Finite Group Databasebeta

← #44417290

Subgroups — #44417290

Order 29248.

330 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 109 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×#20300742C211yesmaximal, characteristic, complemented
2C22×C4×D457C211yesmaximal, characteristic, complemented
3#28410537C212yesmaximal, complemented
4#28410528C211yesmaximal, characteristic, complemented
5#28410527C211yesmaximal, characteristic, complemented
6#28410526C211yesmaximal, characteristic, complemented
7#28410522C212yesmaximal, complemented
8#28410514C212yesmaximal, complemented
9#28410512C212yesmaximal, complemented
10#28410510C211yesmaximal, characteristic, complemented
11C22⋊Q8×C457C211yesmaximal, characteristic, complemented
12C23×D457C2211yescharacteristic
13#20300742—24no—
14C2×D1828C2212yes—
15C2×D1828C2212yes—
16C2×D1828C2211yescharacteristic, complemented
17C2×C22⋊D457C2212yes—
18C22×Dic457C2211yescharacteristic
19C2×C4×D457—24no—
20C2×C4×D457C2211yescharacteristic
21C2×C4×D457C2211yescharacteristic
22C2×C4×D457C2211yescharacteristic
23C2×C4×D457C2211yescharacteristic
24C2×C4×D457C2212yescomplemented
25C4⋊Dic457C2211yescharacteristic, complemented
26#20300720C2212yescomplemented
27#20300718C2212yes—
28#20300718C2212yes—
29#20300718C2212yescomplemented
30#20300718C2212yescomplemented
31C4×Dic457C2212yescomplemented
32C2×Q8×C457C2211yescharacteristic, complemented
33C22×C4×C457C2211yescharacteristic
34C4⋊C4×C457C2211yescharacteristic, complemented
35C4⋊C4×C457C2212yescomplemented
36C22⋊C4×C457C2212yes—
37C22×D457—22no—
38C22×D457—22no—
39C22×D457—22no—
40C22×D457C2312yes—
41C22×D457C2311yescharacteristic
42C22×D457C2311yescharacteristic
43D1828—24no—
44D1828—24no—
45D1828—22no—
46C22⋊D457—24no—
47C2×Dic457—22no—
48C2×Dic457C2312yes—
49C2×Dic457C2311yescharacteristic
50C2×Dic457C2311yescharacteristic
51C4×D457—22no—
52C4×D457—22no—
53C4×D457—22no—
54C4×D457—24no—
55C4×D457D414yescomplemented
56C23×C457C2311yescharacteristic
57Q8×C457—22no—
58C2×C4×C457—22no—
59C2×C4×C457C2312yes—
60C2×C4×C457C2312yes—
61C2×C4×C457C2311yescharacteristic
62C2×C4×C457C2311yescharacteristic
63C22×C457—21no—
64C22×C457—21no—
65C22×C457#4112yescomplemented
66C22×C457C2411yescharacteristic
67C2×D457—28no—
68C2×D457—21no—
69C2×D457—21no—
70C2×D457—21no—
71C2×D457—21no—
72C2×D457—22no—
73C2×D457—22no—
74C2×D457—22no—
75C2×D457C2×D412yes—
76C2×D457#4112yescomplemented
77C4×C457—21no—
78C4×C457—22no—
79C4×C457—22no—
80C4×C457C2×D412yes—
81Dic457—21no—
82Dic457—22no—
83Dic457C2×D412yes—
84C2×C457—22no—
85C2×C457C22×D411yescharacteristic
86C2×C457C2×#4111yescharacteristic
87C2×C457C2×#4111yescharacteristic
88D457—22no—
89D457—22no—
90D457—42no—
91C457#52511yescharacteristic, complemented, Sylow
92#525—4571nomaximal, Sylow
93C2×#41—4571no—
94C23×C4—4571no—
95#123—4572no—
96C22⋊Q8—4571no—
97C22⋊Q8D45711yescharacteristic, complemented
98#121—4571no—
99#121—4571no—
100#120—4572no—
101C4×D4—4572no—
102C4×D4—4572no—
103#117—4571no—
104C24—4571no—
105#41—9144no—
106C2×Q8C2×D45711yescharacteristic, complemented
107C2×D4—4572no—
108C2×D4—4572no—
109C2×D4—4572no—
110C2×D4—4571no—
111C22×C4—9144no—
112C22×C4—4571no—
113C22×C4—4571no—
114C22×C4—4572no—
115C22×C4—4571no—
116C22×C4—4571no—
117C22×C4—4571no—
118C22×C4C2×D45711yescharacteristic
119C4⋊C4—4571no—
120C4⋊C4—4572no—
121C4⋊C4C2×D45711yescharacteristic, complemented
122C4⋊C4C2×D45712yescomplemented
123C22⋊C4—4572no—
124C22⋊C4—4572no—
125C22⋊C4—4572no—
126C22⋊C4—4572no—
127C22⋊C4C2×D45712yes—
128C42—4572no—
129C23—9142no—
130C23—9142no—
131C23—9142no—
132C23—4572no—
133C23—4571no—
134C23—4571no—
135C23C22×D45711yescharacteristic
136Q8—22no—
137D4—9144no—
138D4—9144no—
139D4—9144no—
140D4—9142no—
141C2×C4—22no—
142C2×C4—9142no—
143C2×C4—9142no—
144C2×C4—9144no—
145C2×C4—9142no—
146C2×C4—9142no—
147C2×C4—4572no—
148C2×C4—4571no—
149C2×C4—4571no—
150C2×C4—4574no—
151C2×C4C22×D45712yes—
152C2×C4C22×D45711yescharacteristic
153C2×C4C22×D45712yes—
154C2×C4C22×D45711yescharacteristic
155C22—21no—
156C22—21no—
157C22—9148no—
158C22—9142no—
159C22—9142no—
160C22—9142no—
161C22—9141no—
162C22—9141no—
163C22—9141no—
164C22—9141no—
165C22—4572no—
166C22—4572no—
167C22#2030074212yescomplemented
168C22C23×D45711yescharacteristic
169C4—21no—
170C4—22no—
171C4—22no—
172C4—9141no—
173C4—9142no—
174C4—4572no—
175C4D4×D45712yes—
176C2—22no—
177C2—18282no—
178C2—9142no—
179C2—9142no—
180C2C2×D4×D45711yescharacteristic
181C2#41×D45711yescharacteristic
182C2C2×#2030074211yescharacteristic
subgroupG/H
center, FrattiniC22C23×D457
commutator, socle, cosocleC22×C457C24
FittingC22⋊Q8×C457C2
hypercenterC22⋊Q8D457
hypocenterC457#525
LCSG ▷ C22×C457 ▷ C457
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#20300742C2semidirect
2. C22×C4×D457C2semidirect
3. #28410537C2semidirect
4. #28410528C2semidirect
5. #28410527C2semidirect
6. #28410526C2semidirect
7. #28410522C2semidirect
8. #28410514C2semidirect
9. #28410512C2semidirect
10. #28410510C2semidirect
11. C22⋊Q8×C457C2semidirect
16. C2×D1828C22semidirect
24. C2×C4×D457C22semidirect
25. C4⋊Dic457C22semidirect
26. #20300720C22semidirect
29. #20300718C22semidirect
30. #20300718C22semidirect
31. C4×Dic457C22semidirect
32. C2×Q8×C457C22semidirect
34. C4⋊C4×C457C22semidirect
35. C4⋊C4×C457C22semidirect
55. C4×D457D4semidirect
65. C22×C457#41semidirect
76. C2×D457#41semidirect
91. C457#525semidirect
97. C22⋊Q8D457semidirect
106. C2×Q8C2×D457semidirect
121. C4⋊C4C2×D457semidirect
122. C4⋊C4C2×D457semidirect
167. C22#20300742semidirect

Maximal subgroups

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

subgroupindex
C2×#203007422
C22×C4×D4572
#284105372
#284105282
#284105272
#284105262
#284105222
#284105142
#284105122
#284105102
C22⋊Q8×C4572
#525457