Finite Group Databasebeta

← C2×D4×#11715906 #27158975

Subgroups — C2×D4×#11715906 #27158975

Order 12416.

616 conjugacy classes of subgroups; only the 215 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 215 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×#11715906C212yesmaximal, complemented
2D4×#11715906C218yesmaximal, direct product factor
3C2×C4⋊#11715906C211yesmaximal, characteristic, complemented
4C2×C22⋊#11715906C212yesmaximal, complemented
5C2×C4×#11715906C211yesmaximal, characteristic, complemented
6C2×D4×#87604C211yesmaximal, characteristic
7C22×#11715906—28no—
8C22×#11715906C2211yescharacteristic
9C22×#11715906C2212yes—
10C22×#11715906C2211yescharacteristic
11C22×#11715906C2218yescomplemented
12C4⋊#11715906C2214yescomplemented
13C22⋊#11715906C2218yescomplemented
14C4×#11715906C2214yescomplemented
15C23×#87604C2212yes—
16D4×#87604C414yes—
17D4×#87604C2214yes—
18C2×C4⋊#87604C411yescharacteristic
19C2×C22⋊#87604C412yes—
20C2×C4×#87604C2211yescharacteristic
21C2×D4×D97C411yescharacteristic
22C2×#11715906—24no—
23C2×#11715906—22no—
24C2×#11715906—216no—
25C2×#11715906D414yesdirect product factor
26C2×#11715906C2312yes—
27C2×#11715906C2314yes—
28C2×#11715906C2312yes—
29C22×#87604—24no—
30C22×#87604—24no—
31C22×#87604C2×C411yescharacteristic
32C22×#87604C2×C412yes—
33C22×#87604C2×C414yes—
34C22×#87604C2314yes—
35C22×#87604C2311yescharacteristic
36C4⋊#87604C2×C414yes—
37C22⋊#87604C2×C418yes—
38C4×#87604C2×C412yes—
39C4×#87604C2312yes—
40C23×D97C2×C412yes—
41C2×D388C811yescharacteristic, complemented
42C2×C22⋊D97C812yescomplemented
43D4×D97C814yescomplemented
44D4×D97C2×C414yes—
45C2×C4×D97C2×C411yescharacteristic
46C2×D4×C97C811yescharacteristic, complemented
47C23×C97C2×C812yescomplemented
48C22×D97—24no—
49C22×D97—24no—
50C22×D97C2×C814yescomplemented
51C22×D97C2×C812yescomplemented
52C22×D97C22×C414yes—
53C22×D97C22×C411yescharacteristic
54C2×#87604—24no—
55C2×#87604—22no—
56C2×#87604—28no—
57C2×#87604—24no—
58C2×#87604—24no—
59C2×#87604C22×C412yes—
60C2×#87604C22×C414yes—
61C2×#87604C22×C411yescharacteristic
62C2×#87604C2×D412yes—
63C2×#87604#4112yes—
64C2×#87604C2411yescharacteristic
65#11715906—24no—
66#11715906—22no—
67#11715906C2×D414yesdirect product factor
68D4×C97C2×C814yescomplemented
69C2×C4×C97C2×C811yescharacteristic, complemented
70C22⋊D97C2×C818yescomplemented
71C2×Dic97C2×C811yescharacteristic, complemented
72D388C2×C814yescomplemented
73C4×D97C2×C812yescomplemented
74C4×D97C22×C412yes—
75C22×C97—24no—
76C22×C97C22×C811yescharacteristic
77C22×C97C22×C814yescomplemented
78C2×D97—24no—
79C2×D97—24no—
80C2×D97—24no—
81C2×D97—28no—
82C2×D97C4×D412yes—
83C2×D97C22×C811yescharacteristic
84C2×D97C22×C814yescomplemented
85C2×D97#13112yes—
86C2×D97C23×C411yescharacteristic
87#87604—24no—
88#87604—22no—
89#87604C4×D412yes—
90#87604C22×D411yescharacteristic
91#87604C2×#4111yescharacteristic
92C4×C97C22×C812yescomplemented
93Dic97C22×C812yescomplemented
94C2×C97—24no—
95C2×C97C8×D412yescomplemented
96C2×C97C23×C811yescharacteristic
97D97—24no—
98D97C8×D412yescomplemented
99D97C2×C4×D411yescharacteristic
100D97C2×#13111yescharacteristic
101C2×C8×D4—971nomaximal, Sylow
102C97C2×C8×D411yescharacteristic, complemented, Sylow
103C23×C8—972no—
104C2×C4×D4—971no—
105C8×D4—978no—
106C2×C4⋊C8—971no—
107C2×C22⋊C8—972no—
108C2×C4×C8—971no—
109C22×D4—971no—
110C23×C4—972no—
111C22×C8—971no—
112C22×C8—972no—
113C22×C8—978no—
114C22×C8—971no—
115C22×C8—1948no—
116C4×D4—974no—
117C4×D4—974no—
118C2×C4⋊C4—971no—
119C2×C22⋊C4—972no—
120C2×C42—971no—
121C4⋊C8—974no—
122C22⋊C8—978no—
123C4×C8—974no—
124C24—972no—
125C2×D4—972no—
126C2×D4—974no—
127C2×D4—974no—
128C2×D4—971no—
129C2×D4#1171590611yescharacteristic, direct product factor
130C22×C4—971no—
131C22×C4—971no—
132C22×C4—974no—
133C22×C4—971no—
134C22×C4—972no—
135C22×C4—974no—
136C22×C4—1944no—
137C22×C4—1944no—
138C2×C8—972no—
139C2×C8—974no—
140C2×C8—972no—
141C2×C8—974no—
142C2×C8—19416no—
143C2×C8—1942no—
144C2×C8—1944no—
145C4⋊C4—974no—
146C22⋊C4—978no—
147C42—972no—
148C42—972no—
149C23—971no—
150C23—972no—
151C23—974no—
152C23—974no—
153C23—1944no—
154C23—1944no—
155C23C2×#1171590612yescomplemented
156D4—978no—
157D4—974no—
158D4C2×#1171590614yesdirect product factor
159C2×C4—971no—
160C2×C4—971no—
161C2×C4—972no—
162C2×C4—972no—
163C2×C4—971no—
164C2×C4—972no—
165C2×C4—972no—
166C2×C4—972no—
167C2×C4—974no—
168C2×C4—1944no—
169C2×C4—1944no—
170C2×C4—1948no—
171C2×C4—1944no—
172C2×C4—1942no—
173C2×C4C2×#1171590611yescharacteristic, complemented
174C8—974no—
175C8—1944no—
176C8—1942no—
177C22—24no—
178C22—972no—
179C22—972no—
180C22—971no—
181C22—971no—
182C22—974no—
183C22—1944no—
184C22—1944no—
185C22—1948no—
186C22—1944no—
187C22C22×#1171590611yescharacteristic
188C22C22×#1171590614yescomplemented
189C4—971no—
190C4—971no—
191C4—972no—
192C4—972no—
193C4—1944no—
194C4—1942no—
195C4C22×#1171590612yescomplemented
196C2—24no—
197C2—971no—
198C2—971no—
199C2—972no—
200C2—1944no—
201C2D4×#1171590612yesdirect product factor
202C2C23×#1171590611yescharacteristic
subgroupG/H
centerC22C22×#11715906
commutatorC2×C97C23×C8
FrattiniC2C23×#11715906
FittingC2×D4×C97C8
socleC22×C97C22×C8
prime radicalC2×D4×D97C4
cosocleC2×#87604C24
hypercenterC2×D4#11715906
hypocenterC97C2×C8×D4
LCSG ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C23×#11715906C2semidirect
2. D4×#11715906C2direct (N × H)
3. C2×C4⋊#11715906C2semidirect
4. C2×C22⋊#11715906C2semidirect
5. C2×C4×#11715906C2semidirect
11. C22×#11715906C22semidirect
12. C4⋊#11715906C22semidirect
13. C22⋊#11715906C22semidirect
14. C4×#11715906C22semidirect
25. C2×#11715906D4direct (N × H)
41. C2×D388C8semidirect
42. C2×C22⋊D97C8semidirect
43. D4×D97C8semidirect
46. C2×D4×C97C8semidirect
47. C23×C97C2×C8semidirect
50. C22×D97C2×C8semidirect
51. C22×D97C2×C8semidirect
67. #11715906C2×D4direct (N × H)
68. D4×C97C2×C8semidirect
69. C2×C4×C97C2×C8semidirect
70. C22⋊D97C2×C8semidirect
71. C2×Dic97C2×C8semidirect
72. D388C2×C8semidirect
73. C4×D97C2×C8semidirect
77. C22×C97C22×C8semidirect
84. C2×D97C22×C8semidirect
92. C4×C97C22×C8semidirect
93. Dic97C22×C8semidirect
95. C2×C97C8×D4semidirect
98. D97C8×D4semidirect
102. C97C2×C8×D4semidirect
129. C2×D4#11715906direct (N × H)
155. C23C2×#11715906semidirect
158. D4C2×#11715906direct (N × H)
173. C2×C4C2×#11715906semidirect
188. C22C22×#11715906semidirect
195. C4C22×#11715906semidirect
201. C2D4×#11715906direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117159062
D4×#117159062
C2×C4⋊#117159062
C2×C22⋊#117159062
C2×C4×#117159062
C2×D4×#876042
C2×C8×D497