Finite Group Databasebeta

← #13180659

Subgroups — #13180659

Order 1728.

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

normal subgroups only

This group has 276 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#11724156C212yesmaximal, complemented
3#10593840—31nomaximal
4C32×#41⋊C3C411yescharacteristic, complemented
5C4⋊(C3⋊#157)—41nomaximal
6C3⋊D3×SL(2,𝔽3)C2211yescharacteristic
7C3⋊(C3⋊(#41⋊C3))C411yescharacteristic, complemented
8#41×C3⋊D3D311yescharacteristic
9C2×C4⋊#157—31no—
10#64283—31no—
11D3×#41⋊C3—62no—
12#63784—32no—
13#63782—32no—
14C2×C3⋊#157—42no—
15C3×C4×C3⋊D3—41no—
16C32×SL(2,𝔽3)C2×C411yescharacteristic, complemented
17#5426—91nomaximal
18C22×#157—61no—
19C32×#41Dic311yescharacteristic, complemented
20#3832Dic311yescharacteristic, complemented
21Q8×C3⋊D3C2×D311yescharacteristic
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—61no—
32#3787—32no—
33D3×SL(2,𝔽3)—62no—
34C3⋊(#41⋊C3)—62no—
35C2×C3×C3⋊D3—41no—
36C3⋊#157—82no—
37C33×C4—41no—
38C3×C3⋊Dic3—41no—
39D3×#41—62no—
40C2×#41⋊C3—91no—
41#859—92no—
42C22×C3⋊D3—31no—
43C2×#157—62no—
44C2×#157—122no—
45C32×Q8C2×Dic311yescharacteristic
46C32×D4—31no—
47C2×C32×C4—31no—
48#638—61no—
49C2×C3⋊Dic3—31no—
50C4⋊(C3⋊D3)—31no—
51C4×C3⋊D3—31no—
52C4×C3⋊D3—31no—
53C4×C3⋊D3—31no—
54C4×C3⋊D3S411yescharacteristic
55#634—31no—
56C3×C4×D3—242no—
57C3×SL(2,𝔽3)—22no—
58C3×SL(2,𝔽3)—42no—
59#622—62no—
60#417—271noSylow
61C2×C33—41no—
62C3×C3⋊D3—41no—
63C3×C3⋊D3—41no—
64C3×#41—22no—
65#239—62no—
66D3×Q8—62no—
67#237—182no—
68D3×D4—182no—
69#235—182no—
70C2×C4×D3—182no—
71#41⋊C3—42no—
72#41⋊C3—91no—
73#41⋊C3#15711yescharacteristic, complemented
74C2×SL(2,𝔽3)—91no—
75C4⋊Dic3—361no—
76C22×C32—31no—
77C2×C3⋊D3—31no—
78C2×C3⋊D3—61no—
79C2×C3⋊D3—61no—
80C2×C3⋊D3C2×S411yescharacteristic
81C2×C3×D3—242no—
82#157—122no—
83C32×C4—31no—
84C32×C4—82no—
85C32×C4—162no—
86C32×C4SL(2,ℤ/4)11yescharacteristic
87C3⋊Dic3—31no—
88C3⋊Dic3SL(2,ℤ/4)11yescharacteristic
89C3×Dic3—242no—
90C2×#41—91no—
91C2×M16—271no—
92C2×C4⋊C4—271no—
93#103—272no—
94#102—272no—
95C33—41noSylow
96C22×D3—182no—
97C3×Q8—22no—
98C3×D4—62no—
99C2×C3×C4—62no—
100C2×C3×C4—361no—
101#67—184no—
102C2×Dic3—182no—
103C2×Dic3—362no—
104D12—182no—
105C4×D3—62no—
106C4×D3—182no—
107C4×D3—182no—
108C4×D3—182no—
109Dic6—182no—
110SL(2,𝔽3)—42no—
111SL(2,𝔽3)C2×#15711yescharacteristic, complemented
112C2×C32—61no—
113C2×C32—82no—
114C2×C32—162no—
115C2×C32C2×SL(2,ℤ/4)11yescharacteristic
116C3⋊D3—61no—
117C3⋊D3#98411yescharacteristic
118C3⋊D3#98611yescharacteristic
119C3×D3—242no—
120C3×D3—242no—
121#41—91no—
122#41—271no—
123#41—271no—
124#41C3⋊#15711yescharacteristic, complemented
125C2×Q8—91no—
126C2×D4—271no—
127C22×C4—271no—
128C22×C4—541no—
129M16—541no—
130M16—541no—
131C2×C8—272no—
132C4⋊C4—272no—
133C4⋊C4—541no—
134C22×C3—62no—
135C22×C3—361no—
136C2×D3—62no—
137C2×D3—182no—
138C2×D3—362no—
139C2×D3—362no—
140C3×C4—22no—
141C3×C4—41no—
142C3×C4—62no—
143C3×C4—162no—
144C3×C4—361no—
145Dic3—62no—
146Dic3—182no—
147Dic3—722no—
148C32—82no—
149C32—162no—
150C32#542611yescharacteristic, complemented
151C23—271no—
152Q8—271no—
153Q8C2×C3⋊#15711yescharacteristic
154D4—31no—
155D4—271no—
156D4—541no—
157C2×C4—31no—
158C2×C4—91no—
159C2×C4—271no—
160C2×C4—271no—
161C2×C4—271no—
162C2×C4—271no—
163C2×C4—542no—
164C2×C4—1082no—
165C8—542no—
166C2×C3—22no—
167C2×C3—41no—
168C2×C3—122no—
169C2×C3—162no—
170C2×C3—361no—
171C2×C3—361no—
172D3—62no—
173D3—62no—
174D3—362no—
175C22—31no—
176C22—91no—
177C22—271no—
178C22—541no—
179C22—541no—
180C4—31no—
181C4—91no—
182C4—271no—
183C4—1082no—
184C4A4⋊#15711yescharacteristic
185C3—22no—
186C3—41no—
187C3—162no—
188C2—61no—
189C2—91no—
190C2—91no—
191C2—541no—
192C2C2×A4⋊#15711yescharacteristic
subgroupG/H
center, FrattiniC2C2×A4⋊#157
commutator, hypocenterC32×SL(2,𝔽3)C2×C4
FittingC32×#41Dic3
socleC2×C32C2×SL(2,ℤ/4)
prime radicalC3⋊D3×#41⋊C3C2
cosocleC3⋊D3×SL(2,𝔽3)C22
hypercenterC4A4⋊#157
LCSG ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
2. #11724156C2semidirect
4. C32×#41⋊C3C4semidirect
7. C3⋊(C3⋊(#41⋊C3))C4semidirect
16. C32×SL(2,𝔽3)C2×C4semidirect
19. C32×#41Dic3semidirect
20. #3832Dic3semidirect
73. #41⋊C3#157semidirect
111. SL(2,𝔽3)C2×#157semidirect
124. #41C3⋊#157semidirect
150. C32#5426semidirect

Maximal subgroups

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

subgroupindex
C3⋊D3×#41⋊C32
#117241562
#105938403
C4⋊(C3⋊#157)4
#54269