Finite Group Databasebeta

← #29508360

Subgroups — #29508360

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×#16838771C211yesmaximal, characteristic, complemented
2C2×D4×D241C211yesmaximal, characteristic, complemented
3#21440624C212yesmaximal, complemented
4#21440618C211yesmaximal, characteristic, complemented
5#21440615C212yesmaximal, complemented
6#21440613C211yesmaximal, characteristic, complemented
7#21440612C212yesmaximal, complemented
8C4×D964C211yesmaximal, characteristic, complemented
9C4×D964C211yesmaximal, characteristic, complemented
10C22⋊C4×D241C212yesmaximal, complemented
11#124×C241C211yesmaximal, characteristic, complemented
12C23×D241C2212yes—
13#16838771—24no—
14C2×D964C2211yescharacteristic
15C2×D964C2211yescharacteristic
16C2×D964C2212yescomplemented
17C2×C22⋊D241C2212yes—
18C2×C22⋊D241C2212yescomplemented
19D4×D241—24no—
20C2×C4×D241C2211yescharacteristic
21C2×C4×D241C2212yescomplemented
22C2×C4×D241C2211yescharacteristic, complemented
23C2×C4×D241C2212yescomplemented
24C4⋊Dic241C2211yescharacteristic, complemented
25C4⋊Dic241C2211yescharacteristic, complemented
26#16838749C2212yescomplemented
27C22⋊Dic241C2212yescomplemented
28#16838747C2211yescharacteristic
29#16838747C2212yes—
30#16838747C2212yescomplemented
31#16838747C2211yescharacteristic, complemented
32C2×Q8×C241C2211yescharacteristic, complemented
33C2×D4×C241C2211yescharacteristic
34C22⋊C4×C241C2212yescomplemented
35C22⋊C4×C241C2212yescomplemented
36C42×C241C2211yescharacteristic, complemented
37C22×D241—24no—
38C22×D241—24no—
39C22×D241—42no—
40C22×D241C2311yescharacteristic
41C22×D241C2312yes—
42C22×D241C2311yescharacteristic
43D964—24no—
44D964—22no—
45D964D414yescomplemented
46C22⋊D241—24no—
47C22⋊D241—42no—
48C2×Dic241C2312yes—
49C2×Dic241C2311yescharacteristic
50C2×Dic241C2311yescharacteristic
51C4×D241—24no—
52C4×D241—22no—
53C4×D241—22no—
54C4×D241—42no—
55C23×C241C2312yes—
56Q8×C241—22no—
57D4×C241—22no—
58C2×C4×C241C2311yescharacteristic
59C2×C4×C241C2311yescharacteristic
60C2×C4×C241C2312yes—
61C2×C4×C241C2311yescharacteristic
62C22×C241—22no—
63C22×C241—22no—
64C22×C241—22no—
65C22×C241C2411yescharacteristic
66C2×D241—21no—
67C2×D241—22no—
68C2×D241—22no—
69C2×D241—21no—
70C2×D241—21no—
71C2×D241—21no—
72C2×D241—21no—
73C2×D241—48no—
74C2×D241C2×D414yes—
75C2×D241#4112yescomplemented
76C4×C241—22no—
77C4×C241—21no—
78C4×C241—21no—
79C4×C241C2×D412yes—
80Dic241—21no—
81Dic241—21no—
82Dic241—22no—
83C2×C241—42no—
84C2×C241C22×D411yescharacteristic
85C2×C241C2×#4111yescharacteristic
86C2×C241H(5,𝔽2)11yescharacteristic
87D241—22no—
88D241—24no—
89D241—41no—
90C241#54611yescharacteristic, complemented, Sylow
91#546—2411nomaximal, Sylow
92C2×#41—2411no—
93C22×D4—2411no—
94#124D24111yescharacteristic, complemented
95#123—2412no—
96C22⋊Q8—2411no—
97#121—2411no—
98#121—2412no—
99#120—2412no—
100C4×D4—2411no—
101C4×D4—2411no—
102C2×C22⋊C4—2412no—
103C24—2412no—
104#41—4824no—
105C2×Q8C2×D24111yescharacteristic, complemented
106C2×D4—4824no—
107C2×D4—2412no—
108C2×D4—2412no—
109C2×D4—2411no—
110C2×D4—2412no—
111C2×D4—2411no—
112C2×D4C2×D24111yescharacteristic
113C22×C4—2412no—
114C22×C4—2411no—
115C22×C4—2412no—
116C22×C4—2411no—
117C4⋊C4—2412no—
118C4⋊C4—2411no—
119C4⋊C4—2411no—
120C22⋊C4—2412no—
121C22⋊C4—2411no—
122C22⋊C4—2412no—
123C22⋊C4—2412no—
124C22⋊C4—2411no—
125C22⋊C4C2×D24112yescomplemented
126C22⋊C4C2×D24112yescomplemented
127C42C2×D24111yescharacteristic, complemented
128C23—9642no—
129C23—4824no—
130C23—4824no—
131C23—2411no—
132C23—2411no—
133C23—2412no—
134C23C22×D24112yes—
135Q8—22no—
136D4—22no—
137D4—9642no—
138D4—4824no—
139D4—4822no—
140D4—4824no—
141D4—2414no—
142C2×C4—9642no—
143C2×C4—4824no—
144C2×C4—4822no—
145C2×C4—4822no—
146C2×C4—2412no—
147C2×C4—2411no—
148C2×C4—2411no—
149C2×C4C22×D24111yescharacteristic
150C2×C4C22×D24111yescharacteristic
151C2×C4C22×D24112yes—
152C2×C4C22×D24111yescharacteristic
153C22—22no—
154C22—22no—
155C22—22no—
156C22—9648no—
157C22—4821no—
158C22—4821no—
159C22—4821no—
160C22—4821no—
161C22—4822no—
162C22—4822no—
163C22—4821no—
164C22—2414no—
165C22—2412no—
166C22C23×D24111yescharacteristic
167C4—22no—
168C4—21no—
169C4—21no—
170C4—4821no—
171C4—4821no—
172C4—4822no—
173C4D4×D24112yes—
174C2—42no—
175C2—9641no—
176C2—4822no—
177C2—4824no—
178C2C2×D4×D24111yescharacteristic
179C2#41×D24111yescharacteristic
180C2#2144068811yescharacteristic
subgroupG/H
center, FrattiniC22C23×D241
commutator, socle, cosocleC22×C241C24
Fitting#124×C241C2
hypercenter#124D241
hypocenterC241#546
LCSG ▷ C22×C241 ▷ C241
UCSC1 ◁ C22 ◁ #124

Complements

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

NH ≅ G/Ntype
1. C2×#16838771C2semidirect
2. C2×D4×D241C2semidirect
3. #21440624C2semidirect
4. #21440618C2semidirect
5. #21440615C2semidirect
6. #21440613C2semidirect
7. #21440612C2semidirect
8. C4×D964C2semidirect
9. C4×D964C2semidirect
10. C22⋊C4×D241C2semidirect
11. #124×C241C2semidirect
16. C2×D964C22semidirect
18. C2×C22⋊D241C22semidirect
21. C2×C4×D241C22semidirect
22. C2×C4×D241C22semidirect
23. C2×C4×D241C22semidirect
24. C4⋊Dic241C22semidirect
25. C4⋊Dic241C22semidirect
26. #16838749C22semidirect
27. C22⋊Dic241C22semidirect
30. #16838747C22semidirect
31. #16838747C22semidirect
32. C2×Q8×C241C22semidirect
34. C22⋊C4×C241C22semidirect
35. C22⋊C4×C241C22semidirect
36. C42×C241C22semidirect
45. D964D4semidirect
75. C2×D241#41semidirect
90. C241#546semidirect
94. #124D241semidirect
105. C2×Q8C2×D241semidirect
125. C22⋊C4C2×D241semidirect
126. C22⋊C4C2×D241semidirect
127. C42C2×D241semidirect

Maximal subgroups

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

subgroupindex
C2×#168387712
C2×D4×D2412
#214406242
#214406182
#214406152
#214406132
#214406122
C4×D9642
C22⋊C4×D2412
#124×C2412
#546241