Finite Group Databasebeta

← #13093103

Subgroups — #13093103

Order 1472.

214 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C23×#126C211yesmaximal, characteristic, complemented
2#10625067C211yesmaximal, characteristic, complemented
3#10625063C211yesmaximal, characteristic, complemented
4C4⋊C4×D23C211yesmaximal, characteristic, complemented
5#10625061C211yesmaximal, characteristic, complemented
6#10625060C211yesmaximal, characteristic, complemented
7#10625060C211yesmaximal, characteristic, complemented
8#10625054C211yesmaximal, characteristic
9#10625054C211yesmaximal, characteristic
10(C22⋊D23)⋊C4C211yesmaximal, characteristic
11#10625050C211yesmaximal, characteristic, complemented
12#10625049C211yesmaximal, characteristic, complemented
13#10625048C211yesmaximal, characteristic, complemented
14#10625043C211yesmaximal, characteristic, complemented
15C4×Dic46C211yesmaximal, characteristic, complemented
16C2×C22⋊D23C2211yescharacteristic
17C22×Dic23C2211yescharacteristic
18C2×C4×D23C2211yescharacteristic
19C2×C4×D23C2211yescharacteristic
20C2×C4×D23C2211yescharacteristic
21C2×Dic46C2211yescharacteristic
22C4⋊C4×C23C2211yescharacteristic
23C4⋊C4×C23C2211yescharacteristic
24C4⋊C4×C23C2211yescharacteristic
25C22⋊C4×C23C2211yescharacteristic
26C22⋊C4×C23C2211yescharacteristic
27C22⋊C4×C23C2211yescharacteristic
28C42×C23C2211yescharacteristic
29C22⋊Dic23C2211yescharacteristic
30C22⋊Dic23C2211yescharacteristic
31(C2×D23)⋊C4C2211yescharacteristic
32(C2×D23)⋊C4C2211yescharacteristic
33(C2×D23)⋊C4C2211yescharacteristic
34C4⋊Dic23C2211yescharacteristic
35C4⋊Dic23C2211yescharacteristic
36C4⋊Dic23C2211yescharacteristic
37C4⋊Dic23C2211yescharacteristic
38Dic23⋊C4C2211yescharacteristic
39Dic23⋊C4C2211yescharacteristic
40Dic23⋊C4C2211yescharacteristic
41Dic23⋊C4C2211yescharacteristic
42Dic23⋊C4C2211yescharacteristic
43Dic23⋊C4C2211yescharacteristic
44Dic23⋊C4C2211yescharacteristic
45Dic23⋊C4C2211yescharacteristic
46Dic23⋊C4C2211yescharacteristic
47C4×Dic23C2211yescharacteristic
48C4×Dic23C2211yescharacteristic
49C4×Dic23C2211yescharacteristic
50C4×Dic23C2211yescharacteristic
51C23×C23C2311yescharacteristic
52C22×D23C2311yescharacteristic
53C2×C4×C23C2311yescharacteristic
54C2×C4×C23C2311yescharacteristic
55C2×C4×C23C2311yescharacteristic
56C2×C4×C23C2311yescharacteristic
57C2×C4×C23C2311yescharacteristic
58C2×C4×C23C2311yescharacteristic
59C22⋊D23—22no—
60C2×Dic23—22no—
61C2×Dic23C2311yescharacteristic
62C2×Dic23C2311yescharacteristic
63C2×Dic23C2311yescharacteristic
64C2×Dic23C2311yescharacteristic
65C2×Dic23C2311yescharacteristic
66C2×Dic23C2311yescharacteristic
67C2×Dic23C2311yescharacteristic
68C4×D23—22no—
69C4×D23—22no—
70C4×D23—22no—
71Dic46—22no—
72C22×C23—21no—
73C22×C23—21no—
74C22×C23—21no—
75C22×C23C2411yescharacteristic
76C2×D23—21no—
77C2×D23—21no—
78C2×D23#4112yes—
79C4×C23—21no—
80C4×C23—21no—
81C4×C23—21no—
82C4×C23—21no—
83C4×C23—21no—
84C4×C23—21no—
85Dic23—21no—
86Dic23—21no—
87Dic23—21no—
88Dic23—21no—
89Dic23—21no—
90Dic23—21no—
91Dic23#4112yes—
92#552—231nomaximal, Sylow
93C2×C23—41no—
94C2×C23C2×#4111yescharacteristic
95C2×C23C2×#4111yescharacteristic
96C2×C232−1+411yescharacteristic
97D23—22no—
98#126—231no—
99#126D2311yescharacteristic, complemented
100#125—231no—
101#125—231no—
102#125—231no—
103#123—231no—
104#123—231no—
105C22⋊Q8—231no—
106C22⋊Q8—231no—
107C4×Q8—231no—
108C4×D4—231no—
109#117—231no—
110#117—231no—
111#117—231no—
112C2×C4⋊C4—231no—
113C23#55211yescharacteristic, complemented, Sylow
114C2×Q8—231no—
115C2×D4—231no—
116C22×C4—231no—
117C22×C4—231no—
118C22×C4—231no—
119C22×C4—231no—
120C4⋊C4—231no—
121C4⋊C4—231no—
122C4⋊C4—231no—
123C4⋊C4—231no—
124C4⋊C4—231no—
125C4⋊C4—231no—
126C4⋊C4—231no—
127C4⋊C4—231no—
128C4⋊C4—231no—
129C4⋊C4—231no—
130C4⋊C4—231no—
131C4⋊C4—231no—
132C4⋊C4—231no—
133C4⋊C4C2×D2311yescharacteristic
134C4⋊C4C2×D2311yescharacteristic
135C4⋊C4C2×D2311yescharacteristic
136C22⋊C4—231no—
137C22⋊C4—231no—
138C22⋊C4—231no—
139C22⋊C4—231no—
140C22⋊C4—231no—
141C22⋊C4C2×D2311yescharacteristic
142C22⋊C4C2×D2311yescharacteristic
143C22⋊C4C2×D2311yescharacteristic
144C42—231no—
145C42—231no—
146C42—231no—
147C42—231no—
148C42C2×D2311yescharacteristic
149C23—231no—
150C23C22×D2311yescharacteristic
151Q8—462no—
152D4—462no—
153C2×C4—231no—
154C2×C4—231no—
155C2×C4—231no—
156C2×C4—231no—
157C2×C4—231no—
158C2×C4—231no—
159C2×C4—231no—
160C2×C4—462no—
161C2×C4—462no—
162C2×C4—462no—
163C2×C4—462no—
164C2×C4C22×D2311yescharacteristic
165C2×C4C22×D2311yescharacteristic
166C2×C4C22×D2311yescharacteristic
167C2×C4C22×D2311yescharacteristic
168C2×C4C22×D2311yescharacteristic
169C2×C4C22×D2311yescharacteristic
170C22—21no—
171C22—21no—
172C22—21no—
173C22—232no—
174C22—461no—
175C22—461no—
176C22C23×D2311yescharacteristic
177C4—21no—
178C4—21no—
179C4—21no—
180C4—21no—
181C4—21no—
182C4—21no—
183C4—232no—
184C4—461no—
185C4—461no—
186C4—461no—
187C4—461no—
188C4—461no—
189C4—461no—
190C2—41no—
191C2—462no—
192C2#41×D2311yescharacteristic
193C2#41×D2311yescharacteristic
194C2#1062516111yescharacteristic
subgroupG/H
center, FrattiniC22C23×D23
commutator, socle, cosocleC22×C23C24
FittingC23×#126C2
prime radicalC2×C22⋊D23C22
hypercenter#126D23
hypocenterC23#552
LCSG ▷ C22×C23 ▷ C23
UCSC1 ◁ C22 ◁ #126

Complements

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

NH ≅ G/Ntype
1. C23×#126C2semidirect
2. #10625067C2semidirect
3. #10625063C2semidirect
4. C4⋊C4×D23C2semidirect
5. #10625061C2semidirect
6. #10625060C2semidirect
7. #10625060C2semidirect
11. #10625050C2semidirect
12. #10625049C2semidirect
13. #10625048C2semidirect
14. #10625043C2semidirect
15. C4×Dic46C2semidirect
99. #126D23semidirect
113. C23#552semidirect

Maximal subgroups

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

subgroupindex
C23×#1262
#106250672
#106250632
C4⋊C4×D232
#106250612
#106250602
#106250542
(C22⋊D23)⋊C42
#106250502
#106250492
#106250482
#106250432
C4×Dic462
#55223