Finite Group Databasebeta

← #29505404

Subgroups — #29505404

Order 15296.

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

normal subgroups only

This group has 428 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
1C22×C22⋊D239C212yesmaximal, complemented
2C2×D4×D239C212yesmaximal, complemented
3#21439956C211yesmaximal, characteristic, complemented
4#21439933C212yesmaximal, complemented
5#21439932C212yesmaximal, complemented
6#21439930C212yesmaximal, complemented
7#21439928C211yesmaximal, characteristic, complemented
8(C22⋊D239)⋊C4C212yesmaximal, complemented
9#120×C239C211yesmaximal, characteristic, complemented
10C23×D239C2211yescharacteristic, complemented
11C23×D239C2212yescomplemented
12C2×D956C2212yescomplemented
13C2×D956C2211yescharacteristic, complemented
14C2×C22⋊D239—28no—
15C2×C22⋊D239C2212yes—
16C2×C22⋊D239C2212yescomplemented
17C2×C22⋊D239C2212yescomplemented
18C2×C22⋊D239C2212yescomplemented
19C2×C22⋊D239C2212yescomplemented
20D4×D239—28no—
21C22×Dic239C2212yescomplemented
22C2×C4×D239C2212yescomplemented
23Dic239⋊C4C2212yescomplemented
24C22⋊Dic239C2211yescharacteristic, complemented
25(C2×D239)⋊C4C2212yescomplemented
26(C2×D239)⋊C4C2212yescomplemented
27C4×Dic239C2211yescharacteristic, complemented
28C24×C239C2211yescharacteristic, complemented
29C2×D4×C239C2212yescomplemented
30C2×D4×C239C2211yescharacteristic, complemented
31C22⋊C4×C239C2212yescomplemented
32C22⋊C4×C239C2211yescharacteristic, complemented
33C22×D239—24no—
34C22×D239—24no—
35C22×D239—24no—
36C22×D239—41no—
37C22×D239—42no—
38C22×D239C2312yes—
39C22×D239C2312yes—
40D956—24no—
41D956—41no—
42C22⋊D239—24no—
43C22⋊D239—24no—
44C22⋊D239—24no—
45C22⋊D239—24no—
46C22⋊D239D418yescomplemented
47C2×Dic239—24no—
48C2×Dic239C2312yes—
49C2×Dic239C2312yes—
50C4×D239—24no—
51C23×C239—24no—
52C23×C239—41no—
53C23×C239C2311yescharacteristic
54C23×C239C2311yescharacteristic
55C23×C239C2312yes—
56D4×C239—24no—
57D4×C239—41no—
58C2×C4×C239C2311yescharacteristic
59C2×C4×C239C2312yes—
60C22×C239—22no—
61C22×C239—22no—
62C22×C239—22no—
63C22×C239—22no—
64C22×C239—21no—
65C22×C239—21no—
66C22×C239—44no—
67C22×C239C2×D414yescomplemented
68C22×C239C2411yescharacteristic
69C2×D239—22no—
70C2×D239—22no—
71C2×D239—22no—
72C2×D239—22no—
73C2×D239—22no—
74C2×D239—44no—
75C2×D239—48no—
76C2×D239C2×D414yescomplemented
77C4×C239—22no—
78C4×C239—21no—
79Dic239—22no—
80Dic239C2×D414yescomplemented
81C2×C239—24no—
82C2×C239—41no—
83C2×C239—41no—
84C2×C239C22×D412yes—
85C2×C239H(5,𝔽2)11yescharacteristic
86D239—24no—
87D239—42no—
88C239D4211yescharacteristic, complemented, Sylow
89D42—2391nomaximal, Sylow
90C22×D4—2392no—
91C22×D4—2392no—
92#127—2391no—
93#121—2392no—
94#121—2392no—
95#120—2391no—
96#120—2392no—
97#120D23911yescharacteristic, complemented
98C4×D4—2392no—
99C24—2391no—
100C24—2392no—
101C24C2×D23911yescharacteristic, complemented
102C2×D4—4788no—
103C2×D4—4788no—
104C2×D4—2392no—
105C2×D4—2392no—
106C2×D4—2392no—
107C2×D4—2392no—
108C2×D4—2392no—
109C2×D4—2391no—
110C2×D4—2392no—
111C2×D4C2×D23912yescomplemented
112C2×D4C2×D23911yescharacteristic, complemented
113C22×C4—2392no—
114C22×C4—2392no—
115C4⋊C4—2392no—
116C22⋊C4—2391no—
117C22⋊C4—2392no—
118C22⋊C4—2392no—
119C22⋊C4C2×D23912yescomplemented
120C22⋊C4C2×D23911yescharacteristic, complemented
121C42—2391no—
122C23—24no—
123C23—41no—
124C23—9562no—
125C23—9561no—
126C23—4784no—
127C23—4784no—
128C23—4784no—
129C23—2392no—
130C23—2392no—
131C23C22×D23911yescharacteristic
132C23C22×D23911yescharacteristic
133C23C22×D23912yes—
134D4—24no—
135D4—41no—
136D4—9561no—
137D4—4784no—
138D4—4784no—
139D4—4784no—
140D4—4784no—
141D4—4784no—
142D4—2398no—
143C2×C4—4784no—
144C2×C4—4784no—
145C2×C4—2392no—
146C2×C4—2392no—
147C2×C4C22×D23911yescharacteristic
148C2×C4C22×D23912yes—
149C22—22no—
150C22—22no—
151C22—21no—
152C22—21no—
153C22—22no—
154C22—22no—
155C22—44no—
156C22—9568no—
157C22—9564no—
158C22—4782no—
159C22—4782no—
160C22—4782no—
161C22—4782no—
162C22—4782no—
163C22—2394no—
164C22D4×D23914yescomplemented
165C22C23×D23911yescharacteristic
166C4—21no—
167C4—22no—
168C4—4782no—
169C4—2394no—
170C2—24no—
171C2—41no—
172C2—41no—
173C2—9562no—
174C2—4784no—
175C2C2×D4×D23912yes—
176C2#2144000511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D239
commutator, socle, cosocleC22×C239C24
Fitting#120×C239C2
hypercenter#120D239
hypocenterC239D42
LCSG ▷ C22×C239 ▷ C239
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊D239C2semidirect
2. C2×D4×D239C2semidirect
3. #21439956C2semidirect
4. #21439933C2semidirect
5. #21439932C2semidirect
6. #21439930C2semidirect
7. #21439928C2semidirect
8. (C22⋊D239)⋊C4C2semidirect
9. #120×C239C2semidirect
10. C23×D239C22semidirect
11. C23×D239C22semidirect
12. C2×D956C22semidirect
13. C2×D956C22semidirect
16. C2×C22⋊D239C22semidirect
17. C2×C22⋊D239C22semidirect
18. C2×C22⋊D239C22semidirect
19. C2×C22⋊D239C22semidirect
21. C22×Dic239C22semidirect
22. C2×C4×D239C22semidirect
23. Dic239⋊C4C22semidirect
24. C22⋊Dic239C22semidirect
25. (C2×D239)⋊C4C22semidirect
26. (C2×D239)⋊C4C22semidirect
27. C4×Dic239C22semidirect
28. C24×C239C22semidirect
29. C2×D4×C239C22semidirect
30. C2×D4×C239C22semidirect
31. C22⋊C4×C239C22semidirect
32. C22⋊C4×C239C22semidirect
46. C22⋊D239D4semidirect
67. C22×C239C2×D4semidirect
76. C2×D239C2×D4semidirect
80. Dic239C2×D4semidirect
88. C239D42semidirect
97. #120D239semidirect
101. C24C2×D239semidirect
111. C2×D4C2×D239semidirect
112. C2×D4C2×D239semidirect
119. C22⋊C4C2×D239semidirect
120. C22⋊C4C2×D239semidirect
164. C22D4×D239semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊D2392
C2×D4×D2392
#214399562
#214399332
#214399322
#214399302
#214399282
(C22⋊D239)⋊C42
#120×C2392
D42239