Finite Group Databasebeta

← #13173044

Subgroups — #13173044

Order 1728.

862 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 862 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C2×C3⋊(C3⋊GL(2,𝔽3))C211yesmaximal, characteristic
2C32×C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3⋊(C3⋊#859)C211yesmaximal, characteristic, complemented
4#10593628—312nomaximal
5#10592725—31nomaximal
6C2×C32×SL(2,𝔽3)C2211yescharacteristic
7C3⋊(C3⋊GL(2,𝔽3))C412yescomplemented
8#88859—41nomaximal
9C2×C3⋊GL(2,𝔽3)—312no—
10C32×C4×Q8D311yescharacteristic, complemented
11C2×#3755—31no—
12C4×C4⋊(C3⋊D3)—31no—
13C3×C4×SL(2,𝔽3)D3112yescomplemented
14C3⋊#859—312no—
15#63662—31no—
16#63636—31no—
17#63634—31no—
18#63630—31no—
19C22×C3⋊(C3⋊D3)—41no—
20C2×C33×C4—41no—
21C2×C3⋊(C3⋊Dic3)—41no—
22C32×SL(2,𝔽3)C2×C411yescharacteristic, complemented
23#5394—99no—
24#5026—94no—
25C2×C32×Q8C2×D311yescharacteristic
26C2×C4⋊(C3⋊D3)—31no—
27C2×C4×C3⋊D3—61no—
28C2×C3×SL(2,𝔽3)C2×D3112yes—
29C3⋊GL(2,𝔽3)—324no—
30C32×C4⋊C4—31no—
31C32×C42—31no—
32#3755—32no—
33#3755—32no—
34#3752—61no—
35#3752—1212no—
36C4⋊(C3⋊Dic3)—31no—
37C2×C3⋊(C3⋊C8)—31no—
38C2×C3⋊(C3⋊C8)—31no—
39C22×C33—41no—
40C2×C3⋊(C3⋊D3)—42no—
41C2×C3⋊(C3⋊D3)—81no—
42C2×C3⋊(C3⋊D3)—81no—
43C33×C4—81no—
44C3⋊(C3⋊Dic3)—81no—
45C2×GL(2,𝔽3)—99no—
46C3×C4×Q8C3⋊D314yescomplemented
47C2×#215—94no—
48C4×D12—94no—
49C4×SL(2,𝔽3)C3⋊D319yescomplemented
50#859—99no—
51#835—94no—
52#809—94no—
53#807—94no—
54#803—94no—
55C22×C3⋊D3—61no—
56C22×C3⋊D3—1212no—
57C32×Q8—31no—
58C32×Q8C4×D311yescharacteristic
59C2×C32×C4—31no—
60C2×C32×C4—31no—
61C2×C32×C4—412no—
62C2×C32×C4S411yescharacteristic
63C2×C3⋊Dic3—61no—
64C2×C3⋊Dic3—1212no—
65C4⋊(C3⋊D3)—32no—
66C4⋊(C3⋊D3)—61no—
67C4×C3⋊D3—122no—
68C3×SL(2,𝔽3)C4×D3112yescomplemented
69C3⋊(C3⋊C8)—32no—
70C3⋊(C3⋊C8)—61no—
71#440—271noSylow
72C2×C33—41no—
73C2×C33—41no—
74C2×C33—41no—
75C3⋊(C3⋊D3)—82no—
76C2×C3×Q8C2×C3⋊D314yes—
77C2×D12—94no—
78C2×C4×D3—184no—
79C2×SL(2,𝔽3)C2×C3⋊D319yes—
80GL(2,𝔽3)—918no—
81C3×C4⋊C4—34no—
82C3×C42—34no—
83#215—98no—
84#215—98no—
85(C2×D3)⋊C4—184no—
86(C2×D3)⋊C4—369no—
87C4⋊Dic3—94no—
88C2×C3⋊C8—94no—
89C2×C3⋊C8—94no—
90C22×C32—412no—
91C22×C32C2×S411yescharacteristic
92C2×C3⋊D3—62no—
93C2×C3⋊D3—1224no—
94C2×C3⋊D3—121no—
95C2×C3⋊D3—121no—
96C2×C3⋊D3—2412no—
97C2×C3⋊D3—2412no—
98C32×C4—21no—
99C32×C4—31no—
100C32×C4—31no—
101C32×C4—61no—
102C32×C4—812no—
103C3⋊Dic3—121no—
104C3⋊Dic3—2412no—
105C2×SD16—271no—
106C4×Q8C3⋊(C3⋊D3)11yescharacteristic, complemented
107C4×D4—271no—
108#107—271no—
109#103—271no—
110#102—271no—
111#97—271no—
112C33—41noSylow
113C22×D3—184no—
114C22×D3—369no—
115C3×Q8—34no—
116C3×Q8C4×C3⋊D314yes—
117C2×C3×C4—34no—
118C2×C3×C4—34no—
119C2×C3×C4—49no—
120C2×C3×C4C3⋊S414yes—
121C2×Dic3—184no—
122C2×Dic3—369no—
123D12—98no—
124D12—184no—
125C4×D3—368no—
126SL(2,𝔽3)C4×C3⋊D319yescomplemented
127C3⋊C8—98no—
128C3⋊C8—184no—
129C2×C32—412no—
130C2×C32—412no—
131C2×C32—412no—
132C2×C32C4×S411yescharacteristic
133C2×C32#98311yescharacteristic
134C2×C32#98611yescharacteristic
135C3⋊D3—122no—
136C3⋊D3—2424no—
137C2×Q8C2×C3⋊(C3⋊D3)11yescharacteristic
138C2×D4—271no—
139C22×C4—541no—
140SD16—272no—
141SD16—272no—
142C2×C8—271no—
143C2×C8—271no—
144C4⋊C4—31no—
145C4⋊C4—271no—
146C22⋊C4—541no—
147C42—31no—
148C22×C3—49no—
149C22×C3C2×C3⋊S414yes—
150C2×D3—188no—
151C2×D3—3618no—
152C2×D3—364no—
153C2×D3—364no—
154C2×D3—729no—
155C2×D3—729no—
156C3×C4—24no—
157C3×C4—34no—
158C3×C4—34no—
159C3×C4—64no—
160C3×C4—89no—
161Dic3—364no—
162Dic3—729no—
163C32—412no—
164C32#539411yescharacteristic, complemented
165C23—541no—
166Q8—31no—
167Q8C4×C3⋊(C3⋊D3)11yescharacteristic
168D4—272no—
169D4—541no—
170C2×C4—31no—
171C2×C4—31no—
172C2×C4—541no—
173C2×C4—1082no—
174C2×C4C3⋊(C3⋊S4)11yescharacteristic
175C8—272no—
176C8—541no—
177C2×C3—49no—
178C2×C3—49no—
179C2×C3—49no—
180C2×C3C4×C3⋊S414yes—
181C2×C3#6426414yes—
182C2×C3#6426714yes—
183D3—368no—
184D3—7218no—
185C22—542no—
186C22—1081no—
187C22—1081no—
188C22C2×C3⋊(C3⋊S4)11yescharacteristic
189C4—21no—
190C4—31no—
191C4—31no—
192C4—61no—
193C4—1081no—
194C3—49no—
195C3#1059362814yescomplemented
196C2—1082no—
197C2C4×C3⋊(C3⋊S4)11yescharacteristic
198C2#1172446811yescharacteristic
199C2#1172447111yescharacteristic
subgroupG/H
center, FrattiniC22C2×C3⋊(C3⋊S4)
commutator, hypocenterC32×SL(2,𝔽3)C2×C4
FittingC32×C4×Q8D3
socleC22×C32C2×S4
prime radicalC2×C3⋊(C3⋊GL(2,𝔽3))C2
cosocleC2×C32×SL(2,𝔽3)C22
hypercenterC2×C4C3⋊(C3⋊S4)
LCSG ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C32×C4×SL(2,𝔽3)C2semidirect
3. C3⋊(C3⋊#859)C2semidirect
7. C3⋊(C3⋊GL(2,𝔽3))C4semidirect
10. C32×C4×Q8D3semidirect
13. C3×C4×SL(2,𝔽3)D3semidirect
22. C32×SL(2,𝔽3)C2×C4semidirect
46. C3×C4×Q8C3⋊D3semidirect
49. C4×SL(2,𝔽3)C3⋊D3semidirect
68. C3×SL(2,𝔽3)C4×D3semidirect
106. C4×Q8C3⋊(C3⋊D3)semidirect
126. SL(2,𝔽3)C4×C3⋊D3semidirect
164. C32#5394semidirect
195. C3#10593628semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊(C3⋊GL(2,𝔽3))2
C32×C4×SL(2,𝔽3)2
C3⋊(C3⋊#859)2
#105936283
#105927253
#888594