Finite Group Databasebeta

← C2×#5493 #85694

Subgroups — C2×#5493 #85694

Order 384.

1984 conjugacy classes of subgroups; only the 565 normal subgroups are shown.

normal subgroups only

This group has 1984 conjugacy classes of subgroups — too many to draw. Only its 565 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
1C23×#67C212yesmaximal, complemented
2C22×C3×C22⋊C4C211yesmaximal, characteristic, complemented
3C2×#930C214yesmaximal, complemented
4#5493C2116yesmaximal, direct product factor
5C2×#886C214yesmaximal, complemented
6C2×#882C214yesmaximal, complemented
7C2×#534—31nomaximal, Sylow
8C25×C3C2211yescharacteristic
9C24×D3C2212yescomplemented
10C23×C3×C4C2212yescomplemented
11C22×#67—216no—
12C22×#67C2214yes—
13C22×#67C2214yescomplemented
14C22×#67C22116yescomplemented
15C23×Dic3C2212yescomplemented
16C22×D12C2214yescomplemented
17C2×C3×C22⋊C4C2214yescomplemented
18C2×C3×C22⋊C4C2218yescomplemented
19#930C22132yescomplemented
20C2×(C2×D3)⋊C4C2218yescomplemented
21C2×C4⋊Dic3C2214yescomplemented
22#886C22132yescomplemented
23#882C22132yescomplemented
24C23×D4—32no—
25#534—316no—
26C2×#123—34no—
27C2×#121—34no—
28C2×#120—34no—
29C22×C22⋊C4D311yescharacteristic, complemented
30C24×C3—44no—
31C24×C3D418yescomplemented
32C24×C3C2312yes—
33C24×C3C2311yescharacteristic
34C24×C3C2314yes—
35C23×D3—28no—
36C23×D3—48no—
37C23×D3C2314yes—
38C23×D3C2318yescomplemented
39C22×C3×C4—44no—
40C22×C3×C4C2314yes—
41C22×C3×C4C2318yescomplemented
42C2×#67—216no—
43C2×#67—216no—
44C2×#67—264no—
45C2×#67C23116yes—
46C2×#67C23116yescomplemented
47C22×Dic3—28no—
48C22×Dic3C2314yes—
49C22×Dic3C2318yescomplemented
50C2×D12—48no—
51C2×D12C23116yescomplemented
52C3×C22⋊C4C23116yescomplemented
53(C2×D3)⋊C4C23132yescomplemented
54C4⋊Dic3C23116yescomplemented
55C25—32no—
56C25C2×D311yescharacteristic
57C22×D4—34no—
58C22×D4—34no—
59C22×D4—316no—
60C22×D4—34no—
61C22×D4—616no—
62C23×C4—32no—
63C23×C4C2×D312yescomplemented
64#123—332no—
65#121—332no—
66#120—332no—
67C2×C4⋊C4—34no—
68C2×C22⋊C4—38no—
69C2×C22⋊C4C2×D314yescomplemented
70C2×C22⋊C4C2×D318yescomplemented
71C23×C3—24no—
72C23×C3—28no—
73C23×C3—416no—
74C23×C3—48no—
75C23×C3C2×D418yes—
76C23×C3C2×D414yes—
77C23×C3C2×D4116yescomplemented
78C23×C3C2414yes—
79C23×C3C2412yes—
80C23×C3C2411yescharacteristic
81C22×D3—28no—
82C22×D3—216no—
83C22×D3—216no—
84C22×D3—416no—
85C22×D3—432no—
86C22×D3C2418yes—
87C2×C3×C4—28no—
88C2×C3×C4—48no—
89C2×C3×C4C2418yes—
90#67—232no—
91#67—232no—
92C2×Dic3—28no—
93C2×Dic3—216no—
94C2×Dic3C2418yes—
95D12—416no—
96C24—34no—
97C24—38no—
98C24—44no—
99C24—68no—
100C24—128no—
101C24D1218yescomplemented
102C24C22×D312yes—
103C24C22×D311yescharacteristic
104C24C22×D314yes—
105C2×D4—316no—
106C2×D4—316no—
107C2×D4—316no—
108C2×D4—616no—
109C2×D4—616no—
110C2×D4—664no—
111C2×D4—128no—
112C22×C4—38no—
113C22×C4—34no—
114C22×C4—44no—
115C22×C4—68no—
116C22×C4C22×D314yes—
117C22×C4C22×D318yescomplemented
118C4⋊C4—316no—
119C22⋊C4—332no—
120C22⋊C4C22×D3116yescomplemented
121C22×C3—216no—
122C22×C3—28no—
123C22×C3—24no—
124C22×C3—28no—
125C22×C3—416no—
126C22×C3C22×D418yes—
127C22×C3C22×D414yes—
128C22×C3C22×D412yes—
129C22×C3H(5,𝔽2)14yes—
130C22×C3C2511yescharacteristic
131C2×D3—28no—
132C2×D3—216no—
133C2×D3—432no—
134C2×D3—416no—
135C3×C4—28no—
136Dic3—28no—
137C23—24no—
138C23—28no—
139C23—38no—
140C23—416no—
141C23—48no—
142C23—616no—
143C23—68no—
144C23—616no—
145C23—1216no—
146C23—1232no—
147C23C2×D1218yes—
148C23C2×D1214yes—
149C23C2×D12116yescomplemented
150C23C23×D314yes—
151C23C23×D312yes—
152C23C23×D311yescharacteristic
153D4—632no—
154D4—632no—
155D4—1216no—
156C2×C4—28no—
157C2×C4—38no—
158C2×C4—48no—
159C2×C4—616no—
160C2×C4—68no—
161C2×C4C23×D318yes—
162C2×C3—28no—
163C2×C3—24no—
164C2×C3#53414yescomplemented
165C2×C3C23×D411yescharacteristic
166C2×C3C2×H(5,𝔽2)12yes—
167D3—48no—
168C22—24no—
169C22—28no—
170C22—216no—
171C22—28no—
172C22—416no—
173C22—68no—
174C22—616no—
175C22—1232no—
176C22—1216no—
177C22C22×D1218yes—
178C22C22×D1214yes—
179C22C22×D1212yes—
180C22#100414yes—
181C22C24×D311yescharacteristic
182C4—28no—
183C4—68no—
184C3C2×#53411yescharacteristic, complemented, Sylow
185C2—28no—
186C2—24no—
187C2—128no—
188C2#549314yesdirect product factor
189C2C23×D1211yescharacteristic
190C2C2×#100412yes—
subgroupG/H
centerC23C23×D3
commutator, cosocleC22×C3C25
FrattiniC22C24×D3
FittingC22×C3×C22⋊C4C2
socleC23×C3C24
hypercenterC22×C22⋊C4D3
hypocenterC3C2×#534
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C23×#67C2semidirect
2. C22×C3×C22⋊C4C2semidirect
3. C2×#930C2semidirect
4. #5493C2direct (N × H)
5. C2×#886C2semidirect
6. C2×#882C2semidirect
9. C24×D3C22semidirect
10. C23×C3×C4C22semidirect
13. C22×#67C22semidirect
14. C22×#67C22semidirect
15. C23×Dic3C22semidirect
16. C22×D12C22semidirect
17. C2×C3×C22⋊C4C22semidirect
18. C2×C3×C22⋊C4C22semidirect
19. #930C22semidirect
20. C2×(C2×D3)⋊C4C22semidirect
21. C2×C4⋊Dic3C22semidirect
22. #886C22semidirect
23. #882C22semidirect
29. C22×C22⋊C4D3semidirect
31. C24×C3D4semidirect
38. C23×D3C23semidirect
41. C22×C3×C4C23semidirect
46. C2×#67C23semidirect
49. C22×Dic3C23semidirect
51. C2×D12C23semidirect
52. C3×C22⋊C4C23semidirect
53. (C2×D3)⋊C4C23semidirect
54. C4⋊Dic3C23semidirect
63. C23×C4C2×D3semidirect
69. C2×C22⋊C4C2×D3semidirect
70. C2×C22⋊C4C2×D3semidirect
77. C23×C3C2×D4semidirect
101. C24D12semidirect
117. C22×C4C22×D3semidirect
120. C22⋊C4C22×D3semidirect
149. C23C2×D12semidirect
164. C2×C3#534semidirect
184. C3C2×#534semidirect
188. C2#5493direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#672
C22×C3×C22⋊C42
C2×#9302
#54932
C2×#8862
C2×#8822
C2×#5343