Finite Group Databasebeta

← #16840402

Subgroups — #16840402

Order 3904.

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×#11758222C211yesmaximal, characteristic, complemented
2C22×C22⋊D61C211yesmaximal, characteristic, complemented
3#14519763C211yesmaximal, characteristic, complemented
4#14519760C211yesmaximal, characteristic, complemented
5#14519759C211yesmaximal, characteristic, complemented
6#14519754C211yesmaximal, characteristic, complemented
7#14519751C211yesmaximal, characteristic, complemented
8#14519750C211yesmaximal, characteristic, complemented
9#14519750C211yesmaximal, characteristic, complemented
10#14519747C211yesmaximal, characteristic, complemented
11#14519746C211yesmaximal, characteristic, complemented
12C4×C22⋊D61C211yesmaximal, characteristic, complemented
13#14519737C211yesmaximal, characteristic, complemented
14C22⋊C4×D61C211yesmaximal, characteristic, complemented
15C2×C22⋊C4×C61C211yesmaximal, characteristic, complemented
16C24×C61C2211yescharacteristic
17C23×D61C2211yescharacteristic
18C22×C4×C61C2211yescharacteristic, complemented
19C22×C4×C61C2211yescharacteristic, complemented
20C2×C22⋊D61—24no—
21C2×C22⋊D61C2211yescharacteristic
22C2×C22⋊D61C2211yescharacteristic
23C2×C22⋊D61C2211yescharacteristic
24C2×C22⋊D61C2211yescharacteristic
25C2×C22⋊D61C2211yescharacteristic
26C2×C22⋊D61C2211yescharacteristic, complemented
27C2×C22⋊D61C2211yescharacteristic, complemented
28C22×Dic61C2211yescharacteristic
29#11758222—24no—
30C2×D244C2211yescharacteristic, complemented
31C2×D244C2211yescharacteristic, complemented
32C2×C4×D61C2211yescharacteristic, complemented
33C2×C4×D61C2211yescharacteristic, complemented
34C2×C4×D61C2211yescharacteristic, complemented
35C2×Dic122C2211yescharacteristic, complemented
36C22⋊C4×C61C2211yescharacteristic
37C22⋊C4×C61C2211yescharacteristic
38C22⋊C4×C61C2211yescharacteristic, complemented
39C22⋊C4×C61C2211yescharacteristic, complemented
40C22⋊Dic61C2211yescharacteristic
41C22⋊Dic61C2211yescharacteristic, complemented
42C22⋊Dic61C2211yescharacteristic, complemented
43#11758198C2211yescharacteristic, complemented
44#11758198C2211yescharacteristic, complemented
45#11758198C2211yescharacteristic, complemented
46#11758198C2211yescharacteristic, complemented
47#11758198C2211yescharacteristic, complemented
48C4⋊Dic61C2211yescharacteristic, complemented
49#11758196C2211yescharacteristic, complemented
50#11758196C2211yescharacteristic, complemented
51#11758196C2211yescharacteristic, complemented
52C4×Dic61C2211yescharacteristic, complemented
53C23×C61—22no—
54C23×C61—22no—
55C23×C61—41no—
56C23×C61C2311yescharacteristic
57C23×C61C2311yescharacteristic
58C23×C61C2311yescharacteristic
59C22×D61—22no—
60C22×D61—22no—
61C22×D61—41no—
62C22×D61C2311yescharacteristic
63C22×D61C2311yescharacteristic
64C22×D61C2311yescharacteristic
65C2×C4×C61—22no—
66C2×C4×C61—41no—
67C2×C4×C61C2311yescharacteristic
68C2×C4×C61C2311yescharacteristic
69C2×C4×C61C2311yescharacteristic
70C2×C4×C61C2311yescharacteristic
71C22⋊D61—22no—
72C22⋊D61—22no—
73C22⋊D61—22no—
74C22⋊D61—22no—
75C22⋊D61—22no—
76C22⋊D61—41no—
77C22⋊D61D414yescomplemented
78C2×Dic61—22no—
79C2×Dic61C2311yescharacteristic
80C2×Dic61C2311yescharacteristic
81C2×Dic61C2311yescharacteristic
82C2×Dic61C2311yescharacteristic
83C2×Dic61C2311yescharacteristic
84D244—22no—
85D244—41no—
86C4×D61—22no—
87C4×D61—22no—
88C4×D61—41no—
89Dic122—22no—
90C22×C61—21no—
91C22×C61—21no—
92C22×C61—21no—
93C22×C61—21no—
94C22×C61—21no—
95C22×C61—21no—
96C22×C61—21no—
97C22×C61—44no—
98C22×C61C2×D412yes—
99C22×C61#4112yescomplemented
100C22×C61C2411yescharacteristic
101C2×D61—21no—
102C2×D61—21no—
103C2×D61—21no—
104C2×D61—21no—
105C2×D61—21no—
106C2×D61—21no—
107C2×D61—21no—
108C2×D61—21no—
109C2×D61—44no—
110C2×D61C2×D412yes—
111C4×C61—21no—
112C4×C61—21no—
113C4×C61—21no—
114C4×C61—21no—
115Dic61—21no—
116Dic61—21no—
117Dic61—21no—
118Dic61—21no—
119Dic61C2×D412yes—
120C2×C61—22no—
121C2×C61—22no—
122C2×C61—41no—
123C2×C61C22×D411yescharacteristic
124C2×C61C2×#4111yescharacteristic
125C2×C61H(5,𝔽2)11yescharacteristic
126D61—22no—
127D61—41no—
128D61—41no—
129#546—611nomaximal, Sylow
130C61#54611yescharacteristic, complemented, Sylow
131C2×#41—611no—
132C22×D4—611no—
133#124—611no—
134#123—611no—
135#123—611no—
136C22⋊Q8—611no—
137#121—611no—
138#121—611no—
139#121—611no—
140#120—611no—
141#120—611no—
142C4×D4—611no—
143C4×D4—611no—
144C2×C22⋊C4—611no—
145C2×C22⋊C4D6111yescharacteristic, complemented
146C24—611no—
147C24C2×D6111yescharacteristic
148#41—1224no—
149C2×Q8—611no—
150C2×D4—611no—
151C2×D4—611no—
152C2×D4—611no—
153C2×D4—611no—
154C2×D4—611no—
155C2×D4—611no—
156C2×D4—611no—
157C2×D4—611no—
158C2×D4—611no—
159C2×D4—1224no—
160C22×C4—611no—
161C22×C4—611no—
162C22×C4—611no—
163C22×C4—611no—
164C22×C4C2×D6111yescharacteristic, complemented
165C22×C4C2×D6111yescharacteristic, complemented
166C4⋊C4—611no—
167C4⋊C4—611no—
168C4⋊C4—611no—
169C4⋊C4—611no—
170C22⋊C4—611no—
171C22⋊C4—611no—
172C22⋊C4—611no—
173C22⋊C4—611no—
174C22⋊C4—611no—
175C22⋊C4—611no—
176C22⋊C4—611no—
177C22⋊C4—611no—
178C22⋊C4C2×D6111yescharacteristic
179C22⋊C4C2×D6111yescharacteristic
180C22⋊C4C2×D6111yescharacteristic, complemented
181C22⋊C4C2×D6111yescharacteristic, complemented
182C42—611no—
183C23—22no—
184C23—22no—
185C23—41no—
186C23—611no—
187C23—611no—
188C23—611no—
189C23—1222no—
190C23—1222no—
191C23—2441no—
192C23C22×D6111yescharacteristic
193C23C22×D6111yescharacteristic
194C23C22×D6111yescharacteristic
195Q8—1222no—
196D4—614no—
197D4—1222no—
198D4—1222no—
199D4—1222no—
200D4—1222no—
201D4—1222no—
202D4—1222no—
203D4—2441no—
204D4—2441no—
205C2×C4—22no—
206C2×C4—41no—
207C2×C4—611no—
208C2×C4—611no—
209C2×C4—611no—
210C2×C4—611no—
211C2×C4—611no—
212C2×C4—1222no—
213C2×C4—1222no—
214C2×C4—1222no—
215C2×C4—2441no—
216C2×C4C22×D6111yescharacteristic
217C2×C4C22×D6111yescharacteristic
218C2×C4C22×D6111yescharacteristic
219C2×C4C22×D6111yescharacteristic
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—44no—
228C22—612no—
229C22—1221no—
230C22—1221no—
231C22—1221no—
232C22—1221no—
233C22—1221no—
234C22—1221no—
235C22—1221no—
236C22—1221no—
237C22—2444no—
238C22#1175822212yescomplemented
239C22D4×D6112yes—
240C22C23×D6111yescharacteristic
241C4—21no—
242C4—21no—
243C4—21no—
244C4—21no—
245C4—612no—
246C4—1221no—
247C4—1221no—
248C4—1221no—
249C4—1221no—
250C2—22no—
251C2—22no—
252C2—41no—
253C2—1222no—
254C2—2441no—
255C2—2441no—
256C2C2×D4×D6111yescharacteristic
257C2C2×#1175822211yescharacteristic
258C2#1451982211yescharacteristic
subgroupG/H
center, FrattiniC22C23×D61
commutator, socle, cosocleC22×C61C24
FittingC2×C22⋊C4×C61C2
hypercenterC2×C22⋊C4D61
hypocenterC61#546
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#11758222C2semidirect
2. C22×C22⋊D61C2semidirect
3. #14519763C2semidirect
4. #14519760C2semidirect
5. #14519759C2semidirect
6. #14519754C2semidirect
7. #14519751C2semidirect
8. #14519750C2semidirect
9. #14519750C2semidirect
10. #14519747C2semidirect
11. #14519746C2semidirect
12. C4×C22⋊D61C2semidirect
13. #14519737C2semidirect
14. C22⋊C4×D61C2semidirect
15. C2×C22⋊C4×C61C2semidirect
18. C22×C4×C61C22semidirect
19. C22×C4×C61C22semidirect
26. C2×C22⋊D61C22semidirect
27. C2×C22⋊D61C22semidirect
30. C2×D244C22semidirect
31. C2×D244C22semidirect
32. C2×C4×D61C22semidirect
33. C2×C4×D61C22semidirect
34. C2×C4×D61C22semidirect
35. C2×Dic122C22semidirect
38. C22⋊C4×C61C22semidirect
39. C22⋊C4×C61C22semidirect
41. C22⋊Dic61C22semidirect
42. C22⋊Dic61C22semidirect
43. #11758198C22semidirect
44. #11758198C22semidirect
45. #11758198C22semidirect
46. #11758198C22semidirect
47. #11758198C22semidirect
48. C4⋊Dic61C22semidirect
49. #11758196C22semidirect
50. #11758196C22semidirect
51. #11758196C22semidirect
52. C4×Dic61C22semidirect
77. C22⋊D61D4semidirect
99. C22×C61#41semidirect
130. C61#546semidirect
145. C2×C22⋊C4D61semidirect
164. C22×C4C2×D61semidirect
165. C22×C4C2×D61semidirect
180. C22⋊C4C2×D61semidirect
181. C22⋊C4C2×D61semidirect
238. C22#11758222semidirect

Maximal subgroups

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

subgroupindex
C2×#117582222
C22×C22⋊D612
#145197632
#145197602
#145197592
#145197542
#145197512
#145197502
#145197472
#145197462
C4×C22⋊D612
#145197372
C22⋊C4×D612
C2×C22⋊C4×C612
#54661