Finite Group Databasebeta

← SL(2,𝔽3)×Dic3⋊C4 #11921263

Subgroups — SL(2,𝔽3)×Dic3⋊C4 #11921263

Order 1152.

421 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 421 conjugacy classes of subgroups — too many to draw. Only its 91 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×C4×SL(2,𝔽3)C211yesmaximal, characteristic
2C2×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic
3C2×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic
4Q8×Dic3⋊C4C311yesmaximal, characteristic, complemented
5C4⋊C4×SL(2,𝔽3)—31nomaximal
6C22×C3×SL(2,𝔽3)C2211yescharacteristic
7C2×C3×Dic3⋊C4—41nomaximal
8C3×C4×SL(2,𝔽3)—21no—
9Dic3×SL(2,𝔽3)—21no—
10Dic3×SL(2,𝔽3)C412yescomplemented
11C2×C3×C4×Q8C2×C311yescharacteristic
12C2×Q8×Dic3C2×C311yescharacteristic
13C2×Q8×Dic3C2×C311yescharacteristic
14C2×C4×SL(2,𝔽3)—21no—
15C2×C4×SL(2,𝔽3)—31no—
16C2×C4×SL(2,𝔽3)—31no—
17C2×C4×SL(2,𝔽3)D311yescharacteristic
18#4975—31no—
19#4972—31no—
20C4×Dic3⋊C4—31no—
21#4928—31no—
22C22×C32×C4—41no—
23C22×C3×Dic3—41no—
24C22×C3×Dic3—41no—
25C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
26C2×C3×SL(2,𝔽3)D411yescharacteristic
27C2×C3×SL(2,𝔽3)Q811yescharacteristic
28C3×Dic3⋊C4—44no—
29Q8×C4⋊C4—31noSylow
30C22×C3×Q8C22×C311yescharacteristic
31C22×SL(2,𝔽3)—21no—
32C22×SL(2,𝔽3)C2×D311yescharacteristic
33C3×C4×Q8—21no—
34C3×C4×Q8—61no—
35C2×C3×C4⋊C4—31no—
36C2×C3×C4⋊C4—121no—
37C2×C3×C42—31no—
38Q8×Dic3—21no—
39Q8×Dic3—32no—
40Q8×Dic3—61no—
41Q8×Dic3C3×C412yescomplemented
42C2×C4⋊Dic3—31no—
43C2×C4⋊Dic3—31no—
44C2×Dic3⋊C4—31no—
45C2×Dic3⋊C4—32no—
46C2×Dic3⋊C4A411yescharacteristic
47C2×C4×Dic3—31no—
48C2×C4×Dic3—31no—
49C4×SL(2,𝔽3)—22no—
50C4×SL(2,𝔽3)—21no—
51C4×SL(2,𝔽3)—32no—
52C4×SL(2,𝔽3)—61no—
53C3⋊#95—31no—
54C3⋊#95—31no—
55C23×C32—41no—
56C2×C32×C4—42no—
57C2×C32×C4—81no—
58C2×C32×C4—81no—
59C2×C3×Dic3—42no—
60C2×C3×Dic3—42no—
61C2×C3×Dic3—42no—
62C2×C3×Dic3—42no—
63C2×C3×Dic3—81no—
64C2×C3×Dic3—81no—
65C3×SL(2,𝔽3)C4⋊C411yescharacteristic, complemented
66C2×C4×Q8—31no—
67C2×C4×Q8—31no—
68C2×C4×Q8C3×D311yescharacteristic
69#389—91no—
70#389—91no—
71#384—91no—
72C4×C4⋊C4—91no—
73C2×C3×Q8—31no—
74C2×C3×Q8—31no—
75C2×C3×Q8—31no—
76C2×C3×Q8C2×C3×C411yescharacteristic
77C2×C3×Q8C3×D411yescharacteristic
78C2×C3×Q8C3×Q811yescharacteristic
79C22×C3×C4—31no—
80C22×C3×C4—31no—
81C22×C3×C4—41no—
82C22×C3×C4—81no—
83C22×C3×C4—121no—
84C22×C3×C4—121no—
85C22×C3×C4C2×A411yescharacteristic
86C22×Dic3—31no—
87C22×Dic3—31no—
88C22×Dic3C2×A411yescharacteristic
89C22×Dic3C2×A411yescharacteristic
90C2×SL(2,𝔽3)—21no—
91C2×SL(2,𝔽3)—21no—
92C2×SL(2,𝔽3)—21no—
93C2×SL(2,𝔽3)Dic611yescharacteristic
94C2×SL(2,𝔽3)C4×D311yescharacteristic
95C2×SL(2,𝔽3)#6711yescharacteristic
96C3×C4⋊C4—61no—
97C3×C4⋊C4—61no—
98C3×C4⋊C4—124no—
99C3×C42—61no—
100C3×C42—61no—
101C4⋊Dic3—32no—
102C4⋊Dic3—32no—
103C4⋊Dic3—61no—
104C4⋊Dic3—61no—
105Dic3⋊C4—62no—
106Dic3⋊C4—122no—
107Dic3⋊C4SL(2,𝔽3)14yesdirect product factor
108C4×Dic3—32no—
109C4×Dic3—32no—
110C4×Dic3—61no—
111C4×Dic3—61no—
112C22×C32—41no—
113C22×C32—41no—
114C22×C32—41no—
115C22×C32—41no—
116C22×C32—41no—
117C22×C32—41no—
118C22×C32—41no—
119C32×C4—82no—
120C3×Dic3—44no—
121C3×Dic3—82no—
122C22×Q8C2×C3×D311yescharacteristic
123C4×Q8—21no—
124C4×Q8—32no—
125C4×Q8—61no—
126C4×Q8—61no—
127C4×Q8—92no—
128C4×Q8—181no—
129C2×C4⋊C4—31no—
130C2×C4⋊C4—31no—
131C2×C4⋊C4—91no—
132C2×C4⋊C4—91no—
133C2×C4⋊C4—91no—
134C2×C4⋊C4—92no—
135C2×C42—31no—
136C2×C42—91no—
137C2×C42—91no—
138#95—91no—
139#95—91no—
140C23×C3—41no—
141C23×C3—81no—
142C23×C3C22×A411yescharacteristic
143C3×Q8—31no—
144C3×Q8—31no—
145C3×Q8—31no—
146C3×Q8—32no—
147C3×Q8C3×C4⋊C411yescharacteristic, complemented
148C2×C3×C4—21no—
149C2×C3×C4—21no—
150C2×C3×C4—31no—
151C2×C3×C4—31no—
152C2×C3×C4—31no—
153C2×C3×C4—31no—
154C2×C3×C4—31no—
155C2×C3×C4—31no—
156C2×C3×C4—42no—
157C2×C3×C4—61no—
158C2×C3×C4—61no—
159C2×C3×C4—61no—
160C2×C3×C4—81no—
161C2×C3×C4—81no—
162C2×C3×C4—82no—
163C2×C3×C4—82no—
164C2×C3×C4—82no—
165C2×C3×C4—122no—
166C2×C3×C4—122no—
167C2×C3×C4—122no—
168C2×C3×C4—122no—
169C2×C3×C4—241no—
170C2×C3×C4—241no—
171C2×C3×C4C2×SL(2,𝔽3)12yes—
172C2×Dic3—21no—
173C2×Dic3—21no—
174C2×Dic3—32no—
175C2×Dic3—61no—
176C2×Dic3—61no—
177C2×Dic3—61no—
178C2×Dic3—61no—
179C2×Dic3—61no—
180C2×Dic3C4×A412yes—
181C2×Dic3C2×SL(2,𝔽3)12yes—
182C2×Dic3C2×SL(2,𝔽3)12yes—
183C2×Dic3#41⋊C312yes—
184SL(2,𝔽3)—21no—
185SL(2,𝔽3)Dic3⋊C411yescharacteristic, direct product factor
186C2×C32—41no—
187C2×C32—41no—
188C2×C32—41no—
189C2×C32—41no—
190C2×C32—41no—
191C2×C32—41no—
192C2×C32—41no—
193C2×Q8—31no—
194C2×Q8—31no—
195C2×Q8—31no—
196C2×Q8C3×Dic611yescharacteristic
197C2×Q8C3×C4×D311yescharacteristic
198C2×Q8C3×#6711yescharacteristic
199C22×C4—31no—
200C22×C4—31no—
201C22×C4—31no—
202C22×C4—31no—
203C22×C4—91no—
204C22×C4—91no—
205C22×C4D3×A411yescharacteristic
206C4⋊C4—34no—
207C4⋊C4—61no—
208C4⋊C4—61no—
209C4⋊C4—92no—
210C4⋊C4—92no—
211C4⋊C4—182no—
212C4⋊C4—181no—
213C4⋊C4—181no—
214C4⋊C4—362no—
215C42—61no—
216C42—61no—
217C42—92no—
218C42—92no—
219C42—181no—
220C42—181no—
221C22×C3—41no—
222C22×C3—41no—
223C22×C3—41no—
224C22×C3—41no—
225C22×C3—41no—
226C22×C3—41no—
227C22×C3—41no—
228C22×C3—81no—
229C22×C3—81no—
230C22×C3—81no—
231C22×C3—81no—
232C22×C3—81no—
233C22×C3—81no—
234C22×C3—81no—
235C22×C3C2×C4×A411yescharacteristic
236C22×C3D4×A411yescharacteristic
237C22×C3C22×SL(2,𝔽3)11yescharacteristic
238C22×C3Q8×A411yescharacteristic
239C22×C3C2×#41⋊C311yescharacteristic
240C22×C3#99411yescharacteristic
241C22×C32−1+4⋊C311yescharacteristic
242C3×C4—22no—
243C3×C4—31no—
244C3×C4—31no—
245C3×C4—31no—
246C3×C4—31no—
247C3×C4—82no—
248C3×C4—84no—
249C3×C4—124no—
250C3×C4—121no—
251C3×C4—242no—
252Dic3—22no—
253Dic3—62no—
254Dic3—121no—
255Dic3C4×SL(2,𝔽3)14yescomplemented
256C32—41noSylow
257C23C2×D3×A411yescharacteristic
258Q8—31no—
259Q8—31no—
260Q8—31no—
261Q8—32no—
262Q8C3×Dic3⋊C411yescharacteristic, complemented
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—31no—
266C2×C4—31no—
267C2×C4—31no—
268C2×C4—31no—
269C2×C4—32no—
270C2×C4—32no—
271C2×C4—32no—
272C2×C4—31no—
273C2×C4—31no—
274C2×C4—32no—
275C2×C4—61no—
276C2×C4—61no—
277C2×C4—61no—
278C2×C4—61no—
279C2×C4—61no—
280C2×C4—92no—
281C2×C4—181no—
282C2×C4—181no—
283C2×C4—181no—
284C2×C4—181no—
285C2×C4—181no—
286C2×C4D3×SL(2,𝔽3)12yes—
287C2×C3—41no—
288C2×C3—41no—
289C2×C3—41no—
290C2×C3—41no—
291C2×C3—41no—
292C2×C3—41no—
293C2×C3—41no—
294C2×C3—81no—
295C2×C3—81no—
296C2×C3—81no—
297C2×C3—81no—
298C2×C3—81no—
299C2×C3—81no—
300C2×C3—81no—
301C2×C3A4×C4⋊C411yescharacteristic
302C2×C3C2×C4×SL(2,𝔽3)11yescharacteristic
303C2×C3#520⋊C311yescharacteristic
304C2×C3D4×SL(2,𝔽3)11yescharacteristic
305C2×C3C4⋊(#41⋊C3)11yescharacteristic
306C2×C3#557⋊C311yescharacteristic
307C2×C3Q8×SL(2,𝔽3)11yescharacteristic
308C22A4×Dic611yescharacteristic
309C22C4×D3×A411yescharacteristic
310C22C2×D3×SL(2,𝔽3)11yescharacteristic
311C22#1007⋊C311yescharacteristic
312C22C3⋊#99411yescharacteristic
313C22D3×#41⋊C311yescharacteristic
314C22A4×#6711yescharacteristic
315C4—22no—
316C4—34no—
317C4—31no—
318C4—31no—
319C4—31no—
320C4—31no—
321C4—62no—
322C4—121no—
323C4—182no—
324C4—361no—
325C3—41no—
326C3—81no—
327C3C4⋊C4×SL(2,𝔽3)11yescharacteristic, complemented
328C2SL(2,𝔽3)×Dic611yescharacteristic
329C2#5574⋊C311yescharacteristic
330C2#5569⋊C311yescharacteristic
331C2C4×D3×SL(2,𝔽3)11yescharacteristic
332C2A4×Dic3⋊C411yescharacteristic
333C2SL(2,𝔽3)×#6711yescharacteristic
334C2#5834⋊C311yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×A4
commutatorC2×C3×Q8C2×C3×C4
FittingC2×C3×C4×Q8C2×C3
socleC23×C3C22×A4
prime radicalC22×C3×SL(2,𝔽3)C22
cosocleC22×C3×Q8C22×C3
hypercenterC22×C4D3×A4
hypocenterC3×Q8C3×C4⋊C4
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. Q8×Dic3⋊C4C3semidirect
10. Dic3×SL(2,𝔽3)C4semidirect
41. Q8×Dic3C3×C4semidirect
65. C3×SL(2,𝔽3)C4⋊C4semidirect
107. Dic3⋊C4SL(2,𝔽3)direct (N × H)
147. C3×Q8C3×C4⋊C4semidirect
185. SL(2,𝔽3)Dic3⋊C4direct (N × H)
255. Dic3C4×SL(2,𝔽3)semidirect
262. Q8C3×Dic3⋊C4semidirect
327. C3C4⋊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
C2×C3×C4×SL(2,𝔽3)2
C2×Dic3×SL(2,𝔽3)2
Q8×Dic3⋊C43
C4⋊C4×SL(2,𝔽3)3
C2×C3×Dic3⋊C44