Finite Group Databasebeta

← #6460161

Subgroups — #6460161

Order 512.

1799 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35674C211yesmaximal, characteristic, complemented
2#35557C211yesmaximal, characteristic
3#28904C214yesmaximal, complemented
4#28900C214yesmaximal, complemented
5#28879C214yesmaximal, complemented
6#28875C214yesmaximal, complemented
7#23658C211yesmaximal, characteristic, complemented
8#23615C211yesmaximal, characteristic, complemented
9#23545C211yesmaximal, characteristic
10#23432C211yesmaximal, characteristic
11#21234C211yesmaximal, characteristic
12#21234C211yesmaximal, characteristic
13C2×#1956C211yesmaximal, characteristic, complemented
14#21088C211yesmaximal, characteristic, complemented
15#21083C211yesmaximal, characteristic
16#21064C211yesmaximal, characteristic, complemented
17#20930C211yesmaximal, characteristic
18#20435C211yesmaximal, characteristic, complemented
19#20417C211yesmaximal, characteristic, complemented
20C22×#137C2211yescharacteristic
21C22×#137C2211yescharacteristic
22C22×C4⋊Q8C2211yescharacteristic
23C2×#519C2211yescharacteristic
24C2×#428C2211yescharacteristic
25C2×#428C2211yescharacteristic
26C2×#424C2211yescharacteristic
27C2×#419C2211yescharacteristic
28C2×#419C2211yescharacteristic
29C2×#416C2211yescharacteristic
30C2×#416C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C2×#405C2211yescharacteristic
36C2×#405C2211yescharacteristic
37C22×C4×C8C2211yescharacteristic
38#2512C2214yes—
39#2511C2214yes—
40#2497—22no—
41#2496—22no—
42#2484—24no—
43#2481—24no—
44#2446—22no—
45#2446—22no—
46#2446C2214yes—
47#2446C2214yes—
48#2316C2211yescharacteristic
49#2311C2211yescharacteristic
50#2311C2211yescharacteristic
51#2310C2211yescharacteristic
52#2310C2211yescharacteristic
53#2310C2211yescharacteristic
54#2297C2211yescharacteristic
55#2296C2211yescharacteristic
56#2296C2211yescharacteristic
57C2×#384C2211yescharacteristic
58#2274C2211yescharacteristic
59#2084C2214yes—
60#2083C2214yes—
61#2068C2214yes—
62#2066C2214yes—
63#2056C2214yes—
64#2055C2214yes—
65#2016C2214yes—
66#2013C2214yes—
67#1973C2212yes—
68#1973C2212yes—
69#1971C2212yes—
70#1971C2212yes—
71#1969C2212yes—
72#1969C2212yes—
73#1968C2212yes—
74#1968C2212yes—
75#1956C2214yes—
76#1956C2214yes—
77#1935C2214yes—
78#1934C2212yes—
79#1934C2212yes—
80#1933C2212yes—
81#1933C2212yes—
82#1933C2214yes—
83#1928C2214yes—
84#1927C2214yes—
85#1880C2214yes—
86#1878C2214yes—
87#1771C2214yes—
88C8×C4⋊C4C2214yes—
89#1763C2214yes—
90C4×#103C2214yes—
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1735C2211yescharacteristic
94#1735C2211yescharacteristic
95#1729C2211yescharacteristic
96#1728C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C23×Q8C2311yescharacteristic
100C2×#137—24no—
101C2×#137—24no—
102C2×#137—24no—
103C2×#137—42no—
104C2×#137—42no—
105C2×#137—42no—
106C22×Q16C2312yes—
107C22×Q16C2312yes—
108C22×SD16C2312yes—
109C22×SD16C2312yes—
110C22×M16C2311yescharacteristic
111C22×M16C2311yescharacteristic
112C23×C8C2311yescharacteristic
113C23×C8C2311yescharacteristic
114C2×C4⋊Q8—24no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8—42no—
118C2×C4⋊Q8C2314yes—
119C2×#126—24no—
120#519—24no—
121#519—42no—
122C2×C4×Q8—22no—
123C2×C4×Q8C2314yes—
124C2×#117—22no—
125C2×#117C2311yescharacteristic
126C2×#117C2311yescharacteristic
127C2×#117C2311yescharacteristic
128C2×#117C2311yescharacteristic
129C2×#117C2311yescharacteristic
130C22×C4⋊C4—22no—
131C22×C4⋊C4—22no—
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#428—42no—
136#428—42no—
137C2×#107C2312yes—
138C2×#107C2312yes—
139C2×#106C2312yes—
140C2×#106C2312yes—
141#424—24no—
142C2×C4⋊C8C2314yes—
143#419—24no—
144#419—24no—
145#416—24no—
146#416—42no—
147C2×#103—24no—
148C2×#103—24no—
149C2×#103—42no—
150C2×#103—42no—
151C2×#103C2314yes—
152C2×#103C2312yes—
153C2×#103C2312yes—
154C2×#103C2312yes—
155C2×#103C2312yes—
156C2×#103C2311yescharacteristic
157C2×#103C2311yescharacteristic
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2312yes—
161C2×#103C2314yes—
162C2×#102C2311yescharacteristic
163C2×#102C2311yescharacteristic
164C2×#102C2312yes—
165#405—24no—
166#405—42no—
167C2×#97C2312yes—
168C2×#97C2312yes—
169C2×C4×C8—42no—
170C2×C4×C8C2312yes—
171C2×C4×C8C2312yes—
172C2×C4×C8C2312yes—
173C2×C4×C8C2312yes—
174#400—24no—
175#400—24no—
176#400—24no—
177#400—24no—
178#398—24no—
179#398—24no—
180#397—24no—
181#396—24no—
182#391—22no—
183#391—22no—
184#391—22no—
185#391C2314yes—
186#391C2312yes—
187#391C2312yes—
188#391C2312yes—
189#391C2312yes—
190#391C2312yes—
191#391C2312yes—
192#391C2312yes—
193#390C2312yes—
194#390C2312yes—
195#390C2312yes—
196#389—22no—
197#389D412yes—
198#389D412yes—
199#389C2314yes—
200#389C2312yes—
201#389C2312yes—
202#388—24no—
203#387—24no—
204#387—24no—
205#387—24no—
206#387—24no—
207#387—24no—
208#387—24no—
209#386—22no—
210#386—22no—
211#386—22no—
212#385—22no—
213#385D414yes—
214#384—21no—
215#384—21no—
216#384C2314yes—
217#381—22no—
218#381—22no—
219#381—22no—
220#380—22no—
221#380—22no—
222#380—22no—
223C4×C4⋊C4—22no—
224C4×C4⋊C4—21no—
225C4×C4⋊C4—21no—
226C4×C4⋊C4C2314yes—
227C4×C4⋊C4C2314yes—
228C4×C22⋊C4—22no—
229C4×C22⋊C4—21no—
230C4×C22⋊C4—21no—
231#340C2314yes—
232#340C2312yes—
233#340C2312yes—
234#340C2314yes—
235#340C2312yes—
236#340C2312yes—
237#336C2314yes—
238#336C2314yes—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—22no—
244C2×#41—22no—
245C22×Q8—21no—
246C22×Q8—21no—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8—22no—
250C22×Q8—22no—
251C22×Q8—22no—
252C22×Q8—22no—
253C22×Q8—22no—
254C22×Q8—22no—
255C22×Q8—22no—
256C22×Q8—22no—
257C22×Q8—22no—
258C22×Q8#4114yes—
259C22×Q8C2412yes—
260C22×Q8C2411yescharacteristic
261C22×Q8C2412yes—
262C22×D4—21no—
263C22×D4C2411yescharacteristic
264C23×C4—21no—
265C23×C4—21no—
266C23×C4—21no—
267C23×C4—21no—
268C23×C4—21no—
269C23×C4C2×D412yes—
270C23×C4C2411yescharacteristic
271#137—416no—
272#137—416no—
273C2×Q16—24no—
274C2×Q16—24no—
275C2×Q16—24no—
276C2×Q16—42no—
277C2×Q16—42no—
278C2×Q16—42no—
279C2×SD16—24no—
280C2×SD16—24no—
281C2×SD16—24no—
282C2×SD16—42no—
283C2×SD16—42no—
284C2×SD16—42no—
285C2×M16—22no—
286C2×M16—22no—
287C2×M16—22no—
288C2×M16—22no—
289C2×M16#4114yes—
290C2×M16#4114yes—
291C22×C8—22no—
292C22×C8—22no—
293C22×C8—22no—
294C22×C8—22no—
295C22×C8—22no—
296C22×C8#4114yes—
297C22×C8C2412yes—
298C22×C8C2411yescharacteristic
299C22×C8C2411yescharacteristic
300C22×C8C2412yes—
301C22×C8C2412yes—
302C4⋊Q8—28no—
303C4⋊Q8—44no—
304C4⋊Q8—44no—
305C4⋊Q8—44no—
306#126—88no—
307C4×Q8—28no—
308C4×Q8—44no—
309C4×Q8—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—48no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—48no—
336C2×C4⋊C4—84no—
337C2×C4⋊C4C2×D412yes—
338C2×C4⋊C4C2×D412yes—
339C2×C4⋊C4C2×D414yes—
340C2×C4⋊C4C2×D414yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2411yescharacteristic
346C2×C4⋊C4C2411yescharacteristic
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4C2×D412yes—
355C2×C22⋊C4C2×D412yes—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—24no—
359C2×C42—21no—
360C2×C42—21no—
361C2×C42—21no—
362C2×C42—24no—
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2412yes—
366C2×C42C2412yes—
367C2×C42C2412yes—
368#107—42no—
369#107—42no—
370#106—42no—
371#106—42no—
372C4⋊C8—28no—
373#103—24no—
374#103—24no—
375#103—24no—
376#103—24no—
377#103—22no—
378#103—22no—
379#103—28no—
380#103—28no—
381#103—44no—
382#103—44no—
383#103—42no—
384#102—22no—
385#102—22no—
386#102—42no—
387#97—24no—
388#97—42no—
389C4×C8—24no—
390C4×C8—42no—
391C4×C8—42no—
392C4×C8—42no—
393#95—24no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—24no—
399#95—24no—
400#95—24no—
401#95—24no—
402#95—24no—
403#95—24no—
404#95—24no—
405#95—24no—
406#95—24no—
407#95C2×D414yes—
408#95C2×D414yes—
409C24—21no—
410C24C22×D411yescharacteristic
411#41—44no—
412#41—44no—
413#41—44no—
414#41—44no—
415C2×Q8—24no—
416C2×Q8—22no—
417C2×Q8—28no—
418C2×Q8—28no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—21no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—21no—
433C2×Q8—22no—
434C2×Q8—21no—
435C2×Q8—42no—
436C2×Q8—42no—
437C2×Q8—44no—
438C2×Q8—42no—
439C2×Q8—42no—
440C2×Q8—42no—
441C2×Q8—44no—
442C2×Q8—44no—
443C2×Q8—44no—
444C2×Q8—44no—
445C2×Q8—44no—
446C2×Q8C2×#4114yes—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—42no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—82no—
480C22×C4—82no—
481C22×C4C22×D414yes—
482C22×C4C22×D414yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D411yescharacteristic
486C22×C4C2×#4112yes—
487C22×C4C2×#4112yes—
488C22×C4C2×#4112yes—
489C22×C4C2×#4112yes—
490C22×C4H(5,𝔽2)12yes—
491C22×C4H(5,𝔽2)12yes—
492C22×C4C2511yescharacteristic
493Q16—48no—
494Q16—48no—
495SD16—48no—
496SD16—48no—
497M16—28no—
498M16—28no—
499C2×C8—28no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—22no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—44no—
512C2×C8C2×#4114yes—
513C2×C8C2×#4112yes—
514C2×C8C2×#4112yes—
515C2×C8C2×#4114yes—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—28no—
519C4⋊C4—28no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—48no—
533C4⋊C4—48no—
534C4⋊C4—48no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—44no—
538C4⋊C4—48no—
539C4⋊C4—44no—
540C4⋊C4—84no—
541C4⋊C4—84no—
542C22⋊C4—24no—
543C22⋊C4—24no—
544C22⋊C4—42no—
545C22⋊C4—42no—
546C22⋊C4—42no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—84no—
550C22⋊C4—84no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—48no—
560C42—42no—
561C42—42no—
562C42—42no—
563C42—84no—
564C23—22no—
565C23—22no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—42no—
570C23—42no—
571C23#53212yes—
572C23#54612yes—
573C23#55012yes—
574C23C23×D411yescharacteristic
575Q8—24no—
576Q8—24no—
577Q8—42no—
578Q8—42no—
579Q8—48no—
580Q8—42no—
581Q8—41no—
582Q8—42no—
583Q8—41no—
584Q8—41no—
585Q8—42no—
586Q8—42no—
587Q8—41no—
588Q8—84no—
589D4—41no—
590D4—41no—
591D4—41no—
592D4—41no—
593D4—84no—
594C2×C4—216no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—48no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—84no—
630C2×C4—84no—
631C2×C4—162no—
632C2×C4—162no—
633C2×C4#52914yes—
634C2×C4#53212yes—
635C2×C4#53814yes—
636C2×C4#54612yes—
637C2×C4#55012yes—
638C2×C4#55114yes—
639C2×C4#55314yes—
640C2×C4#57714yes—
641C2×C4#57814yes—
642C2×C4C22×#4111yescharacteristic
643C2×C4C22×#4111yescharacteristic
644C2×C4C22×#4111yescharacteristic
645C2×C4C22×#4111yescharacteristic
646C2×C4C2×H(5,𝔽2)11yescharacteristic
647C2×C4C2×H(5,𝔽2)11yescharacteristic
648C8—24no—
649C8—24no—
650C8—22no—
651C8—22no—
652C8—42no—
653C22—24no—
654C22—22no—
655C22—22no—
656C22—24no—
657C22—81no—
658C22—82no—
659C22—81no—
660C22—81no—
661C22—82no—
662C22#255414yes—
663C22C2×#53211yescharacteristic
664C22C2×#54611yescharacteristic
665C22C2×#55011yescharacteristic
666C22C2×#57712yes—
667C22C2×#57812yes—
668C4—24no—
669C4—24no—
670C4—44no—
671C4—44no—
672C4—44no—
673C4—81no—
674C4—82no—
675C4—82no—
676C4—81no—
677C4—82no—
678C4—82no—
679C4—81no—
680C4—82no—
681C4#255414yes—
682C4C2×#52911yescharacteristic
683C4C2×#53811yescharacteristic
684C4C2×#55111yescharacteristic
685C4C2×#55311yescharacteristic
686C2—24no—
687C2—161no—
688C2C2×#255411yescharacteristic
689C2#6072312yes—
690C2#6116812yes—
691C2#6141011yescharacteristic
692C2#6141111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
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. #35674C2semidirect
3. #28904C2semidirect
4. #28900C2semidirect
5. #28879C2semidirect
6. #28875C2semidirect
7. #23658C2semidirect
8. #23615C2semidirect
13. C2×#1956C2semidirect
14. #21088C2semidirect
16. #21064C2semidirect
18. #20435C2semidirect
19. #20417C2semidirect

Maximal subgroups

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

subgroupindex
#356742
#355572
#289042
#289002
#288792
#288752
#236582
#236152
#235452
#234322
#212342
C2×#19562
#210882
#210832
#210642
#209302
#204352
#204172