Finite Group Databasebeta

← C3⋊(#5934⋊C3) #13180331

Subgroups — C3⋊(#5934⋊C3) #13180331

Order 1728.

230 conjugacy classes of subgroups; only the 18 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 18 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×Ω+(4,𝔽3)C211yesmaximal, characteristic, complemented
2C3⋊#5934C311yesmaximal, characteristic, complemented
3#5934⋊C3—33nomaximal
4C3⋊(B(4,𝔽2)⋊C3)—31nomaximal
5C3×#997D311yescharacteristic, complemented
6C3×#997C2×C311yescharacteristic, complemented
7C3×#994—21no—
8Ω+(4,𝔽3)—23no—
9Ω+(4,𝔽3)D313yescomplemented
10C32×SL(2,𝔽3)—81no—
11#5934—33no—
12#5244—31no—
13B(4,𝔽2)⋊C3—91no—
14C2×D3×A4—121no—
15C2×C3⋊S4—121no—
16C3×#41⋊C3—61no—
17C2×C3×C3⋊D3—161nomaximal
18C3×H(5,𝔽2)C3×D311yescharacteristic, complemented
19#997—21no—
20#997C3×D313yescomplemented
21#997C3⋊D311yescharacteristic, complemented
22#994—23no—
23#937—91no—
24#834—91no—
25C2×C3×A4—121no—
26C2×C3×A4—121no—
27D3×A4—122no—
28C3⋊S4—122no—
29C32×Q8—81no—
30C3×SL(2,𝔽3)—21no—
31C3×SL(2,𝔽3)—83no—
32C3×SL(2,𝔽3)—81no—
33C3×SL(2,𝔽3)—81no—
34C3×SL(2,𝔽3)—86no—
35B(4,𝔽2)—91noSylow
36C2×C33—161no—
37C3×C3⋊D3—162no—
38C23×D3—31no—
39C22×A4—361no—
40C2×S4—363no—
41C3×#41—61no—
42C2×C3×D4—91no—
43C2×#67—181no—
44#41⋊C3—63no—
45#217—91no—
46(C2×D3)⋊C4—181no—
47C2×C3⋊D3—161no—
48C2×C3×D3—481no—
49C2×C3×D3—483no—
50C3×A4—121no—
51C3×A4—121no—
52C32×C4—241no—
53H(5,𝔽2)C3×C3⋊D311yescharacteristic, complemented
54#120—271no—
55#99—271no—
56C33—161noSylow
57C23×C3—31no—
58C23×C3—31no—
59C22×D3—122no—
60C22×D3—181no—
61C2×A4—123no—
62C2×A4—121no—
63C2×A4—241no—
64C2×A4—362no—
65S4—366no—
66C3×Q8—21no—
67C3×Q8—83no—
68C3×D4—181no—
69C2×C3×C4—91no—
70#67—362no—
71C2×Dic3—181no—
72SL(2,𝔽3)—23no—
73SL(2,𝔽3)—83no—
74SL(2,𝔽3)—81no—
75SL(2,𝔽3)—82no—
76C2×C32—81no—
77C2×C32—163no—
78C2×C32—161no—
79C2×C32—161no—
80C2×C32—323no—
81C3⋊D3—162no—
82C3×D3—482no—
83C3×D3—486no—
84C24—91no—
85#41—61no—
86C2×D4—91no—
87C2×D4—541no—
88C22⋊C4—271no—
89C22⋊C4—541no—
90C22×C3—91no—
91C22×C3—121no—
92C22×C3—121no—
93C22×C3—1441no—
94C2×D3—121no—
95C2×D3—182no—
96C2×D3—361no—
97C2×D3—483no—
98A4—123no—
99A4—121no—
100A4—241no—
101C3×C4—61no—
102C3×C4—243no—
103Dic3—361no—
104C32—81no—
105C32—163no—
106C32—161no—
107C32—161no—
108C32—323no—
109C23—31no—
110C23—31no—
111C23—362no—
112C23—541no—
113Q8—21no—
114D4—181no—
115D4—1082no—
116C2×C4—91no—
117C2×C4—541no—
118C2×C3—83no—
119C2×C3—163no—
120C2×C3—161no—
121C2×C3—181no—
122C2×C3—321no—
123C2×C3—1442no—
124C2×C3#1020⋊C311yescharacteristic
125D3—122no—
126D3—486no—
127C22—91no—
128C22—121no—
129C22—121no—
130C22—361no—
131C22—542no—
132C22—1081no—
133C4—61no—
134C4—1081no—
135C3—83no—
136C3—163no—
137C3—161no—
138C3—321no—
139C3#5934⋊C311yescharacteristic, complemented
140C2—181no—
141C2—362no—
142C2C3⋊(#1020⋊C3)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2C3⋊(#1020⋊C3)
commutator, cosocle, hypocenterC3×#997C2×C3
FittingC3×H(5,𝔽2)C3×D3
socleC2×C3#1020⋊C3
LCSG ▷ C3×#997
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C3×Ω+(4,𝔽3)C2semidirect
2. C3⋊#5934C3semidirect
5. C3×#997D3semidirect
6. C3×#997C2×C3semidirect
9. Ω+(4,𝔽3)D3semidirect
18. C3×H(5,𝔽2)C3×D3semidirect
20. #997C3×D3semidirect
21. #997C3⋊D3semidirect
53. H(5,𝔽2)C3×C3⋊D3semidirect
139. C3#5934⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×Ω+(4,𝔽3)2
C3⋊#59343
#5934⋊C33
C3⋊(B(4,𝔽2)⋊C3)3
C2×C3×C3⋊D316