Finite Group Databasebeta

← SL(2,𝔽3)×#638 #13173180

Subgroups — SL(2,𝔽3)×#638 #13173180

Order 1728.

798 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 798 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C22×C32×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C2×C3⋊D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3SL(2,𝔽3)×C3⋊Dic3C211yesmaximal, characteristic, complemented
4Q8×#638C311yesmaximal, characteristic, complemented
5SL(2,𝔽3)×#67—34nomaximal
6C2×C3×#638—41nomaximal
7C2×C32×SL(2,𝔽3)—21no—
8C2×C32×SL(2,𝔽3)C2211yescharacteristic
9C3⋊D3×SL(2,𝔽3)—21no—
10C22×C32×Q8C2×C311yescharacteristic, complemented
11C2×Q8×C3⋊D3C2×C311yescharacteristic, complemented
12C22×C3×SL(2,𝔽3)—24no—
13C22×C3×SL(2,𝔽3)D314yescomplemented
14C2×D3×SL(2,𝔽3)—34no—
15#64155—31no—
16Q8×C3⋊Dic3C2×C311yescharacteristic, complemented
17#64153—31no—
18C4×#638—31no—
19#64133—31no—
20Dic3×SL(2,𝔽3)—34no—
21C23×C33—41no—
22C22×C3×C3⋊D3—41no—
23C3×#638—44no—
24C2×C3×C3⋊Dic3—41no—
25C32×SL(2,𝔽3)D411yescharacteristic, complemented
26Q8×#67—34no—
27D4×SL(2,𝔽3)—91no—
28C2×C32×Q8—21no—
29C2×C32×Q8—31no—
30C2×C32×Q8—61no—
31C2×C32×Q8C22×C311yescharacteristic
32C22×C32×C4—31no—
33C2×#638A411yescharacteristic
34Q8×C3⋊D3—21no—
35Q8×C3⋊D3—61no—
36C2×C4×C3⋊D3—31no—
37C2×#634—31no—
38C2×C3×#67—124no—
39C2×C3×SL(2,𝔽3)—28no—
40C2×C3×SL(2,𝔽3)—24no—
41C2×C3×SL(2,𝔽3)—24no—
42C2×C3×SL(2,𝔽3)C2×D314yes—
43D3×SL(2,𝔽3)—64no—
44#3757—31no—
45#3752—31no—
46C4⋊(C3⋊Dic3)—31no—
47#3750—32no—
48#3750—31no—
49C4×C3⋊Dic3—31no—
50C22×C33—41no—
51C22×C33—42no—
52C22×C33—81no—
53C22×C33—81no—
54C2×C3×C3⋊D3—42no—
55C2×C3×C3⋊D3—81no—
56C2×C3×C3⋊D3—81no—
57C3×C3⋊Dic3—42no—
58C22×C3×Q8C3×D314yescomplemented
59C2×D3×Q8—34no—
60C22×SL(2,𝔽3)—24no—
61C22×SL(2,𝔽3)—91no—
62C22×SL(2,𝔽3)C3⋊D311yescharacteristic, complemented
63#946—94no—
64Q8×Dic3—34no—
65#944—94no—
66C4×#67—94no—
67#924—94no—
68C4×SL(2,𝔽3)—91no—
69C23×C32—44no—
70C23×C32—84no—
71C23×C32C2×A411yescharacteristic
72C22×C3⋊D3C2×A411yescharacteristic
73C22×C3×D3—124no—
74C32×Q8—31no—
75C32×Q8—61no—
76C32×Q8—61no—
77C32×Q8C3×D411yescharacteristic, complemented
78C2×C32×C4—31no—
79C2×C32×C4—31no—
80C2×C32×C4—61no—
81C2×C32×C4—61no—
82#638SL(2,𝔽3)14yesdirect product factor
83C2×C3⋊Dic3—31no—
84C2×C3⋊Dic3—31no—
85C2×C3⋊Dic3C2×A411yescharacteristic
86C4×C3⋊D3—61no—
87C4×C3⋊D3—61no—
88#634—61no—
89#634—61no—
90C3×#67—1216no—
91C2×C3×Dic3—124no—
92C3×SL(2,𝔽3)—24no—
93C3×SL(2,𝔽3)#6714yescomplemented
94D4×Q8—91noSylow
95C2×C33—41no—
96C2×C33—41no—
97C2×C33—41no—
98C2×C33—82no—
99C3×C3⋊D3—82no—
100C2×C3×Q8—24no—
101C2×C3×Q8—34no—
102C2×C3×Q8—64no—
103C2×C3×Q8C2×C3×D314yes—
104C2×C3×D4—361no—
105C22×C3×C4—34no—
106C2×#67—34no—
107D3×Q8—64no—
108D3×Q8—184no—
109C2×C4×D3—94no—
110C2×Dic6—94no—
111C2×SL(2,𝔽3)—28no—
112C2×SL(2,𝔽3)—21no—
113C2×SL(2,𝔽3)—24no—
114C2×SL(2,𝔽3)—181no—
115C2×SL(2,𝔽3)C2×C3⋊D311yescharacteristic
116#217—94no—
117(C2×D3)⋊C4—94no—
118C4⋊Dic3—94no—
119Dic3⋊C4—94no—
120Dic3⋊C4—98no—
121C4×Dic3—94no—
122C22×C32—21no—
123C22×C32—21no—
124C22×C32—44no—
125C22×C32—48no—
126C22×C32—84no—
127C22×C32—84no—
128C22×C32—88no—
129C22×C32—88no—
130C22×C32—84no—
131C22×C32—88no—
132C22×C32C2×SL(2,𝔽3)12yescomplemented
133C22×C32C22×A411yescharacteristic
134C2×C3⋊D3—21no—
135C2×C3⋊D3—21no—
136C2×C3⋊D3C2×SL(2,𝔽3)12yescomplemented
137C2×C3×D3—128no—
138C2×C3×D3—244no—
139C2×C3×D3—244no—
140C32×C4—31no—
141C32×C4—31no—
142C32×C4—61no—
143C3⋊Dic3—61no—
144C3⋊Dic3—61no—
145C3⋊Dic3C2×SL(2,𝔽3)12yescomplemented
146C3×Dic3—128no—
147C22×Q8—91no—
148C22×Q8C3×C3⋊D311yescharacteristic, complemented
149C4⋊Q8—271no—
150C22⋊Q8—271no—
151C22⋊Q8—271no—
152C4×Q8—91no—
153C4×D4—271no—
154C33—41noSylow
155C23×C3—41no—
156C23×C3—84no—
157C23×C3—361no—
158C23×C3D3×A414yes—
159C22×D3—34no—
160C3×Q8—34no—
161C3×Q8—64no—
162C3×Q8—64no—
163C3×Q8C3×#6714yescomplemented
164C3×D4—364no—
165C2×C3×C4—34no—
166C2×C3×C4—34no—
167C2×C3×C4—64no—
168C2×C3×C4—64no—
169C2×C3×C4—361no—
170#67—316no—
171C2×Dic3—34no—
172C2×Dic3—94no—
173C2×Dic3—94no—
174C4×D3—184no—
175C4×D3—184no—
176Dic6—184no—
177Dic6—184no—
178SL(2,𝔽3)—24no—
179SL(2,𝔽3)#63811yescharacteristic, direct product factor
180C2×C32—22no—
181C2×C32—44no—
182C2×C32—44no—
183C2×C32—44no—
184C2×C32—88no—
185C2×C32—816no—
186C2×C32—84no—
187C2×C32—84no—
188C2×C32—84no—
189C2×C32D4×A411yescharacteristic
190C2×C32C22×SL(2,𝔽3)11yescharacteristic
191C2×C322−1+4⋊C311yescharacteristic
192C3⋊D3—22no—
193C3×D3—248no—
194C2×Q8—21no—
195C2×Q8—31no—
196C2×Q8—61no—
197C2×Q8—181no—
198C2×Q8—271no—
199C2×Q8—541no—
200C2×Q8C2×C3×C3⋊D311yescharacteristic
201C2×D4—91no—
202C22×C4—31no—
203C22×C4—271no—
204C4⋊C4—271no—
205C4⋊C4—271no—
206C4⋊C4—272no—
207C22⋊C4—271no—
208C22⋊C4—271no—
209C42—271no—
210C22×C3—24no—
211C22×C3—24no—
212C22×C3—41no—
213C22×C3—42no—
214C22×C3—81no—
215C22×C3—81no—
216C22×C3—88no—
217C22×C3—88no—
218C22×C3—84no—
219C22×C3—88no—
220C22×C3—362no—
221C22×C3—721no—
222C22×C3—721no—
223C22×C3D3×SL(2,𝔽3)18yescomplemented
224C22×C3C2×D3×A414yes—
225C2×D3—38no—
226C2×D3—64no—
227C2×D3—64no—
228C3×C4—34no—
229C3×C4—34no—
230C3×C4—64no—
231C3×C4—362no—
232Dic3—38no—
233Dic3—184no—
234Dic3—184no—
235C32—44no—
236C32—84no—
237C32D4×SL(2,𝔽3)11yescharacteristic, complemented
238C23—91no—
239C23A4×C3⋊D311yescharacteristic
240Q8—31no—
241Q8—61no—
242Q8—61no—
243Q8—541no—
244Q8—541no—
245Q8C3×#63811yescharacteristic, complemented
246D4—94no—
247C2×C4—31no—
248C2×C4—31no—
249C2×C4—61no—
250C2×C4—61no—
251C2×C4—91no—
252C2×C4—271no—
253C2×C4—271no—
254C2×C4—541no—
255C2×C4—541no—
256C2×C3—28no—
257C2×C3—41no—
258C2×C3—41no—
259C2×C3—41no—
260C2×C3—82no—
261C2×C3—816no—
262C2×C3—84no—
263C2×C3—84no—
264C2×C3—84no—
265C2×C3—722no—
266C2×C3C2×D3×SL(2,𝔽3)14yes—
267C2×C3#1007⋊C314yes—
268C2×C3A4×#6714yes—
269D3—68no—
270C22—21no—
271C22—21no—
272C22—92no—
273C22—181no—
274C22—181no—
275C22C3⋊D3×SL(2,𝔽3)12yescomplemented
276C22C2×A4×C3⋊D311yescharacteristic
277C4—31no—
278C4—31no—
279C4—61no—
280C4—92no—
281C4—541no—
282C4—541no—
283C3—41no—
284C3—84no—
285C3SL(2,𝔽3)×#6714yescomplemented
286C2—22no—
287C2—182no—
288C2C2×C3⋊D3×SL(2,𝔽3)11yescharacteristic
289C2#1172449011yescharacteristic
290C2A4×#63811yescharacteristic
subgroupG/H
center, FrattiniC22C2×A4×C3⋊D3
commutator, cosocleC2×C32×Q8C22×C3
FittingC22×C32×Q8C2×C3
socleC22×C32C22×A4
hypercenterC23A4×C3⋊D3
hypocenterC32×Q8C3×D4
LCSG ▷ C2×C32×Q8 ▷ C32×Q8
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×C32×SL(2,𝔽3)C2semidirect
2. C2×C3⋊D3×SL(2,𝔽3)C2semidirect
3. SL(2,𝔽3)×C3⋊Dic3C2semidirect
4. Q8×#638C3semidirect
10. C22×C32×Q8C2×C3semidirect
11. C2×Q8×C3⋊D3C2×C3semidirect
13. C22×C3×SL(2,𝔽3)D3semidirect
16. Q8×C3⋊Dic3C2×C3semidirect
25. C32×SL(2,𝔽3)D4semidirect
58. C22×C3×Q8C3×D3semidirect
62. C22×SL(2,𝔽3)C3⋊D3semidirect
77. C32×Q8C3×D4semidirect
82. #638SL(2,𝔽3)direct (N × H)
93. C3×SL(2,𝔽3)#67semidirect
132. C22×C32C2×SL(2,𝔽3)semidirect
136. C2×C3⋊D3C2×SL(2,𝔽3)semidirect
145. C3⋊Dic3C2×SL(2,𝔽3)semidirect
148. C22×Q8C3×C3⋊D3semidirect
163. C3×Q8C3×#67semidirect
179. SL(2,𝔽3)#638direct (N × H)
223. C22×C3D3×SL(2,𝔽3)semidirect
237. C32D4×SL(2,𝔽3)semidirect
245. Q8C3×#638semidirect
275. C22C3⋊D3×SL(2,𝔽3)semidirect
285. C3SL(2,𝔽3)×#67semidirect

Maximal subgroups

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

subgroupindex
C22×C32×SL(2,𝔽3)2
C2×C3⋊D3×SL(2,𝔽3)2
SL(2,𝔽3)×C3⋊Dic32
Q8×#6383
SL(2,𝔽3)×#673
C2×C3×#6384