401#52511yescharacteristic, complemented, Sylow
92#525—4011nomaximal, Sylow
93C2×#41—4011no—
94C23×C4—4011no—
95#123—4012no—
96C22⋊Q8—4011no—
97C22⋊Q8D40111yescharacteristic, complemented
98#121—4011no—
99#121—4011no—
100#120—4012no—
101C4×D4—4012no—
102C4×D4—4012no—
103#117—4011no—
104C24—4011no—
105#41—8024no—
106C2×Q8C2×D40111yescharacteristic, complemented
107C2×D4—4012no—
108C2×D4—4012no—
109C2×D4—4012no—
110C2×D4—4011no—
111C22×C4—8024no—
112C22×C4—4011no—
113C22×C4—4011no—
114C22×C4—4012no—
115C22×C4—4011no—
116C22×C4—4011no—
117C22×C4—4011no—
118C22×C4C2×D40111yescharacteristic
119C4⋊C4—4011no—
120C4⋊C4—4012no—
121C4⋊C4C2×D40111yescharacteristic, complemented
122C4⋊C4C2×D40112yescomplemented
123C22⋊C4—4012no—
124C22⋊C4—4012no—
125C22⋊C4—4012no—
126C22⋊C4—4012no—
127C22⋊C4C2×D40112yes—
128C42—4012no—
129C23—8022no—
130C23—8022no—
131C23—8022no—
132C23—4012no—
133C23—4011no—
134C23—4011no—
135C23C22×D40111yescharacteristic
136Q8—22no—
137D4—8024no—
138D4—8024no—
139D4—8024no—
140D4—8022no—
141C2×C4—22no—
142C2×C4—8022no—
143C2×C4—8022no—
144C2×C4—8024no—
145C2×C4—8022no—
146C2×C4—8022no—
147C2×C4—4012no—
148C2×C4—4011no—
149C2×C4—4011no—
150C2×C4—4014no—
151C2×C4C22×D40112yes—
152C2×C4C22×D40111yescharacteristic
153C2×C4C22×D40112yes—
154C2×C4C22×D40111yescharacteristic
155C22—21no—
156C22—21no—
157C22—8028no—
158C22—8021no—
159C22—8021no—
160C22—8021no—
161C22—8021no—
162C22—8022no—
163C22—8022no—
164C22—8022no—
165C22—4012no—
166C22—4012no—
167C22#2026436712yescomplemented
168C22C23×D40111yescharacteristic
169C4—21no—
170C4—22no—
171C4—22no—
172C4—8021no—
173C4—8022no—
174C4—4012no—
175C4D4×D40112yes—
176C2—22no—
177C2—8022no—
178C2—8022no—
179C2—16042no—
180C2C2×D4×D40111yescharacteristic
181C2#41×D40111yescharacteristic
182C2C2×#2026436711yescharacteristic
subgroupG/H
center, FrattiniC22C23×D401
commutator, socle, cosocleC22×C401C24
FittingC22⋊Q8×C401C2
hypercenterC22⋊Q8D401
hypocenterC401#525
LCSG ▷ C22×C401 ▷ C401
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#20264367C2semidirect
2. C22×C4×D401C2semidirect
3. #27165722C2semidirect
4. #27165713C2semidirect
5. #27165712C2semidirect
6. #27165711C2semidirect
7. #27165707C2semidirect
8. #27165699C2semidirect
9. #27165697C2semidirect
10. #27165695C2semidirect
11. C22⋊Q8×C401C2semidirect
16. C2×D1604C22semidirect
24. C2×C4×D401C22semidirect
25. C4⋊Dic401C22semidirect
26. #20264345C22semidirect
29. #20264343C22semidirect
30. #20264343C22semidirect
31. C4×Dic401C22semidirect
32. C2×Q8×C401C22semidirect
34. C4⋊C4×C401C22semidirect
35. C4⋊C4×C401C22semidirect
55. C4×D401D4semidirect
65. C22×C401#41semidirect
76. C2×D401#41semidirect
91. C401#525semidirect
97. C22⋊Q8D401semidirect
106. C2×Q8C2×D401semidirect
121. C4⋊C4C2×D401semidirect
122. C4⋊C4C2×D401semidirect
167. C22#20264367semidirect

Maximal subgroups

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

subgroupindex
C2×#202643672
C22×C4×D4012
#271657222
#271657132
#271657122
#271657112
#271657072
#271656992
#271656972
#271656952
C22⋊Q8×C4012
#525401