Finite Group Databasebeta

← #37527088

Subgroups — #37527088

Order 22976.

334 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 113 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×#19153340C211yesmaximal, characteristic, complemented
2C2×D4×D359C211yesmaximal, characteristic, complemented
3#26040003C211yesmaximal, characteristic, complemented
4#26039999C212yesmaximal, complemented
5#26039995C211yesmaximal, characteristic, complemented
6#26039990C211yesmaximal, characteristic, complemented
7#26039989C212yesmaximal, complemented
8#26039985C212yesmaximal, complemented
9C4×D1436C211yesmaximal, characteristic, complemented
10C2×(C2×D359)⋊C4C212yesmaximal, complemented
11C4×D4×C359C211yesmaximal, characteristic, complemented
12C23×D359C2212yes—
13#19153340—24no—
14C2×Dic718C2211yescharacteristic, complemented
15C2×D1436C2211yescharacteristic
16C2×D1436C2211yescharacteristic
17C2×D1436C2212yescomplemented
18C2×C22⋊D359C2212yes—
19C2×C22⋊D359C2212yescomplemented
20D4×D359—24no—
21C22×Dic359C2212yescomplemented
22C2×C4×D359C2211yescharacteristic
23C2×C4×D359C2211yescharacteristic, complemented
24C4⋊Dic359C2211yescharacteristic, complemented
25C4⋊Dic359C2212yescomplemented
26(C2×D359)⋊C4C2211yescharacteristic
27(C2×D359)⋊C4C2212yes—
28(C2×D359)⋊C4C2212yescomplemented
29(C2×D359)⋊C4C2212yescomplemented
30(C2×D359)⋊C4C2211yescharacteristic, complemented
31(C2×D359)⋊C4C2212yescomplemented
32C2×D4×C359C2211yescharacteristic
33C22×C4×C359C2212yescomplemented
34C4⋊C4×C359C2211yescharacteristic, complemented
35C22⋊C4×C359C2212yescomplemented
36C42×C359C2211yescharacteristic, complemented
37C22×D359—24no—
38C22×D359—24no—
39C22×D359—42no—
40C22×D359C2312yes—
41C22×D359C2311yescharacteristic
42C22×D359C2311yescharacteristic
43Dic718—22no—
44D1436—22no—
45D1436—22no—
46D1436—42no—
47C22⋊D359—24no—
48C22⋊D359—24no—
49C2×Dic359—24no—
50C2×Dic359C2311yescharacteristic
51C2×Dic359C2312yes—
52C2×Dic359C2311yescharacteristic
53C4×D359—22no—
54C4×D359—22no—
55C23×C359C2312yes—
56D4×C359D414yescomplemented
57C2×C4×C359—42no—
58C2×C4×C359C2311yescharacteristic
59C2×C4×C359C2311yescharacteristic
60C2×C4×C359C2312yes—
61C2×C4×C359C2311yescharacteristic
62C22×C359—22no—
63C22×C359—22no—
64C22×C359C2×D414yes—
65C22×C359C2411yescharacteristic
66C2×D359—21no—
67C2×D359—21no—
68C2×D359—21no—
69C2×D359—22no—
70C2×D359—21no—
71C2×D359—22no—
72C2×D359—21no—
73C2×D359—22no—
74C2×D359—48no—
75C2×D359#4112yescomplemented
76C4×C359—21no—
77C4×C359—21no—
78C4×C359—22no—
79C4×C359C2×D412yes—
80Dic359—21no—
81Dic359—21no—
82Dic359—22no—
83C2×C359—24no—
84C2×C359C22×D411yescharacteristic
85C2×C359C2×#4111yescharacteristic
86C2×C359H(5,𝔽2)11yescharacteristic
87D359—22no—
88D359—41no—
89D359—42no—
90C359#54611yescharacteristic, complemented, Sylow
91#546—3591nomaximal, Sylow
92C2×#41—3591no—
93C22×D4—3591no—
94#124—3591no—
95#123—3592no—
96C22⋊Q8—3591no—
97#121—3592no—
98#121—3591no—
99#120—3592no—
100C4×D4—3591no—
101C4×D4D35911yescharacteristic, complemented
102C2×C22⋊C4—3592no—
103C24—3592no—
104#41—7184no—
105C2×Q8—3591no—
106C2×D4—3592no—
107C2×D4—3592no—
108C2×D4—3592no—
109C2×D4—3591no—
110C2×D4—3591no—
111C2×D4—7184no—
112C2×D4C2×D35911yescharacteristic
113C22×C4—3592no—
114C22×C4—3591no—
115C22×C4—3591no—
116C22×C4C2×D35912yescomplemented
117C4⋊C4—3591no—
118C4⋊C4—3592no—
119C4⋊C4C2×D35911yescharacteristic, complemented
120C22⋊C4—3591no—
121C22⋊C4—3592no—
122C22⋊C4—3592no—
123C22⋊C4—3592no—
124C22⋊C4—3591no—
125C22⋊C4—3592no—
126C22⋊C4C2×D35912yescomplemented
127C42C2×D35911yescharacteristic, complemented
128C23—3591no—
129C23—3592no—
130C23—3591no—
131C23—14362no—
132C23—7184no—
133C23—7184no—
134C23C22×D35912yes—
135Q8—7182no—
136D4—14362no—
137D4—7182no—
138D4—7184no—
139D4—7184no—
140D4—7182no—
141D4D143614yescomplemented
142C2×C4—42no—
143C2×C4—3591no—
144C2×C4—3592no—
145C2×C4—3591no—
146C2×C4—7184no—
147C2×C4—7182no—
148C2×C4—7182no—
149C2×C4C22×D35911yescharacteristic
150C2×C4C22×D35911yescharacteristic
151C2×C4C22×D35912yes—
152C2×C4C22×D35911yescharacteristic
153C22—22no—
154C22—22no—
155C22—3592no—
156C22—14368no—
157C22—7181no—
158C22—7182no—
159C22—7181no—
160C22—7182no—
161C22—7181no—
162C22—7182no—
163C22—7181no—
164C22—7181no—
165C22C2×D143614yes—
166C22C23×D35911yescharacteristic
167C4—21no—
168C4—21no—
169C4—22no—
170C4—7181no—
171C4—7181no—
172C4—7182no—
173C4C2×D143612yes—
174C2—24no—
175C2—14362no—
176C2—14361no—
177C2—7182no—
178C2C22×D143611yescharacteristic
179C2#41×D35911yescharacteristic
180C2#2604006011yescharacteristic
subgroupG/H
center, FrattiniC22C23×D359
commutator, socle, cosocleC22×C359C24
FittingC4×D4×C359C2
hypercenterC4×D4D359
hypocenterC359#546
LCSG ▷ C22×C359 ▷ C359
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#19153340C2semidirect
2. C2×D4×D359C2semidirect
3. #26040003C2semidirect
4. #26039999C2semidirect
5. #26039995C2semidirect
6. #26039990C2semidirect
7. #26039989C2semidirect
8. #26039985C2semidirect
9. C4×D1436C2semidirect
10. C2×(C2×D359)⋊C4C2semidirect
11. C4×D4×C359C2semidirect
14. C2×Dic718C22semidirect
17. C2×D1436C22semidirect
19. C2×C22⋊D359C22semidirect
21. C22×Dic359C22semidirect
23. C2×C4×D359C22semidirect
24. C4⋊Dic359C22semidirect
25. C4⋊Dic359C22semidirect
28. (C2×D359)⋊C4C22semidirect
29. (C2×D359)⋊C4C22semidirect
30. (C2×D359)⋊C4C22semidirect
31. (C2×D359)⋊C4C22semidirect
33. C22×C4×C359C22semidirect
34. C4⋊C4×C359C22semidirect
35. C22⋊C4×C359C22semidirect
36. C42×C359C22semidirect
56. D4×C359D4semidirect
75. C2×D359#41semidirect
90. C359#546semidirect
101. C4×D4D359semidirect
116. C22×C4C2×D359semidirect
119. C4⋊C4C2×D359semidirect
126. C22⋊C4C2×D359semidirect
127. C42C2×D359semidirect
141. D4D1436semidirect

Maximal subgroups

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

subgroupindex
C2×#191533402
C2×D4×D3592
#260400032
#260399992
#260399952
#260399902
#260399892
#260399852
C4×D14362
C2×(C2×D359)⋊C42
C4×D4×C3592
#546359