Finite Group Databasebeta

← #13093001

Subgroups — #13093001

Order 1472.

250 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 93 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×#124C211yesmaximal, characteristic, complemented
2#10625118C211yesmaximal, characteristic, complemented
3#10625117C211yesmaximal, characteristic, complemented
4#10625110C211yesmaximal, characteristic, complemented
5#10625108C211yesmaximal, characteristic, complemented
6#10625057C211yesmaximal, characteristic, complemented
7#10625055C211yesmaximal, characteristic, complemented
8#10625055C211yesmaximal, characteristic, complemented
9#10625054C211yesmaximal, characteristic, complemented
10#10625053C211yesmaximal, characteristic, complemented
11(C22⋊D23)⋊C4C211yesmaximal, characteristic, complemented
12C22⋊C4×D23C211yesmaximal, characteristic, complemented
13#10625050C211yesmaximal, characteristic, complemented
14#10625047C211yesmaximal, characteristic, complemented
15C4×D92C211yesmaximal, characteristic, complemented
16C23×D23C2211yescharacteristic
17C2×Q8×C23C2211yescharacteristic, complemented
18C2×D4×C23C2211yescharacteristic
19C2×C22⋊D23C2211yescharacteristic
20C2×C22⋊D23C2211yescharacteristic
21C2×C22⋊D23C2211yescharacteristic
22C2×C22⋊D23C2211yescharacteristic, complemented
23C22×Dic23C2211yescharacteristic, complemented
24C2×D92C2211yescharacteristic
25C2×D92C2211yescharacteristic
26C2×C4×D23C2211yescharacteristic
27C2×C4×D23C2211yescharacteristic
28C2×C4×D23C2211yescharacteristic, complemented
29C22⋊C4×C23C2211yescharacteristic
30C22⋊C4×C23C2211yescharacteristic
31C22⋊C4×C23C2211yescharacteristic, complemented
32C22⋊C4×C23C2211yescharacteristic, complemented
33C42×C23C2211yescharacteristic, complemented
34C22⋊Dic23C2211yescharacteristic
35C22⋊Dic23C2211yescharacteristic
36(C2×D23)⋊C4C2211yescharacteristic
37(C2×D23)⋊C4C2211yescharacteristic
38(C2×D23)⋊C4C2211yescharacteristic
39(C2×D23)⋊C4C2211yescharacteristic
40(C2×D23)⋊C4C2211yescharacteristic
41(C2×D23)⋊C4C2211yescharacteristic
42(C2×D23)⋊C4C2211yescharacteristic, complemented
43(C2×D23)⋊C4C2211yescharacteristic, complemented
44C4⋊Dic23C2211yescharacteristic, complemented
45C4⋊Dic23C2211yescharacteristic, complemented
46Dic23⋊C4C2211yescharacteristic, complemented
47Dic23⋊C4C2211yescharacteristic, complemented
48Dic23⋊C4C2211yescharacteristic, complemented
49Dic23⋊C4C2211yescharacteristic, complemented
50C4×Dic23C2211yescharacteristic, complemented
51C23×C23C2311yescharacteristic
52C23×C23C2311yescharacteristic
53C22×D23—22no—
54C22×D23—41no—
55C22×D23—41no—
56C22×D23C2311yescharacteristic
57C22×D23C2311yescharacteristic
58C22×D23C2311yescharacteristic
59Q8×C23—41no—
60D4×C23—41no—
61C2×C4×C23C2311yescharacteristic
62C2×C4×C23C2311yescharacteristic
63C2×C4×C23C2311yescharacteristic
64C2×C4×C23C2311yescharacteristic
65C2×C4×C23C2311yescharacteristic
66C22⋊D23—22no—
67C22⋊D23—41no—
68C22⋊D23—41no—
69C22⋊D23—41no—
70C2×Dic23—41no—
71C2×Dic23C2311yescharacteristic
72C2×Dic23C2311yescharacteristic
73C2×Dic23C2311yescharacteristic
74C2×Dic23C2311yescharacteristic
75C2×Dic23C2311yescharacteristic
76D92—22no—
77D92—41no—
78C4×D23—41no—
79C4×D23—41no—
80C4×D23—41no—
81C22×C23—21no—
82C22×C23—21no—
83C22×C23—21no—
84C22×C23—21no—
85C22×C23—21no—
86C22×C23—21no—
87C22×C23C2411yescharacteristic
88C2×D23—21no—
89C2×D23—21no—
90C2×D23—21no—
91C2×D23—21no—
92C2×D23—21no—
93C2×D23—21no—
94C2×D23—21no—
95C2×D23—21no—
96C2×D23—44no—
97C2×D23#4112yes—
98C4×C23—21no—
99C4×C23—21no—
100C4×C23—21no—
101C4×C23—21no—
102C4×C23—21no—
103Dic23—21no—
104Dic23—21no—
105Dic23—21no—
106Dic23—21no—
107Dic23—21no—
108#538—231nomaximal, Sylow
109C2×C23—41no—
110C2×C23—41no—
111C2×C23C2×#4111yescharacteristic
112C2×C23H(5,𝔽2)11yescharacteristic
113C2×C23H(5,𝔽2)11yescharacteristic
114D23—22no—
115D23—41no—
116D23—41no—
117#126—231no—
118#126—231no—
119#124—231no—
120#124D2311yescharacteristic, complemented
121#123—231no—
122#123—231no—
123C22⋊Q8—231no—
124#121—231no—
125#121—231no—
126#121—231no—
127#120—231no—
128#120—231no—
129C4×D4—231no—
130C4×D4—231no—
131C2×C22⋊C4—231no—
132C23#53811yescharacteristic, complemented, Sylow
133C24—231no—
134C2×Q8C2×D2311yescharacteristic, complemented
135C2×D4—231no—
136C2×D4—231no—
137C2×D4—231no—
138C2×D4—231no—
139C2×D4—231no—
140C2×D4—231no—
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⋊C4C2×D2311yescharacteristic
163C22⋊C4C2×D2311yescharacteristic
164C22⋊C4C2×D2311yescharacteristic, complemented
165C22⋊C4C2×D2311yescharacteristic, complemented
166C42—231no—
167C42C2×D2311yescharacteristic, complemented
168C23—231no—
169C23—231no—
170C23—231no—
171C23—462no—
172C23—921no—
173C23—921no—
174C23C22×D2311yescharacteristic
175C23C22×D2311yescharacteristic
176Q8—41no—
177D4—41no—
178D4—462no—
179D4—462no—
180D4—921no—
181D4—921no—
182D4—921no—
183D4—921no—
184C2×C4—231no—
185C2×C4—231no—
186C2×C4—231no—
187C2×C4—231no—
188C2×C4—231no—
189C2×C4—921no—
190C2×C4—921no—
191C2×C4—921no—
192C2×C4—921no—
193C2×C4C22×D2311yescharacteristic
194C2×C4C22×D2311yescharacteristic
195C2×C4C22×D2311yescharacteristic
196C2×C4C22×D2311yescharacteristic
197C2×C4C22×D2311yescharacteristic
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—21no—
204C22—232no—
205C22—461no—
206C22—461no—
207C22—461no—
208C22—461no—
209C22—461no—
210C22—461no—
211C22—461no—
212C22—461no—
213C22—924no—
214C22C23×D2311yescharacteristic
215C4—21no—
216C4—21no—
217C4—21no—
218C4—21no—
219C4—21no—
220C4—461no—
221C4—461no—
222C4—461no—
223C4—461no—
224C4—461no—
225C2—41no—
226C2—41no—
227C2—462no—
228C2—921no—
229C2—921no—
230C2#1062515511yescharacteristic
231C2#41×D2311yescharacteristic
232C2#1062516011yescharacteristic
subgroupG/H
center, FrattiniC22C23×D23
commutator, socle, cosocleC22×C23C24
FittingC23×#124C2
hypercenter#124D23
hypocenterC23#538
LCSG ▷ C22×C23 ▷ C23
UCSC1 ◁ C22 ◁ #124

Complements

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

NH ≅ G/Ntype
1. C23×#124C2semidirect
2. #10625118C2semidirect
3. #10625117C2semidirect
4. #10625110C2semidirect
5. #10625108C2semidirect
6. #10625057C2semidirect
7. #10625055C2semidirect
8. #10625055C2semidirect
9. #10625054C2semidirect
10. #10625053C2semidirect
11. (C22⋊D23)⋊C4C2semidirect
12. C22⋊C4×D23C2semidirect
13. #10625050C2semidirect
14. #10625047C2semidirect
15. C4×D92C2semidirect
17. C2×Q8×C23C22semidirect
22. C2×C22⋊D23C22semidirect
23. C22×Dic23C22semidirect
28. C2×C4×D23C22semidirect
31. C22⋊C4×C23C22semidirect
32. C22⋊C4×C23C22semidirect
33. C42×C23C22semidirect
42. (C2×D23)⋊C4C22semidirect
43. (C2×D23)⋊C4C22semidirect
44. C4⋊Dic23C22semidirect
45. C4⋊Dic23C22semidirect
46. Dic23⋊C4C22semidirect
47. Dic23⋊C4C22semidirect
48. Dic23⋊C4C22semidirect
49. Dic23⋊C4C22semidirect
50. C4×Dic23C22semidirect
120. #124D23semidirect
132. C23#538semidirect
134. C2×Q8C2×D23semidirect
164. C22⋊C4C2×D23semidirect
165. C22⋊C4C2×D23semidirect
167. C42C2×D23semidirect

Maximal subgroups

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

subgroupindex
C23×#1242
#106251182
#106251172
#106251102
#106251082
#106250572
#106250552
#106250542
#106250532
(C22⋊D23)⋊C42
C22⋊C4×D232
#106250502
#106250472
C4×D922
#53823