Finite Group Databasebeta

← #14581013

Subgroups — #14581013

Order 2624.

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
1#13050159C211yesmaximal, characteristic, complemented
2#13050159C211yesmaximal, characteristic, complemented
3#13050153C211yesmaximal, characteristic, complemented
4#13050149C211yesmaximal, characteristic, complemented
5#13050149C211yesmaximal, characteristic, complemented
6#13050148C211yesmaximal, characteristic, complemented
7#13050140C211yesmaximal, characteristic, complemented
8#13050134C211yesmaximal, characteristic
9#13050126C211yesmaximal, characteristic, complemented
10C4×C22⋊D41C211yesmaximal, characteristic, complemented
11#13050122C211yesmaximal, characteristic, complemented
12#13050120C211yesmaximal, characteristic, complemented
13C2×#10620281C211yesmaximal, characteristic, complemented
14C22⋊C4×D41C211yesmaximal, characteristic, complemented
15C22⋊Q8×C41C211yesmaximal, characteristic, complemented
16C23×D41C2211yescharacteristic
17C2×Q8×C41C2211yescharacteristic, complemented
18C22×C4×C41C2211yescharacteristic
19C2×C22⋊D41C2211yescharacteristic
20C22×Dic41C2211yescharacteristic
21C2×D164C2211yescharacteristic
22C2×D164C2211yescharacteristic
23C2×C4×D41C2211yescharacteristic
24C2×C4×D41C2211yescharacteristic
25C2×C4×D41C2211yescharacteristic, complemented
26C4⋊C4×C41C2211yescharacteristic, complemented
27C4⋊C4×C41C2211yescharacteristic, complemented
28C4⋊C4×C41C2211yescharacteristic, complemented
29C22⋊C4×C41C2211yescharacteristic
30C22⋊C4×C41C2211yescharacteristic
31C22⋊Dic41C2211yescharacteristic
32#10620281C2211yescharacteristic
33#10620281C2211yescharacteristic
34#10620281C2211yescharacteristic
35#10620281C2211yescharacteristic
36#10620281C2211yescharacteristic
37#10620281C2211yescharacteristic
38#10620281C2211yescharacteristic
39#10620281C2211yescharacteristic
40#10620281C2211yescharacteristic, complemented
41#10620281C2211yescharacteristic, complemented
42#10620281C2211yescharacteristic, complemented
43C4⋊Dic41C2211yescharacteristic, complemented
44C4⋊Dic41C2211yescharacteristic, complemented
45#10620279C2211yescharacteristic, complemented
46#10620279C2211yescharacteristic, complemented
47#10620279C2211yescharacteristic, complemented
48C4×Dic41C2211yescharacteristic, complemented
49C4×Dic41C2211yescharacteristic, complemented
50C4×Dic41C2211yescharacteristic, complemented
51C23×C41C2311yescharacteristic
52C22×D41—22no—
53C22×D41—22no—
54C22×D41—41no—
55C22×D41C2311yescharacteristic
56C22×D41C2311yescharacteristic
57C22×D41C2311yescharacteristic
58Q8×C41—41no—
59C2×C4×C41—41no—
60C2×C4×C41C2311yescharacteristic
61C2×C4×C41C2311yescharacteristic
62C2×C4×C41C2311yescharacteristic
63C2×C4×C41C2311yescharacteristic
64C2×C4×C41C2311yescharacteristic
65C2×C4×C41C2311yescharacteristic
66C22⋊D41—22no—
67C2×Dic41—22no—
68C2×Dic41C2311yescharacteristic
69C2×Dic41C2311yescharacteristic
70C2×Dic41C2311yescharacteristic
71C2×Dic41C2311yescharacteristic
72C2×Dic41C2311yescharacteristic
73D164—22no—
74D164—41no—
75C4×D41—22no—
76C4×D41—41no—
77C4×D41—41no—
78C22×C41—21no—
79C22×C41—21no—
80C22×C41#4112yes—
81C22×C41C2411yescharacteristic
82C2×D41—21no—
83C2×D41—21no—
84C2×D41—21no—
85C2×D41—21no—
86C2×D41—21no—
87C2×D41—21no—
88C2×D41—21no—
89C2×D41—21no—
90C2×D41—44no—
91C2×D41#4112yes—
92C4×C41—21no—
93C4×C41—21no—
94C4×C41—21no—
95C4×C41—21no—
96C4×C41—21no—
97C4×C41—21no—
98Dic41—21no—
99Dic41—21no—
100Dic41—21no—
101Dic41—21no—
102Dic41—21no—
103C2×C41—22no—
104C2×C41C2×#4111yescharacteristic
105C2×C41C2×#4111yescharacteristic
106C2×C41H(5,𝔽2)11yescharacteristic
107D41—22no—
108D41—41no—
109D41—41no—
110#551—411nomaximal, Sylow
111C41#55111yescharacteristic, complemented, Sylow
112#126—411no—
113#126—411no—
114#124—411no—
115#123—411no—
116#123—411no—
117#123—411no—
118C22⋊Q8—411no—
119C22⋊Q8D4111yescharacteristic, complemented
120#120—411no—
121C4×D4—411no—
122C4×D4—411no—
123#117—411no—
124#117—411no—
125C2×C22⋊C4—411no—
126C2×C22⋊C4—411no—
127C24—411no—
128C2×Q8C2×D4111yescharacteristic, complemented
129C2×D4—411no—
130C2×D4—411no—
131C2×D4—411no—
132C22×C4—411no—
133C22×C4—411no—
134C22×C4—411no—
135C22×C4—411no—
136C22×C4C2×D4111yescharacteristic
137C4⋊C4—411no—
138C4⋊C4—411no—
139C4⋊C4—411no—
140C4⋊C4—411no—
141C4⋊C4—411no—
142C4⋊C4C2×D4111yescharacteristic, complemented
143C4⋊C4C2×D4111yescharacteristic, complemented
144C4⋊C4C2×D4111yescharacteristic, complemented
145C22⋊C4—411no—
146C22⋊C4—411no—
147C22⋊C4—411no—
148C22⋊C4—411no—
149C22⋊C4—411no—
150C22⋊C4—411no—
151C22⋊C4—411no—
152C22⋊C4—411no—
153C22⋊C4—411no—
154C22⋊C4—411no—
155C22⋊C4—411no—
156C22⋊C4—411no—
157C22⋊C4C2×D4111yescharacteristic
158C22⋊C4C2×D4111yescharacteristic
159C42—411no—
160C42—411no—
161C42—411no—
162C23—411no—
163C23—411no—
164C23—411no—
165C23—822no—
166C23—822no—
167C23—1641no—
168C23C22×D4111yescharacteristic
169Q8—41no—
170D4—822no—
171D4—822no—
172D4—1641no—
173C2×C4—41no—
174C2×C4—411no—
175C2×C4—411no—
176C2×C4—411no—
177C2×C4—411no—
178C2×C4—411no—
179C2×C4—822no—
180C2×C4—822no—
181C2×C4—1641no—
182C2×C4—1641no—
183C2×C4C22×D4111yescharacteristic
184C2×C4C22×D4111yescharacteristic
185C2×C4C22×D4111yescharacteristic
186C2×C4C22×D4111yescharacteristic
187C2×C4C22×D4111yescharacteristic
188C2×C4C22×D4111yescharacteristic
189C22—21no—
190C22—21no—
191C22—412no—
192C22—821no—
193C22—821no—
194C22—821no—
195C22—821no—
196C22—821no—
197C22—821no—
198C22—821no—
199C22—821no—
200C22—1644no—
201C22#1062031012yes—
202C22C23×D4111yescharacteristic
203C4—21no—
204C4—21no—
205C4—21no—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—821no—
210C4—821no—
211C4—821no—
212C4—821no—
213C4—821no—
214C2—22no—
215C2—822no—
216C2—1641no—
217C2—1641no—
218C2#41×D4111yescharacteristic
219C2C2×#1062031011yescharacteristic
220C2#1305020911yescharacteristic
subgroupG/H
center, FrattiniC22C23×D41
commutator, socle, cosocleC22×C41C24
FittingC22⋊Q8×C41C2
prime radical#13050134C2
hypercenterC22⋊Q8D41
hypocenterC41#551
LCSG ▷ C22×C41 ▷ C41
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #13050159C2semidirect
2. #13050159C2semidirect
3. #13050153C2semidirect
4. #13050149C2semidirect
5. #13050149C2semidirect
6. #13050148C2semidirect
7. #13050140C2semidirect
9. #13050126C2semidirect
10. C4×C22⋊D41C2semidirect
11. #13050122C2semidirect
12. #13050120C2semidirect
13. C2×#10620281C2semidirect
14. C22⋊C4×D41C2semidirect
15. C22⋊Q8×C41C2semidirect
17. C2×Q8×C41C22semidirect
25. C2×C4×D41C22semidirect
26. C4⋊C4×C41C22semidirect
27. C4⋊C4×C41C22semidirect
28. C4⋊C4×C41C22semidirect
40. #10620281C22semidirect
41. #10620281C22semidirect
42. #10620281C22semidirect
43. C4⋊Dic41C22semidirect
44. C4⋊Dic41C22semidirect
45. #10620279C22semidirect
46. #10620279C22semidirect
47. #10620279C22semidirect
48. C4×Dic41C22semidirect
49. C4×Dic41C22semidirect
50. C4×Dic41C22semidirect
111. C41#551semidirect
119. C22⋊Q8D41semidirect
128. C2×Q8C2×D41semidirect
142. C4⋊C4C2×D41semidirect
143. C4⋊C4C2×D41semidirect
144. C4⋊C4C2×D41semidirect

Maximal subgroups

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

subgroupindex
#130501592
#130501532
#130501492
#130501482
#130501402
#130501342
#130501262
C4×C22⋊D412
#130501222
#130501202
C2×#106202812
C22⋊C4×D412
C22⋊Q8×C412
#55141