Finite Group Databasebeta

← #42119705

Subgroups — #42119705

Order 27584.

334 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 107 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×#20293800C211yesmaximal, characteristic, complemented
2C22×D1724C211yesmaximal, characteristic, complemented
3#28352177C211yesmaximal, characteristic, complemented
4#28352174C212yesmaximal, complemented
5#28352168C211yesmaximal, characteristic, complemented
6#28352163C211yesmaximal, characteristic, complemented
7#28352162C212yesmaximal, complemented
8#28352157C212yesmaximal, complemented
9C4×D1724C211yesmaximal, characteristic, complemented
10C22⋊C4×D431C212yesmaximal, complemented
11C4×D4×C431C211yesmaximal, characteristic, complemented
12C23×D431C2212yes—
13#20293800—24no—
14C2×Dic862C2211yescharacteristic, complemented
15C2×D1724—24no—
16C2×D1724C2212yes—
17C2×D1724C2211yescharacteristic
18C2×D1724C2211yescharacteristic
19C2×C22⋊D431C2212yescomplemented
20C2×C22⋊D431C2212yescomplemented
21C2×C4×D431C2212yescomplemented
22C2×C4×D431C2212yescomplemented
23C4⋊Dic431C2211yescharacteristic, complemented
24Dic431⋊C4C2212yescomplemented
25C22⋊Dic431C2212yes—
26(C2×D431)⋊C4C2212yescomplemented
27(C2×D431)⋊C4C2212yescomplemented
28(C2×D431)⋊C4C2212yescomplemented
29(C2×D431)⋊C4C2212yescomplemented
30C2×D4×C431C2211yescharacteristic
31C22×C4×C431C2211yescharacteristic
32C22×C4×C431C2211yescharacteristic, complemented
33C4⋊C4×C431C2211yescharacteristic, complemented
34C22⋊C4×C431C2211yescharacteristic
35C22⋊C4×C431C2211yescharacteristic, complemented
36C42×C431C2211yescharacteristic, complemented
37C22×D431—24no—
38C22×D431—24no—
39C22×D431—42no—
40C22×D431C2312yes—
41C22×D431C2312yes—
42Dic862—22no—
43D1724—24no—
44D1724—22no—
45D1724D414yescomplemented
46C22⋊D431—24no—
47C22⋊D431—42no—
48C2×Dic431C2312yes—
49C2×Dic431C2312yes—
50C4×D431—24no—
51C4×D431—42no—
52C23×C431C2311yescharacteristic
53C23×C431C2311yescharacteristic
54D4×C431—22no—
55C2×C4×C431—22no—
56C2×C4×C431—22no—
57C2×C4×C431C2311yescharacteristic
58C2×C4×C431C2311yescharacteristic
59C2×C4×C431C2311yescharacteristic
60C2×C4×C431C2311yescharacteristic
61C2×C4×C431C2311yescharacteristic
62C22×C431—21no—
63C22×C431—21no—
64C22×C431—21no—
65C22×C431—21no—
66C22×C431—21no—
67C22×C431#4112yescomplemented
68C22×C431C2411yescharacteristic
69C2×D431—22no—
70C2×D431—22no—
71C2×D431—22no—
72C2×D431—22no—
73C2×D431—22no—
74C2×D431—48no—
75C2×D431C2×D414yes—
76C4×C431—21no—
77C4×C431—21no—
78C4×C431—21no—
79C4×C431—21no—
80C4×C431C2×D412yes—
81Dic431—22no—
82Dic431—22no—
83C2×C431—22no—
84C2×C431—41no—
85C2×C431C22×D411yescharacteristic
86C2×C431C2×#4111yescharacteristic
87C2×C431H(5,𝔽2)11yescharacteristic
88D431—24no—
89D431—42no—
90C431#54611yescharacteristic, complemented, Sylow
91#546—4311nomaximal, Sylow
92C2×#41—4311no—
93C22×D4—4311no—
94#124—4311no—
95#123—4312no—
96C22⋊Q8—4311no—
97#121—4311no—
98#121—4312no—
99#120—4312no—
100C4×D4—4311no—
101C4×D4D43111yescharacteristic, complemented
102C2×C22⋊C4—4312no—
103C24—4312no—
104#41—8624no—
105C2×Q8—4311no—
106C2×D4—8624no—
107C2×D4—4312no—
108C2×D4—4312no—
109C2×D4—4312no—
110C2×D4—4311no—
111C2×D4—4311no—
112C2×D4C2×D43111yescharacteristic
113C22×C4—4312no—
114C22×C4—4312no—
115C22×C4C2×D43111yescharacteristic
116C22×C4C2×D43111yescharacteristic, complemented
117C4⋊C4—4311no—
118C4⋊C4—4312no—
119C4⋊C4C2×D43111yescharacteristic, complemented
120C22⋊C4—4312no—
121C22⋊C4—4312no—
122C22⋊C4—4312no—
123C22⋊C4—4312no—
124C22⋊C4—4312no—
125C22⋊C4C2×D43111yescharacteristic
126C22⋊C4C2×D43111yescharacteristic, complemented
127C42C2×D43111yescharacteristic, complemented
128C23—8624no—
129C23—8624no—
130C23—4312no—
131C23—4312no—
132C23—17242no—
133C23C22×D43111yescharacteristic
134C23C22×D43111yescharacteristic
135Q8—8622no—
136D4—22no—
137D4—8624no—
138D4—8624no—
139D4—8622no—
140D4—4314no—
141D4—17242no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—8624no—
145C2×C4—4312no—
146C2×C4—4312no—
147C2×C4—17242no—
148C2×C4C22×D43111yescharacteristic
149C2×C4C22×D43111yescharacteristic
150C2×C4C22×D43111yescharacteristic
151C2×C4C22×D43111yescharacteristic
152C2×C4C22×D43111yescharacteristic
153C22—21no—
154C22—21no—
155C22—21no—
156C22—21no—
157C22—21no—
158C22—8622no—
159C22—8622no—
160C22—8622no—
161C22—8622no—
162C22—8622no—
163C22—4314no—
164C22—17248no—
165C22#2029380012yescomplemented
166C22C23×D43111yescharacteristic
167C4—21no—
168C4—21no—
169C4—21no—
170C4—21no—
171C4—8622no—
172C4—8622no—
173C4D4×D43112yes—
174C2—22no—
175C2—41no—
176C2—8624no—
177C2—17242no—
178C2C2×D4×D43111yescharacteristic
179C2C2×#2029380011yescharacteristic
180C2#2835223511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D431
commutator, socle, cosocleC22×C431C24
FittingC4×D4×C431C2
hypercenterC4×D4D431
hypocenterC431#546
LCSG ▷ C22×C431 ▷ C431
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#20293800C2semidirect
2. C22×D1724C2semidirect
3. #28352177C2semidirect
4. #28352174C2semidirect
5. #28352168C2semidirect
6. #28352163C2semidirect
7. #28352162C2semidirect
8. #28352157C2semidirect
9. C4×D1724C2semidirect
10. C22⋊C4×D431C2semidirect
11. C4×D4×C431C2semidirect
14. C2×Dic862C22semidirect
19. C2×C22⋊D431C22semidirect
20. C2×C22⋊D431C22semidirect
21. C2×C4×D431C22semidirect
22. C2×C4×D431C22semidirect
23. C4⋊Dic431C22semidirect
24. Dic431⋊C4C22semidirect
26. (C2×D431)⋊C4C22semidirect
27. (C2×D431)⋊C4C22semidirect
28. (C2×D431)⋊C4C22semidirect
29. (C2×D431)⋊C4C22semidirect
32. C22×C4×C431C22semidirect
33. C4⋊C4×C431C22semidirect
35. C22⋊C4×C431C22semidirect
36. C42×C431C22semidirect
45. D1724D4semidirect
67. C22×C431#41semidirect
90. C431#546semidirect
101. C4×D4D431semidirect
116. C22×C4C2×D431semidirect
119. C4⋊C4C2×D431semidirect
126. C22⋊C4C2×D431semidirect
127. C42C2×D431semidirect
165. C22#20293800semidirect

Maximal subgroups

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

subgroupindex
C2×#202938002
C22×D17242
#283521772
#283521742
#283521682
#283521632
#283521622
#283521572
C4×D17242
C22⋊C4×D4312
C4×D4×C4312
#546431