Finite Group Databasebeta

← #10596644⋊C3 #13180668

Subgroups — #10596644⋊C3 #13180668

Order 1728.

272 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 32 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
1C2×C3⋊(C3⋊(#41⋊C3))C211yesmaximal, characteristic
2#64289⋊C3C212yesmaximal, complemented
3#10596644C311yesmaximal, characteristic, complemented
4C2×C32×SL(2,𝔽3)C411yescharacteristic
5C3×#3792—41nomaximal
6C3⋊(C3⋊(#41⋊C3))C411yescharacteristic
7C3⋊(C3⋊(#41⋊C3))C2211yescharacteristic
8C2×#3832C2×C311yescharacteristic
9#64289—32no—
10#64289C2×C312yescomplemented
11#64283—31no—
12#64283—31no—
13C2×C3⋊(#41⋊C3)—62no—
14C2×C3×C3⋊Dic3—41no—
15C32×SL(2,𝔽3)C2×C411yescharacteristic
16C3×#622—42no—
17#568⋊C3—91nomaximal
18C2×C32×Q8C3×C411yescharacteristic
19#3832—31no—
20#3832—31no—
21#3832C3×C411yescharacteristic
22#3832C22×C311yescharacteristic
23C2×C4⋊(C3⋊D3)—31no—
24C2×C4×C3⋊D3—31no—
25C2×C3×SL(2,𝔽3)—22no—
26C2×C3×SL(2,𝔽3)—42no—
27#3792—31no—
28#3792A411yescharacteristic
29#3788—32no—
30#3788—32no—
31#3787—32no—
32#3787—32no—
33C3⋊(#41⋊C3)—62no—
34C3⋊(#41⋊C3)—62no—
35C22×C33—41no—
36C3×C3⋊Dic3—41no—
37C3×C3⋊Dic3—41no—
38C2×#239—62no—
39C2×#41⋊C3—91no—
40#131⋊C3—92no—
41C22×C3⋊D3—31no—
42C32×Q8—31no—
43C32×Q8C2×C3×C411yescharacteristic
44C2×C32×C4—31no—
45C2×C3⋊Dic3C2×A411yescharacteristic
46C4⋊(C3⋊D3)—31no—
47C4⋊(C3⋊D3)—31no—
48C4⋊(C3⋊D3)—32no—
49C4×C3⋊D3—31no—
50C4×C3⋊D3—31no—
51C4×C3⋊D3—31no—
52C4×C3⋊D3—31no—
53C2×C3×Dic3—242no—
54C3×SL(2,𝔽3)—22no—
55C3×SL(2,𝔽3)—42no—
56#622—32no—
57#622C2×A412yes—
58#568—91noSylow
59C2×C33—41no—
60C2×C33—81no—
61C2×C3×Q8—22no—
62#239—62no—
63#239—62no—
64#239—182no—
65#239—182no—
66C2×D12—182no—
67C2×C4×D3—182no—
68#41⋊C3—91no—
69#41⋊C3—91no—
70C2×SL(2,𝔽3)—42no—
71C2×SL(2,𝔽3)#15711yescharacteristic
72C3×M16—361no—
73C22×C32—82no—
74C22×C32—162no—
75C22×C32C4×A411yescharacteristic
76C2×C3⋊D3—31no—
77C2×C3⋊D3—31no—
78C2×C3⋊D3—121no—
79C32×C4—31no—
80C32×C4—31no—
81C3⋊Dic3C4×A411yescharacteristic
82C3⋊Dic3C22×A411yescharacteristic
83C3×Dic3—242no—
84C3×Dic3—242no—
85C2×#41—91no—
86#131—92no—
87#131—272no—
88C2×M16—271no—
89C2×M16—271no—
90C33—41noSylow
91C22×D3—182no—
92C3×Q8—22no—
93C3×Q8—62no—
94C2×C3×C4—62no—
95C2×C3×C4—361no—
96C2×Dic3—62no—
97D12—182no—
98D12—182no—
99D12—184no—
100C4×D3—182no—
101C4×D3—182no—
102C4×D3—182no—
103C4×D3—182no—
104SL(2,𝔽3)—42no—
105SL(2,𝔽3)C2×#15711yescharacteristic
106C3×C8—362no—
107C2×C32—21no—
108C2×C32—82no—
109C2×C32—84no—
110C2×C32—164no—
111C2×C32—162no—
112C2×C32C2×C4×A411yescharacteristic
113C3⋊D3—61no—
114C3⋊D3—61no—
115#41—91no—
116#41—91no—
117#41—271no—
118#41—271no—
119C2×Q8C3×#15711yescharacteristic
120C2×D4—271no—
121C22×C4—271no—
122M16—91no—
123M16—272no—
124M16—271no—
125M16—272no—
126C2×C8—272no—
127C2×C8—272no—
128C22×C3—22no—
129C22×C3—41no—
130C22×C3—162no—
131C2×D3—182no—
132C2×D3—182no—
133C2×D3—364no—
134C3×C4—62no—
135C3×C4—62no—
136C3×C4—361no—
137C3×C4—361no—
138Dic3—62no—
139Dic3—62no—
140C32—82no—
141C32—162no—
142C32#568⋊C311yescharacteristic, complemented
143C23—271no—
144Q8—31no—
145Q8C2×C3×#15711yescharacteristic
146D4—271no—
147D4—271no—
148D4—272no—
149C2×C4—31no—
150C2×C4—91no—
151C2×C4—271no—
152C2×C4—271no—
153C2×C4—271no—
154C2×C4—271no—
155C8—92no—
156C8—272no—
157C2×C3—24no—
158C2×C3—22no—
159C2×C3—41no—
160C2×C3—81no—
161C2×C3—164no—
162C2×C3—162no—
163D3—362no—
164D3—362no—
165C22—271no—
166C22—271no—
167C22—1081no—
168C22A4×#15711yescharacteristic
169C4—31no—
170C4—31no—
171C4—91no—
172C4—91no—
173C3—22no—
174C3—41no—
175C3—162no—
176C2—21no—
177C2—541no—
178C2—541no—
179C2C2×A4×#15711yescharacteristic
subgroupG/H
center, FrattiniC2C2×A4×#157
commutator, hypocenterC32×Q8C2×C3×C4
FittingC2×C32×Q8C3×C4
socleC2×C32C2×C4×A4
prime radicalC2×C3⋊(C3⋊(#41⋊C3))C2
cosocle#3832C22×C3
hypercenterC22A4×#157
LCSG ▷ C32×Q8
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
2. #64289⋊C3C2semidirect
3. #10596644C3semidirect
10. #64289C2×C3semidirect
142. C32#568⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊(C3⋊(#41⋊C3))2
#64289⋊C32
#105966443
C3×#37924
#568⋊C39