Finite Group Databasebeta

← #45655748

Subgroups — #45655748

Order 32576.

330 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 109 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
2C22×C4×D509C211yesmaximal, characteristic, complemented
3#30654197C212yesmaximal, complemented
4#30654195C211yesmaximal, characteristic, complemented
5#30654194C211yesmaximal, characteristic, complemented
6#30654186C211yesmaximal, characteristic, complemented
7#30654183C212yesmaximal, complemented
8D4×Dic509C212yesmaximal, complemented
9#30654175C212yesmaximal, complemented
10#30654173C211yesmaximal, characteristic, complemented
11#121×C509C211yesmaximal, characteristic, complemented
12C23×D509C2211yescharacteristic
13#22550860—24no—
14C2×Dic1018C2211yescharacteristic, complemented
15C2×C22⋊D509C2212yes—
16C2×C22⋊D509C2212yes—
17C22×Dic509C2211yescharacteristic
18C22×Dic509C2212yescomplemented
19C2×C4×D509—24no—
20C2×C4×D509C2211yescharacteristic
21C2×C4×D509C2211yescharacteristic
22C2×C4×D509C2211yescharacteristic
23C2×C4×D509C2211yescharacteristic
24C4⋊Dic509C2211yescharacteristic, complemented
25C4⋊Dic509C2212yescomplemented
26#22550840C2212yescomplemented
27C22⋊Dic509C2212yes—
28C22⋊Dic509C2212yescomplemented
29#22550838C2212yes—
30#22550838C2212yes—
31C4×Dic509C2212yescomplemented
32C2×D4×C509C2212yes—
33C2×D4×C509C2211yescharacteristic, complemented
34C22×C4×C509C2211yescharacteristic
35C4⋊C4×C509C2211yescharacteristic, complemented
36C22⋊C4×C509C2212yescomplemented
37C22×D509—22no—
38C22×D509—22no—
39C22×D509—22no—
40C22×D509C2311yescharacteristic
41C22×D509C2311yescharacteristic
42Dic1018—22no—
43C22⋊D509—24no—
44C22⋊D509—24no—
45C2×Dic509—22no—
46C2×Dic509—24no—
47C2×Dic509C2312yes—
48C2×Dic509C2312yes—
49C2×Dic509C2311yescharacteristic
50C2×Dic509C2311yescharacteristic
51C4×D509—22no—
52C4×D509—22no—
53C4×D509—22no—
54C4×D509D414yescomplemented
55C23×C509C2312yes—
56C23×C509C2311yescharacteristic
57D4×C509—24no—
58D4×C509—22no—
59C2×C4×C509—22no—
60C2×C4×C509C2311yescharacteristic
61C2×C4×C509C2312yes—
62C2×C4×C509C2311yescharacteristic
63C22×C509—21no—
64C22×C509—21no—
65C22×C509—22no—
66C22×C509—22no—
67C22×C509—22no—
68C22×C509#4112yescomplemented
69C22×C509C2411yescharacteristic
70C2×D509—28no—
71C2×D509—21no—
72C2×D509—21no—
73C2×D509—21no—
74C2×D509—21no—
75C2×D509C2×D412yes—
76C2×D509#4112yescomplemented
77C4×C509—21no—
78C4×C509—22no—
79C4×C509C2×D412yes—
80Dic509—21no—
81Dic509—22no—
82Dic509—22no—
83Dic509C2×D412yes—
84C2×C509—22no—
85C2×C509—42no—
86C2×C509C22×D411yescharacteristic
87C2×C509C2×#4111yescharacteristic
88C2×C509C2×#4111yescharacteristic
89D509—22no—
90D509—22no—
91C509#52511yescharacteristic, complemented, Sylow
92#525—5091nomaximal, Sylow
93C2×#41—5091no—
94C23×C4—5091no—
95#123—5092no—
96C22⋊Q8—5091no—
97C22⋊Q8—5091no—
98#121—5091no—
99#121D50911yescharacteristic, complemented
100#120—5092no—
101C4×D4—5092no—
102C4×D4—5092no—
103#117—5091no—
104C24—5091no—
105#41—10184no—
106C2×Q8—5091no—
107C2×D4—5092no—
108C2×D4—5092no—
109C2×D4C2×D50912yes—
110C2×D4C2×D50911yescharacteristic, complemented
111C22×C4—10184no—
112C22×C4—5091no—
113C22×C4—5092no—
114C22×C4—5091no—
115C22×C4—5091no—
116C22×C4—5091no—
117C22×C4—5091no—
118C22×C4C2×D50911yescharacteristic
119C4⋊C4—5092no—
120C4⋊C4—5091no—
121C4⋊C4—5092no—
122C4⋊C4C2×D50911yescharacteristic, complemented
123C22⋊C4—5092no—
124C22⋊C4—5092no—
125C22⋊C4—5092no—
126C22⋊C4—5092no—
127C22⋊C4C2×D50912yescomplemented
128C42—5092no—
129C23—10182no—
130C23—10182no—
131C23—10182no—
132C23—5091no—
133C23—5091no—
134C23C22×D50912yes—
135C23C22×D50911yescharacteristic
136Q8—10182no—
137D4—24no—
138D4—22no—
139D4—10184no—
140D4—10184no—
141C2×C4—22no—
142C2×C4—10182no—
143C2×C4—10182no—
144C2×C4—10182no—
145C2×C4—10184no—
146C2×C4—10182no—
147C2×C4—5092no—
148C2×C4—5091no—
149C2×C4—5092no—
150C2×C4—5091no—
151C2×C4—5094no—
152C2×C4C22×D50912yes—
153C2×C4C22×D50911yescharacteristic
154C2×C4C22×D50911yescharacteristic
155C22—21no—
156C22—21no—
157C22—22no—
158C22—22no—
159C22—22no—
160C22—10188no—
161C22—10181no—
162C22—10181no—
163C22—10181no—
164C22—10181no—
165C22—5092no—
166C22—5092no—
167C22#2255086012yescomplemented
168C22C23×D50911yescharacteristic
169C4—21no—
170C4—22no—
171C4—10181no—
172C4—10182no—
173C4—10182no—
174C4—5092no—
175C4D4×D50912yes—
176C2—22no—
177C2—42no—
178C2—10182no—
179C2—10182no—
180C2C2×D4×D50911yescharacteristic
181C2#41×D50911yescharacteristic
182C2C2×#2255086011yescharacteristic
subgroupG/H
center, FrattiniC22C23×D509
commutator, socle, cosocleC22×C509C24
Fitting#121×C509C2
hypercenter#121D509
hypocenterC509#525
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. C22×C4×D509C2semidirect
3. #30654197C2semidirect
4. #30654195C2semidirect
5. #30654194C2semidirect
6. #30654186C2semidirect
7. #30654183C2semidirect
8. D4×Dic509C2semidirect
9. #30654175C2semidirect
10. #30654173C2semidirect
11. #121×C509C2semidirect
14. C2×Dic1018C22semidirect
18. C22×Dic509C22semidirect
24. C4⋊Dic509C22semidirect
25. C4⋊Dic509C22semidirect
26. #22550840C22semidirect
28. C22⋊Dic509C22semidirect
31. C4×Dic509C22semidirect
33. C2×D4×C509C22semidirect
35. C4⋊C4×C509C22semidirect
36. C22⋊C4×C509C22semidirect
54. C4×D509D4semidirect
68. C22×C509#41semidirect
76. C2×D509#41semidirect
91. C509#525semidirect
99. #121D509semidirect
110. C2×D4C2×D509semidirect
122. C4⋊C4C2×D509semidirect
127. C22⋊C4C2×D509semidirect
167. C22#22550860semidirect

Maximal subgroups

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

subgroupindex
C2×#225508602
C22×C4×D5092
#306541972
#306541952
#306541942
#306541862
#306541832
D4×Dic5092
#306541752
#306541732
#121×C5092
#525509