Finite Group Databasebeta

← #13166103

Subgroups — #13166103

Order 1728.

344 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 36 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×#64267C212yesmaximal, complemented
2C3⋊(C3⋊(#131⋊C3))C211yesmaximal, characteristic, complemented
3#10593225—32nomaximal
4#10591678—31nomaximal
5C32×#41⋊C3C2211yescharacteristic
6C3×C3⋊GL(2,𝔽3)—22no—
7#88810—41nomaximal
8#64267D312yescomplemented
9C3×#986—32no—
10C3×#986—62no—
11#64157D311yescharacteristic, complemented
12C3×#949—32no—
13C2×#3713—31no—
14C3⋊(#131⋊C3)—32no—
15C3⋊(#131⋊C3)—61no—
16#63578—31no—
17#63558—32no—
18C3×C4⋊(C3⋊D3)—42no—
19C32×SL(2,𝔽3)D411yescharacteristic
20C3×C3⋊(C3⋊C8)—41no—
21#5407—91no—
22#5199—92no—
23C32×#41C2×D311yescharacteristic
24C2×C3×D12—32no—
25C3×#41⋊C3—22no—
26C3×#41⋊C3—21no—
27C3×#41⋊C3—41no—
28C3×#41⋊C3C2×D312yes—
29C3⋊GL(2,𝔽3)—22no—
30C3×GL(2,𝔽3)—64no—
31C3×GL(2,𝔽3)—62no—
32#3748—31no—
33C2×C3⋊(C3⋊C8)—31no—
34C3×#215—62no—
35C3×#213—62no—
36C3×#208—32no—
37#3714—122no—
38#3713—61no—
39#3713—61no—
40C2×C3×C3⋊D3—82no—
41C33×C4—41no—
42#986—32no—
43#986—62no—
44C3×AGL(1,ℤ/8)—92no—
45#949—32no—
46#948—32no—
47#948—61no—
48C2×#213—92no—
49#131⋊C3—91no—
50#833—92no—
51#823—92no—
52#822—92no—
53C22×C3×D3—62no—
54C32×Q8D1211yescharacteristic
55C32×D4—31no—
56C2×C32×C4—31no—
57C4⋊(C3⋊D3)—42no—
58C3×D12—62no—
59C3×D12—62no—
60C3×D12—122no—
61C3×D12—242no—
62C3×SL(2,𝔽3)—22no—
63C3×SL(2,𝔽3)—21no—
64C3×SL(2,𝔽3)—41no—
65C3×SL(2,𝔽3)#6712yes—
66C3⋊(C3⋊C8)—31no—
67C3⋊(C3⋊C8)S411yescharacteristic
68C3×C3⋊C8—62no—
69C3×C3⋊C8—122no—
70C3×C3⋊C8—241no—
71#472—271noSylow
72C2×C33—41no—
73C3×C3⋊D3—162no—
74C3×#41—21no—
75C3×#41D3212yes—
76C2×C3×D4—92no—
77C2×D12—32no—
78#41⋊C3—22no—
79#41⋊C3—41no—
80#41⋊C3D3211yescharacteristic
81GL(2,𝔽3)—64no—
82GL(2,𝔽3)—62no—
83C3×SD16—182no—
84C3×D8—182no—
85C3×M16—92no—
86#215—62no—
87#213—62no—
88#213—182no—
89#213—182no—
90#208—32no—
91#208—92no—
92#208—181no—
93C2×C3⋊C8—92no—
94C2×C3⋊C8—181no—
95D24—361no—
96C22×C32—31no—
97C2×C3⋊D3—82no—
98C2×C3×D3—122no—
99C2×C3×D3—242no—
100C2×C3×D3—244no—
101C2×C3×D3—242no—
102C32×C4—31no—
103C32×C4—42no—
104C32×C4—81no—
105C32×C4—82no—
106C32×C4—161no—
107C32×C4C2×S411yescharacteristic
108AGL(1,ℤ/8)—92no—
109C2×D8—271no—
110#131—91no—
111#108—271no—
112#100—272no—
113C33—41noSylow
114C23×C3—182no—
115C22×D3—62no—
116C3×Q8—21no—
117C3×Q8#62612yes—
118C3×D4—32no—
119C3×D4—61no—
120C3×D4—182no—
121C3×D4—182no—
122C2×C3×C4—32no—
123C2×C3×C4—61no—
124D12—62no—
125D12—62no—
126D12—122no—
127D12—242no—
128SL(2,𝔽3)—22no—
129SL(2,𝔽3)—41no—
130SL(2,𝔽3)#62511yescharacteristic
131C3×C8—182no—
132C3×C8—361no—
133C3⋊C8—32no—
134C3⋊C8—61no—
135C3⋊C8—62no—
136C3⋊C8—92no—
137C3⋊C8—181no—
138C2×C32—42no—
139C2×C32—61no—
140C2×C32—81no—
141C2×C32—82no—
142C2×C32—161no—
143C2×C32C4⋊S411yescharacteristic
144C3⋊D3—162no—
145C3×D3—242no—
146C3×D3—482no—
147C3×D3—484no—
148#41#111311yescharacteristic
149C2×D4—92no—
150SD16—182no—
151D8—182no—
152D8—541no—
153D8—541no—
154M16—92no—
155M16—271no—
156C2×C8—271no—
157C22×C3—32no—
158C22×C3—61no—
159C22×C3—362no—
160C22×C3—722no—
161C2×D3—122no—
162C2×D3—242no—
163C2×D3—244no—
164C2×D3—242no—
165C3×C4—21no—
166C3×C4—32no—
167C3×C4—41no—
168C3×C4—61no—
169C3×C4—82no—
170C3×C4—161no—
171C3×C4D3×S412yes—
172C32—42no—
173C32—81no—
174C32—82no—
175C32—161no—
176C32#540711yescharacteristic, complemented
177C23—182no—
178Q8#629511yescharacteristic
179D4—31no—
180D4—182no—
181D4—182no—
182C2×C4—31no—
183C8—91no—
184C8—182no—
185C8—271no—
186C2×C3—21no—
187C2×C3—41no—
188C2×C3—62no—
189C2×C3—82no—
190C2×C3—121no—
191C2×C3—161no—
192C2×C3—722no—
193C2×C3#6420712yes—
194D3—242no—
195D3—482no—
196D3—484no—
197C22—31no—
198C22—362no—
199C22—722no—
200C4—31no—
201C4#8910111yescharacteristic
202C3—21no—
203C3—41no—
204C3—82no—
205C3—161no—
206C3#1059322512yescomplemented
207C2—61no—
208C2—722no—
209C2#1172421411yescharacteristic
subgroupG/H
centerC2#11724214
commutator, cosocleC32×#41⋊C3C22
Frattini, hypercenterC4#89101
FittingC32×#41C2×D3
socleC2×C32C4⋊S4
hypocenterC32×SL(2,𝔽3)D4
LCSG ▷ C32×#41⋊C3 ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C3×#64267C2semidirect
2. C3⋊(C3⋊(#131⋊C3))C2semidirect
8. #64267D3semidirect
11. #64157D3semidirect
176. C32#5407semidirect
206. C3#10593225semidirect

Maximal subgroups

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

subgroupindex
C3×#642672
C3⋊(C3⋊(#131⋊C3))2
#105932253
#105916783
#888104