Finite Group Databasebeta

← #6459925

Subgroups — #6459925

Order 512.

1764 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35629C211yesmaximal, characteristic, complemented
2#35541C211yesmaximal, characteristic
3#28866C214yesmaximal, complemented
4#28861C214yesmaximal, complemented
5#28849C214yesmaximal, complemented
6#28846C214yesmaximal, complemented
7C2×#2092C211yesmaximal, characteristic, complemented
8#23656C211yesmaximal, characteristic, complemented
9#23640C211yesmaximal, characteristic, complemented
10#23486C211yesmaximal, characteristic
11#21157C211yesmaximal, characteristic, complemented
12#21095C211yesmaximal, characteristic, complemented
13#21037C211yesmaximal, characteristic, complemented
14#20975C211yesmaximal, characteristic
15#20879C211yesmaximal, characteristic, complemented
16#20765C211yesmaximal, characteristic, complemented
17#20663C211yesmaximal, characteristic, complemented
18#20416C211yesmaximal, characteristic, complemented
19#20387C211yesmaximal, characteristic, complemented
20C22×#131C2211yescharacteristic
21C2×#535C2211yescharacteristic
22C2×#518C2211yescharacteristic
23C2×#427C2211yescharacteristic
24C2×#427C2211yescharacteristic
25C22×#107C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C2×#418C2211yescharacteristic
28C2×#418C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C2×#408C2211yescharacteristic
32C2×#405C2211yescharacteristic
33C2×C4×M16C2211yescharacteristic
34#2514—22no—
35#2513—22no—
36#2510—24no—
37#2495C2214yescomplemented
38#2494C2214yescomplemented
39#2447—22no—
40#2445—22no—
41#2422C2214yes—
42#2420C2214yes—
43#2412—24no—
44#2316C2211yescharacteristic
45#2316C2211yescharacteristic
46#2311C2211yescharacteristic
47#2311C2211yescharacteristic
48#2311C2211yescharacteristic
49#2297C2211yescharacteristic
50#2296C2211yescharacteristic
51C2×#390C2211yescharacteristic
52#2285C2211yescharacteristic
53C2×#382C2211yescharacteristic
54#2274C2211yescharacteristic
55#2092C2212yes—
56#2092C2212yes—
57#2092C2212yes—
58#2092C2212yes—
59#2085C2212yescomplemented
60#2085C2212yescomplemented
61#2084C2212yescomplemented
62#2083C2212yescomplemented
63#2077C2214yescomplemented
64#2077C2214yescomplemented
65#2034C2214yes—
66#2033C2214yes—
67#1944C2214yescomplemented
68#1941C2214yescomplemented
69#1936C2214yes—
70#1934C2214yes—
71#1921C2214yescomplemented
72#1918C2214yescomplemented
73#1895C2214yes—
74#1893C2214yes—
75#1864C2214yes—
76#1862C2214yes—
77#1836C2214yescomplemented
78#1835C2214yescomplemented
79#1809C2214yescomplemented
80#1808C2214yescomplemented
81#1762C2214yes—
82C4×#102C2214yes—
83#1753C2212yescomplemented
84#1753C2212yescomplemented
85#1752C2212yescomplemented
86C8×C22⋊C4C2212yescomplemented
87#1736C2211yescharacteristic
88#1736C2211yescharacteristic
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1735C2211yescharacteristic
94#1735C2211yescharacteristic
95#1729C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×D4C2311yescharacteristic
99C2×#131—24no—
100C2×#131—24no—
101C2×#131—42no—
102C22×M16C2311yescharacteristic
103C22×M16C2312yes—
104C22×M16C2311yescharacteristic
105C23×C8C2312yes—
106C23×C8C2311yescharacteristic
107C23×C8C2311yescharacteristic
108#535—24no—
109#535—42no—
110C2×#127—21no—
111C2×#124—21no—
112C2×#123—24no—
113C2×#121C2314yes—
114C2×#120—22no—
115#518—24no—
116#518—42no—
117C2×C4×D4—22no—
118C2×C4×D4C2314yes—
119C2×#117C2311yescharacteristic
120C2×#117C2311yescharacteristic
121C2×#117C2311yescharacteristic
122C2×#117C2311yescharacteristic
123C2×#117C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C22⋊C4—22no—
128C22×C22⋊C4—22no—
129C22×C42C2311yescharacteristic
130#427—42no—
131#427—42no—
132C2×#107—24no—
133C2×#107C2312yes—
134C2×#107C2312yes—
135C2×#107C2312yes—
136C2×#107C2312yes—
137C2×#106—24no—
138C2×#106C2312yes—
139C2×#106C2312yes—
140C2×#106C2312yes—
141C2×#106C2312yes—
142#418—24no—
143#418—24no—
144C2×#102—24no—
145C2×#102—24no—
146C2×#102C2312yes—
147C2×#102C2312yes—
148C2×#102C2312yes—
149C2×#102C2312yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152#408—24no—
153C2×C22⋊C8C2312yes—
154C2×C22⋊C8C2312yes—
155#405—42no—
156C4×M16—24no—
157C2×#97C2312yes—
158C2×#97C2312yes—
159C2×C4×C8C2312yes—
160C2×C4×C8C2312yes—
161#401—24no—
162#400—24no—
163#397—24no—
164#397—24no—
165#397—24no—
166#397—24no—
167#394—24no—
168#394—24no—
169#391—21no—
170#391—21no—
171#391—21no—
172#391D412yes—
173#390—21no—
174#390—21no—
175#390—21no—
176#390—22no—
177#390D412yes—
178#390C2312yes—
179#390C2312yes—
180#389—22no—
181#389—22no—
182#389C2314yes—
183#389C2314yes—
184#389C2312yes—
185#389C2312yes—
186#388—24no—
187#388—24no—
188#387—24no—
189#386—22no—
190#386—22no—
191#386—22no—
192#386—24no—
193#386D414yes—
194#385—22no—
195#385—24no—
196#384C2312yes—
197#383—24no—
198#382—22no—
199#382C2312yes—
200#382C2312yes—
201#382C2312yes—
202#381—24no—
203#380—22no—
204#380—22no—
205#380—22no—
206#380C2314yes—
207#380C2314yes—
208#380C2314yes—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4C2314yes—
213C4×C22⋊C4—22no—
214C4×C22⋊C4C2312yes—
215C4×C22⋊C4C2312yes—
216C2×#95—22no—
217#340C2314yes—
218#340C2314yes—
219#340C2312yes—
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#336C2311yescharacteristic
228#336C2311yescharacteristic
229#336C2311yescharacteristic
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2312yes—
233C25—21no—
234C2×#41—22no—
235C2×#41—22no—
236C2×#41—22no—
237C2×#41—22no—
238C2×#41—22no—
239C2×#41—22no—
240C22×Q8C2×D411yescharacteristic
241C22×D4—24no—
242C22×D4—21no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4C2×D412yes—
249C22×D4C2×D411yescharacteristic
250C22×D4#4114yes—
251C22×D4C2412yes—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—24no—
256C23×C4C2×D412yes—
257C23×C4C2412yes—
258C23×C4C2411yescharacteristic
259#131—416no—
260C2×M16—22no—
261C2×M16—22no—
262C2×M16—22no—
263C2×M16—22no—
264C2×M16—22no—
265C2×M16—24no—
266C2×M16—24no—
267C2×M16—42no—
268C2×M16#4114yes—
269C22×C8—22no—
270C22×C8—22no—
271C22×C8—22no—
272C22×C8—22no—
273C22×C8—24no—
274C22×C8—24no—
275C22×C8—42no—
276C22×C8#4114yes—
277C22×C8#4114yes—
278C22×C8C2412yes—
279C22×C8C2411yescharacteristic
280C22×C8C2411yescharacteristic
281C22×C8C2412yes—
282C22×C8C2411yescharacteristic
283C22×C8C2411yescharacteristic
284#127—44no—
285#124—44no—
286#123—88no—
287#121—28no—
288#121—44no—
289#120—48no—
290C4×D4—28no—
291C4×D4—44no—
292C4×D4—48no—
293#117—22no—
294#117—22no—
295#117—22no—
296#117—22no—
297#117—22no—
298#117—22no—
299#117—22no—
300#117—22no—
301#117—22no—
302#117—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—48no—
338C2×C22⋊C4—84no—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D414yes—
341C2×C22⋊C4C2×D414yes—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—24no—
349C2×C42C2×D412yes—
350C2×C42C2412yes—
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353#107—24no—
354#107—24no—
355#107—42no—
356#107—42no—
357#106—24no—
358#106—24no—
359#106—42no—
360#106—42no—
361#102—28no—
362#102—24no—
363#102—24no—
364#102—24no—
365#102—24no—
366#102—28no—
367C22⋊C8—24no—
368C22⋊C8—24no—
369#97—24no—
370#97—42no—
371C4×C8—24no—
372C4×C8—42no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95C2×D414yes—
386#95C2×D414yes—
387C24—22no—
388C24—24no—
389C24—42no—
390C24—42no—
391C24—82no—
392C24—82no—
393C24C22×D412yes—
394C24C22×D411yescharacteristic
395#41—44no—
396#41—44no—
397#41—44no—
398#41—44no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×D4—24no—
403C2×D4—24no—
404C2×D4—24no—
405C2×D4—28no—
406C2×D4—22no—
407C2×D4—24no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—28no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—48no—
417C2×D4—42no—
418C2×D4—44no—
419C2×D4—42no—
420C2×D4—44no—
421C2×D4—84no—
422C2×D4—84no—
423C2×D4C2×#4114yes—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—48no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—84no—
445C22×C4—84no—
446C22×C4—82no—
447C22×C4—82no—
448C22×C4—82no—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D414yes—
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C2×#4112yes—
455C22×C4C2×#4112yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C4H(5,𝔽2)12yes—
459C22×C42−1+412yes—
460C22×C4C2511yescharacteristic
461M16—28no—
462M16—48no—
463M16—44no—
464C2×C8—28no—
465C2×C8—28no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—22no—
469C2×C8—21no—
470C2×C8—22no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—48no—
480C2×C8C2×#4114yes—
481C2×C8C2×#4112yes—
482C2×C8C2×#4112yes—
483C2×C8C2×#4112yes—
484C2×C8C2×#4112yes—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—84no—
505C4⋊C4—82no—
506C4⋊C4—82no—
507C4⋊C4—82no—
508C4⋊C4—84no—
509C22⋊C4—24no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—84no—
520C22⋊C4—84no—
521C22⋊C4—84no—
522C22⋊C4—84no—
523C42—24no—
524C42—21no—
525C42—21no—
526C42—22no—
527C42—22no—
528C42—21no—
529C42—21no—
530C42—42no—
531C42—42no—
532C42—42no—
533C42—42no—
534C42—84no—
535C23—22no—
536C23—22no—
537C23—24no—
538C23—24no—
539C23—22no—
540C23—24no—
541C23—22no—
542C23—22no—
543C23—44no—
544C23—44no—
545C23—42no—
546C23—42no—
547C23—48no—
548C23—88no—
549C23—162no—
550C23—162no—
551C23—164no—
552C23#53212yes—
553C23#54612yes—
554C23#54712yes—
555C23C23×D411yescharacteristic
556Q8—44no—
557D4—24no—
558D4—24no—
559D4—48no—
560D4—42no—
561D4—42no—
562D4—48no—
563D4—44no—
564D4—168no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—84no—
600C2×C4—168no—
601C2×C4—162no—
602C2×C4—162no—
603C2×C4—162no—
604C2×C4#52914yes—
605C2×C4#53212yes—
606C2×C4#53914yes—
607C2×C4#54612yes—
608C2×C4#54712yes—
609C2×C4#55114yes—
610C2×C4#55214yes—
611C2×C4C22×#4111yescharacteristic
612C2×C4C22×#4111yescharacteristic
613C2×C4C22×#4111yescharacteristic
614C2×C4C22×#4111yescharacteristic
615C2×C4C2×H(5,𝔽2)11yescharacteristic
616C2×C4C2×2−1+411yescharacteristic
617C8—22no—
618C8—22no—
619C8—22no—
620C8—22no—
621C8—24no—
622C8—42no—
623C22—24no—
624C22—22no—
625C22—22no—
626C22—24no—
627C22—42no—
628C22—44no—
629C22—48no—
630C22—44no—
631C22—42no—
632C22—42no—
633C22—44no—
634C22—82no—
635C22—82no—
636C22—84no—
637C22—168no—
638C22#255314yes—
639C22C2×#53211yescharacteristic
640C22C2×#54611yescharacteristic
641C22C2×#54711yescharacteristic
642C22#358412yes—
643C22#358512yes—
644C4—42no—
645C4—44no—
646C4—42no—
647C4—42no—
648C4—82no—
649C4—82no—
650C4—82no—
651C4—82no—
652C4—82no—
653C4—82no—
654C4—82no—
655C4#255314yes—
656C4C2×#52911yescharacteristic
657C4C2×#53911yescharacteristic
658C4C2×#55111yescharacteristic
659C4C2×#55211yescharacteristic
660C2—24no—
661C2—82no—
662C2—84no—
663C2C2×#255311yescharacteristic
664C2#6071311yescharacteristic
665C2#6071911yescharacteristic
666C2#6120112yes—
667C2#6127612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35629C2semidirect
3. #28866C2semidirect
4. #28861C2semidirect
5. #28849C2semidirect
6. #28846C2semidirect
7. C2×#2092C2semidirect
8. #23656C2semidirect
9. #23640C2semidirect
11. #21157C2semidirect
12. #21095C2semidirect
13. #21037C2semidirect
15. #20879C2semidirect
16. #20765C2semidirect
17. #20663C2semidirect
18. #20416C2semidirect
19. #20387C2semidirect
37. #2495C22semidirect
38. #2494C22semidirect
59. #2085C22semidirect
60. #2085C22semidirect
61. #2084C22semidirect
62. #2083C22semidirect
63. #2077C22semidirect
64. #2077C22semidirect
67. #1944C22semidirect
68. #1941C22semidirect
71. #1921C22semidirect
72. #1918C22semidirect
77. #1836C22semidirect
78. #1835C22semidirect
79. #1809C22semidirect
80. #1808C22semidirect
83. #1753C22semidirect
84. #1753C22semidirect
85. #1752C22semidirect
86. C8×C22⋊C4C22semidirect

Maximal subgroups

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

subgroupindex
#356292
#355412
#288662
#288612
#288492
#288462
C2×#20922
#236562
#236402
#234862
#211572
#210952
#210372
#209752
#208792
#207652
#206632
#204162
#203872