Finite Group Databasebeta

← #13165964

Subgroups — #13165964

Order 1728.

696 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 69 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#10593214—34nomaximal
5#10592487—31nomaximal
6C2×C32×SL(2,𝔽3)C2211yescharacteristic
7C2×#6289—41nomaximal
8C3⋊(C3⋊GL(2,𝔽3))C412yescomplemented
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—
18C8×C3⋊Dic3—31no—
19#63635—31no—
20C22×C3⋊(C3⋊D3)—41no—
21C2×C3×C3⋊Dic3—41no—
22C2×C32×Dic3—41no—
23C32×SL(2,𝔽3)C2×C411yescharacteristic, complemented
24#6289—44no—
25C4×GL(2,𝔽3)—91no—
26#4788—94no—
27C2×C32×Q8C2×D311yescharacteristic
28C2×C4⋊(C3⋊D3)—31no—
29C2×C4×C3⋊D3—61no—
30C2×C3×SL(2,𝔽3)—24no—
31C2×C3×SL(2,𝔽3)C2×D314yes—
32C2×#624—124no—
33C3⋊GL(2,𝔽3)—38no—
34C3⋊GL(2,𝔽3)—68no—
35C2×C32×C8—31no—
36C32×C4⋊C4—31no—
37#3755—32no—
38#3755—32no—
39#3752—61no—
40C4⋊(C3⋊Dic3)—31no—
41C4×C3⋊Dic3—31no—
42C2×C3⋊(C3⋊C8)—31no—
43C22×C33—41no—
44C2×C3⋊(C3⋊D3)—42no—
45C2×C3⋊(C3⋊D3)—42no—
46C2×C3⋊(C3⋊D3)—42no—
47C3×C3⋊Dic3—42no—
48C32×Dic3—42no—
49C2×GL(2,𝔽3)—91no—
50C2×GL(2,𝔽3)—184no—
51Q8×Dic3—34no—
52C2×#215—94no—
53#893—94no—
54C4×SL(2,𝔽3)—91no—
55#859C3⋊D311yescharacteristic, complemented
56C3×#103—34no—
57#821—94no—
58C8×Dic3—94no—
59#808—94no—
60C22×C3⋊D3—61no—
61C22×C3⋊D3—124no—
62C22×C3⋊D3—244no—
63C32×Q8—31no—
64C32×Q8C4×D311yescharacteristic, complemented
65C2×C32×C4—31no—
66C2×C32×C4—61no—
67C2×C3⋊Dic3—31no—
68C2×C3⋊Dic3S411yescharacteristic
69C4⋊(C3⋊D3)—32no—
70C4⋊(C3⋊D3)—61no—
71C4×C3⋊D3—122no—
72C2×C3×Dic3—44no—
73C2×C3×Dic3—124no—
74C3×SL(2,𝔽3)—24no—
75C3×SL(2,𝔽3)C4×D314yescomplemented
76#624—1216no—
77C32×C8—61no—
78C3⋊(C3⋊C8)—32no—
79C4×SD16—271noSylow
80C2×C33—41no—
81C2×C33—41no—
82C2×C33—41no—
83C3⋊(C3⋊D3)—44no—
84C2×C3×Q8D3214yes—
85C2×D12—94no—
86C2×C4×D3—184no—
87C2×C4×D3—361no—
88C2×SL(2,𝔽3)—24no—
89C2×SL(2,𝔽3)C2×C3⋊D311yescharacteristic
90GL(2,𝔽3)—92no—
91GL(2,𝔽3)—188no—
92C2×C3×C8—34no—
93C3×C4⋊C4—34no—
94#215—98no—
95#215—98no—
96(C2×D3)⋊C4—184no—
97C4⋊Dic3—94no—
98C4×Dic3—94no—
99C2×C3⋊C8—94no—
100C22×C32—44no—
101C22×C32—84no—
102C22×C32C2×S411yescharacteristic
103C2×C3⋊D3—62no—
104C2×C3⋊D3—128no—
105C2×C3⋊D3—121no—
106C2×C3⋊D3—121no—
107C2×C3⋊D3—128no—
108C2×C3⋊D3—128no—
109C2×C3⋊D3—248no—
110C2×C3⋊D3—248no—
111C2×C3⋊D3—248no—
112C32×C4—31no—
113C32×C4—31no—
114C32×C4—121no—
115C3⋊Dic3—61no—
116C3⋊Dic3GL(2,𝔽3)12yescomplemented
117C3×Dic3—48no—
118C3×Dic3—128no—
119C2×SD16—271no—
120C4×Q8—91no—
121C4×D4—271no—
122#106—271no—
123#103—31no—
124#102—271no—
125C4×C8—271no—
126C33—41noSylow
127C22×D3—184no—
128C22×D3—361no—
129C22×D3—724no—
130C3×Q8—34no—
131C3×Q8#62414yescomplemented
132C2×C3×C4—34no—
133C2×C3×C4—64no—
134C2×C3×C4—361no—
135C2×Dic3—34no—
136C2×Dic3—41no—
137C2×Dic3—94no—
138D12—98no—
139D12—184no—
140C4×D3—368no—
141C4×D3—364no—
142SL(2,𝔽3)—24no—
143SL(2,𝔽3)C4×C3⋊D311yescharacteristic, complemented
144C3×C8—64no—
145C3⋊C8—98no—
146C2×C32—44no—
147C2×C32—44no—
148C2×C32—44no—
149C2×C32—84no—
150C2×C32—84no—
151C2×C32—84no—
152C2×C32C4×S411yescharacteristic
153C2×C32C2×GL(2,𝔽3)11yescharacteristic
154C2×C32#98511yescharacteristic
155C3⋊D3—122no—
156C3⋊D3—1216no—
157C3⋊D3—2416no—
158C2×Q8D3×C3⋊D311yescharacteristic
159C2×D4—271no—
160C22×C4—541no—
161SD16—272no—
162SD16—272no—
163C2×C8—31no—
164C2×C8—271no—
165C4⋊C4—31no—
166C4⋊C4—271no—
167C22⋊C4—541no—
168C42—271no—
169C22×C3—41no—
170C22×C3—84no—
171C22×C3D3×S414yes—
172C2×D3—188no—
173C2×D3—362no—
174C2×D3—362no—
175C2×D3—362no—
176C2×D3—364no—
177C2×D3—364no—
178C2×D3—728no—
179C2×D3—728no—
180C2×D3—728no—
181C3×C4—34no—
182C3×C4—34no—
183C3×C4—124no—
184C3×C4—362no—
185Dic3—38no—
186Dic3—42no—
187Dic3—184no—
188C32—44no—
189C32—84no—
190C32C4×GL(2,𝔽3)11yescharacteristic, complemented
191C23—541no—
192Q8—31no—
193Q8#628911yescharacteristic, complemented
194D4—272no—
195D4—541no—
196C2×C4—31no—
197C2×C4—61no—
198C2×C4—91no—
199C2×C4—271no—
200C2×C4—1082no—
201C8—61no—
202C8—272no—
203C2×C3—41no—
204C2×C3—41no—
205C2×C3—41no—
206C2×C3—84no—
207C2×C3—84no—
208C2×C3—84no—
209C2×C3#6419814yes—
210C2×C3D3×GL(2,𝔽3)14yes—
211C2×C3(C3⋊S4)⋊C414yes—
212D3—364no—
213D3—368no—
214D3—7216no—
215C22—542no—
216C22—1081no—
217C22—1081no—
218C22C3⋊D3×S411yescharacteristic
219C4—31no—
220C4—31no—
221C4—92no—
222C4—121no—
223C4—541no—
224C3—41no—
225C3—84no—
226C3#1059321414yescomplemented
227C2—1082no—
228C2#1172417811yescharacteristic
229C2C3⋊D3×GL(2,𝔽3)11yescharacteristic
230C2#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
8. C3⋊(C3⋊GL(2,𝔽3))C4semidirect
11. Q8×C3⋊Dic3D3semidirect
15. C3×#859D3semidirect
23. C32×SL(2,𝔽3)C2×C4semidirect
55. #859C3⋊D3semidirect
64. C32×Q8C4×D3semidirect
75. C3×SL(2,𝔽3)C4×D3semidirect
116. C3⋊Dic3GL(2,𝔽3)semidirect
131. C3×Q8#624semidirect
143. SL(2,𝔽3)C4×C3⋊D3semidirect
190. C32C4×GL(2,𝔽3)semidirect
193. Q8#6289semidirect
226. C3#10593214semidirect

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
#105932143
#105924873
C2×#62894