Finite Group Databasebeta

← #44537369

Subgroups — #44537369

Order 31424.

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×#21463327C211yesmaximal, characteristic, complemented
2#30628324C211yesmaximal, characteristic, complemented
3#30628318C212yesmaximal, complemented
4#30628316C211yesmaximal, characteristic, complemented
5#30628309C212yesmaximal, complemented
6#30628308C211yesmaximal, characteristic, complemented
7#30628306C212yesmaximal, complemented
8D4×Dic491C211yesmaximal, characteristic, complemented
9C4×C22⋊D491C211yesmaximal, characteristic, complemented
10C2×C22⋊Dic491C212yesmaximal, complemented
11C22×D4×C491C211yesmaximal, characteristic, complemented
12#21463327—24no—
13C2×Dic982C2211yescharacteristic, complemented
14C2×C22⋊D491C2211yescharacteristic
15C2×C22⋊D491C2212yescomplemented
16C2×C22⋊D491C2212yescomplemented
17C22×Dic491C2212yescomplemented
18C22×Dic491C2212yescomplemented
19C2×C4×D491C2211yescharacteristic, complemented
20C4⋊Dic491C2211yescharacteristic, complemented
21Dic491⋊C4C2211yescharacteristic, complemented
22Dic491⋊C4C2212yescomplemented
23C22⋊Dic491C2211yescharacteristic
24C22⋊Dic491C2212yes—
25C22⋊Dic491C2212yescomplemented
26C22⋊Dic491C2212yescomplemented
27C22⋊Dic491C2212yescomplemented
28C22⋊Dic491C2212yescomplemented
29(C2×D491)⋊C4C2211yescharacteristic, complemented
30C4×Dic491C2211yescharacteristic, complemented
31C24×C491C2212yes—
32C2×D4×C491—24no—
33C2×D4×C491C2211yescharacteristic
34C2×D4×C491C2212yes—
35C2×D4×C491C2211yescharacteristic
36C22×C4×C491C2211yescharacteristic
37C22×D491C2311yescharacteristic
38Dic982—22no—
39C22⋊D491—24no—
40C22⋊D491—22no—
41C22⋊D491—42no—
42C2×Dic491—24no—
43C2×Dic491—42no—
44C2×Dic491C2311yescharacteristic
45C2×Dic491C2311yescharacteristic
46C2×Dic491C2312yes—
47C2×Dic491C2311yescharacteristic
48C2×Dic491C2312yes—
49C4×D491—22no—
50C23×C491—24no—
51C23×C491—24no—
52C23×C491—42no—
53C23×C491C2312yes—
54C23×C491C2312yes—
55C23×C491C2311yescharacteristic
56D4×C491—24no—
57D4×C491—22no—
58D4×C491D414yescomplemented
59C2×C4×C491—22no—
60C2×C4×C491C2311yescharacteristic
61C2×C4×C491C2311yescharacteristic
62C22×C491—21no—
63C22×C491—22no—
64C22×C491—21no—
65C22×C491—22no—
66C22×C491—22no—
67C22×C491—22no—
68C22×C491—22no—
69C22×C491—48no—
70C22×C491C2×D414yes—
71C22×C491#4112yescomplemented
72C22×C491C2411yescharacteristic
73C2×D491—21no—
74C2×D491—21no—
75C2×D491—21no—
76C4×C491—21no—
77C4×C491C2×D412yes—
78Dic491—21no—
79Dic491—22no—
80Dic491—21no—
81Dic491—21no—
82Dic491—22no—
83C2×C491—22no—
84C2×C491—24no—
85C2×C491—42no—
86C2×C491C22×D411yescharacteristic
87C2×C491C2×#4111yescharacteristic
88C2×C491H(5,𝔽2)11yescharacteristic
89D491—41no—
90C491#54611yescharacteristic, complemented, Sylow
91#546—4911nomaximal, Sylow
92C2×#41—4911no—
93C22×D4D49111yescharacteristic, complemented
94#124—4911no—
95#123—4912no—
96C22⋊Q8—4911no—
97#121—4911no—
98#121—4912no—
99#120—4912no—
100C4×D4—4911no—
101C4×D4—4911no—
102C2×C22⋊C4—4912no—
103C24C2×D49112yes—
104#41—9824no—
105C2×Q8—4911no—
106C2×D4—24no—
107C2×D4—4912no—
108C2×D4—4911no—
109C2×D4—4912no—
110C2×D4C2×D49112yes—
111C2×D4C2×D49111yescharacteristic
112C2×D4C2×D49111yescharacteristic
113C22×C4—4912no—
114C22×C4—4912no—
115C22×C4—4911no—
116C22×C4C2×D49111yescharacteristic
117C4⋊C4—4911no—
118C4⋊C4—4912no—
119C4⋊C4—4911no—
120C22⋊C4—4911no—
121C22⋊C4—4912no—
122C22⋊C4—4912no—
123C22⋊C4—4912no—
124C22⋊C4—4912no—
125C22⋊C4—4912no—
126C22⋊C4—4911no—
127C42—4911no—
128C23—24no—
129C23—24no—
130C23—42no—
131C23—4911no—
132C23C22×D49112yes—
133C23C22×D49112yes—
134C23C22×D49111yescharacteristic
135Q8—9822no—
136D4—24no—
137D4—22no—
138D4—19642no—
139D4—9824no—
140D4—9822no—
141D4C22⋊D49114yescomplemented
142C2×C4—22no—
143C2×C4—19642no—
144C2×C4—9824no—
145C2×C4—9822no—
146C2×C4—4911no—
147C2×C4—4911no—
148C2×C4—4912no—
149C2×C4—4911no—
150C2×C4—4912no—
151C2×C4C22×D49111yescharacteristic
152C2×C4C22×D49111yescharacteristic
153C22—21no—
154C22—22no—
155C22—22no—
156C22—21no—
157C22—22no—
158C22—22no—
159C22—22no—
160C22—48no—
161C22—9821no—
162C22—9821no—
163C22—9821no—
164C22C2×C22⋊D49114yes—
165C22#2146332712yescomplemented
166C22C23×D49111yescharacteristic
167C4—21no—
168C4—9821no—
169C4—9822no—
170C4—9821no—
171C4—9821no—
172C4—9822no—
173C4C2×C22⋊D49112yes—
174C2—22no—
175C2—24no—
176C2—42no—
177C2—19641no—
178C2C22×C22⋊D49111yescharacteristic
179C2C2×#2146332711yescharacteristic
180C2#3062838211yescharacteristic
subgroupG/H
center, FrattiniC22C23×D491
commutator, socle, cosocleC22×C491C24
FittingC22×D4×C491C2
hypercenterC22×D4D491
hypocenterC491#546
LCSG ▷ C22×C491 ▷ C491
UCSC1 ◁ C22 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C2×#21463327C2semidirect
2. #30628324C2semidirect
3. #30628318C2semidirect
4. #30628316C2semidirect
5. #30628309C2semidirect
6. #30628308C2semidirect
7. #30628306C2semidirect
8. D4×Dic491C2semidirect
9. C4×C22⋊D491C2semidirect
10. C2×C22⋊Dic491C2semidirect
11. C22×D4×C491C2semidirect
13. C2×Dic982C22semidirect
15. C2×C22⋊D491C22semidirect
16. C2×C22⋊D491C22semidirect
17. C22×Dic491C22semidirect
18. C22×Dic491C22semidirect
19. C2×C4×D491C22semidirect
20. C4⋊Dic491C22semidirect
21. Dic491⋊C4C22semidirect
22. Dic491⋊C4C22semidirect
25. C22⋊Dic491C22semidirect
26. C22⋊Dic491C22semidirect
27. C22⋊Dic491C22semidirect
28. C22⋊Dic491C22semidirect
29. (C2×D491)⋊C4C22semidirect
30. C4×Dic491C22semidirect
58. D4×C491D4semidirect
71. C22×C491#41semidirect
90. C491#546semidirect
93. C22×D4D491semidirect
141. D4C22⋊D491semidirect
165. C22#21463327semidirect

Maximal subgroups

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

subgroupindex
C2×#214633272
#306283242
#306283182
#306283162
#306283092
#306283082
#306283062
D4×Dic4912
C4×C22⋊D4912
C2×C22⋊Dic4912
C22×D4×C4912
#546491