Finite Group Databasebeta

← C2×#5717 #86194

Subgroups — C2×#5717 #86194

Order 384.

2536 conjugacy classes of subgroups; only the 581 normal subgroups are shown.

normal subgroups only

This group has 2536 conjugacy classes of subgroups — too many to draw. Only its 581 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
1C22×D3×D4C214yesmaximal, complemented
2C2×C3×#127C211yesmaximal, characteristic, complemented
3C2×#938C214yesmaximal, complemented
4C2×#937C214yesmaximal, complemented
5#5717C2116yesmaximal, direct product factor
6C2×C4×D12C212yesmaximal, complemented
7C2×D42—31nomaximal, Sylow
8C24×D3C2214yescomplemented
9C22×C3×D4C2214yescomplemented
10C22×C3×D4C2212yescomplemented
11C22×#67C2218yescomplemented
12C2×D3×D4—232no—
13C2×D3×D4C22132yescomplemented
14C22×D12C2212yes—
15C22×C4×D3C2214yescomplemented
16C3×#127C2218yescomplemented
17C2×C3×C42C2211yescharacteristic, complemented
18C2×#217C2214yescomplemented
19#938C22132yescomplemented
20#937C22132yescomplemented
21C2×(C2×D3)⋊C4C2214yescomplemented
22C2×C4⋊Dic3C2212yescomplemented
23C4×D12C22116yescomplemented
24C23×D4—34no—
25D42—316no—
26C2×#127D311yescharacteristic, complemented
27C2×#121—34no—
28C2×#120—34no—
29C2×C4×D4—32no—
30C24×C3C2314yes—
31C23×D3—232no—
32C23×D3—48no—
33C23×D3C2314yes—
34C23×D3C23116yescomplemented
35C2×C3×D4—216no—
36C2×C3×D4—44no—
37C2×C3×D4C23116yescomplemented
38C2×C3×D4C2318yescomplemented
39C22×C3×C4C2311yescharacteristic
40C22×C3×C4C2312yes—
41C2×#67—232no—
42C2×#67C23132yescomplemented
43C22×Dic3C2314yes—
44D3×D4—2128no—
45C2×D12D4116yescomplemented
46C2×D12C2318yes—
47C2×C4×D3—216no—
48C2×C4×D3C23116yescomplemented
49C3×C42C2314yescomplemented
50#217C23116yescomplemented
51(C2×D3)⋊C4C23116yescomplemented
52C4⋊Dic3C2318yescomplemented
53C25—34no—
54C22×D4—38no—
55C22×D4—332no—
56C22×D4—32no—
57C22×D4—632no—
58C22×D4C2×D314yescomplemented
59C22×D4C2×D312yescomplemented
60C23×C4—34no—
61#127C2×D318yescomplemented
62#121—332no—
63#120—332no—
64C4×D4—316no—
65C2×C4⋊C4—32no—
66C2×C22⋊C4—34no—
67C2×C22⋊C4—34no—
68C2×C42C2×D311yescharacteristic, complemented
69C23×C3—216no—
70C23×C3—28no—
71C23×C3C2418yes—
72C23×C3C2411yescharacteristic
73C22×D3—28no—
74C22×D3—28no—
75C22×D3—264no—
76C22×D3—416no—
77C22×D3—464no—
78C22×D3C2×D4116yescomplemented
79C22×D3C2418yes—
80C3×D4—232no—
81C3×D4—48no—
82C2×C3×C4—22no—
83C2×C3×C4C2×D418yescomplemented
84C2×C3×C4C2412yes—
85C2×C3×C4C2414yes—
86#67—264no—
87C2×Dic3—28no—
88C2×Dic3C2418yes—
89D12C2×D4132yescomplemented
90C4×D3—232no—
91C24—34no—
92C24—316no—
93C24—632no—
94C24—128no—
95C24C22×D314yes—
96C2×D4—216no—
97C2×D4—332no—
98C2×D4—316no—
99C2×D4—38no—
100C2×D4—44no—
101C2×D4—6128no—
102C2×D4—632no—
103C2×D4C22×D3116yescomplemented
104C2×D4C22×D318yescomplemented
105C22×C4—34no—
106C22×C4—316no—
107C22×C4—616no—
108C22×C4C22×D311yescharacteristic
109C22×C4C22×D312yes—
110C4⋊C4—38no—
111C22⋊C4—316no—
112C22⋊C4—316no—
113C42C22×D314yescomplemented
114C22×C3—216no—
115C22×C3—28no—
116C22×C3—416no—
117C22×C3C22×D414yes—
118C22×C3H(5,𝔽2)12yes—
119C22×C3C2511yescharacteristic
120C2×D3—232no—
121C2×D3—28no—
122C2×D3—28no—
123C2×D3—464no—
124C2×D3C22×D4116yescomplemented
125C3×C4—22no—
126C3×C4C22×D418yescomplemented
127Dic3—28no—
128C23—216no—
129C23—28no—
130C23—316no—
131C23—38no—
132C23—664no—
133C23—68no—
134C23—68no—
135C23—1216no—
136C23—1264no—
137C23C23×D318yes—
138C23C23×D311yescharacteristic
139D4—232no—
140D4—332no—
141D4—48no—
142D4—664no—
143C2×C4—22no—
144C2×C4—38no—
145C2×C4—68no—
146C2×C4—632no—
147C2×C4D3×D418yescomplemented
148C2×C4C23×D312yes—
149C2×C4C23×D314yes—
150C2×C3—48no—
151C2×C3D4214yescomplemented
152C2×C3C23×D412yes—
153C2×C3C2×H(5,𝔽2)11yescharacteristic
154D3—216no—
155C22—216no—
156C22—28no—
157C22—316no—
158C22—416no—
159C22—68no—
160C22—68no—
161C22—632no—
162C22—1264no—
163C22C2×D3×D414yes—
164C22#100412yes—
165C22C24×D311yescharacteristic
166C4—22no—
167C4—68no—
168C4C2×D3×D418yescomplemented
169C3C2×D4211yescharacteristic, complemented, Sylow
170C2—48no—
171C2—616no—
172C2#571714yesdirect product factor
173C2C22×D3×D412yes—
174C2C2×#100411yescharacteristic
subgroupG/H
centerC23C23×D3
commutator, cosocleC22×C3C25
FrattiniC22C24×D3
FittingC2×C3×#127C2
socleC23×C3C24
hypercenterC2×#127D3
hypocenterC3C2×D42
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. C22×D3×D4C2semidirect
2. C2×C3×#127C2semidirect
3. C2×#938C2semidirect
4. C2×#937C2semidirect
5. #5717C2direct (N × H)
6. C2×C4×D12C2semidirect
8. C24×D3C22semidirect
9. C22×C3×D4C22semidirect
10. C22×C3×D4C22semidirect
11. C22×#67C22semidirect
13. C2×D3×D4C22semidirect
15. C22×C4×D3C22semidirect
16. C3×#127C22semidirect
17. C2×C3×C42C22semidirect
18. C2×#217C22semidirect
19. #938C22semidirect
20. #937C22semidirect
21. C2×(C2×D3)⋊C4C22semidirect
22. C2×C4⋊Dic3C22semidirect
23. C4×D12C22semidirect
26. C2×#127D3semidirect
34. C23×D3C23semidirect
37. C2×C3×D4C23semidirect
38. C2×C3×D4C23semidirect
42. C2×#67C23semidirect
45. C2×D12D4semidirect
48. C2×C4×D3C23semidirect
49. C3×C42C23semidirect
50. #217C23semidirect
51. (C2×D3)⋊C4C23semidirect
52. C4⋊Dic3C23semidirect
58. C22×D4C2×D3semidirect
59. C22×D4C2×D3semidirect
61. #127C2×D3semidirect
68. C2×C42C2×D3semidirect
78. C22×D3C2×D4semidirect
83. C2×C3×C4C2×D4semidirect
89. D12C2×D4semidirect
103. C2×D4C22×D3semidirect
104. C2×D4C22×D3semidirect
113. C42C22×D3semidirect
124. C2×D3C22×D4semidirect
126. C3×C4C22×D4semidirect
147. C2×C4D3×D4semidirect
151. C2×C3D42semidirect
168. C4C2×D3×D4semidirect
169. C3C2×D42semidirect
172. C2#5717direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D3×D42
C2×C3×#1272
C2×#9382
C2×#9372
#57172
C2×C4×D122
C2×D423