Finite Group Databasebeta

← #10765324

Subgroups — #10765324

Order 768.

638 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
1#79965C211yesmaximal, characteristic, complemented
2#79651C211yesmaximal, characteristic
3#79426C211yesmaximal, characteristic
4C3×#1585C211yesmaximal, characteristic, complemented
5#70953C211yesmaximal, characteristic, complemented
6SD16⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
7#70842C211yesmaximal, characteristic, complemented
8#68573C211yesmaximal, characteristic, complemented
9#68558C211yesmaximal, characteristic, complemented
10#68235C211yesmaximal, characteristic, complemented
11#68212C211yesmaximal, characteristic, complemented
12#68175C211yesmaximal, characteristic
13#68149C211yesmaximal, characteristic, complemented
14#68080C211yesmaximal, characteristic, complemented
15#68073C211yesmaximal, characteristic, complemented
16#17924—31nomaximal, Sylow
17#5578C2211yescharacteristic
18C4×D3×D4C2211yescharacteristic
19C3×#446C2211yescharacteristic, complemented
20C3×C4×SD16C2211yescharacteristic
21C3×C8×D4C2211yescharacteristic
22#5169C411yescharacteristic
23#5162C411yescharacteristic
24C8×#67—21no—
25C2×(C2×D3)⋊C8—21no—
26C4×#215C2211yescharacteristic
27Q8×C3⋊C8C2211yescharacteristic, complemented
28C4×#214C2211yescharacteristic
29D4×C3⋊C8C2211yescharacteristic
30#4870C411yescharacteristic
31#4862C411yescharacteristic
32#4840—21no—
33#4835C2211yescharacteristic
34#4834C2211yescharacteristic
35#4833C2211yescharacteristic
36#4831—21no—
37#4830C2211yescharacteristic, complemented
38#4818C411yescharacteristic
39#4805C411yescharacteristic
40#4776C411yescharacteristic
41#4761C411yescharacteristic
42#67⋊C8—21no—
43D3×C22⋊C8—21no—
44#4709—21no—
45#4701—21no—
46C4×#204C2211yescharacteristic
47C4×ΓL(1,𝔽25)C2211yescharacteristic
48C8×D12C2211yescharacteristic
49#4685C2211yescharacteristic
50#4685C2211yescharacteristic
51#4679C2211yescharacteristic, complemented
52C3×#335C2211yescharacteristic, complemented
53C3×Q8⋊C8C2211yescharacteristic, complemented
54C3×D4⋊C8C2211yescharacteristic
55C3×#322C2211yescharacteristic, complemented
56#4535C2211yescharacteristic, complemented
57D4⋊(C3⋊C8)C2211yescharacteristic
58#4484C2211yescharacteristic, complemented
59#4483C2211yescharacteristic
60#4480C2211yescharacteristic, complemented
61#4480C2211yescharacteristic, complemented
62#4459C2211yescharacteristic
63#4458C2211yescharacteristic, complemented
64#4456C2211yescharacteristic, complemented
65#4455C2211yescharacteristic, complemented
66#4455C2211yescharacteristic, complemented
67C8×C3⋊C8C2211yescharacteristic, complemented
68#3101—31no—
69#2985—31no—
70#2972—31no—
71#1585—31no—
72#1585D311yescharacteristic, complemented
73#1582—31no—
74#1582—31no—
75SD16⋊C8—31no—
76SD16⋊C8—31no—
77C8×SD16—31no—
78C8×SD16—31no—
79#1572—31no—
80#1490—31no—
81#1485—31no—
82#1453—31no—
83C2×D3×D4C2×C411yescharacteristic
84C22×C4×D3—21no—
85C2×C3×SD16C2×C411yescharacteristic
86C22×C3×C8—21no—
87C3×C4×Q8C2311yescharacteristic
88C3×C4×D4C2311yescharacteristic
89#946C2×C411yescharacteristic
90#944C2×C411yescharacteristic
91C2×#215C2×C411yescharacteristic
92D4×Dic3C2×C411yescharacteristic
93C2×#214C2×C411yescharacteristic
94C4×#67—21no—
95C22×C3⋊C8—21no—
96C2×#204C2×C411yescharacteristic
97C2×ΓL(1,𝔽25)—21no—
98C2×D3×C8—21no—
99C2×D3×C8—21no—
100#897C2×C411yescharacteristic
101#896—21no—
102#893C2×C411yescharacteristic
103D3×C4⋊C4C2×C411yescharacteristic
104#889—21no—
105#888C2×C411yescharacteristic
106#67⋊C4—21no—
107D3×C22⋊C4—21no—
108C4×D12C2311yescharacteristic
109#872—21no—
110C42×D3C2311yescharacteristic
111C4×Dic6—21no—
112C4×Dic6C2311yescharacteristic
113C3×#106C2×C411yescharacteristic
114C3×C4⋊C8—21no—
115C3×C4⋊C8C2311yescharacteristic
116C3×C4⋊C8C2311yescharacteristic
117C3×#103C2×C411yescharacteristic
118C3×#102C2×C411yescharacteristic
119C3×C22⋊C8—21no—
120C3×#97—21no—
121C3×C4×C8C2311yescharacteristic
122C3×C4×C8C2311yescharacteristic
123C3×C4×C8C2311yescharacteristic
124#835C2×C411yescharacteristic
125#832C2×C411yescharacteristic
126#830—21no—
127#821C2×C411yescharacteristic
128(C2×D3)⋊C8—21no—
129(C2×D3)⋊C8—21no—
130(C2×D3)⋊C8—21no—
131(C2×D3)⋊C8—21no—
132(C2×D3)⋊C8C2×C411yescharacteristic
133(C2×D3)⋊C8C2×C411yescharacteristic
134(C2×D3)⋊C8D411yescharacteristic
135(C2×D3)⋊C8D411yescharacteristic
136#817C2×C411yescharacteristic
137#816C2×C411yescharacteristic
138#815—21no—
139#815—21no—
140#815—21no—
141Dic3⋊C8—21no—
142Dic3⋊C8—21no—
143Dic3⋊C8C2×C411yescharacteristic
144Dic3⋊C8C2×C411yescharacteristic
145Dic3⋊C8D411yescharacteristic
146Dic3⋊C8D411yescharacteristic
147C8×Dic3—21no—
148C8×Dic3—21no—
149#810C2×C411yescharacteristic
150#809C2×C411yescharacteristic
151#808C2×C411yescharacteristic
152#808C2×C411yescharacteristic
153C4⋊(C3⋊C8)—21no—
154C4⋊(C3⋊C8)C2311yescharacteristic
155C4⋊(C3⋊C8)C2311yescharacteristic
156C4×C3⋊C8—21no—
157C4×C3⋊C8C2311yescharacteristic
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#533—31no—
161C2×C4×D4—31no—
162#483—31no—
163#483—31no—
164#476—31no—
165#476—31no—
166#461—31no—
167#461—31no—
168#450—31no—
169#450—31no—
170#446—31no—
171#446—61no—
172#446—61no—
173#446C2×D311yescharacteristic, complemented
174C8×Q8—31no—
175C8×Q8—31no—
176C4×SD16—31no—
177C4×SD16—31no—
178C4×SD16—31no—
179C4×SD16C2×D311yescharacteristic
180C8×D4—31no—
181C8×D4—31no—
182C8×D4—31no—
183C8×D4—61no—
184C8×D4—61no—
185C8×D4C2×D311yescharacteristic
186#433—61no—
187#433—61no—
188#432—31no—
189#431—31no—
190#431—31no—
191#431—31no—
192C2×C22⋊C8—61no—
193C2×C22⋊C8—61no—
194C4×M16—31no—
195#335—31no—
196#335C2×D311yescharacteristic, complemented
197#334—31no—
198#334—31no—
199Q8⋊C8—31no—
200Q8⋊C8—31no—
201Q8⋊C8—31no—
202Q8⋊C8C2×D311yescharacteristic, complemented
203D4⋊C8—31no—
204D4⋊C8—31no—
205D4⋊C8—31no—
206D4⋊C8C2×D311yescharacteristic
207#322—31no—
208#322—31no—
209#322C2×D311yescharacteristic, complemented
210C82—31no—
211C23×D3—21no—
212C2×C3×Q8C22×C411yescharacteristic
213C2×C3×D4C22×C411yescharacteristic
214C22×C3×C4—21no—
215C2×#67—21no—
216C22×Dic3—21no—
217D3×D4—22no—
218D3×D4—22no—
219C2×D12C22×C411yescharacteristic
220C2×C4×D3—21no—
221C2×C4×D3—21no—
222C2×C4×D3—21no—
223C2×C4×D3—81no—
224C2×C4×D3—81no—
225C2×C4×D3C22×C411yescharacteristic
226C2×C4×D3C2×D411yescharacteristic
227C2×C4×D3#4111yescharacteristic
228C2×Dic6—21no—
229C2×Dic6C22×C411yescharacteristic
230C3×SD16—22no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—81no—
235C2×C3×C8C22×C411yescharacteristic
236C2×C3×C8C22×C411yescharacteristic
237C2×C3×C8C2×D411yescharacteristic
238C2×C3×C8C2×D411yescharacteristic
239C3×C4⋊C4—21no—
240C3×C4⋊C4C22×C411yescharacteristic
241C3×C4⋊C4C22×C411yescharacteristic
242C3×C22⋊C4—21no—
243C3×C42—21no—
244C3×C42C2411yescharacteristic
245#217—21no—
246#215—22no—
247#214—22no—
248(C2×D3)⋊C4—21no—
249(C2×D3)⋊C4—21no—
250(C2×D3)⋊C4—21no—
251(C2×D3)⋊C4—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3C22×C411yescharacteristic
254C4⋊Dic3C22×C411yescharacteristic
255Dic3⋊C4—21no—
256Dic3⋊C4—21no—
257Dic3⋊C4—21no—
258Dic3⋊C4—21no—
259Dic3⋊C4—21no—
260Dic3⋊C4—21no—
261C4×Dic3—21no—
262C4×Dic3C22×C411yescharacteristic
263C4×Dic3C2×D411yescharacteristic
264C4×Dic3#4111yescharacteristic
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—81no—
269C2×C3⋊C8C22×C411yescharacteristic
270C2×C3⋊C8C22×C411yescharacteristic
271C2×C3⋊C8C2×D411yescharacteristic
272C2×C3⋊C8C2×D411yescharacteristic
273#204—22no—
274ΓL(1,𝔽25)—42no—
275D3×C8—81no—
276D3×C8—81no—
277C22×D4—31no—
278C23×C4—61no—
279C2×SD16—31no—
280C2×SD16—31no—
281C2×SD16—31no—
282C2×SD16C4×D311yescharacteristic
283C2×M16—61no—
284C22×C8—21no—
285C22×C8—61no—
286C22×C8—61no—
287C22×C8—61no—
288C4⋊Q8—31no—
289C4⋊Q8—31no—
290#125—61no—
291C22⋊Q8—31no—
292C22⋊Q8—31no—
293C22⋊Q8—61no—
294C4×Q8—31no—
295C4×Q8—61no—
296C4×Q8C22×D311yescharacteristic
297C4×D4—31no—
298C4×D4—31no—
299C4×D4—31no—
300C4×D4—61no—
301C4×D4—61no—
302C4×D4C22×D311yescharacteristic
303#117—61no—
304C2×C4⋊C4—31no—
305C2×C22⋊C4—61no—
306C2×C42—31no—
307#106—31no—
308#106—31no—
309#106—31no—
310#106C4×D311yescharacteristic
311C4⋊C8—21no—
312C4⋊C8—31no—
313C4⋊C8—31no—
314C4⋊C8—31no—
315C4⋊C8—31no—
316C4⋊C8—31no—
317C4⋊C8—31no—
318C4⋊C8—61no—
319C4⋊C8—61no—
320C4⋊C8—61no—
321C4⋊C8C22×D311yescharacteristic
322C4⋊C8C22×D311yescharacteristic
323#103—31no—
324#103—31no—
325#103—31no—
326#103C4×D311yescharacteristic
327#102—31no—
328#102—31no—
329#102—31no—
330#102C4×D311yescharacteristic
331C22⋊C8—21no—
332C22⋊C8—31no—
333C22⋊C8—31no—
334C22⋊C8—31no—
335C22⋊C8—31no—
336C22⋊C8—61no—
337C22⋊C8—61no—
338C22⋊C8—61no—
339C22⋊C8—61no—
340C22⋊C8—61no—
341#97—21no—
342#97—61no—
343#97—61no—
344#97—61no—
345C4×C8—31no—
346C4×C8—31no—
347C4×C8—31no—
348C4×C8—61no—
349C4×C8—61no—
350C4×C8—61no—
351C4×C8C22×D311yescharacteristic
352C4×C8C22×D311yescharacteristic
353C4×C8C22×D311yescharacteristic
354C23×C3—21no—
355C22×D3—21no—
356C22×D3—24no—
357C22×D3—81no—
358C22×D3—81no—
359C22×D3C4×D411yescharacteristic
360C3×Q8—21no—
361C3×Q8—21no—
362C3×D4—21no—
363C3×D4#13112yes—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4C23×C411yescharacteristic
370C2×C3×C4C22×D411yescharacteristic
371C2×C3×C4C2×#4111yescharacteristic
372#67—24no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—81no—
377C2×Dic3C4×D411yescharacteristic
378C2×Dic3C4×D411yescharacteristic
379C2×Dic3C4×D411yescharacteristic
380D12—21no—
381D12#13112yes—
382C4×D3—21no—
383C4×D3—21no—
384C4×D3—21no—
385C4×D3—21no—
386C4×D3—41no—
387C4×D3—41no—
388C4×D3—41no—
389C4×D3—81no—
390C4×D3—81no—
391Dic6—21no—
392Dic6—21no—
393Dic6—81no—
394C3×C8—22no—
395C3×C8—21no—
396C3×C8—21no—
397C3×C8—21no—
398C3×C8—21no—
399C3×C8—41no—
400C3×C8—41no—
401C3⋊C8—21no—
402C3⋊C8—21no—
403C3⋊C8—21no—
404C3⋊C8—21no—
405C3⋊C8—41no—
406C3⋊C8—41no—
407C3⋊C8—41no—
408C24—61no—
409C2×Q8—31no—
410C2×Q8—61no—
411C2×Q8C2×C4×D311yescharacteristic
412C2×D4—31no—
413C2×D4—61no—
414C2×D4—62no—
415C2×D4—62no—
416C2×D4C2×C4×D311yescharacteristic
417C22×C4—21no—
418C22×C4—31no—
419C22×C4—31no—
420C22×C4—31no—
421C22×C4—61no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—241no—
426C22×C4—241no—
427SD16—22no—
428SD16—62no—
429SD16—62no—
430SD16—62no—
431M16—122no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—31no—
436C2×C8—31no—
437C2×C8—31no—
438C2×C8—31no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8—61no—
442C2×C8—81no—
443C2×C8—241no—
444C2×C8—241no—
445C2×C8—241no—
446C2×C8C2×C4×D311yescharacteristic
447C2×C8C2×C4×D311yescharacteristic
448C2×C8D3×D411yescharacteristic
449C2×C8D3×D411yescharacteristic
450C4⋊C4—21no—
451C4⋊C4—31no—
452C4⋊C4—31no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—61no—
457C4⋊C4—61no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4C2×C4×D311yescharacteristic
461C4⋊C4C2×C4×D311yescharacteristic
462C22⋊C4—21no—
463C22⋊C4—61no—
464C22⋊C4—61no—
465C22⋊C4—61no—
466C22⋊C4—61no—
467C22⋊C4—61no—
468C42—21no—
469C42—31no—
470C42—31no—
471C42—31no—
472C42—61no—
473C42C23×D311yescharacteristic
474C22×C3—22no—
475C22×C3—41no—
476C22×C3—41no—
477C22×C3C2×C4×D411yescharacteristic
478C2×D3—21no—
479C2×D3—22no—
480C2×D3—21no—
481C2×D3—21no—
482C2×D3—41no—
483C2×D3—41no—
484C2×D3—84no—
485C3×C4—21no—
486C3×C4—41no—
487C3×C4—41no—
488C3×C4—41no—
489C3×C4C2×#13111yescharacteristic
490C3×C4C2×#13111yescharacteristic
491C3×C4#57711yescharacteristic
492C3×C4#57711yescharacteristic
493Dic3—21no—
494Dic3—21no—
495Dic3—21no—
496Dic3—41no—
497Dic3—41no—
498Dic3—41no—
499C23—21no—
500C23—31no—
501C23—64no—
502C23—61no—
503C23—241no—
504C23—241no—
505Q8—21no—
506Q8—21no—
507Q8—61no—
508Q8—61no—
509Q8—241no—
510D4—21no—
511D4—32no—
512D4—64no—
513D4—61no—
514D4#90712yes—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—31no—
519C2×C4—31no—
520C2×C4—31no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—61no—
524C2×C4—61no—
525C2×C4—61no—
526C2×C4—61no—
527C2×C4—61no—
528C2×C4—61no—
529C2×C4—61no—
530C2×C4—121no—
531C2×C4—121no—
532C2×C4—121no—
533C2×C4—241no—
534C2×C4—241no—
535C2×C4—241no—
536C2×C4C22×C4×D311yescharacteristic
537C2×C4C2×D3×D411yescharacteristic
538C2×C4D3×#4111yescharacteristic
539C8—22no—
540C8—21no—
541C8—21no—
542C8—21no—
543C8—21no—
544C8—41no—
545C8—41no—
546C8—61no—
547C8—61no—
548C8—61no—
549C8—61no—
550C8—121no—
551C8—121no—
552C8—121no—
553C2×C3—42no—
554C2×C3#292911yescharacteristic
555C2×C3#294911yescharacteristic
556C2×C3#295611yescharacteristic
557D3—41no—
558D3—42no—
559C22—22no—
560C22—41no—
561C22—41no—
562C22—61no—
563C22—61no—
564C22—62no—
565C22—61no—
566C22—121no—
567C22—121no—
568C22—244no—
569C22C4×D3×D411yescharacteristic
570C4—21no—
571C4—41no—
572C4—41no—
573C4—41no—
574C4—61no—
575C4—61no—
576C4—61no—
577C4—121no—
578C4—121no—
579C4—121no—
580C4C2×#90711yescharacteristic
581C4D3×#13111yescharacteristic
582C4#577511yescharacteristic
583C4#577811yescharacteristic
584C3#1792411yescharacteristic, complemented, Sylow
585C2—42no—
586C2—121no—
587C2—122no—
588C2#7947811yescharacteristic
589C2#7977711yescharacteristic
590C2#7988911yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1585C2
socleC22×C3C2×C4×D4
prime radicalC2×D3×D4C2×C4
cosocleC3×C42C24
hypercenter#1585D3
hypocenterC3#17924
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1585

Complements

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

NH ≅ G/Ntype
1. #79965C2semidirect
4. C3×#1585C2semidirect
5. #70953C2semidirect
6. SD16⋊(C3⋊C8)C2semidirect
7. #70842C2semidirect
8. #68573C2semidirect
9. #68558C2semidirect
10. #68235C2semidirect
11. #68212C2semidirect
13. #68149C2semidirect
14. #68080C2semidirect
15. #68073C2semidirect
19. C3×#446C22semidirect
27. Q8×C3⋊C8C22semidirect
37. #4830C22semidirect
51. #4679C22semidirect
52. C3×#335C22semidirect
53. C3×Q8⋊C8C22semidirect
55. C3×#322C22semidirect
56. #4535C22semidirect
58. #4484C22semidirect
60. #4480C22semidirect
61. #4480C22semidirect
63. #4458C22semidirect
64. #4456C22semidirect
65. #4455C22semidirect
66. #4455C22semidirect
67. C8×C3⋊C8C22semidirect
72. #1585D3semidirect
173. #446C2×D3semidirect
196. #335C2×D3semidirect
202. Q8⋊C8C2×D3semidirect
209. #322C2×D3semidirect
584. C3#17924semidirect

Maximal subgroups

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

subgroupindex
#799652
#796512
#794262
C3×#15852
#709532
SD16⋊(C3⋊C8)2
#708422
#685732
#685582
#682352
#682122
#681752
#681492
#680802
#680732
#179243