Finite Group Databasebeta

← #120×#87604 #27158199

Subgroups — #120×#87604 #27158199

Order 12416.

1224 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1224 conjugacy classes of subgroups — too many to draw. Only its 210 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×#87604C211yesmaximal, characteristic, complemented
2C2×D4×#87604C213yesmaximal, complemented
3#20261683C213yesmaximal, complemented
4#20261682C213yesmaximal, complemented
5#20261676C211yesmaximal, characteristic, complemented
6C22⋊C4×#87604C213yesmaximal, complemented
7#120×D97C211yesmaximal, characteristic
8C23×#87604—212no—
9C23×#87604C2213yes—
10C23×#87604C2213yes—
11C23×#87604C2211yescharacteristic, complemented
12D4×#87604—212no—
13C2×C4⋊#87604C2213yescomplemented
14C2×C22⋊#87604C2213yes—
15C2×C22⋊#87604C2213yes—
16C2×C22⋊#87604C2213yescomplemented
17C2×C4×#87604C2213yescomplemented
18(C2×#87604)⋊C4C2216yescomplemented
19C24×D97C2211yescharacteristic
20C2×D4×D97C2213yes—
21#15690780C413yescomplemented
22#15690779C411yescharacteristic, complemented
23#15690778C413yescomplemented
24C22⋊C4×D97C2213yes—
25#120×C97C411yescharacteristic, complemented
26C22×#87604—232no—
27C22×#87604—26no—
28C22×#87604—212no—
29C22×#87604—212no—
30C22×#87604—26no—
31C22×#87604D4112yescomplemented
32C22×#87604C2313yes—
33C22×#87604C2311yescharacteristic
34C22×#87604C2313yes—
35C22×#87604C2311yescharacteristic
36C4⋊#87604—26no—
37C22⋊#87604—26no—
38C22⋊#87604—26no—
39C22⋊#87604—26no—
40C4×#87604—26no—
41C23×D97—26no—
42C23×D97—26no—
43C23×D97C2×C413yescomplemented
44C23×D97C2311yescharacteristic
45C23×D97C2313yes—
46C2×D388C2×C413yescomplemented
47C2×C22⋊D97C2×C413yescomplemented
48C2×C22⋊D97C2×C413yescomplemented
49D4×D97—26no—
50D4×D97—26no—
51C2×C4×D97C2313yes—
52C22⋊Dic97C2×C413yescomplemented
53#13095488C2×C416yescomplemented
54C24×C97C2×C411yescharacteristic, complemented
55C2×D4×C97C2×C413yescomplemented
56C22⋊C4×C97C2×C413yescomplemented
57C23×C97—26no—
58C23×C97C22×C413yes—
59C23×C97C22×C411yescharacteristic, complemented
60C22×D97—224no—
61C22×D97—28no—
62C22×D97—26no—
63C22×D97—212no—
64C22×D97—26no—
65C22×D97—26no—
66C22×D97—23no—
67C22×D97—23no—
68C22×D97C22×C413yes—
69C22×D97C22×C411yescharacteristic, complemented
70C22×D97C2×D416yes—
71C22×D97#4116yes—
72C22×D97C2411yescharacteristic
73C2×#87604—224no—
74C2×#87604—26no—
75C2×#87604—224no—
76C2×#87604—23no—
77C2×#87604—23no—
78C2×#87604—26no—
79C2×#87604—44no—
80C2×#87604C2×D416yes—
81C2×#87604C2×D416yes—
82D4×C97—26no—
83C2×C4×C97C22×C413yescomplemented
84C22⋊D97—26no—
85C22⋊D97—26no—
86C2×Dic97C22×C413yescomplemented
87D388—26no—
88C4×D97—23no—
89C4×D97—23no—
90C22×C97—28no—
91C22×C97—26no—
92C22×C97—23no—
93C22×C97C4×D416yescomplemented
94C22×C97C23×C411yescharacteristic
95C2×D97—224no—
96C2×D97—26no—
97C2×D97—26no—
98C2×D97—26no—
99C2×D97—212no—
100C2×D97—23no—
101C2×D97—41no—
102C2×D97—43no—
103C2×D97C4×D416yescomplemented
104C2×D97C22×D413yes—
105C2×D97C2×#4113yes—
106#87604—26no—
107#87604—41no—
108#87604—43no—
109#87604#12014yesdirect product factor
110C4×C97—23no—
111Dic97—23no—
112C2×C97—26no—
113C2×C97—41no—
114C2×C97C2×C4×D413yes—
115D97—26no—
116D97—41no—
117D97C2×#12011yescharacteristic
118D97#52513yes—
119C4×#120—971nomaximal, Sylow
120C97C4×#12011yescharacteristic, complemented, Sylow
121C24×C4—971no—
122C2×#120—971no—
123C2×C4×D4—973no—
124#386—973no—
125#385—973no—
126#379—971no—
127C4×C22⋊C4—973no—
128C25—971no—
129C22×D4—973no—
130C23×C4—973no—
131C23×C4—971no—
132C23×C4—973no—
133C23×C4—19412no—
134#120—971no—
135#120—973no—
136#120—973no—
137#120#8760411yescharacteristic, direct product factor
138C4×D4—19412no—
139C2×C4⋊C4—973no—
140C2×C22⋊C4—973no—
141C2×C22⋊C4—973no—
142C2×C22⋊C4—973no—
143C2×C22⋊C4—973no—
144C2×C42—973no—
145#95—976no—
146C24—973no—
147C24—973no—
148C24—971no—
149C24—1946no—
150C24—1946no—
151C24C2×#8760411yescharacteristic, complemented
152C2×D4—973no—
153C2×D4—973no—
154C2×D4—973no—
155C2×D4—1946no—
156C2×D4—1946no—
157C2×D4C2×#8760413yescomplemented
158C22×C4—971no—
159C22×C4—973no—
160C22×C4—9712no—
161C22×C4—973no—
162C22×C4—973no—
163C22×C4—971no—
164C22×C4—1946no—
165C22×C4—19412no—
166C22×C4—19412no—
167C22×C4—1946no—
168C22×C4—19432no—
169C4⋊C4—1946no—
170C22⋊C4—973no—
171C22⋊C4—976no—
172C22⋊C4—1946no—
173C22⋊C4—1946no—
174C22⋊C4—1946no—
175C22⋊C4C2×#8760413yescomplemented
176C42—1946no—
177C23—26no—
178C23—971no—
179C23—976no—
180C23—976no—
181C23—973no—
182C23—971no—
183C23—1943no—
184C23—1943no—
185C23—1946no—
186C23—1946no—
187C23—19412no—
188C23—1948no—
189C23—19424no—
190C23—1946no—
191C23C22×#8760413yes—
192C23C22×#8760411yescharacteristic, complemented
193D4—26no—
194D4—1946no—
195D4—1946no—
196D4—1946no—
197C2×C4—976no—
198C2×C4—976no—
199C2×C4—973no—
200C2×C4—3884no—
201C2×C4—1943no—
202C2×C4—1943no—
203C2×C4—19424no—
204C2×C4—1943no—
205C2×C4—1943no—
206C2×C4—1946no—
207C2×C4—1946no—
208C2×C4—19424no—
209C2×C4C22×#8760413yescomplemented
210C22—28no—
211C22—26no—
212C22—23no—
213C22—973no—
214C22—973no—
215C22—976no—
216C22—3881no—
217C22—3883no—
218C22—1946no—
219C22—1946no—
220C22—19412no—
221C22—19424no—
222C22—1943no—
223C22—1946no—
224C22D4×#8760416yescomplemented
225C22C23×#8760411yescharacteristic
226C4—23no—
227C4—974no—
228C4—3881no—
229C4—3883no—
230C4—1946no—
231C4—1943no—
232C2—26no—
233C2—41no—
234C2—971no—
235C2—973no—
236C2—3881no—
237C2—1946no—
238C2C2×D4×#8760413yes—
subgroupG/H
center, FrattiniC22C23×#87604
commutator, socleC22×C97C23×C4
Fitting#120×C97C4
prime radical#120×D97C2
cosocleC22×D97C24
hypercenter#120#87604
hypocenterC97C4×#120
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C24×#87604C2semidirect
2. C2×D4×#87604C2semidirect
3. #20261683C2semidirect
4. #20261682C2semidirect
5. #20261676C2semidirect
6. C22⋊C4×#87604C2semidirect
11. C23×#87604C22semidirect
13. C2×C4⋊#87604C22semidirect
16. C2×C22⋊#87604C22semidirect
17. C2×C4×#87604C22semidirect
18. (C2×#87604)⋊C4C22semidirect
21. #15690780C4semidirect
22. #15690779C4semidirect
23. #15690778C4semidirect
25. #120×C97C4semidirect
31. C22×#87604D4semidirect
43. C23×D97C2×C4semidirect
46. C2×D388C2×C4semidirect
47. C2×C22⋊D97C2×C4semidirect
48. C2×C22⋊D97C2×C4semidirect
52. C22⋊Dic97C2×C4semidirect
53. #13095488C2×C4semidirect
54. C24×C97C2×C4semidirect
55. C2×D4×C97C2×C4semidirect
56. C22⋊C4×C97C2×C4semidirect
59. C23×C97C22×C4semidirect
69. C22×D97C22×C4semidirect
83. C2×C4×C97C22×C4semidirect
86. C2×Dic97C22×C4semidirect
93. C22×C97C4×D4semidirect
103. C2×D97C4×D4semidirect
109. #87604#120direct (N × H)
120. C97C4×#120semidirect
137. #120#87604direct (N × H)
151. C24C2×#87604semidirect
157. C2×D4C2×#87604semidirect
175. C22⋊C4C2×#87604semidirect
192. C23C22×#87604semidirect
209. C2×C4C22×#87604semidirect
224. C22D4×#87604semidirect

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
#202616832
#202616822
#202616762
C22⋊C4×#876042
#120×D972
C4×#12097