Finite Group Databasebeta

← #6615093

Subgroups — #6615093

Order 512.

1693 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36134C211yesmaximal, characteristic, complemented
2C4×#555C211yesmaximal, characteristic
3#33195C211yesmaximal, characteristic, complemented
4#33154C212yesmaximal, complemented
5#29219C211yesmaximal, characteristic, complemented
6#28284C211yesmaximal, characteristic, complemented
7#27173C214yesmaximal, complemented
8#26671C214yesmaximal, complemented
9#24970C212yesmaximal, complemented
10#24700C212yesmaximal, complemented
11#24153C212yesmaximal, complemented
12#24121C212yesmaximal
13#22965C212yesmaximal, complemented
14#22795C211yesmaximal, characteristic, complemented
15#22767C212yesmaximal, complemented
16C4×#461C211yesmaximal, characteristic, complemented
17#21389C212yesmaximal, complemented
18C2×#555C2211yescharacteristic
19C2×#533C2212yes—
20C2×#477C2212yes—
21C2×#474C2212yes—
22C2×#461C2211yescharacteristic
23C2×C4×SD16C2212yes—
24C2×#431C2212yes—
25#2868C2211yescharacteristic
26#2857C2212yes—
27#2552—21no—
28#2542—21no—
29#2538C2212yes—
30#2441—22no—
31#2429—22no—
32#2423C2214yes—
33#2372—22no—
34#2369C2211yescharacteristic
35#2365—21no—
36#2363—21no—
37#2363C2212yes—
38#2354—21no—
39#2354C2211yescharacteristic
40#2354C2211yescharacteristic
41Q8×C4⋊C4C2212yes—
42#2346—21no—
43#2346C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×#124—22no—
47C4×#121—21no—
48C4×#121C2212yes—
49C42×D4C2211yescharacteristic
50C42×D4C2211yescharacteristic
51C4×#117—21no—
52C4×#117C2212yes—
53#2099C2214yes—
54#2075C2214yes—
55#2068C2214yes—
56#2036C2214yes—
57#2033C2214yes—
58#2000C2214yes—
59#1990C2212yes—
60#1988C2212yes—
61#1988C2212yes—
62#1988C2212yes—
63#1981C2212yes—
64#1981C2211yescharacteristic
65#1954C2212yes—
66#1948C2211yescharacteristic
67#1947C2212yes—
68#1947C2212yes—
69#1947C2212yes—
70#1942C2214yes—
71#1934C2214yes—
72#1927C2212yes—
73#1923C2214yes—
74#1921C2214yes—
75#1920C2214yes—
76#1880C2214yes—
77#1877C2212yes—
78#1873C2214yes—
79#1873C2214yes—
80#1863C2214yes—
81#1854C2214yes—
82#1842C2211yescharacteristic
83#1834C2212yes—
84#1825C2214yes—
85#1803C2212yes—
86#1802C2212yes—
87#1802C2212yes—
88#1802C2211yescharacteristic
89#1797C2212yes—
90C8×C4⋊C4C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92C4×#103C2214yes—
93C4×#103C2214yes—
94C4×#103C2212yes—
95C22×SD16C2312yes—
96#555—24no—
97#555—24no—
98#533—28no—
99#533—44no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×#125—22no—
106C2×#124—22no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8C2314yes—
110C2×#121—21no—
111C2×#121C2312yes—
112C2×C4×Q8—22no—
113C2×C4×Q8—21no—
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2314yes—
116C2×C4×D4—22no—
117C2×C4×D4D411yescharacteristic
118C2×C4×D4D411yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×#117—22no—
124C2×#117—21no—
125C2×#117—21no—
126C2×#117C2312yes—
127C2×#117C2312yes—
128C22×C42—21no—
129C22×C42C2312yes—
130#477—28no—
131#474—44no—
132#461—24no—
133C4×SD16—28no—
134#431—44no—
135C2×#107C2314yes—
136C2×#107C2314yes—
137C2×#106C2312yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2311yescharacteristic
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2311yescharacteristic
142C2×C4⋊C8C2312yes—
143C2×#103C2314yes—
144C2×#103C2312yes—
145C2×#103C2312yes—
146C2×#103C2312yes—
147C2×#103C2314yes—
148C2×#102C2312yes—
149C2×#102C2314yes—
150C2×C22⋊C8C2314yes—
151C2×C22⋊C8C2314yes—
152C2×C4×C8C2312yes—
153C2×C4×C8C2314yes—
154#399—24no—
155#398—24no—
156#396—24no—
157#391—21no—
158#391—22no—
159#391—22no—
160#391C2312yes—
161#390—22no—
162#390—21no—
163#390—22no—
164#390C2312yes—
165#389—21no—
166#389—22no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—22no—
171#389—24no—
172#389C2312yes—
173#389C2314yes—
174#389C2312yes—
175#388—22no—
176#388—22no—
177#388—24no—
178#388—22no—
179#388—22no—
180#388—22no—
181#388—22no—
182#387—24no—
183#386—22no—
184#386—22no—
185#384—22no—
186#384—21no—
187#384—21no—
188#384C2311yescharacteristic
189#384C2311yescharacteristic
190#384C2312yes—
191#384C2312yes—
192#382—22no—
193#380—22no—
194#380—21no—
195#380C2312yes—
196#380C2314yes—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C22⋊C4—22no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4—21no—
215C4×C22⋊C4—21no—
216C4×C22⋊C4—22no—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4D412yes—
220C4×C22⋊C4D412yes—
221C4×C22⋊C4C2312yes—
222#376—22no—
223#376—22no—
224C43—22no—
225C43C2311yescharacteristic
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#336C2314yes—
236#336C2312yes—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8C2412yes—
240C22×D4—22no—
241C22×D4C2×D411yescharacteristic
242C22×D4C2411yescharacteristic
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4C2×D412yes—
247C23×C4C2×D412yes—
248C23×C4C2412yes—
249C2×SD16—24no—
250C2×SD16—24no—
251C2×SD16—42no—
252C22×C8C2412yes—
253C22×C8C2412yes—
254C22×C8C2414yes—
255C4⋊Q8—22no—
256C4⋊Q8—24no—
257C4⋊Q8—22no—
258C4⋊Q8—24no—
259C4⋊Q8—44no—
260C4⋊Q8—82no—
261C4⋊Q8#4114yes—
262C4⋊Q8#4114yes—
263#125—48no—
264#124—216no—
265C22⋊Q8—28no—
266C22⋊Q8—44no—
267C22⋊Q8—48no—
268C22⋊Q8—84no—
269#121—28no—
270#121—24no—
271#121—24no—
272C4×Q8—28no—
273C4×Q8—24no—
274C4×Q8—24no—
275C4×Q8—44no—
276C4×Q8—48no—
277C4×Q8—82no—
278C4×D4—24no—
279C4×D4—22no—
280C4×D4—24no—
281C4×D4—22no—
282C4×D4—22no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—41no—
287C4×D4—84no—
288C4×D4#4114yes—
289#117—28no—
290#117—24no—
291#117—24no—
292#117—24no—
293#117—42no—
294#117—48no—
295#117—44no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2414yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4C2×D412yes—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D412yes—
338C2×C42—22no—
339C2×C42—24no—
340C2×C42—24no—
341C2×C42—24no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42—22no—
350C2×C42—44no—
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2411yescharacteristic
358C2×C42C2412yes—
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361#107—28no—
362#107—44no—
363#106—24no—
364C4⋊C8—22no—
365C4⋊C8—24no—
366C4⋊C8—22no—
367C4⋊C8—24no—
368C4⋊C8—42no—
369#103—24no—
370#103—24no—
371#103—28no—
372#103—24no—
373#103—44no—
374#102—24no—
375#102—44no—
376C22⋊C8—44no—
377C22⋊C8—44no—
378C4×C8—24no—
379C4×C8—44no—
380#95—21no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—21no—
388#95—21no—
389#95—24no—
390#95—22no—
391#95—22no—
392#95—24no—
393#95C2×D412yes—
394#95C2×D412yes—
395#95C2×D412yes—
396C24—21no—
397C24C22×D412yes—
398C2×Q8—24no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—81no—
408C2×Q8—81no—
409C2×Q8C2×#4114yes—
410C2×D4—28no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—82no—
420C2×D4—82no—
421C2×D4C2×#4112yes—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—81no—
451C22×C4C22×D412yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C2511yescharacteristic
461SD16—48no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—24no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—24no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—24no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—21no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—82no—
497C4⋊C4—81no—
498C4⋊C4—82no—
499C4⋊C4—84no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C4⋊C4#13714yes—
503C4⋊C4C2×#4112yes—
504C4⋊C4C2×#4114yes—
505C4⋊C42−1+412yes—
506C22⋊C4—28no—
507C22⋊C4—28no—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—24no—
512C22⋊C4—42no—
513C22⋊C4—44no—
514C22⋊C4—48no—
515C22⋊C4—42no—
516C22⋊C4—44no—
517C22⋊C4—82no—
518C22⋊C4—82no—
519C22⋊C4—84no—
520C42—28no—
521C42—24no—
522C42—24no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—21no—
528C42—21no—
529C42—41no—
530C42—44no—
531C42—48no—
532C42—44no—
533C42—42no—
534C42—42no—
535C42—42no—
536C42—44no—
537C42—44no—
538C42—82no—
539C42#13514yes—
540C42C2×#4112yes—
541C42C2×#4112yes—
542C42C2×#4112yes—
543C42H(5,𝔽2)12yes—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23C23×D411yescharacteristic
559Q8—24no—
560Q8—24no—
561Q8—42no—
562Q8—42no—
563Q8—84no—
564Q8—88no—
565D4—22no—
566D4—22no—
567D4—44no—
568D4—41no—
569D4—41no—
570D4—88no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—48no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—48no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—48no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—44no—
619C2×C4—84no—
620C2×C4—84no—
621C2×C4#52512yes—
622C2×C4#52512yes—
623C2×C4#54612yes—
624C2×C4#54612yes—
625C2×C4C2×#13512yes—
626C2×C4C2×#13512yes—
627C2×C4C2×#13512yes—
628C2×C4C2×#13714yes—
629C2×C4C2×#13712yes—
630C2×C4#57612yes—
631C2×C4C22×#4111yescharacteristic
632C2×C4C22×#4111yescharacteristic
633C2×C4C22×#4111yescharacteristic
634C2×C4C2×H(5,𝔽2)11yescharacteristic
635C2×C4C2×2−1+411yescharacteristic
636C2×C4#58511yescharacteristic
637C8—42no—
638C8—42no—
639C8—44no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22C2×#52511yescharacteristic
655C22C2×#54611yescharacteristic
656C22#348911yescharacteristic
657C22C22×#13511yescharacteristic
658C22C22×#13711yescharacteristic
659C22C2×#57611yescharacteristic
660C22#358611yescharacteristic
661C4—22no—
662C4—22no—
663C4—22no—
664C4—24no—
665C4—42no—
666C4—44no—
667C4—42no—
668C4—42no—
669C4—44no—
670C4—42no—
671C4—41no—
672C4—82no—
673C4—81no—
674C4—84no—
675C4—82no—
676C4#310714yes—
677C4#330514yes—
678C4#347511yescharacteristic
679C4#348211yescharacteristic
680C4#351111yescharacteristic
681C4#351711yescharacteristic
682C2—82no—
683C2—82no—
684C2#3640411yescharacteristic
685C2C2×#310711yescharacteristic
686C2#6059611yescharacteristic
687C2C2×#330511yescharacteristic
688C2#6115711yescharacteristic
689C2#6209711yescharacteristic
690C2#6214911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#555C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36134C2semidirect
3. #33195C2semidirect
4. #33154C2semidirect
5. #29219C2semidirect
6. #28284C2semidirect
7. #27173C2semidirect
8. #26671C2semidirect
9. #24970C2semidirect
10. #24700C2semidirect
11. #24153C2semidirect
13. #22965C2semidirect
14. #22795C2semidirect
15. #22767C2semidirect
16. C4×#461C2semidirect
17. #21389C2semidirect

Maximal subgroups

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

subgroupindex
#361342
C4×#5552
#331952
#331542
#292192
#282842
#271732
#266712
#249702
#247002
#241532
#241212
#229652
#227952
#227672
C4×#4612
#213892