Finite Group Databasebeta

← #37526956

Subgroups — #37526956

Order 22976.

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×#19153342C211yesmaximal, characteristic, complemented
2C22×C4×D359C211yesmaximal, characteristic, complemented
3#26040000C212yesmaximal, complemented
4#26039991C211yesmaximal, characteristic, complemented
5#26039990C211yesmaximal, characteristic, complemented
6#26039989C211yesmaximal, characteristic, complemented
7#26039985C212yesmaximal, complemented
8#26039977C212yesmaximal, complemented
9(C22⋊D359)⋊C4C212yesmaximal, complemented
10#26039973C211yesmaximal, characteristic, complemented
11C22⋊Q8×C359C211yesmaximal, characteristic, complemented
12C23×D359C2211yescharacteristic
13#19153342—24no—
14C2×D1436C2212yes—
15C2×D1436C2212yes—
16C2×D1436C2211yescharacteristic, complemented
17C2×C22⋊D359C2212yes—
18C22×Dic359C2211yescharacteristic
19C2×C4×D359—24no—
20C2×C4×D359C2211yescharacteristic
21C2×C4×D359C2211yescharacteristic
22C2×C4×D359C2211yescharacteristic
23C2×C4×D359C2211yescharacteristic
24C2×C4×D359C2212yescomplemented
25C4⋊Dic359C2211yescharacteristic, complemented
26Dic359⋊C4C2212yescomplemented
27(C2×D359)⋊C4C2212yes—
28(C2×D359)⋊C4C2212yes—
29(C2×D359)⋊C4C2212yescomplemented
30(C2×D359)⋊C4C2212yescomplemented
31C4×Dic359C2212yescomplemented
32C2×Q8×C359C2211yescharacteristic, complemented
33C22×C4×C359C2211yescharacteristic
34C4⋊C4×C359C2211yescharacteristic, complemented
35C4⋊C4×C359C2212yescomplemented
36C22⋊C4×C359C2212yes—
37C22×D359—22no—
38C22×D359—22no—
39C22×D359—22no—
40C22×D359C2312yes—
41C22×D359C2311yescharacteristic
42C22×D359C2311yescharacteristic
43D1436—24no—
44D1436—24no—
45D1436—22no—
46C22⋊D359—24no—
47C2×Dic359—22no—
48C2×Dic359C2312yes—
49C2×Dic359C2311yescharacteristic
50C2×Dic359C2311yescharacteristic
51C4×D359—22no—
52C4×D359—22no—
53C4×D359—22no—
54C4×D359—24no—
55C4×D359D414yescomplemented
56C23×C359C2311yescharacteristic
57Q8×C359—22no—
58C2×C4×C359—22no—
59C2×C4×C359C2312yes—
60C2×C4×C359C2312yes—
61C2×C4×C359C2311yescharacteristic
62C2×C4×C359C2311yescharacteristic
63C22×C359—21no—
64C22×C359—21no—
65C22×C359#4112yescomplemented
66C22×C359C2411yescharacteristic
67C2×D359—28no—
68C2×D359—21no—
69C2×D359—21no—
70C2×D359—21no—
71C2×D359—21no—
72C2×D359—22no—
73C2×D359—22no—
74C2×D359—22no—
75C2×D359C2×D412yes—
76C2×D359#4112yescomplemented
77C4×C359—21no—
78C4×C359—22no—
79C4×C359—22no—
80C4×C359C2×D412yes—
81Dic359—21no—
82Dic359—22no—
83Dic359C2×D412yes—
84C2×C359—22no—
85C2×C359C22×D411yescharacteristic
86C2×C359C2×#4111yescharacteristic
87C2×C359C2×#4111yescharacteristic
88D359—22no—
89D359—22no—
90D359—42no—
91C359#52511yescharacteristic, complemented, Sylow
92#525—3591nomaximal, Sylow
93C2×#41—3591no—
94C23×C4—3591no—
95#123—3592no—
96C22⋊Q8—3591no—
97C22⋊Q8D35911yescharacteristic, complemented
98#121—3591no—
99#121—3591no—
100#120—3592no—
101C4×D4—3592no—
102C4×D4—3592no—
103#117—3591no—
104C24—3591no—
105#41—7184no—
106C2×Q8C2×D35911yescharacteristic, complemented
107C2×D4—3592no—
108C2×D4—3592no—
109C2×D4—3592no—
110C2×D4—3591no—
111C22×C4—3591no—
112C22×C4—3591no—
113C22×C4—3591no—
114C22×C4—3592no—
115C22×C4—3591no—
116C22×C4—3591no—
117C22×C4—7184no—
118C22×C4C2×D35911yescharacteristic
119C4⋊C4—3591no—
120C4⋊C4—3592no—
121C4⋊C4C2×D35911yescharacteristic, complemented
122C4⋊C4C2×D35912yescomplemented
123C22⋊C4—3592no—
124C22⋊C4—3592no—
125C22⋊C4—3592no—
126C22⋊C4—3592no—
127C22⋊C4C2×D35912yes—
128C42—3592no—
129C23—3591no—
130C23—3591no—
131C23—3592no—
132C23—7182no—
133C23—7182no—
134C23—7182no—
135C23C22×D35911yescharacteristic
136Q8—22no—
137D4—7184no—
138D4—7184no—
139D4—7184no—
140D4—7182no—
141C2×C4—22no—
142C2×C4—3591no—
143C2×C4—3594no—
144C2×C4—3592no—
145C2×C4—3591no—
146C2×C4—7182no—
147C2×C4—7184no—
148C2×C4—7182no—
149C2×C4—7182no—
150C2×C4—7182no—
151C2×C4C22×D35912yes—
152C2×C4C22×D35911yescharacteristic
153C2×C4C22×D35912yes—
154C2×C4C22×D35911yescharacteristic
155C22—21no—
156C22—21no—
157C22—3592no—
158C22—3592no—
159C22—7188no—
160C22—7182no—
161C22—7182no—
162C22—7182no—
163C22—7181no—
164C22—7181no—
165C22—7181no—
166C22—7181no—
167C22#1915334212yescomplemented
168C22C23×D35911yescharacteristic
169C4—21no—
170C4—22no—
171C4—22no—
172C4—3592no—
173C4—7181no—
174C4—7182no—
175C4D4×D35912yes—
176C2—22no—
177C2—14362no—
178C2—7182no—
179C2—7182no—
180C2C2×D4×D35911yescharacteristic
181C2#41×D35911yescharacteristic
182C2C2×#1915334211yescharacteristic
subgroupG/H
center, FrattiniC22C23×D359
commutator, socle, cosocleC22×C359C24
FittingC22⋊Q8×C359C2
hypercenterC22⋊Q8D359
hypocenterC359#525
LCSG ▷ C22×C359 ▷ C359
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#19153342C2semidirect
2. C22×C4×D359C2semidirect
3. #26040000C2semidirect
4. #26039991C2semidirect
5. #26039990C2semidirect
6. #26039989C2semidirect
7. #26039985C2semidirect
8. #26039977C2semidirect
9. (C22⋊D359)⋊C4C2semidirect
10. #26039973C2semidirect
11. C22⋊Q8×C359C2semidirect
16. C2×D1436C22semidirect
24. C2×C4×D359C22semidirect
25. C4⋊Dic359C22semidirect
26. Dic359⋊C4C22semidirect
29. (C2×D359)⋊C4C22semidirect
30. (C2×D359)⋊C4C22semidirect
31. C4×Dic359C22semidirect
32. C2×Q8×C359C22semidirect
34. C4⋊C4×C359C22semidirect
35. C4⋊C4×C359C22semidirect
55. C4×D359D4semidirect
65. C22×C359#41semidirect
76. C2×D359#41semidirect
91. C359#525semidirect
97. C22⋊Q8D359semidirect
106. C2×Q8C2×D359semidirect
121. C4⋊C4C2×D359semidirect
122. C4⋊C4C2×D359semidirect
167. C22#19153342semidirect

Maximal subgroups

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

subgroupindex
C2×#191533422
C22×C4×D3592
#260400002
#260399912
#260399902
#260399892
#260399852
#260399772
(C22⋊D359)⋊C42
#260399732
C22⋊Q8×C3592
#525359