Finite Group Databasebeta

← #14580976

Subgroups — #14580976

Order 2624.

334 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C2×#10620306C211yesmaximal, characteristic, complemented
2C22×C22⋊D41C211yesmaximal, characteristic, complemented
3#13050150C211yesmaximal, characteristic, complemented
4#13050147C211yesmaximal, characteristic, complemented
5#13050146C211yesmaximal, characteristic, complemented
6#13050141C211yesmaximal, characteristic, complemented
7#13050138C211yesmaximal, characteristic, complemented
8#13050137C211yesmaximal, characteristic, complemented
9#13050137C211yesmaximal, characteristic, complemented
10#13050134C211yesmaximal, characteristic, complemented
11#13050133C211yesmaximal, characteristic, complemented
12C4×C22⋊D41C211yesmaximal, characteristic, complemented
13#13050124C211yesmaximal, characteristic, complemented
14C22⋊C4×D41C211yesmaximal, characteristic, complemented
15C2×C22⋊C4×C41C211yesmaximal, characteristic, complemented
16C24×C41C2211yescharacteristic
17C23×D41C2211yescharacteristic
18C22×C4×C41C2211yescharacteristic, complemented
19C22×C4×C41C2211yescharacteristic, complemented
20C2×C22⋊D41—24no—
21C2×C22⋊D41C2211yescharacteristic
22C2×C22⋊D41C2211yescharacteristic
23C2×C22⋊D41C2211yescharacteristic
24C2×C22⋊D41C2211yescharacteristic
25C2×C22⋊D41C2211yescharacteristic
26C2×C22⋊D41C2211yescharacteristic, complemented
27C2×C22⋊D41C2211yescharacteristic, complemented
28C22×Dic41C2211yescharacteristic
29#10620306—24no—
30C2×D164C2211yescharacteristic, complemented
31C2×D164C2211yescharacteristic, complemented
32C2×C4×D41C2211yescharacteristic, complemented
33C2×C4×D41C2211yescharacteristic, complemented
34C2×C4×D41C2211yescharacteristic, complemented
35C2×Dic82C2211yescharacteristic, complemented
36C22⋊C4×C41C2211yescharacteristic
37C22⋊C4×C41C2211yescharacteristic
38C22⋊C4×C41C2211yescharacteristic, complemented
39C22⋊C4×C41C2211yescharacteristic, complemented
40C22⋊Dic41C2211yescharacteristic
41C22⋊Dic41C2211yescharacteristic, complemented
42C22⋊Dic41C2211yescharacteristic, complemented
43#10620281C2211yescharacteristic, complemented
44#10620281C2211yescharacteristic, complemented
45#10620281C2211yescharacteristic, complemented
46#10620281C2211yescharacteristic, complemented
47#10620281C2211yescharacteristic, complemented
48C4⋊Dic41C2211yescharacteristic, complemented
49#10620279C2211yescharacteristic, complemented
50#10620279C2211yescharacteristic, complemented
51#10620279C2211yescharacteristic, complemented
52C4×Dic41C2211yescharacteristic, complemented
53C23×C41—22no—
54C23×C41—22no—
55C23×C41—41no—
56C23×C41C2311yescharacteristic
57C23×C41C2311yescharacteristic
58C23×C41C2311yescharacteristic
59C22×D41—22no—
60C22×D41—22no—
61C22×D41—41no—
62C22×D41C2311yescharacteristic
63C22×D41C2311yescharacteristic
64C22×D41C2311yescharacteristic
65C2×C4×C41—22no—
66C2×C4×C41—41no—
67C2×C4×C41C2311yescharacteristic
68C2×C4×C41C2311yescharacteristic
69C2×C4×C41C2311yescharacteristic
70C2×C4×C41C2311yescharacteristic
71C22⋊D41—22no—
72C22⋊D41—22no—
73C22⋊D41—22no—
74C22⋊D41—22no—
75C22⋊D41—22no—
76C22⋊D41—41no—
77C22⋊D41D414yescomplemented
78C2×Dic41—22no—
79C2×Dic41C2311yescharacteristic
80C2×Dic41C2311yescharacteristic
81C2×Dic41C2311yescharacteristic
82C2×Dic41C2311yescharacteristic
83C2×Dic41C2311yescharacteristic
84D164—22no—
85D164—41no—
86C4×D41—22no—
87C4×D41—22no—
88C4×D41—41no—
89Dic82—22no—
90C22×C41—21no—
91C22×C41—21no—
92C22×C41—21no—
93C22×C41—21no—
94C22×C41—21no—
95C22×C41—21no—
96C22×C41—21no—
97C22×C41—44no—
98C22×C41C2×D412yes—
99C22×C41#4112yescomplemented
100C22×C41C2411yescharacteristic
101C2×D41—21no—
102C2×D41—21no—
103C2×D41—21no—
104C2×D41—21no—
105C2×D41—21no—
106C2×D41—21no—
107C2×D41—21no—
108C2×D41—21no—
109C2×D41—44no—
110C2×D41C2×D412yes—
111C4×C41—21no—
112C4×C41—21no—
113C4×C41—21no—
114C4×C41—21no—
115Dic41—21no—
116Dic41—21no—
117Dic41—21no—
118Dic41—21no—
119Dic41C2×D412yes—
120C2×C41—22no—
121C2×C41—22no—
122C2×C41—41no—
123C2×C41C22×D411yescharacteristic
124C2×C41C2×#4111yescharacteristic
125C2×C41H(5,𝔽2)11yescharacteristic
126D41—22no—
127D41—41no—
128D41—41no—
129#546—411nomaximal, Sylow
130C41#54611yescharacteristic, complemented, Sylow
131C2×#41—411no—
132C22×D4—411no—
133#124—411no—
134#123—411no—
135#123—411no—
136C22⋊Q8—411no—
137#121—411no—
138#121—411no—
139#121—411no—
140#120—411no—
141#120—411no—
142C4×D4—411no—
143C4×D4—411no—
144C2×C22⋊C4—411no—
145C2×C22⋊C4D4111yescharacteristic, complemented
146C24—411no—
147C24C2×D4111yescharacteristic
148#41—824no—
149C2×Q8—411no—
150C2×D4—411no—
151C2×D4—411no—
152C2×D4—411no—
153C2×D4—411no—
154C2×D4—411no—
155C2×D4—411no—
156C2×D4—411no—
157C2×D4—411no—
158C2×D4—411no—
159C2×D4—824no—
160C22×C4—411no—
161C22×C4—411no—
162C22×C4—411no—
163C22×C4—411no—
164C22×C4C2×D4111yescharacteristic, complemented
165C22×C4C2×D4111yescharacteristic, complemented
166C4⋊C4—411no—
167C4⋊C4—411no—
168C4⋊C4—411no—
169C4⋊C4—411no—
170C22⋊C4—411no—
171C22⋊C4—411no—
172C22⋊C4—411no—
173C22⋊C4—411no—
174C22⋊C4—411no—
175C22⋊C4—411no—
176C22⋊C4—411no—
177C22⋊C4—411no—
178C22⋊C4C2×D4111yescharacteristic
179C22⋊C4C2×D4111yescharacteristic
180C22⋊C4C2×D4111yescharacteristic, complemented
181C22⋊C4C2×D4111yescharacteristic, complemented
182C42—411no—
183C23—22no—
184C23—22no—
185C23—41no—
186C23—411no—
187C23—411no—
188C23—411no—
189C23—822no—
190C23—822no—
191C23—1641no—
192C23C22×D4111yescharacteristic
193C23C22×D4111yescharacteristic
194C23C22×D4111yescharacteristic
195Q8—822no—
196D4—414no—
197D4—822no—
198D4—822no—
199D4—822no—
200D4—822no—
201D4—822no—
202D4—822no—
203D4—1641no—
204D4—1641no—
205C2×C4—22no—
206C2×C4—41no—
207C2×C4—411no—
208C2×C4—411no—
209C2×C4—411no—
210C2×C4—411no—
211C2×C4—411no—
212C2×C4—822no—
213C2×C4—822no—
214C2×C4—822no—
215C2×C4—1641no—
216C2×C4C22×D4111yescharacteristic
217C2×C4C22×D4111yescharacteristic
218C2×C4C22×D4111yescharacteristic
219C2×C4C22×D4111yescharacteristic
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—44no—
228C22—412no—
229C22—821no—
230C22—821no—
231C22—821no—
232C22—821no—
233C22—821no—
234C22—821no—
235C22—821no—
236C22—821no—
237C22—1644no—
238C22#1062030612yescomplemented
239C22D4×D4112yes—
240C22C23×D4111yescharacteristic
241C4—21no—
242C4—21no—
243C4—21no—
244C4—21no—
245C4—412no—
246C4—821no—
247C4—821no—
248C4—821no—
249C4—821no—
250C2—22no—
251C2—22no—
252C2—41no—
253C2—822no—
254C2—1641no—
255C2—1641no—
256C2C2×D4×D4111yescharacteristic
257C2C2×#1062030611yescharacteristic
258C2#1305020911yescharacteristic
subgroupG/H
center, FrattiniC22C23×D41
commutator, socle, cosocleC22×C41C24
FittingC2×C22⋊C4×C41C2
hypercenterC2×C22⋊C4D41
hypocenterC41#546
LCSG ▷ C22×C41 ▷ C41
UCSC1 ◁ C22 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#10620306C2semidirect
2. C22×C22⋊D41C2semidirect
3. #13050150C2semidirect
4. #13050147C2semidirect
5. #13050146C2semidirect
6. #13050141C2semidirect
7. #13050138C2semidirect
8. #13050137C2semidirect
9. #13050137C2semidirect
10. #13050134C2semidirect
11. #13050133C2semidirect
12. C4×C22⋊D41C2semidirect
13. #13050124C2semidirect
14. C22⋊C4×D41C2semidirect
15. C2×C22⋊C4×C41C2semidirect
18. C22×C4×C41C22semidirect
19. C22×C4×C41C22semidirect
26. C2×C22⋊D41C22semidirect
27. C2×C22⋊D41C22semidirect
30. C2×D164C22semidirect
31. C2×D164C22semidirect
32. C2×C4×D41C22semidirect
33. C2×C4×D41C22semidirect
34. C2×C4×D41C22semidirect
35. C2×Dic82C22semidirect
38. C22⋊C4×C41C22semidirect
39. C22⋊C4×C41C22semidirect
41. C22⋊Dic41C22semidirect
42. C22⋊Dic41C22semidirect
43. #10620281C22semidirect
44. #10620281C22semidirect
45. #10620281C22semidirect
46. #10620281C22semidirect
47. #10620281C22semidirect
48. C4⋊Dic41C22semidirect
49. #10620279C22semidirect
50. #10620279C22semidirect
51. #10620279C22semidirect
52. C4×Dic41C22semidirect
77. C22⋊D41D4semidirect
99. C22×C41#41semidirect
130. C41#546semidirect
145. C2×C22⋊C4D41semidirect
164. C22×C4C2×D41semidirect
165. C22×C4C2×D41semidirect
180. C22⋊C4C2×D41semidirect
181. C22⋊C4C2×D41semidirect
238. C22#10620306semidirect

Maximal subgroups

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

subgroupindex
C2×#106203062
C22×C22⋊D412
#130501502
#130501472
#130501462
#130501412
#130501382
#130501372
#130501342
#130501332
C4×C22⋊D412
#130501242
C22⋊C4×D412
C2×C22⋊C4×C412
#54641