Finite Group Databasebeta

← #10596648⋊C3 #13180672

Subgroups — #10596648⋊C3 #13180672

Order 1728.

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

normal subgroups only

This group has 264 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
1C3⋊D3×#41⋊C3C211yesmaximal, characteristic
2#64289⋊C3C212yesmaximal, complemented
3#10596648C311yesmaximal, characteristic, complemented
4C32×#41⋊C3C411yescharacteristic
5C3×#3788—41nomaximal
6C3⋊D3×SL(2,𝔽3)C411yescharacteristic
7C3⋊(C3⋊(#41⋊C3))C2211yescharacteristic
8#41×C3⋊D3C2×C311yescharacteristic
9#64289C2×C312yescomplemented
10#64287—32no—
11#64283—31no—
12C2×#3788—31no—
13D3×#41⋊C3—62no—
14C3×C4×C3⋊D3—41no—
15C32×SL(2,𝔽3)C2×C411yescharacteristic
16C3×#622—42no—
17#568⋊C3—91nomaximal
18C32×#41C3×C411yescharacteristic
19#3832C22×C311yescharacteristic
20Q8×C3⋊D3C3×C411yescharacteristic
21#3830—31no—
22D4×C3⋊D3—31no—
23#3828—31no—
24C2×C4×C3⋊D3—31no—
25C3×#41⋊C3—22no—
26C3×#41⋊C3—42no—
27#3792—32no—
28C2×#622—32no—
29#3788—31no—
30#3788—32no—
31#3788A411yescharacteristic
32#3787—32no—
33D3×SL(2,𝔽3)—62no—
34C3⋊(#41⋊C3)—62no—
35C2×C3×C3⋊D3—41no—
36C33×C4—41no—
37C3×C3⋊Dic3—41no—
38D3×#41—62no—
39C2×#41⋊C3—91no—
40#131⋊C3—92no—
41C22×C3⋊D3—31no—
42C32×Q8C2×C3×C411yescharacteristic
43C32×D4—31no—
44C2×C32×C4—31no—
45#638—32no—
46C2×C3⋊Dic3—31no—
47C4⋊(C3⋊D3)—31no—
48C4×C3⋊D3—31no—
49C4×C3⋊D3—31no—
50C4×C3⋊D3—31no—
51C4×C3⋊D3C2×A411yescharacteristic
52#634—31no—
53C3×C4×D3—242no—
54C3×SL(2,𝔽3)—22no—
55C3×SL(2,𝔽3)—42no—
56#622—32no—
57#622C2×A412yes—
58#568—91noSylow
59C2×C33—41no—
60C3×C3⋊D3—81no—
61C3×#41—22no—
62#239—62no—
63D3×Q8—62no—
64#237—182no—
65D3×D4—182no—
66#235—182no—
67C2×C4×D3—182no—
68#41⋊C3—42no—
69#41⋊C3—91no—
70#41⋊C3#15711yescharacteristic
71C2×SL(2,𝔽3)—91no—
72C3×M16—361no—
73C22×C32—31no—
74C2×C3⋊D3—31no—
75C2×C3⋊D3—121no—
76C2×C3⋊D3C4×A411yescharacteristic
77C2×C3×D3—242no—
78C32×C4—31no—
79C32×C4—82no—
80C32×C4—162no—
81C32×C4C4×A411yescharacteristic
82C3⋊Dic3—31no—
83C3⋊Dic3C22×A411yescharacteristic
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×D4—62no—
94C2×C3×C4—62no—
95C2×C3×C4—361no—
96#67—184no—
97C2×Dic3—182no—
98D12—182no—
99C4×D3—62no—
100C4×D3—182no—
101C4×D3—182no—
102C4×D3—182no—
103Dic6—182no—
104SL(2,𝔽3)—42no—
105SL(2,𝔽3)C2×#15711yescharacteristic
106C3×C8—362no—
107C2×C32—61no—
108C2×C32—82no—
109C2×C32—162no—
110C2×C32C2×C4×A411yescharacteristic
111C3⋊D3—21no—
112C3⋊D3—61no—
113C3×D3—244no—
114#41—91no—
115#41—271no—
116#41—271no—
117#41C3×#15711yescharacteristic
118C2×Q8—91no—
119C2×D4—271no—
120C22×C4—271no—
121M16—91no—
122M16—272no—
123M16—271no—
124M16—272no—
125C2×C8—272no—
126C2×C8—272no—
127C22×C3—62no—
128C22×C3—361no—
129C2×D3—62no—
130C2×D3—182no—
131C2×D3—364no—
132C3×C4—22no—
133C3×C4—41no—
134C3×C4—62no—
135C3×C4—162no—
136C3×C4—361no—
137Dic3—62no—
138Dic3—182no—
139C32—82no—
140C32—162no—
141C32#568⋊C311yescharacteristic, complemented
142C23—271no—
143Q8—271no—
144Q8C2×C3×#15711yescharacteristic
145D4—31no—
146D4—271no—
147D4—272no—
148C2×C4—31no—
149C2×C4—91no—
150C2×C4—271no—
151C2×C4—271no—
152C2×C4—271no—
153C2×C4—271no—
154C8—92no—
155C8—272no—
156C2×C3—22no—
157C2×C3—41no—
158C2×C3—122no—
159C2×C3—162no—
160C2×C3—721no—
161D3—64no—
162D3—362no—
163C22—31no—
164C22—91no—
165C22—271no—
166C22—1081no—
167C4—31no—
168C4—91no—
169C4—271no—
170C4A4×#15711yescharacteristic
171C3—22no—
172C3—41no—
173C3—162no—
174C2—61no—
175C2—181no—
176C2—541no—
177C2C2×A4×#15711yescharacteristic
subgroupG/H
center, FrattiniC2C2×A4×#157
commutator, hypocenterC32×Q8C2×C3×C4
FittingC32×#41C3×C4
socleC2×C32C2×C4×A4
prime radicalC3⋊D3×#41⋊C3C2
cosocle#3832C22×C3
hypercenterC4A4×#157
LCSG ▷ C32×Q8
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
2. #64289⋊C3C2semidirect
3. #10596648C3semidirect
9. #64289C2×C3semidirect
141. C32#568⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3⋊D3×#41⋊C32
#64289⋊C32
#105966483
C3×#37884
#568⋊C39