Finite Group Databasebeta

← D3×#1714 #10782896

Subgroups — D3×#1714 #10782896

Order 768.

914 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×#500C211yesmaximal, characteristic, complemented
2D3×#498C211yesmaximal, characteristic, complemented
3D3×#492C211yesmaximal, characteristic
4D3×#492C211yesmaximal, characteristic
5D3×#487C211yesmaximal, characteristic, complemented
6D3×#486C211yesmaximal, characteristic
7C3×#1714C211yesmaximal, characteristic, complemented
8#71363C211yesmaximal, characteristic, complemented
9#71335C211yesmaximal, characteristic, complemented
10#71258C211yesmaximal, characteristic, complemented
11#71257C211yesmaximal, characteristic, complemented
12#71216C211yesmaximal, characteristic, complemented
13#71184C211yesmaximal, characteristic, complemented
14#68090C211yesmaximal, characteristic, complemented
15D3×#322C211yesmaximal, characteristic, complemented
16C2×#1714—31nomaximal, Sylow
17C2×D3×SD16—22no—
18C2×D3×SD16—22no—
19D3×C4⋊Q8C2211yescharacteristic
20D3×C4⋊Q8C2211yescharacteristic
21D3×C4⋊Q8C2211yescharacteristic
22D3×#127C2211yescharacteristic
23C3×#500C2211yescharacteristic, complemented
24C3×#498C2211yescharacteristic, complemented
25C3×#492C2211yescharacteristic
26C3×#492C2211yescharacteristic
27C3×#487C2211yescharacteristic, complemented
28C3×#486C2211yescharacteristic
29#5175—22no—
30#5175—22no—
31#5093C2211yescharacteristic, complemented
32#5088C2211yescharacteristic
33#5085C2211yescharacteristic
34#5085C2211yescharacteristic
35#5080C2211yescharacteristic, complemented
36#5080C2211yescharacteristic, complemented
37#5079C2211yescharacteristic, complemented
38#5078C2211yescharacteristic, complemented
39#5078C2211yescharacteristic, complemented
40#5076C2211yescharacteristic
41#5076C2211yescharacteristic
42#5070C2211yescharacteristic
43D3×#107—22no—
44#4874—22no—
45D3×#106—22no—
46#4855—22no—
47D3×#103—22no—
48#4792—22no—
49D3×#102—22no—
50#4760—22no—
51#4694C2211yescharacteristic
52#4693C2211yescharacteristic
53#4693C2211yescharacteristic
54C4×D3×C8C2211yescharacteristic
55C4×D3×C8C2211yescharacteristic
56C4×D3×C8C2211yescharacteristic
57#4682C2211yescharacteristic, complemented
58#4681C2211yescharacteristic, complemented
59#4680C2211yescharacteristic, complemented
60C3×#322C2211yescharacteristic, complemented
61#4455C2211yescharacteristic, complemented
62#4454C2211yescharacteristic, complemented
63#4454C2211yescharacteristic, complemented
64C2×#500—31no—
65C2×#498—31no—
66C2×#492—31no—
67C2×#492—31no—
68C2×#487—31no—
69C2×#486—31no—
70#1714—31no—
71#1714—31no—
72#1714—31no—
73#1714—31no—
74#1714—31no—
75#1714—31no—
76#1714—31no—
77#1714D311yescharacteristic, direct product factor
78C2×#322—31no—
79C2×D3×Q8—21no—
80C2×D3×Q8—21no—
81C2×D3×Q8—21no—
82C2×D3×D4—21no—
83C2×D3×D4—21no—
84C2×D3×D4—21no—
85C2×C3×SD16—22no—
86C2×C3×SD16—22no—
87C3×C4⋊Q8C2311yescharacteristic
88C3×C4⋊Q8C2311yescharacteristic
89C3×C4⋊Q8C2311yescharacteristic
90C3×#127C2311yescharacteristic
91#944—21no—
92#944—21no—
93#944—21no—
94C2×#215—22no—
95C2×#215—22no—
96#940—21no—
97#940—21no—
98#940—21no—
99C2×#214—22no—
100C2×#214—22no—
101D3×SD16—48no—
102D3×SD16—48no—
103C2×#204—22no—
104C2×#204—22no—
105C2×D3×C8D412yes—
106C2×D3×C8D412yes—
107C2×D3×C8D412yes—
108D3×C4⋊C4—21no—
109D3×C4⋊C4—21no—
110D3×C4⋊C4—21no—
111D3×C4⋊C4—21no—
112D3×C4⋊C4—21no—
113D3×C4⋊C4—21no—
114#888—21no—
115#888—21no—
116#888—21no—
117#888—21no—
118#888—21no—
119#888—21no—
120#874C2311yescharacteristic
121C42×D3C2311yescharacteristic
122#869C2311yescharacteristic
123#869C2311yescharacteristic
124#869C2311yescharacteristic
125C3×#107—22no—
126C3×#106—22no—
127C3×#103—22no—
128C3×#102—22no—
129C3×C4×C8C2311yescharacteristic
130C3×C4×C8C2311yescharacteristic
131C3×C4×C8C2311yescharacteristic
132#835—22no—
133#832—22no—
134#821—22no—
135#818—22no—
136#817—22no—
137#816—22no—
138C8×Dic3D412yes—
139C8×Dic3D412yes—
140C8×Dic3D412yes—
141#810—22no—
142#809—22no—
143#808—22no—
144#808—22no—
145#807—22no—
146#807—22no—
147C4×C3⋊C8C2311yescharacteristic
148C4×C3⋊C8C2311yescharacteristic
149C4×C3⋊C8C2311yescharacteristic
150C22×SD16—62no—
151C22×SD16—62no—
152C2×C4⋊Q8—31no—
153C2×C4⋊Q8—31no—
154C2×C4⋊Q8—31no—
155C2×#127—31no—
156#500—31no—
157#500—31no—
158#500—31no—
159#500—62no—
160#500C2×D311yescharacteristic, complemented
161#498—31no—
162#498—31no—
163#498—31no—
164#498—62no—
165#498C2×D311yescharacteristic, complemented
166#492—31no—
167#492—31no—
168#492—31no—
169#492—31no—
170#492—31no—
171#492—31no—
172#492—62no—
173#492—62no—
174#492C2×D311yescharacteristic
175#492C2×D311yescharacteristic
176#487—31no—
177#487—31no—
178#487—31no—
179#487—62no—
180#487C2×D311yescharacteristic, complemented
181#486—31no—
182#486—31no—
183#486—31no—
184#486—62no—
185#486C2×D311yescharacteristic
186C2×#107—62no—
187C2×#106—62no—
188C2×#103—62no—
189C2×#102—62no—
190C2×C4×C8—31no—
191C2×C4×C8—31no—
192C2×C4×C8—31no—
193#322—31no—
194#322—31no—
195#322—31no—
196#322C2×D311yescharacteristic, complemented
197C23×D3—41no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×#67—41no—
205C2×#67—41no—
206C2×#67—41no—
207D3×Q8—44no—
208D3×Q8—44no—
209D3×Q8—44no—
210D3×D4—44no—
211D3×D4—44no—
212D3×D4—44no—
213C2×D12—21no—
214C2×D12—21no—
215C2×D12—21no—
216C2×C4×D3—41no—
217C2×C4×D3—41no—
218C2×C4×D3—41no—
219C2×C4×D3C2×D411yescharacteristic
220C2×C4×D3C2×D411yescharacteristic
221C2×C4×D3C2×D411yescharacteristic
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—41no—
226C2×Dic6—41no—
227C2×Dic6—41no—
228C3×SD16—44no—
229C3×SD16—44no—
230C2×C3×C8C2×D412yes—
231C2×C3×C8C2×D412yes—
232C2×C3×C8C2×D412yes—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C42C2411yescharacteristic
240#215—44no—
241#215—44no—
242#214—44no—
243#214—44no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250Dic3⋊C4—41no—
251Dic3⋊C4—41no—
252Dic3⋊C4—41no—
253Dic3⋊C4—41no—
254Dic3⋊C4—41no—
255Dic3⋊C4—41no—
256C4×Dic3C2×D411yescharacteristic
257C4×Dic3C2×D411yescharacteristic
258C4×Dic3C2×D411yescharacteristic
259C2×C3⋊C8C2×D412yes—
260C2×C3⋊C8C2×D412yes—
261C2×C3⋊C8C2×D412yes—
262#204—44no—
263#204—44no—
264D3×C8—24no—
265D3×C8—24no—
266D3×C8SD1618yescomplemented
267C22×Q8—61no—
268C22×Q8—61no—
269C22×Q8—61no—
270C22×D4—61no—
271C22×D4—61no—
272C22×D4—61no—
273C2×SD16—22no—
274C2×SD16—22no—
275C2×SD16—62no—
276C2×SD16—62no—
277C2×SD16—62no—
278C2×SD16—62no—
279C2×SD16—62no—
280C2×SD16—62no—
281C2×SD16—128no—
282C2×SD16—128no—
283C22×C8—32no—
284C22×C8—32no—
285C22×C8—32no—
286C4⋊Q8—31no—
287C4⋊Q8—31no—
288C4⋊Q8—31no—
289C4⋊Q8—61no—
290C4⋊Q8—61no—
291C4⋊Q8—61no—
292C4⋊Q8—61no—
293C4⋊Q8—61no—
294C4⋊Q8—61no—
295C4⋊Q8—61no—
296C4⋊Q8—61no—
297C4⋊Q8—61no—
298C4⋊Q8C22×D311yescharacteristic
299C4⋊Q8C22×D311yescharacteristic
300C4⋊Q8C22×D311yescharacteristic
301#127—31no—
302#127—61no—
303#127—61no—
304#127—61no—
305#127C22×D311yescharacteristic
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—61no—
312C2×C42—31no—
313#107—22no—
314#107—62no—
315#107—62no—
316#107—62no—
317#106—22no—
318#106—62no—
319#106—62no—
320#106—62no—
321#103—22no—
322#103—62no—
323#103—62no—
324#103—62no—
325#102—22no—
326#102—62no—
327#102—62no—
328#102—62no—
329C4×C8—32no—
330C4×C8—32no—
331C4×C8—32no—
332C4×C8—31no—
333C4×C8—31no—
334C4×C8—31no—
335C4×C8C22×D311yescharacteristic
336C4×C8C22×D311yescharacteristic
337C4×C8C22×D311yescharacteristic
338C23×C3—41no—
339C22×D3—41no—
340C22×D3—82no—
341C22×D3—82no—
342C22×D3—82no—
343C22×D3#12711yescharacteristic
344C3×Q8—42no—
345C3×Q8—42no—
346C3×Q8—42no—
347C3×D4—42no—
348C3×D4—42no—
349C3×D4—42no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4C22×D411yescharacteristic
354C2×C3×C4C22×D411yescharacteristic
355C2×C3×C4C22×D411yescharacteristic
356#67—82no—
357#67—82no—
358#67—82no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361C2×Dic3—41no—
362C2×Dic3#12711yescharacteristic
363C2×Dic3#12711yescharacteristic
364C2×Dic3#12711yescharacteristic
365D12—42no—
366D12—42no—
367D12—42no—
368C4×D3—82no—
369C4×D3—82no—
370C4×D3—82no—
371C4×D3C2×SD1614yes—
372C4×D3AGL(1,ℤ/8)14yes—
373C4×D3#13714yes—
374Dic6—42no—
375Dic6—42no—
376Dic6—42no—
377Dic6—82no—
378Dic6—82no—
379Dic6—82no—
380C3×C8—22no—
381C3×C8—22no—
382C3×C8C2×SD1614yescomplemented
383C3⋊C8—22no—
384C3⋊C8—22no—
385C3⋊C8C2×SD1614yescomplemented
386C24—121no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—61no—
391C2×Q8—61no—
392C2×Q8—61no—
393C2×Q8—121no—
394C2×Q8—121no—
395C2×Q8—121no—
396C2×Q8—124no—
397C2×Q8—124no—
398C2×Q8—124no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—61no—
403C2×D4—61no—
404C2×D4—61no—
405C2×D4—124no—
406C2×D4—124no—
407C2×D4—124no—
408C2×D4—121no—
409C2×D4—121no—
410C2×D4—121no—
411C22×C4—31no—
412C22×C4—31no—
413C22×C4—31no—
414C22×C4—121no—
415C22×C4—121no—
416C22×C4—121no—
417SD16—44no—
418SD16—44no—
419SD16—124no—
420SD16—124no—
421SD16—124no—
422SD16—124no—
423SD16—124no—
424SD16—124no—
425C2×C8—32no—
426C2×C8—38no—
427C2×C8—32no—
428C2×C8—32no—
429C2×C8—64no—
430C2×C8—64no—
431C2×C8D3×D412yes—
432C2×C8D3×D412yes—
433C2×C8D3×D412yes—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—121no—
447C4⋊C4—121no—
448C4⋊C4—121no—
449C4⋊C4—121no—
450C4⋊C4—121no—
451C4⋊C4—121no—
452C42—31no—
453C42—31no—
454C42—31no—
455C42C23×D311yescharacteristic
456C22×C3—81no—
457C22×C3—81no—
458C22×C3—81no—
459C22×C3C2×#12711yescharacteristic
460C2×D3—81no—
461C2×D3—81no—
462C2×D3—81no—
463C2×D3—164no—
464C2×D3#49212yes—
465C2×D3#49612yes—
466C2×D3#49712yes—
467C3×C4—81no—
468C3×C4—81no—
469C3×C4—81no—
470C3×C4C22×SD1612yes—
471C3×C4C2×AGL(1,ℤ/8)12yes—
472C3×C4C2×#13712yes—
473Dic3—81no—
474Dic3—81no—
475Dic3—81no—
476Dic3#49212yes—
477Dic3#49612yes—
478Dic3#49712yes—
479C23—31no—
480C23—41no—
481C23—121no—
482C23—242no—
483C23—242no—
484C23—242no—
485Q8—42no—
486Q8—42no—
487Q8—42no—
488Q8—122no—
489Q8—122no—
490Q8—122no—
491Q8—242no—
492Q8—242no—
493Q8—242no—
494D4—42no—
495D4—42no—
496D4—42no—
497D4—122no—
498D4—122no—
499D4—122no—
500D4—242no—
501D4—242no—
502D4—242no—
503C2×C4—31no—
504C2×C4—31no—
505C2×C4—31no—
506C2×C4—34no—
507C2×C4—34no—
508C2×C4—34no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—121no—
513C2×C4—121no—
514C2×C4—121no—
515C2×C4—242no—
516C2×C4—242no—
517C2×C4—242no—
518C2×C4C2×D3×D411yescharacteristic
519C2×C4C2×D3×D411yescharacteristic
520C2×C4C2×D3×D411yescharacteristic
521C8—22no—
522C8—22no—
523C8—34no—
524C8—62no—
525C8—62no—
526C8D3×SD1614yescomplemented
527C2×C3—161no—
528C2×C3C2×#49211yescharacteristic
529C2×C3C2×#49611yescharacteristic
530C2×C3C2×#49711yescharacteristic
531D3—161no—
532D3#171414yesdirect product factor
533C22—32no—
534C22—32no—
535C22—32no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22—241no—
540C22—241no—
541C22—241no—
542C22—484no—
543C22D3×#12711yescharacteristic
544C4—32no—
545C4—32no—
546C4—32no—
547C4—81no—
548C4—81no—
549C4—81no—
550C4—241no—
551C4—241no—
552C4—241no—
553C4C2×D3×SD1612yes—
554C4D3×AGL(1,ℤ/8)12yes—
555C4D3×#13712yes—
556C3C2×#171411yescharacteristic, complemented, Sylow
557C2—34no—
558C2—161no—
559C2—481no—
560C2D3×#49211yescharacteristic
561C2D3×#49611yescharacteristic
562C2D3×#49711yescharacteristic
subgroupG/H
centerC22D3×#127
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1714C2
socleC22×C3C2×#127
prime radicalD3×#127C22
hypercenter#1714D3
hypocenterC3C2×#1714
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #1714

Complements

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

NH ≅ G/Ntype
1. D3×#500C2semidirect
2. D3×#498C2semidirect
5. D3×#487C2semidirect
7. C3×#1714C2semidirect
8. #71363C2semidirect
9. #71335C2semidirect
10. #71258C2semidirect
11. #71257C2semidirect
12. #71216C2semidirect
13. #71184C2semidirect
14. #68090C2semidirect
15. D3×#322C2semidirect
23. C3×#500C22semidirect
24. C3×#498C22semidirect
27. C3×#487C22semidirect
31. #5093C22semidirect
35. #5080C22semidirect
36. #5080C22semidirect
37. #5079C22semidirect
38. #5078C22semidirect
39. #5078C22semidirect
57. #4682C22semidirect
58. #4681C22semidirect
59. #4680C22semidirect
60. C3×#322C22semidirect
61. #4455C22semidirect
62. #4454C22semidirect
63. #4454C22semidirect
77. #1714D3direct (N × H)
160. #500C2×D3semidirect
165. #498C2×D3semidirect
180. #487C2×D3semidirect
196. #322C2×D3semidirect
266. D3×C8SD16semidirect
382. C3×C8C2×SD16semidirect
385. C3⋊C8C2×SD16semidirect
526. C8D3×SD16semidirect
532. D3#1714direct (N × H)
556. C3C2×#1714semidirect

Maximal subgroups

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

subgroupindex
D3×#5002
D3×#4982
D3×#4922
D3×#4872
D3×#4862
C3×#17142
#713632
#713352
#712582
#712572
#712162
#711842
#680902
D3×#3222
C2×#17143