Finite Group Databasebeta

← #6631927

Subgroups — #6631927

Order 512.

1466 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#36635C211yesmaximal, characteristic, complemented
2#36631C211yesmaximal, characteristic
3#32853C212yesmaximal, complemented
4#32840C212yesmaximal, complemented
5#32836C212yesmaximal, complemented
6#32823C212yesmaximal, complemented
7#32787C212yesmaximal, complemented
8#32776C212yesmaximal
9#32742C212yesmaximal, complemented
10#32738C212yesmaximal, complemented
11#23839C211yesmaximal, characteristic, complemented
12#23839C211yesmaximal, characteristic, complemented
13#23822C211yesmaximal, characteristic, complemented
14#23822C211yesmaximal, characteristic, complemented
15#23784C214yesmaximal, complemented
16#23764C214yesmaximal, complemented
17#20778C211yesmaximal, characteristic, complemented
18C2×#529—21no—
19#2864C2212yes—
20#2863—21no—
21#2863—21no—
22#2862C2211yescharacteristic
23#2861—21no—
24#2860—21no—
25#2860C2212yes—
26#2859C2211yescharacteristic
27#2859C2211yescharacteristic
28#2858C2211yescharacteristic
29#2857C2211yescharacteristic
30#2857C2211yescharacteristic
31#2857C2211yescharacteristic
32#2856C2211yescharacteristic
33#2855—21no—
34#2855—21no—
35#2854—21no—
36#2854C2212yes—
37#2851C2212yes—
38#2850—21no—
39C4×C4⋊Q8C2211yescharacteristic
40C4×C4⋊Q8C2211yescharacteristic
41C4×C4⋊Q8C2211yescharacteristic
42C4×#127C2211yescharacteristic
43C4×#125C2212yes—
44C4×#124—21no—
45C4×C22⋊Q8—21no—
46C4×C22⋊Q8—21no—
47C4×#121—21no—
48C4×#121C2212yes—
49#2295—21no—
50#2295—21no—
51#2293—21no—
52#2293C2212yes—
53#2283C2212yes—
54C2×C43C2211yescharacteristic
55#2101C2214yes—
56#2101C2214yes—
57#2100C2212yes—
58#2100C2212yes—
59#2092C2214yes—
60#2092C2214yes—
61#2088C2214yes—
62#2086C2214yes—
63#2085C2214yes—
64#2084C2212yes—
65#2084C2212yes—
66#2083C2212yes—
67#2083C2212yes—
68#2077C2214yes—
69#2075C2214yes—
70#2073C2214yes—
71#2071C2212yes—
72#2071C2212yes—
73#2067C2214yes—
74#2065C2214yes—
75#2059C2214yes—
76#2058C2212yes—
77#2058C2212yes—
78#2057C2212yes—
79#2057C2212yes—
80#2041C2212yes—
81#2041C2212yes—
82#2034C2214yes—
83#2033C2214yes—
84#2025C2214yes—
85#2023C2212yes—
86#2023C2212yes—
87#1839C2212yes—
88#1831C2212yes—
89#1817C2212yes—
90#1797C2212yes—
91#1776C2214yes—
92#1776C2214yes—
93#1767C2212yes—
94#1762C2214yes—
95#1762C2214yes—
96#1749C2212yes—
97#1745C2211yescharacteristic
98#1745C2211yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×#127C2311yescharacteristic
103#529—48no—
104C2×#126—22no—
105C2×#125—21no—
106C2×#125C2312yes—
107C2×#124—21no—
108C2×#124—21no—
109C2×#123—22no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—21no—
113C2×#121—21no—
114C2×#121—21no—
115C2×#121C2312yes—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8—21no—
119C2×C4×D4—22no—
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4C2312yes—
123C2×#117—22no—
124C22×C42D411yescharacteristic
125C22×C42D411yescharacteristic
126C22×C42D411yescharacteristic
127C22×C42D411yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C2×#107C2314yes—
132C2×#107C2314yes—
133C2×#106C2314yes—
134C2×#106C2314yes—
135C2×C4⋊C8C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×C22⋊C8C2314yes—
141C2×#97C2312yes—
142C2×#97C2312yes—
143#391—21no—
144#391—21no—
145#391—21no—
146#391—21no—
147#391—21no—
148#391—21no—
149#391—21no—
150#391—21no—
151#391—21no—
152#391—21no—
153#390—21no—
154#390—21no—
155#390—21no—
156#390—21no—
157#390—21no—
158#390—21no—
159#390C2312yes—
160#390C2312yes—
161#390C2312yes—
162#390C2312yes—
163#389—21no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389C2312yes—
176#389C2312yes—
177#389C2312yes—
178#389C2312yes—
179#389C2312yes—
180#389C2312yes—
181#389C2312yes—
182#389C2312yes—
183#388—22no—
184#388—22no—
185#388—22no—
186#388—22no—
187#388—22no—
188#388—22no—
189#388—22no—
190#388—22no—
191#387—22no—
192#387—22no—
193#387—22no—
194#387—22no—
195#387—22no—
196#387—22no—
197#387—22no—
198#387—22no—
199#386—22no—
200#386—22no—
201#385—22no—
202#385—22no—
203#384C2311yescharacteristic
204#384C2311yescharacteristic
205#384C2311yescharacteristic
206#384C2311yescharacteristic
207#383—22no—
208#382—21no—
209#382C2312yes—
210#382C2312yes—
211#382C2312yes—
212#381—22no—
213#380—21no—
214#380—21no—
215#380—21no—
216#380C2312yes—
217#380C2312yes—
218#380C2312yes—
219C4×C4⋊C4—22no—
220C4×C4⋊C4—22no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4—21no—
223C4×C4⋊C4—21no—
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2312yes—
226C4×C4⋊C4C2312yes—
227C4×C22⋊C4—22no—
228C4×C22⋊C4—22no—
229C4×C22⋊C4—22no—
230#376—22no—
231C43D411yescharacteristic
232C43D411yescharacteristic
233C43D411yescharacteristic
234C43D411yescharacteristic
235C43C2311yescharacteristic
236C43C2311yescharacteristic
237C43C2311yescharacteristic
238C43C2311yescharacteristic
239#340C2314yes—
240#340C2314yes—
241#340C2314yes—
242#340C2314yes—
243#340C2314yes—
244#340C2314yes—
245#340C2314yes—
246#340C2314yes—
247#336C2312yes—
248#336C2312yes—
249#336C2312yes—
250#336C2312yes—
251#336C2314yes—
252#336C2314yes—
253C22×Q8—21no—
254C22×Q8—21no—
255C22×Q8—21no—
256C22×D4—22no—
257C22×D4—21no—
258C22×D4—21no—
259C22×D4C2412yes—
260C23×C4—22no—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2411yescharacteristic
268C22×C8C2414yes—
269C22×C8C2414yes—
270C4⋊Q8—42no—
271C4⋊Q8—42no—
272C4⋊Q8—42no—
273#127—42no—
274#126—48no—
275#125—44no—
276#125—44no—
277#124—44no—
278#124—44no—
279#123—48no—
280C22⋊Q8—44no—
281C22⋊Q8—44no—
282C22⋊Q8—44no—
283#121—44no—
284#121—44no—
285#121—44no—
286C4×Q8—44no—
287C4×Q8—44no—
288C4×Q8—44no—
289C4×D4—28no—
290C4×D4—44no—
291C4×D4—44no—
292C4×D4—48no—
293#117—48no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—42no—
341C2×C42—42no—
342C2×C42—42no—
343C2×C42—42no—
344C2×C42—42no—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2411yescharacteristic
368C2×C42C2411yescharacteristic
369C2×C42C2411yescharacteristic
370C2×C42C2411yescharacteristic
371C2×C42C2411yescharacteristic
372C2×C42C2411yescharacteristic
373#107—44no—
374#107—44no—
375#106—44no—
376#106—44no—
377C4⋊C8—44no—
378#102—44no—
379#102—44no—
380#102—44no—
381#102—44no—
382C22⋊C8—44no—
383#97—24no—
384#97—24no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405C24—22no—
406C24C22×D411yescharacteristic
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q8—42no—
411C2×Q8—42no—
412C2×Q8—42no—
413C2×Q8—42no—
414C2×Q8—42no—
415C2×Q8—42no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—42no—
428C2×D4—42no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4AGL(1,ℤ/8)14yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C2511yescharacteristic
474C2×C8—24no—
475C2×C8—24no—
476C2×C8—24no—
477C2×C8—24no—
478C2×C8—24no—
479C2×C8—24no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—41no—
542C42—41no—
543C42—41no—
544C42—41no—
545C42—41no—
546C42—41no—
547C42—41no—
548C42—41no—
549C42—41no—
550C42—41no—
551C42—44no—
552C42—44no—
553C42—44no—
554C422−1+414yes—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23C2×AGL(1,ℤ/8)12yes—
569C23C23×D411yescharacteristic
570Q8—84no—
571Q8—84no—
572Q8—84no—
573D4—44no—
574D4—44no—
575D4—84no—
576D4—84no—
577D4—88no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
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—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—88no—
639C2×C4#53612yes—
640C2×C4#53612yes—
641C2×C4C2×AGL(1,ℤ/8)12yes—
642C2×C4C2×AGL(1,ℤ/8)12yes—
643C2×C4#57512yes—
644C2×C4#57512yes—
645C2×C4#57512yes—
646C2×C4C2×2−1+412yes—
647C2×C4#58512yes—
648C2×C4#58512yes—
649C8—44no—
650C8—44no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—41no—
655C22—41no—
656C22—41no—
657C22—41no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22C2×#53611yescharacteristic
666C22#344911yescharacteristic
667C22#344911yescharacteristic
668C22C22×AGL(1,ℤ/8)11yescharacteristic
669C22C2×#57511yescharacteristic
670C22C2×#57511yescharacteristic
671C22C2×#57511yescharacteristic
672C4—22no—
673C4—22no—
674C4—22no—
675C4—22no—
676C4—22no—
677C4—22no—
678C4—22no—
679C4—41no—
680C4—41no—
681C4—41no—
682C4—41no—
683C4—41no—
684C4—41no—
685C4—82no—
686C4—82no—
687C4—82no—
688C4—82no—
689C4—82no—
690C4—82no—
691C4—82no—
692C4#353114yes—
693C2—42no—
694C2—162no—
695C2#3669911yescharacteristic
696C2#6086211yescharacteristic
697C2#6086411yescharacteristic
698C2#6095811yescharacteristic
699C2#6096211yescharacteristic
700C2#6097011yescharacteristic
701C2#6097011yescharacteristic
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
1. #36635C2semidirect
3. #32853C2semidirect
4. #32840C2semidirect
5. #32836C2semidirect
6. #32823C2semidirect
7. #32787C2semidirect
9. #32742C2semidirect
10. #32738C2semidirect
11. #23839C2semidirect
12. #23839C2semidirect
13. #23822C2semidirect
14. #23822C2semidirect
15. #23784C2semidirect
16. #23764C2semidirect
17. #20778C2semidirect

Maximal subgroups

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

subgroupindex
#366352
#366312
#328532
#328402
#328362
#328232
#327872
#327762
#327422
#327382
#238392
#238222
#237842
#237642
#207782