Finite Group Databasebeta

← C3×#10596520 #13181039

Subgroups — C3×#10596520 #13181039

Order 1728.

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

normal subgroups only

This group has 864 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×#64316C212yesmaximal, complemented
2C3×#64289C211yesmaximal, characteristic, complemented
3C3×#64228C213yesmaximal, complemented
4C3×#64226C216yesmaximal, complemented
5C3×#64223C213yesmaximal, complemented
6#10596520C311yesmaximal, characteristic, direct product factor
7C3×#3832C2211yescharacteristic
8C32×D3×Q8—22no—
9C3×#3799—26no—
10C3×#3798C2216yes—
11C3×#3797C2216yes—
12C3×#3788C2213yes—
13C3×#3787C2213yes—
14C3×#3776C2213yes—
15C3×#3776C2216yes—
16C3×#3775C2216yes—
17C3×#3774C2211yescharacteristic
18C3×#1007—62no—
19#64316C2×C312yescomplemented
20#64289C2×C311yescharacteristic, complemented
21#64228C2×C313yescomplemented
22#64226C2×C316yescomplemented
23#64223C2×C313yescomplemented
24C33×Q8D411yescharacteristic
25C3×C4⋊(C3⋊D3)D413yes—
26C3×C4×C3⋊D3C2313yes—
27C32×C4×D3—26no—
28C32×Dic6—26no—
29C3×#627C2316yes—
30C3×#626—26no—
31C3×#625C2312yes—
32C3×#624—26no—
33C3×#622C2311yescharacteristic
34C3×#622C2313yes—
35C3×#578—91nomaximal
36C2×C32×Q8—62no—
37#3832C22×C311yescharacteristic
38C3×#239—66no—
39C3×#239—62no—
40C3×D3×Q8—22no—
41C3×D3×Q8—22no—
42C3×D3×Q8—42no—
43C3×D3×Q8—66no—
44C3×#235—66no—
45C3×#235—66no—
46#3799—26no—
47#3798C22×C316yes—
48#3797C22×C316yes—
49#3788C22×C313yes—
50#3787C22×C313yes—
51#3776C22×C313yes—
52#3776C22×C316yes—
53#3775C22×C316yes—
54#3774C22×C311yescharacteristic
55C2×C3×C3⋊D3C2×D413yes—
56C2×C32×D3—22no—
57C33×C4C2×D413yes—
58C3×C3⋊Dic3C2411yescharacteristic
59C32×Dic3—26no—
60C3×2−1+4—92no—
61#1007—62no—
62C3×#137—96no—
63C3×#135—93no—
64C2×C3×Q16—93no—
65C3×#131—91no—
66C32×Q8—22no—
67C32×Q8—42no—
68C32×Q8—66no—
69C32×Q8C3×D411yescharacteristic
70C2×C32×C4—66no—
71C4⋊(C3⋊D3)C3×D413yes—
72C4×C3⋊D3C23×C313yes—
73C3×#67—62no—
74C3×#67—66no—
75C3×D12—66no—
76C3×D12—66no—
77C3×C4×D3—26no—
78C3×C4×D3—26no—
79C3×C4×D3—46no—
80C3×C4×D3—66no—
81C3×C4×D3—612no—
82C3×Dic6—26no—
83C3×Dic6—26no—
84C3×Dic6—46no—
85C3×Dic6—66no—
86#627C23×C316yes—
87#626—26no—
88#625C23×C312yes—
89#624—26no—
90#622C23×C311yescharacteristic
91#622C23×C313yes—
92#578—91noSylow
93C2×C33C22×D411yescharacteristic
94C3×C3⋊D3—23no—
95C32×D3—42no—
96C3×#41—96no—
97C3×#41—91no—
98C3×#41—186no—
99C2×C3×Q8—62no—
100C2×C3×Q8—62no—
101C2×C3×Q8—96no—
102C2×C3×Q8—122no—
103#239—62no—
104#239—66no—
105D3×Q8—22no—
106D3×Q8—66no—
107#235—66no—
108#235—66no—
109C3×Q16—93no—
110C3×Q16—96no—
111C3×SD16—96no—
112C3×D8—91no—
113C3×M16—93no—
114C2×C3×C8—93no—
115C22×C32—62no—
116C2×C3⋊D3C2×C3×D413yes—
117C2×C3×D3—22no—
118C2×C3×D3—22no—
119C2×C3×D3—42no—
120C2×C3×D3—66no—
121C32×C4—26no—
122C32×C4—46no—
123C32×C4—66no—
124C32×C4C2×C3×D413yes—
125C3⋊Dic3C24×C311yescharacteristic
126C3×Dic3—26no—
127C3×Dic3—26no—
128C3×Dic3—46no—
129C3×Dic3—62no—
1302−1+4—92no—
131#137—96no—
132#135—93no—
133C2×Q16—93no—
134#131—91no—
135C33#57811yescharacteristic, complemented, Sylow
136C3×Q8—22no—
137C3×Q8—42no—
138C3×Q8—66no—
139C3×Q8—66no—
140C3×Q8—96no—
141C3×Q8—126no—
142C3×Q8GO+(4,𝔽2)11yescharacteristic
143C3×D4—92no—
144C3×D4—93no—
145C3×D4—186no—
146C2×C3×C4—66no—
147C2×C3×C4—66no—
148C2×C3×C4—93no—
149C2×C3×C4—126no—
150C2×C3×C4—186no—
151#67—62no—
152#67—66no—
153D12—66no—
154D12—66no—
155C4×D3—26no—
156C4×D3—66no—
157C4×D3—612no—
158Dic6—26no—
159Dic6—66no—
160C3×C8—91no—
161C3×C8—93no—
162C2×C32—22no—
163C2×C32—42no—
164C2×C32—122no—
165C2×C32C22×C3×D411yescharacteristic
166C3⋊D3—23no—
167C3×D3—42no—
168C3×D3—44no—
169C3×D3—42no—
170C3×D3—126no—
171#41—96no—
172#41—91no—
173#41—186no—
174C2×Q8—62no—
175C2×Q8—96no—
176Q16—93no—
177Q16—96no—
178SD16—96no—
179D8—91no—
180M16—93no—
181C2×C8—93no—
182C22×C3—62no—
183C22×C3—62no—
184C22×C3—93no—
185C22×C3—122no—
186C2×D3—22no—
187C2×D3—66no—
188C3×C4—26no—
189C3×C4—46no—
190C3×C4—66no—
191C3×C4—66no—
192C3×C4—91no—
193C3×C4—126no—
194C3×C4C2×GO+(4,𝔽2)13yes—
195Dic3—26no—
196Dic3—62no—
197C32—22no—
198C32—42no—
199C32C3×#57811yescharacteristic, complemented
200Q8—66no—
201Q8—96no—
202Q8C3×GO+(4,𝔽2)11yescharacteristic
203D4—92no—
204D4—93no—
205D4—186no—
206C2×C4—66no—
207C2×C4—93no—
208C2×C4—186no—
209C8—91no—
210C8—93no—
211C2×C3—22no—
212C2×C3—42no—
213C2×C3—122no—
214C2×C3—124no—
215C2×C3—122no—
216C2×C3—183no—
217C2×C3C22×GO+(4,𝔽2)11yescharacteristic
218D3—42no—
219D3—126no—
220C22—62no—
221C22—93no—
222C4—66no—
223C4—91no—
224C4C2×C3×GO+(4,𝔽2)13yes—
225C3—22no—
226C3—42no—
227C3#1059652011yescharacteristic, direct product factor
228C2—122no—
229C2—183no—
230C2C22×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×#578
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×#64316C2semidirect
2. C3×#64289C2semidirect
3. C3×#64228C2semidirect
4. C3×#64226C2semidirect
5. C3×#64223C2semidirect
6. #10596520C3direct (N × H)
19. #64316C2×C3semidirect
20. #64289C2×C3semidirect
21. #64228C2×C3semidirect
22. #64226C2×C3semidirect
23. #64223C2×C3semidirect
135. C33#578semidirect
199. C32C3×#578semidirect
227. C3#10596520direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#643162
C3×#642892
C3×#642282
C3×#642262
C3×#642232
#105965203
C3×#5789