Finite Group Databasebeta

← #44537366

Subgroups — #44537366

Order 31424.

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×#21463329C211yesmaximal, characteristic, complemented
2C22×D1964C211yesmaximal, characteristic, complemented
3#30628326C211yesmaximal, characteristic, complemented
4#30628322C212yesmaximal, complemented
5#30628312C211yesmaximal, characteristic, complemented
6#30628312C212yesmaximal, complemented
7#30628307C212yesmaximal, complemented
8#30628299C211yesmaximal, characteristic, complemented
9C4×C22⋊D491C211yesmaximal, characteristic, complemented
10C22⋊C4×D491C212yesmaximal, complemented
11C22⋊Q8×C491C211yesmaximal, characteristic, complemented
12C23×D491C2212yes—
13#21463329—24no—
14C2×D1964—24no—
15C2×D1964C2212yes—
16C2×D1964C2211yescharacteristic
17C2×D1964C2211yescharacteristic
18C2×D1964C2212yescomplemented
19C2×D1964C2212yescomplemented
20C2×C22⋊D491C2211yescharacteristic
21C2×C4×D491C2212yescomplemented
22C2×C4×D491C2211yescharacteristic, complemented
23C2×C4×D491C2212yescomplemented
24Dic491⋊C4C2211yescharacteristic, complemented
25C22⋊Dic491C2211yescharacteristic
26(C2×D491)⋊C4C2212yescomplemented
27(C2×D491)⋊C4C2212yescomplemented
28(C2×D491)⋊C4C2211yescharacteristic, complemented
29(C2×D491)⋊C4C2212yescomplemented
30(C2×D491)⋊C4C2212yescomplemented
31C4×Dic491C2211yescharacteristic, complemented
32C2×Q8×C491C2211yescharacteristic, complemented
33C22×C4×C491C2211yescharacteristic
34C4⋊C4×C491C2211yescharacteristic, complemented
35C4⋊C4×C491C2212yescomplemented
36C22⋊C4×C491C2212yes—
37C22×D491—24no—
38C22×D491—24no—
39C22×D491—42no—
40C22×D491C2312yes—
41C22×D491C2311yescharacteristic
42C22×D491C2312yes—
43D1964—24no—
44D1964—24no—
45D1964—22no—
46D1964—42no—
47D1964D414yescomplemented
48C22⋊D491—22no—
49C2×Dic491C2311yescharacteristic
50C2×Dic491C2311yescharacteristic
51C2×Dic491C2311yescharacteristic
52C4×D491—24no—
53C4×D491—22no—
54C4×D491—42no—
55C23×C491C2311yescharacteristic
56Q8×C491—22no—
57C2×C4×C491—22no—
58C2×C4×C491C2311yescharacteristic
59C2×C4×C491C2312yes—
60C2×C4×C491C2312yes—
61C2×C4×C491C2311yescharacteristic
62C22×C491—21no—
63C22×C491—21no—
64C22×C491#4112yescomplemented
65C22×C491C2411yescharacteristic
66C2×D491—22no—
67C2×D491—22no—
68C2×D491—21no—
69C2×D491—22no—
70C2×D491—21no—
71C2×D491—22no—
72C2×D491—21no—
73C2×D491—22no—
74C2×D491—48no—
75C2×D491C2×D414yes—
76C4×C491—21no—
77C4×C491—22no—
78C4×C491—22no—
79C4×C491C2×D412yes—
80Dic491—21no—
81Dic491—21no—
82Dic491—21no—
83C2×C491—22no—
84C2×C491C22×D411yescharacteristic
85C2×C491C2×#4111yescharacteristic
86C2×C491H(5,𝔽2)11yescharacteristic
87D491—24no—
88D491—41no—
89D491—42no—
90C491#54611yescharacteristic, complemented, Sylow
91#546—4911nomaximal, Sylow
92C2×#41—4911no—
93C22×D4—4911no—
94#124—4911no—
95#123—4912no—
96C22⋊Q8D49111yescharacteristic, complemented
97#121—4911no—
98#121—4912no—
99#120—4912no—
100C4×D4—4911no—
101C4×D4—4911no—
102C2×C22⋊C4—4912no—
103C24—4912no—
104#41—9824no—
105C2×Q8C2×D49111yescharacteristic, complemented
106C2×D4—9824no—
107C2×D4—4911no—
108C2×D4—4912no—
109C2×D4—4912no—
110C2×D4—4911no—
111C2×D4—4912no—
112C2×D4—4911no—
113C22×C4—4912no—
114C22×C4—4912no—
115C22×C4—4911no—
116C22×C4C2×D49111yescharacteristic
117C4⋊C4—4911no—
118C4⋊C4C2×D49111yescharacteristic, complemented
119C4⋊C4C2×D49112yescomplemented
120C22⋊C4—4911no—
121C22⋊C4—4912no—
122C22⋊C4—4912no—
123C22⋊C4—4912no—
124C22⋊C4—4911no—
125C22⋊C4—4912no—
126C22⋊C4C2×D49112yes—
127C42—4911no—
128C23—19642no—
129C23—9824no—
130C23—9824no—
131C23—4912no—
132C23—4911no—
133C23—4912no—
134C23C22×D49111yescharacteristic
135Q8—22no—
136D4—19642no—
137D4—9824no—
138D4—9824no—
139D4—9822no—
140D4—9822no—
141D4—4914no—
142C2×C4—22no—
143C2×C4—19642no—
144C2×C4—9824no—
145C2×C4—9822no—
146C2×C4—4911no—
147C2×C4—4911no—
148C2×C4—4911no—
149C2×C4C22×D49112yes—
150C2×C4C22×D49112yes—
151C2×C4C22×D49111yescharacteristic
152C2×C4C22×D49111yescharacteristic
153C22—21no—
154C22—21no—
155C22—19648no—
156C22—9821no—
157C22—9822no—
158C22—9821no—
159C22—9822no—
160C22—9821no—
161C22—9822no—
162C22—9822no—
163C22—9822no—
164C22—4914no—
165C22#2146332912yescomplemented
166C22C23×D49111yescharacteristic
167C4—21no—
168C4—22no—
169C4—22no—
170C4—9821no—
171C4—9821no—
172C4—9821no—
173C4D4×D49112yes—
174C2—22no—
175C2—19642no—
176C2—19641no—
177C2—9824no—
178C2C2×D4×D49111yescharacteristic
179C2C2×#2146332911yescharacteristic
180C2#3062838311yescharacteristic
subgroupG/H
center, FrattiniC22C23×D491
commutator, socle, cosocleC22×C491C24
FittingC22⋊Q8×C491C2
hypercenterC22⋊Q8D491
hypocenterC491#546
LCSG ▷ C22×C491 ▷ C491
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#21463329C2semidirect
2. C22×D1964C2semidirect
3. #30628326C2semidirect
4. #30628322C2semidirect
5. #30628312C2semidirect
6. #30628312C2semidirect
7. #30628307C2semidirect
8. #30628299C2semidirect
9. C4×C22⋊D491C2semidirect
10. C22⋊C4×D491C2semidirect
11. C22⋊Q8×C491C2semidirect
18. C2×D1964C22semidirect
19. C2×D1964C22semidirect
21. C2×C4×D491C22semidirect
22. C2×C4×D491C22semidirect
23. C2×C4×D491C22semidirect
24. Dic491⋊C4C22semidirect
26. (C2×D491)⋊C4C22semidirect
27. (C2×D491)⋊C4C22semidirect
28. (C2×D491)⋊C4C22semidirect
29. (C2×D491)⋊C4C22semidirect
30. (C2×D491)⋊C4C22semidirect
31. C4×Dic491C22semidirect
32. C2×Q8×C491C22semidirect
34. C4⋊C4×C491C22semidirect
35. C4⋊C4×C491C22semidirect
47. D1964D4semidirect
64. C22×C491#41semidirect
90. C491#546semidirect
96. C22⋊Q8D491semidirect
105. C2×Q8C2×D491semidirect
118. C4⋊C4C2×D491semidirect
119. C4⋊C4C2×D491semidirect
165. C22#21463329semidirect

Maximal subgroups

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

subgroupindex
C2×#214633292
C22×D19642
#306283262
#306283222
#306283122
#306283072
#306282992
C4×C22⋊D4912
C22⋊C4×D4912
C22⋊Q8×C4912
#546491