Finite Group Databasebeta

← #37588916

Subgroups — #37588916

Order 24512.

334 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 103 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×#20259456C211yesmaximal, characteristic, complemented
2C22×C22⋊D383C211yesmaximal, characteristic, complemented
3#27133837C212yesmaximal, complemented
4#27133827C212yesmaximal, complemented
5#27133814C212yesmaximal, complemented
6#27133813C212yesmaximal, complemented
7#27133811C211yesmaximal, characteristic, complemented
8#27133810C211yesmaximal, characteristic, complemented
9#27133809C211yesmaximal, characteristic, complemented
10#27133797C211yesmaximal, characteristic, complemented
11#120×C383C211yesmaximal, characteristic, complemented
12C23×D383C2211yescharacteristic
13#20259456—24no—
14C2×Dic766C2211yescharacteristic, complemented
15C2×D1532C2212yescomplemented
16C2×C22⋊D383—24no—
17C2×C22⋊D383C2211yescharacteristic
18C2×C22⋊D383C2212yes—
19C2×C22⋊D383C2212yes—
20C2×C22⋊D383C2211yescharacteristic, complemented
21C2×C22⋊D383C2212yescomplemented
22C2×C22⋊D383C2212yescomplemented
23C22×Dic383C2211yescharacteristic
24C22×Dic383C2211yescharacteristic, complemented
25C2×C4×D383C2211yescharacteristic, complemented
26Dic383⋊C4C2212yescomplemented
27C22⋊Dic383C2211yescharacteristic
28C22⋊Dic383C2212yescomplemented
29(C2×D383)⋊C4C2212yescomplemented
30(C2×D383)⋊C4C2212yescomplemented
31C4×Dic383C2212yescomplemented
32C24×C383C2211yescharacteristic
33C2×D4×C383C2211yescharacteristic
34C2×D4×C383C2212yescomplemented
35C22⋊C4×C383C2211yescharacteristic
36C22⋊C4×C383C2212yescomplemented
37C22×D383—22no—
38C22×D383—42no—
39C22×D383C2312yes—
40C22×D383C2311yescharacteristic
41Dic766—22no—
42D1532—42no—
43C22⋊D383—22no—
44C22⋊D383—24no—
45C22⋊D383—24no—
46C22⋊D383—22no—
47C22⋊D383—42no—
48C22⋊D383—42no—
49C2×Dic383—22no—
50C2×Dic383D414yescomplemented
51C2×Dic383C2312yes—
52C2×Dic383C2311yescharacteristic
53C2×Dic383C2311yescharacteristic
54C2×Dic383C2311yescharacteristic
55C4×D383—22no—
56C23×C383—22no—
57C23×C383—42no—
58C23×C383C2312yes—
59C23×C383C2311yescharacteristic
60C23×C383C2311yescharacteristic
61D4×C383—22no—
62D4×C383—42no—
63C2×C4×C383C2312yes—
64C2×C4×C383C2311yescharacteristic
65C22×C383—22no—
66C22×C383—21no—
67C22×C383—22no—
68C22×C383—22no—
69C22×C383—22no—
70C22×C383—22no—
71C22×C383—44no—
72C22×C383C2×D412yes—
73C22×C383C2411yescharacteristic
74C2×D383—21no—
75C2×D383—22no—
76C2×D383—22no—
77C2×D383—22no—
78C2×D383—22no—
79C2×D383—44no—
80C4×C383—21no—
81C4×C383—22no—
82Dic383—21no—
83Dic383—22no—
84Dic383C2×D412yes—
85Dic383C2×D412yes—
86C2×C383—22no—
87C2×C383—41no—
88C2×C383—42no—
89C2×C383C22×D411yescharacteristic
90C2×C383H(5,𝔽2)12yes—
91D383—41no—
92D383—42no—
93C383#53511yescharacteristic, complemented, Sylow
94#535—3831nomaximal, Sylow
95C2×#41—3831no—
96C22×D4—3831no—
97#127—3832no—
98#124—3832no—
99#121—3832no—
100#121—3832no—
101#120—3831no—
102#120—3831no—
103#120—3831no—
104#120D38311yescharacteristic, complemented
105#117—3831no—
106C24—3831no—
107C24C2×D38311yescharacteristic
108#41—7664no—
109C2×Q8—3831no—
110C2×D4—3832no—
111C2×D4—3832no—
112C2×D4—3831no—
113C2×D4—3832no—
114C2×D4—3832no—
115C2×D4—3831no—
116C2×D4—3832no—
117C2×D4—7664no—
118C2×D4C2×D38311yescharacteristic
119C2×D4C2×D38312yescomplemented
120C22×C4—3831no—
121C22×C4—3831no—
122C22×C4—3831no—
123C4⋊C4—3832no—
124C22⋊C4—3831no—
125C22⋊C4—3832no—
126C22⋊C4—3832no—
127C22⋊C4—3832no—
128C22⋊C4C2×D38311yescharacteristic
129C22⋊C4C2×D38312yescomplemented
130C42—3832no—
131C23—22no—
132C23—42no—
133C23—3832no—
134C23—3831no—
135C23—15322no—
136C23—7662no—
137C23C22×D38312yes—
138C23C22×D38311yescharacteristic
139C23C22×D38311yescharacteristic
140Q8—7662no—
141D4—22no—
142D4—42no—
143D4—15322no—
144D4—15322no—
145D4—15322no—
146D4—7664no—
147D4—7664no—
148D4—7662no—
149D4—7662no—
150C2×C4—3834no—
151C2×C4—3831no—
152C2×C4—3831no—
153C2×C4—3832no—
154C2×C4—3831no—
155C2×C4—7662no—
156C2×C4—7662no—
157C2×C4C22×D38312yes—
158C2×C4C22×D38311yescharacteristic
159C22—22no—
160C22—21no—
161C22—22no—
162C22—22no—
163C22—22no—
164C22—22no—
165C22—44no—
166C22—15324no—
167C22—7661no—
168C22—7662no—
169C22—7662no—
170C22—7662no—
171C22—7662no—
172C22D4×D38312yes—
173C22C23×D38311yescharacteristic
174C4—21no—
175C4—22no—
176C4—3832no—
177C4—3832no—
178C4—7661no—
179C4—7662no—
180C2—22no—
181C2—42no—
182C2—41no—
183C2—15322no—
184C2—15321no—
185C2C2×D4×D38311yescharacteristic
186C2#2713388612yes—
subgroupG/H
center, FrattiniC22C23×D383
commutator, socle, cosocleC22×C383C24
Fitting#120×C383C2
hypercenter#120D383
hypocenterC383#535
LCSG ▷ C22×C383 ▷ C383
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C2×#20259456C2semidirect
2. C22×C22⋊D383C2semidirect
3. #27133837C2semidirect
4. #27133827C2semidirect
5. #27133814C2semidirect
6. #27133813C2semidirect
7. #27133811C2semidirect
8. #27133810C2semidirect
9. #27133809C2semidirect
10. #27133797C2semidirect
11. #120×C383C2semidirect
14. C2×Dic766C22semidirect
15. C2×D1532C22semidirect
20. C2×C22⋊D383C22semidirect
21. C2×C22⋊D383C22semidirect
22. C2×C22⋊D383C22semidirect
24. C22×Dic383C22semidirect
25. C2×C4×D383C22semidirect
26. Dic383⋊C4C22semidirect
28. C22⋊Dic383C22semidirect
29. (C2×D383)⋊C4C22semidirect
30. (C2×D383)⋊C4C22semidirect
31. C4×Dic383C22semidirect
34. C2×D4×C383C22semidirect
36. C22⋊C4×C383C22semidirect
50. C2×Dic383D4semidirect
93. C383#535semidirect
104. #120D383semidirect
119. C2×D4C2×D383semidirect
129. C22⋊C4C2×D383semidirect

Maximal subgroups

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

subgroupindex
C2×#202594562
C22×C22⋊D3832
#271338372
#271338272
#271338142
#271338132
#271338112
#271338102
#271338092
#271337972
#120×C3832
#535383