Finite Group Databasebeta

← #120×#87938 #27187595

Subgroups — #120×#87938 #27187595

Order 12928.

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×#87938C211yesmaximal, characteristic, complemented
2C2×D4×#87938C213yesmaximal, complemented
3#20266125C213yesmaximal, complemented
4#20266124C213yesmaximal, complemented
5#20266118C211yesmaximal, characteristic, complemented
6C22⋊C4×#87938C213yesmaximal, complemented
7#120×D101C211yesmaximal, characteristic
8C23×#87938—212no—
9C23×#87938C2213yes—
10C23×#87938C2213yes—
11C23×#87938C2211yescharacteristic, complemented
12D4×#87938—212no—
13C2×C4⋊#87938C2213yescomplemented
14C2×C22⋊#87938C2213yes—
15C2×C22⋊#87938C2213yes—
16C2×C22⋊#87938C2213yescomplemented
17C2×C4×#87938C2213yescomplemented
18(C2×#87938)⋊C4C2216yescomplemented
19C24×D101C2211yescharacteristic
20C2×D4×D101C2213yes—
21#15692225C413yescomplemented
22#15692224C411yescharacteristic, complemented
23#15692223C413yescomplemented
24C22⋊C4×D101C2213yes—
25#120×C101C411yescharacteristic, complemented
26C22×#87938—232no—
27C22×#87938—26no—
28C22×#87938—212no—
29C22×#87938—212no—
30C22×#87938—26no—
31C22×#87938D4112yescomplemented
32C22×#87938C2313yes—
33C22×#87938C2311yescharacteristic
34C22×#87938C2313yes—
35C22×#87938C2311yescharacteristic
36C4⋊#87938—26no—
37C22⋊#87938—26no—
38C22⋊#87938—26no—
39C22⋊#87938—26no—
40C4×#87938—26no—
41C23×D101—26no—
42C23×D101—26no—
43C23×D101C2×C413yescomplemented
44C23×D101C2311yescharacteristic
45C23×D101C2313yes—
46C2×D404C2×C413yescomplemented
47C2×C22⋊D101C2×C413yescomplemented
48C2×C22⋊D101C2×C413yescomplemented
49D4×D101—26no—
50D4×D101—26no—
51C2×C4×D101C2313yes—
52C22⋊Dic101C2×C413yescomplemented
53#13108051C2×C416yescomplemented
54C24×C101C2×C411yescharacteristic, complemented
55C2×D4×C101C2×C413yescomplemented
56C22⋊C4×C101C2×C413yescomplemented
57C23×C101—26no—
58C23×C101C22×C413yes—
59C23×C101C22×C411yescharacteristic, complemented
60C22×D101—224no—
61C22×D101—28no—
62C22×D101—26no—
63C22×D101—212no—
64C22×D101—26no—
65C22×D101—26no—
66C22×D101—23no—
67C22×D101—23no—
68C22×D101C22×C413yes—
69C22×D101C22×C411yescharacteristic, complemented
70C22×D101C2×D416yes—
71C22×D101#4116yes—
72C22×D101C2411yescharacteristic
73C2×#87938—224no—
74C2×#87938—26no—
75C2×#87938—224no—
76C2×#87938—23no—
77C2×#87938—23no—
78C2×#87938—26no—
79C2×#87938—44no—
80C2×#87938C2×D416yes—
81C2×#87938C2×D416yes—
82D4×C101—26no—
83C2×C4×C101C22×C413yescomplemented
84C22⋊D101—26no—
85C22⋊D101—26no—
86C2×Dic101C22×C413yescomplemented
87D404—26no—
88C4×D101—23no—
89C4×D101—23no—
90C22×C101—28no—
91C22×C101—26no—
92C22×C101—23no—
93C22×C101C4×D416yescomplemented
94C22×C101C23×C411yescharacteristic
95C2×D101—224no—
96C2×D101—26no—
97C2×D101—26no—
98C2×D101—26no—
99C2×D101—212no—
100C2×D101—23no—
101C2×D101—41no—
102C2×D101—43no—
103C2×D101C4×D416yescomplemented
104C2×D101C22×D413yes—
105C2×D101C2×#4113yes—
106#87938—26no—
107#87938—41no—
108#87938—43no—
109#87938#12014yesdirect product factor
110C4×C101—23no—
111Dic101—23no—
112C2×C101—26no—
113C2×C101—41no—
114C2×C101C2×C4×D413yes—
115D101—26no—
116D101—41no—
117D101C2×#12011yescharacteristic
118D101#52513yes—
119C4×#120—1011nomaximal, Sylow
120C101C4×#12011yescharacteristic, complemented, Sylow
121C24×C4—1011no—
122C2×#120—1011no—
123C2×C4×D4—1013no—
124#386—1013no—
125#385—1013no—
126#379—1011no—
127C4×C22⋊C4—1013no—
128C25—1011no—
129C22×D4—1013no—
130C23×C4—1013no—
131C23×C4—1011no—
132C23×C4—1013no—
133C23×C4—20212no—
134#120—1011no—
135#120—1013no—
136#120—1013no—
137#120#8793811yescharacteristic, direct product factor
138C4×D4—20212no—
139C2×C4⋊C4—1013no—
140C2×C22⋊C4—1013no—
141C2×C22⋊C4—1013no—
142C2×C22⋊C4—1013no—
143C2×C22⋊C4—1013no—
144C2×C42—1013no—
145#95—1016no—
146C24—1013no—
147C24—1013no—
148C24—1011no—
149C24—2026no—
150C24—2026no—
151C24C2×#8793811yescharacteristic, complemented
152C2×D4—1013no—
153C2×D4—1013no—
154C2×D4—1013no—
155C2×D4—2026no—
156C2×D4—2026no—
157C2×D4C2×#8793813yescomplemented
158C22×C4—1011no—
159C22×C4—1013no—
160C22×C4—10112no—
161C22×C4—1013no—
162C22×C4—1013no—
163C22×C4—1011no—
164C22×C4—2026no—
165C22×C4—20212no—
166C22×C4—20212no—
167C22×C4—2026no—
168C22×C4—20232no—
169C4⋊C4—2026no—
170C22⋊C4—1013no—
171C22⋊C4—1016no—
172C22⋊C4—2026no—
173C22⋊C4—2026no—
174C22⋊C4—2026no—
175C22⋊C4C2×#8793813yescomplemented
176C42—2026no—
177C23—26no—
178C23—1011no—
179C23—1016no—
180C23—1016no—
181C23—1013no—
182C23—1011no—
183C23—2023no—
184C23—2023no—
185C23—2026no—
186C23—2026no—
187C23—20212no—
188C23—2028no—
189C23—20224no—
190C23—2026no—
191C23C22×#8793813yes—
192C23C22×#8793811yescharacteristic, complemented
193D4—26no—
194D4—2026no—
195D4—2026no—
196D4—2026no—
197C2×C4—1016no—
198C2×C4—1016no—
199C2×C4—1013no—
200C2×C4—4044no—
201C2×C4—2023no—
202C2×C4—2023no—
203C2×C4—20224no—
204C2×C4—2023no—
205C2×C4—2023no—
206C2×C4—2026no—
207C2×C4—2026no—
208C2×C4—20224no—
209C2×C4C22×#8793813yescomplemented
210C22—28no—
211C22—26no—
212C22—23no—
213C22—1013no—
214C22—1013no—
215C22—1016no—
216C22—4041no—
217C22—4043no—
218C22—2026no—
219C22—2026no—
220C22—20212no—
221C22—20224no—
222C22—2023no—
223C22—2026no—
224C22D4×#8793816yescomplemented
225C22C23×#8793811yescharacteristic
226C4—23no—
227C4—1014no—
228C4—4041no—
229C4—4043no—
230C4—2026no—
231C4—2023no—
232C2—26no—
233C2—41no—
234C2—1011no—
235C2—1013no—
236C2—4041no—
237C2—2026no—
238C2C2×D4×#8793813yes—
subgroupG/H
center, FrattiniC22C23×#87938
commutator, socleC22×C101C23×C4
Fitting#120×C101C4
prime radical#120×D101C2
cosocleC22×D101C24
hypercenter#120#87938
hypocenterC101C4×#120
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C24×#87938C2semidirect
2. C2×D4×#87938C2semidirect
3. #20266125C2semidirect
4. #20266124C2semidirect
5. #20266118C2semidirect
6. C22⋊C4×#87938C2semidirect
11. C23×#87938C22semidirect
13. C2×C4⋊#87938C22semidirect
16. C2×C22⋊#87938C22semidirect
17. C2×C4×#87938C22semidirect
18. (C2×#87938)⋊C4C22semidirect
21. #15692225C4semidirect
22. #15692224C4semidirect
23. #15692223C4semidirect
25. #120×C101C4semidirect
31. C22×#87938D4semidirect
43. C23×D101C2×C4semidirect
46. C2×D404C2×C4semidirect
47. C2×C22⋊D101C2×C4semidirect
48. C2×C22⋊D101C2×C4semidirect
52. C22⋊Dic101C2×C4semidirect
53. #13108051C2×C4semidirect
54. C24×C101C2×C4semidirect
55. C2×D4×C101C2×C4semidirect
56. C22⋊C4×C101C2×C4semidirect
59. C23×C101C22×C4semidirect
69. C22×D101C22×C4semidirect
83. C2×C4×C101C22×C4semidirect
86. C2×Dic101C22×C4semidirect
93. C22×C101C4×D4semidirect
103. C2×D101C4×D4semidirect
109. #87938#120direct (N × H)
120. C101C4×#120semidirect
137. #120#87938direct (N × H)
151. C24C2×#87938semidirect
157. C2×D4C2×#87938semidirect
175. C22⋊C4C2×#87938semidirect
192. C23C22×#87938semidirect
209. C2×C4C22×#87938semidirect
224. C22D4×#87938semidirect

Maximal subgroups

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

subgroupindex
C24×#879382
C2×D4×#879382
#202661252
#202661242
#202661182
C22⋊C4×#879382
#120×D1012
C4×#120101