Finite Group Databasebeta

← C3×#64266 #11724459

Subgroups — C3×#64266 #11724459

Order 864.

253 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 253 conjugacy classes of subgroups — too many to draw. Only its 42 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
1C32×#41⋊C3C211yesmaximal, characteristic, complemented
2C3×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3×C3⋊#226C211yesmaximal, characteristic, complemented
4#64266C311yesmaximal, characteristic, direct product factor
5C3×#985—33nomaximal
6C3×#950—31nomaximal
7C3×C4×C3⋊D3—41nomaximal
8C32×SL(2,𝔽3)C2211yescharacteristic
9C32×#41D311yescharacteristic, complemented
10C3×#235—31no—
11C3×#41⋊C3—21no—
12C3×#41⋊C3—23no—
13C3×#41⋊C3D313yescomplemented
14C3×#41⋊C3C2×C311yescharacteristic, complemented
15C3⋊GL(2,𝔽3)C2×C311yescharacteristic, complemented
16C3⋊#226C2×C311yescharacteristic, complemented
17C3×GL(2,𝔽3)—33no—
18C3×#226—33no—
19C3×#216—31no—
20C3×#215—31no—
21C3×#214—31no—
22C3×#213—31no—
23C2×C3×C3⋊C8—31no—
24C2×C3×C3⋊D3—41no—
25C33×C4—41no—
26C3×C3⋊Dic3—41no—
27#985—33no—
28C3×#135—91no—
29#950—31no—
30C32×Q8C2×D311yescharacteristic
31C32×D4—31no—
32C2×C32×C4—31no—
33C4×C3⋊D3—41no—
34C3×#67—61no—
35C3×D12—31no—
36C3×C4×D3—61no—
37C3×C4×D3—123no—
38C3×Dic6—31no—
39C3×SL(2,𝔽3)—21no—
40C3×SL(2,𝔽3)—23no—
41C3×SL(2,𝔽3)C2×D313yes—
42C3×SL(2,𝔽3)C22×C311yescharacteristic
43C3×C3⋊C8—31no—
44C3×C3⋊C8—31no—
45C2×C33—41no—
46C3×C3⋊D3—42no—
47C3×#41—21no—
48C3×#41—91no—
49C3×#41C3×D311yescharacteristic, complemented
50C3×#41C3⋊D311yescharacteristic, complemented
51#235—31no—
52#41⋊C3—23no—
53#41⋊C3C3×D313yescomplemented
54GL(2,𝔽3)—33no—
55#226—33no—
56C3×Q16—91no—
57C3×SD16—91no—
58C3×SD16—91no—
59C3×D8—91no—
60C2×C3×C8—91no—
61#216—31no—
62#215—31no—
63#214—31no—
64#213—31no—
65C2×C3⋊C8—31no—
66C22×C32—31no—
67C2×C3⋊D3—41no—
68C2×C3×D3—61no—
69C2×C3×D3—123no—
70C32×C4—31no—
71C32×C4—41no—
72C32×C4—43no—
73C32×C4—83no—
74C32×C4—81no—
75C32×C4S411yescharacteristic
76C3⋊Dic3—41no—
77C3×Dic3—61no—
78C3×Dic3—123no—
79#135—91noSylow
80C33—41noSylow
81C3×Q8—21no—
82C3×Q8—91no—
83C3×Q8C2×C3×D311yescharacteristic
84C3×Q8C2×C3⋊D311yescharacteristic
85C3×D4—31no—
86C3×D4—31no—
87C3×D4—61no—
88C3×D4—91no—
89C3×D4—181no—
90C2×C3×C4—31no—
91C2×C3×C4—31no—
92C2×C3×C4—61no—
93C2×C3×C4—181no—
94#67—61no—
95D12—31no—
96C4×D3—61no—
97C4×D3—123no—
98Dic6—31no—
99SL(2,𝔽3)—23no—
100SL(2,𝔽3)C2×C3×D313yes—
101C3×C8—91no—
102C3×C8—91no—
103C3⋊C8—31no—
104C3⋊C8—31no—
105C2×C32—41no—
106C2×C32—43no—
107C2×C32—61no—
108C2×C32—83no—
109C2×C32—81no—
110C2×C32C2×S411yescharacteristic
111C3⋊D3—42no—
112C3×D3—126no—
113C3×D3—121no—
114#41—91no—
115#41C3×C3⋊D311yescharacteristic, complemented
116Q16—91no—
117SD16—91no—
118SD16—91no—
119D8—91no—
120C2×C8—91no—
121C22×C3—31no—
122C22×C3—31no—
123C22×C3—61no—
124C22×C3—181no—
125C2×D3—61no—
126C2×D3—123no—
127C3×C4—21no—
128C3×C4—31no—
129C3×C4—31no—
130C3×C4—43no—
131C3×C4—61no—
132C3×C4—83no—
133C3×C4—181no—
134C3×C4C3×S411yescharacteristic
135C3×C4C3⋊S411yescharacteristic
136Dic3—61no—
137Dic3—123no—
138C32—41no—
139C32—43no—
140C32—83no—
141C32—81no—
142C32#98511yescharacteristic, complemented
143Q8—91no—
144Q8C2×C3×C3⋊D311yescharacteristic
145D4—31no—
146D4—91no—
147D4—181no—
148C2×C4—31no—
149C2×C4—181no—
150C8—91no—
151C8—91no—
152C2×C3—21no—
153C2×C3—43no—
154C2×C3—61no—
155C2×C3—61no—
156C2×C3—83no—
157C2×C3—121no—
158C2×C3—361no—
159C2×C3C2×C3×S411yescharacteristic
160C2×C3C2×C3⋊S411yescharacteristic
161D3—126no—
162D3—121no—
163C22—31no—
164C22—181no—
165C4—31no—
166C4—181no—
167C4C3×C3⋊S411yescharacteristic
168C3—21no—
169C3—43no—
170C3—83no—
171C3C3×#98511yescharacteristic, complemented
172C3#6426611yescharacteristic, direct product factor
173C2—61no—
174C2—361no—
175C2C2×C3×C3⋊S411yescharacteristic
subgroupG/H
center, hypercenterC3×C4C3⋊S4
commutator, cosocle, hypocenterC3×SL(2,𝔽3)C22×C3
FrattiniC2C2×C3×C3⋊S4
FittingC32×#41D3
socleC2×C32C2×S4
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C3×C4

Complements

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

NH ≅ G/Ntype
1. C32×#41⋊C3C2semidirect
2. C3×C3⋊GL(2,𝔽3)C2semidirect
3. C3×C3⋊#226C2semidirect
4. #64266C3direct (N × H)
9. C32×#41D3semidirect
13. C3×#41⋊C3D3semidirect
14. C3×#41⋊C3C2×C3semidirect
15. C3⋊GL(2,𝔽3)C2×C3semidirect
16. C3⋊#226C2×C3semidirect
49. C3×#41C3×D3semidirect
50. C3×#41C3⋊D3semidirect
53. #41⋊C3C3×D3semidirect
115. #41C3×C3⋊D3semidirect
142. C32#985semidirect
171. C3C3×#985semidirect
172. C3#64266direct (N × H)

Maximal subgroups

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

subgroupindex
C32×#41⋊C32
C3×C3⋊GL(2,𝔽3)2
C3×C3⋊#2262
#642663
C3×#9853
C3×#9503
C3×C4×C3⋊D34