Finite Group Databasebeta

← #45655876

Subgroups — #45655876

Order 32576.

334 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 113 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×D4×D509C211yesmaximal, characteristic, complemented
2#30654204C211yesmaximal, characteristic, complemented
3#30654198C212yesmaximal, complemented
4#30654191C211yesmaximal, characteristic, complemented
5#30654189C211yesmaximal, characteristic, complemented
6#30654187C212yesmaximal, complemented
7#30654183C212yesmaximal, complemented
8D4×Dic509C211yesmaximal, characteristic, complemented
9C4×C22⋊D509C211yesmaximal, characteristic, complemented
10C2×#22550838C212yesmaximal, complemented
11C2×#41×C509C211yesmaximal, characteristic, complemented
12C23×D509C2212yes—
13C2×D2036C2211yescharacteristic
14C2×C22⋊D509C2212yes—
15C2×C22⋊D509C2211yescharacteristic
16C2×C22⋊D509C2212yescomplemented
17D4×D509—24no—
18C22×Dic509C2212yescomplemented
19C2×C4×D509C2211yescharacteristic
20C4⋊Dic509C2211yescharacteristic, complemented
21#22550840C2211yescharacteristic, complemented
22#22550840C2212yescomplemented
23C22⋊Dic509C2211yescharacteristic, complemented
24C22⋊Dic509C2212yescomplemented
25#22550838C2211yescharacteristic
26#22550838C2212yes—
27#22550838C2212yescomplemented
28#22550838C2212yescomplemented
29#22550838C2212yescomplemented
30C4×Dic509C2211yescharacteristic, complemented
31#41×C509—24no—
32C2×Q8×C509C2211yescharacteristic, complemented
33C2×D4×C509C2211yescharacteristic
34C2×D4×C509C2212yescomplemented
35C22×C4×C509C2212yescomplemented
36C22×C4×C509C2211yescharacteristic, complemented
37C22×D509—24no—
38C22×D509—24no—
39C22×D509—42no—
40C22×D509C2312yes—
41C22×D509C2311yescharacteristic
42D2036—22no—
43C22⋊D509—24no—
44C22⋊D509—22no—
45C22⋊D509—42no—
46C2×Dic509—42no—
47C2×Dic509C2311yescharacteristic
48C2×Dic509C2311yescharacteristic
49C2×Dic509C2312yes—
50C2×Dic509C2311yescharacteristic
51C4×D509—22no—
52C23×C509C2311yescharacteristic
53C23×C509C2312yes—
54Q8×C509—22no—
55D4×C509—24no—
56D4×C509D414yescomplemented
57C2×C4×C509—24no—
58C2×C4×C509—22no—
59C2×C4×C509C2311yescharacteristic
60C2×C4×C509C2312yes—
61C2×C4×C509C2311yescharacteristic
62C22×C509—22no—
63C22×C509—22no—
64C22×C509—21no—
65C22×C509—21no—
66C22×C509—21no—
67C22×C509C2×D414yes—
68C22×C509C2411yescharacteristic
69C2×D509—21no—
70C2×D509—21no—
71C2×D509—22no—
72C2×D509—22no—
73C2×D509—22no—
74C2×D509—48no—
75C2×D509#4112yescomplemented
76C4×C509—21no—
77C4×C509—22no—
78C4×C509C2×D412yes—
79Dic509—21no—
80Dic509—21no—
81Dic509—21no—
82Dic509—22no—
83C2×C509—24no—
84C2×C509—41no—
85C2×C509C22×D411yescharacteristic
86C2×C509C2×#4111yescharacteristic
87C2×C509H(5,𝔽2)11yescharacteristic
88D509—22no—
89D509—42no—
90C509#54611yescharacteristic, complemented, Sylow
91#546—5091nomaximal, Sylow
92C2×#41D50911yescharacteristic, complemented
93C22×D4—5091no—
94#124—5091no—
95#123—5092no—
96C22⋊Q8—5091no—
97#121—5092no—
98#121—5091no—
99#120—5092no—
100C4×D4—5091no—
101C4×D4—5091no—
102C2×C22⋊C4—5092no—
103C24—5092no—
104#41—24no—
105C2×Q8C2×D50911yescharacteristic, complemented
106C2×D4—10184no—
107C2×D4—5092no—
108C2×D4—5091no—
109C2×D4—5092no—
110C2×D4—5091no—
111C2×D4C2×D50911yescharacteristic
112C2×D4C2×D50912yescomplemented
113C22×C4—5092no—
114C22×C4—5091no—
115C22×C4C2×D50912yescomplemented
116C22×C4C2×D50911yescharacteristic, complemented
117C4⋊C4—5091no—
118C4⋊C4—5092no—
119C4⋊C4—5091no—
120C22⋊C4—5092no—
121C22⋊C4—5091no—
122C22⋊C4—5091no—
123C22⋊C4—5092no—
124C22⋊C4—5092no—
125C22⋊C4—5092no—
126C22⋊C4—5092no—
127C42—5091no—
128C23—20362no—
129C23—10184no—
130C23—10184no—
131C23—5092no—
132C23—5091no—
133C23C22×D50911yescharacteristic
134C23C22×D50912yes—
135Q8—22no—
136D4—24no—
137D4—20362no—
138D4—10182no—
139D4—10184no—
140D4—10182no—
141D4C22⋊D50914yescomplemented
142C2×C4—24no—
143C2×C4—22no—
144C2×C4—20362no—
145C2×C4—10182no—
146C2×C4—5091no—
147C2×C4—5091no—
148C2×C4—5092no—
149C2×C4—5091no—
150C2×C4C22×D50911yescharacteristic
151C2×C4C22×D50912yes—
152C2×C4C22×D50911yescharacteristic
153C22—22no—
154C22—22no—
155C22—21no—
156C22—21no—
157C22—21no—
158C22—20368no—
159C22—10182no—
160C22—10182no—
161C22—10182no—
162C22—10181no—
163C22—10181no—
164C22—5092no—
165C22C2×C22⋊D50914yes—
166C22C23×D50911yescharacteristic
167C4—21no—
168C4—22no—
169C4—10181no—
170C4—10181no—
171C4—10181no—
172C4—10182no—
173C4C2×C22⋊D50912yes—
174C2—24no—
175C2—41no—
176C2—20362no—
177C2—10182no—
178C2C22×C22⋊D50911yescharacteristic
179C2#41×D50911yescharacteristic
180C2#3065426111yescharacteristic
subgroupG/H
center, FrattiniC22C23×D509
commutator, socle, cosocleC22×C509C24
FittingC2×#41×C509C2
hypercenterC2×#41D509
hypocenterC509#546
LCSG ▷ C22×C509 ▷ C509
UCSC1 ◁ C22 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C2×D4×D509C2semidirect
2. #30654204C2semidirect
3. #30654198C2semidirect
4. #30654191C2semidirect
5. #30654189C2semidirect
6. #30654187C2semidirect
7. #30654183C2semidirect
8. D4×Dic509C2semidirect
9. C4×C22⋊D509C2semidirect
10. C2×#22550838C2semidirect
11. C2×#41×C509C2semidirect
16. C2×C22⋊D509C22semidirect
18. C22×Dic509C22semidirect
20. C4⋊Dic509C22semidirect
21. #22550840C22semidirect
22. #22550840C22semidirect
23. C22⋊Dic509C22semidirect
24. C22⋊Dic509C22semidirect
27. #22550838C22semidirect
28. #22550838C22semidirect
29. #22550838C22semidirect
30. C4×Dic509C22semidirect
32. C2×Q8×C509C22semidirect
34. C2×D4×C509C22semidirect
35. C22×C4×C509C22semidirect
36. C22×C4×C509C22semidirect
56. D4×C509D4semidirect
75. C2×D509#41semidirect
90. C509#546semidirect
92. C2×#41D509semidirect
105. C2×Q8C2×D509semidirect
112. C2×D4C2×D509semidirect
115. C22×C4C2×D509semidirect
116. C22×C4C2×D509semidirect
141. D4C22⋊D509semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D5092
#306542042
#306541982
#306541912
#306541892
#306541872
#306541832
D4×Dic5092
C4×C22⋊D5092
C2×#225508382
C2×#41×C5092
#546509