Finite Group Databasebeta

← #38786622

Subgroups — #38786622

Order 25664.

330 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 107 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#27165720C211yesmaximal, characteristic, complemented
4#27165715C212yesmaximal, complemented
5#27165710C212yesmaximal, complemented
6#27165707C211yesmaximal, characteristic, complemented
7#27165705C211yesmaximal, characteristic, complemented
8#27165699C212yesmaximal, complemented
9#27165697C212yesmaximal, complemented
10#27165693C211yesmaximal, characteristic, complemented
11#123×C401C211yesmaximal, characteristic, complemented
12C23×D401C2211yescharacteristic
13#20264365—24no—
14C2×Dic802C2211yescharacteristic, complemented
15C2×D1604C2212yes—
16C2×C22⋊D401C2212yes—
17C2×C22⋊D401C2211yescharacteristic
18C2×C22⋊D401C2211yescharacteristic, complemented
19C22×Dic401C2211yescharacteristic
20C22×Dic401C2211yescharacteristic, complemented
21C2×C4×D401—24no—
22C2×C4×D401C2212yes—
23C2×C4×D401C2212yes—
24C2×C4×D401C2211yescharacteristic, complemented
25#20264345C2212yescomplemented
26#20264345C2212yescomplemented
27C22⋊Dic401C2211yescharacteristic
28#20264343C2212yes—
29#20264343C2212yes—
30#20264343C2212yescomplemented
31C4×Dic401C2212yescomplemented
32C2×D4×C401C2211yescharacteristic
33C22×C4×C401C2211yescharacteristic
34C4⋊C4×C401C2212yescomplemented
35C22⋊C4×C401C2211yescharacteristic
36C22⋊C4×C401C2212yescomplemented
37C22×D401—22no—
38C22×D401—24no—
39C22×D401C2311yescharacteristic
40C22×D401C2312yes—
41Dic802—22no—
42D1604—24no—
43C22⋊D401—22no—
44C22⋊D401—24no—
45C22⋊D401—22no—
46C2×Dic401—22no—
47C2×Dic401D414yescomplemented
48C2×Dic401C2312yes—
49C2×Dic401C2311yescharacteristic
50C2×Dic401C2311yescharacteristic
51C2×Dic401C2311yescharacteristic
52C4×D401—24no—
53C4×D401—24no—
54C4×D401—22no—
55C23×C401C2311yescharacteristic
56C23×C401C2311yescharacteristic
57D4×C401—22no—
58C2×C4×C401—22no—
59C2×C4×C401C2312yes—
60C2×C4×C401C2312yes—
61C2×C4×C401C2311yescharacteristic
62C22×C401—22no—
63C22×C401—21no—
64C22×C401—22no—
65C22×C401C2×D412yes—
66C22×C401C2411yescharacteristic
67C2×D401—28no—
68C2×D401—22no—
69C2×D401—22no—
70C2×D401—21no—
71C2×D401—22no—
72C2×D401#4114yescomplemented
73C4×C401—22no—
74C4×C401—21no—
75C4×C401—22no—
76Dic401—21no—
77Dic401—22no—
78Dic401C2×D412yes—
79Dic401C2×D412yes—
80C2×C401—22no—
81C2×C401—41no—
82C2×C401C22×D411yescharacteristic
83C2×C401C2×#4112yes—
84D401—24no—
85D401—41no—
86C401#52511yescharacteristic, complemented, Sylow
87#525—4011nomaximal, Sylow
88C2×#41—4011no—
89C23×C4—4011no—
90#123—4011no—
91#123D40111yescharacteristic, complemented
92C22⋊Q8—4012no—
93#121—4012no—
94#120—4011no—
95#120—4011no—
96C4×D4—4012no—
97C4×D4—4012no—
98#117—4011no—
99C24—4011no—
100#41—8024no—
101C2×Q8—4011no—
102C2×D4—4012no—
103C2×D4—4011no—
104C2×D4—4011no—
105C2×D4—4012no—
106C2×D4C2×D40111yescharacteristic
107C22×C4—8024no—
108C22×C4—4011no—
109C22×C4—4012no—
110C22×C4—4011no—
111C22×C4—4012no—
112C22×C4—4011no—
113C22×C4C2×D40111yescharacteristic
114C4⋊C4—4012no—
115C4⋊C4—4012no—
116C4⋊C4C2×D40112yescomplemented
117C22⋊C4—4011no—
118C22⋊C4—4012no—
119C22⋊C4—4012no—
120C22⋊C4—4012no—
121C22⋊C4C2×D40111yescharacteristic
122C22⋊C4C2×D40112yescomplemented
123C42—4012no—
124C23—8024no—
125C23—8022no—
126C23—4011no—
127C23—4012no—
128C23C22×D40111yescharacteristic
129C23C22×D40111yescharacteristic
130Q8—8022no—
131D4—22no—
132D4—8024no—
133D4—8024no—
134D4—8022no—
135D4—8022no—
136C2×C4—22no—
137C2×C4—8024no—
138C2×C4—8024no—
139C2×C4—8022no—
140C2×C4—8022no—
141C2×C4—4012no—
142C2×C4—4011no—
143C2×C4—4014no—
144C2×C4—4011no—
145C2×C4—4011no—
146C2×C4C22×D40112yes—
147C2×C4C22×D40111yescharacteristic
148C2×C4C22×D40112yes—
149C22—22no—
150C22—21no—
151C22—22no—
152C22—8028no—
153C22—8022no—
154C22—8022no—
155C22—8021no—
156C22—8022no—
157C22—4014no—
158C22D4×D40112yes—
159C22C23×D40111yescharacteristic
160C4—22no—
161C4—21no—
162C4—22no—
163C4—8021no—
164C4—8022no—
165C4—4012no—
166C4—4012no—
167C2—22no—
168C2—41no—
169C2—8024no—
170C2—16041no—
171C2C2×D4×D40111yescharacteristic
172C2#41×D40112yes—
subgroupG/H
center, FrattiniC22C23×D401
commutator, socle, cosocleC22×C401C24
Fitting#123×C401C2
hypercenter#123D401
hypocenterC401#525
LCSG ▷ C22×C401 ▷ C401
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C2×#20264365C2semidirect
2. C22×C4×D401C2semidirect
3. #27165720C2semidirect
4. #27165715C2semidirect
5. #27165710C2semidirect
6. #27165707C2semidirect
7. #27165705C2semidirect
8. #27165699C2semidirect
9. #27165697C2semidirect
10. #27165693C2semidirect
11. #123×C401C2semidirect
14. C2×Dic802C22semidirect
18. C2×C22⋊D401C22semidirect
20. C22×Dic401C22semidirect
24. C2×C4×D401C22semidirect
25. #20264345C22semidirect
26. #20264345C22semidirect
30. #20264343C22semidirect
31. C4×Dic401C22semidirect
34. C4⋊C4×C401C22semidirect
36. C22⋊C4×C401C22semidirect
47. C2×Dic401D4semidirect
72. C2×D401#41semidirect
86. C401#525semidirect
91. #123D401semidirect
116. C4⋊C4C2×D401semidirect
122. C22⋊C4C2×D401semidirect

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
#271657202
#271657152
#271657102
#271657072
#271657052
#271656992
#271656972
#271656932
#123×C4012
#525401