Finite Group Databasebeta

← C24×#11758224 #30593895

Subgroups — C24×#11758224 #30593895

Order 15616.

65470 conjugacy classes of subgroups; only the 32651 normal subgroups are shown.

normal subgroups only

This group has 65470 conjugacy classes of subgroups — too many to draw. Only its 32651 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
1C23×#11758224C21120yesmaximal, direct product factor
2C24×Dic122C211yesmaximal, characteristic, complemented
3C24×C22⋊D61C212yesmaximal, complemented
4C25×Dic61C212yesmaximal, complemented
5C24×C4×D61C211yesmaximal, characteristic, complemented
6C24×D4×C61C211yesmaximal, characteristic, complemented
7C25×D61C2211yescharacteristic
8C22×#11758224C2212240yesdirect product factor
9C23×Dic122C22160yescomplemented
10C23×C22⋊D61C221120yescomplemented
11C24×Dic61C2211yescharacteristic
12C24×Dic61C2212yes—
13C24×Dic61C221120yescomplemented
14C23×C4×D61C22160yescomplemented
15C26×C61C2212yes—
16C23×D4×C61C22160yescomplemented
17C24×C4×C61C2211yescharacteristic
18C24×D61—216no—
19C24×D61C23130yes—
20C2×#11758224C2317680yesdirect product factor
21C22×Dic122C231560yescomplemented
22C22×C22⋊D61C2311120yescomplemented
23C23×Dic61C23130yes—
24C23×Dic61C23160yes—
25C23×Dic61C2311120yescomplemented
26C22×C4×D61C231560yescomplemented
27C25×C61—232no—
28C25×C61C23160yes—
29C25×C61C2311yescharacteristic
30C22×D4×C61C231560yescomplemented
31C23×C4×C61C23130yes—
32C24×C61—2480no—
33C24×C61#41116yescomplemented
34C24×C61C241280yes—
35C24×C61C24115yes—
36C23×D61—2240no—
37C23×D61C241140yes—
38C2×D4×C61C241960yescomplemented
39C22×C4×C61C241140yes—
40C2×C22⋊D61C2411920yescomplemented
41C22×Dic61C241140yes—
42C22×Dic61C241280yes—
43C22×Dic61C2411920yescomplemented
44#11758224C2414096yesdirect product factor
45C2×C4×D61C241960yescomplemented
46C2×Dic122C241960yescomplemented
47C23×C61—21120no—
48C23×C61C2×#411120yescomplemented
49C23×C61C251240yes—
50C23×C61C25135yes—
51C22×D61—2560no—
52C22×D61C251120yes—
53D4×C61C251256yescomplemented
54C2×C4×C61C251120yes—
55C22⋊D61C251512yescomplemented
56C2×Dic61C251240yes—
57C2×Dic61C251120yes—
58C2×Dic61C251512yescomplemented
59C4×D61C251256yescomplemented
60Dic122C251256yescomplemented
61C24×#41—611nomaximal, Sylow
62C22×C61—2480no—
63C22×C61C22×#411140yescomplemented
64C22×C61C26132yes—
65C22×C61C26115yes—
66C2×D61—2240no—
67C2×D61C26116yes—
68C4×C61C26116yes—
69Dic61C26132yes—
70Dic61C26116yes—
71C23×#41—61120no—
72C24×Q8—611no—
73C24×D4—612no—
74C24×D4D6111yescharacteristic, complemented
75C25×C4—612no—
76C25×C4—611no—
77C2×C61—232no—
78C2×C61C23×#41130yescomplemented
79C2×C61C2711yescharacteristic
80D61—216no—
81C26—611no—
82C26C2×D6112yes—
83C22×#41—612240no—
84C23×Q8—6160no—
85C23×D4—61120no—
86C23×D4C2×D61160yescomplemented
87C24×C4—61120no—
88C24×C4—611no—
89C24×C4—612no—
90C24×C4—6160no—
91C24×C4C2×D6111yescharacteristic
92C61C24×#4111yescharacteristic, complemented, Sylow
93C25—232no—
94C25—6130no—
95C25—12216no—
96C25C22×D61160yes—
97C25C22×D6111yescharacteristic
98C2×#41—617680no—
99C22×Q8—61560no—
100C22×D4—611120no—
101C22×D4C22×D611560yescomplemented
102C23×C4—6130no—
103C23×C4—6160no—
104C23×C4—611120no—
105C23×C4—61560no—
106C23×C4C22×D61130yes—
107C24—2480no—
108C24—61140no—
109C24—122240no—
110C24#11758224116yesdirect product factor
111C24C23×D611280yes—
112C24C23×D61115yes—
113#41—614096no—
114C2×Q8—61960no—
115C2×D4—611920no—
116C2×D4C23×D611960yescomplemented
117C22×C4—61140no—
118C22×C4—61280no—
119C22×C4—611920no—
120C22×C4—61960no—
121C22×C4C23×D611140yes—
122C23—21120no—
123C23—61120no—
124C23—122560no—
125C23C2×#117582241120yesdirect product factor
126C23C24×D611240yes—
127C23C24×D61135yes—
128Q8—61256no—
129D4—61512no—
130D4C24×D611256yescomplemented
131C2×C4—61120no—
132C2×C4—61240no—
133C2×C4—61512no—
134C2×C4—61256no—
135C2×C4C24×D611120yes—
136C22—2480no—
137C22—6116no—
138C22—122240no—
139C22C22×#117582241140yesdirect product factor
140C22C25×D61132yes—
141C22C25×D61115yes—
142C4—6116no—
143C4—6132no—
144C4C25×D61116yes—
145C2—232no—
146C2—12216no—
147C2C23×#11758224130yesdirect product factor
148C2C26×D6111yescharacteristic
subgroupG/H
centerC25C22×D61
commutator, cosocleC2×C61C27
FrattiniC2C26×D61
FittingC24×D4×C61C2
socleC25×C61C23
hypercenterC24×D4D61
hypocenterC61C24×#41
LCSG ▷ C2×C61 ▷ C61
UCSC1 ◁ C25 ◁ C24×D4

Complements

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

NH ≅ G/Ntype
1. C23×#11758224C2direct (N × H)
2. C24×Dic122C2semidirect
3. C24×C22⋊D61C2semidirect
4. C25×Dic61C2semidirect
5. C24×C4×D61C2semidirect
6. C24×D4×C61C2semidirect
8. C22×#11758224C22direct (N × H)
9. C23×Dic122C22semidirect
10. C23×C22⋊D61C22semidirect
13. C24×Dic61C22semidirect
14. C23×C4×D61C22semidirect
16. C23×D4×C61C22semidirect
20. C2×#11758224C23direct (N × H)
21. C22×Dic122C23semidirect
22. C22×C22⋊D61C23semidirect
25. C23×Dic61C23semidirect
26. C22×C4×D61C23semidirect
30. C22×D4×C61C23semidirect
33. C24×C61#41semidirect
38. C2×D4×C61C24semidirect
40. C2×C22⋊D61C24semidirect
43. C22×Dic61C24semidirect
44. #11758224C24direct (N × H)
45. C2×C4×D61C24semidirect
46. C2×Dic122C24semidirect
48. C23×C61C2×#41semidirect
53. D4×C61C25semidirect
55. C22⋊D61C25semidirect
58. C2×Dic61C25semidirect
59. C4×D61C25semidirect
60. Dic122C25semidirect
63. C22×C61C22×#41semidirect
74. C24×D4D61semidirect
78. C2×C61C23×#41semidirect
86. C23×D4C2×D61semidirect
92. C61C24×#41semidirect
101. C22×D4C22×D61semidirect
110. C24#11758224direct (N × H)
116. C2×D4C23×D61semidirect
125. C23C2×#11758224direct (N × H)
130. D4C24×D61semidirect
139. C22C22×#11758224direct (N × H)
147. C2C23×#11758224direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117582242
C24×Dic1222
C24×C22⋊D612
C25×Dic612
C24×C4×D612
C24×D4×C612
C24×#4161