Finite Group Databasebeta

← #29508364

Subgroups — #29508364

Order 15424.

334 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 105 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×#16838770C211yesmaximal, characteristic, complemented
2C2×D4×D241C211yesmaximal, characteristic, complemented
3#21440629C211yesmaximal, characteristic, complemented
4#21440624C212yesmaximal, complemented
5#21440622C211yesmaximal, characteristic, complemented
6#21440613C212yesmaximal, complemented
7#21440610C212yesmaximal, complemented
8#21440604C211yesmaximal, characteristic, complemented
9C4×C22⋊D241C211yesmaximal, characteristic, complemented
10C22⋊C4×D241C212yesmaximal, complemented
11#121×C241C211yesmaximal, characteristic, complemented
12C23×D241C2212yes—
13#16838770—24no—
14C2×Dic482C2211yescharacteristic, complemented
15C2×D964C2211yescharacteristic
16C2×C22⋊D241C2211yescharacteristic
17C2×C22⋊D241C2212yes—
18C2×C22⋊D241C2212yescomplemented
19D4×D241—24no—
20C2×C4×D241C2211yescharacteristic
21C2×C4×D241C2212yescomplemented
22C2×C4×D241C2212yescomplemented
23C4⋊Dic241C2212yescomplemented
24#16838749C2211yescharacteristic, complemented
25C22⋊Dic241C2212yescomplemented
26C22⋊Dic241C2211yescharacteristic, complemented
27C22⋊Dic241C2212yescomplemented
28#16838747C2211yescharacteristic
29#16838747C2212yes—
30#16838747C2212yescomplemented
31C4×Dic241C2211yescharacteristic, complemented
32C2×D4×C241C2211yescharacteristic
33C2×D4×C241C2212yescomplemented
34C22×C4×C241C2211yescharacteristic, complemented
35C4⋊C4×C241C2211yescharacteristic, complemented
36C22⋊C4×C241C2212yescomplemented
37C22×D241—24no—
38C22×D241—24no—
39C22×D241—42no—
40C22×D241C2312yes—
41C22×D241C2311yescharacteristic
42Dic482—22no—
43D964D414yescomplemented
44C22⋊D241—24no—
45C22⋊D241—24no—
46C22⋊D241—22no—
47C2×Dic241C2311yescharacteristic
48C2×Dic241C2311yescharacteristic
49C2×Dic241C2312yes—
50C2×Dic241C2311yescharacteristic
51C4×D241—24no—
52C4×D241—22no—
53C4×D241—42no—
54C23×C241C2312yes—
55C23×C241C2311yescharacteristic
56D4×C241—22no—
57D4×C241—42no—
58C2×C4×C241—22no—
59C2×C4×C241C2311yescharacteristic
60C2×C4×C241C2312yes—
61C2×C4×C241C2311yescharacteristic
62C22×C241—21no—
63C22×C241—22no—
64C22×C241—21no—
65C22×C241—22no—
66C22×C241—21no—
67C22×C241—22no—
68C22×C241C2411yescharacteristic
69C2×D241—21no—
70C2×D241—22no—
71C2×D241—22no—
72C2×D241—21no—
73C2×D241—48no—
74C2×D241C2×D414yes—
75C2×D241#4112yescomplemented
76C4×C241—21no—
77C4×C241—22no—
78C4×C241C2×D412yes—
79Dic241—21no—
80Dic241—22no—
81Dic241—21no—
82Dic241—21no—
83C2×C241—42no—
84C2×C241—41no—
85C2×C241C22×D411yescharacteristic
86C2×C241C2×#4111yescharacteristic
87C2×C241H(5,𝔽2)11yescharacteristic
88D241—22no—
89D241—24no—
90C241#54611yescharacteristic, complemented, Sylow
91#546—2411nomaximal, Sylow
92C2×#41—2411no—
93C22×D4—2411no—
94#124—2411no—
95#123—2412no—
96C22⋊Q8—2411no—
97#121—2412no—
98#121D24111yescharacteristic, complemented
99#120—2412no—
100C4×D4—2411no—
101C4×D4—2411no—
102C2×C22⋊C4—2412no—
103C24—2412no—
104#41—4824no—
105C2×Q8—2411no—
106C2×D4—4824no—
107C2×D4—2411no—
108C2×D4—2412no—
109C2×D4—2412no—
110C2×D4—2411no—
111C2×D4C2×D24111yescharacteristic
112C2×D4C2×D24112yescomplemented
113C22×C4—2412no—
114C22×C4—2411no—
115C22×C4—2412no—
116C22×C4C2×D24111yescharacteristic, complemented
117C4⋊C4—2411no—
118C4⋊C4—2412no—
119C4⋊C4C2×D24111yescharacteristic, complemented
120C22⋊C4—2412no—
121C22⋊C4—2411no—
122C22⋊C4—2412no—
123C22⋊C4—2411no—
124C22⋊C4—2412no—
125C22⋊C4—2412no—
126C22⋊C4C2×D24112yescomplemented
127C42—2411no—
128C23—9642no—
129C23—4824no—
130C23—4824no—
131C23—2411no—
132C23—2412no—
133C23C22×D24112yes—
134C23C22×D24111yescharacteristic
135Q8—4822no—
136D4—22no—
137D4—42no—
138D4—4824no—
139D4—4824no—
140D4—4822no—
141D4—2414no—
142C2×C4—22no—
143C2×C4—9642no—
144C2×C4—4824no—
145C2×C4—4822no—
146C2×C4—2411no—
147C2×C4—2411no—
148C2×C4—2412no—
149C2×C4—2411no—
150C2×C4C22×D24112yes—
151C2×C4C22×D24111yescharacteristic
152C2×C4C22×D24111yescharacteristic
153C22—21no—
154C22—22no—
155C22—21no—
156C22—22no—
157C22—21no—
158C22—22no—
159C22—9648no—
160C22—4821no—
161C22—4822no—
162C22—4822no—
163C22—4821no—
164C22—2414no—
165C22—2412no—
166C22C23×D24111yescharacteristic
167C4—21no—
168C4—22no—
169C4—4821no—
170C4—4822no—
171C4—4821no—
172C4—4821no—
173C4D4×D24112yes—
174C2—42no—
175C2—41no—
176C2—4822no—
177C2—4824no—
178C2C2×D4×D24111yescharacteristic
179C2#41×D24111yescharacteristic
180C2#2144068711yescharacteristic
subgroupG/H
center, FrattiniC22C23×D241
commutator, socle, cosocleC22×C241C24
Fitting#121×C241C2
hypercenter#121D241
hypocenterC241#546
LCSG ▷ C22×C241 ▷ C241
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#16838770C2semidirect
2. C2×D4×D241C2semidirect
3. #21440629C2semidirect
4. #21440624C2semidirect
5. #21440622C2semidirect
6. #21440613C2semidirect
7. #21440610C2semidirect
8. #21440604C2semidirect
9. C4×C22⋊D241C2semidirect
10. C22⋊C4×D241C2semidirect
11. #121×C241C2semidirect
14. C2×Dic482C22semidirect
18. C2×C22⋊D241C22semidirect
21. C2×C4×D241C22semidirect
22. C2×C4×D241C22semidirect
23. C4⋊Dic241C22semidirect
24. #16838749C22semidirect
25. C22⋊Dic241C22semidirect
26. C22⋊Dic241C22semidirect
27. C22⋊Dic241C22semidirect
30. #16838747C22semidirect
31. C4×Dic241C22semidirect
33. C2×D4×C241C22semidirect
34. C22×C4×C241C22semidirect
35. C4⋊C4×C241C22semidirect
36. C22⋊C4×C241C22semidirect
43. D964D4semidirect
75. C2×D241#41semidirect
90. C241#546semidirect
98. #121D241semidirect
112. C2×D4C2×D241semidirect
116. C22×C4C2×D241semidirect
119. C4⋊C4C2×D241semidirect
126. C22⋊C4C2×D241semidirect

Maximal subgroups

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

subgroupindex
C2×#168387702
C2×D4×D2412
#214406292
#214406242
#214406222
#214406132
#214406102
#214406042
C4×C22⋊D2412
C22⋊C4×D2412
#121×C2412
#546241