Finite Group Databasebeta

← #10784792

Subgroups — #10784792

Order 768.

710 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83527C211yesmaximal, characteristic, complemented
2#83524C211yesmaximal, characteristic, complemented
3#80352C211yesmaximal, characteristic, complemented
4#80352C211yesmaximal, characteristic, complemented
5C3×#1625C211yesmaximal, characteristic, complemented
6#71198C211yesmaximal, characteristic, complemented
7#71191C211yesmaximal, characteristic, complemented
8#71011C211yesmaximal, characteristic, complemented
9#71007C211yesmaximal, characteristic, complemented
10#68445C211yesmaximal, characteristic, complemented
11#68441C211yesmaximal, characteristic, complemented
12#68211C211yesmaximal, characteristic, complemented
13#68211C211yesmaximal, characteristic, complemented
14#68153C211yesmaximal, characteristic, complemented
15#68151C211yesmaximal, characteristic, complemented
16#19772—31nomaximal, Sylow
17C3×#547C2211yescharacteristic, complemented
18C2×#950—21no—
19#5717C2211yescharacteristic
20#5551C2211yescharacteristic
21#5551C2211yescharacteristic
22C3×#486C2211yescharacteristic
23C3×#459C2211yescharacteristic
24C3×#459C2211yescharacteristic
25#5239—21no—
26#5157—21no—
27#5157—21no—
28#5156—21no—
29#5156—21no—
30#5093C2211yescharacteristic, complemented
31#5092C2211yescharacteristic, complemented
32#5087C2211yescharacteristic
33#5083C2211yescharacteristic
34#5070C2211yescharacteristic
35#5070C2211yescharacteristic
36#5051—21no—
37#5047—21no—
38#5042—21no—
39#5037—21no—
40#5019C2211yescharacteristic, complemented
41#5019C2211yescharacteristic, complemented
42C4×#213C2211yescharacteristic
43C4×#213C2211yescharacteristic
44#5012C2211yescharacteristic, complemented
45D4×C3⋊C8C2211yescharacteristic, complemented
46#4970—21no—
47#4966—21no—
48#4846C2211yescharacteristic
49#4846C2211yescharacteristic
50#4834C2211yescharacteristic
51#4834C2211yescharacteristic
52#4774—21no—
53#4774—21no—
54D3×#102—21no—
55D3×#102—21no—
56#4680C2211yescharacteristic, complemented
57C3×#332C2211yescharacteristic, complemented
58C3×D4⋊C8C2211yescharacteristic, complemented
59C3×D4⋊C8C2211yescharacteristic, complemented
60#4544C2211yescharacteristic
61#4482C2211yescharacteristic, complemented
62#4482C2211yescharacteristic, complemented
63#4480C2211yescharacteristic, complemented
64#4479C2211yescharacteristic, complemented
65#4459C2211yescharacteristic
66#4459C2211yescharacteristic
67#4454C2211yescharacteristic, complemented
68#3330—31no—
69#3294—31no—
70#3294—31no—
71#3283—31no—
72#1720—31no—
73#1701—31no—
74#1681—31no—
75#1679—31no—
76#1669—31no—
77#1654—31no—
78#1654—31no—
79#1639—31no—
80#1625D311yescharacteristic, complemented
81#1581—31no—
82#1580—31no—
83C2×C3×#41—21no—
84C2×D3×D4—21no—
85C2×D3×D4—21no—
86C2×#235—21no—
87C2×#235—21no—
88C2×C3×D8—21no—
89C2×C3×D8—21no—
90C3×C4⋊Q8C2311yescharacteristic
91C3×#127C2311yescharacteristic
92C3×#123—41no—
93C3×C22⋊Q8—21no—
94C3×#121—21no—
95C3×C4×D4C2311yescharacteristic
96C3×C4×D4C2311yescharacteristic
97C2×C3×C4⋊C4—21no—
98#950—44no—
99C2×#216—21no—
100C2×#216—21no—
101C2×#215—21no—
102#938—21no—
103#938—21no—
104#938—21no—
105#938—21no—
106#937—41no—
107C2×#214—21no—
108C2×#214—21no—
109C2×#213D411yescharacteristic
110C2×#213D411yescharacteristic
111#924—21no—
112#924—21no—
113C2×#208—21no—
114C22×C3⋊C8—21no—
115C2×D3×C8—21no—
116C2×D3×C8—21no—
117D3×C4⋊C4—21no—
118D3×C4⋊C4—21no—
119#883—41no—
120#883—41no—
121C4×D12C2311yescharacteristic
122C4×D12C2311yescharacteristic
123#869C2311yescharacteristic
124#869C2311yescharacteristic
125C3×C4⋊C8C2311yescharacteristic
126C3×C4⋊C8C2311yescharacteristic
127C3×#102—21no—
128C3×#102—21no—
129C3×#102—41no—
130C3×C4×C8C2311yescharacteristic
131#835—21no—
132#832—21no—
133#832—21no—
134#832—21no—
135#832—21no—
136#832D411yescharacteristic
137#832D411yescharacteristic
138#830—21no—
139#830—21no—
140#821—21no—
141#821—21no—
142(C2×D3)⋊C8—21no—
143(C2×D3)⋊C8—21no—
144#818—21no—
145#818—21no—
146#817—41no—
147#810—21no—
148#810—21no—
149#810—21no—
150#810—21no—
151#809—21no—
152#809D411yescharacteristic
153#809D411yescharacteristic
154#807—21no—
155#807D411yescharacteristic
156#807D411yescharacteristic
157C4⋊(C3⋊C8)C2311yescharacteristic
158C4⋊(C3⋊C8)C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160C4×C3⋊C8C2311yescharacteristic
161C2×#135—61no—
162#547—31no—
163#547—31no—
164#547C2×D311yescharacteristic, complemented
165D42—31no—
166#498—31no—
167#494—31no—
168#487—31no—
169#486—31no—
170#486—31no—
171#486C2×D311yescharacteristic
172#484—61no—
173#480—61no—
174#480—61no—
175#474—31no—
176#474—31no—
177#474—31no—
178#468—61no—
179#467—61no—
180#466—61no—
181#466—61no—
182#461—31no—
183#461—31no—
184#461—31no—
185#459C2×D311yescharacteristic
186#459C2×D311yescharacteristic
187#452—61no—
188#452—61no—
189#449—61no—
190#447—61no—
191#447—61no—
192C4×D8—31no—
193C4×D8—31no—
194#436—31no—
195C8×D4—31no—
196C8×D4—31no—
197C8×D4—31no—
198#417—61no—
199C2×#102—61no—
200C2×#102—61no—
201#335—31no—
202#334—31no—
203#332—31no—
204#332—31no—
205#332C2×D311yescharacteristic, complemented
206#331—31no—
207D4⋊C8—31no—
208D4⋊C8—31no—
209D4⋊C8C2×D311yescharacteristic, complemented
210D4⋊C8C2×D311yescharacteristic, complemented
211#322—31no—
212C23×D3—41no—
213C3×#41—42no—
214C3×#41—81no—
215C2×C3×Q8—21no—
216C2×C3×D4—21no—
217C2×C3×D4—21no—
218C2×C3×D4—21no—
219C2×C3×D4—41no—
220C2×C3×D4C2×D411yescharacteristic
221C2×C3×D4C2×D411yescharacteristic
222C22×C3×C4—21no—
223C22×C3×C4—21no—
224C22×C3×C4—41no—
225C2×#67—41no—
226C2×#67—41no—
227C2×#67—41no—
228C2×#67—41no—
229D3×D4—42no—
230D3×D4—42no—
231D3×D4—81no—
232D3×D4—81no—
233#235—42no—
234#235—42no—
235#235—81no—
236#235—81no—
237C2×D12C2×D411yescharacteristic
238C2×D12C2×D411yescharacteristic
239C2×C4×D3—21no—
240C2×C4×D3—21no—
241C2×C4×D3—41no—
242C2×C4×D3—41no—
243C2×Dic6—21no—
244C2×Dic6—21no—
245C3×D8—81no—
246C3×D8—81no—
247C2×C3×C8—21no—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—41no—
253C3×C4⋊C4C2×D411yescharacteristic
254C3×C4⋊C4C2×D411yescharacteristic
255C3×C22⋊C4—21no—
256C3×C22⋊C4—21no—
257C3×C22⋊C4—41no—
258C3×C42C2411yescharacteristic
259#217—41no—
260#217—41no—
261#217—41no—
262#216—42no—
263#216—81no—
264#215—42no—
265#214—42no—
266#214—81no—
267#213—22no—
268#213—22no—
269(C2×D3)⋊C4—21no—
270(C2×D3)⋊C4—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274C4⋊Dic3—21no—
275C4⋊Dic3C2×D411yescharacteristic
276C4⋊Dic3C2×D411yescharacteristic
277Dic3⋊C4—41no—
278Dic3⋊C4—41no—
279#208—81no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—42no—
283C2×C3⋊C8C2×D411yescharacteristic
284C2×C3⋊C8C2×D411yescharacteristic
285C2×C3⋊C8C2×D411yescharacteristic
286C2×C3⋊C8C2×D411yescharacteristic
287D3×C8—81no—
288D3×C8—81no—
289C2×#41—21no—
290C2×#41—61no—
291C2×#41—61no—
292C22×D4—61no—
293C22×D4—61no—
294#135—124no—
295C2×Q16—61no—
296C2×Q16—61no—
297C2×SD16—61no—
298C2×SD16—61no—
299C2×SD16—61no—
300C2×D8—21no—
301C2×D8—21no—
302C2×D8—31no—
303C2×D8—31no—
304C2×M16—61no—
305C22×C8—61no—
306C22×C8—61no—
307C22×C8—61no—
308C4⋊Q8—31no—
309C4⋊Q8—31no—
310C4⋊Q8C22×D311yescharacteristic
311#127C22×D311yescharacteristic
312#123—41no—
313#123—121no—
314#123—121no—
315C22⋊Q8—21no—
316C22⋊Q8—61no—
317C22⋊Q8—61no—
318#121—21no—
319#121—61no—
320#121—61no—
321#121—61no—
322#121—61no—
323#120—121no—
324C4×D4—31no—
325C4×D4—31no—
326C4×D4C22×D311yescharacteristic
327C4×D4C22×D311yescharacteristic
328C2×C4⋊C4—21no—
329C2×C4⋊C4—61no—
330C2×C4⋊C4—61no—
331#107—31no—
332#107—31no—
333#107—61no—
334#107—61no—
335#107—61no—
336#106—121no—
337C4⋊C8—31no—
338C4⋊C8—31no—
339C4⋊C8C22×D311yescharacteristic
340C4⋊C8C22×D311yescharacteristic
341#103—61no—
342#103—61no—
343#103—61no—
344#103—61no—
345#103—61no—
346#102—21no—
347#102—21no—
348#102—31no—
349#102—31no—
350#102—31no—
351#102—31no—
352#102—41no—
353#102—61no—
354#102—61no—
355#102—61no—
356#102—61no—
357#102—61no—
358#102—61no—
359#102—61no—
360C22⋊C8—61no—
361C22⋊C8—61no—
362C22⋊C8—61no—
363C22⋊C8—61no—
364C4×C8—31no—
365C4×C8—31no—
366C4×C8C22×D311yescharacteristic
367C23×C3—21no—
368C23×C3—21no—
369C23×C3—41no—
370C22×D3—21no—
371C22×D3—21no—
372C22×D3—82no—
373C22×D3—82no—
374C22×D3—161no—
375C3×Q8—41no—
376C3×Q8—41no—
377C3×D4—21no—
378C3×D4—21no—
379C3×D4—21no—
380C3×D4—21no—
381C3×D4—41no—
382C3×D4—41no—
383C3×D4—41no—
384C3×D4—41no—
385C3×D4—81no—
386C3×D4—82no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—82no—
395C2×C3×C4C22×D411yescharacteristic
396C2×C3×C4C22×D411yescharacteristic
397C2×C3×C4H(5,𝔽2)11yescharacteristic
398#67—82no—
399#67—82no—
400#67—82no—
401#67—82no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406D12—21no—
407D12—21no—
408D12#13512yescomplemented
409D12AGL(1,ℤ/8)12yes—
410C4×D3—41no—
411C4×D3—41no—
412C4×D3—41no—
413C4×D3—41no—
414C4×D3—82no—
415C4×D3—82no—
416Dic6—41no—
417Dic6—41no—
418Dic6—41no—
419Dic6—41no—
420C3×C8—41no—
421C3×C8—41no—
422C3×C8—41no—
423C3⋊C8—21no—
424C3⋊C8—21no—
425C3⋊C8—21no—
426C3⋊C8—21no—
427C3⋊C8—41no—
428C3⋊C8—41no—
429C24—121no—
430#41—42no—
431#41—81no—
432#41—122no—
433#41—122no—
434#41—241no—
435#41—241no—
436C2×Q8—21no—
437C2×Q8—61no—
438C2×Q8—61no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—31no—
443C2×D4—31no—
444C2×D4—41no—
445C2×D4—121no—
446C2×D4—121no—
447C2×D4—122no—
448C2×D4—122no—
449C2×D4—121no—
450C2×D4—121no—
451C2×D4—241no—
452C2×D4—241no—
453C2×D4D3×D411yescharacteristic
454C2×D4D3×D411yescharacteristic
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—41no—
458C22×C4—61no—
459C22×C4—61no—
460C22×C4—121no—
461C22×C4—121no—
462Q16—122no—
463Q16—241no—
464SD16—122no—
465SD16—122no—
466SD16—241no—
467D8—62no—
468D8—62no—
469D8—81no—
470D8—81no—
471M16—241no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—31no—
476C2×C8—31no—
477C2×C8—31no—
478C2×C8—31no—
479C2×C8—61no—
480C2×C8—61no—
481C2×C8—122no—
482C2×C8—241no—
483C2×C8—241no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—31no—
487C4⋊C4—31no—
488C4⋊C4—41no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C4⋊C4—61no—
492C4⋊C4—61no—
493C4⋊C4—121no—
494C4⋊C4—121no—
495C4⋊C4D3×D411yescharacteristic
496C4⋊C4D3×D411yescharacteristic
497C22⋊C4—21no—
498C22⋊C4—21no—
499C22⋊C4—41no—
500C22⋊C4—61no—
501C22⋊C4—61no—
502C22⋊C4—121no—
503C22⋊C4—121no—
504C22⋊C4—121no—
505C42C23×D311yescharacteristic
506C22×C3—41no—
507C22×C3—41no—
508C22×C3—41no—
509C22×C3—41no—
510C22×C3—41no—
511C22×C3—41no—
512C22×C3—81no—
513C22×C3—81no—
514C22×C3—81no—
515C22×C3D4211yescharacteristic
516C2×D3—22no—
517C2×D3—22no—
518C2×D3—41no—
519C2×D3—41no—
520C2×D3—41no—
521C2×D3—41no—
522C2×D3—164no—
523C3×C4—21no—
524C3×C4—41no—
525C3×C4—41no—
526C3×C4—81no—
527C3×C4C2×#13511yescharacteristic
528C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
529C3×C4#57611yescharacteristic
530C3×C4#57611yescharacteristic
531Dic3—41no—
532Dic3—41no—
533Dic3—81no—
534Dic3—81no—
535C23—21no—
536C23—21no—
537C23—41no—
538C23—61no—
539C23—61no—
540C23—242no—
541C23—242no—
542C23—481no—
543Q8—41no—
544Q8—41no—
545Q8—121no—
546Q8—121no—
547Q8—121no—
548Q8—121no—
549D4—21no—
550D4—21no—
551D4—21no—
552D4—21no—
553D4—32no—
554D4—32no—
555D4—41no—
556D4—41no—
557D4—41no—
558D4—41no—
559D4—61no—
560D4—61no—
561D4—81no—
562D4—82no—
563D4—242no—
564D4—242no—
565D4—242no—
566D4—242no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—61no—
575C2×C4—61no—
576C2×C4—82no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—242no—
584C2×C4—242no—
585C2×C4C2×D3×D411yescharacteristic
586C2×C4C2×D3×D411yescharacteristic
587C2×C4#100411yescharacteristic
588C8—41no—
589C8—41no—
590C8—41no—
591C8—61no—
592C8—61no—
593C8—61no—
594C8—61no—
595C8—121no—
596C8—121no—
597C2×C3—81no—
598C2×C3—81no—
599C2×C3—161no—
600C2×C3#326411yescharacteristic
601C2×C3#328011yescharacteristic
602C2×C3#328911yescharacteristic
603D3—42no—
604D3—42no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—62no—
612C22—62no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—121no—
617C22—121no—
618C22—121no—
619C22—121no—
620C22—484no—
621C22#559011yescharacteristic
622C4—21no—
623C4—41no—
624C4—41no—
625C4—81no—
626C4—121no—
627C4—121no—
628C4—241no—
629C4—241no—
630C4#575711yescharacteristic
631C4D3×#13511yescharacteristic
632C4D3×AGL(1,ℤ/8)11yescharacteristic
633C4#577411yescharacteristic
634C3#1977211yescharacteristic, complemented, Sylow
635C2—81no—
636C2—81no—
637C2—122no—
638C2—122no—
639C2—161no—
640C2#8001211yescharacteristic
641C2#8010111yescharacteristic
642C2#8028811yescharacteristic
subgroupG/H
centerC22#5590
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1625C2
socleC22×C3D42
hypercenter#1625D3
hypocenterC3#19772
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #1625

Complements

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

NH ≅ G/Ntype
1. #83527C2semidirect
2. #83524C2semidirect
3. #80352C2semidirect
4. #80352C2semidirect
5. C3×#1625C2semidirect
6. #71198C2semidirect
7. #71191C2semidirect
8. #71011C2semidirect
9. #71007C2semidirect
10. #68445C2semidirect
11. #68441C2semidirect
12. #68211C2semidirect
13. #68211C2semidirect
14. #68153C2semidirect
15. #68151C2semidirect
17. C3×#547C22semidirect
30. #5093C22semidirect
31. #5092C22semidirect
40. #5019C22semidirect
41. #5019C22semidirect
44. #5012C22semidirect
45. D4×C3⋊C8C22semidirect
56. #4680C22semidirect
57. C3×#332C22semidirect
58. C3×D4⋊C8C22semidirect
59. C3×D4⋊C8C22semidirect
61. #4482C22semidirect
62. #4482C22semidirect
63. #4480C22semidirect
64. #4479C22semidirect
67. #4454C22semidirect
80. #1625D3semidirect
164. #547C2×D3semidirect
205. #332C2×D3semidirect
209. D4⋊C8C2×D3semidirect
210. D4⋊C8C2×D3semidirect
408. D12#135semidirect
634. C3#19772semidirect

Maximal subgroups

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

subgroupindex
#835272
#835242
#803522
C3×#16252
#711982
#711912
#710112
#710072
#684452
#684412
#682112
#681532
#681512
#197723