Finite Group Databasebeta

← SL(2,𝔽3)×#237 #11923260

Subgroups — SL(2,𝔽3)×#237 #11923260

Order 1152.

731 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 731 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C3×D4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2SL(2,𝔽3)×#67C212yesmaximal, complemented
3C2×Dic3×SL(2,𝔽3)C212yesmaximal, complemented
4C4×D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
5SL(2,𝔽3)×Dic6C211yesmaximal, characteristic, complemented
6Q8×#237C311yesmaximal, characteristic, complemented
7#41×SL(2,𝔽3)—31nomaximal
8C2×C3×#237—41nomaximal
9C22×C3×SL(2,𝔽3)C2212yes—
10C2×D3×SL(2,𝔽3)C2211yescharacteristic
11C3×C4×SL(2,𝔽3)C2211yescharacteristic
12Dic3×SL(2,𝔽3)C2212yes—
13Dic3×SL(2,𝔽3)C2211yescharacteristic
14C3×D4×Q8C2×C311yescharacteristic, complemented
15Q8×#67C2×C312yescomplemented
16C2×Q8×Dic3C2×C312yescomplemented
17#5728—31no—
18#5721—31no—
19#5635—32no—
20#5622—32no—
21C4×D3×Q8C2×C311yescharacteristic, complemented
22Q8×Dic6C2×C311yescharacteristic, complemented
23#5543—31no—
24C4×#237—31no—
25Q8×SL(2,𝔽3)—31no—
26D4×SL(2,𝔽3)—21no—
27D4×SL(2,𝔽3)—32no—
28D4×SL(2,𝔽3)D311yescharacteristic, complemented
29C2×C4×SL(2,𝔽3)—32no—
30C2×C4×SL(2,𝔽3)—31no—
31C2×C32×D4—41no—
32C2×C3×#67—42no—
33C22×C3×Dic3—42no—
34C3×#237—48no—
35C2×C3×C4×D3—41no—
36C2×C3×Dic6—41no—
37C2×C3×SL(2,𝔽3)—22no—
38C2×C3×SL(2,𝔽3)C2311yescharacteristic
39D3×SL(2,𝔽3)—21no—
40Q8×#41—31noSylow
41C2×C3×#41—121no—
42C22×C3×Q8C22×C312yes—
43C2×D3×Q8C22×C311yescharacteristic
44C2×#237A411yescharacteristic
45C22×SL(2,𝔽3)—22no—
46C22×SL(2,𝔽3)—31no—
47C22×SL(2,𝔽3)C2×D312yes—
48C3×C4⋊Q8—31no—
49C3×C22⋊Q8—32no—
50C3×C4×Q8C22×C311yescharacteristic
51C3×C4×D4—31no—
52#946—32no—
53Q8×Dic3—32no—
54Q8×Dic3—32no—
55Q8×Dic3C22×C311yescharacteristic
56Q8×Dic3C22×C312yes—
57#944—32no—
58D4×Dic3—31no—
59C4×#67—32no—
60C2×C4⋊Dic3—32no—
61#924—32no—
62C2×C4×Dic3—32no—
63#897—32no—
64#892—31no—
65D3×C4⋊C4—31no—
66#890—32no—
67#889—34no—
68#888—32no—
69#887—32no—
70#887—31no—
71#887—31no—
72#67⋊C4—32no—
73#878—34no—
74#877—32no—
75C42×D3—31no—
76#869—31no—
77C4×Dic6—31no—
78C4×Dic6—31no—
79C4×SL(2,𝔽3)—21no—
80C4×SL(2,𝔽3)—31no—
81C4×SL(2,𝔽3)—32no—
82C4×SL(2,𝔽3)C2×D311yescharacteristic
83C23×C32—42no—
84C22×C3×D3—41no—
85C32×D4—44no—
86C2×C32×C4—41no—
87C3×#67—48no—
88C2×C3×Dic3—48no—
89C2×C3×Dic3—42no—
90C2×C3×Dic3—41no—
91C3×C4×D3—44no—
92C3×Dic6—44no—
93C3×SL(2,𝔽3)#4111yescharacteristic, complemented
94Q82—31no—
95#557—91no—
96#554—91no—
97#554—92no—
98D4×Q8—32no—
99D4×Q8C3×D311yescharacteristic, complemented
100#533—92no—
101#533—91no—
102C4×#41—91no—
103C2×C4×Q8—32no—
104C2×C4×Q8—31no—
105C3×#41—128no—
106C2×C3×Q8—22no—
107C2×C3×Q8—32no—
108C2×C3×Q8—62no—
109C2×C3×Q8—121no—
110C2×C3×Q8C23×C311yescharacteristic
111C2×C3×D4—41no—
112C2×C3×D4—81no—
113C2×C3×D4—122no—
114C2×C3×D4C2×A411yescharacteristic
115C22×C3×C4—32no—
116C22×C3×C4—121no—
117C22×C3×C4—122no—
118C2×#67C2×A412yes—
119C22×Dic3—32no—
120C22×Dic3C2×A412yes—
121D3×Q8—21no—
122D3×Q8—61no—
123#237SL(2,𝔽3)18yesdirect product factor
124C2×C4×D3—31no—
125C2×C4×D3—31no—
126C2×C4×D3C2×A411yescharacteristic
127C2×Dic6—31no—
128C2×Dic6—32no—
129C2×Dic6C2×A411yescharacteristic
130C2×SL(2,𝔽3)—22no—
131C2×SL(2,𝔽3)—21no—
132C2×SL(2,𝔽3)—42no—
133C2×SL(2,𝔽3)—61no—
134C2×SL(2,𝔽3)C22×D311yescharacteristic
135C3×C4⋊C4—31no—
136C3×C4⋊C4—32no—
137C3×C4⋊C4—31no—
138C3×C22⋊C4—32no—
139C3×C42—31no—
140#217—32no—
141(C2×D3)⋊C4—32no—
142C4⋊Dic3—34no—
143C4⋊Dic3—31no—
144C4⋊Dic3—31no—
145C4⋊Dic3—32no—
146Dic3⋊C4—32no—
147Dic3⋊C4—34no—
148Dic3⋊C4—32no—
149Dic3⋊C4—32no—
150C4×Dic3—31no—
151C4×Dic3—32no—
152C4×Dic3—31no—
153C4×Dic3—31no—
154C4×Dic3—32no—
155C4×Dic3—32no—
156C22×C32—44no—
157C22×C32—41no—
158C22×C32—82no—
159C22×C32—82no—
160C2×C3×D3—42no—
161C2×C3×D3—81no—
162C2×C3×D3—81no—
163C32×C4—42no—
164C3×Dic3—44no—
165C3×Dic3—42no—
166C2×#41—31no—
167C22×Q8—31no—
168C22×Q8C2×C3×D312yes—
169C4⋊Q8—31no—
170C4⋊Q8—91no—
171C4⋊Q8—92no—
172C4⋊Q8—92no—
173#125—94no—
174#125—92no—
175C22⋊Q8—32no—
176C22⋊Q8—94no—
177C22⋊Q8—92no—
178C22⋊Q8—92no—
179C22⋊Q8—92no—
180C4×Q8—31no—
181C4×Q8—32no—
182C4×Q8—91no—
183C4×Q8—91no—
184C4×Q8—91no—
185C4×Q8—92no—
186C4×Q8—91no—
187C4×Q8—92no—
188C4×Q8—92no—
189C4×Q8C2×C3×D311yescharacteristic
190C4×D4—31no—
191C4×D4—92no—
192C4×D4—92no—
193C4×D4—91no—
194#117—92no—
195#117—91no—
196C2×C4⋊C4—91no—
197C2×C4⋊C4—92no—
198C2×C42—91no—
199C2×C42—92no—
200C23×C3—42no—
201C23×C3—82no—
202C23×C3—121no—
203C23×C3C22×A412yes—
204C22×D3C22×A411yescharacteristic
205C3×Q8—31no—
206C3×Q8—62no—
207C3×Q8—62no—
208C3×Q8—124no—
209C3×Q8C3×#4111yescharacteristic, complemented
210C3×D4—44no—
211C3×D4—84no—
212C3×D4—128no—
213C3×D4C2×SL(2,𝔽3)14yescomplemented
214C2×C3×C4—31no—
215C2×C3×C4—32no—
216C2×C3×C4—31no—
217C2×C3×C4—41no—
218C2×C3×C4—62no—
219C2×C3×C4—62no—
220C2×C3×C4—81no—
221C2×C3×C4—124no—
222C2×C3×C4—121no—
223C2×C3×C4—122no—
224C2×C3×C4—128no—
225C2×C3×C4C22×A411yescharacteristic
226#67C2×SL(2,𝔽3)18yescomplemented
227C2×Dic3—31no—
228C2×Dic3—31no—
229C2×Dic3—32no—
230C2×Dic3—122no—
231C2×Dic3C2×SL(2,𝔽3)18yescomplemented
232C2×Dic3C22×A411yescharacteristic
233C2×Dic3C22×A412yes—
234C4×D3—61no—
235C4×D3—61no—
236C4×D3—121no—
237C4×D3C2×SL(2,𝔽3)14yescomplemented
238Dic6—61no—
239Dic6—61no—
240Dic6—122no—
241Dic6C2×SL(2,𝔽3)14yescomplemented
242SL(2,𝔽3)—21no—
243SL(2,𝔽3)#23711yescharacteristic, direct product factor
244C2×C32—41no—
245C2×C32—41no—
246C2×C32—41no—
247C2×C32—84no—
248C3×D3—82no—
249#41—38no—
250C2×Q8—22no—
251C2×Q8—31no—
252C2×Q8—32no—
253C2×Q8—62no—
254C2×Q8—61no—
255C2×Q8—92no—
256C2×Q8—91no—
257C2×Q8—181no—
258C2×Q8C22×C3×D311yescharacteristic
259C2×D4—32no—
260C2×D4D3×A411yescharacteristic
261C22×C4—32no—
262C22×C4—31no—
263C22×C4—32no—
264C22×C4—91no—
265C22×C4—91no—
266C22×C4—92no—
267C4⋊C4—31no—
268C4⋊C4—32no—
269C4⋊C4—31no—
270C4⋊C4—92no—
271C4⋊C4—92no—
272C4⋊C4—91no—
273C4⋊C4—92no—
274C4⋊C4—92no—
275C4⋊C4—94no—
276C4⋊C4—94no—
277C4⋊C4—91no—
278C22⋊C4—32no—
279C22⋊C4—92no—
280C22⋊C4—92no—
281C42—31no—
282C42—91no—
283C42—91no—
284C42—92no—
285C42—92no—
286C42—92no—
287C42—91no—
288C22×C3—22no—
289C22×C3—22no—
290C22×C3—44no—
291C22×C3—41no—
292C22×C3—82no—
293C22×C3—82no—
294C22×C3—84no—
295C22×C3—81no—
296C22×C3—122no—
297C22×C3—162no—
298C22×C3—162no—
299C22×C3—241no—
300C22×C3—241no—
301C22×C3C22×SL(2,𝔽3)14yes—
302C22×C3C23×A411yescharacteristic
303C2×D3—21no—
304C2×D3—21no—
305C2×D3C22×SL(2,𝔽3)12yes—
306C3×C4—31no—
307C3×C4—31no—
308C3×C4—42no—
309C3×C4—62no—
310C3×C4—61no—
311C3×C4—82no—
312C3×C4—122no—
313C3×C4—124no—
314C3×C4C22×SL(2,𝔽3)12yes—
315Dic3—61no—
316Dic3—62no—
317Dic3—61no—
318Dic3C22×SL(2,𝔽3)14yes—
319Dic3C22×SL(2,𝔽3)12yes—
320C32—41noSylow
321C23—31no—
322C23C2×D3×A412yes—
323Q8—31no—
324Q8—34no—
325Q8—62no—
326Q8—62no—
327Q8—181no—
328Q8—181no—
329Q8—362no—
330Q8C3×#23711yescharacteristic, complemented
331D4—38no—
332D4D3×SL(2,𝔽3)14yescomplemented
333C2×C4—38no—
334C2×C4—31no—
335C2×C4—31no—
336C2×C4—32no—
337C2×C4—34no—
338C2×C4—31no—
339C2×C4—32no—
340C2×C4—62no—
341C2×C4—62no—
342C2×C4—91no—
343C2×C4—91no—
344C2×C4—92no—
345C2×C4—181no—
346C2×C4—181no—
347C2×C4—361no—
348C2×C4—362no—
349C2×C4C2×D3×A411yescharacteristic
350C2×C3—24no—
351C2×C3—41no—
352C2×C3—41no—
353C2×C3—41no—
354C2×C3—84no—
355C2×C3—81no—
356C2×C3—81no—
357C2×C3—81no—
358C2×C3—164no—
359C2×C3—242no—
360C2×C3C23×SL(2,𝔽3)11yescharacteristic
361C2×C3A4×#4111yescharacteristic
362C2×C3#594511yescharacteristic
363D3—22no—
364C22—22no—
365C22—22no—
366C22—32no—
367C22—61no—
368C22—61no—
369C22C2×D3×SL(2,𝔽3)14yes—
370C22C22×D3×A411yescharacteristic
371C4—31no—
372C4—31no—
373C4—32no—
374C4—34no—
375C4—62no—
376C4—61no—
377C4—181no—
378C4—181no—
379C4—182no—
380C4C2×D3×SL(2,𝔽3)12yes—
381C3—41no—
382C3—81no—
383C3#41×SL(2,𝔽3)11yescharacteristic, complemented
384C2—24no—
385C2—62no—
386C2A4×#23711yescharacteristic
387C2C22×D3×SL(2,𝔽3)11yescharacteristic
388C2#5968⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC3×D4×Q8C2×C3
socleC22×C3C23×A4
hypercenterC2×D4D3×A4
hypocenterC3×Q8C3×#41
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C3×D4×SL(2,𝔽3)C2semidirect
2. SL(2,𝔽3)×#67C2semidirect
3. C2×Dic3×SL(2,𝔽3)C2semidirect
4. C4×D3×SL(2,𝔽3)C2semidirect
5. SL(2,𝔽3)×Dic6C2semidirect
6. Q8×#237C3semidirect
14. C3×D4×Q8C2×C3semidirect
15. Q8×#67C2×C3semidirect
16. C2×Q8×Dic3C2×C3semidirect
21. C4×D3×Q8C2×C3semidirect
22. Q8×Dic6C2×C3semidirect
28. D4×SL(2,𝔽3)D3semidirect
93. C3×SL(2,𝔽3)#41semidirect
99. D4×Q8C3×D3semidirect
123. #237SL(2,𝔽3)direct (N × H)
209. C3×Q8C3×#41semidirect
213. C3×D4C2×SL(2,𝔽3)semidirect
226. #67C2×SL(2,𝔽3)semidirect
231. C2×Dic3C2×SL(2,𝔽3)semidirect
237. C4×D3C2×SL(2,𝔽3)semidirect
241. Dic6C2×SL(2,𝔽3)semidirect
243. SL(2,𝔽3)#237direct (N × H)
330. Q8C3×#237semidirect
332. D4D3×SL(2,𝔽3)semidirect
383. C3#41×SL(2,𝔽3)semidirect

Maximal subgroups

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

subgroupindex
C3×D4×SL(2,𝔽3)2
SL(2,𝔽3)×#672
C2×Dic3×SL(2,𝔽3)2
C4×D3×SL(2,𝔽3)2
SL(2,𝔽3)×Dic62
Q8×#2373
#41×SL(2,𝔽3)3
C2×C3×#2374