Finite Group Databasebeta

← #13180656

Subgroups — #13180656

Order 1728.

280 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 28 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#11724155C212yesmaximal, complemented
3#10593839—31nomaximal
4C32×#41⋊C3C411yescharacteristic, complemented
5#88988—41nomaximal
6C3⋊D3×SL(2,𝔽3)C411yescharacteristic, complemented
7C3⋊(C3⋊(#41⋊C3))C2211yescharacteristic
8#41×C3⋊D3D311yescharacteristic
9C2×C4×#157—31no—
10#64283—31no—
11D3×#41⋊C3—62no—
12#63785—32no—
13#63783—32no—
14C3×C4×C3⋊D3—41no—
15C32×SL(2,𝔽3)C2×C411yescharacteristic, complemented
16C3⋊#622—42no—
17C2×GU(2,𝔽3)—91nomaximal
18C22×#157—61no—
19C32×#41Dic311yescharacteristic
20#3832C2×D311yescharacteristic
21Q8×C3⋊D3Dic311yescharacteristic
22#3830—31no—
23D4×C3⋊D3—31no—
24#3828—31no—
25C2×C4×C3⋊D3—31no—
26C3×#41⋊C3—22no—
27C3×#41⋊C3—42no—
28#3792—61no—
29C4×#157—32no—
30C4×#157—61no—
31#3788—32no—
32#3787—61no—
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—
40GU(2,𝔽3)—92no—
41C22×C3⋊D3—31no—
42C2×#157—62no—
43C2×#157—64no—
44C32×Q8C2×Dic311yescharacteristic
45C32×D4—31no—
46C2×C32×C4—31no—
47#638—61no—
48C2×C3⋊Dic3—31no—
49C4⋊(C3⋊D3)—31no—
50C4×C3⋊D3—31no—
51C4×C3⋊D3—31no—
52C4×C3⋊D3—31no—
53C4×C3⋊D3S411yescharacteristic
54#634—31no—
55C3×C4×D3—242no—
56C3×SL(2,𝔽3)—22no—
57C3×SL(2,𝔽3)—42no—
58#622—62no—
59C2×#104—271noSylow
60C2×C33—41no—
61C3×C3⋊D3—42no—
62C3×#41—22no—
63#239—62no—
64D3×Q8—62no—
65#237—182no—
66D3×D4—182no—
67#235—182no—
68C2×C4×D3—182no—
69#41⋊C3—42no—
70#41⋊C3—91no—
71#41⋊C3#15711yescharacteristic, complemented
72C2×SL(2,𝔽3)—91no—
73C2×C3⋊C8—361no—
74C22×C32—31no—
75C2×C3⋊D3—31no—
76C2×C3⋊D3—62no—
77C2×C3⋊D3SL(2,ℤ/4)11yescharacteristic
78C2×C3×D3—242no—
79#157—64no—
80C32×C4—31no—
81C32×C4—82no—
82C32×C4—162no—
83C32×C4SL(2,ℤ/4)11yescharacteristic
84C3⋊Dic3—31no—
85C3⋊Dic3C2×S411yescharacteristic
86C3×Dic3—242no—
87C2×#41—91no—
88C2×M16—271no—
89C2×C42—271no—
90#104—272no—
91#104—272no—
92C33—41noSylow
93C22×D3—182no—
94C3×Q8—22no—
95C3×D4—62no—
96C2×C3×C4—62no—
97C2×C3×C4—361no—
98#67—184no—
99C2×Dic3—182no—
100D12—182no—
101C4×D3—62no—
102C4×D3—182no—
103C4×D3—182no—
104C4×D3—182no—
105Dic6—182no—
106SL(2,𝔽3)—42no—
107SL(2,𝔽3)C2×#15711yescharacteristic, complemented
108C3⋊C8—362no—
109C2×C32—61no—
110C2×C32—82no—
111C2×C32—162no—
112C2×C32C2×SL(2,ℤ/4)11yescharacteristic
113C3⋊D3—61no—
114C3⋊D3GU(2,𝔽3)12yescomplemented
115C3×D3—244no—
116#41—91no—
117#41—271no—
118#41—271no—
119#41C3⋊#15711yescharacteristic
120C2×Q8—91no—
121C2×D4—271no—
122C22×C4—271no—
123C22×C4—541no—
124M16—272no—
125M16—541no—
126C2×C8—541no—
127C42—272no—
128C42—541no—
129C22×C3—62no—
130C22×C3—361no—
131C2×D3—62no—
132C2×D3—182no—
133C2×D3—364no—
134C3×C4—22no—
135C3×C4—41no—
136C3×C4—62no—
137C3×C4—162no—
138C3×C4—361no—
139Dic3—62no—
140Dic3—182no—
141C32—82no—
142C32—162no—
143C32C2×GU(2,𝔽3)11yescharacteristic, complemented
144C23—271no—
145Q8—271no—
146Q8C2×C3⋊#15711yescharacteristic
147D4—31no—
148D4—271no—
149D4—541no—
150C2×C4—31no—
151C2×C4—91no—
152C2×C4—271no—
153C2×C4—271no—
154C2×C4—271no—
155C2×C4—271no—
156C2×C4—542no—
157C2×C4—544no—
158C8—542no—
159C2×C3—22no—
160C2×C3—41no—
161C2×C3—122no—
162C2×C3—162no—
163C2×C3—362no—
164D3—64no—
165D3—362no—
166C22—31no—
167C22—91no—
168C22—271no—
169C22—542no—
170C4—31no—
171C4—91no—
172C4—271no—
173C4—544no—
174C4A4⋊#15711yescharacteristic
175C3—22no—
176C3—41no—
177C3—162no—
178C2—61no—
179C2—92no—
180C2—541no—
181C2C2×A4⋊#15711yescharacteristic
subgroupG/H
center, hypercenterC4A4⋊#157
commutator, hypocenterC32×SL(2,𝔽3)C2×C4
FrattiniC2C2×A4⋊#157
FittingC32×#41Dic3
socleC2×C32C2×SL(2,ℤ/4)
prime radicalC3⋊D3×#41⋊C3C2
cosocleC3⋊(C3⋊(#41⋊C3))C22
LCSG ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
2. #11724155C2semidirect
4. C32×#41⋊C3C4semidirect
6. C3⋊D3×SL(2,𝔽3)C4semidirect
15. C32×SL(2,𝔽3)C2×C4semidirect
71. #41⋊C3#157semidirect
107. SL(2,𝔽3)C2×#157semidirect
114. C3⋊D3GU(2,𝔽3)semidirect
143. C32C2×GU(2,𝔽3)semidirect

Maximal subgroups

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

subgroupindex
C3⋊D3×#41⋊C32
#117241552
#105938393
#889884
C2×GU(2,𝔽3)9