Finite Group Databasebeta

← C3×#10596522 #13181041

Subgroups — C3×#10596522 #13181041

Order 1728.

948 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 948 conjugacy classes of subgroups — too many to draw. Only its 170 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
1C3×#64320C212yesmaximal, complemented
2C3×#64289C211yesmaximal, characteristic, complemented
3C3×#64225C213yesmaximal, complemented
4C3×#64224C216yesmaximal, complemented
5C3×#64223C213yesmaximal, complemented
6#10596522C311yesmaximal, characteristic, direct product factor
7C3×#3832C2211yescharacteristic
8C32×#239—22no—
9C3×#3802C2216yescomplemented
10C3×D3×D12—26no—
11C3×#3798C2216yescomplemented
12C3×#3788C2213yescomplemented
13C3×#3787C2213yescomplemented
14C3×#3776C2211yescharacteristic, complemented
15C3×#3775C2216yescomplemented
16C3×#3774C2213yescomplemented
17C3×#3774C2216yescomplemented
18C3×#1009—62no—
19#64320C2×C312yescomplemented
20#64289C2×C311yescharacteristic, complemented
21#64225C2×C313yescomplemented
22#64224C2×C316yescomplemented
23#64223C2×C313yescomplemented
24C2×C3×D32—26no—
25C33×Q8D411yescharacteristic
26C3×C4⋊(C3⋊D3)D413yes—
27C3×C4×C3⋊D3C2313yes—
28C32×D12—26no—
29C32×C4×D3—26no—
30C3×#627C2312yes—
31C3×#626—26no—
32C3×#625C2316yes—
33C3×#622C2311yescharacteristic
34C3×#622C2313yes—
35C3×#576—91nomaximal
36C32×#41—62no—
37#3832C22×C311yescharacteristic
38C3×#239—22no—
39C3×#239—22no—
40C3×#239—42no—
41C3×#239—62no—
42C3×D3×D4—66no—
43C3×D3×D4—66no—
44C3×#235—66no—
45C2×C3×D12—66no—
46#3802C22×C316yescomplemented
47D3×D12—26no—
48#3798C22×C316yescomplemented
49#3788C22×C313yescomplemented
50#3787C22×C313yescomplemented
51#3776C22×C311yescharacteristic, complemented
52#3775C22×C316yescomplemented
53#3774C22×C313yescomplemented
54#3774C22×C316yescomplemented
55C2×C3×C3⋊D3C2×D413yes—
56C2×C32×D3—26no—
57C3×D32—86no—
58C33×C4C2×D413yes—
59C3×C3⋊Dic3C2411yescharacteristic
60C32×Dic3—22no—
61C3×H(5,𝔽2)—92no—
62#1009—62no—
63C3×AGL(1,ℤ/8)—96no—
64C3×#135—93no—
65C2×C3×D8—93no—
66C3×#131—91no—
67C22×C3×D3—612no—
68C2×D32—26no—
69C32×Q8—22no—
70C32×Q8—42no—
71C32×Q8C3×D411yescharacteristic
72C32×D4—66no—
73C2×C32×C4—66no—
74C4⋊(C3⋊D3)C3×D413yes—
75C4×C3⋊D3C23×C313yes—
76C3×#67—66no—
77C3×#67—66no—
78C3×D12—26no—
79C3×D12—26no—
80C3×D12—46no—
81C3×D12—66no—
82C3×D12—66no—
83C3×C4×D3—26no—
84C3×C4×D3—26no—
85C3×C4×D3—46no—
86C3×C4×D3—66no—
87C3×Dic6—62no—
88#627C23×C312yes—
89#626—26no—
90#625C23×C316yes—
91#622C23×C311yescharacteristic
92#622C23×C313yes—
93#576—91noSylow
94C2×C33C22×D411yescharacteristic
95C3×C3⋊D3—23no—
96C32×D3—46no—
97C3×#41—62no—
98C3×#41—62no—
99C3×#41—96no—
100C3×#41—91no—
101C3×#41—122no—
102C2×C3×D4—96no—
103C2×C3×D4—186no—
104#239—22no—
105#239—62no—
106D3×D4—66no—
107D3×D4—66no—
108#235—66no—
109C2×D12—66no—
110C3×Q16—91no—
111C3×SD16—96no—
112C3×D8—93no—
113C3×D8—96no—
114C3×M16—93no—
115C2×C3×C8—93no—
116C22×C32—66no—
117C2×C3⋊D3C2×C3×D413yes—
118C2×C3×D3—26no—
119C2×C3×D3—26no—
120C2×C3×D3—46no—
121C2×C3×D3—66no—
122C2×C3×D3—2412no—
123D32—86no—
124C32×C4—26no—
125C32×C4—46no—
126C32×C4—62no—
127C32×C4C2×C3×D413yes—
128C3⋊Dic3C24×C311yescharacteristic
129C3×Dic3—22no—
130C3×Dic3—22no—
131C3×Dic3—42no—
132C3×Dic3—62no—
133H(5,𝔽2)—92no—
134AGL(1,ℤ/8)—96no—
135#135—93no—
136C2×D8—93no—
137#131—91no—
138C33#57611yescharacteristic, complemented, Sylow
139C23×C3—186no—
140C22×D3—612no—
141C3×Q8—22no—
142C3×Q8—42no—
143C3×Q8—92no—
144C3×Q8GO+(4,𝔽2)11yescharacteristic
145C3×D4—66no—
146C3×D4—66no—
147C3×D4—96no—
148C3×D4—93no—
149C3×D4—126no—
150C3×D4—186no—
151C2×C3×C4—66no—
152C2×C3×C4—66no—
153C2×C3×C4—93no—
154C2×C3×C4—126no—
155#67—66no—
156#67—66no—
157D12—26no—
158D12—66no—
159D12—66no—
160C4×D3—26no—
161C4×D3—66no—
162Dic6—62no—
163C3×C8—91no—
164C3×C8—93no—
165C2×C32—22no—
166C2×C32—42no—
167C2×C32—126no—
168C2×C32C22×C3×D411yescharacteristic
169C3⋊D3—23no—
170C3×D3—46no—
171C3×D3—46no—
172C3×D3—86no—
173C3×D3—126no—
174#41—62no—
175#41—96no—
176#41—91no—
177C2×D4—96no—
178C2×D4—186no—
179Q16—91no—
180SD16—96no—
181D8—93no—
182D8—96no—
183M16—93no—
184C2×C8—93no—
185C22×C3—66no—
186C22×C3—66no—
187C22×C3—93no—
188C22×C3—126no—
189C22×C3—726no—
190C2×D3—26no—
191C2×D3—66no—
192C2×D3—2412no—
193C3×C4—26no—
194C3×C4—46no—
195C3×C4—62no—
196C3×C4—62no—
197C3×C4—91no—
198C3×C4—122no—
199C3×C4C2×GO+(4,𝔽2)13yes—
200Dic3—22no—
201Dic3—62no—
202C32—22no—
203C32—42no—
204C32C3×#57611yescharacteristic, complemented
205C23—186no—
206Q8—92no—
207Q8C3×GO+(4,𝔽2)11yescharacteristic
208D4—66no—
209D4—96no—
210D4—93no—
211D4—186no—
212C2×C4—66no—
213C2×C4—93no—
214C8—91no—
215C8—93no—
216C2×C3—22no—
217C2×C3—42no—
218C2×C3—126no—
219C2×C3—126no—
220C2×C3—183no—
221C2×C3—246no—
222C2×C3C22×GO+(4,𝔽2)11yescharacteristic
223D3—46no—
224D3—126no—
225C22—66no—
226C22—93no—
227C22—726no—
228C4—62no—
229C4—91no—
230C4C2×C3×GO+(4,𝔽2)13yes—
231C3—22no—
232C3—42no—
233C3#1059652211yescharacteristic, direct product factor
234C2—126no—
235C2—183no—
236C2C22×C3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
centerC2×C3C22×GO+(4,𝔽2)
commutator, cosocleC3⋊Dic3C24×C3
FrattiniC2C22×C3×GO+(4,𝔽2)
FittingC33×Q8D4
socleC2×C33C22×D4
hypercenterC3×Q8GO+(4,𝔽2)
hypocenterC32C3×#576
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C3×Q8

Complements

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

NH ≅ G/Ntype
1. C3×#64320C2semidirect
2. C3×#64289C2semidirect
3. C3×#64225C2semidirect
4. C3×#64224C2semidirect
5. C3×#64223C2semidirect
6. #10596522C3direct (N × H)
9. C3×#3802C22semidirect
11. C3×#3798C22semidirect
12. C3×#3788C22semidirect
13. C3×#3787C22semidirect
14. C3×#3776C22semidirect
15. C3×#3775C22semidirect
16. C3×#3774C22semidirect
17. C3×#3774C22semidirect
19. #64320C2×C3semidirect
20. #64289C2×C3semidirect
21. #64225C2×C3semidirect
22. #64224C2×C3semidirect
23. #64223C2×C3semidirect
46. #3802C22×C3semidirect
48. #3798C22×C3semidirect
49. #3788C22×C3semidirect
50. #3787C22×C3semidirect
51. #3776C22×C3semidirect
52. #3775C22×C3semidirect
53. #3774C22×C3semidirect
54. #3774C22×C3semidirect
138. C33#576semidirect
204. C32C3×#576semidirect
233. C3#10596522direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#643202
C3×#642892
C3×#642252
C3×#642242
C3×#642232
#105965223
C3×#5769