Finite Group Databasebeta

← #37560068

Subgroups — #37560068

Order 23872.

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

normal subgroups only

This group has 334 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×#20238610C211yesmaximal, characteristic, complemented
2C22×D1492C211yesmaximal, characteristic, complemented
3#26045780C211yesmaximal, characteristic, complemented
4#26045777C212yesmaximal, complemented
5#26045771C211yesmaximal, characteristic, complemented
6#26045766C211yesmaximal, characteristic, complemented
7#26045765C212yesmaximal, complemented
8#26045760C212yesmaximal, complemented
9C4×D1492C211yesmaximal, characteristic, complemented
10C22⋊C4×D373C212yesmaximal, complemented
11C4×D4×C373C211yesmaximal, characteristic, complemented
12C23×D373C2212yes—
13#20238610—24no—
14C2×Dic746C2211yescharacteristic, complemented
15C2×D1492—24no—
16C2×D1492C2212yes—
17C2×D1492C2211yescharacteristic
18C2×D1492C2211yescharacteristic
19C2×C22⋊D373C2212yescomplemented
20C2×C22⋊D373C2212yescomplemented
21C2×C4×D373C2212yescomplemented
22C2×C4×D373C2212yescomplemented
23C4⋊Dic373C2211yescharacteristic, complemented
24#20238589C2212yescomplemented
25C22⋊Dic373C2212yes—
26#20238587C2212yescomplemented
27#20238587C2212yescomplemented
28#20238587C2212yescomplemented
29#20238587C2212yescomplemented
30C2×D4×C373C2211yescharacteristic
31C22×C4×C373C2211yescharacteristic
32C22×C4×C373C2211yescharacteristic, complemented
33C4⋊C4×C373C2211yescharacteristic, complemented
34C22⋊C4×C373C2211yescharacteristic
35C22⋊C4×C373C2211yescharacteristic, complemented
36C42×C373C2211yescharacteristic, complemented
37C22×D373—24no—
38C22×D373—24no—
39C22×D373—42no—
40C22×D373C2312yes—
41C22×D373C2312yes—
42Dic746—22no—
43D1492—24no—
44D1492—22no—
45D1492D414yescomplemented
46C22⋊D373—24no—
47C22⋊D373—42no—
48C2×Dic373C2312yes—
49C2×Dic373C2312yes—
50C4×D373—24no—
51C4×D373—42no—
52C23×C373C2311yescharacteristic
53C23×C373C2311yescharacteristic
54D4×C373—22no—
55C2×C4×C373—22no—
56C2×C4×C373—22no—
57C2×C4×C373C2311yescharacteristic
58C2×C4×C373C2311yescharacteristic
59C2×C4×C373C2311yescharacteristic
60C2×C4×C373C2311yescharacteristic
61C2×C4×C373C2311yescharacteristic
62C22×C373—21no—
63C22×C373—21no—
64C22×C373—21no—
65C22×C373—21no—
66C22×C373—21no—
67C22×C373#4112yescomplemented
68C22×C373C2411yescharacteristic
69C2×D373—22no—
70C2×D373—22no—
71C2×D373—22no—
72C2×D373—22no—
73C2×D373—22no—
74C2×D373—48no—
75C2×D373C2×D414yes—
76C4×C373—21no—
77C4×C373—21no—
78C4×C373—21no—
79C4×C373—21no—
80C4×C373C2×D412yes—
81Dic373—22no—
82Dic373—22no—
83C2×C373—22no—
84C2×C373—41no—
85C2×C373C22×D411yescharacteristic
86C2×C373C2×#4111yescharacteristic
87C2×C373H(5,𝔽2)11yescharacteristic
88D373—24no—
89D373—42no—
90C373#54611yescharacteristic, complemented, Sylow
91#546—3731nomaximal, Sylow
92C2×#41—3731no—
93C22×D4—3731no—
94#124—3731no—
95#123—3732no—
96C22⋊Q8—3731no—
97#121—3731no—
98#121—3732no—
99#120—3732no—
100C4×D4—3731no—
101C4×D4D37311yescharacteristic, complemented
102C2×C22⋊C4—3732no—
103C24—3732no—
104#41—7464no—
105C2×Q8—3731no—
106C2×D4—3732no—
107C2×D4—3732no—
108C2×D4—3731no—
109C2×D4—3732no—
110C2×D4—3731no—
111C2×D4—7464no—
112C2×D4C2×D37311yescharacteristic
113C22×C4—3732no—
114C22×C4—3732no—
115C22×C4C2×D37311yescharacteristic
116C22×C4C2×D37311yescharacteristic, complemented
117C4⋊C4—3731no—
118C4⋊C4—3732no—
119C4⋊C4C2×D37311yescharacteristic, complemented
120C22⋊C4—3732no—
121C22⋊C4—3732no—
122C22⋊C4—3732no—
123C22⋊C4—3732no—
124C22⋊C4—3732no—
125C22⋊C4C2×D37311yescharacteristic
126C22⋊C4C2×D37311yescharacteristic, complemented
127C42C2×D37311yescharacteristic, complemented
128C23—3732no—
129C23—3732no—
130C23—14922no—
131C23—7464no—
132C23—7464no—
133C23C22×D37311yescharacteristic
134C23C22×D37311yescharacteristic
135Q8—7462no—
136D4—22no—
137D4—3734no—
138D4—14922no—
139D4—7464no—
140D4—7464no—
141D4—7462no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—3732no—
145C2×C4—3732no—
146C2×C4—14922no—
147C2×C4—7464no—
148C2×C4C22×D37311yescharacteristic
149C2×C4C22×D37311yescharacteristic
150C2×C4C22×D37311yescharacteristic
151C2×C4C22×D37311yescharacteristic
152C2×C4C22×D37311yescharacteristic
153C22—21no—
154C22—21no—
155C22—21no—
156C22—21no—
157C22—21no—
158C22—3734no—
159C22—14928no—
160C22—7462no—
161C22—7462no—
162C22—7462no—
163C22—7462no—
164C22—7462no—
165C22#2023861012yescomplemented
166C22C23×D37311yescharacteristic
167C4—21no—
168C4—21no—
169C4—21no—
170C4—21no—
171C4—7462no—
172C4—7462no—
173C4D4×D37312yes—
174C2—22no—
175C2—41no—
176C2—14922no—
177C2—7464no—
178C2C2×D4×D37311yescharacteristic
179C2C2×#2023861011yescharacteristic
180C2#2604583811yescharacteristic
subgroupG/H
center, FrattiniC22C23×D373
commutator, socle, cosocleC22×C373C24
FittingC4×D4×C373C2
hypercenterC4×D4D373
hypocenterC373#546
LCSG ▷ C22×C373 ▷ C373
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#20238610C2semidirect
2. C22×D1492C2semidirect
3. #26045780C2semidirect
4. #26045777C2semidirect
5. #26045771C2semidirect
6. #26045766C2semidirect
7. #26045765C2semidirect
8. #26045760C2semidirect
9. C4×D1492C2semidirect
10. C22⋊C4×D373C2semidirect
11. C4×D4×C373C2semidirect
14. C2×Dic746C22semidirect
19. C2×C22⋊D373C22semidirect
20. C2×C22⋊D373C22semidirect
21. C2×C4×D373C22semidirect
22. C2×C4×D373C22semidirect
23. C4⋊Dic373C22semidirect
24. #20238589C22semidirect
26. #20238587C22semidirect
27. #20238587C22semidirect
28. #20238587C22semidirect
29. #20238587C22semidirect
32. C22×C4×C373C22semidirect
33. C4⋊C4×C373C22semidirect
35. C22⋊C4×C373C22semidirect
36. C42×C373C22semidirect
45. D1492D4semidirect
67. C22×C373#41semidirect
90. C373#546semidirect
101. C4×D4D373semidirect
116. C22×C4C2×D373semidirect
119. C4⋊C4C2×D373semidirect
126. C22⋊C4C2×D373semidirect
127. C42C2×D373semidirect
165. C22#20238610semidirect

Maximal subgroups

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

subgroupindex
C2×#202386102
C22×D14922
#260457802
#260457772
#260457712
#260457662
#260457652
#260457602
C4×D14922
C22⋊C4×D3732
C4×D4×C3732
#546373