Finite Group Databasebeta

← SL(2,𝔽3)×#624 #13166149

Subgroups — SL(2,𝔽3)×#624 #13166149

Order 1728.

551 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 551 conjugacy classes of subgroups — too many to draw. Only its 80 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⋊D3×SL(2,𝔽3)C211yesmaximal, characteristic
2C3×Dic3×SL(2,𝔽3)C212yesmaximal, complemented
3Q8×#624C311yesmaximal, characteristic, complemented
4C4×D3×SL(2,𝔽3)—32nomaximal
5C2×C32×SL(2,𝔽3)C2211yescharacteristic
6C2×C3×#624—41nomaximal
7C3⋊D3×SL(2,𝔽3)C412yescomplemented
8C2×Q8×C3⋊D3C2×C311yescharacteristic
9C2×D3×SL(2,𝔽3)—32no—
10C2×D3×SL(2,𝔽3)—61no—
11C3×Q8×Dic3C2×C312yescomplemented
12C3×C4×SL(2,𝔽3)—32no—
13#63900—31no—
14C4×#624—31no—
15#63844—32no—
16Dic3×SL(2,𝔽3)—22no—
17Dic3×SL(2,𝔽3)D312yescomplemented
18C22×C3×C3⋊D3—41no—
19C2×C32×Dic3—42no—
20C32×SL(2,𝔽3)C2×C411yescharacteristic, complemented
21C3×#624—44no—
22C4×D3×Q8—32no—
23C2×C4×SL(2,𝔽3)—91no—
24C2×C32×Q8C22×C311yescharacteristic
25Q8×C3⋊D3—32no—
26Q8×C3⋊D3C3×C412yescomplemented
27C2×C4×C3⋊D3—31no—
28C2×#634—31no—
29C2×C3×C4×D3—122no—
30C2×C3×SL(2,𝔽3)—22no—
31C2×C3×SL(2,𝔽3)—21no—
32C2×C3×SL(2,𝔽3)—41no—
33C2×C3×SL(2,𝔽3)C2×D312yes—
34C2×#624—31no—
35C2×#624A411yescharacteristic
36D3×SL(2,𝔽3)—34no—
37D3×SL(2,𝔽3)—62no—
38C3×C4⋊Dic3—32no—
39C3×C4×Dic3—32no—
40#3724—32no—
41Dic32—32no—
42C22×C33—41no—
43C2×C3×C3⋊D3—42no—
44C2×C3×C3⋊D3—42no—
45C2×C3×C3⋊D3—42no—
46C32×Dic3—44no—
47C2×D3×Q8—32no—
48C2×D3×Q8—61no—
49C22×SL(2,𝔽3)—91no—
50C3×C4×Q8—32no—
51Q8×Dic3C3×D312yescomplemented
52D3×C4⋊C4—92no—
53#887—92no—
54C42×D3—92no—
55C4×Dic6—92no—
56C4×SL(2,𝔽3)—32no—
57C4×SL(2,𝔽3)—62no—
58C22×C3⋊D3C2×A411yescharacteristic
59C22×C3×D3—122no—
60C22×C3×D3—241no—
61C32×Q8—31no—
62C32×Q8C2×C3×C411yescharacteristic, complemented
63C2×C32×C4—31no—
64C2×C32×C4—122no—
65C2×C3⋊Dic3—31no—
66C4×C3⋊D3—32no—
67C4×C3⋊D3—32no—
68#634—32no—
69#634—32no—
70C2×C3×Dic3—32no—
71C2×C3×Dic3—42no—
72C2×C3×Dic3—82no—
73C2×C3×Dic3C2×A412yes—
74C3×C4×D3—128no—
75C3×SL(2,𝔽3)—22no—
76C3×SL(2,𝔽3)—21no—
77C3×SL(2,𝔽3)—41no—
78C3×SL(2,𝔽3)C4×D312yescomplemented
79#624—62no—
80#624SL(2,𝔽3)14yesdirect product factor
81C2×C4×Q8—91noSylow
82C2×C33—41no—
83C2×C33—41no—
84C2×C33—41no—
85C3×C3⋊D3—44no—
86C2×C3×Q8—21no—
87C2×C3×Q8C2×C3×D312yes—
88C22×C3×C4—361no—
89D3×Q8—34no—
90D3×Q8—62no—
91D3×Q8—94no—
92D3×Q8—182no—
93C2×C4×D3—32no—
94C2×C4×D3—92no—
95C2×C4×D3—92no—
96C2×C4×D3—181no—
97C2×Dic6—92no—
98C2×Dic6—181no—
99C2×SL(2,𝔽3)—22no—
100C2×SL(2,𝔽3)—41no—
101C2×SL(2,𝔽3)—92no—
102C2×SL(2,𝔽3)D3211yescharacteristic
103C3×C4⋊C4—92no—
104C3×C42—92no—
105C4⋊Dic3—32no—
106Dic3⋊C4—94no—
107C4×Dic3—32no—
108C4×Dic3—92no—
109C4×Dic3—92no—
110C22×C32—42no—
111C22×C32—81no—
112C22×C32—82no—
113C22×C32—161no—
114C22×C32C22×A411yescharacteristic
115C2×C3⋊D3C4×A412yes—
116C2×C3⋊D3C2×SL(2,𝔽3)12yes—
117C2×C3⋊D3#41⋊C312yes—
118C2×C3×D3—124no—
119C2×C3×D3—124no—
120C2×C3×D3—124no—
121C2×C3×D3—242no—
122C2×C3×D3—242no—
123C2×C3×D3—242no—
124C32×C4—31no—
125C32×C4—31no—
126C32×C4—124no—
127C3⋊Dic3—32no—
128C3×Dic3—44no—
129C3×Dic3—62no—
130C3×Dic3—84no—
131C3×Dic3C2×SL(2,𝔽3)14yescomplemented
132C22×Q8—91no—
133C4×Q8—32no—
134C4×Q8—272no—
135C2×C4⋊C4—271no—
136C2×C42—271no—
137C33—41noSylow
138C23×C3—361no—
139C22×D3—32no—
140C22×D3—61no—
141C3×Q8—21no—
142C3×Q8—32no—
143C3×Q8—61no—
144C3×Q8C3×C4×D312yescomplemented
145C2×C3×C4—32no—
146C2×C3×C4—32no—
147C2×C3×C4—61no—
148C2×C3×C4—92no—
149C2×C3×C4—122no—
150C2×C3×C4—242no—
151C2×C3×C4—364no—
152C2×Dic3—32no—
153C2×Dic3—92no—
154C2×Dic3—181no—
155C2×Dic3D3×A412yes—
156C4×D3—38no—
157C4×D3—94no—
158C4×D3—94no—
159C4×D3—182no—
160C4×D3—182no—
161C4×D3—184no—
162Dic6—94no—
163Dic6—94no—
164Dic6—182no—
165Dic6—182no—
166SL(2,𝔽3)—22no—
167SL(2,𝔽3)—41no—
168SL(2,𝔽3)#62411yescharacteristic, direct product factor
169C2×C32—42no—
170C2×C32—42no—
171C2×C32—42no—
172C2×C32—81no—
173C2×C32—81no—
174C2×C32—81no—
175C2×C32—82no—
176C2×C32—82no—
177C2×C32—82no—
178C2×C32—161no—
179C2×C32—161no—
180C2×C32—161no—
181C2×C32C2×C4×A411yescharacteristic
182C2×C32C22×SL(2,𝔽3)11yescharacteristic
183C2×C32C2×#41⋊C311yescharacteristic
184C3⋊D3C4×SL(2,𝔽3)14yescomplemented
185C3×D3—128no—
186C3×D3—244no—
187C2×Q8—92no—
188C2×Q8—271no—
189C2×Q8—272no—
190C2×Q8C3×D3211yescharacteristic
191C22×C4—91no—
192C22×C4—271no—
193C22×C4—271no—
194C4⋊C4—92no—
195C4⋊C4—272no—
196C42—92no—
197C42—272no—
198C22×C3—21no—
199C22×C3—41no—
200C22×C3—82no—
201C22×C3—161no—
202C22×C3—362no—
203C22×C3—362no—
204C22×C3—362no—
205C22×C3C2×D3×A412yes—
206C2×D3—34no—
207C2×D3—34no—
208C2×D3—34no—
209C2×D3—62no—
210C2×D3—62no—
211C2×D3—62no—
212C3×C4—32no—
213C3×C4—32no—
214C3×C4—34no—
215C3×C4—61no—
216C3×C4—61no—
217C3×C4—124no—
218C3×C4—182no—
219C3×C4—244no—
220Dic3—62no—
221Dic3—94no—
222Dic3—182no—
223Dic3D3×SL(2,𝔽3)14yescomplemented
224C32—42no—
225C32—81no—
226C32—82no—
227C32—161no—
228C32C2×C4×SL(2,𝔽3)11yescharacteristic, complemented
229C23—91no—
230Q8—31no—
231Q8—272no—
232Q8—272no—
233Q8C3×#62411yescharacteristic, complemented
234C2×C4—31no—
235C2×C4—32no—
236C2×C4—94no—
237C2×C4—92no—
238C2×C4—272no—
239C2×C4—272no—
240C2×C4—271no—
241C2×C4—542no—
242C2×C3—21no—
243C2×C3—21no—
244C2×C3—21no—
245C2×C3—41no—
246C2×C3—41no—
247C2×C3—41no—
248C2×C3—82no—
249C2×C3—82no—
250C2×C3—82no—
251C2×C3—161no—
252C2×C3—161no—
253C2×C3—161no—
254C2×C3—364no—
255C2×C3C4×D3×A412yes—
256C2×C3C2×D3×SL(2,𝔽3)12yes—
257C2×C3D3×#41⋊C312yes—
258D3—38no—
259D3—64no—
260C22—92no—
261C22—92no—
262C22—92no—
263C22D32×A411yescharacteristic
264C4—34no—
265C4—31no—
266C4—31no—
267C4—182no—
268C4—272no—
269C3—21no—
270C3—41no—
271C3—82no—
272C3—161no—
273C3C4×D3×SL(2,𝔽3)12yescomplemented
274C2—94no—
275C2#64319⋊C311yescharacteristic
276C2D32×SL(2,𝔽3)11yescharacteristic
277C2A4×#62411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D32×A4
commutator, hypocenterC32×Q8C2×C3×C4
Fitting, cosocleC2×C32×Q8C22×C3
socleC22×C32C22×A4
prime radicalC2×C3⋊D3×SL(2,𝔽3)C2
LCSG ▷ C32×Q8
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C3×Dic3×SL(2,𝔽3)C2semidirect
3. Q8×#624C3semidirect
7. C3⋊D3×SL(2,𝔽3)C4semidirect
11. C3×Q8×Dic3C2×C3semidirect
17. Dic3×SL(2,𝔽3)D3semidirect
20. C32×SL(2,𝔽3)C2×C4semidirect
26. Q8×C3⋊D3C3×C4semidirect
51. Q8×Dic3C3×D3semidirect
62. C32×Q8C2×C3×C4semidirect
78. C3×SL(2,𝔽3)C4×D3semidirect
80. #624SL(2,𝔽3)direct (N × H)
131. C3×Dic3C2×SL(2,𝔽3)semidirect
144. C3×Q8C3×C4×D3semidirect
168. SL(2,𝔽3)#624direct (N × H)
184. C3⋊D3C4×SL(2,𝔽3)semidirect
223. Dic3D3×SL(2,𝔽3)semidirect
228. C32C2×C4×SL(2,𝔽3)semidirect
233. Q8C3×#624semidirect
273. C3C4×D3×SL(2,𝔽3)semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×SL(2,𝔽3)2
C3×Dic3×SL(2,𝔽3)2
Q8×#6243
C4×D3×SL(2,𝔽3)3
C2×C3×#6244