Finite Group Databasebeta

← #29508225

Subgroups — #29508225

Order 15424.

330 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 330 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×#16838769C211yesmaximal, characteristic, complemented
2C22×C4×D241C211yesmaximal, characteristic, complemented
3#21440625C211yesmaximal, characteristic, complemented
4#21440620C212yesmaximal, complemented
5#21440615C212yesmaximal, complemented
6#21440612C211yesmaximal, characteristic, complemented
7#21440610C211yesmaximal, characteristic, complemented
8#21440604C212yesmaximal, complemented
9#21440602C212yesmaximal, complemented
10#21440598C211yesmaximal, characteristic, complemented
11#123×C241C211yesmaximal, characteristic, complemented
12C23×D241C2211yescharacteristic
13#16838769—24no—
14C2×Dic482C2211yescharacteristic, complemented
15C2×D964C2212yes—
16C2×C22⋊D241C2212yes—
17C2×C22⋊D241C2211yescharacteristic
18C2×C22⋊D241C2211yescharacteristic, complemented
19C22×Dic241C2211yescharacteristic
20C22×Dic241C2211yescharacteristic, complemented
21C2×C4×D241—24no—
22C2×C4×D241C2212yes—
23C2×C4×D241C2212yes—
24C2×C4×D241C2211yescharacteristic, complemented
25#16838749C2212yescomplemented
26#16838749C2212yescomplemented
27C22⋊Dic241C2211yescharacteristic
28#16838747C2212yes—
29#16838747C2212yes—
30#16838747C2212yescomplemented
31C4×Dic241C2212yescomplemented
32C2×D4×C241C2211yescharacteristic
33C22×C4×C241C2211yescharacteristic
34C4⋊C4×C241C2212yescomplemented
35C22⋊C4×C241C2211yescharacteristic
36C22⋊C4×C241C2212yescomplemented
37C22×D241—22no—
38C22×D241—24no—
39C22×D241C2311yescharacteristic
40C22×D241C2312yes—
41Dic482—22no—
42D964—24no—
43C22⋊D241—22no—
44C22⋊D241—24no—
45C22⋊D241—22no—
46C2×Dic241—22no—
47C2×Dic241D414yescomplemented
48C2×Dic241C2312yes—
49C2×Dic241C2311yescharacteristic
50C2×Dic241C2311yescharacteristic
51C2×Dic241C2311yescharacteristic
52C4×D241—24no—
53C4×D241—24no—
54C4×D241—22no—
55C23×C241C2311yescharacteristic
56C23×C241C2311yescharacteristic
57D4×C241—22no—
58C2×C4×C241—22no—
59C2×C4×C241C2312yes—
60C2×C4×C241C2312yes—
61C2×C4×C241C2311yescharacteristic
62C22×C241—22no—
63C22×C241—21no—
64C22×C241—22no—
65C22×C241C2×D412yes—
66C22×C241C2411yescharacteristic
67C2×D241—28no—
68C2×D241—22no—
69C2×D241—22no—
70C2×D241—21no—
71C2×D241—22no—
72C2×D241#4114yescomplemented
73C4×C241—22no—
74C4×C241—21no—
75C4×C241—22no—
76Dic241—21no—
77Dic241—22no—
78Dic241C2×D412yes—
79Dic241C2×D412yes—
80C2×C241—22no—
81C2×C241—41no—
82C2×C241C22×D411yescharacteristic
83C2×C241C2×#4112yes—
84D241—24no—
85D241—41no—
86C241#52511yescharacteristic, complemented, Sylow
87#525—2411nomaximal, Sylow
88C2×#41—2411no—
89C23×C4—2411no—
90#123—2411no—
91#123D24111yescharacteristic, complemented
92C22⋊Q8—2412no—
93#121—2412no—
94#120—2411no—
95#120—2411no—
96C4×D4—2412no—
97C4×D4—2412no—
98#117—2411no—
99C24—2411no—
100#41—4824no—
101C2×Q8—2411no—
102C2×D4—2412no—
103C2×D4—2411no—
104C2×D4—2411no—
105C2×D4—2412no—
106C2×D4C2×D24111yescharacteristic
107C22×C4—4824no—
108C22×C4—2411no—
109C22×C4—2412no—
110C22×C4—2412no—
111C22×C4—2411no—
112C22×C4—2411no—
113C22×C4C2×D24111yescharacteristic
114C4⋊C4—2412no—
115C4⋊C4—2412no—
116C4⋊C4C2×D24112yescomplemented
117C22⋊C4—2411no—
118C22⋊C4—2412no—
119C22⋊C4—2412no—
120C22⋊C4—2412no—
121C22⋊C4C2×D24111yescharacteristic
122C22⋊C4C2×D24112yescomplemented
123C42—2412no—
124C23—4824no—
125C23—4822no—
126C23—2411no—
127C23—2412no—
128C23C22×D24111yescharacteristic
129C23C22×D24111yescharacteristic
130Q8—4822no—
131D4—22no—
132D4—4824no—
133D4—4824no—
134D4—4822no—
135D4—4822no—
136C2×C4—22no—
137C2×C4—4824no—
138C2×C4—4824no—
139C2×C4—4822no—
140C2×C4—4822no—
141C2×C4—2412no—
142C2×C4—2411no—
143C2×C4—2414no—
144C2×C4—2411no—
145C2×C4—2411no—
146C2×C4C22×D24112yes—
147C2×C4C22×D24111yescharacteristic
148C2×C4C22×D24112yes—
149C22—22no—
150C22—21no—
151C22—22no—
152C22—4828no—
153C22—4821no—
154C22—4822no—
155C22—4822no—
156C22—4822no—
157C22—2414no—
158C22D4×D24112yes—
159C22C23×D24111yescharacteristic
160C4—22no—
161C4—21no—
162C4—22no—
163C4—4821no—
164C4—4822no—
165C4—2412no—
166C4—2412no—
167C2—22no—
168C2—41no—
169C2—9641no—
170C2—4824no—
171C2C2×D4×D24111yescharacteristic
172C2#41×D24112yes—
subgroupG/H
center, FrattiniC22C23×D241
commutator, socle, cosocleC22×C241C24
Fitting#123×C241C2
hypercenter#123D241
hypocenterC241#525
LCSG ▷ C22×C241 ▷ C241
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C2×#16838769C2semidirect
2. C22×C4×D241C2semidirect
3. #21440625C2semidirect
4. #21440620C2semidirect
5. #21440615C2semidirect
6. #21440612C2semidirect
7. #21440610C2semidirect
8. #21440604C2semidirect
9. #21440602C2semidirect
10. #21440598C2semidirect
11. #123×C241C2semidirect
14. C2×Dic482C22semidirect
18. C2×C22⋊D241C22semidirect
20. C22×Dic241C22semidirect
24. C2×C4×D241C22semidirect
25. #16838749C22semidirect
26. #16838749C22semidirect
30. #16838747C22semidirect
31. C4×Dic241C22semidirect
34. C4⋊C4×C241C22semidirect
36. C22⋊C4×C241C22semidirect
47. C2×Dic241D4semidirect
72. C2×D241#41semidirect
86. C241#525semidirect
91. #123D241semidirect
116. C4⋊C4C2×D241semidirect
122. C22⋊C4C2×D241semidirect

Maximal subgroups

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

subgroupindex
C2×#168387692
C22×C4×D2412
#214406252
#214406202
#214406152
#214406122
#214406102
#214406042
#214406022
#214405982
#123×C2412
#525241