Finite Group Databasebeta

← SL(2,𝔽3)×#235 #11923183

Subgroups — SL(2,𝔽3)×#235 #11923183

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
1C2×C3×C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2SL(2,𝔽3)×#67C212yesmaximal, complemented
3SL(2,𝔽3)×D12C211yesmaximal, characteristic, complemented
4C4×D3×SL(2,𝔽3)C212yesmaximal, complemented
5SL(2,𝔽3)×Dic6C211yesmaximal, characteristic, complemented
6Q8×#235C311yesmaximal, characteristic, complemented
7#41×SL(2,𝔽3)—31nomaximal
8C2×C3×#235—41nomaximal
9C22×C3×SL(2,𝔽3)C2211yescharacteristic
10C2×D3×SL(2,𝔽3)C2212yes—
11C3×C4×SL(2,𝔽3)C2211yescharacteristic
12C3×C4×SL(2,𝔽3)C2211yescharacteristic
13Dic3×SL(2,𝔽3)C2212yes—
14C2×C3×C4×Q8C2×C311yescharacteristic, complemented
15Q8×#67C2×C312yescomplemented
16#5579—31no—
17#5578—32no—
18Q8×D12C2×C311yescharacteristic, complemented
19C4×D3×Q8C2×C312yescomplemented
20#5567—31no—
21Q8×Dic6C2×C311yescharacteristic, complemented
22#5511—32no—
23C4×#235—31no—
24#5470—31no—
25Q8×SL(2,𝔽3)—31no—
26D4×SL(2,𝔽3)—32no—
27D4×SL(2,𝔽3)—31no—
28C2×C4×SL(2,𝔽3)—21no—
29C2×C4×SL(2,𝔽3)—32no—
30C2×C4×SL(2,𝔽3)D311yescharacteristic, complemented
31C22×C32×C4—41no—
32C2×C3×#67—42no—
33C3×#235—48no—
34C2×C3×D12—41no—
35C2×C3×C4×D3—42no—
36C2×C3×Dic6—41no—
37C2×C3×SL(2,𝔽3)—21no—
38C2×C3×SL(2,𝔽3)C2311yescharacteristic
39D3×SL(2,𝔽3)—22no—
40Q8×#41—31noSylow
41C2×C3×#41—121no—
42C22×C3×Q8C22×C311yescharacteristic
43C2×D3×Q8C22×C312yes—
44C2×#235A411yescharacteristic
45C22×SL(2,𝔽3)—21no—
46C22×SL(2,𝔽3)—32no—
47C22×SL(2,𝔽3)C2×D311yescharacteristic
48C3×C4×Q8—31no—
49C3×C4×Q8—31no—
50C3×C4×Q8C22×C311yescharacteristic
51C3×C4×Q8C22×C311yescharacteristic
52C2×C3×C4⋊C4—31no—
53C2×C3×C42—31no—
54#946—32no—
55Q8×Dic3C22×C312yes—
56#944—32no—
57C4×#67—32no—
58C4×#67—32no—
59#926—31no—
60#924—32no—
61#924—32no—
62#897—32no—
63#896—34no—
64D3×C4⋊C4—32no—
65#889—34no—
66#888—32no—
67#887—32no—
68C4×D12—31no—
69C4×D12—31no—
70#872—32no—
71C42×D3—32no—
72#870—32no—
73#869—31no—
74#869—31no—
75C4×Dic6—32no—
76C4×Dic6—31no—
77C4×Dic6—31no—
78C4×Dic6—32no—
79C4×SL(2,𝔽3)—21no—
80C4×SL(2,𝔽3)—21no—
81C4×SL(2,𝔽3)—32no—
82C4×SL(2,𝔽3)C2×D311yescharacteristic
83C4×SL(2,𝔽3)C2×D311yescharacteristic
84C23×C32—41no—
85C22×C3×D3—42no—
86C2×C32×C4—44no—
87C2×C32×C4—41no—
88C2×C32×C4—41no—
89C3×#67—48no—
90C2×C3×Dic3—42no—
91C3×D12—44no—
92C3×C4×D3—48no—
93C3×Dic6—44no—
94C3×SL(2,𝔽3)#4111yescharacteristic, complemented
95Q82—31no—
96#557—91no—
97#554—92no—
98#554—91no—
99D4×Q8—32no—
100D4×Q8—31no—
101#533—91no—
102#533—92no—
103C4×#41—91no—
104C2×C4×Q8—32no—
105C2×C4×Q8C3×D311yescharacteristic, complemented
106C3×#41—128no—
107C2×C3×Q8—21no—
108C2×C3×Q8—31no—
109C2×C3×Q8—61no—
110C2×C3×Q8—121no—
111C2×C3×Q8C23×C311yescharacteristic
112C2×C3×D4—121no—
113C2×C3×D4—122no—
114C22×C3×C4—31no—
115C22×C3×C4—31no—
116C22×C3×C4—41no—
117C22×C3×C4—81no—
118C22×C3×C4—122no—
119C22×C3×C4C2×A411yescharacteristic
120C2×#67C2×A412yes—
121D3×Q8—22no—
122D3×Q8—62no—
123#235SL(2,𝔽3)18yesdirect product factor
124C2×D12C2×A411yescharacteristic
125C2×C4×D3—32no—
126C2×C4×D3—32no—
127C2×C4×D3C2×A412yes—
128C2×Dic6—32no—
129C2×Dic6—32no—
130C2×Dic6C2×A411yescharacteristic
131C2×SL(2,𝔽3)—22no—
132C2×SL(2,𝔽3)—21no—
133C2×SL(2,𝔽3)—21no—
134C2×SL(2,𝔽3)—62no—
135C2×SL(2,𝔽3)C22×D311yescharacteristic
136C3×C4⋊C4—31no—
137C3×C4⋊C4—31no—
138C3×C4⋊C4—32no—
139C3×C42—31no—
140C3×C42—31no—
141C3×C42—31no—
142C3×C42—31no—
143#217—32no—
144(C2×D3)⋊C4—32no—
145(C2×D3)⋊C4—32no—
146C4⋊Dic3—32no—
147C4⋊Dic3—31no—
148C4⋊Dic3—31no—
149C4⋊Dic3—32no—
150Dic3⋊C4—34no—
151Dic3⋊C4—34no—
152Dic3⋊C4—32no—
153Dic3⋊C4—32no—
154C4×Dic3—32no—
155C4×Dic3—32no—
156C4×Dic3—32no—
157C22×C32—42no—
158C22×C32—41no—
159C22×C32—81no—
160C22×C32—81no—
161C2×C3×D3—44no—
162C2×C3×D3—82no—
163C2×C3×D3—82no—
164C32×C4—42no—
165C32×C4—42no—
166C3×Dic3—44no—
167C2×#41—31no—
168C22×Q8—32no—
169C22×Q8C2×C3×D311yescharacteristic
170C4⋊Q8—91no—
171C4⋊Q8—91no—
172C4⋊Q8—92no—
173C4⋊Q8—92no—
174#125—92no—
175#125—94no—
176C22⋊Q8—94no—
177C22⋊Q8—92no—
178C22⋊Q8—92no—
179C22⋊Q8—92no—
180C22⋊Q8—92no—
181C4×Q8—31no—
182C4×Q8—31no—
183C4×Q8—32no—
184C4×Q8—91no—
185C4×Q8—91no—
186C4×Q8—92no—
187C4×Q8—92no—
188C4×Q8—92no—
189C4×Q8C2×C3×D311yescharacteristic
190C4×Q8C2×C3×D311yescharacteristic
191C4×D4—91no—
192C4×D4—91no—
193C4×D4—92no—
194C4×D4—92no—
195#117—92no—
196#117—91no—
197C2×C4⋊C4—31no—
198C2×C4⋊C4—92no—
199C2×C42—31no—
200C2×C42—92no—
201C23×C3—41no—
202C23×C3—81no—
203C23×C3—122no—
204C23×C3C22×A411yescharacteristic
205C22×D3C22×A412yes—
206C3×Q8—31no—
207C3×Q8—61no—
208C3×Q8—61no—
209C3×Q8—124no—
210C3×Q8C3×#4111yescharacteristic, complemented
211C3×D4—124no—
212C3×D4—128no—
213C2×C3×C4—31no—
214C2×C3×C4—31no—
215C2×C3×C4—31no—
216C2×C3×C4—31no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—44no—
220C2×C3×C4—61no—
221C2×C3×C4—61no—
222C2×C3×C4—81no—
223C2×C3×C4—81no—
224C2×C3×C4—84no—
225C2×C3×C4—128no—
226C2×C3×C4—122no—
227C2×C3×C4—121no—
228C2×C3×C4C2×SL(2,𝔽3)14yescomplemented
229C2×C3×C4C22×A411yescharacteristic
230C2×C3×C4C22×A411yescharacteristic
231#67C2×SL(2,𝔽3)18yescomplemented
232C2×Dic3—32no—
233C2×Dic3—32no—
234C2×Dic3C22×A412yes—
235D12C2×SL(2,𝔽3)14yescomplemented
236C4×D3—62no—
237C4×D3—62no—
238C4×D3—122no—
239C4×D3C2×SL(2,𝔽3)18yescomplemented
240Dic6—62no—
241Dic6—62no—
242Dic6—122no—
243Dic6C2×SL(2,𝔽3)14yescomplemented
244SL(2,𝔽3)—21no—
245SL(2,𝔽3)#23511yescharacteristic, direct product factor
246C2×C32—41no—
247C2×C32—41no—
248C2×C32—41no—
249C2×C32—82no—
250C3×D3—84no—
251#41—38no—
252C2×Q8—21no—
253C2×Q8—31no—
254C2×Q8—31no—
255C2×Q8—61no—
256C2×Q8—62no—
257C2×Q8—92no—
258C2×Q8—92no—
259C2×Q8—182no—
260C2×Q8C22×C3×D311yescharacteristic
261C2×D4—32no—
262C2×D4—31no—
263C22×C4—32no—
264C22×C4—31no—
265C22×C4—31no—
266C22×C4—92no—
267C22×C4—92no—
268C22×C4D3×A411yescharacteristic
269C4⋊C4—31no—
270C4⋊C4—31no—
271C4⋊C4—32no—
272C4⋊C4—92no—
273C4⋊C4—92no—
274C4⋊C4—92no—
275C4⋊C4—94no—
276C4⋊C4—94no—
277C4⋊C4—92no—
278C4⋊C4—91no—
279C4⋊C4—91no—
280C22⋊C4—92no—
281C22⋊C4—92no—
282C22⋊C4—92no—
283C42—31no—
284C42—31no—
285C42—31no—
286C42—31no—
287C42—92no—
288C42—92no—
289C42—92no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—42no—
293C22×C3—41no—
294C22×C3—81no—
295C22×C3—81no—
296C22×C3—82no—
297C22×C3—82no—
298C22×C3—82no—
299C22×C3—81no—
300C22×C3—124no—
301C22×C3—242no—
302C22×C3—242no—
303C22×C3C22×SL(2,𝔽3)12yes—
304C22×C3C23×A411yescharacteristic
305C2×D3—22no—
306C2×D3—22no—
307C2×D3C22×SL(2,𝔽3)14yes—
308C3×C4—31no—
309C3×C4—31no—
310C3×C4—42no—
311C3×C4—42no—
312C3×C4—61no—
313C3×C4—61no—
314C3×C4—61no—
315C3×C4—82no—
316C3×C4—82no—
317C3×C4—124no—
318C3×C4C22×SL(2,𝔽3)12yes—
319C3×C4C22×SL(2,𝔽3)12yes—
320Dic3—62no—
321Dic3—62no—
322Dic3C22×SL(2,𝔽3)14yes—
323C32—41noSylow
324C23—32no—
325C23C2×D3×A411yescharacteristic
326Q8—31no—
327Q8—34no—
328Q8—61no—
329Q8—61no—
330Q8—182no—
331Q8—182no—
332Q8—362no—
333Q8C3×#23511yescharacteristic, complemented
334D4—38no—
335D4—34no—
336C2×C4—31no—
337C2×C4—32no—
338C2×C4—38no—
339C2×C4—31no—
340C2×C4—31no—
341C2×C4—31no—
342C2×C4—61no—
343C2×C4—61no—
344C2×C4—92no—
345C2×C4—92no—
346C2×C4—121no—
347C2×C4—182no—
348C2×C4—182no—
349C2×C4—362no—
350C2×C4D3×SL(2,𝔽3)14yescomplemented
351C2×C4C2×D3×A411yescharacteristic
352C2×C4C2×D3×A411yescharacteristic
353C2×C3—22no—
354C2×C3—41no—
355C2×C3—41no—
356C2×C3—41no—
357C2×C3—82no—
358C2×C3—84no—
359C2×C3—81no—
360C2×C3—81no—
361C2×C3—81no—
362C2×C3—244no—
363C2×C3C23×SL(2,𝔽3)11yescharacteristic
364C2×C3A4×#4111yescharacteristic
365C2×C3#594511yescharacteristic
366D3—24no—
367C22—21no—
368C22—21no—
369C22—34no—
370C22—62no—
371C22—62no—
372C22C2×D3×SL(2,𝔽3)12yes—
373C22C22×D3×A411yescharacteristic
374C4—31no—
375C4—31no—
376C4—34no—
377C4—61no—
378C4—61no—
379C4—61no—
380C4—182no—
381C4—182no—
382C4C2×D3×SL(2,𝔽3)12yes—
383C4C2×D3×SL(2,𝔽3)12yes—
384C3—41no—
385C3—81no—
386C3#41×SL(2,𝔽3)11yescharacteristic, complemented
387C2—22no—
388C2—64no—
389C2A4×#23511yescharacteristic
390C2C22×D3×SL(2,𝔽3)11yescharacteristic
391C2#5964⋊C311yescharacteristic
subgroupG/H
centerC2×C4C2×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FrattiniC22C22×D3×A4
FittingC2×C3×C4×Q8C2×C3
socleC22×C3C23×A4
hypercenterC22×C4D3×A4
hypocenterC3×Q8C3×#41
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×SL(2,𝔽3)C2semidirect
2. SL(2,𝔽3)×#67C2semidirect
3. SL(2,𝔽3)×D12C2semidirect
4. C4×D3×SL(2,𝔽3)C2semidirect
5. SL(2,𝔽3)×Dic6C2semidirect
6. Q8×#235C3semidirect
14. C2×C3×C4×Q8C2×C3semidirect
15. Q8×#67C2×C3semidirect
18. Q8×D12C2×C3semidirect
19. C4×D3×Q8C2×C3semidirect
21. Q8×Dic6C2×C3semidirect
30. C2×C4×SL(2,𝔽3)D3semidirect
94. C3×SL(2,𝔽3)#41semidirect
105. C2×C4×Q8C3×D3semidirect
123. #235SL(2,𝔽3)direct (N × H)
210. C3×Q8C3×#41semidirect
228. C2×C3×C4C2×SL(2,𝔽3)semidirect
231. #67C2×SL(2,𝔽3)semidirect
235. D12C2×SL(2,𝔽3)semidirect
239. C4×D3C2×SL(2,𝔽3)semidirect
243. Dic6C2×SL(2,𝔽3)semidirect
245. SL(2,𝔽3)#235direct (N × H)
333. Q8C3×#235semidirect
350. C2×C4D3×SL(2,𝔽3)semidirect
386. 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
C2×C3×C4×SL(2,𝔽3)2
SL(2,𝔽3)×#672
SL(2,𝔽3)×D122
C4×D3×SL(2,𝔽3)2
SL(2,𝔽3)×Dic62
Q8×#2353
#41×SL(2,𝔽3)3
C2×C3×#2354