Finite Group Databasebeta

← SL(2,𝔽3)×#217 #11921364

Subgroups — SL(2,𝔽3)×#217 #11921364

Order 1152.

633 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 633 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C23×C3×SL(2,𝔽3)C211yesmaximal, characteristic
2C2×Dic3×SL(2,𝔽3)C212yesmaximal, complemented
3Q8×#217C311yesmaximal, characteristic, complemented
4C22⋊C4×SL(2,𝔽3)—31nomaximal
5C22×C3×SL(2,𝔽3)—22no—
6C22×C3×SL(2,𝔽3)C412yescomplemented
7C22×C3×SL(2,𝔽3)C2211yescharacteristic
8C2×C3×#217—41nomaximal
9Dic3×SL(2,𝔽3)—22no—
10C23×C3×Q8C2×C311yescharacteristic
11C23×SL(2,𝔽3)—21no—
12C23×SL(2,𝔽3)D311yescharacteristic
13C2×Q8×Dic3C2×C312yescomplemented
14C2×C4×SL(2,𝔽3)—32no—
15#5229—32no—
16#5212—31no—
17C4×#217—31no—
18C24×C32—41no—
19C22×C3×Dic3—42no—
20C2×C3×SL(2,𝔽3)—22no—
21C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
22C2×C3×SL(2,𝔽3)D412yes—
23C3×#217—44no—
24Q8×C22⋊C4—31noSylow
25C22×C3×Q8—22no—
26C22×C3×Q8—32no—
27C22×C3×Q8—31no—
28C22×C3×Q8—62no—
29C22×C3×Q8C3×C412yescomplemented
30C22×C3×Q8C22×C311yescharacteristic
31C23×C3×C4—31no—
32C22×SL(2,𝔽3)—24no—
33C22×SL(2,𝔽3)—22no—
34C22×SL(2,𝔽3)—22no—
35C22×SL(2,𝔽3)—21no—
36C22×SL(2,𝔽3)Dic312yescomplemented
37C22×SL(2,𝔽3)C2×D311yescharacteristic
38C2×C3×C22⋊C4—121no—
39C2×#217—31no—
40C2×#217A411yescharacteristic
41Q8×Dic3—22no—
42Q8×Dic3—62no—
43C2×C4⋊Dic3—32no—
44C2×C4×Dic3—32no—
45C4×SL(2,𝔽3)—62no—
46C3⋊#95—32no—
47C3⋊#95—32no—
48C23×C32—42no—
49C23×C32—42no—
50C23×C32—42no—
51C23×C32—41no—
52C23×C32—82no—
53C23×C32—82no—
54C2×C3×Dic3—44no—
55C2×C3×Dic3—82no—
56C2×C3×Dic3—82no—
57C3×SL(2,𝔽3)C22⋊C411yescharacteristic, complemented
58C23×Q8C3×D311yescharacteristic
59C2×C4×Q8—32no—
60#391—92no—
61#380—91no—
62C4×C22⋊C4—91no—
63C24×C3—41no—
64C24×C3—81no—
65C24×C3C2×A411yescharacteristic
66C2×C3×Q8—22no—
67C2×C3×Q8—32no—
68C2×C3×Q8—32no—
69C2×C3×Q8—31no—
70C2×C3×Q8—32no—
71C2×C3×Q8—62no—
72C2×C3×Q8—62no—
73C2×C3×Q8—64no—
74C2×C3×Q8—62no—
75C2×C3×Q8—64no—
76C2×C3×Q8C2×C3×C411yescharacteristic
77C2×C3×Q8C3×D412yes—
78C22×C3×C4—31no—
79C22×C3×C4—32no—
80C22×C3×C4—32no—
81C22×C3×C4—31no—
82C22×C3×C4—62no—
83C22×C3×C4—62no—
84C22×C3×C4—122no—
85C22×Dic3—32no—
86C22×Dic3C2×A412yes—
87C2×SL(2,𝔽3)—24no—
88C2×SL(2,𝔽3)—22no—
89C2×SL(2,𝔽3)—21no—
90C2×SL(2,𝔽3)—22no—
91C2×SL(2,𝔽3)C2×Dic311yescharacteristic
92C2×SL(2,𝔽3)#6712yes—
93C3×C22⋊C4—124no—
94#217—62no—
95#217SL(2,𝔽3)14yesdirect product factor
96C4⋊Dic3—62no—
97C4⋊Dic3—62no—
98C4×Dic3—62no—
99C4×Dic3—62no—
100C22×C32—44no—
101C22×C32—41no—
102C22×C32—42no—
103C22×C32—41no—
104C22×C32—41no—
105C22×C32—42no—
106C22×C32—82no—
107C22×C32—84no—
108C22×C32—84no—
109C22×C32—82no—
110C3×Dic3—84no—
111C22×Q8—22no—
112C22×Q8—31no—
113C22×Q8—32no—
114C22×Q8—62no—
115C22×Q8C3×Dic312yescomplemented
116C22×Q8C2×C3×D311yescharacteristic
117C23×C4—31no—
118C4×Q8—62no—
119C4×Q8—182no—
120C2×C4⋊C4—92no—
121C2×C22⋊C4—31no—
122C2×C22⋊C4—91no—
123C2×C42—92no—
124#95—92no—
125#95—92no—
126C23×C3—22no—
127C23×C3—22no—
128C23×C3—41no—
129C23×C3—42no—
130C23×C3—42no—
131C23×C3—42no—
132C23×C3—82no—
133C23×C3—82no—
134C23×C3—84no—
135C23×C3—84no—
136C23×C3—81no—
137C23×C3—82no—
138C23×C3—82no—
139C23×C3—82no—
140C23×C3C4×A412yes—
141C23×C3C2×SL(2,𝔽3)12yes—
142C23×C3#41⋊C312yes—
143C23×C3C22×A411yescharacteristic
144C3×Q8—31no—
145C3×Q8—32no—
146C3×Q8—32no—
147C3×Q8—62no—
148C3×Q8—62no—
149C3×Q8—64no—
150C3×Q8C3×C22⋊C411yescharacteristic, complemented
151C2×C3×C4—31no—
152C2×C3×C4—31no—
153C2×C3×C4—32no—
154C2×C3×C4—32no—
155C2×C3×C4—32no—
156C2×C3×C4—62no—
157C2×C3×C4—64no—
158C2×C3×C4—62no—
159C2×C3×C4—62no—
160C2×C3×C4—124no—
161C2×C3×C4—242no—
162C2×C3×C4—242no—
163C2×Dic3—22no—
164C2×Dic3—22no—
165C2×Dic3—62no—
166C2×Dic3—62no—
167C2×Dic3—62no—
168C2×Dic3C2×SL(2,𝔽3)14yescomplemented
169SL(2,𝔽3)—21no—
170SL(2,𝔽3)#21711yescharacteristic, direct product factor
171C2×C32—41no—
172C2×C32—41no—
173C2×C32—41no—
174C2×C32—42no—
175C2×C32—42no—
176C2×C32—84no—
177C24D3×A411yescharacteristic
178C2×Q8—22no—
179C2×Q8—32no—
180C2×Q8—32no—
181C2×Q8—31no—
182C2×Q8—32no—
183C2×Q8—62no—
184C2×Q8—62no—
185C2×Q8—64no—
186C2×Q8—62no—
187C2×Q8—64no—
188C2×Q8C2×C3×Dic311yescharacteristic
189C2×Q8C3×#6712yes—
190C22×C4—31no—
191C22×C4—31no—
192C22×C4—32no—
193C22×C4—32no—
194C22×C4—32no—
195C22×C4—62no—
196C22×C4—62no—
197C22×C4—92no—
198C4⋊C4—182no—
199C4⋊C4—182no—
200C22⋊C4—34no—
201C22⋊C4—182no—
202C42—182no—
203C42—182no—
204C22×C3—22no—
205C22×C3—22no—
206C22×C3—24no—
207C22×C3—24no—
208C22×C3—44no—
209C22×C3—41no—
210C22×C3—41no—
211C22×C3—42no—
212C22×C3—41no—
213C22×C3—42no—
214C22×C3—84no—
215C22×C3—82no—
216C22×C3—84no—
217C22×C3—82no—
218C22×C3—88no—
219C22×C3—84no—
220C22×C3—88no—
221C22×C3—84no—
222C22×C3—84no—
223C22×C3—81no—
224C22×C3—81no—
225C22×C3—82no—
226C22×C3—81no—
227C22×C3—82no—
228C22×C3C4×SL(2,𝔽3)14yescomplemented
229C22×C3C2×C4×A411yescharacteristic
230C22×C3D4×A412yes—
231C22×C3C22×SL(2,𝔽3)11yescharacteristic
232C22×C3C2×#41⋊C311yescharacteristic
233C22×C32−1+4⋊C312yes—
234C3×C4—32no—
235C3×C4—31no—
236C3×C4—31no—
237C3×C4—62no—
238C3×C4—244no—
239Dic3—24no—
240Dic3—122no—
241C32—41noSylow
242C23—22no—
243C23—22no—
244C23C3⋊(#41⋊C3)12yes—
245C23D3×SL(2,𝔽3)12yes—
246C23Dic3×A412yes—
247C23C2×D3×A411yescharacteristic
248Q8—31no—
249Q8—32no—
250Q8—32no—
251Q8—62no—
252Q8—62no—
253Q8—64no—
254Q8C3×#21711yescharacteristic, complemented
255C2×C4—31no—
256C2×C4—31no—
257C2×C4—32no—
258C2×C4—32no—
259C2×C4—34no—
260C2×C4—32no—
261C2×C4—62no—
262C2×C4—62no—
263C2×C4—64no—
264C2×C4—62no—
265C2×C4—62no—
266C2×C4—62no—
267C2×C4—182no—
268C2×C4—182no—
269C2×C4—182no—
270C2×C3—24no—
271C2×C3—41no—
272C2×C3—42no—
273C2×C3—41no—
274C2×C3—42no—
275C2×C3—41no—
276C2×C3—84no—
277C2×C3—88no—
278C2×C3—81no—
279C2×C3—82no—
280C2×C3—81no—
281C2×C3—82no—
282C2×C3—81no—
283C2×C3A4×C22⋊C411yescharacteristic
284C2×C3C2×C4×SL(2,𝔽3)11yescharacteristic
285C2×C3#519⋊C311yescharacteristic
286C2×C3#548⋊C312yes—
287C2×C3D4×SL(2,𝔽3)12yes—
288C22—24no—
289C22—22no—
290C22—24no—
291C22—22no—
292C22Dic3×SL(2,𝔽3)14yescomplemented
293C22C2×C3⋊(#41⋊C3)11yescharacteristic
294C22C2×D3×SL(2,𝔽3)11yescharacteristic
295C22#1007⋊C312yes—
296C22C2×Dic3×A411yescharacteristic
297C22A4×#6712yes—
298C4—31no—
299C4—31no—
300C4—32no—
301C4—64no—
302C4—62no—
303C4—362no—
304C3—41no—
305C3—81no—
306C3C22⋊C4×SL(2,𝔽3)11yescharacteristic, complemented
307C2—24no—
308C2C2×Dic3×SL(2,𝔽3)11yescharacteristic
309C2#5817⋊C312yes—
310C2#5812⋊C311yescharacteristic
311C2SL(2,𝔽3)×#6712yes—
312C2A4×#21711yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×A4
commutatorC2×C3×Q8C2×C3×C4
FittingC23×C3×Q8C2×C3
socleC23×C3C22×A4
prime radicalC23×C3×SL(2,𝔽3)C2
cosocleC22×C3×Q8C22×C3
hypercenterC24D3×A4
hypocenterC3×Q8C3×C22⋊C4
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C2×Dic3×SL(2,𝔽3)C2semidirect
3. Q8×#217C3semidirect
6. C22×C3×SL(2,𝔽3)C4semidirect
13. C2×Q8×Dic3C2×C3semidirect
29. C22×C3×Q8C3×C4semidirect
36. C22×SL(2,𝔽3)Dic3semidirect
57. C3×SL(2,𝔽3)C22⋊C4semidirect
95. #217SL(2,𝔽3)direct (N × H)
115. C22×Q8C3×Dic3semidirect
150. C3×Q8C3×C22⋊C4semidirect
168. C2×Dic3C2×SL(2,𝔽3)semidirect
170. SL(2,𝔽3)#217direct (N × H)
228. C22×C3C4×SL(2,𝔽3)semidirect
254. Q8C3×#217semidirect
292. C22Dic3×SL(2,𝔽3)semidirect
306. C3C22⋊C4×SL(2,𝔽3)semidirect

Maximal subgroups

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

subgroupindex
C23×C3×SL(2,𝔽3)2
C2×Dic3×SL(2,𝔽3)2
Q8×#2173
C22⋊C4×SL(2,𝔽3)3
C2×C3×#2174