Finite Group Databasebeta

← #27187611

Subgroups — #27187611

Order 12928.

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

normal subgroups only

This group has 816 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
1C22×C22⋊#87938C212yesmaximal
2#20266124C218yesmaximal, complemented
3#20266119C214yesmaximal, complemented
4C2×C22⋊C4×D101C211yesmaximal, characteristic
5C23×#87938C2214yes—
6C2×C4⋊#87938C2218yescomplemented
7C2×C22⋊#87938—28no—
8C2×C22⋊#87938C2214yes—
9C2×C22⋊#87938C2214yes—
10(C2×#87938)⋊C4C2218yescomplemented
11C24×D101C2211yescharacteristic
12C22×C4×D101C2212yes—
13C2×C22⋊Dic101C411yescharacteristic, complemented
14C2×#13108051C412yescomplemented
15C22⋊C4×D101C414yescomplemented
16C22⋊C4×D101C2214yes—
17C2×C22⋊C4×C101C411yescharacteristic, complemented
18C22×#87938—28no—
19C22×#87938—48no—
20C22×#87938C2318yes—
21C4⋊#87938—48no—
22C22⋊#87938—28no—
23C22⋊#87938—28no—
24C23×D101—44no—
25C23×D101C2×C412yes—
26C23×D101C2×C411yescharacteristic
27C23×D101D414yes—
28C23×D101Q814yes—
29C23×D101C2311yescharacteristic
30C23×D101C2312yes—
31C22×Dic101C2×C412yescomplemented
32C2×C4×D101—44no—
33C2×C4×D101C2×C414yescomplemented
34C2×C4×D101C2314yes—
35C22⋊Dic101C2×C414yescomplemented
36#13108051C2×C418yescomplemented
37C24×C101C2×C411yescharacteristic
38C22×C4×C101C2×C412yescomplemented
39C22⋊C4×C101C2×C414yescomplemented
40C23×C101—42no—
41C23×C101C4⋊C414yescomplemented
42C23×C101C22×C412yes—
43C23×C101C22×C411yescharacteristic
44C22×D101—24no—
45C22×D101—28no—
46C22×D101—48no—
47C22×D101—48no—
48C22×D101—44no—
49C22×D101—42no—
50C22×D101C4⋊C418yescomplemented
51C22×D101C4⋊C414yescomplemented
52C22×D101C22×C412yes—
53C22×D101C22×C411yescharacteristic
54C22×D101C2×D412yes—
55C22×D101C2×D414yes—
56C22×D101C2×Q812yes—
57C22×D101C2×Q814yes—
58C22×D101C2411yescharacteristic
59C2×#87938—28no—
60C2×#87938—216no—
61C2×#87938—416no—
62C2×C4×C101—42no—
63C2×C4×C101C22×C414yes—
64C2×Dic101—42no—
65C2×Dic101C22×C414yes—
66C4×D101—28no—
67C4×D101—44no—
68C22×C101—22no—
69C22×C101—24no—
70C22×C101—44no—
71C22×C101C2×C4⋊C412yes—
72C22×C101C2×C4⋊C414yes—
73C22×C101C23×C411yescharacteristic
74C2×D101—22no—
75C2×D101—24no—
76C2×D101—28no—
77C2×D101—24no—
78C2×D101—24no—
79C2×D101—28no—
80C2×D101—48no—
81C2×D101—44no—
82C2×D101C2×C4⋊C414yes—
83C2×D101C2×C4⋊C412yes—
84C2×D101C22×D411yescharacteristic
85C2×D101C22×Q811yescharacteristic
86C2×D101H(5,𝔽2)14yes—
87#87938—48no—
88C4×C101—24no—
89Dic101—24no—
90C2×C101—24no—
91C2×C101—22no—
92C2×C101C22×C4⋊C411yescharacteristic
93C2×C101#51812yes—
94D101—22no—
95D101—24no—
96D101#53412yes—
97D101#54312yes—
98#2315—1011nomaximal, Sylow
99C101#231511yescharacteristic, complemented, Sylow
100C22×C22⋊C4—1011no—
101C22×C22⋊C4—1012no—
102#385—1018no—
103#380—1014no—
104C25—1011no—
105C23×C4—1012no—
106C23×C4—1014no—
107C2×C4⋊C4—1018no—
108C2×C22⋊C4—1011no—
109C2×C22⋊C4—1012no—
110C2×C22⋊C4—1014no—
111C2×C22⋊C4—1014no—
112C2×C22⋊C4—1014no—
113C2×C22⋊C4—1014no—
114C2×C22⋊C4—2028no—
115C2×C22⋊C4#8793811yescharacteristic, complemented
116#95—1018no—
117C24—1014no—
118C24—1014no—
119C24—1011no—
120C24—1012no—
121C24—1011no—
122C24—1012no—
123C24—4044no—
124C24C2×#8793811yescharacteristic
125C22×C4—1012no—
126C22×C4—1014no—
127C22×C4—1014no—
128C22×C4—1018no—
129C22×C4—4044no—
130C22×C4—4048no—
131C22×C4—2028no—
132C22×C4C2×#8793812yescomplemented
133C4⋊C4—4048no—
134C22⋊C4—1014no—
135C22⋊C4—1018no—
136C22⋊C4—2028no—
137C22⋊C4—2028no—
138C22⋊C4C2×#8793814yescomplemented
139C23—42no—
140C23—1011no—
141C23—1018no—
142C23—1014no—
143C23—1012no—
144C23—1014no—
145C23—1012no—
146C23—1014no—
147C23—1012no—
148C23—1011no—
149C23—4042no—
150C23—4044no—
151C23—4048no—
152C23—4048no—
153C23—2024no—
154C23—2028no—
155C23C4⋊#8793814yescomplemented
156C23C22×#8793812yes—
157C23C22×#8793811yescharacteristic
158C2×C4—42no—
159C2×C4—1014no—
160C2×C4—4044no—
161C2×C4—40416no—
162C2×C4—4042no—
163C2×C4—2028no—
164C2×C4—2028no—
165C2×C4—20216no—
166C2×C4C22×#8793814yes—
167C22—22no—
168C22—24no—
169C22—44no—
170C22—1014no—
171C22—1011no—
172C22—1011no—
173C22—1012no—
174C22—1014no—
175C22—4044no—
176C22—4048no—
177C22—2028no—
178C22—2024no—
179C22—2024no—
180C22—2028no—
181C22—2022no—
182C22—2024no—
183C22C2×C4⋊#8793812yes—
184C22C2×C4⋊#8793814yes—
185C22C23×#8793811yescharacteristic
186C4—24no—
187C4—4048no—
188C4—2024no—
189C2—22no—
190C2—24no—
191C2—1012no—
192C2—1012no—
193C2—2024no—
194C2—2022no—
195C2C22×C4⋊#8793811yescharacteristic
196C2#2026624412yes—
subgroupG/H
center, FrattiniC22C23×#87938
commutator, socleC22×C101C23×C4
FittingC2×C22⋊C4×C101C4
prime radicalC24×D101C22
cosocleC22×D101C24
hypercenterC2×C22⋊C4#87938
hypocenterC101#2315
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #20266124C2semidirect
3. #20266119C2semidirect
6. C2×C4⋊#87938C22semidirect
10. (C2×#87938)⋊C4C22semidirect
13. C2×C22⋊Dic101C4semidirect
14. C2×#13108051C4semidirect
15. C22⋊C4×D101C4semidirect
17. C2×C22⋊C4×C101C4semidirect
31. C22×Dic101C2×C4semidirect
33. C2×C4×D101C2×C4semidirect
35. C22⋊Dic101C2×C4semidirect
36. #13108051C2×C4semidirect
38. C22×C4×C101C2×C4semidirect
39. C22⋊C4×C101C2×C4semidirect
41. C23×C101C4⋊C4semidirect
50. C22×D101C4⋊C4semidirect
51. C22×D101C4⋊C4semidirect
99. C101#2315semidirect
115. C2×C22⋊C4#87938semidirect
132. C22×C4C2×#87938semidirect
138. C22⋊C4C2×#87938semidirect
155. C23C4⋊#87938semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊#879382
#202661242
#202661192
C2×C22⋊C4×D1012
#2315101