Finite Group Databasebeta

← #64361⋊C3 #11722615

Subgroups — #64361⋊C3 #11722615

Order 864.

635 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 635 conjugacy classes of subgroups — too many to draw. Only its 187 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×D4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#638⋊C3C214yesmaximal, complemented
3#3830⋊C3C214yesmaximal, complemented
4D4×(C3⋊D3)⋊C3C214yesmaximal, complemented
5#3828⋊C3C212yesmaximal, complemented
6#64361C311yesmaximal, characteristic, complemented
7C3×#1004—31nomaximal
8C23×H(3,𝔽3)C2212yescomplemented
9C22×(C3⋊D3)⋊C3C2214yescomplemented
10D4×H(3,𝔽3)C2214yescomplemented
11C2×C4×H(3,𝔽3)C2211yescharacteristic, complemented
12#638⋊C3C2214yescomplemented
13#638⋊C3C2218yescomplemented
14C2×(C3⋊Dic3)⋊C3C2214yescomplemented
15C4⋊((C3⋊D3)⋊C3)C2212yescomplemented
16C4×(C3⋊D3)⋊C3C2214yescomplemented
17#634⋊C3C2212yescomplemented
18C2×C32×D4—21no—
19C2×C32×D4D311yescharacteristic, complemented
20C2×C32×D4C2×C311yescharacteristic, complemented
21C2×#638C2×C314yescomplemented
22#3830C2×C314yescomplemented
23D4×C3⋊D3C2×C314yescomplemented
24#3828C2×C312yescomplemented
25C2×C3×#67—34no—
26C3×#237—34no—
27C3×D3×D4—34no—
28C3×#235—32no—
29C22×H(3,𝔽3)—42no—
30C22×H(3,𝔽3)C2314yes—
31C22×H(3,𝔽3)C2311yescharacteristic
32C2×(C3⋊D3)⋊C3—44no—
33C2×(C3⋊D3)⋊C3C2314yes—
34C4×H(3,𝔽3)C2312yes—
35(C3⋊Dic3)⋊C3C2314yes—
36C3×H(5,𝔽2)—91no—
37#1004—31no—
38#1004—91no—
39C23×C32—22no—
40C23×C32C2×D312yescomplemented
41C23×C32C22×C312yescomplemented
42C22×C3⋊D3C22×C314yescomplemented
43C22×C3×D3—34no—
44C32×D4—24no—
45C32×D4C2×D314yescomplemented
46C32×D4C22×C314yescomplemented
47C2×C32×C4—21no—
48C2×C32×C4C2×D311yescharacteristic, complemented
49C2×C32×C4C22×C311yescharacteristic, complemented
50#638C22×C314yescomplemented
51#638C22×C318yescomplemented
52C2×C3⋊Dic3C22×C314yescomplemented
53C4⋊(C3⋊D3)C22×C312yescomplemented
54C4×C3⋊D3C22×C314yescomplemented
55#634C22×C312yescomplemented
56C3×#67—34no—
57C3×#67—38no—
58C2×C3×Dic3—34no—
59C3×D12—32no—
60C3×C4×D3—34no—
61C3×Dic6—32no—
62C2×H(3,𝔽3)—21no—
63C2×H(3,𝔽3)—24no—
64C2×H(3,𝔽3)C2411yescharacteristic
65(C3⋊D3)⋊C3—24no—
66C3×#41—92no—
67C3×#41—94no—
68C2×C3×D4—31no—
69C2×C3×D4—31no—
70C2×C3×D4—61no—
71C2×C3×D4—94no—
72C2×C3×D4—94no—
73C2×C3×D4C3×D311yescharacteristic
74C2×#67—34no—
75C2×#67—94no—
76#237—34no—
77#237—94no—
78D3×D4—34no—
79D3×D4—94no—
80#235—32no—
81#235—92no—
82C22×C32—21no—
83C22×C32—24no—
84C22×C32—42no—
85C22×C32—42no—
86C22×C32—44no—
87C22×C32C22×D314yes—
88C22×C32C22×D311yescharacteristic
89C22×C32C23×C314yes—
90C22×C32C23×C311yescharacteristic
91C2×C3⋊D3—44no—
92C2×C3⋊D3C23×C314yes—
93C2×C3×D3—34no—
94C2×C3×D3—124no—
95C32×C4—22no—
96C32×C4C22×D312yes—
97C32×C4C23×C312yes—
98C3⋊Dic3C23×C314yes—
99C3×Dic3—34no—
100H(5,𝔽2)—91noSylow
101H(3,𝔽3)H(5,𝔽2)11yescharacteristic, complemented, Sylow
102C23×C3—32no—
103C23×C3—32no—
104C23×C3—62no—
105C23×C3—94no—
106C23×C3C2×C3×D312yes—
107C22×D3—34no—
108C22×D3—94no—
109C3×Q8—92no—
110C3×D4—34no—
111C3×D4—34no—
112C3×D4—64no—
113C3×D4—92no—
114C3×D4—94no—
115C3×D4—98no—
116C3×D4C2×C3×D314yes—
117C2×C3×C4—31no—
118C2×C3×C4—31no—
119C2×C3×C4—61no—
120C2×C3×C4—94no—
121C2×C3×C4—94no—
122C2×C3×C4C2×C3×D311yescharacteristic
123#67—34no—
124#67—38no—
125#67—94no—
126#67—98no—
127C2×Dic3—34no—
128C2×Dic3—94no—
129D12—32no—
130D12—92no—
131C4×D3—34no—
132C4×D3—94no—
133Dic6—32no—
134Dic6—92no—
135C2×C32—22no—
136C2×C32—24no—
137C2×C32—21no—
138C2×C32—24no—
139C2×C32—21no—
140C2×C32—21no—
141C2×C32—44no—
142C2×C32C23×D311yescharacteristic
143C2×C32C24×C311yescharacteristic
144C3⋊D3—24no—
145C3×D3—64no—
146#41—94no—
147#41—92no—
148C2×D4—94no—
149C2×D4—94no—
150C2×D4(C3⋊D3)⋊C311yescharacteristic, complemented
151C22×C3—34no—
152C22×C3—31no—
153C22×C3—31no—
154C22×C3—34no—
155C22×C3—42no—
156C22×C3—61no—
157C22×C3—64no—
158C22×C3—94no—
159C22×C3—122no—
160C22×C3—122no—
161C22×C3—124no—
162C22×C3—364no—
163C22×C3C22×C3×D314yes—
164C22×C3C22×C3×D311yescharacteristic
165C2×D3—34no—
166C2×D3—94no—
167C2×D3—124no—
168C2×D3—364no—
169C3×C4—32no—
170C3×C4—32no—
171C3×C4—62no—
172C3×C4—94no—
173C3×C4C22×C3×D312yes—
174Dic3—34no—
175Dic3—94no—
176C32—21no—
177C32#100411yescharacteristic, complemented
178C32C3×H(5,𝔽2)11yescharacteristic, complemented
179C23—94no—
180C23C2×(C3⋊D3)⋊C312yescomplemented
181Q8—92no—
182D4—98no—
183D4—94no—
184D4—92no—
185D4C2×(C3⋊D3)⋊C314yescomplemented
186C2×C4—94no—
187C2×C4—94no—
188C2×C4C2×(C3⋊D3)⋊C311yescharacteristic, complemented
189C2×C3—24no—
190C2×C3—21no—
191C2×C3—31no—
192C2×C3—31no—
193C2×C3—64no—
194C2×C3—61no—
195C2×C3—61no—
196C2×C3—64no—
197C2×C3—62no—
198C2×C3—61no—
199C2×C3—124no—
200C2×C3—184no—
201C2×C3C23×C3×D311yescharacteristic
202D3—64no—
203D3—184no—
204C22—42no—
205C22—94no—
206C22—364no—
207C22C22×(C3⋊D3)⋊C314yes—
208C22C22×(C3⋊D3)⋊C311yescharacteristic
209C4—94no—
210C4C22×(C3⋊D3)⋊C312yes—
211C3—31no—
212C3—31no—
213C3—61no—
214C3C3×#100411yescharacteristic
215C2—21no—
216C2—24no—
217C2—184no—
218C2C23×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC2C23×(C3⋊D3)⋊C3
commutator, cosocleC2×C32C24×C3
Frattini, socleC2×C3C23×C3×D3
FittingC2×D4×H(3,𝔽3)C2
hypercenterC2×D4(C3⋊D3)⋊C3
hypocenterC32C3×H(5,𝔽2)
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×D4×H(3,𝔽3)C2semidirect
2. C2×#638⋊C3C2semidirect
3. #3830⋊C3C2semidirect
4. D4×(C3⋊D3)⋊C3C2semidirect
5. #3828⋊C3C2semidirect
6. #64361C3semidirect
8. C23×H(3,𝔽3)C22semidirect
9. C22×(C3⋊D3)⋊C3C22semidirect
10. D4×H(3,𝔽3)C22semidirect
11. C2×C4×H(3,𝔽3)C22semidirect
12. #638⋊C3C22semidirect
13. #638⋊C3C22semidirect
14. C2×(C3⋊Dic3)⋊C3C22semidirect
15. C4⋊((C3⋊D3)⋊C3)C22semidirect
16. C4×(C3⋊D3)⋊C3C22semidirect
17. #634⋊C3C22semidirect
19. C2×C32×D4D3semidirect
20. C2×C32×D4C2×C3semidirect
21. C2×#638C2×C3semidirect
22. #3830C2×C3semidirect
23. D4×C3⋊D3C2×C3semidirect
24. #3828C2×C3semidirect
40. C23×C32C2×D3semidirect
41. C23×C32C22×C3semidirect
42. C22×C3⋊D3C22×C3semidirect
45. C32×D4C2×D3semidirect
46. C32×D4C22×C3semidirect
48. C2×C32×C4C2×D3semidirect
49. C2×C32×C4C22×C3semidirect
50. #638C22×C3semidirect
51. #638C22×C3semidirect
52. C2×C3⋊Dic3C22×C3semidirect
53. C4⋊(C3⋊D3)C22×C3semidirect
54. C4×C3⋊D3C22×C3semidirect
55. #634C22×C3semidirect
101. H(3,𝔽3)H(5,𝔽2)semidirect
150. C2×D4(C3⋊D3)⋊C3semidirect
177. C32#1004semidirect
178. C32C3×H(5,𝔽2)semidirect
180. C23C2×(C3⋊D3)⋊C3semidirect
185. D4C2×(C3⋊D3)⋊C3semidirect
188. C2×C4C2×(C3⋊D3)⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×D4×H(3,𝔽3)2
C2×#638⋊C32
#3830⋊C32
D4×(C3⋊D3)⋊C32
#3828⋊C32
#643613
C3×#10043