Finite Group Databasebeta

← #10782847

Subgroups — #10782847

Order 768.

814 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3167C211yesmaximal, characteristic, complemented
2#83343C211yesmaximal, characteristic, complemented
3#83339C211yesmaximal, characteristic, complemented
4#83290C211yesmaximal, characteristic, complemented
5#83285C211yesmaximal, characteristic
6#78468C211yesmaximal, characteristic, complemented
7#71333C211yesmaximal, characteristic, complemented
8#71332C211yesmaximal, characteristic, complemented
9#71255C211yesmaximal, characteristic, complemented
10#71254C211yesmaximal, characteristic, complemented
11#71213C211yesmaximal, characteristic, complemented
12#71212C211yesmaximal, characteristic, complemented
13#71181C211yesmaximal, characteristic, complemented
14#71180C211yesmaximal, characteristic, complemented
15C2×C8×C3⋊C8C211yesmaximal, characteristic, complemented
16#19596—31nomaximal, Sylow
17C3×#533C2211yescharacteristic
18C3×#532C2211yescharacteristic
19C2×#950—22no—
20#5477C2211yescharacteristic
21#5470C2211yescharacteristic
22C3×#498C2211yescharacteristic
23C3×#492C2211yescharacteristic
24C3×#487C2211yescharacteristic
25C3×#486C2211yescharacteristic
26C3×#465—22no—
27C3×C4×SD16—22no—
28C2×C3×C4×C8C2211yescharacteristic
29#5114—22no—
30#5093C2211yescharacteristic
31#5092C2211yescharacteristic
32#5088C2211yescharacteristic
33#5085C2211yescharacteristic
34#5085C2211yescharacteristic
35#5080C2211yescharacteristic
36#5080C2211yescharacteristic
37#5080C2211yescharacteristic
38#5080C2211yescharacteristic
39#5079C2211yescharacteristic
40#5078C2211yescharacteristic
41#5078C2211yescharacteristic
42#5076C2211yescharacteristic
43#5076C2211yescharacteristic
44#5073C2211yescharacteristic
45#5070C2211yescharacteristic
46#5065—22no—
47#5059—22no—
48#5048—22no—
49#5041—22no—
50C4×#215—22no—
51C4×#214—22no—
52#4998—22no—
53C2×C4×C3⋊C8C2211yescharacteristic
54C2×C4×C3⋊C8C2211yescharacteristic
55#4694C2211yescharacteristic
56#4693C2211yescharacteristic
57C4×#204—22no—
58#4681C2211yescharacteristic
59#4680C2211yescharacteristic
60C8×C3⋊C8C2211yescharacteristic
61C8×C3⋊C8C2211yescharacteristic
62C8×C3⋊C8C2211yescharacteristic
63C8×C3⋊C8C2211yescharacteristic
64#3167—31no—
65#3167—31no—
66#3167—31no—
67#3167D311yescharacteristic, complemented
68#3160—31no—
69#3147—31no—
70#1708—31no—
71#1708—31no—
72#1707—31no—
73#1707—31no—
74#1706—31no—
75#1706—31no—
76#1705—31no—
77#1705—31no—
78C2×C82—31no—
79C2×C3×#41—21no—
80C2×#235—21no—
81C2×C3×SD16—22no—
82C22×C3×C8D412yes—
83C3×C4⋊Q8C2311yescharacteristic
84C3×C4⋊Q8C2311yescharacteristic
85C3×C4⋊Q8C2311yescharacteristic
86C3×#127C2311yescharacteristic
87C3×#125—21no—
88C3×#124—21no—
89C3×C22⋊Q8—21no—
90C3×C22⋊Q8—21no—
91C3×#121—21no—
92C3×#121—21no—
93C3×C4×Q8—21no—
94C3×C4×Q8—21no—
95C3×C4×D4—21no—
96C3×C4×D4—21no—
97C3×#117—21no—
98C2×C3×C42C2311yescharacteristic
99#950—48no—
100C2×#216—22no—
101C2×#215—22no—
102C2×#215—22no—
103C2×#214—22no—
104C2×#214—22no—
105C2×#213—22no—
106#930—21no—
107#930—21no—
108#926—21no—
109#924—21no—
110#924—21no—
111C22×C3⋊C8D412yes—
112C22×C3⋊C8D412yes—
113C2×#204—22no—
114#875—21no—
115#874C2311yescharacteristic
116C4×D12—21no—
117C4×D12—21no—
118#870—21no—
119#869C2311yescharacteristic
120#869C2311yescharacteristic
121#869C2311yescharacteristic
122C4×Dic6—21no—
123C4×Dic6—21no—
124C3×#106—22no—
125C3×#103—22no—
126C3×#102—22no—
127C3×C4×C8D412yes—
128C3×C4×C8C2311yescharacteristic
129C3×C4×C8C2311yescharacteristic
130#835—22no—
131#832—22no—
132#821—22no—
133#817—22no—
134#816—22no—
135#810—22no—
136#809—22no—
137#808—22no—
138#808—22no—
139#808—22no—
140#808—22no—
141#807—22no—
142#807—22no—
143C4×C3⋊C8D412yes—
144C4×C3⋊C8D412yes—
145C4×C3⋊C8C2311yescharacteristic
146C4×C3⋊C8C2311yescharacteristic
147C4×C3⋊C8C2311yescharacteristic
148C4×C3⋊C8C2311yescharacteristic
149C2×#135—62no—
150#533—31no—
151#533C2×D311yescharacteristic
152#532—31no—
153#532C2×D311yescharacteristic
154#500—31no—
155#500—31no—
156#499—62no—
157#498—31no—
158#498—31no—
159#498—31no—
160#498—31no—
161#498—31no—
162#498C2×D311yescharacteristic
163#495—62no—
164#494—31no—
165#493—31no—
166#492—31no—
167#492—31no—
168#492—31no—
169#492—31no—
170#492—31no—
171#492C2×D311yescharacteristic
172#487—31no—
173#487—31no—
174#487—31no—
175#487C2×D311yescharacteristic
176#486—31no—
177#486—31no—
178#486—31no—
179#486C2×D311yescharacteristic
180#465—22no—
181#465—62no—
182#465—62no—
183#465—62no—
184C4×SD16—22no—
185C4×SD16—62no—
186C4×SD16—62no—
187C4×SD16—62no—
188#427—62no—
189C2×C4×C8—31no—
190C2×C4×C8—31no—
191C2×C4×C8C2×D311yescharacteristic
192C82—31no—
193C82—31no—
194C82—31no—
195C82—31no—
196C3×#41—44no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—41no—
204C22×C3×C4—41no—
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C2×#67—41no—
209#235—44no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12—21no—
213C2×C4×D3—41no—
214C2×Dic6—21no—
215C2×Dic6—21no—
216C2×Dic6—21no—
217C3×SD16—44no—
218C2×C3×C8—24no—
219C2×C3×C8C2×D412yes—
220C2×C3×C8C2×D412yes—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—41no—
228C3×C4⋊C4—41no—
229C3×C22⋊C4—41no—
230C3×C22⋊C4—41no—
231C3×C22⋊C4—41no—
232C3×C42—41no—
233C3×C42C2×D411yescharacteristic
234C3×C42C2×D411yescharacteristic
235C3×C42C2×D411yescharacteristic
236C3×C42C2411yescharacteristic
237#217—41no—
238#216—44no—
239#215—44no—
240#215—44no—
241#214—44no—
242#214—44no—
243#213—44no—
244(C2×D3)⋊C4—41no—
245(C2×D3)⋊C4—41no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252Dic3⋊C4—41no—
253Dic3⋊C4—41no—
254C4×Dic3—41no—
255C2×C3⋊C8—24no—
256C2×C3⋊C8SD1618yescomplemented
257C2×C3⋊C8C2×D412yes—
258C2×C3⋊C8C2×D412yes—
259C2×C3⋊C8C2×D412yes—
260C2×C3⋊C8C2×D412yes—
261#204—44no—
262C2×#41—21no—
263C2×#41—61no—
264#135—128no—
265C2×Q16—62no—
266C2×SD16—22no—
267C2×SD16—62no—
268C2×SD16—62no—
269C2×SD16—62no—
270C2×SD16—62no—
271C2×SD16—62no—
272C2×D8—62no—
273C22×C8—32no—
274C22×C8—32no—
275C22×C8#6712yes—
276C4⋊Q8—31no—
277C4⋊Q8—31no—
278C4⋊Q8—31no—
279C4⋊Q8C22×D311yescharacteristic
280C4⋊Q8C22×D311yescharacteristic
281C4⋊Q8C22×D311yescharacteristic
282#127—31no—
283#127C22×D311yescharacteristic
284#125—21no—
285#125—61no—
286#124—21no—
287#124—61no—
288C22⋊Q8—21no—
289C22⋊Q8—21no—
290C22⋊Q8—61no—
291C22⋊Q8—61no—
292#121—21no—
293#121—21no—
294#121—61no—
295#121—61no—
296C4×Q8—21no—
297C4×Q8—21no—
298C4×Q8—61no—
299C4×Q8—61no—
300C4×D4—21no—
301C4×D4—21no—
302C4×D4—61no—
303C4×D4—61no—
304#117—21no—
305#117—61no—
306C2×C42C22×D311yescharacteristic
307#107—62no—
308#107—62no—
309#106—22no—
310#106—62no—
311#106—62no—
312#106—62no—
313#106—62no—
314#106—62no—
315#103—22no—
316#103—62no—
317#103—62no—
318#103—62no—
319#102—22no—
320#102—62no—
321#102—62no—
322#102—62no—
323C4×C8—32no—
324C4×C8—32no—
325C4×C8—31no—
326C4×C8—31no—
327C4×C8—31no—
328C4×C8—31no—
329C4×C8#6712yes—
330C4×C8C22×D311yescharacteristic
331C4×C8C22×D311yescharacteristic
332C23×C3—41no—
333C23×C3#12711yescharacteristic
334C22×D3—41no—
335C3×Q8—42no—
336C3×Q8—42no—
337C3×Q8—42no—
338C3×D4—42no—
339C3×D4—42no—
340C3×D4—42no—
341C3×D4—82no—
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#12711yescharacteristic
349C2×C3×C4#12711yescharacteristic
350C2×C3×C4#12711yescharacteristic
351C2×C3×C4C2×SD1614yes—
352C2×C3×C4C22×D411yescharacteristic
353C2×C3×C4C22×D411yescharacteristic
354C2×C3×C4C22×D411yescharacteristic
355#67—82no—
356C2×Dic3—41no—
357C2×Dic3—41no—
358C2×Dic3—41no—
359D12—42no—
360D12—42no—
361D12—42no—
362C4×D3—82no—
363Dic6—42no—
364Dic6—42no—
365Dic6—42no—
366C3×C8—22no—
367C3×C8#13514yescomplemented
368C3⋊C8—22no—
369C3⋊C8C2×SD1614yes—
370C3⋊C8C2×SD1614yes—
371C3⋊C8#13514yescomplemented
372#41—44no—
373#41—124no—
374C2×Q8—21no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—61no—
378C2×Q8—61no—
379C2×Q8—61no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—41no—
384C2×D4—61no—
385C2×D4—61no—
386C2×D4—61no—
387C2×D4—121no—
388C22×C4—41no—
389C22×C4—121no—
390C22×C4D3×D411yescharacteristic
391C22×C4D3×D411yescharacteristic
392C22×C4C2×#6711yescharacteristic
393Q16—124no—
394SD16—44no—
395SD16—124no—
396SD16—124no—
397SD16—124no—
398SD16—124no—
399SD16—124no—
400D8—124no—
401C2×C8—24no—
402C2×C8—38no—
403C2×C8—32no—
404C2×C8—32no—
405C2×C8—32no—
406C2×C8—32no—
407C2×C8—64no—
408C2×C8C2×#6712yes—
409C2×C8C2×#6712yes—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—61no—
419C4⋊C4—61no—
420C4⋊C4—61no—
421C4⋊C4—61no—
422C4⋊C4—61no—
423C4⋊C4—61no—
424C4⋊C4—121no—
425C4⋊C4—121no—
426C22⋊C4—41no—
427C22⋊C4—41no—
428C22⋊C4—41no—
429C22⋊C4—121no—
430C22⋊C4—121no—
431C22⋊C4—121no—
432C42—41no—
433C42—121no—
434C42D3×D411yescharacteristic
435C42D3×D411yescharacteristic
436C42C2×#6711yescharacteristic
437C42C23×D311yescharacteristic
438C22×C3—21no—
439C22×C3—21no—
440C22×C3—81no—
441C22×C3—81no—
442C22×C3—81no—
443C22×C3#49212yes—
444C22×C3C2×#12711yescharacteristic
445C2×D3—81no—
446C2×D3—81no—
447C2×D3—81no—
448C3×C4—21no—
449C3×C4—21no—
450C3×C4—81no—
451C3×C4—81no—
452C3×C4—81no—
453C3×C4#49212yes—
454C3×C4C22×SD1612yes—
455C3×C4C2×#13512yes—
456C3×C4C2×#13512yes—
457Dic3—81no—
458Dic3—81no—
459Dic3—81no—
460C23—41no—
461C23—121no—
462C23#94011yescharacteristic
463Q8—42no—
464Q8—42no—
465Q8—42no—
466Q8—122no—
467Q8—122no—
468Q8—122no—
469D4—42no—
470D4—42no—
471D4—42no—
472D4—82no—
473D4—122no—
474D4—122no—
475D4—122no—
476D4—242no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—82no—
483C2×C4—121no—
484C2×C4—121no—
485C2×C4—121no—
486C2×C4—242no—
487C2×C4D3×SD1614yes—
488C2×C4#94011yescharacteristic
489C2×C4#94011yescharacteristic
490C2×C4#94011yescharacteristic
491C2×C4C2×D3×D411yescharacteristic
492C2×C4C2×D3×D411yescharacteristic
493C2×C4C22×#6711yescharacteristic
494C8—22no—
495C8—34no—
496C8—34no—
497C8—34no—
498C8—62no—
499C8#95014yescomplemented
500C2×C3—22no—
501C2×C3—161no—
502C2×C3C2×#49211yescharacteristic
503C2×C3#314711yescharacteristic
504C2×C3#314711yescharacteristic
505D3—161no—
506C22—21no—
507C22—21no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22—241no—
512C22—241no—
513C22—241no—
514C22#517512yes—
515C22C2×#94011yescharacteristic
516C4—21no—
517C4—21no—
518C4—81no—
519C4—81no—
520C4—81no—
521C4—241no—
522C4—241no—
523C4—241no—
524C4#517512yes—
525C4C2×D3×SD1612yes—
526C4D3×#13512yes—
527C4C2×#95012yes—
528C3#1959611yescharacteristic, complemented, Sylow
529C2—22no—
530C2—161no—
531C2—481no—
532C2#8328511yescharacteristic
533C2C2×#517511yescharacteristic
534C2#8390311yescharacteristic
subgroupG/H
centerC2×C4#940
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#3167C2
socleC22×C3C2×#127
prime radical#83285C2
hypercenter#3167D3
hypocenterC3#19596
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3167

Complements

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

NH ≅ G/Ntype
1. C3×#3167C2semidirect
2. #83343C2semidirect
3. #83339C2semidirect
4. #83290C2semidirect
6. #78468C2semidirect
7. #71333C2semidirect
8. #71332C2semidirect
9. #71255C2semidirect
10. #71254C2semidirect
11. #71213C2semidirect
12. #71212C2semidirect
13. #71181C2semidirect
14. #71180C2semidirect
15. C2×C8×C3⋊C8C2semidirect
67. #3167D3semidirect
256. C2×C3⋊C8SD16semidirect
367. C3×C8#135semidirect
371. C3⋊C8#135semidirect
499. C8#950semidirect
528. C3#19596semidirect

Maximal subgroups

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

subgroupindex
C3×#31672
#833432
#833392
#832902
#832852
#784682
#713332
#713322
#712552
#712542
#712132
#712122
#711812
#711802
C2×C8×C3⋊C82
#195963