Finite Group Databasebeta

← C3×#10593275 #13166644

Subgroups — C3×#10593275 #13166644

Order 1728.

332 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 54 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×#63809C212yesmaximal, complemented
2C3×#63764C211yesmaximal, characteristic, complemented
3C3×#63736C211yesmaximal, characteristic, complemented
4C3×#63735C212yesmaximal, complemented
5C3×#63734C211yesmaximal, characteristic, complemented
6#10593275C311yesmaximal, characteristic, direct product factor
7C3×#3796—22no—
8C3×C8×C3⋊D3C2211yescharacteristic
9C32×#204—22no—
10C3×#3718C2212yes—
11C3×#3713C2212yes—
12C3×#3708C2211yescharacteristic
13C3×#3708C2211yescharacteristic
14C3×#916—62no—
15#63809C2×C312yescomplemented
16#63764C2×C311yescharacteristic, complemented
17#63736C2×C311yescharacteristic, complemented
18#63735C2×C312yescomplemented
19#63734C2×C311yescharacteristic, complemented
20C3×C4×C3⋊D3D411yescharacteristic
21C32×D12—22no—
22C32×Dic6—22no—
23C3×#626—42no—
24C3×D3×Dic3—42no—
25C33×C8D411yescharacteristic
26C3×C3⋊(C3⋊C8)C2311yescharacteristic
27C3×#508—91nomaximal
28C3×#239—62no—
29C3×#237—62no—
30#3796—22no—
31C32×SD16—62no—
32C8×C3⋊D3C22×C311yescharacteristic
33C3×#216—62no—
34C3×#213—62no—
35C3×#204—22no—
36C3×#204—22no—
37C3×#204—42no—
38C3×D3×C8—62no—
39#3718C22×C312yes—
40#3713C22×C312yes—
41#3708C22×C311yescharacteristic
42#3708C22×C311yescharacteristic
43C2×C3×C3⋊D3D811yescharacteristic
44C2×C32×D3—42no—
45C33×C4C2×D411yescharacteristic
46C3×C3⋊Dic3D811yescharacteristic
47C32×Dic3—42no—
48C3×#135—92no—
49#916—62no—
50C3×Q32—91no—
51C3×SD32—92no—
52C3×D16—91no—
53C2×C3×C16—91no—
54C32×Q8—62no—
55C32×D4—62no—
56C4×C3⋊D3C3×D411yescharacteristic
57C3×#67—122no—
58C2×C3×Dic3—122no—
59C3×D12—22no—
60C3×D12—22no—
61C3×D12—42no—
62C3×D12—122no—
63C3×C4×D3—62no—
64C3×C4×D3—122no—
65C3×Dic6—22no—
66C3×Dic6—22no—
67C3×Dic6—42no—
68#626—42no—
69D3×Dic3—42no—
70C32×C8—22no—
71C32×C8—42no—
72C32×C8C3×D411yescharacteristic
73C3⋊(C3⋊C8)C23×C311yescharacteristic
74C3×C3⋊C8—62no—
75#508—91noSylow
76C2×C33C2×D811yescharacteristic
77C3×C3⋊D3—21no—
78C32×D3—82no—
79C3×#41—182no—
80#239—62no—
81#237—62no—
82C3×Q16—92no—
83C3×SD16—62no—
84C3×SD16—62no—
85C3×SD16—122no—
86C3×D8—92no—
87C2×C3×C8—91no—
88#216—62no—
89#213—62no—
90#204—22no—
91D3×C8—62no—
92C3×C16—91no—
93C3×C16—91no—
94C22×C32—122no—
95C2×C3⋊D3C3×D811yescharacteristic
96C2×C3×D3—42no—
97C2×C3×D3—42no—
98C2×C3×D3—62no—
99C2×C3×D3—82no—
100C32×C4—22no—
101C32×C4—42no—
102C32×C4—122no—
103C32×C4C2×C3×D411yescharacteristic
104C3⋊Dic3C3×D811yescharacteristic
105C3×Dic3—42no—
106C3×Dic3—42no—
107C3×Dic3—62no—
108C3×Dic3—82no—
109#135—92no—
110Q32—91no—
111SD32—92no—
112D16—91no—
113C2×C16—91no—
114C33#50811yescharacteristic, complemented, Sylow
115C3×Q8—62no—
116C3×Q8—62no—
117C3×Q8—122no—
118C3×D4—62no—
119C3×D4—62no—
120C3×D4—122no—
121C3×D4—362no—
122C2×C3×C4—91no—
123C2×C3×C4—362no—
124#67—122no—
125C2×Dic3—122no—
126D12—22no—
127D12—122no—
128C4×D3—62no—
129C4×D3—122no—
130Dic6—22no—
131C3×C8—22no—
132C3×C8—42no—
133C3×C8—91no—
134C3×C8GO+(4,𝔽2)11yescharacteristic
135C3⋊C8—62no—
136C2×C32—22no—
137C2×C32—42no—
138C2×C32—242no—
139C2×C32C2×C3×D811yescharacteristic
140C3⋊D3—21no—
141C3×D3—82no—
142C3×D3—82no—
143C3×D3—122no—
144C3×D3—162no—
145#41—182no—
146Q16—92no—
147SD16—62no—
148D8—92no—
149C2×C8—91no—
150C16—91no—
151C16—91no—
152C22×C3—91no—
153C22×C3—122no—
154C22×C3—122no—
155C22×C3—242no—
156C2×D3—42no—
157C2×D3—62no—
158C3×C4—22no—
159C3×C4—42no—
160C3×C4—91no—
161C3×C4—122no—
162C3×C4—122no—
163C3×C4—242no—
164C3×C4C2×GO+(4,𝔽2)11yescharacteristic
165Dic3—42no—
166Dic3—62no—
167C32—22no—
168C32—42no—
169C32C3×#50811yescharacteristic, complemented
170Q8—62no—
171D4—62no—
172D4—362no—
173C2×C4—91no—
174C2×C4—362no—
175C8—91no—
176C8C3×GO+(4,𝔽2)11yescharacteristic
177C2×C3—22no—
178C2×C3—42no—
179C2×C3—181no—
180C2×C3—242no—
181C2×C3—242no—
182C2×C3—482no—
183C2×C3#6422511yescharacteristic
184D3—82no—
185D3—122no—
186C22—91no—
187C22—122no—
188C4—91no—
189C4—122no—
190C4C2×C3×GO+(4,𝔽2)11yescharacteristic
191C3—22no—
192C3—42no—
193C3#1059327511yescharacteristic, direct product factor
194C2—181no—
195C2—242no—
196C2C3×#6422511yescharacteristic
subgroupG/H
centerC2×C3#64225
commutator, cosocleC3⋊(C3⋊C8)C23×C3
FrattiniC4C2×C3×GO+(4,𝔽2)
FittingC33×C8D4
socleC2×C33C2×D8
hypercenterC3×C8GO+(4,𝔽2)
hypocenterC32C3×#508
LCSG ▷ C3⋊(C3⋊C8) ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×C8

Complements

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

NH ≅ G/Ntype
1. C3×#63809C2semidirect
2. C3×#63764C2semidirect
3. C3×#63736C2semidirect
4. C3×#63735C2semidirect
5. C3×#63734C2semidirect
6. #10593275C3direct (N × H)
15. #63809C2×C3semidirect
16. #63764C2×C3semidirect
17. #63736C2×C3semidirect
18. #63735C2×C3semidirect
19. #63734C2×C3semidirect
114. C33#508semidirect
169. C32C3×#508semidirect
193. C3#10593275direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#638092
C3×#637642
C3×#637362
C3×#637352
C3×#637342
#105932753
C3×#5089