Finite Group Databasebeta

← #6495688

Subgroups — #6495688

Order 512.

1852 conjugacy classes of subgroups.

The subgroup lattice has 657 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35278C211yesmaximal, characteristic, complemented
2#34255C211yesmaximal, characteristic, complemented
3#33118C212yesmaximal, complemented
4#32911C212yesmaximal, complemented
5#29987C212yesmaximal, complemented
6#29794C212yesmaximal, complemented
7#29564C214yesmaximal, complemented
8#29365C212yesmaximal, complemented
9#29364C212yesmaximal, complemented
10#29337C211yesmaximal, characteristic, complemented
11#29331C211yesmaximal, characteristic, complemented
12#28310C211yesmaximal, characteristic, complemented
13#28301C211yesmaximal, characteristic, complemented
14#24069C212yesmaximal, complemented
15#23939C212yesmaximal, complemented
16#22158C214yesmaximal, complemented
17#20784C211yesmaximal, characteristic, complemented
18C2×#540C2212yes—
19C2×#518C2211yescharacteristic
20C2×#481C2212yes—
21C2×#480C2212yes—
22C2×#459C2214yes—
23C2×#447C2212yes—
24C2×#447C2212yes—
25#2867C2211yescharacteristic, complemented
26#2859C2211yescharacteristic, complemented
27#2856C2211yescharacteristic
28#2856C2211yescharacteristic
29#2641C2212yescomplemented
30#2619C2212yes—
31#2445—22no—
32#2445—22no—
33#2428—22no—
34#2428—22no—
35#2357—22no—
36#2354—22no—
37#2342—22no—
38#2342—22no—
39#2328C2212yescomplemented
40#2319C2212yes—
41#2319C2212yes—
42#2317C2211yescharacteristic
43#2309C2211yescharacteristic
44C4×#127C2211yescharacteristic, complemented
45C4×#127C2211yescharacteristic, complemented
46#2296C2211yescharacteristic
47#2293C2212yes—
48#2293C2212yes—
49C2×#390C2211yescharacteristic
50#2274C2211yescharacteristic
51C4×#117C2211yescharacteristic
52#2100C2212yescomplemented
53#2099C2212yescomplemented
54#2094C2212yes—
55#2093C2212yes—
56#2059C2214yescomplemented
57#2041C2214yescomplemented
58#2041C2212yescomplemented
59#2041C2212yescomplemented
60#2025C2214yes—
61#2011C2214yes—
62#2011C2212yes—
63#2011C2212yes—
64#1982C2214yescomplemented
65#1972C2214yes—
66#1970C2214yes—
67#1949C2214yescomplemented
68#1936C2214yescomplemented
69#1934C2214yescomplemented
70#1926C2214yescomplemented
71#1925C2214yescomplemented
72#1879C2214yes—
73#1879C2214yes—
74#1879C2214yes—
75#1879C2214yes—
76#1875C2212yes—
77#1875C2212yes—
78#1875C2212yes—
79#1875C2212yes—
80#1864C2214yescomplemented
81#1862C2214yescomplemented
82#1839C2212yescomplemented
83#1831C2212yescomplemented
84#1818C2212yes—
85#1804C2214yes—
86#1798C2212yes—
87#1768C2212yescomplemented
88#1749C2211yescharacteristic
89C4×C22⋊C8C2211yescharacteristic
90#1746C2212yescomplemented
91C23×D4C2311yescharacteristic
92C22×D8C2314yes—
93C22×D8C2314yes—
94#540—44no—
95#540—44no—
96C2×#127—22no—
97C2×#127—22no—
98C2×#127C2312yes—
99C2×#127C2311yescharacteristic
100C2×#127C2311yescharacteristic
101C2×#127C2312yes—
102C2×#125C2312yes—
103C2×#123—22no—
104C2×#123—22no—
105C2×#121—22no—
106C2×#121—22no—
107C2×#121—22no—
108C2×#121—22no—
109C2×#121C2312yes—
110C2×#121C2312yes—
111#518—28no—
112C2×C4×D4—22no—
113C2×C4×D4—22no—
114C2×C4×D4—22no—
115C2×C4×D4C2312yes—
116C2×C4×D4C2312yes—
117C2×C4×D4C2312yes—
118C2×C4×D4C2312yes—
119C2×#117—22no—
120C2×#117—22no—
121C2×#117C2311yescharacteristic
122C2×#117C2311yescharacteristic
123C2×#117C2311yescharacteristic
124C2×#117C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C22⋊C4—22no—
127C22×C22⋊C4—22no—
128C22×C42C2311yescharacteristic
129#481—44no—
130#480—44no—
131#459—216no—
132#447—44no—
133#447—44no—
134C2×#107C2314yes—
135C2×#106C2314yes—
136C2×C4⋊C8C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2312yes—
147C2×#97C2312yes—
148C2×C4×C8C2312yes—
149#394—24no—
150#394—24no—
151#394—24no—
152#392—24no—
153#390—21no—
154#390—21no—
155#390—22no—
156#390—22no—
157#390—22no—
158#390—22no—
159#390—22no—
160#390C2312yes—
161#390C2312yes—
162#390C2312yes—
163#390C2312yes—
164#390C2312yes—
165#390C2312yes—
166#390C2312yes—
167#390C2312yes—
168#389—22no—
169#389—22no—
170#389C2312yes—
171#389C2312yes—
172#389C2312yes—
173#389C2312yes—
174#388—22no—
175#388—22no—
176#388—22no—
177#388—22no—
178#388—22no—
179#388—22no—
180#388—22no—
181#388—22no—
182#386—24no—
183#386—24no—
184#385—22no—
185#385—21no—
186#385—21no—
187#384—21no—
188#384C2311yescharacteristic
189#384C2311yescharacteristic
190#382—21no—
191#382C2312yes—
192#381—22no—
193#381—22no—
194#380—22no—
195#380—22no—
196#379—21no—
197#379—21no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4D412yes—
201C4×C4⋊C4D412yes—
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2312yes—
204C4×C22⋊C4—22no—
205C4×C22⋊C4—22no—
206C4×C22⋊C4—22no—
207C4×C22⋊C4D412yes—
208C4×C22⋊C4D412yes—
209#376C2311yescharacteristic
210#376C2311yescharacteristic
211C43C2311yescharacteristic
212C43C2311yescharacteristic
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#336C2314yes—
218#336C2314yes—
219#336C2314yes—
220#336C2312yes—
221#336C2312yes—
222C25—21no—
223C22×D4—24no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—21no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4—22no—
230C22×D4—22no—
231C22×D4—24no—
232C22×D4—24no—
233C22×D4—42no—
234C22×D4C2412yes—
235C22×D4C2412yes—
236C22×D4C2412yes—
237C23×C4—22no—
238C23×C4—22no—
239C23×C4—21no—
240C23×C4—22no—
241C23×C4—22no—
242C23×C4C2×D412yes—
243C23×C4C2411yescharacteristic
244C2×D8—28no—
245C2×D8—28no—
246C2×D8—44no—
247C2×D8—44no—
248C2×D8—44no—
249C2×D8—44no—
250C22×C8C2414yes—
251C22×C8C2414yes—
252#127—28no—
253#127—42no—
254#127—42no—
255#127—42no—
256#127—42no—
257#127—84no—
258#127—84no—
259#125—42no—
260#125—42no—
261#123—48no—
262#123—48no—
263#121—24no—
264#121—24no—
265#121—24no—
266#121—24no—
267#121—48no—
268#121—48no—
269#121—84no—
270#121—84no—
271C4×D4—216no—
272C4×D4—24no—
273C4×D4—24no—
274C4×D4—24no—
275C4×D4—42no—
276C4×D4—42no—
277C4×D4—42no—
278C4×D4—42no—
279C4×D4—84no—
280C4×D4—84no—
281C4×D4#4118yes—
282#117—24no—
283#117—24no—
284#117—24no—
285#117—42no—
286#117—84no—
287#117—84no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C22⋊C4—216no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—84no—
328C2×C22⋊C4C2×D412yes—
329C2×C22⋊C4C2×D411yescharacteristic
330C2×C22⋊C4C2×D411yescharacteristic
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—42no—
336C2×C42C2×D412yes—
337C2×C42C2×D412yes—
338C2×C42C2×D412yes—
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2411yescharacteristic
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347#107—44no—
348#106—44no—
349C4⋊C8—28no—
350#102—28no—
351#102—28no—
352#102—44no—
353#102—44no—
354#102—44no—
355#102—44no—
356#102—44no—
357#102—44no—
358C22⋊C8—24no—
359C22⋊C8—24no—
360#97—42no—
361C4×C8—42no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—24no—
369#95C2×D412yes—
370#95C2×D412yes—
371#95C2×D412yes—
372#95C2×D412yes—
373C24—24no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—42no—
378C24—42no—
379C24—82no—
380C24—82no—
381C24C22×D411yescharacteristic
382C2×D4—24no—
383C2×D4—22no—
384C2×D4—24no—
385C2×D4—24no—
386C2×D4—24no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—24no—
392C2×D4—24no—
393C2×D4—24no—
394C2×D4—416no—
395C2×D4—42no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4—44no—
403C2×D4—44no—
404C2×D4—48no—
405C2×D4—42no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—84no—
409C2×D4—84no—
410C2×D4—84no—
411C2×D4—84no—
412C2×D4—84no—
413C2×D4—84no—
414C2×D4C2×#4114yes—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—28no—
419C22×C4—28no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—42no—
430C22×C4—82no—
431C22×C4—82no—
432C22×C4—82no—
433C22×C4—82no—
434C22×C4—84no—
435C22×C4—84no—
436C22×C4—82no—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D412yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D412yes—
446C22×C4H(5,𝔽2)14yes—
447C22×C4C2511yescharacteristic
448D8—88no—
449D8—88no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—24no—
454C2×C8—24no—
455C2×C8—24no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—82no—
473C4⋊C4—82no—
474C4⋊C4—82no—
475C4⋊C4C2×#4114yes—
476C22⋊C4—24no—
477C22⋊C4—28no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—24no—
481C22⋊C4—28no—
482C22⋊C4—44no—
483C22⋊C4—44no—
484C22⋊C4—42no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—82no—
490C22⋊C4—82no—
491C22⋊C4—82no—
492C22⋊C4—82no—
493C22⋊C4—82no—
494C22⋊C4—82no—
495C22⋊C4—84no—
496C22⋊C4—84no—
497C22⋊C4—84no—
498C22⋊C4—84no—
499C42—24no—
500C42—24no—
501C42—24no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—42no—
508C42—42no—
509C42—42no—
510C42—42no—
511C42—82no—
512C42—82no—
513C42C2×#4114yes—
514C23—22no—
515C23—21no—
516C23—21no—
517C23—22no—
518C23—24no—
519C23—48no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—44no—
527C23—44no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—44no—
535C23—44no—
536C23—88no—
537C23—162no—
538C23—162no—
539C23—164no—
540C23#53512yes—
541C23C23×D411yescharacteristic
542D4—28no—
543D4—48no—
544D4—44no—
545D4—44no—
546D4—44no—
547D4—44no—
548D4—44no—
549D4—88no—
550D4—88no—
551D4—168no—
552D4—164no—
553D4—164no—
554D4—164no—
555D4—164no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—28no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—84no—
591C2×C4—88no—
592C2×C4—88no—
593C2×C4—164no—
594C2×C4—164no—
595C2×C4—162no—
596C2×C4#52514yes—
597C2×C4#53512yes—
598C2×C4#57614yes—
599C2×C4C22×#4112yes—
600C2×C4C2×H(5,𝔽2)12yes—
601C2×C4#58512yes—
602C8—44no—
603C8—44no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—42no—
608C22—42no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—84no—
617C22—84no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—84no—
623C22—84no—
624C22—82no—
625C22—82no—
626C22—168no—
627C22C2×#52511yescharacteristic
628C22C2×#53511yescharacteristic
629C22#344911yescharacteristic
630C22C2×#57612yes—
631C22#358512yes—
632C4—24no—
633C4—24no—
634C4—22no—
635C4—42no—
636C4—41no—
637C4—41no—
638C4—41no—
639C4—41no—
640C4—42no—
641C4—44no—
642C4—82no—
643C4—82no—
644C4—82no—
645C4—82no—
646C4#349414yes—
647C2—42no—
648C2—84no—
649C2—162no—
650C2—162no—
651C2#3530111yescharacteristic
652C2#6059012yes—
653C2#6081711yescharacteristic
654C2#6082711yescharacteristic
655C2#6097812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35278C2semidirect
2. #34255C2semidirect
3. #33118C2semidirect
4. #32911C2semidirect
5. #29987C2semidirect
6. #29794C2semidirect
7. #29564C2semidirect
8. #29365C2semidirect
9. #29364C2semidirect
10. #29337C2semidirect
11. #29331C2semidirect
12. #28310C2semidirect
13. #28301C2semidirect
14. #24069C2semidirect
15. #23939C2semidirect
16. #22158C2semidirect
17. #20784C2semidirect
25. #2867C22semidirect
26. #2859C22semidirect
29. #2641C22semidirect
39. #2328C22semidirect
44. C4×#127C22semidirect
45. C4×#127C22semidirect
52. #2100C22semidirect
53. #2099C22semidirect
56. #2059C22semidirect
57. #2041C22semidirect
58. #2041C22semidirect
59. #2041C22semidirect
64. #1982C22semidirect
67. #1949C22semidirect
68. #1936C22semidirect
69. #1934C22semidirect
70. #1926C22semidirect
71. #1925C22semidirect
80. #1864C22semidirect
81. #1862C22semidirect
82. #1839C22semidirect
83. #1831C22semidirect
87. #1768C22semidirect
90. #1746C22semidirect

Maximal subgroups

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

subgroupindex
#352782
#342552
#331182
#329112
#299872
#297942
#295642
#293652
#293642
#293372
#293312
#283102
#283012
#240692
#239392
#221582
#207842