Finite Group Databasebeta

← #38786624

Subgroups — #38786624

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×#20264365C211yesmaximal, characteristic, complemented
2C22×C4×D401C211yesmaximal, characteristic, complemented
3#27165719C212yesmaximal, complemented
4#27165717C211yesmaximal, characteristic, complemented
5#27165716C211yesmaximal, characteristic, complemented
6#27165708C211yesmaximal, characteristic, complemented
7#27165705C212yesmaximal, complemented
8D4×Dic401C212yesmaximal, complemented
9#27165697C212yesmaximal, complemented
10#27165695C211yesmaximal, characteristic, complemented
11#121×C401C211yesmaximal, characteristic, complemented
12C23×D401C2211yescharacteristic
13#20264365—24no—
14C2×Dic802C2211yescharacteristic, complemented
15C2×C22⋊D401C2212yes—
16C2×C22⋊D401C2212yes—
17C22×Dic401C2211yescharacteristic
18C22×Dic401C2212yescomplemented
19C2×C4×D401—24no—
20C2×C4×D401C2211yescharacteristic
21C2×C4×D401C2211yescharacteristic
22C2×C4×D401C2211yescharacteristic
23C2×C4×D401C2211yescharacteristic
24C4⋊Dic401C2212yescomplemented
25C4⋊Dic401C2211yescharacteristic, complemented
26#20264345C2212yescomplemented
27C22⋊Dic401C2212yes—
28C22⋊Dic401C2212yescomplemented
29#20264343C2212yes—
30#20264343C2212yes—
31C4×Dic401C2212yescomplemented
32C2×D4×C401C2212yes—
33C2×D4×C401C2211yescharacteristic, complemented
34C22×C4×C401C2211yescharacteristic
35C4⋊C4×C401C2211yescharacteristic, complemented
36C22⋊C4×C401C2212yescomplemented
37C22×D401—22no—
38C22×D401—22no—
39C22×D401—22no—
40C22×D401C2311yescharacteristic
41C22×D401C2311yescharacteristic
42Dic802—22no—
43C22⋊D401—24no—
44C22⋊D401—24no—
45C2×Dic401—22no—
46C2×Dic401—24no—
47C2×Dic401C2312yes—
48C2×Dic401C2312yes—
49C2×Dic401C2311yescharacteristic
50C2×Dic401C2311yescharacteristic
51C4×D401—22no—
52C4×D401—22no—
53C4×D401—22no—
54C4×D401D414yescomplemented
55C23×C401C2312yes—
56C23×C401C2311yescharacteristic
57D4×C401—24no—
58D4×C401—22no—
59C2×C4×C401—22no—
60C2×C4×C401C2311yescharacteristic
61C2×C4×C401C2312yes—
62C2×C4×C401C2311yescharacteristic
63C22×C401—21no—
64C22×C401—21no—
65C22×C401—22no—
66C22×C401—22no—
67C22×C401—22no—
68C22×C401#4112yescomplemented
69C22×C401C2411yescharacteristic
70C2×D401—28no—
71C2×D401—21no—
72C2×D401—21no—
73C2×D401—21no—
74C2×D401—21no—
75C2×D401C2×D412yes—
76C2×D401#4112yescomplemented
77C4×C401—21no—
78C4×C401—22no—
79C4×C401C2×D412yes—
80Dic401—21no—
81Dic401—22no—
82Dic401—22no—
83Dic401C2×D412yes—
84C2×C401—22no—
85C2×C401—42no—
86C2×C401C22×D411yescharacteristic
87C2×C401C2×#4111yescharacteristic
88C2×C401C2×#4111yescharacteristic
89D401—22no—
90D401—22no—
91C401#52511yescharacteristic, complemented, Sylow
92#525—4011nomaximal, Sylow
93C2×#41—4011no—
94C23×C4—4011no—
95#123—4012no—
96C22⋊Q8—4011no—
97C22⋊Q8—4011no—
98#121—4011no—
99#121D40111yescharacteristic, complemented
100#120—4012no—
101C4×D4—4012no—
102C4×D4—4012no—
103#117—4011no—
104C24—4011no—
105#41—8024no—
106C2×Q8—4011no—
107C2×D4—4012no—
108C2×D4—4012no—
109C2×D4C2×D40112yes—
110C2×D4C2×D40111yescharacteristic, complemented
111C22×C4—8024no—
112C22×C4—4011no—
113C22×C4—4012no—
114C22×C4—4011no—
115C22×C4—4011no—
116C22×C4—4011no—
117C22×C4—4011no—
118C22×C4C2×D40111yescharacteristic
119C4⋊C4—4011no—
120C4⋊C4—4012no—
121C4⋊C4—4012no—
122C4⋊C4C2×D40111yescharacteristic, complemented
123C22⋊C4—4012no—
124C22⋊C4—4012no—
125C22⋊C4—4012no—
126C22⋊C4—4012no—
127C22⋊C4C2×D40112yescomplemented
128C42—4012no—
129C23—8022no—
130C23—8022no—
131C23—8022no—
132C23—4011no—
133C23—4011no—
134C23C22×D40112yes—
135C23C22×D40111yescharacteristic
136Q8—8022no—
137D4—24no—
138D4—22no—
139D4—8024no—
140D4—8024no—
141C2×C4—22no—
142C2×C4—8022no—
143C2×C4—8022no—
144C2×C4—8022no—
145C2×C4—8024no—
146C2×C4—8022no—
147C2×C4—4012no—
148C2×C4—4011no—
149C2×C4—4012no—
150C2×C4—4011no—
151C2×C4—4014no—
152C2×C4C22×D40112yes—
153C2×C4C22×D40111yescharacteristic
154C2×C4C22×D40111yescharacteristic
155C22—21no—
156C22—21no—
157C22—22no—
158C22—22no—
159C22—22no—
160C22—8028no—
161C22—8021no—
162C22—8021no—
163C22—8021no—
164C22—8021no—
165C22—4012no—
166C22—4012no—
167C22#2026436512yescomplemented
168C22C23×D40111yescharacteristic
169C4—21no—
170C4—22no—
171C4—8021no—
172C4—8022no—
173C4—8022no—
174C4—4012no—
175C4D4×D40112yes—
176C2—22no—
177C2—42no—
178C2—8022no—
179C2—8022no—
180C2C2×D4×D40111yescharacteristic
181C2#41×D40111yescharacteristic
182C2C2×#2026436511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D401
commutator, socle, cosocleC22×C401C24
Fitting#121×C401C2
hypercenter#121D401
hypocenterC401#525
LCSG ▷ C22×C401 ▷ C401
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#20264365C2semidirect
2. C22×C4×D401C2semidirect
3. #27165719C2semidirect
4. #27165717C2semidirect
5. #27165716C2semidirect
6. #27165708C2semidirect
7. #27165705C2semidirect
8. D4×Dic401C2semidirect
9. #27165697C2semidirect
10. #27165695C2semidirect
11. #121×C401C2semidirect
14. C2×Dic802C22semidirect
18. C22×Dic401C22semidirect
24. C4⋊Dic401C22semidirect
25. C4⋊Dic401C22semidirect
26. #20264345C22semidirect
28. C22⋊Dic401C22semidirect
31. C4×Dic401C22semidirect
33. C2×D4×C401C22semidirect
35. C4⋊C4×C401C22semidirect
36. C22⋊C4×C401C22semidirect
54. C4×D401D4semidirect
68. C22×C401#41semidirect
76. C2×D401#41semidirect
91. C401#525semidirect
99. #121D401semidirect
110. C2×D4C2×D401semidirect
122. C4⋊C4C2×D401semidirect
127. C22⋊C4C2×D401semidirect
167. C22#20264365semidirect

Maximal subgroups

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

subgroupindex
C2×#202643652
C22×C4×D4012
#271657192
#271657172
#271657162
#271657082
#271657052
D4×Dic4012
#271656972
#271656952
#121×C4012
#525401