Finite Group Databasebeta

← #42119703

Subgroups — #42119703

Order 27584.

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

normal subgroups only

This group has 334 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
1C2×#20293799C211yesmaximal, characteristic, complemented
2#28352176C211yesmaximal, characteristic, complemented
3#28352170C212yesmaximal, complemented
4#28352168C211yesmaximal, characteristic, complemented
5#28352161C212yesmaximal, complemented
6#28352160C211yesmaximal, characteristic, complemented
7#28352158C212yesmaximal, complemented
8D4×Dic431C211yesmaximal, characteristic, complemented
9C4×C22⋊D431C211yesmaximal, characteristic, complemented
10C2×C22⋊Dic431C212yesmaximal, complemented
11C22×D4×C431C211yesmaximal, characteristic, complemented
12#20293799—24no—
13C2×Dic862C2211yescharacteristic, complemented
14C2×C22⋊D431C2211yescharacteristic
15C2×C22⋊D431C2212yescomplemented
16C2×C22⋊D431C2212yescomplemented
17C22×Dic431C2212yescomplemented
18C22×Dic431C2212yescomplemented
19C2×C4×D431C2211yescharacteristic, complemented
20C4⋊Dic431C2211yescharacteristic, complemented
21Dic431⋊C4C2211yescharacteristic, complemented
22Dic431⋊C4C2212yescomplemented
23C22⋊Dic431C2211yescharacteristic
24C22⋊Dic431C2212yes—
25C22⋊Dic431C2212yescomplemented
26C22⋊Dic431C2212yescomplemented
27C22⋊Dic431C2212yescomplemented
28C22⋊Dic431C2212yescomplemented
29(C2×D431)⋊C4C2211yescharacteristic, complemented
30C4×Dic431C2211yescharacteristic, complemented
31C24×C431C2212yes—
32C2×D4×C431—24no—
33C2×D4×C431C2211yescharacteristic
34C2×D4×C431C2212yes—
35C2×D4×C431C2211yescharacteristic
36C22×C4×C431C2211yescharacteristic
37C22×D431C2311yescharacteristic
38Dic862—22no—
39C22⋊D431—24no—
40C22⋊D431—22no—
41C22⋊D431—42no—
42C2×Dic431—24no—
43C2×Dic431—42no—
44C2×Dic431C2311yescharacteristic
45C2×Dic431C2311yescharacteristic
46C2×Dic431C2312yes—
47C2×Dic431C2311yescharacteristic
48C2×Dic431C2312yes—
49C4×D431—22no—
50C23×C431—24no—
51C23×C431—24no—
52C23×C431—42no—
53C23×C431C2312yes—
54C23×C431C2312yes—
55C23×C431C2311yescharacteristic
56D4×C431—24no—
57D4×C431—22no—
58D4×C431D414yescomplemented
59C2×C4×C431—22no—
60C2×C4×C431C2311yescharacteristic
61C2×C4×C431C2311yescharacteristic
62C22×C431—21no—
63C22×C431—22no—
64C22×C431—21no—
65C22×C431—22no—
66C22×C431—22no—
67C22×C431—22no—
68C22×C431—22no—
69C22×C431—48no—
70C22×C431C2×D414yes—
71C22×C431#4112yescomplemented
72C22×C431C2411yescharacteristic
73C2×D431—21no—
74C2×D431—21no—
75C2×D431—21no—
76C4×C431—21no—
77C4×C431C2×D412yes—
78Dic431—21no—
79Dic431—22no—
80Dic431—21no—
81Dic431—21no—
82Dic431—22no—
83C2×C431—22no—
84C2×C431—24no—
85C2×C431—42no—
86C2×C431C22×D411yescharacteristic
87C2×C431C2×#4111yescharacteristic
88C2×C431H(5,𝔽2)11yescharacteristic
89D431—41no—
90C431#54611yescharacteristic, complemented, Sylow
91#546—4311nomaximal, Sylow
92C2×#41—4311no—
93C22×D4D43111yescharacteristic, complemented
94#124—4311no—
95#123—4312no—
96C22⋊Q8—4311no—
97#121—4311no—
98#121—4312no—
99#120—4312no—
100C4×D4—4311no—
101C4×D4—4311no—
102C2×C22⋊C4—4312no—
103C24C2×D43112yes—
104#41—8624no—
105C2×Q8—4311no—
106C2×D4—24no—
107C2×D4—4311no—
108C2×D4—4312no—
109C2×D4—4312no—
110C2×D4C2×D43112yes—
111C2×D4C2×D43111yescharacteristic
112C2×D4C2×D43111yescharacteristic
113C22×C4—4312no—
114C22×C4—4311no—
115C22×C4—4312no—
116C22×C4C2×D43111yescharacteristic
117C4⋊C4—4311no—
118C4⋊C4—4311no—
119C4⋊C4—4312no—
120C22⋊C4—4311no—
121C22⋊C4—4312no—
122C22⋊C4—4312no—
123C22⋊C4—4312no—
124C22⋊C4—4312no—
125C22⋊C4—4312no—
126C22⋊C4—4311no—
127C42—4311no—
128C23—24no—
129C23—24no—
130C23—42no—
131C23—4311no—
132C23C22×D43112yes—
133C23C22×D43112yes—
134C23C22×D43111yescharacteristic
135Q8—8622no—
136D4—24no—
137D4—22no—
138D4—8624no—
139D4—8622no—
140D4—17242no—
141D4C22⋊D43114yescomplemented
142C2×C4—22no—
143C2×C4—8624no—
144C2×C4—8622no—
145C2×C4—4311no—
146C2×C4—4311no—
147C2×C4—4312no—
148C2×C4—4311no—
149C2×C4—4312no—
150C2×C4—17242no—
151C2×C4C22×D43111yescharacteristic
152C2×C4C22×D43111yescharacteristic
153C22—21no—
154C22—22no—
155C22—22no—
156C22—21no—
157C22—22no—
158C22—22no—
159C22—22no—
160C22—48no—
161C22—8621no—
162C22—8621no—
163C22—8621no—
164C22C2×C22⋊D43114yes—
165C22#2029379912yescomplemented
166C22C23×D43111yescharacteristic
167C4—21no—
168C4—8621no—
169C4—8622no—
170C4—8621no—
171C4—8621no—
172C4—8622no—
173C4C2×C22⋊D43112yes—
174C2—22no—
175C2—24no—
176C2—42no—
177C2—17241no—
178C2C22×C22⋊D43111yescharacteristic
179C2C2×#2029379911yescharacteristic
180C2#2835223411yescharacteristic
subgroupG/H
center, FrattiniC22C23×D431
commutator, socle, cosocleC22×C431C24
FittingC22×D4×C431C2
hypercenterC22×D4D431
hypocenterC431#546
LCSG ▷ C22×C431 ▷ C431
UCSC1 ◁ C22 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C2×#20293799C2semidirect
2. #28352176C2semidirect
3. #28352170C2semidirect
4. #28352168C2semidirect
5. #28352161C2semidirect
6. #28352160C2semidirect
7. #28352158C2semidirect
8. D4×Dic431C2semidirect
9. C4×C22⋊D431C2semidirect
10. C2×C22⋊Dic431C2semidirect
11. C22×D4×C431C2semidirect
13. C2×Dic862C22semidirect
15. C2×C22⋊D431C22semidirect
16. C2×C22⋊D431C22semidirect
17. C22×Dic431C22semidirect
18. C22×Dic431C22semidirect
19. C2×C4×D431C22semidirect
20. C4⋊Dic431C22semidirect
21. Dic431⋊C4C22semidirect
22. Dic431⋊C4C22semidirect
25. C22⋊Dic431C22semidirect
26. C22⋊Dic431C22semidirect
27. C22⋊Dic431C22semidirect
28. C22⋊Dic431C22semidirect
29. (C2×D431)⋊C4C22semidirect
30. C4×Dic431C22semidirect
58. D4×C431D4semidirect
71. C22×C431#41semidirect
90. C431#546semidirect
93. C22×D4D431semidirect
141. D4C22⋊D431semidirect
165. C22#20293799semidirect

Maximal subgroups

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

subgroupindex
C2×#202937992
#283521762
#283521702
#283521682
#283521612
#283521602
#283521582
D4×Dic4312
C4×C22⋊D4312
C2×C22⋊Dic4312
C22×D4×C4312
#546431