Finite Group Databasebeta

← #13093094

Subgroups — #13093094

Order 1472.

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

normal subgroups only

This group has 248 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
1C23×C22⋊Q8C211yesmaximal, characteristic, complemented
2#10625118C211yesmaximal, characteristic, complemented
3#10625117C211yesmaximal, characteristic, complemented
4C4×C22⋊D23C211yesmaximal, characteristic, complemented
5C2×(C2×D23)⋊C4C211yesmaximal, characteristic, complemented
6#10625069C211yesmaximal, characteristic, complemented
7#10625069C211yesmaximal, characteristic, complemented
8#10625065C211yesmaximal, characteristic, complemented
9#10625065C211yesmaximal, characteristic, complemented
10#10625064C211yesmaximal, characteristic, complemented
11#10625063C211yesmaximal, characteristic, complemented
12#10625057C211yesmaximal, characteristic, complemented
13#10625053C211yesmaximal, characteristic
14C22⋊C4×D23C211yesmaximal, characteristic, complemented
15#10625048C211yesmaximal, characteristic, complemented
16C23×D23C2211yescharacteristic
17C2×Q8×C23C2211yescharacteristic, complemented
18C22×C4×C23C2211yescharacteristic
19C2×C22⋊D23C2211yescharacteristic
20C22×Dic23C2211yescharacteristic
21C2×D92C2211yescharacteristic
22C2×D92C2211yescharacteristic
23C2×C4×D23C2211yescharacteristic
24C2×C4×D23C2211yescharacteristic
25C2×C4×D23C2211yescharacteristic, complemented
26C4⋊C4×C23C2211yescharacteristic, complemented
27C4⋊C4×C23C2211yescharacteristic, complemented
28C4⋊C4×C23C2211yescharacteristic, complemented
29C22⋊C4×C23C2211yescharacteristic
30C22⋊C4×C23C2211yescharacteristic
31C22⋊Dic23C2211yescharacteristic
32(C2×D23)⋊C4C2211yescharacteristic
33(C2×D23)⋊C4C2211yescharacteristic
34(C2×D23)⋊C4C2211yescharacteristic
35(C2×D23)⋊C4C2211yescharacteristic
36(C2×D23)⋊C4C2211yescharacteristic
37(C2×D23)⋊C4C2211yescharacteristic
38(C2×D23)⋊C4C2211yescharacteristic
39(C2×D23)⋊C4C2211yescharacteristic
40(C2×D23)⋊C4C2211yescharacteristic, 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
48C4×Dic23C2211yescharacteristic, complemented
49C4×Dic23C2211yescharacteristic, complemented
50C4×Dic23C2211yescharacteristic, complemented
51C23×C23C2311yescharacteristic
52C22×D23—22no—
53C22×D23—22no—
54C22×D23—41no—
55C22×D23C2311yescharacteristic
56C22×D23C2311yescharacteristic
57C22×D23C2311yescharacteristic
58Q8×C23—41no—
59C2×C4×C23—41no—
60C2×C4×C23C2311yescharacteristic
61C2×C4×C23C2311yescharacteristic
62C2×C4×C23C2311yescharacteristic
63C2×C4×C23C2311yescharacteristic
64C2×C4×C23C2311yescharacteristic
65C2×C4×C23C2311yescharacteristic
66C22⋊D23—22no—
67C2×Dic23—22no—
68C2×Dic23C2311yescharacteristic
69C2×Dic23C2311yescharacteristic
70C2×Dic23C2311yescharacteristic
71C2×Dic23C2311yescharacteristic
72C2×Dic23C2311yescharacteristic
73D92—22no—
74D92—41no—
75C4×D23—22no—
76C4×D23—41no—
77C4×D23—41no—
78C22×C23—21no—
79C22×C23—21no—
80C22×C23#4112yes—
81C22×C23C2411yescharacteristic
82C2×D23—21no—
83C2×D23—21no—
84C2×D23—21no—
85C2×D23—21no—
86C2×D23—21no—
87C2×D23—21no—
88C2×D23—21no—
89C2×D23—21no—
90C2×D23—44no—
91C2×D23#4112yes—
92C4×C23—21no—
93C4×C23—21no—
94C4×C23—21no—
95C4×C23—21no—
96C4×C23—21no—
97C4×C23—21no—
98Dic23—21no—
99Dic23—21no—
100Dic23—21no—
101Dic23—21no—
102Dic23—21no—
103#551—231nomaximal, Sylow
104C2×C23—22no—
105C2×C23C2×#4111yescharacteristic
106C2×C23C2×#4111yescharacteristic
107C2×C23H(5,𝔽2)11yescharacteristic
108D23—22no—
109D23—41no—
110D23—41no—
111#126—231no—
112#126—231no—
113#124—231no—
114#123—231no—
115#123—231no—
116#123—231no—
117C22⋊Q8—231no—
118C22⋊Q8D2311yescharacteristic, complemented
119#120—231no—
120C4×D4—231no—
121C4×D4—231no—
122#117—231no—
123#117—231no—
124C2×C22⋊C4—231no—
125C2×C22⋊C4—231no—
126C23#55111yescharacteristic, complemented, Sylow
127C24—231no—
128C2×Q8C2×D2311yescharacteristic, complemented
129C2×D4—231no—
130C2×D4—231no—
131C2×D4—231no—
132C22×C4—231no—
133C22×C4—231no—
134C22×C4—231no—
135C22×C4—231no—
136C22×C4C2×D2311yescharacteristic
137C4⋊C4—231no—
138C4⋊C4—231no—
139C4⋊C4—231no—
140C4⋊C4—231no—
141C4⋊C4—231no—
142C4⋊C4C2×D2311yescharacteristic, complemented
143C4⋊C4C2×D2311yescharacteristic, complemented
144C4⋊C4C2×D2311yescharacteristic, complemented
145C22⋊C4—231no—
146C22⋊C4—231no—
147C22⋊C4—231no—
148C22⋊C4—231no—
149C22⋊C4—231no—
150C22⋊C4—231no—
151C22⋊C4—231no—
152C22⋊C4—231no—
153C22⋊C4—231no—
154C22⋊C4—231no—
155C22⋊C4—231no—
156C22⋊C4—231no—
157C22⋊C4C2×D2311yescharacteristic
158C22⋊C4C2×D2311yescharacteristic
159C42—231no—
160C42—231no—
161C42—231no—
162C23—231no—
163C23—231no—
164C23—231no—
165C23—462no—
166C23—462no—
167C23—921no—
168C23C22×D2311yescharacteristic
169Q8—41no—
170D4—462no—
171D4—462no—
172D4—921no—
173C2×C4—41no—
174C2×C4—231no—
175C2×C4—231no—
176C2×C4—231no—
177C2×C4—231no—
178C2×C4—231no—
179C2×C4—462no—
180C2×C4—462no—
181C2×C4—921no—
182C2×C4—921no—
183C2×C4C22×D2311yescharacteristic
184C2×C4C22×D2311yescharacteristic
185C2×C4C22×D2311yescharacteristic
186C2×C4C22×D2311yescharacteristic
187C2×C4C22×D2311yescharacteristic
188C2×C4C22×D2311yescharacteristic
189C22—21no—
190C22—21no—
191C22—232no—
192C22—461no—
193C22—461no—
194C22—461no—
195C22—461no—
196C22—461no—
197C22—461no—
198C22—461no—
199C22—461no—
200C22—924no—
201C22#6729212yes—
202C22C23×D2311yescharacteristic
203C4—21no—
204C4—21no—
205C4—21no—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—461no—
210C4—461no—
211C4—461no—
212C4—461no—
213C4—461no—
214C2—22no—
215C2—462no—
216C2—921no—
217C2—921no—
218C2#1062515511yescharacteristic
219C2C2×#6729211yescharacteristic
220C2#41×D2311yescharacteristic
subgroupG/H
center, FrattiniC22C23×D23
commutator, socle, cosocleC22×C23C24
FittingC23×C22⋊Q8C2
prime radical#10625053C2
hypercenterC22⋊Q8D23
hypocenterC23#551
LCSG ▷ C22×C23 ▷ C23
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊Q8C2semidirect
2. #10625118C2semidirect
3. #10625117C2semidirect
4. C4×C22⋊D23C2semidirect
5. C2×(C2×D23)⋊C4C2semidirect
6. #10625069C2semidirect
7. #10625069C2semidirect
8. #10625065C2semidirect
9. #10625065C2semidirect
10. #10625064C2semidirect
11. #10625063C2semidirect
12. #10625057C2semidirect
14. C22⋊C4×D23C2semidirect
15. #10625048C2semidirect
17. C2×Q8×C23C22semidirect
25. C2×C4×D23C22semidirect
26. C4⋊C4×C23C22semidirect
27. C4⋊C4×C23C22semidirect
28. C4⋊C4×C23C22semidirect
40. (C2×D23)⋊C4C22semidirect
41. (C2×D23)⋊C4C22semidirect
42. (C2×D23)⋊C4C22semidirect
43. C4⋊Dic23C22semidirect
44. C4⋊Dic23C22semidirect
45. Dic23⋊C4C22semidirect
46. Dic23⋊C4C22semidirect
47. Dic23⋊C4C22semidirect
48. C4×Dic23C22semidirect
49. C4×Dic23C22semidirect
50. C4×Dic23C22semidirect
118. C22⋊Q8D23semidirect
126. C23#551semidirect
128. C2×Q8C2×D23semidirect
142. C4⋊C4C2×D23semidirect
143. C4⋊C4C2×D23semidirect
144. C4⋊C4C2×D23semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊Q82
#106251182
#106251172
C4×C22⋊D232
C2×(C2×D23)⋊C42
#106250692
#106250652
#106250642
#106250632
#106250572
#106250532
C22⋊C4×D232
#106250482
#55123