Finite Group Databasebeta

← #15671917

Subgroups — #15671917

Order 2944.

294 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 85 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
1#13092381C211yesmaximal, characteristic
2C4×C22⋊Dic23C211yesmaximal, 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—
16C2×#67267—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—24no—
41C4×Dic23—42no—
42#67267—42no—
43C2×C23⋊C8—22no—
44C23×C23—22no—
45C23×C23—22no—
46C23×C23—41no—
47C23×C23C22⋊C411yescharacteristic
48C23×C23C22⋊C411yescharacteristic
49C23×C23C2×D411yescharacteristic
50Q8×C23—42no—
51Q8×C23—42no—
52C2×C4×C23—22no—
53C2×C4×C23—22no—
54C2×C4×C23—22no—
55C2×C4×C23—22no—
56C2×C4×C23—22no—
57C2×C4×C23—41no—
58C2×C4×C23—41no—
59C2×C4×C23—41no—
60C2×C4×C23—42no—
61C2×C4×C23—42no—
62C2×C4×C23C22⋊C411yescharacteristic
63C2×C4×C23C22⋊C411yescharacteristic
64C2×C4×C23C22×C411yescharacteristic
65C2×C4×C23C2×D411yescharacteristic
66C2×Dic23—24no—
67C2×Dic23—42no—
68C2×Dic23—42no—
69C23⋊C8—42no—
70#1508—231nomaximal, Sylow
71C22×C23—21no—
72C22×C23—21no—
73C22×C23—21no—
74C22×C23—21no—
75C22×C23—21no—
76C22×C23—21no—
77C22×C23—21no—
78C22×C23—44no—
79C22×C23#10412yes—
80C22×C23C2×C22⋊C411yescharacteristic
81C22×C23#13712yes—
82C4×C23—21no—
83C4×C23—21no—
84C4×C23—21no—
85C4×C23—42no—
86C4×C23—42no—
87C4×C23#9912yes—
88Dic23—44no—
89C2×C22⋊Q8D2311yescharacteristic
90#407—231no—
91C4×C22⋊C4—231no—
92#328—234no—
93C2×C23—22no—
94C2×C23—22no—
95C2×C23—41no—
96C2×C23C2×#9911yescharacteristic
97C2×C23#41911yescharacteristic
98C2×C23C2×#10411yescharacteristic
99C22×Q8Dic2311yescharacteristic, complemented
100C23×C4C2×D2311yescharacteristic
101C2×M16—461no—
102C22⋊Q8—21no—
103C22⋊Q8—21no—
104C22⋊Q8Dic2312yescomplemented
105C22⋊Q8C2×D2312yes—
106C2×C4⋊C4—21no—
107C2×C4⋊C4Dic2311yescharacteristic, complemented
108C2×C22⋊C4—21no—
109C2×C22⋊C4—461no—
110C2×C42—461no—
111C22⋊C8—232no—
112C22⋊C8—461no—
113#95—232no—
114C23#150811yescharacteristic, complemented, Sylow
115C24C22⋊D2311yescharacteristic
116C2×Q8—21no—
117C2×Q8—42no—
118C2×Q8C2×Dic2312yes—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—42no—
122C22×C4—462no—
123C22×C4C2×Dic2311yescharacteristic
124C22×C4C2×Dic2311yescharacteristic
125C22×C4C22⋊D2311yescharacteristic
126C22×C4C22⋊D2311yescharacteristic
127C22×C4C22⋊D2311yescharacteristic
128C22×C4C22×D2311yescharacteristic
129M16—922no—
130C2×C8—462no—
131C4⋊C4—22no—
132C4⋊C4—21no—
133C4⋊C4—22no—
134C4⋊C4C2×Dic2312yes—
135C22⋊C4—22no—
136C22⋊C4—22no—
137C22⋊C4—464no—
138C42—922no—
139C23—22no—
140C23—22no—
141C23—41no—
142C23C22⋊Dic2311yescharacteristic
143C23C22⋊Dic2311yescharacteristic
144C23C2×C22⋊D2311yescharacteristic
145Q8—42no—
146Q8—42no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—464no—
158C2×C4—922no—
159C2×C4—922no—
160C2×C4C22⋊Dic2311yescharacteristic
161C2×C4C22⋊Dic2311yescharacteristic
162C2×C4C22×Dic2311yescharacteristic
163C2×C4C2×C22⋊D2311yescharacteristic
164C8—922no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—21no—
169C22—21no—
170C22—21no—
171C22—21no—
172C22—44no—
173C22#1062501912yes—
174C22#1062511312yes—
175C22C2×C22⋊Dic2311yescharacteristic
176C4—21no—
177C4—21no—
178C4—21no—
179C4—42no—
180C4—42no—
181C4—924no—
182C4#1062501612yes—
183C2—22no—
184C2—22no—
185C2—41no—
186C2C2×#1062501611yescharacteristic
187C2#1309242611yescharacteristic
188C2C2×#1062501911yescharacteristic
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#1508
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
99. C22×Q8Dic23semidirect
104. C22⋊Q8Dic23semidirect
107. C2×C4⋊C4Dic23semidirect
114. C23#1508semidirect

Maximal subgroups

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

subgroupindex
#130923812
C4×C22⋊Dic232
#130921542
C2×C23×C22⋊Q82
#150823