Finite Group Databasebeta

← C22×D4×#87604 #27158710

Subgroups — C22×D4×#87604 #27158710

Order 12416.

2528 conjugacy classes of subgroups; only the 850 normal subgroups are shown.

normal subgroups only

This group has 2528 conjugacy classes of subgroups — too many to draw. Only its 850 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×#87604C212yesmaximal, complemented
2C2×D4×#87604C2124yesmaximal, direct product factor
3C22×C4⋊#87604C211yesmaximal, characteristic, complemented
4C22×C22⋊#87604C212yesmaximal, complemented
5C22×C4×#87604C211yesmaximal, characteristic, complemented
6C22×D4×D97C211yesmaximal, characteristic
7C23×#87604—216no—
8C23×#87604C2211yescharacteristic
9C23×#87604C2212yes—
10C23×#87604C2211yescharacteristic
11C23×#87604C22124yescomplemented
12D4×#87604C22164yesdirect product factor
13C2×C4⋊#87604C22112yescomplemented
14C2×C22⋊#87604C22124yescomplemented
15C2×C4×#87604C22112yescomplemented
16C24×D97C2212yes—
17C22×D388C411yescharacteristic, complemented
18C22×C22⋊D97C412yescomplemented
19C2×D4×D97C4112yescomplemented
20C2×D4×D97C22112yes—
21C22×C4×D97C2211yescharacteristic
22C22×D4×C97C411yescharacteristic, complemented
23C22×#87604—28no—
24C22×#87604—24no—
25C22×#87604—296no—
26C22×#87604D418yesdirect product factor
27C22×#87604C2316yes—
28C22×#87604C23112yes—
29C22×#87604C2316yes—
30C22×#87604C23132yescomplemented
31C4⋊#87604C23116yescomplemented
32C22⋊#87604C23132yescomplemented
33C4×#87604C23116yescomplemented
34C23×D97—28no—
35C23×D97—28no—
36C23×D97C2×C4112yescomplemented
37C23×D97C2×C412yescomplemented
38C23×D97C23112yes—
39C23×D97C2311yescharacteristic
40C2×D388C2×C4112yescomplemented
41C2×C22⋊D97C2×C4124yescomplemented
42D4×D97C2×C4148yescomplemented
43D4×D97C23116yes—
44C22×Dic97C2×C411yescharacteristic, complemented
45C2×C4×D97C2×C416yescomplemented
46C2×C4×D97C2316yes—
47C24×C97C2×C412yescomplemented
48C2×D4×C97C2×C4112yescomplemented
49C22×C4×C97C2×C411yescharacteristic, complemented
50C23×C97—28no—
51C23×C97C22×C411yescharacteristic
52C23×C97C22×C4112yescomplemented
53C22×D97—28no—
54C22×D97—248no—
55C22×D97—224no—
56C22×D97—224no—
57C22×D97C22×C413yes—
58C22×D97C22×C4124yescomplemented
59C22×D97C22×C4112yescomplemented
60C22×D97C2×D414yes—
61C22×D97#4114yes—
62C22×D97C2418yes—
63C22×D97C2413yes—
64C2×#87604—224no—
65C2×#87604—212no—
66C2×#87604—264no—
67C2×#87604C2×D4124yesdirect product factor
68C2×#87604C2414yes—
69C2×#87604C2418yes—
70C2×#87604C2414yes—
71D4×C97C22×C4116yescomplemented
72C2×C4×C97C22×C416yescomplemented
73C22⋊D97C22×C4132yescomplemented
74C2×Dic97C22×C416yescomplemented
75D388C22×C4116yescomplemented
76C4×D97C22×C4112yescomplemented
77C4×D97C2414yes—
78C22×C97—224no—
79C22×C97C4×D414yescomplemented
80C22×C97C23×C413yes—
81C22×C97C23×C418yescomplemented
82C2×D97—224no—
83C2×D97—248no—
84C2×D97—28no—
85C2×D97—28no—
86C2×D97C4×D4112yescomplemented
87C2×D97C23×C413yes—
88C2×D97C23×C418yescomplemented
89C2×D97C22×D416yes—
90C2×D97C2×#4116yes—
91C2×D97C2511yescharacteristic
92#87604—28no—
93#87604—24no—
94#87604C22×D418yesdirect product factor
95C4×C97C23×C414yescomplemented
96Dic97C23×C414yescomplemented
97C2×C97—28no—
98C2×C97C2×C4×D416yescomplemented
99C2×C97C24×C411yescharacteristic
100D97—28no—
101D97C2×C4×D416yescomplemented
102D97C23×D411yescharacteristic
103D97C22×#4111yescharacteristic
104C22×C4×D4—971nomaximal, Sylow
105C97C22×C4×D411yescharacteristic, complemented, Sylow
106C23×D4—971no—
107C24×C4—972no—
108C2×C4×D4—9724no—
109C22×C4⋊C4—971no—
110C22×C22⋊C4—972no—
111C22×C42—971no—
112C25—972no—
113C22×D4—972no—
114C22×D4—9712no—
115C22×D4—9712no—
116C22×D4—971no—
117C22×D4#8760411yescharacteristic, direct product factor
118C23×C4—971no—
119C23×C4—971no—
120C23×C4—971no—
121C23×C4—972no—
122C23×C4—9724no—
123C23×C4—19416no—
124C4×D4—9764no—
125C2×C4⋊C4—9712no—
126C2×C22⋊C4—9724no—
127C2×C42—9712no—
128C24—971no—
129C24—972no—
130C24—9712no—
131C24—9712no—
132C24—1948no—
133C24—1948no—
134C24C2×#8760412yescomplemented
135C2×D4—9724no—
136C2×D4—9748no—
137C2×D4—9716no—
138C2×D4—9712no—
139C2×D4C2×#87604112yesdirect product factor
140C22×C4—978no—
141C22×C4—976no—
142C22×C4—971no—
143C22×C4—976no—
144C22×C4—976no—
145C22×C4—976no—
146C22×C4—9712no—
147C22×C4—9732no—
148C22×C4—19496no—
149C22×C4—1944no—
150C22×C4—1948no—
151C22×C4C2×#8760411yescharacteristic, complemented
152C4⋊C4—9716no—
153C22⋊C4—9732no—
154C42—9716no—
155C23—28no—
156C23—974no—
157C23—974no—
158C23—973no—
159C23—973no—
160C23—9712no—
161C23—978no—
162C23—9724no—
163C23—19424no—
164C23—19424no—
165C23—19448no—
166C23—1948no—
167C23C22×#8760411yescharacteristic
168C23C22×#87604112yescomplemented
169D4—9732no—
170D4—9716no—
171D4C22×#87604116yesdirect product factor
172C2×C4—974no—
173C2×C4—9724no—
174C2×C4—976no—
175C2×C4—974no—
176C2×C4—9712no—
177C2×C4—974no—
178C2×C4—978no—
179C2×C4—19464no—
180C2×C4—19424no—
181C2×C4—19412no—
182C2×C4C22×#8760416yescomplemented
183C22—224no—
184C22—976no—
185C22—976no—
186C22—9712no—
187C22—971no—
188C22—973no—
189C22—978no—
190C22—1948no—
191C22—1948no—
192C22—19448no—
193C22—19424no—
194C22D4×#8760414yesdirect product factor
195C22C23×#8760413yes—
196C22C23×#8760418yescomplemented
197C4—978no—
198C4—974no—
199C4—1948no—
200C4—1944no—
201C4C23×#8760414yescomplemented
202C2—28no—
203C2—971no—
204C2—971no—
205C2—976no—
206C2—1948no—
207C2C2×D4×#8760416yesdirect product factor
208C2C24×#8760411yescharacteristic
subgroupG/H
centerC23C22×#87604
commutatorC2×C97C24×C4
FrattiniC2C24×#87604
FittingC22×D4×C97C4
socleC23×C97C22×C4
prime radicalC22×D4×D97C2
cosocleC2×D97C25
hypercenterC22×D4#87604
hypocenterC97C22×C4×D4
LCSG ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C24×#87604C2semidirect
2. C2×D4×#87604C2direct (N × H)
3. C22×C4⋊#87604C2semidirect
4. C22×C22⋊#87604C2semidirect
5. C22×C4×#87604C2semidirect
11. C23×#87604C22semidirect
12. D4×#87604C22direct (N × H)
13. C2×C4⋊#87604C22semidirect
14. C2×C22⋊#87604C22semidirect
15. C2×C4×#87604C22semidirect
17. C22×D388C4semidirect
18. C22×C22⋊D97C4semidirect
19. C2×D4×D97C4semidirect
22. C22×D4×C97C4semidirect
26. C22×#87604D4direct (N × H)
30. C22×#87604C23semidirect
31. C4⋊#87604C23semidirect
32. C22⋊#87604C23semidirect
33. C4×#87604C23semidirect
36. C23×D97C2×C4semidirect
37. C23×D97C2×C4semidirect
40. C2×D388C2×C4semidirect
41. C2×C22⋊D97C2×C4semidirect
42. D4×D97C2×C4semidirect
44. C22×Dic97C2×C4semidirect
45. C2×C4×D97C2×C4semidirect
47. C24×C97C2×C4semidirect
48. C2×D4×C97C2×C4semidirect
49. C22×C4×C97C2×C4semidirect
52. C23×C97C22×C4semidirect
58. C22×D97C22×C4semidirect
59. C22×D97C22×C4semidirect
67. C2×#87604C2×D4direct (N × H)
71. D4×C97C22×C4semidirect
72. C2×C4×C97C22×C4semidirect
73. C22⋊D97C22×C4semidirect
74. C2×Dic97C22×C4semidirect
75. D388C22×C4semidirect
76. C4×D97C22×C4semidirect
79. C22×C97C4×D4semidirect
81. C22×C97C23×C4semidirect
86. C2×D97C4×D4semidirect
88. C2×D97C23×C4semidirect
94. #87604C22×D4direct (N × H)
95. C4×C97C23×C4semidirect
96. Dic97C23×C4semidirect
98. C2×C97C2×C4×D4semidirect
101. D97C2×C4×D4semidirect
105. C97C22×C4×D4semidirect
117. C22×D4#87604direct (N × H)
134. C24C2×#87604semidirect
139. C2×D4C2×#87604direct (N × H)
151. C22×C4C2×#87604semidirect
168. C23C22×#87604semidirect
171. D4C22×#87604direct (N × H)
182. C2×C4C22×#87604semidirect
194. C22D4×#87604direct (N × H)
196. C22C23×#87604semidirect
201. C4C23×#87604semidirect
207. C2C2×D4×#87604direct (N × H)

Maximal subgroups

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

subgroupindex
C24×#876042
C2×D4×#876042
C22×C4⋊#876042
C22×C22⋊#876042
C22×C4×#876042
C22×D4×D972
C22×C4×D497