Finite Group Databasebeta

← #13173081

Subgroups — #13173081

Order 1728.

736 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C32×2−1+4⋊C3C211yesmaximal, characteristic, complemented
2#11724471C211yesmaximal, characteristic, complemented
3C3⋊(C3⋊GU(2,𝔽3))C211yesmaximal, characteristic, complemented
4#10593665—312nomaximal
5#10592944—31nomaximal
6C32×#41⋊C3C2211yescharacteristic
7C2×C32×SL(2,𝔽3)—21no—
8C3⋊(C3⋊GL(2,𝔽3))—21no—
9#88860—41nomaximal
10C32×2−1+4D311yescharacteristic, complemented
11C3×2−1+4⋊C3D3112yescomplemented
12#64267—312no—
13#64158—31no—
14#64158—31no—
15#64156—31no—
16C3⋊GU(2,𝔽3)—312no—
17#63664—31no—
18#63664—31no—
19#63663—31no—
20C33×D4—41no—
21C4⋊(C3⋊(C3⋊D3))—41no—
22C32×SL(2,𝔽3)D411yescharacteristic
23C3⋊(C3⋊(C3⋊C8))—41no—
24#5431—99no—
25#5245—94no—
26C32×#41—31no—
27C32×#41—61no—
28C32×#41C2×D311yescharacteristic
29C2×C32×Q8—21no—
30C2×C32×Q8—31no—
31C2×C4⋊(C3⋊D3)—31no—
32C3×#41⋊C3C2×D3112yes—
33C2×C3×SL(2,𝔽3)—212no—
34C3⋊GL(2,𝔽3)—612no—
35#3755—61no—
36#3755—61no—
37#3753—61no—
38#3753—61no—
39#3753—1212no—
40#3752—62no—
41C4×C3⋊Dic3—31no—
42#3748—31no—
43#3748—31no—
44C22×C33—81no—
45C2×C3⋊(C3⋊D3)—81no—
46C33×C4—41no—
47C3×2−1+4C3⋊D314yescomplemented
482−1+4⋊C3C3⋊D319yescomplemented
49#986—99no—
50#949—94no—
51#949—94no—
52#947—94no—
53GU(2,𝔽3)—99no—
54#837—94no—
55#837—94no—
56#836—94no—
57C22×C3⋊D3—61no—
58C32×Q8—31no—
59C32×Q8—61no—
60C32×Q8#6711yescharacteristic
61C32×D4—31no—
62C32×D4—412no—
63C32×D4—61no—
64C32×D4S411yescharacteristic
65C2×C32×C4—31no—
66C2×C32×C4—32no—
67C2×C32×C4—61no—
68C2×C3⋊Dic3—61no—
69C4⋊(C3⋊D3)—61no—
70C4⋊(C3⋊D3)—61no—
71C4⋊(C3⋊D3)—1212no—
72C3×SL(2,𝔽3)#67112yes—
73C3⋊(C3⋊C8)—61no—
74C3⋊(C3⋊C8)—61no—
75C3⋊(C3⋊C8)—1212no—
76#454—271noSylow
77C2×C33—41no—
78C2×C33—161no—
79C3⋊(C3⋊D3)—161no—
80C3×#41—34no—
81C3×#41—64no—
82C3×#41C2×C3⋊D314yes—
83C2×C3×Q8—24no—
84C2×C3×Q8—34no—
85C2×D12—94no—
86#41⋊C3C2×C3⋊D319yes—
87C2×SL(2,𝔽3)—29no—
88GL(2,𝔽3)—189no—
89#215—184no—
90#215—184no—
91#213—184no—
92#213—184no—
93#213—369no—
94(C2×D3)⋊C4—188no—
95C4×Dic3—94no—
96#208—94no—
97#208—94no—
98C22×C32—21no—
99C22×C32—31no—
100C22×C32—812no—
101C2×C3⋊D3—122no—
102C2×C3⋊D3—121no—
103C2×C3⋊D3—2412no—
104C32×C4—31no—
105C32×C4—412no—
106C32×C4—61no—
107C32×C4C2×S411yescharacteristic
108C3⋊Dic3—121no—
1092−1+4C3⋊(C3⋊D3)11yescharacteristic, complemented
110AGL(1,ℤ/8)—271no—
111AGL(1,ℤ/8)—271no—
112#124—271no—
113#104—271no—
114#104—271no—
115#101—271no—
116C33—41noSylow
117C22×D3—184no—
118C3×Q8—34no—
119C3×Q8—64no—
120C3×Q8#63814yes—
121C3×D4—34no—
122C3×D4—49no—
123C3×D4—64no—
124C3×D4C3⋊S414yes—
125C2×C3×C4—34no—
126C2×C3×C4—38no—
127C2×C3×C4—64no—
128C2×Dic3—184no—
129D12—184no—
130D12—184no—
131D12—369no—
132SL(2,𝔽3)#63819yes—
133C3⋊C8—184no—
134C3⋊C8—184no—
135C3⋊C8—369no—
136C2×C32—41no—
137C2×C32—412no—
138C2×C32—61no—
139C2×C32—1612no—
140C2×C32GL(2,ℤ/4)11yescharacteristic
141C3⋊D3—241no—
142C3⋊D3—4812no—
143#41—31no—
144#41—61no—
145#41C2×C3⋊(C3⋊D3)11yescharacteristic
146C2×Q8—21no—
147C2×Q8—31no—
148C2×D4—271no—
149SD16—541no—
150SD16—541no—
151D8—541no—
152D8—541no—
153M16—271no—
154M16—271no—
155C22⋊C4—542no—
156C42—271no—
157C22×C3—24no—
158C22×C3—34no—
159C22×C3—89no—
160C2×D3—368no—
161C2×D3—364no—
162C2×D3—729no—
163C3×C4—34no—
164C3×C4—49no—
165C3×C4—64no—
166C3×C4C2×C3⋊S414yes—
167Dic3—364no—
168C32—412no—
169C32#543111yescharacteristic, complemented
170C23—541no—
171Q8—31no—
172Q8—61no—
173Q8#631211yescharacteristic
174D4—31no—
175D4—61no—
176D4—541no—
177D4—541no—
178D4C3⋊(C3⋊S4)11yescharacteristic
179C2×C4—31no—
180C2×C4—32no—
181C2×C4—61no—
182C2×C4—541no—
183C8—541no—
184C8—541no—
185C2×C3—44no—
186C2×C3—49no—
187C2×C3—64no—
188C2×C3—169no—
189C2×C3#6426914yes—
190D3—724no—
191D3—1449no—
192C22—21no—
193C22—31no—
194C22—1082no—
195C22—1081no—
196C4—31no—
197C4—61no—
198C4—1081no—
199C4C2×C3⋊(C3⋊S4)11yescharacteristic
200C3—49no—
201C3#1059366514yescomplemented
202C2—41no—
203C2—61no—
204C2—2161no—
205C2#1172447311yescharacteristic
subgroupG/H
centerC2#11724473
commutator, cosocleC32×#41⋊C3C22
FrattiniC4C2×C3⋊(C3⋊S4)
FittingC32×2−1+4D3
socleC2×C32GL(2,ℤ/4)
hypercenterD4C3⋊(C3⋊S4)
hypocenterC32×SL(2,𝔽3)D4
LCSG ▷ C32×#41⋊C3 ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C4 ◁ D4

Complements

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

NH ≅ G/Ntype
1. C32×2−1+4⋊C3C2semidirect
2. #11724471C2semidirect
3. C3⋊(C3⋊GU(2,𝔽3))C2semidirect
10. C32×2−1+4D3semidirect
11. C3×2−1+4⋊C3D3semidirect
47. C3×2−1+4C3⋊D3semidirect
48. 2−1+4⋊C3C3⋊D3semidirect
109. 2−1+4C3⋊(C3⋊D3)semidirect
169. C32#5431semidirect
201. C3#10593665semidirect

Maximal subgroups

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

subgroupindex
C32×2−1+4⋊C32
#117244712
C3⋊(C3⋊GU(2,𝔽3))2
#105936653
#105929443
#888604