Finite Group Databasebeta

← C2×D8×#87604 #27158483

Subgroups — C2×D8×#87604 #27158483

Order 12416.

616 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 178 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×D4×#87604C212yesmaximal, complemented
2D8×#87604C218yesmaximal, direct product factor
3C2×#15690879C211yesmaximal, characteristic, complemented
4C2×#15690872C212yesmaximal, complemented
5C2×C8×#87604C211yesmaximal, characteristic, complemented
6C2×D8×D97C211yesmaximal, characteristic
7C23×#87604—22no—
8C2×#13095519C2211yescharacteristic
9D4×#87604—24no—
10D4×#87604C2218yescomplemented
11C2×C4⋊#87604C2212yes—
12C2×C22⋊#87604—22no—
13C2×C4×#87604C2211yescharacteristic
14#15690879C2214yescomplemented
15#15690872C2218yescomplemented
16C8×#87604C2214yescomplemented
17C2×D4×D97C2212yes—
18C2×D776C411yescharacteristic, complemented
19C2×#13095496C412yescomplemented
20D8×D97C414yescomplemented
21D8×D97C2214yes—
22C2×C8×D97C2211yescharacteristic
23C2×D8×C97C411yescharacteristic, complemented
24C22×#87604—22no—
25C22×#87604—28no—
26C22×#87604—48no—
27C22×#87604D411yescharacteristic
28C22×#87604D411yescharacteristic
29#13095519—21no—
30#13095519C2312yes—
31C2×#11715897—21no—
32C4⋊#87604—22no—
33C4⋊#87604C2314yes—
34C22⋊#87604—28no—
35C4×#87604D412yes—
36C4×#87604C2312yes—
37C23×D97—22no—
38C2×D388C2×C412yescomplemented
39C2×C22⋊D97—22no—
40D4×D97—22no—
41D4×D97—22no—
42D4×D97C2×C414yescomplemented
43D4×D97C2314yes—
44C2×C4×D97C2311yescharacteristic
45D776C2×C414yescomplemented
46#13095496C2×C418yescomplemented
47C2×#11715895C2×C411yescharacteristic, complemented
48C8×D97C2×C412yescomplemented
49C8×D97C2312yes—
50C2×D4×C97C2×C412yescomplemented
51D8×C97C2×C414yescomplemented
52C2×C8×C97C2×C411yescharacteristic, complemented
53C23×C97—22no—
54C22×D97—22no—
55C22×D97—24no—
56C22×D97—24no—
57C22×D97—44no—
58C22×D97—44no—
59C22×D97C2×D411yescharacteristic
60C2×#87604—22no—
61C2×#87604—24no—
62C2×#87604—44no—
63C2×#87604—416no—
64C2×#87604D814yesdirect product factor
65C2×#87604C2×D412yes—
66C2×#87604C2×D412yes—
67D4×C97—22no—
68D4×C97C22×C414yescomplemented
69C2×C4×C97C22×C411yescharacteristic
70C22⋊D97—24no—
71C22⋊D97—24no—
72C2×Dic97#4111yescharacteristic
73D388—22no—
74D388C22×C414yescomplemented
75C4×D97C22×C411yescharacteristic
76C4×D97C2×D411yescharacteristic
77C4×D97#4111yescharacteristic
78C4×D97C2411yescharacteristic
79#11715897—22no—
80C8×C97C22×C412yescomplemented
81#11715895C22×C412yescomplemented
82C22×C97—24no—
83C22×C97—44no—
84C22×C97C4×D411yescharacteristic
85C2×D97—24no—
86C2×D97—44no—
87C2×D97—44no—
88C2×D97—44no—
89C2×D97—48no—
90C2×D97C4×D411yescharacteristic
91C2×D97C2×D812yes—
92C2×D97#13512yes—
93C2×D97C22×D411yescharacteristic
94#87604—22no—
95#87604—44no—
96#87604C2×D814yesdirect product factor
97C4×C97C4×D411yescharacteristic
98C4×C97C23×C411yescharacteristic
99Dic97C4×D411yescharacteristic
100Dic97C2×#4111yescharacteristic
101C2×C97—44no—
102C2×C97C4×D812yescomplemented
103C2×C97C2×C4×D411yescharacteristic
104D97—44no—
105D97C4×D812yescomplemented
106D97C22×D811yescharacteristic
107D97C2×#13511yescharacteristic
108C2×C4×D8—971nomaximal, Sylow
109C97C2×C4×D811yescharacteristic, complemented, Sylow
110C22×D8—971no—
111C2×C4×D4—972no—
112C4×D8—978no—
113C2×#107—971no—
114C2×#102—972no—
115C2×C4×C8—971no—
116C22×D4—972no—
117C23×C4—1942no—
118C2×D8—972no—
119C2×D8—974no—
120C2×D8—974no—
121C2×D8—971no—
122C2×D8#8760411yescharacteristic, direct product factor
123C22×C8—971no—
124C22×C8—971no—
125C4×D4—978no—
126C4×D4—1944no—
127C2×C4⋊C4—972no—
128C2×C22⋊C4—1942no—
129C2×C42—971no—
130#107—974no—
131#102—978no—
132C4×C8—974no—
133C24—1942no—
134C2×D4—974no—
135C2×D4—974no—
136C2×D4—972no—
137C2×D4—1942no—
138C2×D4—1942no—
139C2×D4—1942no—
140C2×D4C2×#8760412yescomplemented
141C22×C4—971no—
142C22×C4—971no—
143C22×C4—971no—
144C22×C4—3888no—
145C22×C4—1948no—
146C22×C4—1942no—
147D8—978no—
148D8—974no—
149D8C2×#8760414yesdirect product factor
150C2×C8—971no—
151C2×C8—972no—
152C2×C8—972no—
153C2×C8—972no—
154C2×C8—1941no—
155C2×C8—1941no—
156C2×C8C2×#8760411yescharacteristic, complemented
157C4⋊C4—974no—
158C4⋊C4—1942no—
159C22⋊C4—1948no—
160C42—972no—
161C42—972no—
162C23—22no—
163C23—971no—
164C23—3884no—
165C23—3884no—
166C23—1944no—
167C23—1944no—
168C23—1942no—
169D4—22no—
170D4—974no—
171D4—1944no—
172D4—1944no—
173D4—1942no—
174D4C22×#8760414yescomplemented
175C2×C4—972no—
176C2×C4—974no—
177C2×C4—971no—
178C2×C4—971no—
179C2×C4—971no—
180C2×C4—971no—
181C2×C4—971no—
182C2×C4—972no—
183C2×C4—38816no—
184C2×C4—3884no—
185C2×C4—1944no—
186C2×C4—1942no—
187C2×C4C22×#8760411yescharacteristic
188C8—972no—
189C8—1942no—
190C8C22×#8760412yescomplemented
191C22—24no—
192C22—44no—
193C22—972no—
194C22—972no—
195C22—971no—
196C22—971no—
197C22—3884no—
198C22—3884no—
199C22—3888no—
200C22—3884no—
201C22—1944no—
202C22D4×#8760411yescharacteristic
203C4—974no—
204C4—971no—
205C4—971no—
206C4—3884no—
207C4—1942no—
208C4D4×#8760411yescharacteristic
209C4C23×#8760411yescharacteristic
210C2—44no—
211C2—971no—
212C2—971no—
213C2—972no—
214C2—3884no—
215C2D8×#8760412yesdirect product factor
216C2C2×D4×#8760411yescharacteristic
subgroupG/H
centerC22D4×#87604
commutatorC4×C97C23×C4
FrattiniC4C23×#87604
FittingC2×D8×C97C4
socleC22×C97C4×D4
prime radicalC2×D8×D97C2
cosocleC4×D97C24
hypercenterC2×D8#87604
hypocenterC97C2×C4×D8
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D8

Complements

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

NH ≅ G/Ntype
1. C2×D4×#87604C2semidirect
2. D8×#87604C2direct (N × H)
3. C2×#15690879C2semidirect
4. C2×#15690872C2semidirect
5. C2×C8×#87604C2semidirect
10. D4×#87604C22semidirect
14. #15690879C22semidirect
15. #15690872C22semidirect
16. C8×#87604C22semidirect
18. C2×D776C4semidirect
19. C2×#13095496C4semidirect
20. D8×D97C4semidirect
23. C2×D8×C97C4semidirect
38. C2×D388C2×C4semidirect
42. D4×D97C2×C4semidirect
45. D776C2×C4semidirect
46. #13095496C2×C4semidirect
47. C2×#11715895C2×C4semidirect
48. C8×D97C2×C4semidirect
50. C2×D4×C97C2×C4semidirect
51. D8×C97C2×C4semidirect
52. C2×C8×C97C2×C4semidirect
64. C2×#87604D8direct (N × H)
68. D4×C97C22×C4semidirect
74. D388C22×C4semidirect
80. C8×C97C22×C4semidirect
81. #11715895C22×C4semidirect
96. #87604C2×D8direct (N × H)
102. C2×C97C4×D8semidirect
105. D97C4×D8semidirect
109. C97C2×C4×D8semidirect
122. C2×D8#87604direct (N × H)
140. C2×D4C2×#87604semidirect
149. D8C2×#87604direct (N × H)
156. C2×C8C2×#87604semidirect
174. D4C22×#87604semidirect
190. C8C22×#87604semidirect
215. C2D8×#87604direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D4×#876042
D8×#876042
C2×#156908792
C2×#156908722
C2×C8×#876042
C2×D8×D972
C2×C4×D897