Finite Group Databasebeta

← #6437022

Subgroups — #6437022

Order 512.

1678 conjugacy classes of subgroups.

The subgroup lattice has 723 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36631C211yesmaximal, characteristic
2C4×#533C211yesmaximal, characteristic, complemented
3#30169C212yesmaximal, complemented
4#30150C212yesmaximal, complemented
5#30130C212yesmaximal, complemented
6#30108C212yesmaximal
7#30091C211yesmaximal, characteristic, complemented
8#30078C211yesmaximal, characteristic, complemented
9#30059C211yesmaximal, characteristic, complemented
10#30041C211yesmaximal, characteristic, complemented
11#30019C214yesmaximal, complemented
12#23783C212yesmaximal, complemented
13#23761C212yesmaximal, complemented
14#23684C212yesmaximal, complemented
15#23574C212yesmaximal, complemented
16#21124C214yesmaximal, complemented
17#20778C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#529—21no—
20C2×#440C2214yes—
21#2861—21no—
22#2860—21no—
23#2860C2212yes—
24#2859C2211yescharacteristic
25#2858C2211yescharacteristic
26#2857C2211yescharacteristic
27#2856C2211yescharacteristic
28#2854—21no—
29#2854C2212yes—
30#2850—21no—
31C4×C4⋊Q8C2211yescharacteristic
32C4×C4⋊Q8C2211yescharacteristic
33C4×C4⋊Q8C2211yescharacteristic
34C4×C4⋊Q8C2211yescharacteristic
35C4×C4⋊Q8C2211yescharacteristic
36C4×#127C2211yescharacteristic
37C4×#125—21no—
38C4×#125—21no—
39C4×#124—21no—
40C4×C22⋊Q8—21no—
41C4×C22⋊Q8—21no—
42C4×C22⋊Q8C2212yes—
43C4×C22⋊Q8C2212yes—
44C4×#121—21no—
45C4×#121C2212yes—
46#2296—21no—
47#2295—21no—
48#2295C2212yes—
49#2293—21no—
50#2293C2212yes—
51#2285C2211yescharacteristic
52C42×Q8—21no—
53C42×Q8C2212yes—
54C4×#117—21no—
55C2×C43C2211yescharacteristic
56#2099C2212yes—
57#2099C2212yes—
58#2096C2212yes—
59#2093C2212yes—
60#2088C2214yes—
61#2084C2212yes—
62#2083C2212yes—
63#2076C2214yes—
64#2072C2212yes—
65#2071C2212yes—
66#2068C2214yes—
67#2067C2214yes—
68#2057C2212yes—
69#2057C2212yes—
70#2055C2214yes—
71#2043C2212yes—
72#2041C2212yes—
73#2036C2214yes—
74#2034C2214yes—
75#2024C2212yes—
76#2023C2212yes—
77#2014C2212yes—
78#2013C2212yes—
79#2000C2214yes—
80#1926C2214yes—
81#1925C2214yes—
82#1921C2214yes—
83#1918C2214yes—
84#1880C2214yes—
85#1880C2214yes—
86#1865C2214yes—
87#1863C2214yes—
88#1839C2211yescharacteristic
89#1839C2211yescharacteristic
90#1831C2212yes—
91#1817C2212yes—
92#1797C2212yes—
93#1776C2214yes—
94#1767C2212yes—
95#1763C2214yes—
96#1762C2214yes—
97#1749C2212yes—
98#1745C2212yes—
99C22×SD16C2314yes—
100#533—28no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×#127C2311yescharacteristic
105#529—48no—
106C2×#126—22no—
107C2×#125—21no—
108C2×#125—21no—
109C2×#124—21no—
110C2×#124—21no—
111C2×#123—22no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8C2312yes—
115C2×#121—21no—
116C2×#121—21no—
117C2×#121C2312yes—
118C2×C4×Q8—21no—
119C2×C4×Q8—21no—
120C2×C4×Q8—21no—
121C2×C4×Q8—21no—
122C2×C4×Q8C2312yes—
123C2×C4×Q8C2312yes—
124C2×C4×Q8C2312yes—
125C2×C4×D4—22no—
126C2×C4×D4—21no—
127C2×C4×D4—21no—
128C2×C4×D4C2312yes—
129C2×#117—21no—
130C2×#117—21no—
131C2×#117—22no—
132C22×C42D411yescharacteristic
133C22×C42D411yescharacteristic
134C22×C42D411yescharacteristic
135C22×C42D411yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139#440—216no—
140C2×#107C2314yes—
141C2×#107C2314yes—
142C2×#106C2314yes—
143C2×C4⋊C8C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×C22⋊C8C2314yes—
151C2×#97C2314yes—
152#391—21no—
153#391—21no—
154#391—21no—
155#391—21no—
156#391C2312yes—
157#391C2312yes—
158#390—21no—
159#390—21no—
160#390—21no—
161#390—21no—
162#390—21no—
163#390—21no—
164#390C2312yes—
165#390C2312yes—
166#390C2312yes—
167#390C2312yes—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—21no—
178#389C2312yes—
179#389C2312yes—
180#389C2312yes—
181#389C2312yes—
182#389C2312yes—
183#389C2312yes—
184#388—22no—
185#388—22no—
186#388—22no—
187#388—22no—
188#388—22no—
189#388—22no—
190#388—22no—
191#388—22no—
192#387—22no—
193#387—22no—
194#387—22no—
195#387—22no—
196#386—22no—
197#386—22no—
198#385—22no—
199#385—22no—
200#384C2311yescharacteristic
201#384C2311yescharacteristic
202#384C2311yescharacteristic
203#384C2311yescharacteristic
204#382—21no—
205#382—21no—
206#380—21no—
207#380—21no—
208#380C2312yes—
209#380C2312yes—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4—22no—
223C4×C4⋊C4—22no—
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2312yes—
226C4×C4⋊C4C2312yes—
227C4×C4⋊C4C2312yes—
228C4×C22⋊C4—22no—
229C4×C22⋊C4—22no—
230C4×C22⋊C4—22no—
231C4×C22⋊C4—22no—
232#376—22no—
233C43—22no—
234C43D411yescharacteristic
235C43D411yescharacteristic
236C43D411yescharacteristic
237C43D411yescharacteristic
238C43C2311yescharacteristic
239C43C2311yescharacteristic
240C43C2311yescharacteristic
241C43C2311yescharacteristic
242#340C2314yes—
243#340C2314yes—
244#340C2314yes—
245#340C2314yes—
246#340C2314yes—
247#340C2314yes—
248#336C2312yes—
249#336C2312yes—
250#336C2312yes—
251#336C2312yes—
252#336C2312yes—
253#336C2314yes—
254#336C2312yes—
255C22×Q8—21no—
256C22×Q8—21no—
257C22×Q8C2412yes—
258C22×D4—22no—
259C22×D4—21no—
260C22×D4—21no—
261C22×D4C2412yes—
262C23×C4—22no—
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D411yescharacteristic
269C23×C4C2411yescharacteristic
270C2×SD16—28no—
271C2×SD16—44no—
272C2×SD16—44no—
273C22×C8C2414yes—
274C22×C8C2414yes—
275C4⋊Q8—24no—
276C4⋊Q8—24no—
277C4⋊Q8—42no—
278#127—42no—
279#126—48no—
280#125—28no—
281#125—44no—
282#124—44no—
283#124—44no—
284#123—48no—
285C22⋊Q8—28no—
286C22⋊Q8—24no—
287C22⋊Q8—24no—
288C22⋊Q8—44no—
289#121—42no—
290#121—42no—
291#121—44no—
292#121—44no—
293C4×Q8—28no—
294C4×Q8—24no—
295C4×Q8—44no—
296C4×Q8—44no—
297C4×Q8—44no—
298C4×Q8—44no—
299C4×Q8—44no—
300C4×Q8#4118yes—
301C4×D4—28no—
302C4×D4—44no—
303C4×D4—44no—
304C4×D4—48no—
305#117—28no—
306#117—48no—
307#117—44no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
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—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
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⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C42—24no—
350C2×C42—22no—
351C2×C42—24no—
352C2×C42—24no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—42no—
360C2×C42—42no—
361C2×C42—42no—
362C2×C42—42no—
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2411yescharacteristic
386C2×C42C2411yescharacteristic
387C2×C42C2411yescharacteristic
388C2×C42C2411yescharacteristic
389C2×C42C2411yescharacteristic
390C2×C42C2411yescharacteristic
391#107—28no—
392#107—44no—
393#106—44no—
394C4⋊C8—44no—
395#103—28no—
396#103—44no—
397#103—44no—
398#102—28no—
399#102—44no—
400#102—44no—
401C22⋊C8—44no—
402#97—28no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419C24—22no—
420C24C22×D411yescharacteristic
421C2×Q8—24no—
422C2×Q8—22no—
423C2×Q8—24no—
424C2×Q8—24no—
425C2×Q8—42no—
426C2×Q8—42no—
427C2×Q8—42no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8C2×#4114yes—
431C2×D4—24no—
432C2×D4—22no—
433C2×D4—24no—
434C2×D4—22no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—44no—
440C2×D4—44no—
441C2×D4—44no—
442C2×D4—42no—
443C2×D4—42no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4#13714yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C2511yescharacteristic
489SD16—416no—
490C2×C8—24no—
491C2×C8—24no—
492C2×C8—24no—
493C2×C8—24no—
494C2×C8—24no—
495C2×C8—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—28no—
499C4⋊C4—24no—
500C4⋊C4—28no—
501C4⋊C4—22no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—24no—
507C4⋊C4—22no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—44no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—44no—
525C4⋊C4C2×#4114yes—
526C22⋊C4—28no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C42—22no—
540C42—22no—
541C42—28no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—41no—
559C42—41no—
560C42—41no—
561C42—41no—
562C42—41no—
563C42—41no—
564C42—41no—
565C42—41no—
566C42—44no—
567C42—44no—
568C42—44no—
569C42—44no—
570C42—44no—
571C42—44no—
572C42AGL(1,ℤ/8)14yes—
573C42C2×#4114yes—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23C2×#13712yes—
588C23C23×D411yescharacteristic
589Q8—28no—
590Q8—48no—
591Q8—44no—
592Q8—84no—
593D4—44no—
594D4—44no—
595D4—84no—
596D4—84no—
597D4—88no—
598C2×C4—28no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—88no—
657C2×C4#52512yes—
658C2×C4#52512yes—
659C2×C4C2×AGL(1,ℤ/8)12yes—
660C2×C4C2×AGL(1,ℤ/8)12yes—
661C2×C4C2×AGL(1,ℤ/8)12yes—
662C2×C4C2×#13712yes—
663C2×C4C2×#13712yes—
664C2×C4#57512yes—
665C2×C4#57512yes—
666C2×C4C22×#4112yes—
667C2×C4#58512yes—
668C2×C4#58512yes—
669C8—44no—
670C8—44no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—41no—
675C22—41no—
676C22—41no—
677C22—41no—
678C22—82no—
679C22—82no—
680C22—82no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22C2×#52511yescharacteristic
686C22#344911yescharacteristic
687C22#344911yescharacteristic
688C22C22×AGL(1,ℤ/8)11yescharacteristic
689C22C22×#13711yescharacteristic
690C22C2×#57511yescharacteristic
691C22C2×#57511yescharacteristic
692C4—22no—
693C4—22no—
694C4—22no—
695C4—22no—
696C4—22no—
697C4—22no—
698C4—22no—
699C4—22no—
700C4—22no—
701C4—41no—
702C4—41no—
703C4—41no—
704C4—41no—
705C4—44no—
706C4—44no—
707C4—82no—
708C4—82no—
709C4—82no—
710C4—82no—
711C4—82no—
712C4#345514yes—
713C2—42no—
714C2—162no—
715C2#3463611yescharacteristic
716C2#6056011yescharacteristic
717C2#6056111yescharacteristic
718C2#6096011yescharacteristic
719C2#6096111yescharacteristic
720C2#6096511yescharacteristic
721C2#6096911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#533C2semidirect
3. #30169C2semidirect
4. #30150C2semidirect
5. #30130C2semidirect
7. #30091C2semidirect
8. #30078C2semidirect
9. #30059C2semidirect
10. #30041C2semidirect
11. #30019C2semidirect
12. #23783C2semidirect
13. #23761C2semidirect
14. #23684C2semidirect
15. #23574C2semidirect
16. #21124C2semidirect
17. #20778C2semidirect

Maximal subgroups

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

subgroupindex
#366312
C4×#5332
#301692
#301502
#301302
#301082
#300912
#300782
#300592
#300412
#300192
#237832
#237612
#236842
#235742
#211242
#207782