Finite Group Databasebeta

← C2×D4×#157 #10596641

Subgroups — C2×D4×#157 #10596641

Order 576.

662 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 662 conjugacy classes of subgroups — too many to draw. Only its 156 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×#157C212yesmaximal, complemented
2C2×D4×C3⋊D3C211yesmaximal, characteristic
3C2×C22⋊#157C212yesmaximal, complemented
4D4×#157C218yesmaximal, direct product factor
5C2×C4⋊#157C211yesmaximal, characteristic, complemented
6C2×C4×#157C211yesmaximal, characteristic, complemented
7C23×C3⋊D3C2212yes—
8C22×#157—28no—
9C22×#157C2211yescharacteristic
10C22×#157C2212yes—
11C22×#157C2211yescharacteristic
12C22×#157C2218yescomplemented
13C2×C32×D4C411yescharacteristic, complemented
14C2×#638C412yescomplemented
15D4×C3⋊D3C414yescomplemented
16D4×C3⋊D3C2214yes—
17C2×C4⋊(C3⋊D3)C411yescharacteristic, complemented
18C2×C4×C3⋊D3C2211yescharacteristic
19C22⋊#157C2218yescomplemented
20C4⋊#157C2214yescomplemented
21C4×#157C2214yescomplemented
22C2×D3×D4—62no—
23C23×C32C2×C412yescomplemented
24C22×C3⋊D3—24no—
25C22×C3⋊D3—24no—
26C22×C3⋊D3C2×C414yescomplemented
27C22×C3⋊D3C2×C412yescomplemented
28C22×C3⋊D3C2314yes—
29C22×C3⋊D3C2311yescharacteristic
30C2×#157—24no—
31C2×#157—22no—
32C2×#157—216no—
33C2×#157D414yesdirect product factor
34C2×#157C2314yes—
35C2×#157C2312yes—
36C2×#157C2312yes—
37C32×D4C2×C414yescomplemented
38C2×C32×C4C2×C411yescharacteristic, complemented
39#638C2×C418yescomplemented
40C2×C3⋊Dic3C2×C411yescharacteristic, complemented
41C4⋊(C3⋊D3)C2×C414yescomplemented
42C4×C3⋊D3C2×C412yescomplemented
43C4×C3⋊D3C2312yes—
44C2×C4×D4—91nomaximal, Sylow
45C23×D3—64no—
46C2×C3×D4—22no—
47C2×#67—64no—
48D3×D4—68no—
49D3×D4—68no—
50C2×D12—62no—
51C2×C4×D3—62no—
52C22×C32—24no—
53C22×C32C22×C411yescharacteristic
54C22×C32C22×C414yescomplemented
55C2×C3⋊D3—24no—
56C2×C3⋊D3—28no—
57C2×C3⋊D3—24no—
58C2×C3⋊D3—24no—
59C2×C3⋊D3C22×C411yescharacteristic
60C2×C3⋊D3C22×C414yescomplemented
61C2×C3⋊D3C2×D412yes—
62C2×C3⋊D3#4112yes—
63C2×C3⋊D3C2411yescharacteristic
64#157—24no—
65#157—22no—
66#157C2×D414yesdirect product factor
67C32×C4C22×C412yescomplemented
68C3⋊Dic3C22×C412yescomplemented
69C22×D4—91no—
70C23×C4—92no—
71C4×D4—98no—
72C2×C4⋊C4—91no—
73C2×C22⋊C4—92no—
74C2×C42—91no—
75C23×C3—24no—
76C22×D3—62no—
77C22×D3—64no—
78C22×D3—68no—
79C22×D3—68no—
80C22×D3—128no—
81C22×D3—128no—
82C3×D4—28no—
83C2×C3×C4—22no—
84#67—616no—
85C2×Dic3—62no—
86D12—68no—
87C4×D3—64no—
88C4×D3—64no—
89C2×C32—24no—
90C2×C32C4×D412yescomplemented
91C2×C32C23×C411yescharacteristic
92C3⋊D3—24no—
93C3⋊D3C4×D412yescomplemented
94C3⋊D3C22×D411yescharacteristic
95C3⋊D3C2×#4111yescharacteristic
96C24—92no—
97C2×D4—92no—
98C2×D4—94no—
99C2×D4—94no—
100C2×D4—91no—
101C2×D4#15711yescharacteristic, direct product factor
102C22×C4—91no—
103C22×C4—91no—
104C22×C4—91no—
105C22×C4—92no—
106C22×C4—98no—
107C22×C4—188no—
108C4⋊C4—94no—
109C22⋊C4—98no—
110C42—94no—
111C22×C3—22no—
112C22×C3—28no—
113C22×C3—48no—
114C2×D3—62no—
115C2×D3—62no—
116C2×D3—64no—
117C2×D3—64no—
118C2×D3—68no—
119C2×D3—128no—
120C2×D3—1216no—
121C2×D3—128no—
122C2×D3—128no—
123C3×C4—24no—
124Dic3—64no—
125C32C2×C4×D411yescharacteristic, complemented, Sylow
126C23—91no—
127C23—92no—
128C23—94no—
129C23—94no—
130C23—184no—
131C23—184no—
132C23C2×#15712yescomplemented
133D4—98no—
134D4—94no—
135D4C2×#15714yesdirect product factor
136C2×C4—92no—
137C2×C4—94no—
138C2×C4—91no—
139C2×C4—92no—
140C2×C4—92no—
141C2×C4—92no—
142C2×C4—94no—
143C2×C4—1816no—
144C2×C4—184no—
145C2×C4—182no—
146C2×C4C2×#15711yescharacteristic, complemented
147C2×C3—24no—
148C2×C3—22no—
149C2×C3—48no—
150D3—64no—
151D3—62no—
152D3—62no—
153D3—128no—
154C22—24no—
155C22—92no—
156C22—92no—
157C22—91no—
158C22—91no—
159C22—94no—
160C22—184no—
161C22—184no—
162C22—188no—
163C22—184no—
164C22C22×#15711yescharacteristic
165C22C22×#15714yescomplemented
166C4—94no—
167C4—92no—
168C4—184no—
169C4—182no—
170C4C22×#15712yescomplemented
171C3—22no—
172C2—24no—
173C2—91no—
174C2—91no—
175C2—92no—
176C2—184no—
177C2D4×#15712yesdirect product factor
178C2C23×#15711yescharacteristic
subgroupG/H
centerC22C22×#157
commutatorC2×C32C23×C4
FrattiniC2C23×#157
FittingC2×C32×D4C4
socleC22×C32C22×C4
prime radicalC2×D4×C3⋊D3C2
cosocleC2×C3⋊D3C24
hypercenterC2×D4#157
hypocenterC32C2×C4×D4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C23×#157C2semidirect
3. C2×C22⋊#157C2semidirect
4. D4×#157C2direct (N × H)
5. C2×C4⋊#157C2semidirect
6. C2×C4×#157C2semidirect
12. C22×#157C22semidirect
13. C2×C32×D4C4semidirect
14. C2×#638C4semidirect
15. D4×C3⋊D3C4semidirect
17. C2×C4⋊(C3⋊D3)C4semidirect
19. C22⋊#157C22semidirect
20. C4⋊#157C22semidirect
21. C4×#157C22semidirect
23. C23×C32C2×C4semidirect
26. C22×C3⋊D3C2×C4semidirect
27. C22×C3⋊D3C2×C4semidirect
33. C2×#157D4direct (N × H)
37. C32×D4C2×C4semidirect
38. C2×C32×C4C2×C4semidirect
39. #638C2×C4semidirect
40. C2×C3⋊Dic3C2×C4semidirect
41. C4⋊(C3⋊D3)C2×C4semidirect
42. C4×C3⋊D3C2×C4semidirect
54. C22×C32C22×C4semidirect
60. C2×C3⋊D3C22×C4semidirect
66. #157C2×D4direct (N × H)
67. C32×C4C22×C4semidirect
68. C3⋊Dic3C22×C4semidirect
90. C2×C32C4×D4semidirect
93. C3⋊D3C4×D4semidirect
101. C2×D4#157direct (N × H)
125. C32C2×C4×D4semidirect
132. C23C2×#157semidirect
135. D4C2×#157direct (N × H)
146. C2×C4C2×#157semidirect
165. C22C22×#157semidirect
170. C4C22×#157semidirect
177. C2D4×#157direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#1572
C2×D4×C3⋊D32
C2×C22⋊#1572
D4×#1572
C2×C4⋊#1572
C2×C4×#1572
C2×C4×D49