Finite Group Databasebeta

← #13165957

Subgroups — #13165957

Order 1728.

630 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 67 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
2SL(2,𝔽3)×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C32×#859C211yesmaximal, characteristic, complemented
4#10593207—34nomaximal
5#10592516—31nomaximal
6C2×C32×SL(2,𝔽3)C2211yescharacteristic
7C3⋊(C3⋊GL(2,𝔽3))C412yescomplemented
8#88804—41nomaximal
9C2×C3⋊GL(2,𝔽3)—34no—
10C2×C3⋊GL(2,𝔽3)—64no—
11Q8×C3⋊Dic3D311yescharacteristic, complemented
12C2×#3755—31no—
13#64102—31no—
14Dic3×SL(2,𝔽3)—34no—
15C3×#859D314yescomplemented
16C32×#103—31no—
17#63648—31no—
18#63642—31no—
19#63634—31no—
20C22×C3⋊(C3⋊D3)—41no—
21C2×C3×C3⋊Dic3—41no—
22C2×C32×Dic3—41no—
23C32×SL(2,𝔽3)C2×C411yescharacteristic, complemented
24#5394—91no—
25#4817—94no—
26C2×C32×Q8C2×D311yescharacteristic
27C2×C4⋊(C3⋊D3)—31no—
28C2×C4×C3⋊D3—61no—
29C2×C3×SL(2,𝔽3)—24no—
30C2×C3×SL(2,𝔽3)C2×D314yes—
31C3⋊GL(2,𝔽3)—38no—
32C3⋊GL(2,𝔽3)—68no—
33C2×C32×C8—31no—
34C32×C4⋊C4—31no—
35#3755—32no—
36#3755—32no—
37#3752—61no—
38C4⋊(C3⋊Dic3)—31no—
39C4×C3⋊Dic3—31no—
40C2×C3⋊(C3⋊C8)—31no—
41#3722—124no—
42C22×C33—41no—
43C2×C3⋊(C3⋊D3)—42no—
44C2×C3⋊(C3⋊D3)—81no—
45C2×C3⋊(C3⋊D3)—81no—
46C3×C3⋊Dic3—81no—
47C32×Dic3—81no—
48C2×GL(2,𝔽3)—91no—
49C2×GL(2,𝔽3)—184no—
50Q8×Dic3—34no—
51C2×#215—94no—
52#893—94no—
53C4×SL(2,𝔽3)—91no—
54#859C3⋊D311yescharacteristic, complemented
55C3×#103—34no—
56#821—94no—
57#815—94no—
58#807—94no—
59C22×C3⋊D3—61no—
60C22×C3⋊D3—124no—
61C22×C3⋊D3—244no—
62C32×Q8—31no—
63C32×Q8C4×D311yescharacteristic
64C2×C32×C4—31no—
65C2×C32×C4—61no—
66C2×C3⋊Dic3—31no—
67C2×C3⋊Dic3S411yescharacteristic
68C4⋊(C3⋊D3)—32no—
69C4⋊(C3⋊D3)—61no—
70C4×C3⋊D3—122no—
71C2×C3×Dic3—44no—
72C2×C3×Dic3—124no—
73C3×SL(2,𝔽3)—24no—
74C3×SL(2,𝔽3)C4×D314yescomplemented
75C32×C8—61no—
76C3⋊(C3⋊C8)—32no—
77#440—271noSylow
78C2×C33—41no—
79C2×C33—41no—
80C2×C33—41no—
81C3⋊(C3⋊D3)—82no—
82C2×C3×Q8D3214yes—
83C2×D12—94no—
84C2×C4×D3—184no—
85C2×SL(2,𝔽3)—24no—
86C2×SL(2,𝔽3)C2×C3⋊D311yescharacteristic
87GL(2,𝔽3)—92no—
88GL(2,𝔽3)—188no—
89C2×C3×C8—34no—
90C3×C4⋊C4—34no—
91#215—98no—
92#215—98no—
93(C2×D3)⋊C4—184no—
94(C2×D3)⋊C4—361no—
95C4⋊Dic3—94no—
96C4×Dic3—94no—
97C2×C3⋊C8—94no—
98C22×C32—44no—
99C22×C32—84no—
100C22×C32C2×S411yescharacteristic
101C2×C3⋊D3—62no—
102C2×C3⋊D3—128no—
103C2×C3⋊D3—121no—
104C2×C3⋊D3—121no—
105C2×C3⋊D3—248no—
106C2×C3⋊D3—244no—
107C2×C3⋊D3—244no—
108C2×C3⋊D3—248no—
109C2×C3⋊D3—248no—
110C32×C4—31no—
111C32×C4—31no—
112C32×C4—121no—
113C3⋊Dic3—21no—
114C3⋊Dic3—61no—
115C3×Dic3—84no—
116C3×Dic3—244no—
117C2×SD16—271no—
118C4×Q8—91no—
119C4×D4—271no—
120#107—271no—
121#103—31no—
122#102—271no—
123#97—271no—
124C33—41noSylow
125C22×D3—184no—
126C22×D3—361no—
127C22×D3—724no—
128C3×Q8—34no—
129C3×Q8#62414yes—
130C2×C3×C4—34no—
131C2×C3×C4—64no—
132C2×C3×C4—361no—
133C2×Dic3—34no—
134C2×Dic3—41no—
135C2×Dic3—94no—
136D12—98no—
137D12—184no—
138C4×D3—368no—
139SL(2,𝔽3)—24no—
140SL(2,𝔽3)C4×C3⋊D311yescharacteristic, complemented
141C3×C8—64no—
142C3⋊C8—98no—
143C2×C32—44no—
144C2×C32—44no—
145C2×C32—44no—
146C2×C32—84no—
147C2×C32—84no—
148C2×C32—84no—
149C2×C32C4×S411yescharacteristic
150C2×C32#98311yescharacteristic
151C2×C32#98611yescharacteristic
152C3⋊D3—122no—
153C3⋊D3—248no—
154C3⋊D3—2416no—
155C2×Q8D3×C3⋊D311yescharacteristic
156C2×D4—271no—
157C22×C4—541no—
158SD16—272no—
159SD16—272no—
160C2×C8—31no—
161C2×C8—271no—
162C4⋊C4—31no—
163C4⋊C4—271no—
164C22⋊C4—541no—
165C42—271no—
166C22×C3—41no—
167C22×C3—84no—
168C22×C3D3×S414yes—
169C2×D3—188no—
170C2×D3—362no—
171C2×D3—364no—
172C2×D3—364no—
173C2×D3—728no—
174C2×D3—721no—
175C2×D3—721no—
176C2×D3—728no—
177C2×D3—728no—
178C3×C4—34no—
179C3×C4—34no—
180C3×C4—124no—
181C3×C4—721no—
182Dic3—64no—
183Dic3—81no—
184Dic3—184no—
185C32—44no—
186C32—84no—
187C32#539411yescharacteristic, complemented
188C23—541no—
189Q8—31no—
190Q8#628911yescharacteristic
191D4—272no—
192D4—541no—
193C2×C4—31no—
194C2×C4—61no—
195C2×C4—91no—
196C2×C4—271no—
197C2×C4—1082no—
198C8—61no—
199C8—272no—
200C2×C3—41no—
201C2×C3—41no—
202C2×C3—41no—
203C2×C3—84no—
204C2×C3—84no—
205C2×C3—84no—
206C2×C3#6419914yes—
207C2×C3#6420214yes—
208C2×C3(C3⋊S4)⋊C414yes—
209D3—368no—
210D3—722no—
211D3—7216no—
212C22—542no—
213C22—1081no—
214C22—1081no—
215C22C3⋊D3×S411yescharacteristic
216C4—31no—
217C4—31no—
218C4—121no—
219C4—181no—
220C4—541no—
221C3—41no—
222C3—84no—
223C3#1059320714yescomplemented
224C2—1082no—
225C2#1172417911yescharacteristic
226C2#1172418211yescharacteristic
227C2#1172418611yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22C3⋊D3×S4
commutator, hypocenterC32×SL(2,𝔽3)C2×C4
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
prime radicalC2×C3⋊(C3⋊GL(2,𝔽3))C2
cosocleC2×C32×SL(2,𝔽3)C22
LCSG ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. SL(2,𝔽3)×C3⋊Dic3C2semidirect
3. C32×#859C2semidirect
7. C3⋊(C3⋊GL(2,𝔽3))C4semidirect
11. Q8×C3⋊Dic3D3semidirect
15. C3×#859D3semidirect
23. C32×SL(2,𝔽3)C2×C4semidirect
54. #859C3⋊D3semidirect
74. C3×SL(2,𝔽3)C4×D3semidirect
140. SL(2,𝔽3)C4×C3⋊D3semidirect
187. C32#5394semidirect
223. C3#10593207semidirect

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
SL(2,𝔽3)×C3⋊Dic32
C32×#8592
#105932073
#105925163
#888044