Finite Group Databasebeta

← #6623453

Subgroups — #6623453

Order 512.

1737 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36559C211yesmaximal, characteristic, complemented
2C4×#556C211yesmaximal, characteristic, complemented
3#33195C211yesmaximal, characteristic, complemented
4#33069C212yesmaximal, complemented
5#29141C211yesmaximal, characteristic, complemented
6#29131C212yesmaximal, complemented
7#28252C211yesmaximal, characteristic, complemented
8#28158C212yesmaximal
9#27446C214yesmaximal, complemented
10#26916C214yesmaximal, complemented
11#24993C212yesmaximal, complemented
12#24185C212yesmaximal, complemented
13#23002C212yesmaximal, complemented
14#22825C211yesmaximal, characteristic, complemented
15#22758C212yesmaximal, complemented
16#21505C212yesmaximal, complemented
17C4×#459C211yesmaximal, characteristic
18C2×#556C2211yescharacteristic
19C2×#554—21no—
20C2×#474C2212yes—
21C2×#460C2212yes—
22C2×#461C2212yes—
23C2×#459C2211yescharacteristic
24C2×#440C2214yes—
25C2×C4×D8C2212yes—
26#2868C2211yescharacteristic
27#2859C2212yes—
28#2566—22no—
29#2563—21no—
30#2538—21no—
31#2538C2212yes—
32#2452—22no—
33#2452C2214yes—
34#2373—22no—
35#2367C2212yes—
36#2365C2211yescharacteristic
37#2362—21no—
38#2356C2212yes—
39#2355C2211yescharacteristic
40#2354C2211yescharacteristic
41D4×C4⋊C4—21no—
42D4×C4⋊C4C2212yes—
43#2347—21no—
44C4×C4⋊Q8C2211yescharacteristic
45C4×#127C2212yes—
46C4×#127C2211yescharacteristic
47C4×#126—22no—
48C4×#124C2212yes—
49C4×C22⋊Q8—21no—
50C42×Q8C2212yes—
51C42×Q8C2211yescharacteristic
52C42×D4C2211yescharacteristic
53C4×#117—21no—
54#2101C2214yes—
55#2073C2214yes—
56#2055C2214yes—
57#2042C2214yes—
58#2024C2214yes—
59#2012C2214yes—
60#1988C2212yes—
61#1984C2212yes—
62#1982C2211yescharacteristic
63#1956C2212yes—
64#1947C2212yes—
65#1947C2211yescharacteristic
66#1947C2212yes—
67#1936C2214yes—
68#1935C2214yes—
69#1925C2214yes—
70#1925C2212yes—
71#1921C2214yes—
72#1921C2214yes—
73#1917C2214yes—
74#1880C2214yes—
75#1879C2212yes—
76#1879C2214yes—
77#1877C2214yes—
78#1863C2214yes—
79#1863C2214yes—
80#1842C2211yescharacteristic
81#1834C2212yes—
82#1831C2212yes—
83#1803C2212yes—
84#1803C2212yes—
85#1803C2212yes—
86#1802C2212yes—
87#1802C2212yes—
88#1802C2212yes—
89#1802C2211yescharacteristic
90#1770C2212yes—
91#1767C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93#1762C2214yes—
94C4×#102C2212yes—
95C22×SD16C2314yes—
96C22×D8C2312yes—
97#556—24no—
98#556—24no—
99#554—48no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×#127C2311yescharacteristic
105C2×#126—22no—
106C2×#125—22no—
107C2×#124C2312yes—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—24no—
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×D4—21no—
118C2×C4×D4—22no—
119C2×C4×D4—21no—
120C2×C4×D4—22no—
121C2×C4×D4C2312yes—
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×C4×D4C2314yes—
126C2×#117—21no—
127C2×#117—21no—
128C22×C4⋊C4—22no—
129C22×C42—21no—
130#474—44no—
131#461—28no—
132#460—28no—
133#459—24no—
134#440—48no—
135C4×D8—28no—
136C2×#107C2312yes—
137C2×#107C2314yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2311yescharacteristic
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2311yescharacteristic
142C2×C4⋊C8C2312yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#102C2314yes—
147C2×#102C2312yes—
148C2×#102C2312yes—
149C2×#102C2312yes—
150C2×#102C2312yes—
151C2×#102C2314yes—
152C2×#97C2314yes—
153C2×C4×C8C2312yes—
154#396—24no—
155#396—24no—
156#393—24no—
157#391—21no—
158#391C2312yes—
159#391C2312yes—
160#390—22no—
161#390—21no—
162#390—21no—
163#390—22no—
164#390C2312yes—
165#390C2314yes—
166#390C2312yes—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—22no—
171#389—22no—
172#389C2314yes—
173#389C2314yes—
174#388—22no—
175#388—22no—
176#388—22no—
177#388—22no—
178#388—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#385—22no—
184#385—24no—
185#385—24no—
186#384—21no—
187#384—21no—
188#384C2311yescharacteristic
189#384C2312yes—
190#384C2311yescharacteristic
191#384C2312yes—
192#383—22no—
193#382—22no—
194#382C2312yes—
195#380—22no—
196#380—22no—
197#380—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4D412yes—
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D412yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C22⋊C4—21no—
217C4×C22⋊C4—22no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—21no—
221C4×C22⋊C4—22no—
222#376—22no—
223C43—22no—
224C43C2311yescharacteristic
225C43C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2312yes—
233#336C2314yes—
234#336C2314yes—
235#336C2314yes—
236#336C2312yes—
237C22×Q8—22no—
238C22×Q8C2×D411yescharacteristic
239C22×Q8C2412yes—
240C22×D4—21no—
241C22×D4—21no—
242C22×D4C2411yescharacteristic
243C22×D4C2412yes—
244C23×C4—21no—
245C23×C4—22no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—24no—
249C2×SD16—28no—
250C2×SD16—28no—
251C2×SD16—44no—
252C2×D8—24no—
253C2×D8—24no—
254C2×D8—42no—
255C22×C8C2412yes—
256C22×C8C2414yes—
257C22×C8C2412yes—
258C4⋊Q8—22no—
259C4⋊Q8—42no—
260C4⋊Q8—42no—
261C4⋊Q8—44no—
262C4⋊Q8—82no—
263C4⋊Q8#4114yes—
264#127—22no—
265#127#4114yes—
266#126—216no—
267#125—48no—
268#124—24no—
269#124—24no—
270C22⋊Q8—28no—
271C22⋊Q8—416no—
272C22⋊Q8—48no—
273C4×Q8—24no—
274C4×Q8—24no—
275C4×Q8—24no—
276C4×Q8—22no—
277C4×Q8—22no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—24no—
281C4×Q8—42no—
282C4×Q8—42no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—24no—
286C4×D4—22no—
287C4×D4—24no—
288C4×D4—22no—
289C4×D4—41no—
290C4×D4—44no—
291C4×D4—48no—
292C4×D4—44no—
293C4×D4—44no—
294C4×D4—44no—
295C4×D4—84no—
296C4×D4#4114yes—
297#117—28no—
298#117—44no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—48no—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2414yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2411yescharacteristic
328C2×C22⋊C4—22no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—22no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—22no—
349C2×C42—44no—
350C2×C42C2×D412yes—
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D412yes—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D412yes—
355C2×C42C2×D412yes—
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2412yes—
359C2×C42C2411yescharacteristic
360C2×C42C2412yes—
361C2×C42C2411yescharacteristic
362C2×C42C2411yescharacteristic
363#107—24no—
364#107—28no—
365C4⋊C8—22no—
366C4⋊C8—24no—
367C4⋊C8—24no—
368C4⋊C8—22no—
369C4⋊C8—42no—
370#103—28no—
371#103—44no—
372#103—44no—
373#102—24no—
374#102—24no—
375#102—28no—
376#102—24no—
377#102—24no—
378#102—44no—
379#97—44no—
380C4×C8—24no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—21no—
385#95—21no—
386#95—21no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—24no—
393#95—24no—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397C24—21no—
398C24—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—44no—
408C2×Q8—82no—
409C2×Q8—82no—
410C2×Q8C2×#4114yes—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—22no—
414C2×D4—21no—
415C2×D4—22no—
416C2×D4—21no—
417C2×D4—22no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—21no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4C2×#4114yes—
428C2×D4C2×#4112yes—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—42no—
438C22×C4—48no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—44no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—42no—
448C22×C4—81no—
449C22×C4—84no—
450C22×C4—84no—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C2511yescharacteristic
462SD16—416no—
463D8—48no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—22no—
470C2×C8—24no—
471C2×C8—22no—
472C2×C8—22no—
473C4⋊C4—28no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—21no—
489C4⋊C4—22no—
490C4⋊C4—24no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—48no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—48no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—81no—
503C4⋊C4—82no—
504C4⋊C4—82no—
505C4⋊C4C2×#4112yes—
506C4⋊C4H(5,𝔽2)12yes—
507C22⋊C4—28no—
508C22⋊C4—28no—
509C22⋊C4—24no—
510C22⋊C4—42no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—48no—
514C22⋊C4—44no—
515C22⋊C4—42no—
516C22⋊C4—44no—
517C22⋊C4—82no—
518C22⋊C4—84no—
519C22⋊C4—84no—
520C42—28no—
521C42—24no—
522C42—22no—
523C42—22no—
524C42—24no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—21no—
529C42—22no—
530C42—21no—
531C42—42no—
532C42—42no—
533C42—41no—
534C42—44no—
535C42—44no—
536C42—44no—
537C42—42no—
538C42—42no—
539C42—42no—
540C42—82no—
541C42AGL(1,ℤ/8)14yes—
542C42C2×#4112yes—
543C42C2×#4112yes—
544C42C2×#4112yes—
545C422−1+412yes—
546C23—22no—
547C23—24no—
548C23—42no—
549C23—41no—
550C23—42no—
551C23—41no—
552C23—41no—
553C23—42no—
554C23—42no—
555C23—41no—
556C23—41no—
557C23—42no—
558C23—41no—
559C23—42no—
560C23C23×D411yescharacteristic
561Q8—24no—
562Q8—24no—
563Q8—44no—
564Q8—42no—
565Q8—42no—
566Q8—164no—
567D4—22no—
568D4—24no—
569D4—24no—
570D4—22no—
571D4—48no—
572D4—48no—
573D4—41no—
574D4—42no—
575D4—42no—
576D4—41no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—48no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—44no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—84no—
621C2×C4—84no—
622C2×C4—88no—
623C2×C4—168no—
624C2×C4#52512yes—
625C2×C4#52512yes—
626C2×C4#54812yes—
627C2×C4#54812yes—
628C2×C4C2×AGL(1,ℤ/8)12yes—
629C2×C4C2×AGL(1,ℤ/8)12yes—
630C2×C4C2×AGL(1,ℤ/8)12yes—
631C2×C4#57512yes—
632C2×C4#57812yes—
633C2×C4C22×#4111yescharacteristic
634C2×C4C22×#4111yescharacteristic
635C2×C4C22×#4111yescharacteristic
636C2×C4C2×H(5,𝔽2)11yescharacteristic
637C2×C4C2×2−1+411yescharacteristic
638C2×C4#58511yescharacteristic
639C8—42no—
640C8—44no—
641C8—42no—
642C22—42no—
643C22—44no—
644C22—44no—
645C22—42no—
646C22—44no—
647C22—42no—
648C22—81no—
649C22—82no—
650C22—81no—
651C22—82no—
652C22—82no—
653C22—81no—
654C22—81no—
655C22—82no—
656C22C2×#52511yescharacteristic
657C22C2×#54811yescharacteristic
658C22#348811yescharacteristic
659C22C22×AGL(1,ℤ/8)11yescharacteristic
660C22C2×#57511yescharacteristic
661C22C2×#57811yescharacteristic
662C22#358511yescharacteristic
663C4—22no—
664C4—22no—
665C4—22no—
666C4—42no—
667C4—42no—
668C4—42no—
669C4—41no—
670C4—42no—
671C4—42no—
672C4—44no—
673C4—42no—
674C4—84no—
675C4—82no—
676C4—81no—
677C4—82no—
678C4#311514yes—
679C4#336714yes—
680C4#347511yescharacteristic
681C4#348111yescharacteristic
682C4#351111yescharacteristic
683C4#351511yescharacteristic
684C2—82no—
685C2—84no—
686C2#3649211yescharacteristic
687C2C2×#311511yescharacteristic
688C2#6060011yescharacteristic
689C2C2×#336711yescharacteristic
690C2#6131811yescharacteristic
691C2#6195911yescharacteristic
692C2#6199811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#459C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36559C2semidirect
2. C4×#556C2semidirect
3. #33195C2semidirect
4. #33069C2semidirect
5. #29141C2semidirect
6. #29131C2semidirect
7. #28252C2semidirect
9. #27446C2semidirect
10. #26916C2semidirect
11. #24993C2semidirect
12. #24185C2semidirect
13. #23002C2semidirect
14. #22825C2semidirect
15. #22758C2semidirect
16. #21505C2semidirect

Maximal subgroups

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

subgroupindex
#365592
C4×#5562
#331952
#330692
#291412
#291312
#282522
#281582
#274462
#269162
#249932
#241852
#230022
#228252
#227582
#215052
C4×#4592