Finite Group Databasebeta

← #15671716

Subgroups — #15671716

Order 2944.

312 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 81 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
1C2×(C22⋊D23)⋊C4C211yesmaximal, characteristic
2#13092467C212yesmaximal, complemented
3#13092141C211yesmaximal, characteristic, complemented
4#13092140C212yesmaximal, complemented
5C23×#323C211yesmaximal, characteristic, complemented
6C22×C22⋊D23C411yescharacteristic
7C23×Dic23C411yescharacteristic
8C22×C4×D23C411yescharacteristic
9C2×C22⋊C4×C23C2211yescharacteristic
10C2×(C2×D23)⋊C4C2211yescharacteristic
11C2×Dic23⋊C4C411yescharacteristic
12C2×C4×Dic23C2211yescharacteristic
13(C22⋊D23)⋊C4—22no—
14(C22⋊D23)⋊C4—22no—
15C23×C22⋊C8C2212yes—
16C22⋊(C23⋊C8)C2212yes—
17Dic23⋊C8—22no—
18C8×Dic23—22no—
19C24×C23C2×C411yescharacteristic
20C23×D23C2×C411yescharacteristic
21C22×C4×C23C2×C411yescharacteristic
22C22×C4×C23C2311yescharacteristic
23C2×C22⋊D23—22no—
24C2×C22⋊D23—22no—
25C2×C22⋊D23C814yescomplemented
26C22×Dic23—21no—
27C22×Dic23—21no—
28C22×Dic23—42no—
29C22×Dic23C2×C411yescharacteristic
30C22×Dic23C2×C411yescharacteristic
31C22×Dic23C2×C411yescharacteristic
32C2×C4×D23—21no—
33C2×C4×D23—21no—
34C2×C4×D23—42no—
35C2×C8×C23—22no—
36C22⋊C4×C23—22no—
37(C2×D23)⋊C4—22no—
38Dic23⋊C4—22no—
39C4×Dic23D412yes—
40C4×Dic23D412yes—
41C2×C23⋊C8—22no—
42C23×C23—21no—
43C23×C23—21no—
44C23×C23—41no—
45C23×C23—41no—
46C23×C23C2×C812yes—
47C23×C23C22×C411yescharacteristic
48C22×D23—21no—
49C22×D23—21no—
50C22×D23—41no—
51C22×D23—41no—
52C22×D23C2×C812yes—
53C2×C4×C23—21no—
54C2×C4×C23—41no—
55C2×C4×C23—41no—
56C2×C4×C23C2×D412yes—
57C22⋊D23—28no—
58C2×Dic23—22no—
59C2×Dic23—21no—
60C2×Dic23—42no—
61C2×Dic23—42no—
62C2×Dic23—41no—
63C2×Dic23C22⋊C411yescharacteristic
64C2×Dic23C22⋊C412yes—
65C2×Dic23C22⋊C411yescharacteristic
66C2×Dic23C2×C812yes—
67C2×Dic23M1612yes—
68C4×D23—42no—
69C4×D23—42no—
70C8×C23—42no—
71C23⋊C8—42no—
72#1460—231nomaximal, Sylow
73C22×C23—21no—
74C22×C23—21no—
75C22×C23—22no—
76C22×C23—42no—
77C22×C23—42no—
78C22×C23—41no—
79C22×C23—41no—
80C22×C23C2×C22⋊C411yescharacteristic
81C22×C23C22×C811yescharacteristic
82C22×C23C2×M1611yescharacteristic
83C2×D23—22no—
84C2×D23—42no—
85C2×D23—41no—
86C2×D23—41no—
87C2×D23—42no—
88C4×C23—22no—
89C4×C23—41no—
90Dic23—21no—
91Dic23—41no—
92Dic23C22⋊C814yes—
93Dic23#10012yes—
94C2×C4×D4—231no—
95#431—232no—
96#323—231no—
97#323—232no—
98#323D2311yescharacteristic, complemented
99C2×C23—21no—
100C2×C23—21no—
101C2×C23—42no—
102C2×C23C2×C22⋊C811yescharacteristic
103C2×C23#41011yescharacteristic
104C2×C23C2×#10011yescharacteristic
105D23—42no—
106C22×D4—231no—
107C23×C4—231no—
108C23×C4—231no—
109C4×D4—462no—
110C4×D4—462no—
111C2×C4⋊C4—231no—
112C2×C22⋊C4—231no—
113C2×C22⋊C4C2×D2311yescharacteristic
114C2×C42—231no—
115C4⋊C8—462no—
116C22⋊C8—232no—
117C22⋊C8C2×D2312yes—
118C4×C8—462no—
119C23#146011yescharacteristic, complemented, Sylow
120C24—231no—
121C24C4×D2311yescharacteristic
122C2×D4—234no—
123C2×D4—462no—
124C2×D4—462no—
125C22×C4—231no—
126C22×C4—231no—
127C22×C4—231no—
128C22×C4—461no—
129C22×C4—461no—
130C22×C4—461no—
131C22×C4—461no—
132C22×C4—922no—
133C22×C4—922no—
134C22×C4C4×D2311yescharacteristic
135C22×C4C22×D2311yescharacteristic
136C2×C8—22no—
137C2×C8—462no—
138C4⋊C4—462no—
139C22⋊C4—22no—
140C22⋊C4—462no—
141C42—232no—
142C42—232no—
143C23—21no—
144C23—21no—
145C23—41no—
146C23—41no—
147C23—232no—
148C23—461no—
149C23—461no—
150C23—921no—
151C23—921no—
152C23C8×D2312yes—
153C23C2×C4×D2311yescharacteristic
154D4—468no—
155C2×C4—21no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—232no—
159C2×C4—231no—
160C2×C4—232no—
161C2×C4—232no—
162C2×C4—231no—
163C2×C4—462no—
164C2×C4—461no—
165C2×C4—922no—
166C2×C4—922no—
167C2×C4—922no—
168C2×C4—922no—
169C2×C4—921no—
170C2×C4D4×D2312yes—
171C8—42no—
172C8—922no—
173C22—21no—
174C22—21no—
175C22—22no—
176C22—42no—
177C22—42no—
178C22—41no—
179C22—41no—
180C22—462no—
181C22—922no—
182C22—922no—
183C22—921no—
184C22—921no—
185C22C22⋊C4×D2311yescharacteristic
186C22C2×C8×D2311yescharacteristic
187C22M16×D2311yescharacteristic
188C4—22no—
189C4—41no—
190C4—234no—
191C4—232no—
192C4—461no—
193C4—921no—
194C2—21no—
195C2—21no—
196C2—42no—
197C2—922no—
198C2C22⋊C8×D2311yescharacteristic
199C2#1309238911yescharacteristic
200C2#100×D2311yescharacteristic
subgroupG/H
centerC22C22⋊C4×D23
commutatorC22×C23C22×C8
FrattiniC22×C4C22×D23
FittingC23×#323C2
socleC22×C23C2×C22⋊C4
prime radicalC22×C22⋊D23C4
cosocleC22×C4×C23C23
hypercenter#323D23
hypocenterC23#1460
LCSG ▷ C22×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #13092467C2semidirect
3. #13092141C2semidirect
4. #13092140C2semidirect
5. C23×#323C2semidirect
25. C2×C22⋊D23C8semidirect
98. #323D23semidirect
119. C23#1460semidirect

Maximal subgroups

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

subgroupindex
C2×(C22⋊D23)⋊C42
#130924672
#130921412
#130921402
C23×#3232
#146023