Finite Group Databasebeta

← #6630883

Subgroups — #6630883

Order 512.

1722 conjugacy classes of subgroups.

The subgroup lattice has 764 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35298C211yesmaximal, characteristic
2#35294C211yesmaximal, characteristic, complemented
3#33294C211yesmaximal, characteristic, complemented
4#33289C211yesmaximal, characteristic, complemented
5#33287C211yesmaximal, characteristic, complemented
6#33286C211yesmaximal, characteristic, complemented
7#33278C211yesmaximal, characteristic, complemented
8#33277C211yesmaximal, characteristic, complemented
9#33230C211yesmaximal, characteristic, complemented
10#33226C211yesmaximal, characteristic, complemented
11#33054C214yesmaximal, complemented
12#33047C214yesmaximal, complemented
13#21654C214yesmaximal, complemented
14#21643C214yesmaximal, complemented
15#21410C211yesmaximal, characteristic
16#21409C211yesmaximal, characteristic, complemented
17#21401C211yesmaximal, characteristic
18#21400C211yesmaximal, characteristic, complemented
19C4×#405C211yesmaximal, characteristic, complemented
20C2×#533C2211yescharacteristic
21C2×#532C2211yescharacteristic
22C2×#489C2214yes—
23C2×#488C2214yes—
24C2×#405C2212yes—
25C2×#405C2212yes—
26C2×C4×M16C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28#2868C2211yescharacteristic
29#2868C2211yescharacteristic
30#2862C2211yescharacteristic
31#2859C2211yescharacteristic
32#2859C2211yescharacteristic
33#2859C2211yescharacteristic
34#2858C2211yescharacteristic
35#2858C2211yescharacteristic
36#2629C2212yes—
37#2628C2212yes—
38#2620C2212yes—
39#2620C2212yes—
40#2570—21no—
41#2570—21no—
42#2566—21no—
43#2566—21no—
44#2550—21no—
45#2550—21no—
46#2545—21no—
47#2545—21no—
48#2500—21no—
49#2500—21no—
50#2441—21no—
51#2441—21no—
52#2439—21no—
53#2439—21no—
54#2432—21no—
55#2432—21no—
56#2331C2212yes—
57#2330C2212yes—
58#2326C2212yes—
59#2326C2212yes—
60#2285C2211yescharacteristic
61#2285C2211yescharacteristic
62C4×#117C2211yescharacteristic
63#1992C2214yes—
64#1990C2211yescharacteristic
65#1990C2211yescharacteristic
66#1990C2211yescharacteristic
67#1990C2211yescharacteristic
68#1990C2214yes—
69#1990C2211yescharacteristic
70#1990C2211yescharacteristic
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1959C2214yes—
74#1958C2214yes—
75#1957C2211yescharacteristic
76#1957C2211yescharacteristic
77#1957C2211yescharacteristic
78#1957C2211yescharacteristic
79#1956C2211yescharacteristic
80#1956C2211yescharacteristic
81#1956C2211yescharacteristic
82#1956C2211yescharacteristic
83#1936C2214yes—
84#1936C2214yes—
85#1935C2214yes—
86#1935C2214yes—
87#1934C2214yes—
88#1934C2214yes—
89#1933C2214yes—
90#1933C2214yes—
91#1895C2214yes—
92#1894C2214yes—
93#1893C2214yes—
94#1892C2214yes—
95#1836C2214yes—
96#1834C2214yes—
97#1763C2214yes—
98#1762C2214yes—
99C4×#103C2214yes—
100C4×#102C2214yes—
101#1726C2212yes—
102#1726C2212yes—
103C4×#97C2211yescharacteristic
104C4×#97C2211yescharacteristic
105C42×C8C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C22×#41—21no—
108C22×M16C2312yes—
109C23×C8C2312yes—
110#533—24no—
111#533—24no—
112#532—24no—
113#532—24no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8C2311yescharacteristic
119C2×C4⋊Q8C2312yes—
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2311yescharacteristic
122C2×#127C2311yescharacteristic
123C2×#125—21no—
124C2×#125—21no—
125C2×#125—21no—
126C2×#125C2312yes—
127C2×#124—21no—
128C2×#124—21no—
129C2×#124—21no—
130C2×#124C2312yes—
131C2×#124C2312yes—
132C2×C22⋊Q8—21no—
133C2×C22⋊Q8C2312yes—
134C2×#121—21no—
135C2×#121C2312yes—
136C2×C4×Q8—21no—
137C2×C4×Q8C2312yes—
138C2×C4×D4—21no—
139C2×C4×D4C2312yes—
140C2×#117—21no—
141C2×#117—21no—
142C2×#117—21no—
143C2×#117C2311yescharacteristic
144C2×#117C2311yescharacteristic
145C22×C42C2311yescharacteristic
146#489—216no—
147#488—216no—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156#405—28no—
157#405—28no—
158C4×M16—24no—
159C2×#97C2312yes—
160C2×#97C2312yes—
161C2×#97C2312yes—
162C2×#97C2312yes—
163C2×#97C2312yes—
164C2×C4×C8—24no—
165C2×C4×C8C2311yescharacteristic
166C2×C4×C8C2311yescharacteristic
167C2×C4×C8C2312yes—
168C2×C4×C8C2311yescharacteristic
169C2×C4×C8C2311yescharacteristic
170C2×C4×C8C2312yes—
171C2×C4×C8C2312yes—
172C2×C4×C8C2312yes—
173C2×C4×C8C2312yes—
174#400—22no—
175#399—22no—
176#398—22no—
177#398—22no—
178#396—22no—
179#396—22no—
180#396—22no—
181#396—22no—
182#396—22no—
183#396—22no—
184#395—22no—
185#394—22no—
186#394—22no—
187#394—22no—
188#394—22no—
189#393—22no—
190#391—21no—
191#391—21no—
192#391—22no—
193#391—22no—
194#391C2312yes—
195#391C2312yes—
196#390—21no—
197#390—21no—
198#390—21no—
199#390—21no—
200#390—21no—
201#390—21no—
202#390C2312yes—
203#390C2312yes—
204#389—21no—
205#389—21no—
206#389—21no—
207#389—21no—
208#389—21no—
209#389—21no—
210#389—21no—
211#389—21no—
212#389—22no—
213#389—22no—
214#389C2312yes—
215#389C2312yes—
216#389C2312yes—
217#389C2312yes—
218#388—22no—
219#388—22no—
220#388—22no—
221#388—22no—
222#387—22no—
223#387—22no—
224#387—22no—
225#387—22no—
226#385—22no—
227#385—22no—
228#385—22no—
229#385—22no—
230#384—21no—
231#384—21no—
232#384—21no—
233#384—21no—
234#384C2312yes—
235#384C2312yes—
236#384C2312yes—
237#384C2311yescharacteristic
238#384C2311yescharacteristic
239#384C2311yescharacteristic
240#384C2311yescharacteristic
241#382—21no—
242#382—21no—
243#382—21no—
244#382—21no—
245#382—21no—
246#382—21no—
247#382C2312yes—
248#380—21no—
249#380—21no—
250#380C2312yes—
251#380C2312yes—
252C4×C4⋊C4—21no—
253C4×C4⋊C4—21no—
254C4×C4⋊C4D411yescharacteristic
255C4×C4⋊C4D411yescharacteristic
256C4×C4⋊C4D411yescharacteristic
257C4×C4⋊C4D411yescharacteristic
258C4×C4⋊C4C2312yes—
259C4×C4⋊C4C2312yes—
260C4×C22⋊C4D411yescharacteristic
261C4×C22⋊C4D411yescharacteristic
262C4×C22⋊C4D411yescharacteristic
263C4×C22⋊C4D411yescharacteristic
264#376C2311yescharacteristic
265#376C2311yescharacteristic
266C43C2311yescharacteristic
267C43C2311yescharacteristic
268#340C2314yes—
269#340C2314yes—
270#340C2314yes—
271#340C2314yes—
272#340C2314yes—
273#340C2314yes—
274#340C2314yes—
275#340C2314yes—
276C2×#41—44no—
277C2×#41—82no—
278C2×#41—82no—
279C22×Q8—21no—
280C22×Q8—21no—
281C22×Q8—22no—
282C22×Q8C2412yes—
283C22×D4—22no—
284C22×D4—21no—
285C22×D4—21no—
286C22×D4C2412yes—
287C23×C4—22no—
288C23×C4C2×D411yescharacteristic
289C23×C4C2×D411yescharacteristic
290C23×C4C2411yescharacteristic
291C2×M16—24no—
292C2×M16—24no—
293C2×M16—24no—
294C22×C8—24no—
295C22×C8—24no—
296C22×C8—24no—
297C22×C8C2412yes—
298C22×C8C2412yes—
299C22×C8C2412yes—
300C22×C8C2412yes—
301C4⋊Q8—22no—
302C4⋊Q8—22no—
303C4⋊Q8—22no—
304C4⋊Q8—22no—
305C4⋊Q8—28no—
306C4⋊Q8—22no—
307C4⋊Q8—22no—
308C4⋊Q8—82no—
309C4⋊Q8—82no—
310C4⋊Q8—82no—
311C4⋊Q8—82no—
312#127—22no—
313#127—22no—
314#125—28no—
315#125—44no—
316#125—82no—
317#125—82no—
318#124—28no—
319#124—28no—
320#124—44no—
321#124—82no—
322#124—82no—
323C22⋊Q8—24no—
324C22⋊Q8—24no—
325C22⋊Q8—44no—
326#121—24no—
327#121—24no—
328#121—44no—
329C4×Q8—24no—
330C4×Q8—24no—
331C4×Q8—44no—
332C4×D4—24no—
333C4×D4—24no—
334C4×D4—44no—
335#117—24no—
336#117—24no—
337#117—44no—
338#117—82no—
339#117—82no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4C2×D411yescharacteristic
368C2×C4⋊C4C2×D411yescharacteristic
369C2×C4⋊C4C2×D411yescharacteristic
370C2×C4⋊C4C2×D411yescharacteristic
371C2×C4⋊C4C2412yes—
372C2×C4⋊C4C2412yes—
373C2×C4⋊C4C2412yes—
374C2×C4⋊C4C2412yes—
375C2×C4⋊C4C2412yes—
376C2×C4⋊C4C2412yes—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—22no—
386C2×C22⋊C4—22no—
387C2×C22⋊C4—22no—
388C2×C22⋊C4—22no—
389C2×C22⋊C4—22no—
390C2×C22⋊C4C2×D411yescharacteristic
391C2×C22⋊C4C2×D411yescharacteristic
392C2×C22⋊C4C2×D411yescharacteristic
393C2×C22⋊C4C2×D411yescharacteristic
394C2×C42—22no—
395C2×C42—22no—
396C2×C42—22no—
397C2×C42C2×D411yescharacteristic
398C2×C42C2×D411yescharacteristic
399C2×C42C2×D411yescharacteristic
400C2×C42C2×D411yescharacteristic
401C2×C42C2×D411yescharacteristic
402C2×C42C2×D411yescharacteristic
403C2×C42C2×D411yescharacteristic
404C2×C42C2×D411yescharacteristic
405C2×C42C2×D411yescharacteristic
406C2×C42C2×D411yescharacteristic
407C2×C42#4114yes—
408C2×C42#4114yes—
409C2×C42C2411yescharacteristic
410C2×C42C2411yescharacteristic
411C2×C42C2411yescharacteristic
412C2×C42C2411yescharacteristic
413C2×C42C2411yescharacteristic
414C2×C42C2411yescharacteristic
415#103—28no—
416#103—28no—
417#103—28no—
418#103—28no—
419#102—28no—
420#102—28no—
421#102—28no—
422#102—28no—
423#97—24no—
424#97—24no—
425#97—24no—
426#97—24no—
427#97#4118yescomplemented
428C4×C8—24no—
429C4×C8—24no—
430C4×C8—24no—
431C4×C8—22no—
432C4×C8—22no—
433C4×C8—22no—
434C4×C8—22no—
435C4×C8—24no—
436C4×C8#4118yescomplemented
437#95—22no—
438#95—22no—
439#95—22no—
440#95—22no—
441#95—22no—
442#95—22no—
443#95—22no—
444#95—22no—
445#95—22no—
446#95—22no—
447#95—22no—
448#95—22no—
449#95—22no—
450#95—22no—
451#95C2×D411yescharacteristic
452#95C2×D411yescharacteristic
453#95C2×D411yescharacteristic
454#95C2×D411yescharacteristic
455#95C2×D411yescharacteristic
456#95C2×D411yescharacteristic
457#95C2×D411yescharacteristic
458#95C2×D411yescharacteristic
459C24—22no—
460C24C22×D411yescharacteristic
461#41—816no—
462C2×Q8—24no—
463C2×Q8—24no—
464C2×Q8—22no—
465C2×Q8—22no—
466C2×Q8—42no—
467C2×Q8—42no—
468C2×Q8—44no—
469C2×Q8—42no—
470C2×Q8—42no—
471C2×Q8—42no—
472C2×Q8—42no—
473C2×Q8—82no—
474C2×Q8—82no—
475C2×D4—24no—
476C2×D4—24no—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—44no—
480C2×D4—42no—
481C2×D4—42no—
482C2×D4—42no—
483C2×D4—42no—
484C2×D4—42no—
485C2×D4—42no—
486C2×D4—82no—
487C2×D4—82no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—44no—
502C22×C4—82no—
503C22×C4—82no—
504C22×C4#12414yes—
505C22×C4#12414yes—
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D411yescharacteristic
517C22×C4C22×D411yescharacteristic
518C22×C4C22×D411yescharacteristic
519C22×C4C2×#4112yes—
520C22×C4C2×#4112yes—
521C22×C4C2511yescharacteristic
522M16—216no—
523C2×C8—216no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8C2×#4114yes—
533C2×C8C2×#4114yes—
534C2×C8C2×#4114yes—
535C2×C8C2×#4114yes—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—24no—
541C4⋊C4—24no—
542C4⋊C4—24no—
543C4⋊C4—24no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4—44no—
552C4⋊C4—44no—
553C4⋊C4—42no—
554C4⋊C4—44no—
555C4⋊C4—44no—
556C4⋊C4—44no—
557C4⋊C4—42no—
558C4⋊C4—42no—
559C4⋊C4—42no—
560C4⋊C4—42no—
561C4⋊C4—42no—
562C4⋊C4—42no—
563C4⋊C4—42no—
564C4⋊C4—42no—
565C4⋊C4—82no—
566C4⋊C4—82no—
567C4⋊C4—82no—
568C4⋊C4—82no—
569C4⋊C4—82no—
570C4⋊C4—82no—
571C4⋊C4—82no—
572C4⋊C4—82no—
573C4⋊C4—82no—
574C4⋊C4—82no—
575C22⋊C4—22no—
576C22⋊C4—22no—
577C22⋊C4—22no—
578C22⋊C4—22no—
579C22⋊C4—44no—
580C22⋊C4—44no—
581C22⋊C4—44no—
582C22⋊C4—44no—
583C22⋊C4—44no—
584C22⋊C4—44no—
585C22⋊C4—44no—
586C22⋊C4—82no—
587C22⋊C4—82no—
588C22⋊C4—82no—
589C22⋊C4—82no—
590C22⋊C4—82no—
591C22⋊C4—82no—
592C42—22no—
593C42—22no—
594C42—22no—
595C42—22no—
596C42—22no—
597C42—22no—
598C42—22no—
599C42—22no—
600C42—22no—
601C42—22no—
602C42—44no—
603C42—82no—
604C42—82no—
605C42#12414yes—
606C42#12414yes—
607C42C2×#4112yes—
608C42C2×#4112yes—
609C42C2×#4112yes—
610C42C2×#4112yes—
611C42C2×#4114yes—
612C42C2×#4114yes—
613C23—21no—
614C23—21no—
615C23—21no—
616C23—21no—
617C23—21no—
618C23—21no—
619C23—42no—
620C23—42no—
621C23—42no—
622C23—42no—
623C23—42no—
624C23—42no—
625C23—42no—
626C23C2×#12412yes—
627C23C23×D411yescharacteristic
628Q8—44no—
629Q8—44no—
630Q8—84no—
631Q8—84no—
632Q8—164no—
633D4—44no—
634D4—44no—
635D4—88no—
636D4—84no—
637D4—84no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—28no—
653C2×C4—28no—
654C2×C4—24no—
655C2×C4—24no—
656C2×C4—21no—
657C2×C4—21no—
658C2×C4—21no—
659C2×C4—21no—
660C2×C4—21no—
661C2×C4—21no—
662C2×C4—21no—
663C2×C4—21no—
664C2×C4—21no—
665C2×C4—21no—
666C2×C4—21no—
667C2×C4—21no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—88no—
690C2×C4C2×#12412yes—
691C2×C4C2×#12412yes—
692C2×C4C2×#12412yes—
693C2×C4C2×#12412yes—
694C2×C4C2×#12412yes—
695C2×C4#53212yes—
696C2×C4#53212yes—
697C2×C4#53212yes—
698C2×C4#53212yes—
699C2×C4#57612yes—
700C2×C4#57712yes—
701C2×C4#57712yes—
702C2×C4#57812yes—
703C2×C4C22×#4111yescharacteristic
704C2×C4C22×#4111yescharacteristic
705C2×C4C22×#4112yes—
706C2×C4C22×#4112yes—
707C8—24no—
708C8—24no—
709C8—24no—
710C8—24no—
711C22—22no—
712C22—22no—
713C22—22no—
714C22—41no—
715C22—41no—
716C22—41no—
717C22—41no—
718C22—82no—
719C22—82no—
720C22—82no—
721C22—82no—
722C22—82no—
723C22—82no—
724C22—82no—
725C22C22×#12411yescharacteristic
726C22C2×#53211yescharacteristic
727C22C2×#53211yescharacteristic
728C22C2×#57611yescharacteristic
729C22C2×#57711yescharacteristic
730C22C2×#57711yescharacteristic
731C22C2×#57811yescharacteristic
732C4—22no—
733C4—22no—
734C4—22no—
735C4—22no—
736C4—22no—
737C4—22no—
738C4—22no—
739C4—41no—
740C4—41no—
741C4—41no—
742C4—41no—
743C4—82no—
744C4—82no—
745C4—82no—
746C4—82no—
747C4—82no—
748C4—82no—
749C4—82no—
750C4#313614yes—
751C4#313714yes—
752C4#344412yes—
753C4#344412yes—
754C2—42no—
755C2—162no—
756C2#3663911yescharacteristic
757C2C2×#313611yescharacteristic
758C2C2×#313711yescharacteristic
759C2#6071611yescharacteristic
760C2#6071811yescharacteristic
761C2#6072211yescharacteristic
762C2#6072311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #35294C2semidirect
3. #33294C2semidirect
4. #33289C2semidirect
5. #33287C2semidirect
6. #33286C2semidirect
7. #33278C2semidirect
8. #33277C2semidirect
9. #33230C2semidirect
10. #33226C2semidirect
11. #33054C2semidirect
12. #33047C2semidirect
13. #21654C2semidirect
14. #21643C2semidirect
16. #21409C2semidirect
18. #21400C2semidirect
19. C4×#405C2semidirect
427. #97#41semidirect
436. C4×C8#41semidirect

Maximal subgroups

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

subgroupindex
#352982
#352942
#332942
#332892
#332872
#332862
#332782
#332772
#332302
#332262
#330542
#330472
#216542
#216432
#214102
#214092
#214012
#214002
C4×#4052