Finite Group Databasebeta

← #6433530

Subgroups — #6433530

Order 512.

1772 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35280C211yesmaximal, characteristic, complemented
2#34267C211yesmaximal, characteristic
3#30182C214yesmaximal, complemented
4#30180C214yesmaximal, complemented
5#30176C212yesmaximal, complemented
6#30172C212yesmaximal, complemented
7#30168C212yesmaximal, complemented
8#30166C212yesmaximal, complemented
9C2×#2084C211yesmaximal, characteristic, complemented
10#23643C212yesmaximal, complemented
11#23630C211yesmaximal, characteristic, complemented
12#23554C211yesmaximal, characteristic, complemented
13#23489C212yesmaximal
14#23451C211yesmaximal, characteristic, complemented
15#21038C212yesmaximal, complemented
16#20878C212yesmaximal, complemented
17#20684C211yesmaximal, characteristic, complemented
18C2×#537C2211yescharacteristic
19C2×#533C2211yescharacteristic
20C2×#520C2211yescharacteristic
21C2×C4×#41C2211yescharacteristic
22C2×#428C2212yes—
23C22×#106C2212yes—
24C2×#417C2212yes—
25C2×#416C2212yes—
26C2×#408C2212yes—
27#2857C2212yes—
28#2857C2212yes—
29#2629C2214yes—
30#2582—21no—
31#2582—21no—
32#2538—21no—
33#2538—21no—
34#2417—22no—
35#2417—22no—
36#2364—22no—
37#2363—21no—
38#2362—21no—
39#2361—22no—
40#2361—21no—
41#2360—21no—
42#2324C2212yes—
43#2323C2212yes—
44#2316C2212yes—
45#2311C2211yescharacteristic
46#2311C2212yes—
47C4×C22⋊Q8C2212yes—
48C4×#121C2212yes—
49#2297C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#384C2211yescharacteristic
52#2274C2211yescharacteristic
53#2084C2212yes—
54#2084C2212yes—
55#2084C2214yes—
56#2077C2214yes—
57#2077C2214yes—
58#2076C2214yes—
59#2075C2214yes—
60#2072C2212yes—
61#2072C2212yes—
62#2071C2212yes—
63#2071C2212yes—
64#2059C2214yes—
65#2058C2212yes—
66#2057C2212yes—
67#2035C2214yes—
68#2035C2214yes—
69#2034C2214yes—
70#2034C2214yes—
71#2026C2212yes—
72#2025C2212yes—
73#2024C2211yescharacteristic
74#2024C2211yescharacteristic
75#2023C2211yescharacteristic
76#2023C2211yescharacteristic
77#1920C2214yes—
78#1920C2214yes—
79#1919C2214yes—
80#1919C2214yes—
81#1865C2214yes—
82#1864C2214yes—
83#1863C2214yes—
84#1862C2214yes—
85#1818C2214yes—
86#1817C2212yes—
87#1817C2212yes—
88#1736C2212yes—
89#1736C2212yes—
90#1735C2212yes—
91C2×#340C2212yes—
92#1729C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C22×#41C2311yescharacteristic
97C22×M16C2312yes—
98C23×C8C2312yes—
99#537—24no—
100#537—24no—
101#533—44no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×#125C2312yes—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8C2312yes—
111C2×#121—21no—
112C2×#121—22no—
113C2×#121C2312yes—
114#520—44no—
115C4×#41—44no—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8D411yescharacteristic
119C2×C4×Q8D411yescharacteristic
120C2×C4×Q8C2312yes—
121C2×C4×D4—22no—
122C2×C4×D4—22no—
123C2×C4×D4D412yes—
124C2×C4×D4D411yescharacteristic
125C2×C4×D4D411yescharacteristic
126C2×C4×D4D412yes—
127C2×C4×D4C2312yes—
128C2×#117—21no—
129C2×#117C2312yes—
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C2×#117C2312yes—
133C22×C4⋊C4—21no—
134C22×C4⋊C4—21no—
135C22×C4⋊C4—21no—
136C22×C4⋊C4—21no—
137C22×C4⋊C4C2312yes—
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2311yescharacteristic
140C22×C42C2311yescharacteristic
141C22×C42C2312yes—
142#428—44no—
143C2×#107C2314yes—
144C2×#106—28no—
145C2×#106C2314yes—
146C2×#106C2314yes—
147C2×#106C2314yes—
148#417—28no—
149#416—44no—
150C2×#103C2314yes—
151C2×#103C2312yes—
152C2×#103C2312yes—
153C2×#102C2314yes—
154C2×#102C2312yes—
155C2×#102C2312yes—
156#408—44no—
157C2×C22⋊C8C2314yes—
158C2×C22⋊C8C2314yes—
159#398—24no—
160#398—24no—
161#396—24no—
162#396—24no—
163#391—22no—
164#391—21no—
165#391—22no—
166#391—22no—
167#391C2311yescharacteristic
168#391C2311yescharacteristic
169#390—21no—
170#390—22no—
171#390—22no—
172#390C2311yescharacteristic
173#390C2311yescharacteristic
174#389—22no—
175#389—22no—
176#389—22no—
177#389—22no—
178#389—22no—
179#389—21no—
180#389—21no—
181#389C2312yes—
182#389C2312yes—
183#389C2314yes—
184#389C2314yes—
185#388—22no—
186#388—22no—
187#388—22no—
188#388—22no—
189#387—22no—
190#387—22no—
191#386—22no—
192#386—24no—
193#386—21no—
194#386—22no—
195#386—21no—
196#386—22no—
197#385—22no—
198#385—22no—
199#384—21no—
200#384—21no—
201#384C2312yes—
202#384C2312yes—
203#380—22no—
204#380—22no—
205#380—24no—
206#380C2312yes—
207#380C2312yes—
208#380C2314yes—
209#380C2314yes—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C22⋊C4—22no—
216C4×C22⋊C4—22no—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—21no—
220C2×#95—21no—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#336C2312yes—
232#336C2312yes—
233#336C2311yescharacteristic
234#336C2311yescharacteristic
235#336C2314yes—
236#336C2314yes—
237#336C2312yes—
238C2×#41—22no—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—24no—
242C2×#41—24no—
243C2×#41—24no—
244C2×#41—24no—
245C2×#41#4114yes—
246C22×Q8—21no—
247C22×Q8—21no—
248C22×Q8C2×D411yescharacteristic
249C22×Q8C2411yescharacteristic
250C22×D4—21no—
251C22×D4—22no—
252C22×D4C2×D412yes—
253C22×D4C2×D411yescharacteristic
254C22×D4C2411yescharacteristic
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—22no—
258C23×C4—21no—
259C23×C4—22no—
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D412yes—
263C23×C4C2×D412yes—
264C23×C4C2412yes—
265C23×C4C2411yescharacteristic
266C2×M16—24no—
267C2×M16—24no—
268C2×M16—24no—
269C22×C8—24no—
270C22×C8—24no—
271C22×C8—24no—
272C22×C8C2412yes—
273C22×C8C2412yes—
274C22×C8C2414yes—
275C4⋊Q8—42no—
276C4⋊Q8—42no—
277#125—42no—
278#125—42no—
279C22⋊Q8—28no—
280C22⋊Q8—24no—
281C22⋊Q8—24no—
282C22⋊Q8—44no—
283C22⋊Q8—44no—
284C22⋊Q8—84no—
285C22⋊Q8—84no—
286#121—216no—
287#121—28no—
288#121—24no—
289#121—24no—
290C4×Q8—42no—
291C4×Q8—42no—
292C4×Q8—44no—
293C4×Q8—44no—
294C4×Q8—44no—
295C4×D4—42no—
296C4×D4—42no—
297C4×D4—44no—
298C4×D4—44no—
299C4×D4—44no—
300C4×D4—48no—
301C4×D4—84no—
302#117—24no—
303#117—24no—
304#117—24no—
305#117—24no—
306#117—44no—
307#117—44no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—44no—
330C2×C4⋊C4—82no—
331C2×C4⋊C4—82no—
332C2×C4⋊C4—82no—
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2×D412yes—
336C2×C4⋊C4C2×D412yes—
337C2×C4⋊C4#4114yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2414yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2411yescharacteristic
343C2×C4⋊C4C2411yescharacteristic
344C2×C4⋊C4C2412yes—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4C2×D412yes—
359C2×C22⋊C4C2×D412yes—
360C2×C22⋊C4C2×D412yes—
361C2×C42—24no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—21no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—44no—
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D412yes—
371C2×C42C2×D412yes—
372C2×C42C2412yes—
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375#107—44no—
376#106—28no—
377#106—28no—
378#106—44no—
379#103—28no—
380#103—42no—
381#103—42no—
382#102—28no—
383#102—42no—
384#102—42no—
385C22⋊C8—44no—
386C22⋊C8—44no—
387#95—22no—
388#95—22no—
389#95—21no—
390#95—21no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—21no—
395#95—21no—
396#95—22no—
397#95—24no—
398#95—24no—
399#95—24no—
400#95—24no—
401C24—21no—
402C24C22×D412yes—
403C24C22×D411yescharacteristic
404#41—216no—
405#41—48no—
406#41—48no—
407#41—48no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—21no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—42no—
415C2×Q8—42no—
416C2×Q8—42no—
417C2×Q8—42no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q8C2×#4112yes—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—28no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—21no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—82no—
435C2×D4—82no—
436C2×D4C2×#4112yes—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—44no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—44no—
467C22×C4—42no—
468C22×C4—82no—
469C22×C4—82no—
470C22×C4—81no—
471C22×C4—81no—
472C22×C4—82no—
473C22×C4—82no—
474C22×C4—81no—
475C22×C4—81no—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4H(5,𝔽2)14yes—
487C22×C42−1+414yes—
488C22×C4C2511yescharacteristic
489M16—48no—
490C2×C8—24no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—24no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—24no—
499C2×C8—48no—
500C4⋊C4—28no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—24no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—44no—
523C4⋊C4—42no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—82no—
527C4⋊C4—82no—
528C4⋊C4—82no—
529C4⋊C4—84no—
530C4⋊C4—84no—
531C4⋊C4C2×#4114yes—
532C22⋊C4—28no—
533C22⋊C4—28no—
534C22⋊C4—24no—
535C22⋊C4—42no—
536C22⋊C4—42no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—42no—
540C22⋊C4—42no—
541C22⋊C4—42no—
542C22⋊C4—81no—
543C22⋊C4—81no—
544C22⋊C4—82no—
545C22⋊C4—82no—
546C22⋊C4—84no—
547C22⋊C4—84no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—24no—
553C42—42no—
554C42—42no—
555C42—44no—
556C42—44no—
557C42—42no—
558C42—82no—
559C42—82no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23#52512yes—
579C23#53412yes—
580C23#53612yes—
581C23C23×D411yescharacteristic
582Q8—24no—
583Q8—44no—
584Q8—42no—
585Q8—84no—
586Q8—84no—
587D4—24no—
588D4—48no—
589D4—48no—
590D4—44no—
591D4—42no—
592D4—88no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—21no—
612C2×C4—22no—
613C2×C4—21no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—48no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—84no—
642C2×C4—84no—
643C2×C4—84no—
644C2×C4—164no—
645C2×C4—162no—
646C2×C4—164no—
647C2×C4#52512yes—
648C2×C4#53412yes—
649C2×C4#53612yes—
650C2×C4#53918yes—
651C2×C4#54218yes—
652C2×C4#57712yes—
653C2×C4#57712yes—
654C2×C4C22×#4111yescharacteristic
655C2×C4C22×#4111yescharacteristic
656C2×C4C2×H(5,𝔽2)12yes—
657C2×C4C2×2−1+412yes—
658C8—44no—
659C8—42no—
660C8—42no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22—42no—
675C22—42no—
676C22—42no—
677C22—81no—
678C22—81no—
679C22—81no—
680C22—81no—
681C22#263814yes—
682C22C2×#52511yescharacteristic
683C22C2×#53411yescharacteristic
684C22C2×#53611yescharacteristic
685C22C2×#57711yescharacteristic
686C22C2×#57711yescharacteristic
687C22#358511yescharacteristic
688C22#358611yescharacteristic
689C4—22no—
690C4—22no—
691C4—42no—
692C4—42no—
693C4—41no—
694C4—42no—
695C4—41no—
696C4—42no—
697C4—44no—
698C4—82no—
699C4—82no—
700C4—82no—
701C4—84no—
702C4—81no—
703C4—81no—
704C4#263814yes—
705C4C2×#53912yes—
706C4C2×#54212yes—
707C2—24no—
708C2—82no—
709C2—82no—
710C2C2×#263811yescharacteristic
711C2#6059111yescharacteristic
712C2#6059311yescharacteristic
713C2#6076311yescharacteristic
714C2#6076511yescharacteristic
715C2#6088111yescharacteristic
716C2#6088711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#537C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35280C2semidirect
3. #30182C2semidirect
4. #30180C2semidirect
5. #30176C2semidirect
6. #30172C2semidirect
7. #30168C2semidirect
8. #30166C2semidirect
9. C2×#2084C2semidirect
10. #23643C2semidirect
11. #23630C2semidirect
12. #23554C2semidirect
14. #23451C2semidirect
15. #21038C2semidirect
16. #20878C2semidirect
17. #20684C2semidirect

Maximal subgroups

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

subgroupindex
#352802
#342672
#301822
#301802
#301762
#301722
#301682
#301662
C2×#20842
#236432
#236302
#235542
#234892
#234512
#210382
#208782
#206842