Finite Group Databasebeta

← #45655877

Subgroups — #45655877

Order 32576.

334 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 105 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×#22550860C211yesmaximal, characteristic, complemented
2C2×D4×D509C211yesmaximal, characteristic, complemented
3#30654201C211yesmaximal, characteristic, complemented
4#30654200C211yesmaximal, characteristic, complemented
5#30654196C211yesmaximal, characteristic, complemented
6#30654193C211yesmaximal, characteristic, complemented
7#30654188C211yesmaximal, characteristic, complemented
8#30654187C211yesmaximal, characteristic, complemented
9#30654183C211yesmaximal, characteristic, complemented
10#30654183C211yesmaximal, characteristic, complemented
11C4×C22⋊D509C211yesmaximal, characteristic, complemented
12#30654175C211yesmaximal, characteristic, complemented
13C2×#22550838C211yesmaximal, characteristic, complemented
14C22⋊C4×D509C211yesmaximal, characteristic, complemented
15#121×C509C211yesmaximal, characteristic, complemented
16C23×D509C2211yescharacteristic
17C23×D509C2211yescharacteristic
18#22550860—24no—
19C2×Dic1018C2211yescharacteristic, complemented
20C2×D2036C2211yescharacteristic
21C2×C22⋊D509C2211yescharacteristic
22C2×C22⋊D509C2211yescharacteristic
23C2×C22⋊D509C2211yescharacteristic
24C2×C22⋊D509C2211yescharacteristic, complemented
25C2×C22⋊D509C2211yescharacteristic, complemented
26D4×D509—24no—
27C22×Dic509C2211yescharacteristic, complemented
28C22×Dic509C2211yescharacteristic, complemented
29C2×C4×D509C2211yescharacteristic
30C2×C4×D509C2211yescharacteristic, complemented
31C2×C4×D509C2211yescharacteristic, complemented
32#22550840C2211yescharacteristic, complemented
33#22550840C2211yescharacteristic, complemented
34#22550840C2211yescharacteristic, complemented
35C22⋊Dic509C2211yescharacteristic, complemented
36C22⋊Dic509C2211yescharacteristic, complemented
37C22⋊Dic509C2211yescharacteristic, complemented
38#22550838C2211yescharacteristic
39#22550838C2211yescharacteristic
40#22550838C2211yescharacteristic
41#22550838C2211yescharacteristic, complemented
42#22550838C2211yescharacteristic, complemented
43#22550838C2211yescharacteristic, complemented
44#22550838C2211yescharacteristic, complemented
45C4×Dic509C2211yescharacteristic, complemented
46C2×D4×C509C2211yescharacteristic
47C2×D4×C509C2211yescharacteristic, complemented
48C2×D4×C509C2211yescharacteristic, complemented
49C22×C4×C509C2211yescharacteristic, complemented
50C4⋊C4×C509C2211yescharacteristic, complemented
51C22⋊C4×C509C2211yescharacteristic, complemented
52C22⋊C4×C509C2211yescharacteristic, complemented
53C22×D509—22no—
54C22×D509—22no—
55C22×D509—22no—
56C22×D509—22no—
57C22×D509—41no—
58C22×D509—41no—
59C22×D509C2311yescharacteristic
60C22×D509C2311yescharacteristic
61C22×D509C2311yescharacteristic
62Dic1018—22no—
63D2036—22no—
64C22⋊D509—22no—
65C22⋊D509—22no—
66C22⋊D509—22no—
67C22⋊D509—41no—
68C22⋊D509D414yescomplemented
69C2×Dic509—22no—
70C2×Dic509—22no—
71C2×Dic509C2311yescharacteristic
72C2×Dic509C2311yescharacteristic
73C2×Dic509C2311yescharacteristic
74C2×Dic509C2311yescharacteristic
75C2×Dic509C2311yescharacteristic
76C4×D509—22no—
77C4×D509—22no—
78C4×D509—41no—
79C23×C509C2311yescharacteristic
80C23×C509C2311yescharacteristic
81C23×C509C2311yescharacteristic
82D4×C509—22no—
83D4×C509—22no—
84D4×C509—41no—
85C2×C4×C509—41no—
86C2×C4×C509C2311yescharacteristic
87C2×C4×C509C2311yescharacteristic
88C2×C4×C509C2311yescharacteristic
89C2×C4×C509C2311yescharacteristic
90C22×C509—21no—
91C22×C509—21no—
92C22×C509—21no—
93C22×C509—21no—
94C22×C509—21no—
95C22×C509—21no—
96C22×C509—21no—
97C22×C509—21no—
98C22×C509C2×D412yes—
99C22×C509C2411yescharacteristic
100C2×D509—21no—
101C2×D509—21no—
102C2×D509—21no—
103C2×D509—21no—
104C2×D509—21no—
105C2×D509—21no—
106C2×D509—21no—
107C2×D509—44no—
108C2×D509—44no—
109C2×D509C2×D412yes—
110C2×D509#4112yescomplemented
111C4×C509—21no—
112C4×C509—21no—
113C4×C509—21no—
114C4×C509—21no—
115Dic509—21no—
116Dic509—21no—
117Dic509—21no—
118Dic509—21no—
119Dic509C2×D412yes—
120C2×C509—22no—
121C2×C509—41no—
122C2×C509—41no—
123C2×C509C22×D411yescharacteristic
124C2×C509C2×#4111yescharacteristic
125C2×C509H(5,𝔽2)11yescharacteristic
126D509—22no—
127D509—22no—
128D509—41no—
129C509#54611yescharacteristic, complemented, Sylow
130#546—5091nomaximal, Sylow
131C2×#41—5091no—
132C22×D4—5091no—
133#124—5091no—
134#123—5091no—
135#123—5091no—
136C22⋊Q8—5091no—
137#121—5091no—
138#121—5091no—
139#121D50911yescharacteristic, complemented
140#120—5091no—
141#120—5091no—
142C4×D4—5091no—
143C4×D4—5091no—
144C2×C22⋊C4—5091no—
145C2×C22⋊C4—5091no—
146C24—5091no—
147C24—5091no—
148#41—10184no—
149C2×Q8—5091no—
150C2×D4—10184no—
151C2×D4—5091no—
152C2×D4—5091no—
153C2×D4—5091no—
154C2×D4—5091no—
155C2×D4—5091no—
156C2×D4—5091no—
157C2×D4C2×D50911yescharacteristic
158C2×D4C2×D50911yescharacteristic, complemented
159C2×D4C2×D50911yescharacteristic, complemented
160C22×C4—5091no—
161C22×C4—5091no—
162C22×C4—5091no—
163C22×C4—5091no—
164C22×C4—5091no—
165C22×C4C2×D50911yescharacteristic, complemented
166C4⋊C4—5091no—
167C4⋊C4—5091no—
168C4⋊C4—5091no—
169C4⋊C4C2×D50911yescharacteristic, complemented
170C22⋊C4—5091no—
171C22⋊C4—5091no—
172C22⋊C4—5091no—
173C22⋊C4—5091no—
174C22⋊C4—5091no—
175C22⋊C4—5091no—
176C22⋊C4—5091no—
177C22⋊C4—5091no—
178C22⋊C4—5091no—
179C22⋊C4—5091no—
180C22⋊C4C2×D50911yescharacteristic, complemented
181C22⋊C4C2×D50911yescharacteristic, complemented
182C42—5091no—
183C23—20361no—
184C23—20361no—
185C23—10182no—
186C23—10182no—
187C23—10182no—
188C23—10182no—
189C23—5091no—
190C23—5091no—
191C23—5091no—
192C23C22×D50911yescharacteristic
193C23C22×D50911yescharacteristic
194C23C22×D50911yescharacteristic
195Q8—10182no—
196D4—22no—
197D4—22no—
198D4—41no—
199D4—20361no—
200D4—10182no—
201D4—10182no—
202D4—10182no—
203D4—10182no—
204D4—5094no—
205C2×C4—41no—
206C2×C4—20361no—
207C2×C4—10182no—
208C2×C4—10182no—
209C2×C4—10182no—
210C2×C4—10182no—
211C2×C4—5091no—
212C2×C4—5091no—
213C2×C4—5091no—
214C2×C4—5091no—
215C2×C4—5091no—
216C2×C4C22×D50911yescharacteristic
217C2×C4C22×D50911yescharacteristic
218C2×C4C22×D50911yescharacteristic
219C2×C4C22×D50911yescharacteristic
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—20364no—
229C22—20364no—
230C22—10181no—
231C22—10181no—
232C22—10181no—
233C22—10181no—
234C22—10181no—
235C22—10181no—
236C22—10181no—
237C22—5092no—
238C22—5092no—
239C22D4×D50912yes—
240C22C23×D50911yescharacteristic
241C4—21no—
242C4—21no—
243C4—21no—
244C4—21no—
245C4—10181no—
246C4—10181no—
247C4—10181no—
248C4—10181no—
249C4—5092no—
250C2—22no—
251C2—41no—
252C2—41no—
253C2—20361no—
254C2—10182no—
255C2—10182no—
256C2C2×D4×D50911yescharacteristic
257C2#41×D50911yescharacteristic
258C2#3065426011yescharacteristic
subgroupG/H
center, FrattiniC22C23×D509
commutator, socle, cosocleC22×C509C24
Fitting#121×C509C2
hypercenter#121D509
hypocenterC509#546
LCSG ▷ C22×C509 ▷ C509
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#22550860C2semidirect
2. C2×D4×D509C2semidirect
3. #30654201C2semidirect
4. #30654200C2semidirect
5. #30654196C2semidirect
6. #30654193C2semidirect
7. #30654188C2semidirect
8. #30654187C2semidirect
9. #30654183C2semidirect
10. #30654183C2semidirect
11. C4×C22⋊D509C2semidirect
12. #30654175C2semidirect
13. C2×#22550838C2semidirect
14. C22⋊C4×D509C2semidirect
15. #121×C509C2semidirect
19. C2×Dic1018C22semidirect
24. C2×C22⋊D509C22semidirect
25. C2×C22⋊D509C22semidirect
27. C22×Dic509C22semidirect
28. C22×Dic509C22semidirect
30. C2×C4×D509C22semidirect
31. C2×C4×D509C22semidirect
32. #22550840C22semidirect
33. #22550840C22semidirect
34. #22550840C22semidirect
35. C22⋊Dic509C22semidirect
36. C22⋊Dic509C22semidirect
37. C22⋊Dic509C22semidirect
41. #22550838C22semidirect
42. #22550838C22semidirect
43. #22550838C22semidirect
44. #22550838C22semidirect
45. C4×Dic509C22semidirect
47. C2×D4×C509C22semidirect
48. C2×D4×C509C22semidirect
49. C22×C4×C509C22semidirect
50. C4⋊C4×C509C22semidirect
51. C22⋊C4×C509C22semidirect
52. C22⋊C4×C509C22semidirect
68. C22⋊D509D4semidirect
110. C2×D509#41semidirect
129. C509#546semidirect
139. #121D509semidirect
158. C2×D4C2×D509semidirect
159. C2×D4C2×D509semidirect
165. C22×C4C2×D509semidirect
169. C4⋊C4C2×D509semidirect
180. C22⋊C4C2×D509semidirect
181. C22⋊C4C2×D509semidirect

Maximal subgroups

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

subgroupindex
C2×#225508602
C2×D4×D5092
#306542012
#306542002
#306541962
#306541932
#306541882
#306541872
#306541832
C4×C22⋊D5092
#306541752
C2×#225508382
C22⋊C4×D5092
#121×C5092
#546509