Finite Group Databasebeta

← D3×D4×SL(2,𝔽3) #11923259

Subgroups — D3×D4×SL(2,𝔽3) #11923259

Order 1152.

1041 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 1041 conjugacy classes of subgroups — too many to draw. Only its 147 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×D3×SL(2,𝔽3)C212yesmaximal, complemented
2C3×D4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3SL(2,𝔽3)×#67C212yesmaximal, complemented
4SL(2,𝔽3)×D12C211yesmaximal, characteristic, complemented
5C4×D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
6D3×D4×Q8C311yesmaximal, characteristic, complemented
7C2×D4×SL(2,𝔽3)—31nomaximal
8C2×C3×D3×D4—41nomaximal
9C22×C3×SL(2,𝔽3)C2212yescomplemented
10C2×D3×SL(2,𝔽3)—24no—
11C2×D3×SL(2,𝔽3)C2211yescharacteristic
12C2×D3×SL(2,𝔽3)C2212yescomplemented
13C3×C4×SL(2,𝔽3)C2211yescharacteristic, complemented
14Dic3×SL(2,𝔽3)C2211yescharacteristic, complemented
15C22×D3×Q8C2×C312yescomplemented
16C23×SL(2,𝔽3)—32no—
17C3×D4×Q8C2×C311yescharacteristic, complemented
18Q8×#67C2×C312yescomplemented
19#5727—31no—
20D3×C4⋊Q8—31no—
21#5632—32no—
22D3×C22⋊Q8—32no—
23Q8×D12C2×C311yescharacteristic, complemented
24C4×D3×Q8C2×C311yescharacteristic, complemented
25C4×D3×D4—31no—
26D4×Dic6—31no—
27D4×SL(2,𝔽3)—21no—
28D4×SL(2,𝔽3)—32no—
29D4×SL(2,𝔽3)—31no—
30D4×SL(2,𝔽3)D311yescharacteristic, direct product factor
31C2×C4×SL(2,𝔽3)—31no—
32C23×C3×D3—42no—
33C2×C32×D4—41no—
34C2×C3×#67—42no—
35C3×D3×D4—48no—
36C2×C3×D12—41no—
37C2×C3×C4×D3—41no—
38C2×C3×SL(2,𝔽3)—22no—
39C2×C3×SL(2,𝔽3)C2311yescharacteristic
40D3×SL(2,𝔽3)—22no—
41D3×SL(2,𝔽3)D412yesdirect product factor
42C2×D4×Q8—31noSylow
43C22×C3×Q8C22×C312yescomplemented
44C22×C3×D4—121no—
45C2×D3×Q8—24no—
46C2×D3×Q8—32no—
47C2×D3×Q8—32no—
48C2×D3×Q8—64no—
49C2×D3×Q8C22×C311yescharacteristic
50C2×D3×Q8C22×C312yescomplemented
51C2×D3×D4A411yescharacteristic
52C22×C4×D3—32no—
53C22×Dic6—32no—
54C22×SL(2,𝔽3)—22no—
55C22×SL(2,𝔽3)—31no—
56C22×SL(2,𝔽3)—32no—
57C22×SL(2,𝔽3)—64no—
58C22×SL(2,𝔽3)C2×D312yescomplemented
59C3×C4⋊Q8—31no—
60C3×C22⋊Q8—32no—
61C3×C4×Q8C22×C311yescharacteristic, complemented
62C3×C4×D4—31no—
63#946—32no—
64Q8×Dic3C22×C311yescharacteristic, complemented
65#944—32no—
66D4×Dic3—31no—
67C4×#67—32no—
68#924—32no—
69#897—32no—
70#896—34no—
71#893—31no—
72D3×C4⋊C4—31no—
73D3×C4⋊C4—32no—
74D3×C4⋊C4—31no—
75#888—32no—
76#888—32no—
77#887—31no—
78#67⋊C4—32no—
79D3×C22⋊C4—32no—
80#878—34no—
81C4×D12—31no—
82C42×D3—31no—
83#869—31no—
84C4×Dic6—31no—
85C4×SL(2,𝔽3)—21no—
86C4×SL(2,𝔽3)—31no—
87C4×SL(2,𝔽3)C2×D311yescharacteristic, complemented
88C23×C32—42no—
89C22×C3×D3—48no—
90C22×C3×D3—41no—
91C22×C3×D3—42no—
92C22×C3×D3—84no—
93C22×C3×D3—84no—
94C32×D4—44no—
95C2×C32×C4—41no—
96C3×#67—48no—
97C2×C3×Dic3—41no—
98C3×D12—44no—
99C3×C4×D3—44no—
100C3×SL(2,𝔽3)C2×D411yescharacteristic, complemented
101C23×Q8—32no—
102D4×Q8—32no—
103D4×Q8—31no—
104D4×Q8—91no—
105D4×Q8—92no—
106D4×Q8—91no—
107D4×Q8C3×D311yescharacteristic, complemented
108C2×C4⋊Q8—91no—
109C2×C22⋊Q8—92no—
110C2×C4×Q8—31no—
111C2×C4×D4—91no—
112C24×C3—122no—
113C23×D3C2×A412yes—
114C2×C3×Q8—22no—
115C2×C3×Q8—32no—
116C2×C3×Q8—62no—
117C2×C3×Q8C23×C311yescharacteristic
118C2×C3×D4—41no—
119C2×C3×D4—81no—
120C2×C3×D4—121no—
121C2×C3×D4—128no—
122C2×C3×D4—122no—
123C2×C3×D4C2×A411yescharacteristic
124C22×C3×C4—32no—
125C22×C3×C4—121no—
126C2×#67C2×A412yes—
127C22×Dic3—32no—
128D3×Q8—22no—
129D3×Q8—32no—
130D3×Q8—64no—
131D3×Q8—64no—
132D3×Q8—64no—
133D3×Q8—64no—
134D3×Q8—62no—
135D3×Q8C3×D412yescomplemented
136D3×D4SL(2,𝔽3)18yesdirect product factor
137C2×D12C2×A411yescharacteristic
138C2×C4×D3—31no—
139C2×C4×D3—31no—
140C2×C4×D3—32no—
141C2×C4×D3—32no—
142C2×C4×D3—32no—
143C2×C4×D3—64no—
144C2×C4×D3—64no—
145C2×C4×D3C2×A411yescharacteristic
146C2×Dic6—31no—
147C2×Dic6—34no—
148C2×Dic6—32no—
149C2×Dic6—64no—
150C2×Dic6—64no—
151C2×SL(2,𝔽3)—22no—
152C2×SL(2,𝔽3)—21no—
153C2×SL(2,𝔽3)—32no—
154C2×SL(2,𝔽3)—42no—
155C2×SL(2,𝔽3)—62no—
156C2×SL(2,𝔽3)C22×D311yescharacteristic
157C3×C4⋊C4—31no—
158C3×C4⋊C4—32no—
159C3×C4⋊C4—31no—
160C3×C22⋊C4—32no—
161C3×C42—31no—
162#217—32no—
163(C2×D3)⋊C4—32no—
164(C2×D3)⋊C4—32no—
165C4⋊Dic3—32no—
166C4⋊Dic3—31no—
167C4⋊Dic3—31no—
168Dic3⋊C4—32no—
169Dic3⋊C4—34no—
170Dic3⋊C4—32no—
171C4×Dic3—31no—
172C4×Dic3—31no—
173C4×Dic3—31no—
174C22×C32—44no—
175C22×C32—41no—
176C22×C32—82no—
177C22×C32—82no—
178C2×C3×D3—42no—
179C2×C3×D3—42no—
180C2×C3×D3—44no—
181C2×C3×D3—42no—
182C2×C3×D3—816no—
183C2×C3×D3—82no—
184C2×C3×D3—82no—
185C32×C4—42no—
186C3×Dic3—42no—
187C22×Q8—31no—
188C22×Q8—32no—
189C22×Q8—64no—
190C22×Q8—92no—
191C22×Q8—92no—
192C22×Q8—92no—
193C22×Q8—184no—
194C22×Q8C2×C3×D312yescomplemented
195C22×D4—31no—
196C23×C4—92no—
197C4⋊Q8—31no—
198C4⋊Q8—91no—
199C4⋊Q8—92no—
200C4⋊Q8—92no—
201C4⋊Q8—92no—
202C22⋊Q8—32no—
203C22⋊Q8—94no—
204C22⋊Q8—94no—
205C22⋊Q8—92no—
206C22⋊Q8—92no—
207C22⋊Q8—92no—
208C4×Q8—31no—
209C4×Q8—91no—
210C4×Q8—91no—
211C4×Q8C2×C3×D311yescharacteristic, complemented
212C4×D4—31no—
213C4×D4—91no—
214C4×D4—92no—
215C4×D4—91no—
216C4×D4—92no—
217C4×D4—91no—
218C2×C4⋊C4—91no—
219C2×C4⋊C4—91no—
220C2×C4⋊C4—92no—
221C2×C22⋊C4—92no—
222C2×C42—91no—
223C23×C3—42no—
224C23×C3—82no—
225C23×C3—121no—
226C23×C3—122no—
227C23×C3—128no—
228C23×C3—244no—
229C23×C3—244no—
230C23×C3C22×A412yes—
231C22×D3—24no—
232C22×D3—24no—
233C22×D3C2×SL(2,𝔽3)18yescomplemented
234C22×D3C22×A412yes—
235C22×D3C22×A411yescharacteristic
236C3×Q8—31no—
237C3×Q8—62no—
238C3×Q8—62no—
239C3×Q8C2×C3×D411yescharacteristic, complemented
240C3×D4—44no—
241C3×D4—84no—
242C3×D4—124no—
243C3×D4—128no—
244C3×D4C2×SL(2,𝔽3)14yescomplemented
245C2×C3×C4—31no—
246C2×C3×C4—31no—
247C2×C3×C4—32no—
248C2×C3×C4—41no—
249C2×C3×C4—62no—
250C2×C3×C4—62no—
251C2×C3×C4—81no—
252C2×C3×C4—124no—
253C2×C3×C4—121no—
254C2×C3×C4C22×A411yescharacteristic
255#67C2×SL(2,𝔽3)18yescomplemented
256C2×Dic3—31no—
257C2×Dic3—31no—
258C2×Dic3—32no—
259C2×Dic3—64no—
260C2×Dic3C22×A411yescharacteristic
261D12C2×SL(2,𝔽3)14yescomplemented
262C4×D3—32no—
263C4×D3—32no—
264C4×D3—64no—
265C4×D3—62no—
266C4×D3—62no—
267C4×D3—64no—
268C4×D3—62no—
269C4×D3C2×SL(2,𝔽3)14yescomplemented
270Dic6—32no—
271Dic6—32no—
272Dic6—68no—
273Dic6—62no—
274Dic6—62no—
275SL(2,𝔽3)—21no—
276SL(2,𝔽3)D3×D411yescharacteristic, direct product factor
277C2×C32—41no—
278C2×C32—41no—
279C2×C32—41no—
280C2×C32—84no—
281C3×D3—44no—
282C3×D3—84no—
283C24—32no—
284C2×Q8—22no—
285C2×Q8—32no—
286C2×Q8—32no—
287C2×Q8—62no—
288C2×Q8—62no—
289C2×Q8—94no—
290C2×Q8—91no—
291C2×Q8—92no—
292C2×Q8—92no—
293C2×Q8—184no—
294C2×Q8—184no—
295C2×Q8—184no—
296C2×Q8—184no—
297C2×Q8—184no—
298C2×Q8—182no—
299C2×Q8—184no—
300C2×Q8C22×C3×D311yescharacteristic
301C2×D4—32no—
302C2×D4—38no—
303C2×D4—31no—
304C2×D4D3×A411yescharacteristic
305C22×C4—31no—
306C22×C4—32no—
307C22×C4—91no—
308C22×C4—91no—
309C22×C4—92no—
310C22×C4—92no—
311C22×C4—92no—
312C22×C4—92no—
313C22×C4—184no—
314C22×C4—184no—
315C4⋊C4—31no—
316C4⋊C4—32no—
317C4⋊C4—31no—
318C4⋊C4—92no—
319C4⋊C4—91no—
320C4⋊C4—92no—
321C4⋊C4—94no—
322C4⋊C4—92no—
323C4⋊C4—91no—
324C22⋊C4—32no—
325C22⋊C4—92no—
326C22⋊C4—92no—
327C22⋊C4—92no—
328C42—31no—
329C42—91no—
330C42—91no—
331C42—91no—
332C22×C3—22no—
333C22×C3—22no—
334C22×C3—44no—
335C22×C3—41no—
336C22×C3—82no—
337C22×C3—82no—
338C22×C3—84no—
339C22×C3—81no—
340C22×C3—122no—
341C22×C3—122no—
342C22×C3—122no—
343C22×C3—124no—
344C22×C3—162no—
345C22×C3—162no—
346C22×C3—242no—
347C22×C3—242no—
348C22×C3—2416no—
349C22×C3C22×SL(2,𝔽3)14yescomplemented
350C22×C3C23×A411yescharacteristic
351C2×D3—22no—
352C2×D3—22no—
353C2×D3—216no—
354C2×D3D4×A412yes—
355C2×D3C22×SL(2,𝔽3)12yes—
356C2×D3C22×SL(2,𝔽3)14yescomplemented
357C2×D32−1+4⋊C312yes—
358C3×C4—31no—
359C3×C4—31no—
360C3×C4—42no—
361C3×C4—62no—
362C3×C4—61no—
363C3×C4—82no—
364C3×C4—122no—
365C3×C4C22×SL(2,𝔽3)12yescomplemented
366Dic3—32no—
367Dic3—62no—
368Dic3—61no—
369Dic3C22×SL(2,𝔽3)12yescomplemented
370C32—41noSylow
371C23—31no—
372C23—38no—
373C23—32no—
374C23—64no—
375C23—64no—
376C23C2×D3×A412yes—
377Q8—31no—
378Q8—62no—
379Q8—62no—
380Q8—92no—
381Q8—92no—
382Q8—182no—
383Q8—182no—
384Q8—188no—
385Q8C3×D3×D411yescharacteristic, complemented
386D4—38no—
387D4—34no—
388D4D3×SL(2,𝔽3)14yesdirect product factor
389C2×C4—31no—
390C2×C4—31no—
391C2×C4—34no—
392C2×C4—31no—
393C2×C4—32no—
394C2×C4—62no—
395C2×C4—62no—
396C2×C4—92no—
397C2×C4—92no—
398C2×C4—91no—
399C2×C4—91no—
400C2×C4—92no—
401C2×C4—182no—
402C2×C4—182no—
403C2×C4—184no—
404C2×C4—184no—
405C2×C4—182no—
406C2×C4—184no—
407C2×C4C2×D3×A411yescharacteristic
408C2×C3—24no—
409C2×C3—41no—
410C2×C3—41no—
411C2×C3—41no—
412C2×C3—84no—
413C2×C3—81no—
414C2×C3—81no—
415C2×C3—81no—
416C2×C3—124no—
417C2×C3—164no—
418C2×C3—244no—
419C2×C3C2×D4×A411yescharacteristic
420C2×C3C23×SL(2,𝔽3)11yescharacteristic
421C2×C3C2×2−1+4⋊C311yescharacteristic
422D3—24no—
423D3D4×SL(2,𝔽3)14yesdirect product factor
424C22—22no—
425C22—22no—
426C22—32no—
427C22—32no—
428C22—32no—
429C22—34no—
430C22—616no—
431C22—62no—
432C22—62no—
433C22C2×D3×SL(2,𝔽3)14yescomplemented
434C22C22×D3×A411yescharacteristic
435C4—31no—
436C4—31no—
437C4—32no—
438C4—62no—
439C4—61no—
440C4—92no—
441C4—182no—
442C4—181no—
443C4C2×D3×SL(2,𝔽3)12yescomplemented
444C3—41no—
445C3—81no—
446C3C2×D4×SL(2,𝔽3)11yescharacteristic, complemented
447C2—24no—
448C2—34no—
449C2—64no—
450C2D3×D4×A411yescharacteristic
451C2C22×D3×SL(2,𝔽3)11yescharacteristic
452C2D3×2−1+4⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC3×D4×Q8C2×C3
socleC22×C3C23×A4
hypercenterC2×D4D3×A4
hypocenterC3×Q8C2×C3×D4
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. C22×D3×SL(2,𝔽3)C2semidirect
2. C3×D4×SL(2,𝔽3)C2semidirect
3. SL(2,𝔽3)×#67C2semidirect
4. SL(2,𝔽3)×D12C2semidirect
5. C4×D3×SL(2,𝔽3)C2semidirect
6. D3×D4×Q8C3semidirect
9. C22×C3×SL(2,𝔽3)C22semidirect
12. C2×D3×SL(2,𝔽3)C22semidirect
13. C3×C4×SL(2,𝔽3)C22semidirect
14. Dic3×SL(2,𝔽3)C22semidirect
15. C22×D3×Q8C2×C3semidirect
17. C3×D4×Q8C2×C3semidirect
18. Q8×#67C2×C3semidirect
23. Q8×D12C2×C3semidirect
24. C4×D3×Q8C2×C3semidirect
30. D4×SL(2,𝔽3)D3direct (N × H)
41. D3×SL(2,𝔽3)D4direct (N × H)
43. C22×C3×Q8C22×C3semidirect
50. C2×D3×Q8C22×C3semidirect
58. C22×SL(2,𝔽3)C2×D3semidirect
61. C3×C4×Q8C22×C3semidirect
64. Q8×Dic3C22×C3semidirect
87. C4×SL(2,𝔽3)C2×D3semidirect
100. C3×SL(2,𝔽3)C2×D4semidirect
107. D4×Q8C3×D3semidirect
135. D3×Q8C3×D4semidirect
136. D3×D4SL(2,𝔽3)direct (N × H)
194. C22×Q8C2×C3×D3semidirect
211. C4×Q8C2×C3×D3semidirect
233. C22×D3C2×SL(2,𝔽3)semidirect
239. C3×Q8C2×C3×D4semidirect
244. C3×D4C2×SL(2,𝔽3)semidirect
255. #67C2×SL(2,𝔽3)semidirect
261. D12C2×SL(2,𝔽3)semidirect
269. C4×D3C2×SL(2,𝔽3)semidirect
276. SL(2,𝔽3)D3×D4direct (N × H)
349. C22×C3C22×SL(2,𝔽3)semidirect
356. C2×D3C22×SL(2,𝔽3)semidirect
365. C3×C4C22×SL(2,𝔽3)semidirect
369. Dic3C22×SL(2,𝔽3)semidirect
385. Q8C3×D3×D4semidirect
388. D4D3×SL(2,𝔽3)direct (N × H)
423. D3D4×SL(2,𝔽3)direct (N × H)
433. C22C2×D3×SL(2,𝔽3)semidirect
443. C4C2×D3×SL(2,𝔽3)semidirect
446. C3C2×D4×SL(2,𝔽3)semidirect

Maximal subgroups

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

subgroupindex
C22×D3×SL(2,𝔽3)2
C3×D4×SL(2,𝔽3)2
SL(2,𝔽3)×#672
SL(2,𝔽3)×D122
C4×D3×SL(2,𝔽3)2
D3×D4×Q83
C2×D4×SL(2,𝔽3)3
C2×C3×D3×D44