Finite Group Databasebeta

← #10784404

Subgroups — #10784404

Order 768.

814 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3205C211yesmaximal, characteristic, complemented
2#83304C211yesmaximal, characteristic, complemented
3#83303C211yesmaximal, characteristic, complemented
4#83166C211yesmaximal, characteristic, complemented
5#83166C211yesmaximal, characteristic, complemented
6#79304C211yesmaximal, characteristic, complemented
7#71280C211yesmaximal, characteristic, complemented
8#71279C211yesmaximal, characteristic, complemented
9#71278C211yesmaximal, characteristic, complemented
10#71276C211yesmaximal, characteristic, complemented
11#71194C211yesmaximal, characteristic, complemented
12#71188C211yesmaximal, characteristic, complemented
13#71186C211yesmaximal, characteristic, complemented
14#71183C211yesmaximal, characteristic, complemented
15#70744C211yesmaximal, characteristic, complemented
16#19743—31nomaximal, Sylow
17C3×#532C2211yescharacteristic
18C2×C3×#127C2211yescharacteristic
19#5477C2211yescharacteristic
20#5477C2211yescharacteristic
21C3×#493C2211yescharacteristic
22C3×#493C2211yescharacteristic
23C3×#486C2211yescharacteristic
24C3×#486C2211yescharacteristic
25C3×#459—22no—
26C3×#447—22no—
27C3×#431C2211yescharacteristic
28#5238—22no—
29#5217—22no—
30#5217—22no—
31#5093C2211yescharacteristic, complemented
32#5088C2211yescharacteristic, complemented
33#5085C2211yescharacteristic, complemented
34#5079C2211yescharacteristic, complemented
35#5076C2211yescharacteristic, complemented
36#5073C2211yescharacteristic
37#5073C2211yescharacteristic
38#5073C2211yescharacteristic
39#5073C2211yescharacteristic
40#5073C2211yescharacteristic, complemented
41#5073C2211yescharacteristic, complemented
42#5072—22no—
43#5072—22no—
44#5070C2211yescharacteristic
45#5070C2211yescharacteristic
46#5070C2211yescharacteristic
47#5070C2211yescharacteristic
48#5070C2211yescharacteristic, complemented
49#5057—22no—
50#5037—22no—
51#5016—22no—
52#4925C2211yescharacteristic
53#4925C2211yescharacteristic
54#4844—22no—
55#4734—22no—
56#4695C2211yescharacteristic, complemented
57#4694C2211yescharacteristic, complemented
58#4693C2211yescharacteristic, complemented
59#4681C2211yescharacteristic, complemented
60#4455C2211yescharacteristic, complemented
61#4454C2211yescharacteristic, complemented
62#4454C2211yescharacteristic, complemented
63C8×C3⋊C8C2211yescharacteristic, complemented
64#3207—31no—
65#3207—31no—
66#3207—31no—
67#3205—31no—
68#3205—31no—
69#3205D311yescharacteristic, complemented
70#1721—31no—
71#1721—31no—
72#1721—31no—
73#1717—31no—
74#1717—31no—
75#1717—31no—
76#1712—31no—
77#1711—31no—
78#1454—31no—
79C2×C3×#41—21no—
80C22×C3×D4—21no—
81C22×C3×D4—21no—
82C22×C3×D4—21no—
83C2×#235—21no—
84C2×#235—21no—
85C2×C3×D8—22no—
86C2×C3×D8—22no—
87C3×C4⋊Q8C2311yescharacteristic
88C3×#127—21no—
89C3×#127—21no—
90C3×#127—21no—
91C3×#127C2311yescharacteristic
92C3×#127C2311yescharacteristic
93C3×#127C2311yescharacteristic
94C3×#124—21no—
95C3×#121—21no—
96C3×#121—21no—
97C3×C4×D4—21no—
98C3×C4×D4—21no—
99C2×C3×C42C2311yescharacteristic
100C2×#215—22no—
101C2×#214—22no—
102C2×#213—22no—
103C2×#213—22no—
104C2×#213—22no—
105C2×#213—22no—
106C2×#213—22no—
107C2×#213—22no—
108#930—21no—
109#930—21no—
110#930—21no—
111#930—21no—
112C2×D24—22no—
113C2×#204—22no—
114#875—21no—
115#875—21no—
116#874C2311yescharacteristic
117#874C2311yescharacteristic
118C4×D12—21no—
119C4×D12—21no—
120C4×D12—21no—
121C4×D12—21no—
122#869C2311yescharacteristic
123#869C2311yescharacteristic
124C3×C4⋊C8D412yes—
125C3×#102—22no—
126C3×#102—22no—
127C3×C22⋊C8D412yes—
128C3×C4×C8C2311yescharacteristic
129C3×C4×C8C2311yescharacteristic
130#835—22no—
131#832—22no—
132#832—22no—
133#832—22no—
134#832—22no—
135#832—22no—
136#830D412yes—
137#830D412yes—
138#821—22no—
139#816—22no—
140#810—22no—
141#809—22no—
142C4⋊(C3⋊C8)D412yes—
143C4⋊(C3⋊C8)D412yes—
144C4×C3⋊C8C2311yescharacteristic
145C4×C3⋊C8C2311yescharacteristic
146C4×C3⋊C8C2311yescharacteristic
147C4×C3⋊C8C2311yescharacteristic
148#532—31no—
149#532—31no—
150#532C2×D311yescharacteristic
151C2×#127C2×D311yescharacteristic
152#493—31no—
153#493—31no—
154#493—31no—
155#493—31no—
156#493—31no—
157#493—31no—
158#493—31no—
159#493C2×D311yescharacteristic
160#493C2×D311yescharacteristic
161#492—31no—
162#492—31no—
163#492—31no—
164#487—31no—
165#487—31no—
166#487—31no—
167#486—31no—
168#486—31no—
169#486—31no—
170#486—31no—
171#486—31no—
172#486—31no—
173#486—31no—
174#486C2×D311yescharacteristic
175#486C2×D311yescharacteristic
176#463—62no—
177#463—62no—
178#463—62no—
179#459—22no—
180#459—62no—
181#459—62no—
182#449—62no—
183#449—62no—
184#449—62no—
185#447—22no—
186#447—62no—
187#447—62no—
188#431—31no—
189#431—31no—
190#431C2×D311yescharacteristic
191#322—31no—
192#322—31no—
193#322—31no—
194C82—31no—
195C24×C3—41no—
196C3×#41—82no—
197C2×C3×Q8—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—41no—
208C2×C3×D4—41no—
209C2×C3×D4—41no—
210C2×C3×D4—41no—
211C2×C3×D4—82no—
212C2×C3×D4—82no—
213C2×C3×D4—82no—
214C22×C3×C4—41no—
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C2×#67—41no—
219C2×#67—41no—
220#235—82no—
221#235—82no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—21no—
227C2×D12—21no—
228C2×C4×D3—41no—
229C2×C4×D3—41no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C3×D8—82no—
233C3×D8—82no—
234C2×C3×C8C2×D412yes—
235C2×C3×C8C2×D412yes—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C22⋊C4—41no—
239C3×C22⋊C4—41no—
240C3×C42C2×D411yescharacteristic
241C3×C42C2×D411yescharacteristic
242C3×C42C2×D411yescharacteristic
243C3×C42C2411yescharacteristic
244#215—82no—
245#214—82no—
246#213—82no—
247#213—82no—
248#213—82no—
249#213—82no—
250#213—82no—
251#213—82no—
252(C2×D3)⋊C4—41no—
253(C2×D3)⋊C4—41no—
254(C2×D3)⋊C4—41no—
255(C2×D3)⋊C4—41no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C2×C3⋊C8C2×D412yes—
261C2×C3⋊C8C2×D412yes—
262C2×C3⋊C8C2×D412yes—
263C2×C3⋊C8C2×D412yes—
264D24—82no—
265#204—82no—
266C2×#41—21no—
267C2×#41—61no—
268C2×#41—61no—
269C22×D4—21no—
270C22×D4—21no—
271C22×D4—21no—
272C2×SD16—62no—
273C2×SD16—62no—
274C2×SD16—62no—
275C2×D8—22no—
276C2×D8—22no—
277C2×D8—62no—
278C2×D8—62no—
279C2×D8—62no—
280C2×D8—62no—
281C2×D8—62no—
282C2×D8—62no—
283C2×D8—62no—
284C4⋊Q8—31no—
285C4⋊Q8—31no—
286C4⋊Q8C22×D311yescharacteristic
287#127—21no—
288#127—21no—
289#127—21no—
290#127—31no—
291#127—31no—
292#127C22×D311yescharacteristic
293#127C22×D311yescharacteristic
294#127C22×D311yescharacteristic
295#124—21no—
296#124—61no—
297#124—61no—
298#121—21no—
299#121—21no—
300#121—61no—
301#121—61no—
302#121—61no—
303#121—61no—
304C4×D4—21no—
305C4×D4—21no—
306C4×D4—61no—
307C4×D4—61no—
308C4×D4—61no—
309C4×D4—61no—
310C2×C42C22×D311yescharacteristic
311C4⋊C8—32no—
312C4⋊C8—32no—
313C4⋊C8#6712yes—
314#103—62no—
315#103—62no—
316#103—62no—
317#102—22no—
318#102—22no—
319#102—62no—
320#102—62no—
321#102—62no—
322#102—62no—
323#102—62no—
324#102—62no—
325#102—62no—
326C22⋊C8—32no—
327C22⋊C8—32no—
328C22⋊C8#6712yes—
329C4×C8—31no—
330C4×C8—31no—
331C4×C8—31no—
332C4×C8—31no—
333C4×C8C22×D311yescharacteristic
334C4×C8C22×D311yescharacteristic
335C23×C3—41no—
336C23×C3—41no—
337C23×C3—41no—
338C23×C3—82no—
339C23×C3—82no—
340C23×C3—82no—
341C23×C3#12711yescharacteristic
342C22×D3—41no—
343C22×D3—41no—
344C3×Q8—42no—
345C3×D4—42no—
346C3×D4—42no—
347C3×D4—42no—
348C3×D4—42no—
349C3×D4—42no—
350C3×D4—42no—
351C3×D4—42no—
352C3×D4—42no—
353C3×D4—42no—
354C3×D4—82no—
355C3×D4—82no—
356C3×D4—82no—
357C3×D4—82no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—41no—
362C2×C3×C4—82no—
363C2×C3×C4#12711yescharacteristic
364C2×C3×C4#12711yescharacteristic
365C2×C3×C4#12711yescharacteristic
366C2×C3×C4C22×D411yescharacteristic
367C2×C3×C4C22×D411yescharacteristic
368C2×C3×C4C22×D411yescharacteristic
369#67—82no—
370#67—82no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373D12—42no—
374D12—42no—
375D12—42no—
376D12—42no—
377D12—42no—
378D12—42no—
379C4×D3—82no—
380C4×D3—82no—
381Dic6—42no—
382Dic6—42no—
383C3×C8—22no—
384C3×C8—22no—
385C3⋊C8—22no—
386C3⋊C8—22no—
387C3⋊C8—22no—
388C3⋊C8—22no—
389C24—41no—
390#41—82no—
391#41—242no—
392#41—242no—
393C2×Q8—21no—
394C2×Q8—61no—
395C2×Q8—61no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4—41no—
408C2×D4—41no—
409C2×D4—61no—
410C2×D4—61no—
411C2×D4—61no—
412C2×D4—61no—
413C2×D4—61no—
414C2×D4—61no—
415C2×D4—82no—
416C2×D4—82no—
417C2×D4—82no—
418C2×D4—121no—
419C2×D4—121no—
420C22×C4—41no—
421C22×C4—121no—
422C22×C4—121no—
423C22×C4D3×D411yescharacteristic
424C22×C4D3×D411yescharacteristic
425C22×C4C2×#6711yescharacteristic
426SD16—242no—
427SD16—242no—
428SD16—242no—
429D8—82no—
430D8—82no—
431D8—242no—
432D8—242no—
433D8—242no—
434D8—242no—
435D8—242no—
436D8—242no—
437D8—242no—
438C2×C8—32no—
439C2×C8—32no—
440C2×C8—32no—
441C2×C8—32no—
442C2×C8C2×#6712yes—
443C2×C8C2×#6712yes—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C22⋊C4—41no—
451C22⋊C4—41no—
452C22⋊C4—121no—
453C22⋊C4—121no—
454C22⋊C4—121no—
455C22⋊C4—121no—
456C42D3×D411yescharacteristic
457C42D3×D411yescharacteristic
458C42C2×#6711yescharacteristic
459C42C23×D311yescharacteristic
460C22×C3—21no—
461C22×C3—21no—
462C22×C3—21no—
463C22×C3—81no—
464C22×C3—81no—
465C22×C3—81no—
466C22×C3—81no—
467C22×C3—81no—
468C22×C3—81no—
469C22×C3—81no—
470C22×C3—81no—
471C22×C3—81no—
472C22×C3—164no—
473C22×C3C2×#12711yescharacteristic
474C2×D3—81no—
475C2×D3—81no—
476C2×D3—81no—
477C2×D3—81no—
478C2×D3—81no—
479C2×D3—81no—
480C3×C4—21no—
481C3×C4—21no—
482C3×C4—21no—
483C3×C4—81no—
484C3×C4#57612yes—
485C3×C4#57612yes—
486C3×C4#57612yes—
487Dic3—81no—
488Dic3—81no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—82no—
493C23—82no—
494C23—82no—
495C23—121no—
496C23—121no—
497C23#94011yescharacteristic
498Q8—42no—
499Q8—122no—
500Q8—122no—
501D4—42no—
502D4—42no—
503D4—42no—
504D4—42no—
505D4—42no—
506D4—42no—
507D4—42no—
508D4—42no—
509D4—42no—
510D4—82no—
511D4—82no—
512D4—82no—
513D4—82no—
514D4—122no—
515D4—122no—
516D4—122no—
517D4—122no—
518D4—122no—
519D4—122no—
520D4—242no—
521D4—242no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—41no—
526C2×C4—82no—
527C2×C4—121no—
528C2×C4—121no—
529C2×C4—242no—
530C2×C4—242no—
531C2×C4#94011yescharacteristic
532C2×C4#94011yescharacteristic
533C2×C4#94011yescharacteristic
534C2×C4C2×D3×D411yescharacteristic
535C2×C4C2×D3×D411yescharacteristic
536C2×C4C22×#6711yescharacteristic
537C8—22no—
538C8—22no—
539C8—62no—
540C8—62no—
541C8—62no—
542C8—62no—
543C2×C3—41no—
544C2×C3—161no—
545C2×C3—161no—
546C2×C3—161no—
547C2×C3#315511yescharacteristic
548C2×C3#315511yescharacteristic
549C2×C3#315511yescharacteristic
550D3—161no—
551D3—161no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—164no—
565C22—241no—
566C22—241no—
567C22—241no—
568C22—241no—
569C22—241no—
570C22—241no—
571C22C2×#94011yescharacteristic
572C4—21no—
573C4—21no—
574C4—21no—
575C4—81no—
576C4—241no—
577C4—241no—
578C4#575712yes—
579C4#577412yes—
580C4#583512yes—
581C3#1974311yescharacteristic, complemented, Sylow
582C2—41no—
583C2—161no—
584C2—161no—
585C2—161no—
586C2—481no—
587C2—481no—
588C2#8338211yescharacteristic
589C2#8383011yescharacteristic
590C2#8394011yescharacteristic
subgroupG/H
centerC22C2×#940
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#3205C2
socleC22×C3C2×#127
hypercenter#3205D3
hypocenterC3#19743
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3205

Complements

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

NH ≅ G/Ntype
1. C3×#3205C2semidirect
2. #83304C2semidirect
3. #83303C2semidirect
4. #83166C2semidirect
5. #83166C2semidirect
6. #79304C2semidirect
7. #71280C2semidirect
8. #71279C2semidirect
9. #71278C2semidirect
10. #71276C2semidirect
11. #71194C2semidirect
12. #71188C2semidirect
13. #71186C2semidirect
14. #71183C2semidirect
15. #70744C2semidirect
31. #5093C22semidirect
32. #5088C22semidirect
33. #5085C22semidirect
34. #5079C22semidirect
35. #5076C22semidirect
40. #5073C22semidirect
41. #5073C22semidirect
48. #5070C22semidirect
56. #4695C22semidirect
57. #4694C22semidirect
58. #4693C22semidirect
59. #4681C22semidirect
60. #4455C22semidirect
61. #4454C22semidirect
62. #4454C22semidirect
63. C8×C3⋊C8C22semidirect
69. #3205D3semidirect
581. C3#19743semidirect

Maximal subgroups

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

subgroupindex
C3×#32052
#833042
#833032
#831662
#793042
#712802
#712792
#712782
#712762
#711942
#711882
#711862
#711832
#707442
#197433