Finite Group Databasebeta

← #6593962

Subgroups — #6593962

Order 512.

2098 conjugacy classes of subgroups.

The subgroup lattice has 715 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36176C211yesmaximal, characteristic, complemented
2C2×D4×C4⋊C4C211yesmaximal, characteristic, complemented
3#33002C212yesmaximal
4#32956C211yesmaximal, characteristic, complemented
5#27960C211yesmaximal, characteristic, complemented
6#27644C211yesmaximal, characteristic, complemented
7#27616C212yesmaximal, complemented
8#27568C214yesmaximal, complemented
9#27310C214yesmaximal, complemented
10#26111C212yesmaximal, complemented
11#24812C212yesmaximal, complemented
12#24608C212yesmaximal, complemented
13#22969C212yesmaximal, complemented
14#22758C212yesmaximal, complemented
15#22603C211yesmaximal, characteristic, complemented
16C4×#447C211yesmaximal, characteristic
17#20739C212yesmaximal, complemented
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2212yes—
20C23×C4⋊C4—21no—
21C2×#480C2212yes—
22C2×#476C2212yes—
23C2×#447C2211yescharacteristic
24C2×C4×D8C2212yes—
25C2×#432C2212yes—
26#2857C2211yescharacteristic
27#2854C2212yes—
28#2571C2211yescharacteristic, complemented
29#2563C2211yescharacteristic, complemented
30#2532—22no—
31#2513C2212yescomplemented
32#2508C2212yescomplemented
33#2495—21no—
34#2488—22no—
35#2486—21no—
36#2369C2211yescharacteristic, complemented
37#2354C2211yescharacteristic, complemented
38#2352—21no—
39D4×C4⋊C4—21no—
40D4×C4⋊C4—21no—
41D4×C4⋊C4—22no—
42D4×C4⋊C4C2214yes—
43D4×C4⋊C4C2212yes—
44D4×C4⋊C4C2212yes—
45#2346—21no—
46C4×#121C2211yescharacteristic
47C2×#389C2212yes—
48C2×#385—22no—
49C2×#384C2211yescharacteristic
50#2283C2212yes—
51C2×#380—21no—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2101C2214yescomplemented
54#2099C2214yescomplemented
55#2087C2214yescomplemented
56#2085C2214yescomplemented
57#2083C2212yescomplemented
58#2071C2212yescomplemented
59#2065C2214yescomplemented
60#1990C2212yescomplemented
61#1947C2212yescomplemented
62#1944C2214yescomplemented
63#1939C2212yescomplemented
64#1936C2214yescomplemented
65#1921C2214yescomplemented
66#1920C2214yescomplemented
67#1917C2214yescomplemented
68#1879C2212yes—
69#1879C2214yes—
70#1879C2214yes—
71#1874C2211yescharacteristic
72#1864C2214yes—
73#1862C2212yes—
74#1862C2214yes—
75#1862C2214yes—
76#1853C2212yes—
77#1852C2211yescharacteristic
78#1834C2212yescomplemented
79#1823C2212yes—
80#1823C2212yes—
81#1823C2212yes—
82#1817C2212yes—
83#1804C2214yes—
84#1804C2214yes—
85#1802C2212yes—
86#1802C2212yescomplemented
87#1797C2211yescharacteristic
88#1786C2211yescharacteristic
89#1786C2212yes—
90#1770C2212yescomplemented
91#1762C2214yes—
92#1762C2214yes—
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×D4C2311yescharacteristic
96C24×C4—21no—
97C24×C4—22no—
98C22×D8C2312yes—
99C2×C4⋊Q8C2311yescharacteristic
100C2×#123—22no—
101C2×C22⋊Q8—21no—
102C2×#121C2311yescharacteristic
103C2×C4×D4—28no—
104C2×C4×D4—21no—
105C2×C4×D4—28no—
106C2×C4×D4—21no—
107C2×C4×D4—21no—
108C2×C4×D4—22no—
109C2×C4×D4—22no—
110C2×C4×D4—44no—
111C2×C4×D4C2311yescharacteristic
112C2×C4×D4C2312yes—
113C2×C4×D4C2314yes—
114C2×C4×D4C2314yes—
115C2×C4×D4C2312yes—
116C2×#117—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—22no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—22no—
121C22×C4⋊C4—22no—
122C22×C4⋊C4—24no—
123C22×C4⋊C4—48no—
124C22×C4⋊C4D411yescharacteristic
125C22×C4⋊C4D411yescharacteristic
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2311yescharacteristic
128C22×C22⋊C4—21no—
129C22×C22⋊C4—22no—
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132#480—44no—
133#476—28no—
134#447—24no—
135C4×D8—28no—
136#432—44no—
137C2×#107C2312yes—
138C2×#107C2314yes—
139C2×#106C2314yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2314yes—
142C2×#102C2314yes—
143C2×#102C2312yes—
144C2×#102C2312yes—
145C2×#102C2314yes—
146C2×#102C2312yes—
147C2×#102C2312yes—
148C2×#102C2314yes—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2311yescharacteristic
151C2×C22⋊C8C2311yescharacteristic
152C2×#97C2314yes—
153C2×C4×C8C2312yes—
154#400—22no—
155#400—22no—
156#398—22no—
157#397—22no—
158#397—22no—
159#393—22no—
160#391—21no—
161#391—22no—
162#390C2312yes—
163#390C2311yescharacteristic
164#390C2312yes—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—22no—
169#389—22no—
170#389C2312yes—
171#389C2314yes—
172#389C2311yescharacteristic
173#389C2312yes—
174#389C2312yes—
175#389C2314yes—
176#388—22no—
177#388—22no—
178#388—22no—
179#388—21no—
180#388—22no—
181#388—22no—
182#387—22no—
183#386—22no—
184#386—21no—
185#385—24no—
186#385—24no—
187#385—24no—
188#385—24no—
189#384—21no—
190#384—21no—
191#384C2311yescharacteristic
192#384C2312yes—
193#384C2312yes—
194#383—22no—
195#382—21no—
196#382—22no—
197#382C2312yes—
198#382C2312yes—
199#381—22no—
200#380—21no—
201#380—21no—
202#380—21no—
203#380—22no—
204#380—22no—
205#380—24no—
206#380—21no—
207#380C2312yes—
208#380C2311yescharacteristic
209C4×C4⋊C4—21no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D412yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—21no—
221#376—22no—
222C2×#95—22no—
223C2×#95D412yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#336C2312yes—
234#336C2314yes—
235#336C2314yes—
236#336C2312yes—
237C25—21no—
238C22×Q8—21no—
239C22×D4—24no—
240C22×D4—24no—
241C22×D4—21no—
242C22×D4—21no—
243C22×D4—22no—
244C22×D4#4114yes—
245C22×D4C2411yescharacteristic
246C22×D4C2412yes—
247C23×C4—28no—
248C23×C4—21no—
249C23×C4—22no—
250C23×C4—22no—
251C23×C4—22no—
252C23×C4—21no—
253C23×C4—24no—
254C23×C4—24no—
255C23×C4—44no—
256C23×C4—48no—
257C23×C4—48no—
258C23×C4—44no—
259C23×C4—84no—
260C23×C4C2×D412yes—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D412yes—
264C23×C4C2411yescharacteristic
265C2×D8—24no—
266C2×D8—24no—
267C2×D8—24no—
268C22×C8C2412yes—
269C22×C8C2414yes—
270C22×C8C2412yes—
271C4⋊Q8—24no—
272#123—84no—
273C22⋊Q8—82no—
274#121—22no—
275#121—22no—
276C4×D4—28no—
277C4×D4—28no—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—28no—
281C4×D4—28no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—44no—
285C4×D4—44no—
286C4×D4—48no—
287C4×D4—48no—
288C4×D4—44no—
289C4×D4#4118yes—
290#117—44no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—42no—
311C2×C4⋊C4—416no—
312C2×C4⋊C4—48no—
313C2×C4⋊C4—48no—
314C2×C4⋊C4—48no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—82no—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2414yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C22⋊C4—28no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—48no—
346C2×C42—24no—
347C2×C42—24no—
348C2×C42—24no—
349C2×C42—22no—
350C2×C42—21no—
351C2×C42—22no—
352C2×C42C2×D412yes—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D412yes—
355C2×C42C2×D411yescharacteristic
356C2×C42C2411yescharacteristic
357C2×C42C2412yes—
358C2×C42C2412yes—
359C2×C42C2411yescharacteristic
360#107—24no—
361#107—44no—
362#106—28no—
363C4⋊C8—24no—
364C4⋊C8—44no—
365#102—24no—
366#102—24no—
367#102—28no—
368#102—28no—
369#102—24no—
370#102—24no—
371#102—44no—
372C22⋊C8—22no—
373C22⋊C8—22no—
374C22⋊C8—24no—
375#97—44no—
376C4×C8—24no—
377#95—24no—
378#95—24no—
379#95—21no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—21no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—21no—
390#95—22no—
391#95—22no—
392#95C2×D412yes—
393#95C2×D412yes—
394#95C2×D412yes—
395#95C2×D412yes—
396C24—24no—
397C24—24no—
398C24—24no—
399C24—22no—
400C24—21no—
401C24—42no—
402C24—42no—
403C24—42no—
404C24—42no—
405C24C22×D411yescharacteristic
406C2×Q8—42no—
407C2×Q8—81no—
408C2×Q8—81no—
409C2×D4—24no—
410C2×D4—24no—
411C2×D4—216no—
412C2×D4—216no—
413C2×D4—24no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—21no—
417C2×D4—22no—
418C2×D4—21no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4C2×#4114yes—
425C2×D4C2×#4114yes—
426C2×D42−1+414yes—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—28no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—28no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—24no—
447C22×C4—41no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—44no—
451C22×C4—432no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—84no—
468C22×C4—84no—
469C22×C4—832no—
470C22×C4AGL(1,ℤ/8)14yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C2×#4112yes—
481C22×C4H(5,𝔽2)12yes—
482C22×C4C2511yescharacteristic
483D8—48no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—24no—
488C2×C8—22no—
489C2×C8—24no—
490C2×C8—24no—
491C2×C8—22no—
492C2×C8—22no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—21no—
506C4⋊C4—22no—
507C4⋊C4—24no—
508C4⋊C4—42no—
509C4⋊C4—48no—
510C4⋊C4—48no—
511C4⋊C4—42no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—82no—
518C4⋊C4—82no—
519C4⋊C4—81no—
520C4⋊C4—82no—
521C4⋊C4—82no—
522C4⋊C4—82no—
523C4⋊C4C2×#4114yes—
524C22⋊C4—28no—
525C22⋊C4—28no—
526C22⋊C4—48no—
527C22⋊C4—48no—
528C22⋊C4—44no—
529C22⋊C4—42no—
530C22⋊C4—44no—
531C22⋊C4—42no—
532C22⋊C4—42no—
533C22⋊C4—42no—
534C22⋊C4—81no—
535C22⋊C4—82no—
536C22⋊C4—82no—
537C22⋊C4—82no—
538C42—22no—
539C42—22no—
540C42—24no—
541C42—24no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—44no—
548C42—42no—
549C42—82no—
550C42C2×#4114yes—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—216no—
556C23—24no—
557C23—24no—
558C23—24no—
559C23—48no—
560C23—48no—
561C23—48no—
562C23—48no—
563C23—48no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—48no—
568C23—42no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23#54612yes—
577C23C2×AGL(1,ℤ/8)12yes—
578C23#57612yes—
579C23C23×D411yescharacteristic
580Q8—162no—
581D4—216no—
582D4—24no—
583D4—24no—
584D4—24no—
585D4—42no—
586D4—42no—
587D4—84no—
588D4#55218yes—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—28no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—21no—
614C2×C4—44no—
615C2×C4—416no—
616C2×C4—48no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—48no—
620C2×C4—416no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—44no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—41no—
638C2×C4—44no—
639C2×C4—84no—
640C2×C4—816no—
641C2×C4—816no—
642C2×C4—88no—
643C2×C4—84no—
644C2×C4#52512yes—
645C2×C4#52512yes—
646C2×C4#54612yes—
647C2×C4C2×AGL(1,ℤ/8)12yes—
648C2×C4C2×AGL(1,ℤ/8)12yes—
649C2×C4#57512yes—
650C2×C4C22×#4111yescharacteristic
651C2×C4C22×#4112yes—
652C2×C4C2×H(5,𝔽2)11yescharacteristic
653C2×C4C2×2−1+412yes—
654C8—42no—
655C8—42no—
656C8—44no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—28no—
664C22—432no—
665C22—44no—
666C22—44no—
667C22—44no—
668C22—44no—
669C22—44no—
670C22—44no—
671C22—81no—
672C22—81no—
673C22—81no—
674C22—81no—
675C22—81no—
676C22—81no—
677C22—81no—
678C22#332514yes—
679C22C2×#52511yescharacteristic
680C22C2×#54611yescharacteristic
681C22#348611yescharacteristic
682C22C22×AGL(1,ℤ/8)11yescharacteristic
683C22C2×#57511yescharacteristic
684C22C2×#57611yescharacteristic
685C22#358611yescharacteristic
686C4—22no—
687C4—22no—
688C4—22no—
689C4—44no—
690C4—41no—
691C4—42no—
692C4—42no—
693C4—42no—
694C4—42no—
695C4—81no—
696C4—81no—
697C4—82no—
698C4—82no—
699C4—84no—
700C4—81no—
701C4#311414yes—
702C4C2×#55212yes—
703C4#349212yes—
704C2—24no—
705C2—48no—
706C2—161no—
707C2#3616311yescharacteristic
708C2C2×#311411yescharacteristic
709C2#6058411yescharacteristic
710C2C2×#332511yescharacteristic
711C2#6115911yescharacteristic
712C2#6182711yescharacteristic
713C2#6183511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#447C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36176C2semidirect
2. C2×D4×C4⋊C4C2semidirect
4. #32956C2semidirect
5. #27960C2semidirect
6. #27644C2semidirect
7. #27616C2semidirect
8. #27568C2semidirect
9. #27310C2semidirect
10. #26111C2semidirect
11. #24812C2semidirect
12. #24608C2semidirect
13. #22969C2semidirect
14. #22758C2semidirect
15. #22603C2semidirect
17. #20739C2semidirect
28. #2571C22semidirect
29. #2563C22semidirect
31. #2513C22semidirect
32. #2508C22semidirect
36. #2369C22semidirect
37. #2354C22semidirect
53. #2101C22semidirect
54. #2099C22semidirect
55. #2087C22semidirect
56. #2085C22semidirect
57. #2083C22semidirect
58. #2071C22semidirect
59. #2065C22semidirect
60. #1990C22semidirect
61. #1947C22semidirect
62. #1944C22semidirect
63. #1939C22semidirect
64. #1936C22semidirect
65. #1921C22semidirect
66. #1920C22semidirect
67. #1917C22semidirect
78. #1834C22semidirect
86. #1802C22semidirect
90. #1770C22semidirect

Maximal subgroups

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

subgroupindex
#361762
C2×D4×C4⋊C42
#330022
#329562
#279602
#276442
#276162
#275682
#273102
#261112
#248122
#246082
#229692
#227582
#226032
C4×#4472
#207392