Finite Group Databasebeta

← #10784397

Subgroups — #10784397

Order 768.

738 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3227C211yesmaximal, characteristic, complemented
2#83304C211yesmaximal, characteristic, complemented
3#83303C211yesmaximal, characteristic, complemented
4#83215C211yesmaximal, characteristic, complemented
5#83213C211yesmaximal, characteristic, complemented
6#79304C211yesmaximal, characteristic, complemented
7#71339C211yesmaximal, characteristic, complemented
8#71336C211yesmaximal, characteristic, complemented
9#71335C211yesmaximal, characteristic, complemented
10#71334C211yesmaximal, characteristic, complemented
11#71194C211yesmaximal, characteristic, complemented
12#71189C211yesmaximal, characteristic, complemented
13#71186C211yesmaximal, characteristic, complemented
14#71180C211yesmaximal, characteristic, complemented
15#70744C211yesmaximal, characteristic, complemented
16#19742—31nomaximal, Sylow
17C3×#532C2211yescharacteristic
18C2×C3×C4⋊Q8C2211yescharacteristic
19#5477C2211yescharacteristic
20#5477C2211yescharacteristic
21C3×#498C2211yescharacteristic
22C3×#498C2211yescharacteristic
23C3×#486C2211yescharacteristic
24C3×#486C2211yescharacteristic
25C3×#481—22no—
26C3×#476—22no—
27C3×#431C2211yescharacteristic
28#5238—22no—
29#5227—22no—
30#5093C2211yescharacteristic
31#5088C2211yescharacteristic
32#5088C2211yescharacteristic
33#5088C2211yescharacteristic
34#5087—22no—
35#5085C2211yescharacteristic
36#5085C2211yescharacteristic
37#5085C2211yescharacteristic
38#5083—22no—
39#5079C2211yescharacteristic
40#5079C2211yescharacteristic
41#5079C2211yescharacteristic
42#5078C2211yescharacteristic
43#5078C2211yescharacteristic
44#5076C2211yescharacteristic
45#5073C2211yescharacteristic
46#5073C2211yescharacteristic
47#5070C2211yescharacteristic
48#5057—22no—
49#5037—22no—
50#5016—22no—
51#4967—22no—
52#4925C2211yescharacteristic
53#4925C2211yescharacteristic
54#4844—22no—
55#4734—22no—
56#4695C2211yescharacteristic
57#4694C2211yescharacteristic
58#4693C2211yescharacteristic
59#4681C2211yescharacteristic
60#4455C2211yescharacteristic
61#4454C2211yescharacteristic
62#4454C2211yescharacteristic
63C8×C3⋊C8C2211yescharacteristic
64#3227D311yescharacteristic, complemented
65#3226—31no—
66#3207—31no—
67#3207—31no—
68#3207—31no—
69#3206—31no—
70#1721—31no—
71#1719—31no—
72#1718—31no—
73#1717—31no—
74#1715—31no—
75#1714—31no—
76#1708—31no—
77#1706—31no—
78#1454—31no—
79C2×C3×#41—21no—
80C22×C3×Q8—21no—
81C2×#235—21no—
82C2×#235—21no—
83C3×C4⋊Q8—21no—
84C3×C4⋊Q8—21no—
85C3×C4⋊Q8—21no—
86C3×C4⋊Q8C2311yescharacteristic
87C3×C4⋊Q8C2311yescharacteristic
88C3×C4⋊Q8C2311yescharacteristic
89C3×#127C2311yescharacteristic
90C3×#124—21no—
91C3×#121—21no—
92C3×#121—21no—
93C3×C4×D4—21no—
94C3×C4×D4—21no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C42C2311yescharacteristic
98C2×#215—22no—
99C2×#215—22no—
100C2×#215—22no—
101C2×#214—22no—
102C2×#213—22no—
103C2×#213—22no—
104#930—21no—
105#930—21no—
106#930—21no—
107#930—21no—
108C2×D24—22no—
109C2×#204—22no—
110#875—21no—
111#875—21no—
112#874C2311yescharacteristic
113#874C2311yescharacteristic
114C4×D12—21no—
115C4×D12—21no—
116C4×D12—21no—
117C4×D12—21no—
118#869C2311yescharacteristic
119#869C2311yescharacteristic
120C3×#106—22no—
121C3×#106—22no—
122C3×C4⋊C8D412yes—
123C3×#102—22no—
124C3×#102—22no—
125C3×C22⋊C8D412yes—
126C3×C4×C8C2311yescharacteristic
127C3×C4×C8C2311yescharacteristic
128#835—22no—
129#835—22no—
130#835—22no—
131#832—22no—
132#830D412yes—
133#830D412yes—
134#821—22no—
135#816—22no—
136#810—22no—
137#809—22no—
138#809—22no—
139#809—22no—
140#807—22no—
141#807—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×C4⋊Q8C2×D311yescharacteristic
152#500—31no—
153#500—31no—
154#498C2×D311yescharacteristic
155#498C2×D311yescharacteristic
156#493—31no—
157#493—31no—
158#493—31no—
159#492—31no—
160#492—31no—
161#492—31no—
162#492—31no—
163#492—31no—
164#487—31no—
165#487—31no—
166#487—31no—
167#487—31no—
168#487—31no—
169#486—31no—
170#486—31no—
171#486—31no—
172#486—31no—
173#486—31no—
174#486C2×D311yescharacteristic
175#486C2×D311yescharacteristic
176#481—22no—
177#480—62no—
178#476—22no—
179#474—62no—
180#463—62no—
181#463—62no—
182#463—62no—
183#461—62no—
184#449—62no—
185#449—62no—
186#449—62no—
187#448—62no—
188#431—31no—
189#431—31no—
190#431C2×D311yescharacteristic
191#322—31no—
192#322—31no—
193#322—31no—
194C82—31no—
195C3×#41—82no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—41no—
200C2×C3×Q8—82no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—41no—
205C22×C3×C4—41no—
206C22×C3×C4—41no—
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C2×#67—41no—
211C2×#67—41no—
212#235—82no—
213#235—82no—
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12—21no—
220C2×C4×D3—41no—
221C2×C4×D3—41no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×C3×C8C2×D412yes—
225C2×C3×C8C2×D412yes—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—41no—
233C3×C4⋊C4—41no—
234C3×C22⋊C4—41no—
235C3×C22⋊C4—41no—
236C3×C42C2×D411yescharacteristic
237C3×C42C2×D411yescharacteristic
238C3×C42C2×D411yescharacteristic
239C3×C42C2411yescharacteristic
240#215—82no—
241#215—82no—
242#215—82no—
243#214—82no—
244#213—82no—
245#213—82no—
246(C2×D3)⋊C4—41no—
247(C2×D3)⋊C4—41no—
248(C2×D3)⋊C4—41no—
249(C2×D3)⋊C4—41no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C2×C3⋊C8C2×D412yes—
255C2×C3⋊C8C2×D412yes—
256C2×C3⋊C8C2×D412yes—
257C2×C3⋊C8C2×D412yes—
258D24—82no—
259#204—82no—
260C2×#41—21no—
261C2×#41—61no—
262C2×#41—61no—
263C22×Q8—21no—
264C2×SD16—62no—
265C2×SD16—62no—
266C2×SD16—62no—
267C2×SD16—62no—
268C2×SD16—62no—
269C2×D8—62no—
270C2×D8—62no—
271C2×D8—62no—
272C4⋊Q8—21no—
273C4⋊Q8—21no—
274C4⋊Q8—21no—
275C4⋊Q8—31no—
276C4⋊Q8—31no—
277C4⋊Q8C22×D311yescharacteristic
278C4⋊Q8C22×D311yescharacteristic
279C4⋊Q8C22×D311yescharacteristic
280#127—31no—
281#127—31no—
282#127C22×D311yescharacteristic
283#124—21no—
284#124—61no—
285#124—61no—
286#121—21no—
287#121—21no—
288#121—61no—
289#121—61no—
290#121—61no—
291#121—61no—
292C4×D4—21no—
293C4×D4—21no—
294C4×D4—61no—
295C4×D4—61no—
296C4×D4—61no—
297C4×D4—61no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C42C22×D311yescharacteristic
301#107—62no—
302#107—62no—
303#106—22no—
304#106—22no—
305C4⋊C8—32no—
306C4⋊C8—32no—
307C4⋊C8#6712yes—
308#103—62no—
309#103—62no—
310#103—62no—
311#103—62no—
312#103—62no—
313#102—22no—
314#102—22no—
315#102—62no—
316#102—62no—
317#102—62no—
318#102—62no—
319#102—62no—
320C22⋊C8—32no—
321C22⋊C8—32no—
322C22⋊C8#6712yes—
323C4×C8—31no—
324C4×C8—31no—
325C4×C8—31no—
326C4×C8—31no—
327C4×C8C22×D311yescharacteristic
328C4×C8C22×D311yescharacteristic
329C23×C3—41no—
330C23×C3#12711yescharacteristic
331C22×D3—41no—
332C22×D3—41no—
333C3×Q8—42no—
334C3×Q8—42no—
335C3×Q8—42no—
336C3×Q8—82no—
337C3×D4—42no—
338C3×D4—42no—
339C3×D4—42no—
340C3×D4—82no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—82no—
348C2×C3×C4—82no—
349C2×C3×C4#12711yescharacteristic
350C2×C3×C4#12711yescharacteristic
351C2×C3×C4#12711yescharacteristic
352C2×C3×C4C22×D411yescharacteristic
353C2×C3×C4C22×D411yescharacteristic
354C2×C3×C4C22×D411yescharacteristic
355#67—82no—
356#67—82no—
357C2×Dic3—41no—
358C2×Dic3—41no—
359D12—42no—
360D12—42no—
361D12—42no—
362D12—42no—
363D12—42no—
364D12—42no—
365C4×D3—82no—
366C4×D3—82no—
367Dic6—42no—
368Dic6—42no—
369C3×C8—22no—
370C3×C8—22no—
371C3⋊C8—22no—
372C3⋊C8—22no—
373C3⋊C8—22no—
374C3⋊C8—22no—
375#41—82no—
376#41—242no—
377#41—242no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—41no—
382C2×Q8—61no—
383C2×Q8—61no—
384C2×Q8—82no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—41no—
389C2×D4—61no—
390C2×D4—61no—
391C2×D4—61no—
392C2×D4—61no—
393C2×D4—61no—
394C2×D4—61no—
395C2×D4—121no—
396C2×D4—121no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—121no—
400C22×C4—121no—
401C22×C4D3×D411yescharacteristic
402C22×C4D3×D411yescharacteristic
403C22×C4C2×#6711yescharacteristic
404SD16—242no—
405SD16—242no—
406SD16—242no—
407SD16—242no—
408SD16—242no—
409D8—242no—
410D8—242no—
411D8—242no—
412C2×C8—32no—
413C2×C8—32no—
414C2×C8—32no—
415C2×C8—32no—
416C2×C8C2×#6712yes—
417C2×C8C2×#6712yes—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—61no—
427C4⋊C4—61no—
428C4⋊C4—61no—
429C4⋊C4—61no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—121no—
433C22⋊C4—121no—
434C22⋊C4—121no—
435C22⋊C4—121no—
436C42D3×D411yescharacteristic
437C42D3×D411yescharacteristic
438C42C2×#6711yescharacteristic
439C42C23×D311yescharacteristic
440C22×C3—21no—
441C22×C3—21no—
442C22×C3—21no—
443C22×C3—81no—
444C22×C3—81no—
445C22×C3—81no—
446C22×C3C2×#12711yescharacteristic
447C2×D3—81no—
448C2×D3—81no—
449C2×D3—81no—
450C2×D3—81no—
451C2×D3—81no—
452C2×D3—81no—
453C3×C4—21no—
454C3×C4—21no—
455C3×C4—21no—
456C3×C4—81no—
457C3×C4—81no—
458C3×C4—81no—
459C3×C4#57612yes—
460C3×C4#57712yes—
461C3×C4#57712yes—
462Dic3—81no—
463Dic3—81no—
464C23—41no—
465C23—121no—
466C23—121no—
467C23#94011yescharacteristic
468Q8—42no—
469Q8—42no—
470Q8—42no—
471Q8—82no—
472Q8—122no—
473Q8—122no—
474D4—42no—
475D4—42no—
476D4—42no—
477D4—82no—
478D4—122no—
479D4—122no—
480D4—122no—
481D4—122no—
482D4—122no—
483D4—122no—
484D4—242no—
485D4—242no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—121no—
495C2×C4—121no—
496C2×C4—242no—
497C2×C4—242no—
498C2×C4#94011yescharacteristic
499C2×C4#94011yescharacteristic
500C2×C4#94011yescharacteristic
501C2×C4C2×D3×D411yescharacteristic
502C2×C4C2×D3×D411yescharacteristic
503C2×C4C22×#6711yescharacteristic
504C8—22no—
505C8—22no—
506C8—62no—
507C8—62no—
508C8—62no—
509C8—62no—
510C2×C3—41no—
511C2×C3—161no—
512C2×C3#315411yescharacteristic
513C2×C3#315411yescharacteristic
514C2×C3#315511yescharacteristic
515D3—161no—
516D3—161no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—81no—
521C22—81no—
522C22—81no—
523C22—241no—
524C22—241no—
525C22—241no—
526C22—241no—
527C22—241no—
528C22—241no—
529C22C2×#94011yescharacteristic
530C4—21no—
531C4—21no—
532C4—21no—
533C4—81no—
534C4—81no—
535C4—81no—
536C4—241no—
537C4—241no—
538C4#576212yes—
539C4#577412yes—
540C4#583712yes—
541C3#1974211yescharacteristic, complemented, Sylow
542C2—41no—
543C2—161no—
544C2—481no—
545C2—481no—
546C2#8340211yescharacteristic
547C2#8385411yescharacteristic
548C2#8394011yescharacteristic
subgroupG/H
centerC22C2×#940
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#3227C2
socleC22×C3C2×#127
hypercenter#3227D3
hypocenterC3#19742
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3227

Complements

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

NH ≅ G/Ntype
1. C3×#3227C2semidirect
2. #83304C2semidirect
3. #83303C2semidirect
4. #83215C2semidirect
5. #83213C2semidirect
6. #79304C2semidirect
7. #71339C2semidirect
8. #71336C2semidirect
9. #71335C2semidirect
10. #71334C2semidirect
11. #71194C2semidirect
12. #71189C2semidirect
13. #71186C2semidirect
14. #71180C2semidirect
15. #70744C2semidirect
64. #3227D3semidirect
541. C3#19742semidirect

Maximal subgroups

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

subgroupindex
C3×#32272
#833042
#833032
#832152
#832132
#793042
#713392
#713362
#713352
#713342
#711942
#711892
#711862
#711802
#707442
#197423