Finite Group Databasebeta

← #13165958

Subgroups — #13165958

Order 1728.

606 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 606 conjugacy classes of subgroups — too many to draw. Only its 65 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
2C2×C32×#226C211yesmaximal, characteristic, complemented
3SL(2,𝔽3)×C3⋊Dic3C211yesmaximal, characteristic, complemented
4#10593208—34nomaximal
5#10592884—31nomaximal
6C2×C32×SL(2,𝔽3)C2211yescharacteristic
7C3⋊(C3⋊GL(2,𝔽3))—21no—
8C32×#226—21no—
9#88804—41nomaximal
10C2×C3⋊GL(2,𝔽3)—34no—
11C2×C3⋊GL(2,𝔽3)—64no—
12C2×C3×#226D314yescomplemented
13C2×C32×Q16—31no—
14#64156—31no—
15Q8×C3⋊Dic3D311yescharacteristic, complemented
16C2×#3755—31no—
17Dic3×SL(2,𝔽3)—34no—
18#63662—31no—
19#63648—31no—
20#63641—31no—
21C22×C3⋊(C3⋊D3)—41no—
22C2×C3×C3⋊Dic3—41no—
23C2×C32×Dic3—41no—
24C32×SL(2,𝔽3)D411yescharacteristic
25#5395—91no—
26#5185—94no—
27C2×C32×Q8—31no—
28C2×C32×Q8C2×D311yescharacteristic
29C2×C4⋊(C3⋊D3)—31no—
30C2×C3×SL(2,𝔽3)—24no—
31C2×C3×SL(2,𝔽3)C2×D314yes—
32C3⋊GL(2,𝔽3)—68no—
33C3⋊GL(2,𝔽3)—64no—
34C3×#226—24no—
35C32×Q16—61no—
36C32×Q16—61no—
37C2×C32×C8—31no—
38#3755—61no—
39#3755—61no—
40#3752—61no—
41#3752—61no—
42C4⋊(C3⋊Dic3)—31no—
43C4×C3⋊Dic3—31no—
44C2×C3⋊(C3⋊C8)—31no—
45#3722—124no—
46C22×C33—41no—
47C2×C3⋊(C3⋊D3)—42no—
48C2×C3⋊(C3⋊D3)—81no—
49C2×C3⋊(C3⋊D3)—81no—
50C3×C3⋊Dic3—81no—
51C32×Dic3—81no—
52C2×GL(2,𝔽3)—91no—
53C2×GL(2,𝔽3)—184no—
54C2×#226C3⋊D311yescharacteristic, complemented
55C2×C3×Q16—34no—
56#947—94no—
57Q8×Dic3—34no—
58C2×#215—94no—
59C4×SL(2,𝔽3)—91no—
60#835—94no—
61#821—94no—
62Dic3⋊C8—94no—
63C22×C3⋊D3—61no—
64C22×C3⋊D3—124no—
65C22×C3⋊D3—244no—
66C32×Q8—31no—
67C32×Q8—61no—
68C32×Q8—61no—
69C32×Q8D1211yescharacteristic
70C2×C32×C4—31no—
71C2×C32×C4—61no—
72C2×C3⋊Dic3—31no—
73C2×C3⋊Dic3S411yescharacteristic
74C4⋊(C3⋊D3)—61no—
75C4⋊(C3⋊D3)—61no—
76C2×C3×Dic3—44no—
77C2×C3×Dic3—124no—
78C3×SL(2,𝔽3)—24no—
79C3×SL(2,𝔽3)#6714yes—
80C32×C8—61no—
81C3⋊(C3⋊C8)—61no—
82#464—271noSylow
83C2×C33—41no—
84C2×C33—41no—
85C2×C33—41no—
86C3⋊(C3⋊D3)—82no—
87C2×C3×Q8—34no—
88C2×C3×Q8D3214yes—
89C2×D12—94no—
90C2×SL(2,𝔽3)—24no—
91C2×SL(2,𝔽3)C2×C3⋊D311yescharacteristic
92GL(2,𝔽3)—188no—
93GL(2,𝔽3)—181no—
94#226—21no—
95C3×Q16—64no—
96C3×Q16—64no—
97C2×C3×C8—34no—
98#215—184no—
99#215—184no—
100(C2×D3)⋊C4—184no—
101(C2×D3)⋊C4—184no—
102(C2×D3)⋊C4—361no—
103C4⋊Dic3—94no—
104C4×Dic3—94no—
105C2×C3⋊C8—94no—
106C22×C32—44no—
107C22×C32—84no—
108C22×C32C2×S411yescharacteristic
109C2×C3⋊D3—121no—
110C2×C3⋊D3—121no—
111C2×C3⋊D3—121no—
112C2×C3⋊D3—128no—
113C2×C3⋊D3—244no—
114C2×C3⋊D3—244no—
115C2×C3⋊D3—248no—
116C2×C3⋊D3—248no—
117C2×C3⋊D3—248no—
118C32×C4—31no—
119C32×C4—31no—
120C32×C4—121no—
121C3⋊Dic3—21no—
122C3⋊Dic3—61no—
123C3×Dic3—84no—
124C3×Dic3—244no—
125C2×Q16—31no—
126C2×SD16—271no—
127#124—271no—
128C4×Q8—91no—
129C4⋊C8—271no—
130#103—271no—
131#102—271no—
132C33—41noSylow
133C22×D3—184no—
134C22×D3—361no—
135C22×D3—724no—
136C3×Q8—34no—
137C3×Q8—64no—
138C3×Q8—64no—
139C3×Q8#62614yes—
140C2×C3×C4—34no—
141C2×C3×C4—64no—
142C2×C3×C4—361no—
143C2×Dic3—34no—
144C2×Dic3—41no—
145C2×Dic3—94no—
146D12—184no—
147D12—184no—
148SL(2,𝔽3)—24no—
149SL(2,𝔽3)#63811yescharacteristic
150C3×C8—64no—
151C3⋊C8—184no—
152C2×C32—44no—
153C2×C32—44no—
154C2×C32—44no—
155C2×C32—84no—
156C2×C32—84no—
157C2×C32—84no—
158C2×C32C4⋊S411yescharacteristic
159C2×C32#98311yescharacteristic
160C2×C32#98511yescharacteristic
161C3⋊D3—248no—
162C3⋊D3—2416no—
163C3⋊D3—241no—
164C2×Q8—31no—
165C2×Q8D3×C3⋊D311yescharacteristic
166C2×D4—271no—
167Q16—61no—
168Q16—61no—
169SD16—541no—
170SD16—541no—
171C2×C8—31no—
172C2×C8—271no—
173C4⋊C4—271no—
174C22⋊C4—541no—
175C22⋊C4—541no—
176C42—271no—
177C22×C3—41no—
178C22×C3—84no—
179C22×C3D3×S414yes—
180C2×D3—362no—
181C2×D3—364no—
182C2×D3—364no—
183C2×D3—364no—
184C2×D3—721no—
185C2×D3—721no—
186C2×D3—728no—
187C2×D3—728no—
188C2×D3—728no—
189C3×C4—34no—
190C3×C4—34no—
191C3×C4—124no—
192C3×C4—721no—
193Dic3—64no—
194Dic3—81no—
195Dic3—184no—
196C32—44no—
197C32—84no—
198C32#539511yescharacteristic, complemented
199C23—541no—
200Q8—31no—
201Q8—61no—
202Q8—61no—
203Q8#629111yescharacteristic
204D4—541no—
205D4—541no—
206C2×C4—31no—
207C2×C4—61no—
208C2×C4—91no—
209C2×C4—271no—
210C8—61no—
211C8—541no—
212C2×C3—41no—
213C2×C3—41no—
214C2×C3—41no—
215C2×C3—84no—
216C2×C3—84no—
217C2×C3—84no—
218C2×C3#6419814yes—
219C2×C3#6420214yes—
220C2×C3#6420714yes—
221D3—722no—
222D3—7216no—
223D3—724no—
224C22—1081no—
225C22—1081no—
226C22—1081no—
227C22C3⋊D3×S411yescharacteristic
228C4—31no—
229C4—31no—
230C4—121no—
231C4—181no—
232C4—541no—
233C3—41no—
234C3—84no—
235C3#1059320814yescomplemented
236C2—2161no—
237C2#1172417811yescharacteristic
238C2#1172418211yescharacteristic
239C2#1172418711yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22C3⋊D3×S4
commutator, cosocleC2×C32×SL(2,𝔽3)C22
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
prime radicalC2×C3⋊(C3⋊GL(2,𝔽3))C2
hypocenterC32×SL(2,𝔽3)D4
LCSG ▷ C2×C32×SL(2,𝔽3) ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×C32×#226C2semidirect
3. SL(2,𝔽3)×C3⋊Dic3C2semidirect
12. C2×C3×#226D3semidirect
15. Q8×C3⋊Dic3D3semidirect
54. C2×#226C3⋊D3semidirect
198. C32#5395semidirect
235. C3#10593208semidirect

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
C2×C32×#2262
SL(2,𝔽3)×C3⋊Dic32
#105932083
#105928843
#888044