Finite Group Databasebeta

← #15671969

Subgroups — #15671969

Order 2944.

288 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 97 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⋊(C23⋊C8)C211yesmaximal, characteristic
2#13092277C211yesmaximal, characteristic
3#13092154C214yesmaximal, complemented
4C2×C23×C22⋊Q8C211yesmaximal, characteristic
5C22×Q8×C23C411yescharacteristic, complemented
6C23×C4×C23C2211yescharacteristic
7C23×C22⋊Q8—21no—
8C23×C22⋊Q8—21no—
9C23×C22⋊Q8C412yescomplemented
10C23×C22⋊Q8C2212yes—
11C2×C4⋊C4×C23—21no—
12C2×C4⋊C4×C23C411yescharacteristic, complemented
13C2×C22⋊C4×C23—21no—
14C2×C22⋊Dic23—21no—
15C2×C4⋊Dic23—21no—
16C22×C23⋊C8—21no—
17C23⋊#95C2212yes—
18C22⋊(C23⋊C8)—21no—
19C22⋊(C23⋊C8)C2212yes—
20C24×C23D411yescharacteristic
21C2×Q8×C23—21no—
22C2×Q8×C23—42no—
23C2×Q8×C23C2×C412yes—
24C22×C4×C23—21no—
25C22×C4×C23—21no—
26C22×C4×C23—42no—
27C22×C4×C23C2×C411yescharacteristic
28C22×C4×C23C2×C411yescharacteristic
29C22×C4×C23D411yescharacteristic
30C22×C4×C23D411yescharacteristic
31C22×C4×C23D411yescharacteristic
32C22×C4×C23C2311yescharacteristic
33C22×Dic23—22no—
34C4⋊C4×C23—22no—
35C4⋊C4×C23—21no—
36C4⋊C4×C23—22no—
37C4⋊C4×C23C2×C412yes—
38C22⋊C4×C23—22no—
39C22⋊C4×C23—22no—
40C22⋊Dic23—42no—
41C4⋊Dic23—42no—
42C2×C23⋊C8—22no—
43C2×C23⋊C8—42no—
44C23×C23—41no—
45C23×C23—41no—
46C23×C23C22⋊C411yescharacteristic
47C23×C23C22⋊C411yescharacteristic
48C23×C23SD1612yes—
49C23×C23Q1612yes—
50C23×C23C2×D411yescharacteristic
51Q8×C23—42no—
52Q8×C23—42no—
53C2×C4×C23—22no—
54C2×C4×C23—22no—
55C2×C4×C23—22no—
56C2×C4×C23—22no—
57C2×C4×C23—22no—
58C2×C4×C23—41no—
59C2×C4×C23—41no—
60C2×C4×C23—41no—
61C2×C4×C23—42no—
62C2×C4×C23—42no—
63C2×C4×C23C22⋊C411yescharacteristic
64C2×C4×C23C22⋊C411yescharacteristic
65C2×C4×C23C22×C411yescharacteristic
66C2×C4×C23C2×D411yescharacteristic
67C2×Dic23—42no—
68C2×Dic23—42no—
69C2×Dic23—42no—
70C23⋊C8—42no—
71#1520—231nomaximal, Sylow
72C22×C23—21no—
73C22×C23—21no—
74C22×C23—21no—
75C22×C23—21no—
76C22×C23—21no—
77C22×C23—21no—
78C22×C23—42no—
79C22×C23—42no—
80C22×C23#10312yes—
81C22×C23#10312yes—
82C22×C23C2×C22⋊C411yescharacteristic
83C22×C23C2×SD1611yescharacteristic
84C22×C23C2×Q1611yescharacteristic
85C4×C23—21no—
86C4×C23—21no—
87C4×C23—21no—
88C4×C23—42no—
89C4×C23—42no—
90C4×C23#9912yes—
91Dic23—82no—
92C2×C22⋊Q8D2311yescharacteristic
93C2×C22⋊C8—231no—
94#380—231no—
95#328—234no—
96C2×C23—21no—
97C2×C23—22no—
98C2×C23—22no—
99C2×C23—21no—
100C2×C23C2×#9911yescharacteristic
101C2×C23C2×#10311yescharacteristic
102C2×C23#42111yescharacteristic
103C22×Q8Dic2311yescharacteristic, complemented
104C23×C4C2×D2311yescharacteristic
105C22×C8—461no—
106C22⋊Q8—21no—
107C22⋊Q8—21no—
108C22⋊Q8Dic2312yescomplemented
109C22⋊Q8C2×D2312yes—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—461no—
112C2×C4⋊C4Dic2311yescharacteristic, complemented
113C2×C22⋊C4—21no—
114C2×C22⋊C4—461no—
115C22⋊C8—232no—
116C22⋊C8—461no—
117#95—232no—
118C23#152011yescharacteristic, complemented, Sylow
119C24C22⋊D2311yescharacteristic
120C2×Q8—21no—
121C2×Q8—42no—
122C2×Q8C2×Dic2312yes—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—42no—
126C22×C4—462no—
127C22×C4C2×Dic2311yescharacteristic
128C22×C4C2×Dic2311yescharacteristic
129C22×C4C22⋊D2311yescharacteristic
130C22×C4C22⋊D2311yescharacteristic
131C22×C4C22⋊D2311yescharacteristic
132C22×C4C22×D2311yescharacteristic
133C2×C8—462no—
134C2×C8—922no—
135C4⋊C4—22no—
136C4⋊C4—21no—
137C4⋊C4—22no—
138C4⋊C4—922no—
139C4⋊C4C2×Dic2312yes—
140C22⋊C4—22no—
141C22⋊C4—22no—
142C22⋊C4—922no—
143C23—41no—
144C23—41no—
145C23#6727412yes—
146C23#6727512yes—
147C23C22⋊Dic2311yescharacteristic
148C23C22⋊Dic2311yescharacteristic
149C23C2×C22⋊D2311yescharacteristic
150Q8—42no—
151Q8—42no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—922no—
163C2×C4—922no—
164C2×C4—922no—
165C2×C4C22⋊Dic2311yescharacteristic
166C2×C4C22⋊Dic2311yescharacteristic
167C2×C4C22×Dic2311yescharacteristic
168C2×C4C2×C22⋊D2311yescharacteristic
169C8—922no—
170C22—21no—
171C22—21no—
172C22—21no—
173C22—21no—
174C22—21no—
175C22—21no—
176C22—42no—
177C22—42no—
178C22#1062501712yes—
179C22#1062501712yes—
180C22C2×#6727411yescharacteristic
181C22C2×#6727511yescharacteristic
182C22C2×C22⋊Dic2311yescharacteristic
183C4—21no—
184C4—21no—
185C4—21no—
186C4—42no—
187C4—42no—
188C4—1842no—
189C4#1062501612yes—
190C2—21no—
191C2—21no—
192C2—22no—
193C2—22no—
194C2C2×#1062501611yescharacteristic
195C2C2×#1062501711yescharacteristic
196C2#1309243411yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic23
commutatorC2×C4×C23C22×C4
FrattiniC22×C4C22×D23
FittingC2×C23×C22⋊Q8C2
socleC22×C23C2×C22⋊C4
prime radicalC24×C23D4
cosocleC22×C4×C23C23
hypercenterC2×C22⋊Q8D23
hypocenterC23#1520
LCSG ▷ C2×C4×C23 ▷ C22×C23 ▷ C23
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #13092154C2semidirect
5. C22×Q8×C23C4semidirect
9. C23×C22⋊Q8C4semidirect
12. C2×C4⋊C4×C23C4semidirect
103. C22×Q8Dic23semidirect
108. C22⋊Q8Dic23semidirect
112. C2×C4⋊C4Dic23semidirect
118. C23#1520semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊(C23⋊C8)2
#130922772
#130921542
C2×C23×C22⋊Q82
#152023