Finite Group Databasebeta

← C24×#104 #10574041

Subgroups — C24×#104 #10574041

Order 512.

26434 conjugacy classes of subgroups; only the 8034 normal subgroups are shown.

normal subgroups only

This group has 26434 conjugacy classes of subgroups — too many to draw. Only its 8034 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
1C24×#41C211yesmaximal, characteristic
2C24×M16C211yesmaximal, characteristic, complemented
3C24×C42C211yesmaximal, characteristic, complemented
4C23×#104C2160yesmaximal, direct product factor
5C23×#41—215no—
6C23×#41—215no—
7C23×#41C4130yescomplemented
8C23×#41C22130yes—
9C24×Q8C411yescharacteristic, complemented
10C24×D4—21no—
11C24×D4C411yescharacteristic, complemented
12C25×C4—21no—
13C25×C4—21no—
14C25×C4C2211yescharacteristic
15C23×M16—215no—
16C23×M16C22130yescomplemented
17C24×C8—21no—
18C23×C42—215no—
19C23×C42C22130yescomplemented
20C22×#104C221560yesdirect product factor
21C26—21no—
22C26D411yescharacteristic
23C22×#41—2420no—
24C22×#41—2210no—
25C22×#41—2210no—
26C22×#41C2×C41420yescomplemented
27C22×#41C231140yes—
28C23×Q8—215no—
29C23×Q8C2×C4130yescomplemented
30C23×D4—230no—
31C23×D4—230no—
32C23×D4—215no—
33C23×D4C2×C4130yescomplemented
34C24×C4—232no—
35C24×C4—21no—
36C24×C4—230no—
37C24×C4—230no—
38C24×C4—230no—
39C24×C4C2×C411yescharacteristic
40C24×C4C2×C4115yes—
41C24×C4D4115yes—
42C24×C4D4115yes—
43C24×C4D411yescharacteristic
44C24×C4C23115yes—
45C22×M16—2210no—
46C22×M16C231140yescomplemented
47C23×C8—230no—
48C22×C42—2210no—
49C22×C42C231140yescomplemented
50C2×#104C231960yesdirect product factor
51C25—216no—
52C25—230no—
53C25—416no—
54C25C22⋊C4115yes—
55C25C22⋊C411yescharacteristic
56C25C2×D4115yes—
57C2×#41—22520no—
58C2×#41—2420no—
59C2×#41—2420no—
60C2×#41C22×C41840yescomplemented
61C2×#41C241120yes—
62C22×Q8—2210no—
63C22×Q8C22×C41140yescomplemented
64C22×D4—2420no—
65C22×D4—2140no—
66C22×D4—2210no—
67C22×D4C22×C41140yescomplemented
68C23×C4—2480no—
69C23×C4—230no—
70C23×C4—2140no—
71C23×C4—2420no—
72C23×C4—2140no—
73C23×C4C22⋊C4115yes—
74C23×C4C22⋊C4115yes—
75C23×C4C22⋊C41210yes—
76C23×C4C22×C4115yes—
77C23×C4C22×C41105yes—
78C23×C4C2×D41105yes—
79C23×C4C2×D4115yes—
80C23×C4C2×D41105yes—
81C23×C4C24135yes—
82C2×M16—2420no—
83C2×M16C241120yescomplemented
84C22×C8—2140no—
85C2×C42—2420no—
86C2×C42C241120yescomplemented
87#104C241256yesdirect product factor
88C24—2120no—
89C24—2120no—
90C24—2140no—
91C24—4240no—
92C24#104116yesdirect product factor
93C24C2×C22⋊C41105yes—
94C24C2×C22⋊C4115yes—
95C24C22×D4135yes—
96#41—21680no—
97#41—2120no—
98#41—2120no—
99#41C23×C41240yescomplemented
100#41C25116yes—
101C2×Q8—2420no—
102C2×Q8C23×C41120yescomplemented
103C2×D4—2840no—
104C2×D4—2120no—
105C2×D4—2420no—
106C2×D4C23×C41120yescomplemented
107C22×C4—21120no—
108C22×C4—2140no—
109C22×C4—2120no—
110C22×C4—2840no—
111C22×C4—2120no—
112C22×C4C2×C22⋊C41105yes—
113C22×C4C2×C22⋊C41105yes—
114C22×C4C2×C22⋊C41630yes—
115C22×C4C23×C4135yes—
116C22×C4C23×C41105yes—
117C22×C4C22×D41105yes—
118C22×C4C22×D4135yes—
119C22×C4C22×D41105yes—
120C22×C4C25115yes—
121M16—2120no—
122M16C25116yescomplemented
123C2×C8—2120no—
124C42—2120no—
125C42C25116yescomplemented
126C23—2140no—
127C23—2420no—
128C23—2120no—
129C23—4560no—
130C23C2×#1041120yesdirect product factor
131C23C22×C22⋊C41105yes—
132C23C22×C22⋊C4135yes—
133C23C23×D4115yes—
134Q8—2120no—
135Q8C24×C4116yescomplemented
136D4—216no—
137D4—2240no—
138D4—2120no—
139D4C24×C4116yescomplemented
140C2×C4—2480no—
141C2×C4—216no—
142C2×C4—2120no—
143C2×C4—216no—
144C2×C4—2240no—
145C2×C4C22×C22⋊C41105yes—
146C2×C4C22×C22⋊C41105yes—
147C2×C4C22×C22⋊C41210yes—
148C2×C4C24×C4115yes—
149C2×C4C24×C4115yes—
150C2×C4C23×D4115yes—
151C2×C4C23×D4115yes—
152C2×C4C23×D4115yes—
153C2×C4C2611yescharacteristic
154C8—216no—
155C22—230no—
156C22—2210no—
157C22—216no—
158C22—4240no—
159C22C22×#1041140yesdirect product factor
160C22C23×C22⋊C4115yes—
161C22C23×C22⋊C4115yes—
162C22C24×D411yescharacteristic
163C4—232no—
164C4—216no—
165C4C23×C22⋊C4115yes—
166C4C23×C22⋊C4115yes—
167C4C25×C411yescharacteristic
168C4C24×D411yescharacteristic
169C2—21no—
170C2—215no—
171C2—416no—
172C2C23×#104130yesdirect product factor
173C2C24×C22⋊C411yescharacteristic
subgroupG/H
centerC24×C4D4
commutatorC4C25×C4
Frattini, cosocleC2×C4C26
socleC25C22⋊C4
prime radicalC24×#41C2
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C24×C4 ◁ C25×C4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C24×M16C2semidirect
3. C24×C42C2semidirect
4. C23×#104C2direct (N × H)
7. C23×#41C4semidirect
9. C24×Q8C4semidirect
11. C24×D4C4semidirect
16. C23×M16C22semidirect
19. C23×C42C22semidirect
20. C22×#104C22direct (N × H)
26. C22×#41C2×C4semidirect
29. C23×Q8C2×C4semidirect
33. C23×D4C2×C4semidirect
46. C22×M16C23semidirect
49. C22×C42C23semidirect
50. C2×#104C23direct (N × H)
60. C2×#41C22×C4semidirect
63. C22×Q8C22×C4semidirect
67. C22×D4C22×C4semidirect
83. C2×M16C24semidirect
86. C2×C42C24semidirect
87. #104C24direct (N × H)
92. C24#104direct (N × H)
99. #41C23×C4semidirect
102. C2×Q8C23×C4semidirect
106. C2×D4C23×C4semidirect
122. M16C25semidirect
125. C42C25semidirect
130. C23C2×#104direct (N × H)
135. Q8C24×C4semidirect
139. D4C24×C4semidirect
159. C22C22×#104direct (N × H)
172. C2C23×#104direct (N × H)

Maximal subgroups

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

subgroupindex
C24×#412
C24×M162
C24×C422
C23×#1042