Finite Group Databasebeta

← #6849041

Subgroups — #6849041

Order 512.

1695 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36180C211yesmaximal, characteristic, complemented
2#34934C211yesmaximal, characteristic
3#33198C211yesmaximal, characteristic, complemented
4#31754C214yesmaximal, complemented
5#31132C214yesmaximal, complemented
6#29692C212yesmaximal, complemented
7#29139C211yesmaximal, characteristic, complemented
8#28798C212yesmaximal, complemented
9#28280C211yesmaximal, characteristic, complemented
10#27758C212yesmaximal, complemented
11#27644C212yesmaximal, complemented
12#26230C212yesmaximal
13#24969C212yesmaximal, complemented
14#24708C212yesmaximal, complemented
15#22799C211yesmaximal, characteristic, complemented
16#21867C212yesmaximal, complemented
17C4×#477C211yesmaximal, characteristic, complemented
18C2×#554—21no—
19C2×#529C2212yes—
20C2×#480C2214yes—
21C2×#477C2211yescharacteristic
22C2×#477C2212yes—
23C2×#476C2212yes—
24C2×#432C2212yes—
25#2868C2211yescharacteristic
26#2857C2212yes—
27#2567—21no—
28#2554—22no—
29#2552—21no—
30#2552C2212yes—
31#2552C2211yescharacteristic
32#2548C2211yescharacteristic
33#2541C2211yescharacteristic
34#2538C2211yescharacteristic
35#2512—21no—
36#2502—22no—
37#2490C2212yes—
38#2445—21no—
39#2440—22no—
40#2440C2214yes—
41#2426C2212yes—
42#2392—22no—
43D4×C4⋊C4—21no—
44D4×C4⋊C4C2212yes—
45C4×C4⋊Q8C2211yescharacteristic
46C4×#127—21no—
47C4×#127C2211yescharacteristic
48C4×#121C2212yes—
49#2293—21no—
50C2×#384C2212yes—
51C42×Q8C2211yescharacteristic
52C42×D4C2211yescharacteristic
53#2083C2214yes—
54#2077C2214yes—
55#2071C2214yes—
56#2065C2214yes—
57#2059C2214yes—
58#2042C2214yes—
59#2035C2214yes—
60#2033C2214yes—
61#2033C2214yes—
62#2023C2214yes—
63#1992C2214yes—
64#1987C2212yes—
65#1987C2211yescharacteristic
66#1982C2212yes—
67#1959C2214yes—
68#1957C2212yes—
69#1954C2212yes—
70#1948C2212yes—
71#1947C2211yescharacteristic
72#1947C2212yes—
73#1947C2212yes—
74#1943C2214yes—
75#1936C2214yes—
76#1924C2214yes—
77#1921C2212yes—
78#1874C2214yes—
79#1872C2214yes—
80#1863C2212yes—
81#1842C2211yescharacteristic
82#1834C2212yes—
83#1823C2214yes—
84#1803C2211yescharacteristic
85#1803C2212yes—
86#1803C2212yes—
87#1802C2212yes—
88#1802C2212yes—
89#1802C2212yes—
90#1797C2212yes—
91C4×#107C2212yes—
92#1770C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95#554—48no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2312yes—
100C2×#127—21no—
101C2×#127C2311yescharacteristic
102#529—28no—
103#529—44no—
104C2×#126—22no—
105C2×#125—22no—
106C2×#125C2312yes—
107C2×#124C2312yes—
108C2×#123—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—24no—
111C2×C22⋊Q8C2312yes—
112C2×#121C2312yes—
113C2×#121C2312yes—
114C2×C4×Q8—21no—
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2311yescharacteristic
118C2×C4×D4—22no—
119C2×C4×D4—22no—
120C2×C4×D4—22no—
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4D411yescharacteristic
124C2×C4×D4D411yescharacteristic
125C2×C4×D4C2312yes—
126C2×C4×D4C2311yescharacteristic
127C2×#117—22no—
128C22×C4⋊C4—22no—
129C22×C4⋊C4—22no—
130C22×C4⋊C4C2314yes—
131C22×C42C2312yes—
132#480—48no—
133#477—24no—
134#477—44no—
135#476—44no—
136#432—44no—
137C2×#107C2312yes—
138C2×#107C2314yes—
139C2×#107C2312yes—
140C2×#106C2314yes—
141C2×C4⋊C8C2311yescharacteristic
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2311yescharacteristic
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2312yes—
146C2×#103C2312yes—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×C22⋊C8C2314yes—
154C2×C22⋊C8C2314yes—
155C2×#97C2314yes—
156C2×C4×C8C2312yes—
157#398—22no—
158#398—24no—
159#398—24no—
160#397—22no—
161#396—22no—
162#396—22no—
163#395—24no—
164#394—22no—
165#394—22no—
166#394—22no—
167#392—22no—
168#391—21no—
169#391—22no—
170#390—21no—
171#390—21no—
172#390—21no—
173#390—21no—
174#390—22no—
175#390—22no—
176#390—21no—
177#390—21no—
178#390C2312yes—
179#390C2312yes—
180#389—22no—
181#389—22no—
182#389—21no—
183#389—21no—
184#389—21no—
185#389—22no—
186#389C2312yes—
187#389C2314yes—
188#389C2312yes—
189#388—22no—
190#388—22no—
191#388—22no—
192#386—22no—
193#385—24no—
194#385—24no—
195#384—21no—
196#384—22no—
197#384—21no—
198#384—21no—
199#384—22no—
200#384C2311yescharacteristic
201#384C2311yescharacteristic
202#384C2312yes—
203#384C2312yes—
204#384C2314yes—
205#380—22no—
206#380C2312yes—
207#380C2314yes—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C22⋊C4—22no—
220C4×C22⋊C4D412yes—
221C4×C22⋊C4D412yes—
222C4×C22⋊C4C2312yes—
223C43—21no—
224C43C2311yescharacteristic
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#340C2312yes—
233#340C2312yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8C2411yescharacteristic
239C22×D4—22no—
240C22×D4—22no—
241C22×D4—21no—
242C22×D4—21no—
243C22×D4C2×D411yescharacteristic
244C22×D4C2412yes—
245C23×C4—24no—
246C23×C4—22no—
247C23×C4C2×D412yes—
248C23×C4C2×D412yes—
249C23×C4C2412yes—
250C22×C8C2412yes—
251C22×C8C2412yes—
252C22×C8C2414yes—
253C4⋊Q8—22no—
254C4⋊Q8—22no—
255C4⋊Q8—42no—
256C4⋊Q8—42no—
257C4⋊Q8—44no—
258C4⋊Q8—82no—
259#127—22no—
260#127—22no—
261#127—44no—
262#126—48no—
263#125—24no—
264#125—42no—
265#125—48no—
266#124—24no—
267#124—42no—
268#123—48no—
269C22⋊Q8—24no—
270C22⋊Q8—416no—
271C22⋊Q8—42no—
272C22⋊Q8—48no—
273#121—24no—
274#121—24no—
275#121—42no—
276#121—42no—
277C4×Q8—22no—
278C4×Q8—22no—
279C4×Q8—22no—
280C4×Q8—22no—
281C4×Q8—41no—
282C4×Q8—44no—
283C4×Q8#4114yes—
284C4×D4—24no—
285C4×D4—24no—
286C4×D4—24no—
287C4×D4—24no—
288C4×D4—22no—
289C4×D4—22no—
290C4×D4—48no—
291C4×D4—48no—
292C4×D4—44no—
293C4×D4—82no—
294C4×D4—84no—
295#117—48no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—48no—
321C2×C4⋊C4—84no—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2414yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2411yescharacteristic
332C2×C22⋊C4—24no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4C2×D412yes—
345C2×C22⋊C4C2×D412yes—
346C2×C22⋊C4C2×D412yes—
347C2×C42—21no—
348C2×C42—24no—
349C2×C42—24no—
350C2×C42—21no—
351C2×C42—22no—
352C2×C42—21no—
353C2×C42—21no—
354C2×C42—21no—
355C2×C42—21no—
356C2×C42C2×D412yes—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D412yes—
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2411yescharacteristic
363C2×C42C2412yes—
364C2×C42C2411yescharacteristic
365C2×C42C2411yescharacteristic
366#107—24no—
367#107—28no—
368#107—24no—
369#106—44no—
370C4⋊C8—22no—
371C4⋊C8—22no—
372C4⋊C8—24no—
373C4⋊C8—24no—
374C4⋊C8—42no—
375#103—24no—
376#103—42no—
377#103—44no—
378#102—44no—
379#102—44no—
380#102—44no—
381#102—44no—
382C22⋊C8—44no—
383C22⋊C8—44no—
384#97—44no—
385C4×C8—24no—
386#95—22no—
387#95—22no—
388#95—21no—
389#95—21no—
390#95—21no—
391#95—22no—
392#95—22no—
393#95—24no—
394#95—24no—
395#95C2×D412yes—
396#95C2×D412yes—
397#95C2×D412yes—
398C24—22no—
399C24C22×D412yes—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—44no—
408C2×Q8—42no—
409C2×Q8—82no—
410C2×Q8—82no—
411C2×Q8C2×#4112yes—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—81no—
429C2×D4—81no—
430C2×D4—82no—
431C2×D4—82no—
432C2×D4AGL(1,ℤ/8)14yes—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—42no—
453C22×C4—48no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—84no—
458C22×C4—84no—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C2511yescharacteristic
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—24no—
472C2×C8—22no—
473C2×C8—24no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—24no—
477C2×C8—22no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—48no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—48no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—44no—
509C4⋊C4—84no—
510C4⋊C4—84no—
511C4⋊C4—81no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C4—82no—
515C4⋊C4C2×#4112yes—
516C4⋊C4H(5,𝔽2)12yes—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—48no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—82no—
529C22⋊C4—82no—
530C22⋊C4—82no—
531C22⋊C4—82no—
532C42—24no—
533C42—24no—
534C42—24no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—21no—
540C42—21no—
541C42—41no—
542C42—42no—
543C42—42no—
544C42—44no—
545C42—42no—
546C42—42no—
547C42—81no—
548C42#13714yes—
549C42C2×#4112yes—
550C42H(5,𝔽2)12yes—
551C422−1+412yes—
552C422−1+412yes—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23C2×AGL(1,ℤ/8)14yes—
567C23C23×D411yescharacteristic
568Q8—22no—
569Q8—22no—
570Q8—41no—
571Q8—41no—
572Q8—84no—
573Q8—164no—
574D4—28no—
575D4—42no—
576D4—42no—
577D4—42no—
578D4—42no—
579D4—84no—
580D4—88no—
581D4—88no—
582D4—84no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—48no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—48no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—88no—
632C2×C4—168no—
633C2×C4#53612yes—
634C2×C4#53612yes—
635C2×C4#54612yes—
636C2×C4#54612yes—
637C2×C4C2×AGL(1,ℤ/8)12yes—
638C2×C4C2×#13712yes—
639C2×C4C2×#13712yes—
640C2×C4C2×#13712yes—
641C2×C4#57612yes—
642C2×C4C22×#4111yescharacteristic
643C2×C4C2×H(5,𝔽2)11yescharacteristic
644C2×C4C2×H(5,𝔽2)11yescharacteristic
645C2×C4C2×2−1+411yescharacteristic
646C2×C4C2×2−1+411yescharacteristic
647C2×C4#58511yescharacteristic
648C8—42no—
649C8—42no—
650C8—44no—
651C22—24no—
652C22—24no—
653C22—24no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22C2×#53611yescharacteristic
666C22C2×#54611yescharacteristic
667C22#348811yescharacteristic
668C22C22×AGL(1,ℤ/8)11yescharacteristic
669C22C22×#13711yescharacteristic
670C22C2×#57611yescharacteristic
671C22#358611yescharacteristic
672C4—22no—
673C4—22no—
674C4—22no—
675C4—41no—
676C4—42no—
677C4—42no—
678C4—42no—
679C4—42no—
680C4—42no—
681C4—42no—
682C4—82no—
683C4—84no—
684C4—81no—
685C4—82no—
686C4—82no—
687C4#323814yes—
688C4#333314yes—
689C4#350711yescharacteristic
690C4#351111yescharacteristic
691C4#354711yescharacteristic
692C4#354811yescharacteristic
693C2—44no—
694C2—162no—
695C2#3947011yescharacteristic
696C2C2×#323811yescharacteristic
697C2#6088811yescharacteristic
698C2C2×#333311yescharacteristic
699C2#6116111yescharacteristic
700C2#6196411yescharacteristic
701C2#6197911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2541C22
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. #36180C2semidirect
3. #33198C2semidirect
4. #31754C2semidirect
5. #31132C2semidirect
6. #29692C2semidirect
7. #29139C2semidirect
8. #28798C2semidirect
9. #28280C2semidirect
10. #27758C2semidirect
11. #27644C2semidirect
13. #24969C2semidirect
14. #24708C2semidirect
15. #22799C2semidirect
16. #21867C2semidirect
17. C4×#477C2semidirect

Maximal subgroups

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

subgroupindex
#361802
#349342
#331982
#317542
#311322
#296922
#291392
#287982
#282802
#277582
#276442
#262302
#249692
#247082
#227992
#218672
C4×#4772