Finite Group Databasebeta

← #45655879

Subgroups — #45655879

Order 32576.

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×#22550862C211yesmaximal, characteristic, complemented
2C22×D2036C211yesmaximal, characteristic, complemented
3#30654204C211yesmaximal, characteristic, complemented
4#30654200C212yesmaximal, complemented
5#30654190C211yesmaximal, characteristic, complemented
6#30654190C212yesmaximal, complemented
7#30654185C212yesmaximal, complemented
8#30654177C211yesmaximal, characteristic, complemented
9C4×C22⋊D509C211yesmaximal, characteristic, complemented
10C22⋊C4×D509C212yesmaximal, complemented
11C22⋊Q8×C509C211yesmaximal, characteristic, complemented
12C23×D509C2212yes—
13#22550862—24no—
14C2×D2036—24no—
15C2×D2036C2212yes—
16C2×D2036C2211yescharacteristic
17C2×D2036C2211yescharacteristic
18C2×D2036C2212yescomplemented
19C2×D2036C2212yescomplemented
20C2×C22⋊D509C2211yescharacteristic
21C2×C4×D509C2212yescomplemented
22C2×C4×D509C2211yescharacteristic, complemented
23C2×C4×D509C2212yescomplemented
24#22550840C2211yescharacteristic, complemented
25C22⋊Dic509C2211yescharacteristic
26#22550838C2212yescomplemented
27#22550838C2212yescomplemented
28#22550838C2211yescharacteristic, complemented
29#22550838C2212yescomplemented
30#22550838C2212yescomplemented
31C4×Dic509C2211yescharacteristic, complemented
32C2×Q8×C509C2211yescharacteristic, complemented
33C22×C4×C509C2211yescharacteristic
34C4⋊C4×C509C2211yescharacteristic, complemented
35C4⋊C4×C509C2212yescomplemented
36C22⋊C4×C509C2212yes—
37C22×D509—24no—
38C22×D509—24no—
39C22×D509—42no—
40C22×D509C2312yes—
41C22×D509C2311yescharacteristic
42C22×D509C2312yes—
43D2036—24no—
44D2036—24no—
45D2036—22no—
46D2036—42no—
47D2036D414yescomplemented
48C22⋊D509—22no—
49C2×Dic509C2311yescharacteristic
50C2×Dic509C2311yescharacteristic
51C2×Dic509C2311yescharacteristic
52C4×D509—24no—
53C4×D509—22no—
54C4×D509—42no—
55C23×C509C2311yescharacteristic
56Q8×C509—22no—
57C2×C4×C509—22no—
58C2×C4×C509C2312yes—
59C2×C4×C509C2312yes—
60C2×C4×C509C2311yescharacteristic
61C2×C4×C509C2311yescharacteristic
62C22×C509—21no—
63C22×C509—21no—
64C22×C509#4112yescomplemented
65C22×C509C2411yescharacteristic
66C2×D509—22no—
67C2×D509—22no—
68C2×D509—21no—
69C2×D509—22no—
70C2×D509—21no—
71C2×D509—22no—
72C2×D509—21no—
73C2×D509—22no—
74C2×D509—48no—
75C2×D509C2×D414yes—
76C4×C509—21no—
77C4×C509—22no—
78C4×C509—22no—
79C4×C509C2×D412yes—
80Dic509—21no—
81Dic509—21no—
82Dic509—21no—
83C2×C509—22no—
84C2×C509C22×D411yescharacteristic
85C2×C509C2×#4111yescharacteristic
86C2×C509H(5,𝔽2)11yescharacteristic
87D509—24no—
88D509—41no—
89D509—42no—
90C509#54611yescharacteristic, complemented, Sylow
91#546—5091nomaximal, Sylow
92C2×#41—5091no—
93C22×D4—5091no—
94#124—5091no—
95#123—5092no—
96C22⋊Q8D50911yescharacteristic, complemented
97#121—5091no—
98#121—5092no—
99#120—5092no—
100C4×D4—5091no—
101C4×D4—5091no—
102C2×C22⋊C4—5092no—
103C24—5092no—
104#41—10184no—
105C2×Q8C2×D50911yescharacteristic, complemented
106C2×D4—10184no—
107C2×D4—5091no—
108C2×D4—5092no—
109C2×D4—5092no—
110C2×D4—5091no—
111C2×D4—5092no—
112C2×D4—5091no—
113C22×C4—5092no—
114C22×C4—5092no—
115C22×C4—5091no—
116C22×C4C2×D50911yescharacteristic
117C4⋊C4—5091no—
118C4⋊C4C2×D50911yescharacteristic, complemented
119C4⋊C4C2×D50912yescomplemented
120C22⋊C4—5091no—
121C22⋊C4—5092no—
122C22⋊C4—5092no—
123C22⋊C4—5092no—
124C22⋊C4—5091no—
125C22⋊C4—5092no—
126C22⋊C4C2×D50912yes—
127C42—5091no—
128C23—20362no—
129C23—10184no—
130C23—10184no—
131C23—5092no—
132C23—5091no—
133C23—5092no—
134C23C22×D50911yescharacteristic
135Q8—22no—
136D4—20362no—
137D4—10184no—
138D4—10184no—
139D4—10182no—
140D4—10182no—
141D4—5094no—
142C2×C4—22no—
143C2×C4—20362no—
144C2×C4—10184no—
145C2×C4—10182no—
146C2×C4—5091no—
147C2×C4—5091no—
148C2×C4—5091no—
149C2×C4C22×D50912yes—
150C2×C4C22×D50912yes—
151C2×C4C22×D50911yescharacteristic
152C2×C4C22×D50911yescharacteristic
153C22—21no—
154C22—21no—
155C22—20368no—
156C22—10181no—
157C22—10182no—
158C22—10181no—
159C22—10182no—
160C22—10181no—
161C22—10182no—
162C22—10182no—
163C22—10182no—
164C22—5094no—
165C22#2255086212yescomplemented
166C22C23×D50911yescharacteristic
167C4—21no—
168C4—22no—
169C4—22no—
170C4—10181no—
171C4—10181no—
172C4—10181no—
173C4D4×D50912yes—
174C2—22no—
175C2—20362no—
176C2—20361no—
177C2—10184no—
178C2C2×D4×D50911yescharacteristic
179C2C2×#2255086211yescharacteristic
180C2#3065426111yescharacteristic
subgroupG/H
center, FrattiniC22C23×D509
commutator, socle, cosocleC22×C509C24
FittingC22⋊Q8×C509C2
hypercenterC22⋊Q8D509
hypocenterC509#546
LCSG ▷ C22×C509 ▷ C509
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#22550862C2semidirect
2. C22×D2036C2semidirect
3. #30654204C2semidirect
4. #30654200C2semidirect
5. #30654190C2semidirect
6. #30654190C2semidirect
7. #30654185C2semidirect
8. #30654177C2semidirect
9. C4×C22⋊D509C2semidirect
10. C22⋊C4×D509C2semidirect
11. C22⋊Q8×C509C2semidirect
18. C2×D2036C22semidirect
19. C2×D2036C22semidirect
21. C2×C4×D509C22semidirect
22. C2×C4×D509C22semidirect
23. C2×C4×D509C22semidirect
24. #22550840C22semidirect
26. #22550838C22semidirect
27. #22550838C22semidirect
28. #22550838C22semidirect
29. #22550838C22semidirect
30. #22550838C22semidirect
31. C4×Dic509C22semidirect
32. C2×Q8×C509C22semidirect
34. C4⋊C4×C509C22semidirect
35. C4⋊C4×C509C22semidirect
47. D2036D4semidirect
64. C22×C509#41semidirect
90. C509#546semidirect
96. C22⋊Q8D509semidirect
105. C2×Q8C2×D509semidirect
118. C4⋊C4C2×D509semidirect
119. C4⋊C4C2×D509semidirect
165. C22#22550862semidirect

Maximal subgroups

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

subgroupindex
C2×#225508622
C22×D20362
#306542042
#306542002
#306541902
#306541852
#306541772
C4×C22⋊D5092
C22⋊C4×D5092
C22⋊Q8×C5092
#546509