Finite Group Databasebeta

← #13093002

Subgroups — #13093002

Order 1472.

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

normal subgroups only

This group has 250 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×#120C211yesmaximal, characteristic, complemented
2#10625124C211yesmaximal, characteristic, complemented
3C2×C22⋊Dic23C211yesmaximal, characteristic, complemented
4#10625110C211yesmaximal, characteristic, complemented
5#10625110C211yesmaximal, characteristic, complemented
6#10625109C211yesmaximal, characteristic, complemented
7#10625107C211yesmaximal, characteristic, complemented
8#10625106C211yesmaximal, characteristic, complemented
9D4×Dic23C211yesmaximal, characteristic, complemented
10#10625056C211yesmaximal, characteristic, complemented
11#10625054C211yesmaximal, characteristic, complemented
12(C22⋊D23)⋊C4C211yesmaximal, characteristic, complemented
13#10625050C211yesmaximal, characteristic, complemented
14#10625050C211yesmaximal, characteristic, complemented
15#10625049C211yesmaximal, characteristic, complemented
16C24×C23C2211yescharacteristic
17C2×D4×C23C2211yescharacteristic
18C2×D4×C23C2211yescharacteristic
19C2×D4×C23C2211yescharacteristic
20C2×C22⋊D23C2211yescharacteristic
21C2×C22⋊D23C2211yescharacteristic
22C2×C22⋊D23C2211yescharacteristic
23C2×C22⋊D23C2211yescharacteristic, complemented
24C22×Dic23C2211yescharacteristic
25C22×Dic23C2211yescharacteristic
26C22×Dic23C2211yescharacteristic, complemented
27C2×C4×D23C2211yescharacteristic, complemented
28C2×Dic46C2211yescharacteristic, complemented
29C22⋊C4×C23C2211yescharacteristic
30C22⋊C4×C23C2211yescharacteristic
31C22⋊C4×C23C2211yescharacteristic
32C22⋊Dic23C2211yescharacteristic
33C22⋊Dic23C2211yescharacteristic
34C22⋊Dic23C2211yescharacteristic
35C22⋊Dic23C2211yescharacteristic
36C22⋊Dic23C2211yescharacteristic
37C22⋊Dic23C2211yescharacteristic
38C22⋊Dic23C2211yescharacteristic
39C22⋊Dic23C2211yescharacteristic, complemented
40C22⋊Dic23C2211yescharacteristic, complemented
41(C2×D23)⋊C4C2211yescharacteristic, complemented
42(C2×D23)⋊C4C2211yescharacteristic, complemented
43C4⋊Dic23C2211yescharacteristic, complemented
44C4⋊Dic23C2211yescharacteristic, complemented
45Dic23⋊C4C2211yescharacteristic, complemented
46Dic23⋊C4C2211yescharacteristic, complemented
47Dic23⋊C4C2211yescharacteristic, complemented
48Dic23⋊C4C2211yescharacteristic, complemented
49C4×Dic23C2211yescharacteristic, complemented
50C4×Dic23C2211yescharacteristic, complemented
51C23×C23—22no—
52C23×C23—41no—
53C23×C23—41no—
54C23×C23C2311yescharacteristic
55C23×C23C2311yescharacteristic
56C23×C23C2311yescharacteristic
57C23×C23C2311yescharacteristic
58C22×D23C2311yescharacteristic
59D4×C23—22no—
60D4×C23—41no—
61D4×C23—41no—
62C2×C4×C23C2311yescharacteristic
63C2×C4×C23C2311yescharacteristic
64C2×C4×C23C2311yescharacteristic
65C22⋊D23—22no—
66C22⋊D23—41no—
67C22⋊D23—41no—
68C22⋊D23—41no—
69C2×Dic23—41no—
70C2×Dic23—41no—
71C2×Dic23—41no—
72C2×Dic23C2311yescharacteristic
73C2×Dic23C2311yescharacteristic
74C2×Dic23C2311yescharacteristic
75C2×Dic23C2311yescharacteristic
76C2×Dic23C2311yescharacteristic
77C2×Dic23C2311yescharacteristic
78C2×Dic23C2311yescharacteristic
79C4×D23—41no—
80Dic46—41no—
81C22×C23—21no—
82C22×C23—21no—
83C22×C23—21no—
84C22×C23—21no—
85C22×C23—21no—
86C22×C23—21no—
87C22×C23—21no—
88C22×C23—21no—
89C22×C23—21no—
90C22×C23—21no—
91C22×C23—21no—
92C22×C23—44no—
93C22×C23#4112yes—
94C22×C23C2411yescharacteristic
95C2×D23—21no—
96C2×D23—21no—
97C2×D23—21no—
98C4×C23—21no—
99C4×C23—21no—
100C4×C23—21no—
101Dic23—21no—
102Dic23—21no—
103Dic23—21no—
104Dic23—21no—
105Dic23—21no—
106Dic23—21no—
107Dic23—21no—
108#538—231nomaximal, Sylow
109C2×C23—22no—
110C2×C23—41no—
111C2×C23—41no—
112C2×C23—41no—
113C2×C23C2×#4111yescharacteristic
114C2×C23H(5,𝔽2)11yescharacteristic
115C2×C23H(5,𝔽2)11yescharacteristic
116D23—41no—
117#126—231no—
118#126—231no—
119#124—231no—
120#124—231no—
121#123—231no—
122#123—231no—
123C22⋊Q8—231no—
124#121—231no—
125#121—231no—
126#121—231no—
127#120—231no—
128#120D2311yescharacteristic, complemented
129C4×D4—231no—
130C4×D4—231no—
131C2×C22⋊C4—231no—
132C23#53811yescharacteristic, complemented, Sylow
133C24C2×D2311yescharacteristic
134C2×Q8—231no—
135C2×D4—231no—
136C2×D4—231no—
137C2×D4—231no—
138C2×D4—231no—
139C2×D4C2×D2311yescharacteristic
140C2×D4C2×D2311yescharacteristic
141C2×D4C2×D2311yescharacteristic
142C22×C4—231no—
143C22×C4—231no—
144C22×C4—231no—
145C22×C4—231no—
146C4⋊C4—231no—
147C4⋊C4—231no—
148C4⋊C4—231no—
149C4⋊C4—231no—
150C4⋊C4—231no—
151C4⋊C4—231no—
152C22⋊C4—231no—
153C22⋊C4—231no—
154C22⋊C4—231no—
155C22⋊C4—231no—
156C22⋊C4—231no—
157C22⋊C4—231no—
158C22⋊C4—231no—
159C22⋊C4—231no—
160C22⋊C4—231no—
161C22⋊C4—231no—
162C22⋊C4—231no—
163C22⋊C4C2×D2311yescharacteristic
164C22⋊C4C2×D2311yescharacteristic
165C22⋊C4C2×D2311yescharacteristic
166C42—231no—
167C42—231no—
168C23—22no—
169C23—41no—
170C23—41no—
171C23—231no—
172C23C22×D2311yescharacteristic
173C23C22×D2311yescharacteristic
174C23C22×D2311yescharacteristic
175C23C22×D2311yescharacteristic
176Q8—921no—
177D4—22no—
178D4—41no—
179D4—41no—
180D4—462no—
181D4—921no—
182D4—921no—
183D4—921no—
184C2×C4—231no—
185C2×C4—231no—
186C2×C4—231no—
187C2×C4—231no—
188C2×C4—231no—
189C2×C4—231no—
190C2×C4—231no—
191C2×C4—921no—
192C2×C4—921no—
193C2×C4—921no—
194C2×C4—921no—
195C2×C4C22×D2311yescharacteristic
196C2×C4C22×D2311yescharacteristic
197C2×C4C22×D2311yescharacteristic
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—44no—
210C22—461no—
211C22—461no—
212C22—461no—
213C22#6729012yes—
214C22C23×D2311yescharacteristic
215C4—21no—
216C4—21no—
217C4—21no—
218C4—461no—
219C4—461no—
220C4—461no—
221C4—461no—
222C4—461no—
223C4—461no—
224C4—461no—
225C2—22no—
226C2—41no—
227C2—41no—
228C2—41no—
229C2—921no—
230C2C2×#6729011yescharacteristic
231C2#1062515511yescharacteristic
232C2#1062515511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D23
commutator, socle, cosocleC22×C23C24
FittingC23×#120C2
hypercenter#120D23
hypocenterC23#538
LCSG ▷ C22×C23 ▷ C23
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C23×#120C2semidirect
2. #10625124C2semidirect
3. C2×C22⋊Dic23C2semidirect
4. #10625110C2semidirect
5. #10625110C2semidirect
6. #10625109C2semidirect
7. #10625107C2semidirect
8. #10625106C2semidirect
9. D4×Dic23C2semidirect
10. #10625056C2semidirect
11. #10625054C2semidirect
12. (C22⋊D23)⋊C4C2semidirect
13. #10625050C2semidirect
14. #10625050C2semidirect
15. #10625049C2semidirect
23. C2×C22⋊D23C22semidirect
26. C22×Dic23C22semidirect
27. C2×C4×D23C22semidirect
28. C2×Dic46C22semidirect
39. C22⋊Dic23C22semidirect
40. C22⋊Dic23C22semidirect
41. (C2×D23)⋊C4C22semidirect
42. (C2×D23)⋊C4C22semidirect
43. C4⋊Dic23C22semidirect
44. C4⋊Dic23C22semidirect
45. Dic23⋊C4C22semidirect
46. Dic23⋊C4C22semidirect
47. Dic23⋊C4C22semidirect
48. Dic23⋊C4C22semidirect
49. C4×Dic23C22semidirect
50. C4×Dic23C22semidirect
128. #120D23semidirect
132. C23#538semidirect

Maximal subgroups

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

subgroupindex
C23×#1202
#106251242
C2×C22⋊Dic232
#106251102
#106251092
#106251072
#106251062
D4×Dic232
#106250562
#106250542
(C22⋊D23)⋊C42
#106250502
#106250492
#53823