Finite Group Databasebeta

← C2×#30632429 #44564485

Subgroups — C2×#30632429 #44564485

Order 32128.

1408 conjugacy classes of subgroups; only the 469 normal subgroups are shown.

normal subgroups only

This group has 1408 conjugacy classes of subgroups — too many to draw. Only its 469 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
1#30632429C2116yesmaximal, direct product factor
2C2×#22548704C212yesmaximal, complemented
3C2×#22548703C212yesmaximal, complemented
4C2×#22548701C214yesmaximal, complemented
5C2×#22548685C214yesmaximal, complemented
6C2×#22548683C212yesmaximal, complemented
7C2×#124×C251C211yesmaximal, characteristic, complemented
8C24×D251C2212yescomplemented
9C22×Dic502C2212yescomplemented
10C22×D1004C2212yes—
11C22×D1004C2212yes—
12C22×C22⋊D251C2214yes—
13#22548704C22116yescomplemented
14#22548703C22116yescomplemented
15#22548701C22132yescomplemented
16#22548685C22132yescomplemented
17#22548683C22116yescomplemented
18C2×C22⋊Dic251C2212yescomplemented
19C2×(C2×D251)⋊C4C2214yescomplemented
20C2×(C2×D251)⋊C4C2214yescomplemented
21C2×(C2×D251)⋊C4C2214yescomplemented
22C2×C4×Dic251C2212yescomplemented
23C22×Q8×C251C2211yescharacteristic, complemented
24C22×D4×C251C2211yescharacteristic
25#124×C251C2218yescomplemented
26C2×C22⋊C4×C251C2214yescomplemented
27C2×C42×C251C2211yescharacteristic, complemented
28C23×D251—44no—
29C23×D251—48no—
30C23×D251C2314yes—
31C23×D251C2318yescomplemented
32C2×Dic502—44no—
33C2×Dic502C2318yescomplemented
34C2×D1004—44no—
35C2×D1004—44no—
36C2×D1004C2318yes—
37C2×D1004C2318yes—
38C2×C22⋊D251—48no—
39C2×C22⋊D251C23116yes—
40C22×Dic251C2314yes—
41C22⋊Dic251C2318yescomplemented
42(C2×D251)⋊C4C23116yescomplemented
43(C2×D251)⋊C4C23116yescomplemented
44(C2×D251)⋊C4C23116yescomplemented
45C4×Dic251C2318yescomplemented
46C24×C251C2312yes—
47C2×Q8×C251—42no—
48C2×Q8×C251C2314yescomplemented
49C2×D4×C251—42no—
50C2×D4×C251C2314yes—
51C22×C4×C251C2312yes—
52C22×C4×C251C2312yes—
53C22×C4×C251C2311yescharacteristic
54C22⋊C4×C251C23116yescomplemented
55C42×C251C2314yescomplemented
56C22×D251—28no—
57C22×D251—28no—
58C22×D251—28no—
59C22×D251—416no—
60C22×D251—48no—
61C22×D251—432no—
62C22×D251C2418yes—
63Dic502—48no—
64D1004—48no—
65D1004—48no—
66C22⋊D251—416no—
67C2×Dic251—28no—
68C2×Dic251C2418yes—
69C23×C251—24no—
70C23×C251—28no—
71C23×C251C2414yes—
72C23×C251C2411yescharacteristic
73Q8×C251—44no—
74D4×C251—44no—
75C2×C4×C251—22no—
76C2×C4×C251—24no—
77C2×C4×C251—24no—
78C2×C4×C251C2414yes—
79C2×C4×C251C2412yes—
80C2×C4×C251C2414yes—
81C22×C251—24no—
82C22×C251—28no—
83C22×C251—48no—
84C22×C251H(5,𝔽2)12yes—
85C22×C251H(5,𝔽2)14yes—
86C22×C251C2511yescharacteristic
87C2×D251—28no—
88C2×D251—28no—
89C2×D251—28no—
90C2×D251—432no—
91C2×D251—416no—
92C4×C251—22no—
93C4×C251—24no—
94C4×C251—24no—
95Dic251—28no—
96C2×C251—44no—
97C2×C251H(3,𝔽4)14yescomplemented
98C2×C251C2×H(5,𝔽2)11yescharacteristic
99C2×C251C2×H(5,𝔽2)12yes—
100D251—48no—
101C251C2×H(3,𝔽4)11yescharacteristic, complemented, Sylow
102C2×H(3,𝔽4)—2511nomaximal, Sylow
103H(3,𝔽4)—25116no—
104C2×#124—2512no—
105C2×#124—2514no—
106C2×#124—2512no—
107C2×#124D25111yescharacteristic, complemented
108C2×#120—2512no—
109C2×#120—2514no—
110C25—2512no—
111C22×Q8—2512no—
112C22×Q8C2×D25111yescharacteristic, complemented
113C22×D4—2514no—
114C22×D4—2512no—
115C22×D4—2512no—
116C22×D4C2×D25111yescharacteristic
117#124—25116no—
118#124—25132no—
119#124—25116no—
120#124C2×D25118yescomplemented
121#120—25116no—
122#120—25132no—
123C2×C22⋊C4—2512no—
124C2×C22⋊C4—2514no—
125C2×C22⋊C4—2514no—
126C2×C22⋊C4—2514no—
127C2×C22⋊C4C2×D25114yescomplemented
128C2×C42—2512no—
129C2×C42C2×D25111yescharacteristic, complemented
130C24—10044no—
131C24—10048no—
132C24—2518no—
133C24—2514no—
134C24C22×D25112yes—
135C2×Q8—42no—
136C2×Q8—10044no—
137C2×Q8—2518no—
138C2×Q8C22×D25114yescomplemented
139C2×D4—42no—
140C2×D4—10044no—
141C2×D4—10044no—
142C2×D4—10048no—
143C2×D4—25116no—
144C2×D4—2518no—
145C2×D4—2518no—
146C2×D4C22×D25114yes—
147C22×C4—2514no—
148C22×C4C22×D25112yes—
149C22×C4C22×D25111yescharacteristic
150C22×C4C22×D25112yes—
151C22⋊C4—2518no—
152C22⋊C4—25116no—
153C22⋊C4—25116no—
154C22⋊C4—25116no—
155C22⋊C4C22×D251116yescomplemented
156C42—2518no—
157C42C22×D25114yescomplemented
158C23—24no—
159C23—28no—
160C23—10048no—
161C23—100416no—
162C23—100432no—
163C23—5028no—
164C23—5028no—
165C23—5028no—
166C23—2518no—
167C23C23×D25114yes—
168C23C23×D25111yescharacteristic
169Q8—44no—
170Q8—10048no—
171D4—44no—
172D4—10048no—
173D4—10048no—
174D4—100416no—
175C2×C4—22no—
176C2×C4—24no—
177C2×C4—24no—
178C2×C4—5028no—
179C2×C4—2518no—
180C2×C4C23×D25114yes—
181C2×C4C23×D25112yes—
182C2×C4C23×D25114yes—
183C22—24no—
184C22—28no—
185C22—48no—
186C22—100432no—
187C22—100416no—
188C22—5028no—
189C22—5028no—
190C22—5028no—
191C22#2254876012yes—
192C22#2254876114yes—
193C22C24×D25111yescharacteristic
194C4—22no—
195C4—24no—
196C4—24no—
197C4—5028no—
198C2—44no—
199C2—10048no—
200C2#3063242914yesdirect product factor
201C2C2×#2254876011yescharacteristic
202C2C2×#2254876112yes—
subgroupG/H
centerC23C23×D251
commutator, cosocleC22×C251C25
FrattiniC22C24×D251
FittingC2×#124×C251C2
socleC23×C251C24
hypercenterC2×#124D251
hypocenterC251C2×H(3,𝔽4)
LCSG ▷ C22×C251 ▷ C251
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #30632429C2direct (N × H)
2. C2×#22548704C2semidirect
3. C2×#22548703C2semidirect
4. C2×#22548701C2semidirect
5. C2×#22548685C2semidirect
6. C2×#22548683C2semidirect
7. C2×#124×C251C2semidirect
8. C24×D251C22semidirect
9. C22×Dic502C22semidirect
13. #22548704C22semidirect
14. #22548703C22semidirect
15. #22548701C22semidirect
16. #22548685C22semidirect
17. #22548683C22semidirect
18. C2×C22⋊Dic251C22semidirect
19. C2×(C2×D251)⋊C4C22semidirect
20. C2×(C2×D251)⋊C4C22semidirect
21. C2×(C2×D251)⋊C4C22semidirect
22. C2×C4×Dic251C22semidirect
23. C22×Q8×C251C22semidirect
25. #124×C251C22semidirect
26. C2×C22⋊C4×C251C22semidirect
27. C2×C42×C251C22semidirect
31. C23×D251C23semidirect
33. C2×Dic502C23semidirect
41. C22⋊Dic251C23semidirect
42. (C2×D251)⋊C4C23semidirect
43. (C2×D251)⋊C4C23semidirect
44. (C2×D251)⋊C4C23semidirect
45. C4×Dic251C23semidirect
48. C2×Q8×C251C23semidirect
54. C22⋊C4×C251C23semidirect
55. C42×C251C23semidirect
97. C2×C251H(3,𝔽4)semidirect
101. C251C2×H(3,𝔽4)semidirect
107. C2×#124D251semidirect
112. C22×Q8C2×D251semidirect
120. #124C2×D251semidirect
127. C2×C22⋊C4C2×D251semidirect
129. C2×C42C2×D251semidirect
138. C2×Q8C22×D251semidirect
155. C22⋊C4C22×D251semidirect
157. C42C22×D251semidirect
200. C2#30632429direct (N × H)

Maximal subgroups

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

subgroupindex
#306324292
C2×#225487042
C2×#225487032
C2×#225487012
C2×#225486852
C2×#225486832
C2×#124×C2512
C2×H(3,𝔽4)251