Finite Group Databasebeta

← #37560066

Subgroups — #37560066

Order 23872.

334 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 115 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×#20238609C211yesmaximal, characteristic, complemented
2#26045779C211yesmaximal, characteristic, complemented
3#26045773C212yesmaximal, complemented
4#26045771C211yesmaximal, characteristic, complemented
5#26045764C212yesmaximal, complemented
6#26045763C211yesmaximal, characteristic, complemented
7#26045761C212yesmaximal, complemented
8D4×Dic373C211yesmaximal, characteristic, complemented
9C4×C22⋊D373C211yesmaximal, characteristic, complemented
10C2×C22⋊Dic373C212yesmaximal, complemented
11C22×D4×C373C211yesmaximal, characteristic, complemented
12#20238609—24no—
13C2×Dic746C2211yescharacteristic, complemented
14C2×C22⋊D373C2211yescharacteristic
15C2×C22⋊D373C2212yescomplemented
16C2×C22⋊D373C2212yescomplemented
17C22×Dic373C2212yescomplemented
18C22×Dic373C2212yescomplemented
19C2×C4×D373C2211yescharacteristic, complemented
20C4⋊Dic373C2211yescharacteristic, complemented
21#20238589C2211yescharacteristic, complemented
22#20238589C2212yescomplemented
23C22⋊Dic373C2211yescharacteristic
24C22⋊Dic373C2212yes—
25C22⋊Dic373C2212yescomplemented
26C22⋊Dic373C2212yescomplemented
27C22⋊Dic373C2212yescomplemented
28C22⋊Dic373C2212yescomplemented
29#20238587C2211yescharacteristic, complemented
30C4×Dic373C2211yescharacteristic, complemented
31C24×C373C2212yes—
32C2×D4×C373—24no—
33C2×D4×C373C2211yescharacteristic
34C2×D4×C373C2212yes—
35C2×D4×C373C2211yescharacteristic
36C22×C4×C373C2211yescharacteristic
37C22×D373C2311yescharacteristic
38Dic746—22no—
39C22⋊D373—24no—
40C22⋊D373—22no—
41C22⋊D373—42no—
42C2×Dic373—24no—
43C2×Dic373—42no—
44C2×Dic373C2311yescharacteristic
45C2×Dic373C2311yescharacteristic
46C2×Dic373C2312yes—
47C2×Dic373C2312yes—
48C2×Dic373C2311yescharacteristic
49C4×D373—22no—
50C23×C373—24no—
51C23×C373—24no—
52C23×C373—42no—
53C23×C373C2312yes—
54C23×C373C2312yes—
55C23×C373C2311yescharacteristic
56D4×C373—24no—
57D4×C373—22no—
58D4×C373D414yescomplemented
59C2×C4×C373—22no—
60C2×C4×C373C2311yescharacteristic
61C2×C4×C373C2311yescharacteristic
62C22×C373—21no—
63C22×C373—22no—
64C22×C373—21no—
65C22×C373—22no—
66C22×C373—22no—
67C22×C373—22no—
68C22×C373—22no—
69C22×C373—48no—
70C22×C373C2×D414yes—
71C22×C373#4112yescomplemented
72C22×C373C2411yescharacteristic
73C2×D373—21no—
74C2×D373—21no—
75C2×D373—21no—
76C4×C373—21no—
77C4×C373C2×D412yes—
78Dic373—21no—
79Dic373—22no—
80Dic373—21no—
81Dic373—21no—
82Dic373—22no—
83C2×C373—22no—
84C2×C373—24no—
85C2×C373—42no—
86C2×C373C22×D411yescharacteristic
87C2×C373C2×#4111yescharacteristic
88C2×C373H(5,𝔽2)11yescharacteristic
89D373—41no—
90C373#54611yescharacteristic, complemented, Sylow
91#546—3731nomaximal, Sylow
92C2×#41—3731no—
93C22×D4D37311yescharacteristic, complemented
94#124—3731no—
95#123—3732no—
96C22⋊Q8—3731no—
97#121—3731no—
98#121—3732no—
99#120—3732no—
100C4×D4—3731no—
101C4×D4—3731no—
102C2×C22⋊C4—3732no—
103C24C2×D37312yes—
104#41—7464no—
105C2×Q8—3731no—
106C2×D4—24no—
107C2×D4—3732no—
108C2×D4—3731no—
109C2×D4—3732no—
110C2×D4C2×D37312yes—
111C2×D4C2×D37311yescharacteristic
112C2×D4C2×D37311yescharacteristic
113C22×C4—3732no—
114C22×C4—3732no—
115C22×C4—3731no—
116C22×C4C2×D37311yescharacteristic
117C4⋊C4—3731no—
118C4⋊C4—3731no—
119C4⋊C4—3732no—
120C22⋊C4—3731no—
121C22⋊C4—3732no—
122C22⋊C4—3732no—
123C22⋊C4—3732no—
124C22⋊C4—3732no—
125C22⋊C4—3732no—
126C22⋊C4—3731no—
127C42—3731no—
128C23—24no—
129C23—24no—
130C23—42no—
131C23—3731no—
132C23C22×D37312yes—
133C23C22×D37312yes—
134C23C22×D37311yescharacteristic
135Q8—7462no—
136D4—24no—
137D4—22no—
138D4—14922no—
139D4—7464no—
140D4—7462no—
141D4C22⋊D37314yescomplemented
142C2×C4—22no—
143C2×C4—3731no—
144C2×C4—3731no—
145C2×C4—3732no—
146C2×C4—3731no—
147C2×C4—3732no—
148C2×C4—14922no—
149C2×C4—7464no—
150C2×C4—7462no—
151C2×C4C22×D37311yescharacteristic
152C2×C4C22×D37311yescharacteristic
153C22—21no—
154C22—22no—
155C22—22no—
156C22—21no—
157C22—22no—
158C22—22no—
159C22—22no—
160C22—48no—
161C22—7461no—
162C22—7461no—
163C22—7461no—
164C22C2×C22⋊D37314yes—
165C22#2023860912yescomplemented
166C22C23×D37311yescharacteristic
167C4—21no—
168C4—7461no—
169C4—7462no—
170C4—7461no—
171C4—7461no—
172C4—7462no—
173C4C2×C22⋊D37312yes—
174C2—22no—
175C2—24no—
176C2—42no—
177C2—14921no—
178C2C22×C22⋊D37311yescharacteristic
179C2C2×#2023860911yescharacteristic
180C2#2604583711yescharacteristic
subgroupG/H
center, FrattiniC22C23×D373
commutator, socle, cosocleC22×C373C24
FittingC22×D4×C373C2
hypercenterC22×D4D373
hypocenterC373#546
LCSG ▷ C22×C373 ▷ C373
UCSC1 ◁ C22 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C2×#20238609C2semidirect
2. #26045779C2semidirect
3. #26045773C2semidirect
4. #26045771C2semidirect
5. #26045764C2semidirect
6. #26045763C2semidirect
7. #26045761C2semidirect
8. D4×Dic373C2semidirect
9. C4×C22⋊D373C2semidirect
10. C2×C22⋊Dic373C2semidirect
11. C22×D4×C373C2semidirect
13. C2×Dic746C22semidirect
15. C2×C22⋊D373C22semidirect
16. C2×C22⋊D373C22semidirect
17. C22×Dic373C22semidirect
18. C22×Dic373C22semidirect
19. C2×C4×D373C22semidirect
20. C4⋊Dic373C22semidirect
21. #20238589C22semidirect
22. #20238589C22semidirect
25. C22⋊Dic373C22semidirect
26. C22⋊Dic373C22semidirect
27. C22⋊Dic373C22semidirect
28. C22⋊Dic373C22semidirect
29. #20238587C22semidirect
30. C4×Dic373C22semidirect
58. D4×C373D4semidirect
71. C22×C373#41semidirect
90. C373#546semidirect
93. C22×D4D373semidirect
141. D4C22⋊D373semidirect
165. C22#20238609semidirect

Maximal subgroups

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

subgroupindex
C2×#202386092
#260457792
#260457732
#260457712
#260457642
#260457632
#260457612
D4×Dic3732
C4×C22⋊D3732
C2×C22⋊Dic3732
C22×D4×C3732
#546373