Finite Group Databasebeta

← #6460146

Subgroups — #6460146

Order 512.

1794 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35630C211yesmaximal, characteristic, complemented
2#35626C211yesmaximal, characteristic
3#28901C214yesmaximal, complemented
4#28889C214yesmaximal, complemented
5#28886C214yesmaximal, complemented
6#28874C214yesmaximal, complemented
7#23656C211yesmaximal, characteristic, complemented
8#23612C211yesmaximal, characteristic
9#23542C211yesmaximal, characteristic, complemented
10C2×#2015C211yesmaximal, characteristic
11#21232C211yesmaximal, characteristic, complemented
12#21231C211yesmaximal, characteristic
13#21204C211yesmaximal, characteristic
14#21094C211yesmaximal, characteristic, complemented
15#21090C211yesmaximal, characteristic, complemented
16#21066C211yesmaximal, characteristic, complemented
17#20933C211yesmaximal, characteristic
18#20439C211yesmaximal, characteristic, complemented
19#20416C211yesmaximal, characteristic
20C23×Q16C2211yescharacteristic
21C23×SD16C2211yescharacteristic
22C22×#124C2211yescharacteristic
23C2×#518C2211yescharacteristic
24C2×#427C2211yescharacteristic
25C2×#427C2211yescharacteristic
26C2×#424C2211yescharacteristic
27C2×#419C2211yescharacteristic
28C2×#419C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#419C2211yescharacteristic
31C2×#418C2211yescharacteristic
32C2×#418C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C2×#405C2211yescharacteristic
36C2×#405C2211yescharacteristic
37C2×C4×M16C2211yescharacteristic
38#2514—22no—
39#2514C2214yescomplemented
40#2511—22no—
41#2511C2214yescomplemented
42#2493—24no—
43#2483—24no—
44#2447—22no—
45#2447—22no—
46#2447C2214yes—
47#2447C2214yes—
48#2317C2211yescharacteristic
49#2310C2211yescharacteristic
50#2310C2211yescharacteristic
51#2310C2211yescharacteristic
52#2310C2211yescharacteristic
53#2309C2211yescharacteristic
54#2296C2211yescharacteristic
55#2296C2211yescharacteristic
56C2×#391C2211yescharacteristic
57#2285C2211yescharacteristic
58#2274C2211yescharacteristic
59#2085C2212yescomplemented
60#2085C2212yescomplemented
61#2084C2212yescomplemented
62#2083C2212yescomplemented
63#2067C2214yes—
64#2065C2214yes—
65#2056C2214yescomplemented
66#2056C2214yescomplemented
67#2015C2212yes—
68#2015C2212yes—
69#2015C2212yes—
70#2015C2212yes—
71#1973C2214yes—
72#1971C2214yescomplemented
73#1969C2214yescomplemented
74#1968C2214yes—
75#1959C2214yes—
76#1958C2214yes—
77#1935C2214yescomplemented
78#1935C2214yescomplemented
79#1933C2214yescomplemented
80#1933C2214yescomplemented
81#1928C2214yescomplemented
82#1927C2214yescomplemented
83#1877C2214yes—
84#1877C2214yes—
85#1771C2212yescomplemented
86#1771C2212yescomplemented
87#1770C2212yescomplemented
88C8×C4⋊C4C2212yescomplemented
89#1762C2214yes—
90C4×#102C2214yes—
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1735C2211yescharacteristic
94#1735C2211yescharacteristic
95#1729C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C23×Q8C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C23×D4C2311yescharacteristic
100C22×Q16—24no—
101C22×Q16—42no—
102C22×Q16—42no—
103C22×Q16C2312yes—
104C22×Q16C2312yes—
105C22×SD16—24no—
106C22×SD16—24no—
107C22×SD16—42no—
108C22×SD16C2312yes—
109C22×SD16C2312yes—
110C22×M16C2311yescharacteristic
111C22×M16C2311yescharacteristic
112C23×C8C2311yescharacteristic
113C23×C8C2311yescharacteristic
114C2×#124—24no—
115C2×#124—22no—
116C2×#124—42no—
117C2×#124C2312yes—
118C2×#124C2312yes—
119#518—24no—
120#518—42no—
121C2×C4×D4—22no—
122C2×C4×D4C2314yes—
123C2×#117C2311yescharacteristic
124C2×#117C2311yescharacteristic
125C2×#117C2311yescharacteristic
126C2×#117C2311yescharacteristic
127C2×#117C2311yescharacteristic
128C2×#117C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C22⋊C4—22no—
131C22×C22⋊C4—22no—
132C22×C42C2311yescharacteristic
133#427—42no—
134#427—42no—
135C2×#107C2312yes—
136C2×#107C2312yes—
137C2×#106C2312yes—
138C2×#106C2312yes—
139#424—24no—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2312yes—
142#419—24no—
143#419—24no—
144#419—42no—
145#419—42no—
146#418—24no—
147#418—42no—
148C2×#103—24no—
149C2×#103C2312yes—
150C2×#103C2312yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#102—24no—
156C2×#102C2312yes—
157C2×#102C2312yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160#405—24no—
161#405—42no—
162C4×M16—42no—
163C2×#97C2312yes—
164C2×#97C2312yes—
165C2×#97C2312yes—
166C2×C4×C8C2312yes—
167C2×C4×C8C2312yes—
168C2×C4×C8C2312yes—
169#401—24no—
170#400—24no—
171#400—24no—
172#400—24no—
173#397—24no—
174#397—24no—
175#397—24no—
176#397—24no—
177#391—21no—
178#391—21no—
179#391—22no—
180#391—21no—
181#391—22no—
182#391—21no—
183#391—22no—
184#391C2314yes—
185#391C2314yes—
186#391C2314yes—
187#391C2314yes—
188#391C2312yes—
189#391C2312yes—
190#390—21no—
191#390—21no—
192#390C2314yes—
193#389D412yes—
194#389D412yes—
195#389C2314yes—
196#389C2312yes—
197#389C2312yes—
198#387—24no—
199#387—24no—
200#387—24no—
201#385—22no—
202#385—24no—
203#385—24no—
204#385—22no—
205#385D414yes—
206#384—21no—
207#384C2312yes—
208#383—24no—
209#382—21no—
210#382C2312yes—
211#381—22no—
212#381—22no—
213#381—22no—
214#381—22no—
215#381—24no—
216#381—24no—
217#380—22no—
218#379—22no—
219C4×C4⋊C4—22no—
220C4×C4⋊C4—22no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4C2314yes—
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2312yes—
225C4×C22⋊C4—22no—
226C2×#95—22no—
227#340C2312yes—
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#340C2314yes—
233#336C2311yescharacteristic
234#336C2311yescharacteristic
235#336C2311yescharacteristic
236#336C2311yescharacteristic
237#336C2312yes—
238#336C2312yes—
239C25—21no—
240C22×Q8—21no—
241C22×Q8—21no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8—22no—
250C22×Q8—22no—
251C22×Q8—22no—
252C22×Q8—22no—
253C22×Q8—22no—
254C22×Q8C2412yes—
255C22×Q8C2412yes—
256C22×D4—21no—
257C22×D4—22no—
258C22×D4—22no—
259C22×D4—22no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4#4114yes—
263C22×D4C2412yes—
264C23×C4—21no—
265C23×C4—21no—
266C23×C4—21no—
267C23×C4—24no—
268C23×C4C2×D412yes—
269C23×C4C2411yescharacteristic
270C2×Q16—24no—
271C2×Q16—24no—
272C2×Q16—42no—
273C2×Q16—42no—
274C2×Q16—42no—
275C2×Q16—42no—
276C2×Q16—416no—
277C2×SD16—24no—
278C2×SD16—24no—
279C2×SD16—24no—
280C2×SD16—24no—
281C2×SD16—42no—
282C2×SD16—42no—
283C2×SD16—416no—
284C2×M16—22no—
285C2×M16—22no—
286C2×M16—22no—
287C2×M16—22no—
288C2×M16—22no—
289C2×M16#4114yes—
290C22×C8—22no—
291C22×C8—22no—
292C22×C8—22no—
293C22×C8—22no—
294C22×C8#4114yes—
295C22×C8#4114yes—
296C22×C8C2412yes—
297C22×C8C2411yescharacteristic
298C22×C8C2411yescharacteristic
299C22×C8C2411yescharacteristic
300C22×C8C2411yescharacteristic
301C22×C8C2412yes—
302#124—24no—
303#124—24no—
304#124—42no—
305#124—42no—
306#124—48no—
307C4×D4—28no—
308C4×D4—44no—
309C4×D4—48no—
310#117—22no—
311#117—22no—
312#117—22no—
313#117—22no—
314#117—22no—
315#117—22no—
316#117—22no—
317#117—22no—
318#117—22no—
319#117—22no—
320#117—22no—
321#117—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2×D414yes—
334C2×C4⋊C4C2×D414yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2412yes—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—48no—
350C2×C22⋊C4—48no—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—21no—
356C2×C42—21no—
357C2×C42—21no—
358C2×C42—21no—
359C2×C42—24no—
360C2×C42C2×D412yes—
361C2×C42C2412yes—
362C2×C42C2412yes—
363C2×C42C2411yescharacteristic
364C2×C42C2411yescharacteristic
365#107—42no—
366#107—42no—
367#106—42no—
368#106—42no—
369C4⋊C8—24no—
370C4⋊C8—24no—
371#103—24no—
372#103—24no—
373#103—28no—
374#103—28no—
375#103—44no—
376#103—44no—
377#102—24no—
378#102—24no—
379#102—28no—
380#102—44no—
381#97—24no—
382#97—42no—
383#97—42no—
384C4×C8—24no—
385C4×C8—42no—
386C4×C8—42no—
387#95—24no—
388#95—24no—
389#95—24no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—24no—
395#95—24no—
396#95—24no—
397#95—24no—
398#95—24no—
399#95—24no—
400#95—24no—
401#95—24no—
402#95—24no—
403#95C2×D414yes—
404#95C2×D414yes—
405C24—22no—
406C24—24no—
407C24—42no—
408C24—42no—
409C24—82no—
410C24—82no—
411C24C22×D411yescharacteristic
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—42no—
425C2×Q8—42no—
426C2×Q8—42no—
427C2×Q8—42no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8—44no—
431C2×Q8—44no—
432C2×Q8—44no—
433C2×Q8—44no—
434C2×Q8—44no—
435C2×Q8—44no—
436C2×Q8—44no—
437C2×Q8—44no—
438C2×D4—28no—
439C2×D4—24no—
440C2×D4—28no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—42no—
447C2×D4—44no—
448C2×D4—42no—
449C2×D4—44no—
450C2×D4C2×#4114yes—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—48no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—84no—
470C22×C4—84no—
471C22×C4—82no—
472C22×C4C22×D412yes—
473C22×C4C22×D414yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C2×#4112yes—
479C22×C4C2×#4112yes—
480C22×C4C2×#4112yes—
481C22×C4C2×#4112yes—
482C22×C4H(5,𝔽2)12yes—
483C22×C4H(5,𝔽2)12yes—
484C22×C4C2511yescharacteristic
485Q16—48no—
486Q16—48no—
487SD16—48no—
488SD16—48no—
489M16—28no—
490M16—44no—
491C2×C8—28no—
492C2×C8—28no—
493C2×C8—22no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—22no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8C2×#4112yes—
507C2×C8C2×#4112yes—
508C2×C8C2×#4114yes—
509C2×C8C2×#4112yes—
510C2×C8C2×#4112yes—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—24no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—42no—
528C4⋊C4—84no—
529C4⋊C4—82no—
530C4⋊C4—82no—
531C4⋊C4—82no—
532C22⋊C4—24no—
533C22⋊C4—24no—
534C22⋊C4—48no—
535C22⋊C4—48no—
536C22⋊C4—48no—
537C22⋊C4—42no—
538C22⋊C4—42no—
539C22⋊C4—42no—
540C22⋊C4—42no—
541C22⋊C4—84no—
542C22⋊C4—84no—
543C42—24no—
544C42—22no—
545C42—22no—
546C42—21no—
547C42—21no—
548C42—22no—
549C42—21no—
550C42—22no—
551C42—21no—
552C42—42no—
553C42—42no—
554C42—42no—
555C42—42no—
556C42—84no—
557C23—22no—
558C23—22no—
559C23—24no—
560C23—44no—
561C23—44no—
562C23—42no—
563C23—42no—
564C23—48no—
565C23—88no—
566C23—162no—
567C23—162no—
568C23—164no—
569C23#53212yes—
570C23#54612yes—
571C23#55012yes—
572C23C23×D411yescharacteristic
573Q8—42no—
574Q8—42no—
575Q8—42no—
576Q8—42no—
577Q8—42no—
578Q8—42no—
579Q8—42no—
580Q8—42no—
581Q8—84no—
582Q8—84no—
583D4—24no—
584D4—24no—
585D4—42no—
586D4—42no—
587D4—48no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—84no—
622C2×C4—84no—
623C2×C4—84no—
624C2×C4—168no—
625C2×C4—162no—
626C2×C4#52914yes—
627C2×C4#53212yes—
628C2×C4#53814yes—
629C2×C4#54612yes—
630C2×C4#55012yes—
631C2×C4#55114yes—
632C2×C4#55314yes—
633C2×C4C22×#4111yescharacteristic
634C2×C4C22×#4111yescharacteristic
635C2×C4C22×#4111yescharacteristic
636C2×C4C22×#4111yescharacteristic
637C2×C4C2×H(5,𝔽2)11yescharacteristic
638C2×C4C2×H(5,𝔽2)11yescharacteristic
639C8—22no—
640C8—22no—
641C8—22no—
642C8—22no—
643C8—24no—
644C8—42no—
645C22—24no—
646C22—22no—
647C22—22no—
648C22—24no—
649C22—48no—
650C22—44no—
651C22—82no—
652C22—82no—
653C22—84no—
654C22—168no—
655C22#255414yes—
656C22C2×#53211yescharacteristic
657C22C2×#54611yescharacteristic
658C22C2×#55011yescharacteristic
659C22#358412yes—
660C22#358612yes—
661C4—42no—
662C4—42no—
663C4—44no—
664C4—42no—
665C4—42no—
666C4—82no—
667C4—82no—
668C4—82no—
669C4—82no—
670C4—82no—
671C4—82no—
672C4—82no—
673C4#255414yes—
674C4C2×#52911yescharacteristic
675C4C2×#53811yescharacteristic
676C4C2×#55111yescharacteristic
677C4C2×#55311yescharacteristic
678C2—24no—
679C2—84no—
680C2C2×#255411yescharacteristic
681C2#6071311yescharacteristic
682C2#6072011yescharacteristic
683C2#6118812yes—
684C2#6142312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35630C2semidirect
3. #28901C2semidirect
4. #28889C2semidirect
5. #28886C2semidirect
6. #28874C2semidirect
7. #23656C2semidirect
9. #23542C2semidirect
11. #21232C2semidirect
14. #21094C2semidirect
15. #21090C2semidirect
16. #21066C2semidirect
18. #20439C2semidirect
39. #2514C22semidirect
41. #2511C22semidirect
59. #2085C22semidirect
60. #2085C22semidirect
61. #2084C22semidirect
62. #2083C22semidirect
65. #2056C22semidirect
66. #2056C22semidirect
72. #1971C22semidirect
73. #1969C22semidirect
77. #1935C22semidirect
78. #1935C22semidirect
79. #1933C22semidirect
80. #1933C22semidirect
81. #1928C22semidirect
82. #1927C22semidirect
85. #1771C22semidirect
86. #1771C22semidirect
87. #1770C22semidirect
88. C8×C4⋊C4C22semidirect

Maximal subgroups

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

subgroupindex
#356302
#356262
#289012
#288892
#288862
#288742
#236562
#236122
#235422
C2×#20152
#212322
#212312
#212042
#210942
#210902
#210662
#209332
#204392
#204162