Finite Group Databasebeta

← #6513782

Subgroups — #6513782

Order 512.

1648 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34952C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic
3#27682C212yesmaximal, complemented
4#27649C212yesmaximal, complemented
5#27631C212yesmaximal, complemented
6#27630C212yesmaximal, complemented
7#25063C212yesmaximal, complemented
8#25017C212yesmaximal, complemented
9#24997C212yesmaximal, complemented
10#24972C212yesmaximal, complemented
11#24943C214yesmaximal, complemented
12#24871C214yesmaximal, complemented
13#21982C211yesmaximal, characteristic, complemented
14#21959C211yesmaximal, characteristic, complemented
15#21847C211yesmaximal, characteristic
16#20734C211yesmaximal, characteristic, complemented
17C2×#1803C211yesmaximal, characteristic, complemented
18C2×C4×#41C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C2×#485C2214yes—
21C2×#431C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C22×#103C2212yes—
24#2582C2211yescharacteristic
25#2582C2211yescharacteristic
26#2581—21no—
27#2580—21no—
28#2579—22no—
29#2538C2211yescharacteristic
30#2538C2211yescharacteristic
31#2487C2212yes—
32#2409C2212yes—
33#2396—22no—
34#2355—21no—
35#2355C2212yes—
36#2354—21no—
37#2354C2212yes—
38#2352—22no—
39#2352C2214yes—
40#2314—22no—
41#2311—21no—
42C4×C22⋊Q8C2212yes—
43C4×C22⋊Q8C2212yes—
44#2297—21no—
45#2296—21no—
46#2285C2211yescharacteristic
47#2285C2211yescharacteristic
48#2285C2212yes—
49#2285C2212yes—
50C2×#384C2211yescharacteristic
51#2274—21no—
52#2101C2214yes—
53#2097C2214yes—
54#2085C2214yes—
55#2077C2214yes—
56#2075C2214yes—
57#2074C2214yes—
58#2036C2214yes—
59#2035C2214yes—
60#1984C2212yes—
61#1954C2212yes—
62#1948C2212yes—
63#1948C2212yes—
64#1948C2212yes—
65#1948C2212yes—
66#1947C2212yes—
67#1947C2212yes—
68#1928C2214yes—
69#1927C2214yes—
70#1919C2214yes—
71#1918C2214yes—
72#1873C2214yes—
73#1869C2214yes—
74#1854C2214yes—
75#1852C2214yes—
76#1845C2212yes—
77#1834C2212yes—
78#1825C2214yes—
79#1824C2212yes—
80#1807C2214yes—
81#1803C2211yescharacteristic
82#1803C2212yes—
83#1803C2211yescharacteristic
84#1803C2212yes—
85#1803C2212yes—
86#1803C2212yes—
87#1803C2211yescharacteristic
88#1803C2211yescharacteristic
89#1797C2212yes—
90#1797C2212yes—
91#1787C2212yes—
92C4×#103C2212yes—
93C4×#103C2212yes—
94C2×#340C2212yes—
95C22×#41D411yescharacteristic
96C23×Q8C2311yescharacteristic
97C23×C8C2312yes—
98C2×C22⋊Q8C2312yes—
99C2×C22⋊Q8C2312yes—
100C4×#41—24no—
101C4×#41—24no—
102C2×C4×Q8—21no—
103C2×C4×Q8—21no—
104C2×C4×Q8—24no—
105C2×C4×Q8—24no—
106C2×C4×Q8D411yescharacteristic
107C2×C4×Q8C2311yescharacteristic
108C2×C4×Q8C2311yescharacteristic
109C2×C4×Q8C2311yescharacteristic
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2311yescharacteristic
112C2×C4×D4D412yes—
113C2×C4×D4D411yescharacteristic
114C2×C4×D4C2312yes—
115C2×C4×D4C2311yescharacteristic
116C2×#117—21no—
117C2×#117—21no—
118C2×#117—22no—
119C2×#117D412yes—
120C2×#117D411yescharacteristic
121C2×#117C2314yes—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2312yes—
127C22×C42—21no—
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130#485—48no—
131#431—24no—
132C2×#107C2314yes—
133C2×#107C2314yes—
134C2×#106C2314yes—
135C2×#106C2314yes—
136C2×C4⋊C8—24no—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2311yescharacteristic
139C2×C4⋊C8C2311yescharacteristic
140C2×C4⋊C8C2312yes—
141C2×#103—44no—
142C2×#103C2314yes—
143C2×#103C2312yes—
144C2×#103C2312yes—
145C2×#103C2314yes—
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×C22⋊C8C2312yes—
149C2×C22⋊C8C2314yes—
150C2×C22⋊C8C2314yes—
151C2×C4×C8C2312yes—
152#400—22no—
153#399—22no—
154#398—22no—
155#398—22no—
156#396—22no—
157#396—22no—
158#395—22no—
159#393—22no—
160#391—22no—
161#391—21no—
162#391—21no—
163#391C2312yes—
164#391C2312yes—
165#390—22no—
166#390—21no—
167#390—21no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389C2312yes—
173#389C2312yes—
174#388—24no—
175#388—24no—
176#387—24no—
177#387—24no—
178#387—22no—
179#387—22no—
180#386—24no—
181#386—22no—
182#386—22no—
183#385—22no—
184#385—22no—
185#385—22no—
186#385—22no—
187#384—21no—
188#384—21no—
189#384C2312yes—
190#384C2312yes—
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#384C2311yescharacteristic
194#384C2311yescharacteristic
195#382—21no—
196#382—21no—
197#382—22no—
198#382C2314yes—
199#380—22no—
200#380—22no—
201#380—21no—
202#380—21no—
203#380—22no—
204#380C2312yes—
205#380C2312yes—
206#380C2312yes—
207#380C2314yes—
208#380C2312yes—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4C2314yes—
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2312yes—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—24no—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#336C2314yes—
235#336C2314yes—
236C2×#41—24no—
237C2×#41—24no—
238C2×#41—24no—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—41no—
245C22×Q8—41no—
246C22×Q8C2×D411yescharacteristic
247C22×Q8C2411yescharacteristic
248C22×Q8C2411yescharacteristic
249C22×D4C2×D412yes—
250C22×D4C2×D411yescharacteristic
251C23×C4—21no—
252C23×C4—22no—
253C23×C4—21no—
254C23×C4C2×D412yes—
255C23×C4C2×D412yes—
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2412yes—
259C23×C4C2411yescharacteristic
260C22×C8—24no—
261C22×C8—42no—
262C22×C8—42no—
263C22×C8C2414yes—
264C22×C8C2412yes—
265C22×C8C2412yes—
266C22⋊Q8—24no—
267C22⋊Q8—24no—
268C22⋊Q8—42no—
269C22⋊Q8—42no—
270C4×Q8—24no—
271C4×Q8—22no—
272C4×Q8—22no—
273C4×Q8—22no—
274C4×Q8—22no—
275C4×Q8—22no—
276C4×Q8—22no—
277C4×Q8—22no—
278C4×Q8—22no—
279C4×Q8—22no—
280C4×Q8—22no—
281C4×Q8—44no—
282C4×Q8—44no—
283C4×D4—28no—
284C4×D4—24no—
285C4×D4—24no—
286C4×D4—24no—
287C4×D4—22no—
288C4×D4—22no—
289#117—24no—
290#117—28no—
291#117—44no—
292#117—44no—
293#117—44no—
294#117—44no—
295#117—84no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
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—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—42no—
321C2×C4⋊C4—42no—
322C2×C4⋊C4—44no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4—82no—
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2414yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2411yescharacteristic
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—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4C2×D412yes—
345C2×C22⋊C4C2×D412yes—
346C2×C22⋊C4C2×D412yes—
347C2×C42—24no—
348C2×C42—24no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—21no—
352C2×C42—21no—
353C2×C42—21no—
354C2×C42—21no—
355C2×C42—24no—
356C2×C42—24no—
357C2×C42—44no—
358C2×C42C2×D412yes—
359C2×C42C2×D412yes—
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42#4114yes—
363C2×C42#4114yes—
364C2×C42C2412yes—
365C2×C42C2411yescharacteristic
366C2×C42C2411yescharacteristic
367#107—44no—
368#107—44no—
369#106—44no—
370#106—44no—
371C4⋊C8—24no—
372C4⋊C8—24no—
373C4⋊C8—22no—
374C4⋊C8—22no—
375#103—28no—
376#103—24no—
377#103—24no—
378#103—44no—
379#103—42no—
380#103—42no—
381C22⋊C8—24no—
382C22⋊C8—44no—
383C22⋊C8—44no—
384C4×C8—24no—
385#95—24no—
386#95—24no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—24no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—24no—
400#95—22no—
401C24C22×D412yes—
402C24C22×D411yescharacteristic
403#41—416no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—28no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q8—44no—
421C2×Q8—44no—
422C2×Q8—44no—
423C2×Q8—42no—
424C2×Q8H(5,𝔽2)12yes—
425C2×Q82−1+412yes—
426C2×D4—24no—
427C2×D4—24no—
428C2×D4—24no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
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—24no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—41no—
456C22×C4—42no—
457C22×C4—41no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—44no—
464C22×C4—81no—
465C22×C4—81no—
466C22×C4—82no—
467C22×C4—82no—
468C22×C4#12314yes—
469C22×C4#12314yes—
470C22×C4#13514yes—
471C22×C4#13714yes—
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×C4C2×#4112yes—
482C22×C4C2511yescharacteristic
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—24no—
486C2×C8—22no—
487C2×C8—24no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—24no—
491C2×C8—22no—
492C2×C8—84no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—24no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—44no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—81no—
528C4⋊C4—81no—
529C4⋊C4—82no—
530C4⋊C4—82no—
531C4⋊C4—82no—
532C4⋊C4—82no—
533C22⋊C4—24no—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C22⋊C4—82no—
541C22⋊C4—82no—
542C22⋊C4—84no—
543C22⋊C4—84no—
544C42—24no—
545C42—24no—
546C42—22no—
547C42—22no—
548C42—42no—
549C42—42no—
550C42—42no—
551C42—42no—
552C42—44no—
553C42—44no—
554C42—84no—
555C42—84no—
556C42#12314yes—
557C42#12314yes—
558C42C2×#4112yes—
559C42C2×#4112yes—
560C42C2×#4112yes—
561C42C2×#4112yes—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23C2×#12312yes—
574C23C2×#13512yes—
575C23C2×#13712yes—
576C23C23×D411yescharacteristic
577Q8—22no—
578Q8—22no—
579Q8—22no—
580Q8—22no—
581Q8—48no—
582Q8—44no—
583Q8—41no—
584Q8—41no—
585Q8—41no—
586Q8—41no—
587D4—48no—
588D4—44no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—44no—
611C2×C4—48no—
612C2×C4—48no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—48no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—84no—
644C2×C4—164no—
645C2×C4#485116yescomplemented
646C2×C4C2×#12312yes—
647C2×C4C2×#12312yes—
648C2×C4C2×#12312yes—
649C2×C4C2×#12312yes—
650C2×C4C2×#12312yes—
651C2×C4C2×#13512yes—
652C2×C4C2×#13512yes—
653C2×C4C2×#13712yes—
654C2×C4C2×#13712yes—
655C2×C4C22×#4111yescharacteristic
656C2×C4C22×#4111yescharacteristic
657C2×C4C2×H(5,𝔽2)11yescharacteristic
658C2×C4C2×2−1+411yescharacteristic
659C2×C4#58511yescharacteristic
660C2×C4#58511yescharacteristic
661C8—42no—
662C8—42no—
663C8—44no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22—42no—
675C22—42no—
676C22—42no—
677C22C2×#48514yes—
678C22C22×#12311yescharacteristic
679C22#348811yescharacteristic
680C22#348911yescharacteristic
681C22C22×#13511yescharacteristic
682C22C22×#13711yescharacteristic
683C22#358511yescharacteristic
684C22#358611yescharacteristic
685C4—22no—
686C4—22no—
687C4—22no—
688C4—42no—
689C4—42no—
690C4—42no—
691C4—42no—
692C4—42no—
693C4—82no—
694C4—84no—
695C4—82no—
696C4—81no—
697C4—81no—
698C4—82no—
699C4—82no—
700C4C2×#48514yes—
701C4C2×#48514yes—
702C4#349211yescharacteristic
703C4#349311yescharacteristic
704C4#349411yescharacteristic
705C4#349511yescharacteristic
706C2—24no—
707C2—82no—
708C2#3551611yescharacteristic
709C2C22×#48511yescharacteristic
710C2#6051711yescharacteristic
711C2#6195311yescharacteristic
712C2#6196611yescharacteristic
713C2#6210111yescharacteristic
714C2#6211411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #27682C2semidirect
4. #27649C2semidirect
5. #27631C2semidirect
6. #27630C2semidirect
7. #25063C2semidirect
8. #25017C2semidirect
9. #24997C2semidirect
10. #24972C2semidirect
11. #24943C2semidirect
12. #24871C2semidirect
13. #21982C2semidirect
14. #21959C2semidirect
16. #20734C2semidirect
17. C2×#1803C2semidirect
645. C2×C4#485semidirect

Maximal subgroups

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

subgroupindex
#349522
#341882
#276822
#276492
#276312
#276302
#250632
#250172
#249972
#249722
#249432
#248712
#219822
#219592
#218472
#207342
C2×#18032