Finite Group Databasebeta

← C3×#64361 #11724664

Subgroups — C3×#64361 #11724664

Order 864.

1212 conjugacy classes of subgroups; only the 306 normal subgroups are shown.

normal subgroups only

This group has 1212 conjugacy classes of subgroups — too many to draw. Only its 306 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×C33×D4C211yesmaximal, characteristic, complemented
2C2×C3×#638C214yesmaximal, complemented
3C3×#3830C214yesmaximal, complemented
4C3×D4×C3⋊D3C214yesmaximal, complemented
5C3×#3828C212yesmaximal, complemented
6#64361C311yesmaximal, characteristic, direct product factor
7C3×#1004—34nomaximal
8C23×C33C2212yescomplemented
9C22×C3×C3⋊D3C2214yescomplemented
10C33×D4C2214yescomplemented
11C2×C33×C4C2211yescharacteristic, complemented
12C3×#638C2214yescomplemented
13C3×#638C2218yescomplemented
14C2×C3×C3⋊Dic3C2214yescomplemented
15C3×C4⋊(C3⋊D3)C2212yescomplemented
16C3×C4×C3⋊D3C2214yescomplemented
17C3×#634C2212yescomplemented
18C2×C32×D4—24no—
19C2×C32×D4D314yescomplemented
20C2×C32×D4C2×C311yescharacteristic, complemented
21C2×#638C2×C314yescomplemented
22#3830C2×C314yescomplemented
23D4×C3⋊D3C2×C314yescomplemented
24#3828C2×C312yescomplemented
25C2×C3×#67—316no—
26C3×#237—316no—
27C3×D3×D4—316no—
28C3×#235—38no—
29C22×C33—42no—
30C22×C33C2314yes—
31C22×C33C2311yescharacteristic
32C2×C3×C3⋊D3—44no—
33C2×C3×C3⋊D3C2314yes—
34C33×C4C2312yes—
35C3×C3⋊Dic3C2314yes—
36C3×H(5,𝔽2)—91no—
37#1004—34no—
38C23×C32—28no—
39C23×C32C2×D318yescomplemented
40C23×C32C22×C312yescomplemented
41C22×C3⋊D3C22×C314yescomplemented
42C22×C3×D3—316no—
43C32×D4—216no—
44C32×D4C2×D3116yescomplemented
45C32×D4C22×C314yescomplemented
46C2×C32×C4—24no—
47C2×C32×C4C2×D314yescomplemented
48C2×C32×C4C22×C311yescharacteristic, complemented
49#638C22×C314yescomplemented
50#638C22×C318yescomplemented
51C2×C3⋊Dic3C22×C314yescomplemented
52C4⋊(C3⋊D3)C22×C312yescomplemented
53C4×C3⋊D3C22×C314yescomplemented
54#634C22×C312yescomplemented
55C3×#67—316no—
56C3×#67—332no—
57C2×C3×Dic3—316no—
58C3×D12—38no—
59C3×C4×D3—316no—
60C3×Dic6—38no—
61C2×C33—21no—
62C2×C33—24no—
63C2×C33C2411yescharacteristic
64C3×C3⋊D3—24no—
65C3×#41—94no—
66C3×#41—92no—
67C2×C3×D4—24no—
68C2×C3×D4—94no—
69C2×C3×D4—94no—
70C2×C3×D4C3×D314yescomplemented
71C2×C3×D4C3⋊D311yescharacteristic, complemented
72C2×#67—316no—
73#237—316no—
74D3×D4—316no—
75#235—38no—
76C22×C32—24no—
77C22×C32—216no—
78C22×C32—48no—
79C22×C32—42no—
80C22×C32—416no—
81C22×C32C22×D3116yes—
82C22×C32C22×D314yes—
83C22×C32C23×C314yes—
84C22×C32C23×C311yescharacteristic
85C2×C3⋊D3—44no—
86C2×C3⋊D3C23×C314yes—
87C2×C3×D3—316no—
88C2×C3×D3—1216no—
89C32×C4—28no—
90C32×C4C22×D318yes—
91C32×C4C23×C312yes—
92C3⋊Dic3C23×C314yes—
93C3×Dic3—316no—
94H(5,𝔽2)—91noSylow
95C33H(5,𝔽2)11yescharacteristic, complemented, Sylow
96C23×C3—28no—
97C23×C3—94no—
98C23×C3C2×C3×D318yescomplemented
99C23×C3C2×C3⋊D312yescomplemented
100C22×D3—316no—
101C3×Q8—92no—
102C3×D4—216no—
103C3×D4—98no—
104C3×D4—94no—
105C3×D4—92no—
106C3×D4C2×C3×D3116yescomplemented
107C3×D4C2×C3⋊D314yescomplemented
108C2×C3×C4—24no—
109C2×C3×C4—94no—
110C2×C3×C4—94no—
111C2×C3×C4C2×C3×D314yescomplemented
112C2×C3×C4C2×C3⋊D311yescharacteristic, complemented
113#67—316no—
114#67—332no—
115C2×Dic3—316no—
116D12—38no—
117C4×D3—316no—
118Dic6—38no—
119C2×C32—28no—
120C2×C32—24no—
121C2×C32—24no—
122C2×C32—21no—
123C2×C32—24no—
124C2×C32—216no—
125C2×C32—416no—
126C2×C32C23×D314yes—
127C2×C32C24×C311yescharacteristic
128C3⋊D3—24no—
129C3×D3—616no—
130#41—94no—
131#41—92no—
132C2×D4—94no—
133C2×D4—94no—
134C2×D4C3×C3⋊D311yescharacteristic, complemented
135C22×C3—24no—
136C22×C3—216no—
137C22×C3—48no—
138C22×C3—42no—
139C22×C3—416no—
140C22×C3—94no—
141C22×C3—364no—
142C22×C3C22×C3×D3116yes—
143C22×C3C22×C3×D314yes—
144C22×C3C22×C3⋊D314yes—
145C22×C3C22×C3⋊D311yescharacteristic
146C2×D3—316no—
147C2×D3—1216no—
148C3×C4—28no—
149C3×C4—94no—
150C3×C4C22×C3×D318yes—
151C3×C4C22×C3⋊D312yes—
152Dic3—316no—
153C32—24no—
154C32#100414yescomplemented
155C32C3×H(5,𝔽2)11yescharacteristic, complemented
156C23—94no—
157C23C2×C3×C3⋊D312yescomplemented
158Q8—92no—
159D4—98no—
160D4—94no—
161D4—92no—
162D4C2×C3×C3⋊D314yescomplemented
163C2×C4—94no—
164C2×C4—94no—
165C2×C4C2×C3×C3⋊D311yescharacteristic, complemented
166C2×C3—28no—
167C2×C3—216no—
168C2×C3—24no—
169C2×C3—21no—
170C2×C3—24no—
171C2×C3—24no—
172C2×C3—416no—
173C2×C3—184no—
174C2×C3C23×C3×D314yes—
175C2×C3C23×C3⋊D311yescharacteristic
176D3—616no—
177C22—42no—
178C22—94no—
179C22—364no—
180C22C22×C3×C3⋊D314yes—
181C22C22×C3×C3⋊D311yescharacteristic
182C4—94no—
183C4C22×C3×C3⋊D312yes—
184C3—24no—
185C3C3×#100414yescomplemented
186C3#6436111yescharacteristic, direct product factor
187C2—21no—
188C2—24no—
189C2—184no—
190C2C23×C3×C3⋊D311yescharacteristic
subgroupG/H
centerC2×C3C23×C3⋊D3
commutator, cosocleC2×C32C24×C3
FrattiniC2C23×C3×C3⋊D3
FittingC2×C33×D4C2
socleC2×C33C24
hypercenterC2×C3×D4C3⋊D3
hypocenterC32C3×H(5,𝔽2)
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
1. C2×C33×D4C2semidirect
2. C2×C3×#638C2semidirect
3. C3×#3830C2semidirect
4. C3×D4×C3⋊D3C2semidirect
5. C3×#3828C2semidirect
6. #64361C3direct (N × H)
8. C23×C33C22semidirect
9. C22×C3×C3⋊D3C22semidirect
10. C33×D4C22semidirect
11. C2×C33×C4C22semidirect
12. C3×#638C22semidirect
13. C3×#638C22semidirect
14. C2×C3×C3⋊Dic3C22semidirect
15. C3×C4⋊(C3⋊D3)C22semidirect
16. C3×C4×C3⋊D3C22semidirect
17. C3×#634C22semidirect
19. C2×C32×D4D3semidirect
20. C2×C32×D4C2×C3semidirect
21. C2×#638C2×C3semidirect
22. #3830C2×C3semidirect
23. D4×C3⋊D3C2×C3semidirect
24. #3828C2×C3semidirect
39. C23×C32C2×D3semidirect
40. C23×C32C22×C3semidirect
41. C22×C3⋊D3C22×C3semidirect
44. C32×D4C2×D3semidirect
45. C32×D4C22×C3semidirect
47. C2×C32×C4C2×D3semidirect
48. C2×C32×C4C22×C3semidirect
49. #638C22×C3semidirect
50. #638C22×C3semidirect
51. C2×C3⋊Dic3C22×C3semidirect
52. C4⋊(C3⋊D3)C22×C3semidirect
53. C4×C3⋊D3C22×C3semidirect
54. #634C22×C3semidirect
70. C2×C3×D4C3×D3semidirect
71. C2×C3×D4C3⋊D3semidirect
95. C33H(5,𝔽2)semidirect
98. C23×C3C2×C3×D3semidirect
99. C23×C3C2×C3⋊D3semidirect
106. C3×D4C2×C3×D3semidirect
107. C3×D4C2×C3⋊D3semidirect
111. C2×C3×C4C2×C3×D3semidirect
112. C2×C3×C4C2×C3⋊D3semidirect
134. C2×D4C3×C3⋊D3semidirect
154. C32#1004semidirect
155. C32C3×H(5,𝔽2)semidirect
157. C23C2×C3×C3⋊D3semidirect
162. D4C2×C3×C3⋊D3semidirect
165. C2×C4C2×C3×C3⋊D3semidirect
185. C3C3×#1004semidirect
186. C3#64361direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C33×D42
C2×C3×#6382
C3×#38302
C3×D4×C3⋊D32
C3×#38282
#643613
C3×#10043