Finite Group Databasebeta

← #6514004

Subgroups — #6514004

Order 512.

1596 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34979C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic
3#27682C212yesmaximal, complemented
4#27649C212yesmaximal, complemented
5#27629C212yesmaximal, complemented
6#27628C212yesmaximal, complemented
7#26319C212yesmaximal, complemented
8#26301C212yesmaximal, complemented
9#26283C212yesmaximal, complemented
10#26231C212yesmaximal, complemented
11#26226C214yesmaximal, complemented
12#26154C214yesmaximal, complemented
13#21894C211yesmaximal, characteristic, complemented
14#21860C211yesmaximal, characteristic, complemented
15#21849C211yesmaximal, characteristic
16#21362C211yesmaximal, characteristic, complemented
17#21171C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×C4×#41C2211yescharacteristic
20C2×#485C2214yes—
21C2×#431C2211yescharacteristic
22C2×#428C2212yes—
23C2×#423C2211yescharacteristic
24#2611—22no—
25#2568C2211yescharacteristic
26#2547C2211yescharacteristic
27#2546C2211yescharacteristic
28#2542C2211yescharacteristic
29#2499—21no—
30#2432—21no—
31#2415C2212yes—
32#2415C2212yes—
33#2395—22no—
34#2355—21no—
35#2355C2212yes—
36#2354—21no—
37#2354C2212yes—
38#2352—22no—
39#2352C2214yes—
40#2314—22no—
41#2311—21no—
42C4×C4⋊Q8—21no—
43C4×#124—21no—
44C4×C22⋊Q8C2212yes—
45C4×C22⋊Q8C2212yes—
46#2285C2211yescharacteristic
47#2285C2212yes—
48#2285C2212yes—
49C2×#384C2211yescharacteristic
50#2274—21no—
51C4×#117C2211yescharacteristic
52#2101C2214yes—
53#2097C2214yes—
54#2085C2214yes—
55#2077C2214yes—
56#2076C2214yes—
57#2074C2214yes—
58#2036C2214yes—
59#2035C2214yes—
60#1990C2212yes—
61#1988C2212yes—
62#1984C2212yes—
63#1981C2212yes—
64#1956C2212yes—
65#1954C2212yes—
66#1948C2212yes—
67#1948C2211yescharacteristic
68#1948C2211yescharacteristic
69#1948C2212yes—
70#1947C2211yescharacteristic
71#1947C2212yes—
72#1947C2211yescharacteristic
73#1947C2212yes—
74#1928C2214yes—
75#1927C2214yes—
76#1921C2214yes—
77#1920C2214yes—
78#1894C2214yes—
79#1873C2214yes—
80#1872C2214yes—
81#1870C2214yes—
82#1854C2214yes—
83#1854C2212yes—
84#1854C2212yes—
85#1825C2214yes—
86#1803C2212yes—
87#1803C2212yes—
88#1803C2212yes—
89#1803C2212yes—
90#1797C2212yes—
91#1797C2212yes—
92C4×#107C2212yes—
93C4×#106C2212yes—
94#1736C2212yes—
95C22×#41D411yescharacteristic
96C22×M16C2312yes—
97#533—24no—
98#533—24no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×#125—21no—
103C2×#124—21no—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8C2312yes—
106C2×C22⋊Q8C2312yes—
107C2×C22⋊Q8C2312yes—
108C4×#41—24no—
109C4×#41—24no—
110C2×C4×Q8—21no—
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2311yescharacteristic
114C2×C4×D4D412yes—
115C2×C4×D4D411yescharacteristic
116C2×C4×D4C2312yes—
117C2×C4×D4C2311yescharacteristic
118C2×#117—21no—
119C2×#117—21no—
120C2×#117—22no—
121C2×#117D412yes—
122C2×#117D411yescharacteristic
123C2×#117C2311yescharacteristic
124C2×#117C2311yescharacteristic
125C2×#117C2314yes—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4C2312yes—
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131#485—48no—
132#431—24no—
133#428—44no—
134C2×#107C2312yes—
135C2×#107C2312yes—
136C2×#107C2314yes—
137C2×#106C2314yes—
138C2×#106C2312yes—
139C2×#106C2312yes—
140#423—24no—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2311yescharacteristic
143C2×C4⋊C8C2311yescharacteristic
144C2×C4⋊C8C2312yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2314yes—
151C2×C22⋊C8C2314yes—
152C2×C4×C8C2312yes—
153#400—22no—
154#399—22no—
155#398—22no—
156#397—22no—
157#397—22no—
158#397—22no—
159#394—22no—
160#394—22no—
161#391—21no—
162#391—22no—
163#391—21no—
164#391—21no—
165#391C2312yes—
166#391C2312yes—
167#390—22no—
168#390—21no—
169#389—22no—
170#389—22no—
171#389C2312yes—
172#389C2312yes—
173#388—24no—
174#388—24no—
175#388—22no—
176#388—22no—
177#387—24no—
178#387—24no—
179#387—22no—
180#387—22no—
181#386—24no—
182#385—22no—
183#385—22no—
184#384—21no—
185#384—21no—
186#384—21no—
187#384C2312yes—
188#384C2311yescharacteristic
189#384C2311yescharacteristic
190#384C2312yes—
191#382—21no—
192#382—21no—
193#382—22no—
194#382C2314yes—
195#380—22no—
196#380—22no—
197#380—21no—
198#380—22no—
199#380C2312yes—
200#380C2314yes—
201#380C2312yes—
202#380C2312yes—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2314yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C22⋊C4—21no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—21no—
221C4×C22⋊C4—24no—
222#376—21no—
223C43—21no—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#340C2314yes—
233#340C2312yes—
234#336C2314yes—
235#336C2314yes—
236C2×#41—24no—
237C2×#41—42no—
238C2×#41—42no—
239C22×Q8—21no—
240C22×Q8C2×D411yescharacteristic
241C22×Q8C2412yes—
242C22×D4C2×D411yescharacteristic
243C22×D4C2×D412yes—
244C23×C4—22no—
245C23×C4C2×D412yes—
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D412yes—
249C23×C4C2412yes—
250C23×C4C2411yescharacteristic
251C2×M16—24no—
252C2×M16—42no—
253C2×M16—42no—
254C22×C8C2414yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C4⋊Q8—22no—
258C4⋊Q8—22no—
259C4⋊Q8—22no—
260C4⋊Q8—22no—
261C4⋊Q8—44no—
262#125—44no—
263#124—44no—
264C22⋊Q8—24no—
265C22⋊Q8—24no—
266C22⋊Q8—24no—
267C22⋊Q8—24no—
268C22⋊Q8—42no—
269C22⋊Q8—42no—
270C22⋊Q8—44no—
271C4×Q8—24no—
272C4×Q8—22no—
273C4×Q8—22no—
274C4×Q8—24no—
275C4×Q8—24no—
276C4×Q8—44no—
277C4×D4—28no—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—24no—
281C4×D4—22no—
282C4×D4—22no—
283#117—24no—
284#117—28no—
285#117—22no—
286#117—22no—
287#117—41no—
288#117—41no—
289#117—44no—
290#117—44no—
291#117—44no—
292#117—44no—
293#117—84no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
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—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—42no—
313C2×C4⋊C4—42no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—82no—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2414yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2411yescharacteristic
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C42—24no—
342C2×C42—24no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—24no—
352C2×C42—24no—
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42#4114yes—
358C2×C42#4114yes—
359C2×C42C2412yes—
360C2×C42C2411yescharacteristic
361C2×C42C2411yescharacteristic
362#107—24no—
363#107—24no—
364#107—44no—
365#106—28no—
366#106—42no—
367#106—42no—
368C4⋊C8—24no—
369C4⋊C8—24no—
370C4⋊C8—22no—
371C4⋊C8—22no—
372#103—44no—
373#103—44no—
374#103—44no—
375#103—44no—
376C22⋊C8—24no—
377C22⋊C8—44no—
378C22⋊C8—44no—
379C4×C8—24no—
380#95—21no—
381#95—21no—
382#95—21no—
383#95—21no—
384#95—24no—
385#95—24no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—24no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—24no—
397#95—22no—
398C24C22×D412yes—
399C24C22×D411yescharacteristic
400#41—416no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—42no—
411C2×Q8—42no—
412C2×Q8—42no—
413C2×D4—24no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—42no—
418C2×D4—42no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—44no—
449C22×C4—82no—
450C22×C4—82no—
451C22×C4#12314yes—
452C22×C4#12314yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C2×#4112yes—
462C22×C4C2×#4112yes—
463C22×C4C2511yescharacteristic
464M16—48no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—24no—
468C2×C8—22no—
469C2×C8—24no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—24no—
473C2×C8—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—24no—
484C4⋊C4—21no—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—82no—
509C4⋊C4—82no—
510C4⋊C4H(5,𝔽2)12yes—
511C4⋊C42−1+412yes—
512C22⋊C4—24no—
513C22⋊C4—24no—
514C22⋊C4—24no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—84no—
524C22⋊C4—84no—
525C42—24no—
526C42—24no—
527C42—22no—
528C42—22no—
529C42—42no—
530C42—42no—
531C42—44no—
532C42—44no—
533C42—42no—
534C42—42no—
535C42—42no—
536C42—42no—
537C42—84no—
538C42—84no—
539C42#12314yes—
540C42#12314yes—
541C42C2×#4112yes—
542C42C2×#4112yes—
543C42C2×#4112yes—
544C42C2×#4112yes—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23C2×#12312yes—
559C23C23×D411yescharacteristic
560Q8—44no—
561Q8—42no—
562Q8—42no—
563Q8—42no—
564Q8—42no—
565Q8—84no—
566D4—48no—
567D4—44no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—48no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—48no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—164no—
623C2×C4C2×#12312yes—
624C2×C4C2×#12312yes—
625C2×C4C2×#12312yes—
626C2×C4C2×#12312yes—
627C2×C4C2×#12312yes—
628C2×C4#57512yes—
629C2×C4#57512yes—
630C2×C4#57712yes—
631C2×C4#57812yes—
632C2×C4C22×#4111yescharacteristic
633C2×C4C22×#4111yescharacteristic
634C2×C4C2×H(5,𝔽2)11yescharacteristic
635C2×C4C2×2−1+411yescharacteristic
636C2×C4#58511yescharacteristic
637C2×C4#58511yescharacteristic
638C8—44no—
639C8—42no—
640C8—42no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22C22×#12311yescharacteristic
656C22#348811yescharacteristic
657C22#348911yescharacteristic
658C22C2×#57511yescharacteristic
659C22C2×#57511yescharacteristic
660C22C2×#57711yescharacteristic
661C22C2×#57811yescharacteristic
662C4—22no—
663C4—22no—
664C4—22no—
665C4—42no—
666C4—42no—
667C4—42no—
668C4—42no—
669C4—42no—
670C4—82no—
671C4—81no—
672C4—84no—
673C4—82no—
674C4—82no—
675C4—82no—
676C4—81no—
677C4#309314yes—
678C4#309714yes—
679C4#350611yescharacteristic
680C4#350711yescharacteristic
681C4#351111yescharacteristic
682C4#351211yescharacteristic
683C2—42no—
684C2—82no—
685C2#3551711yescharacteristic
686C2C2×#309311yescharacteristic
687C2C2×#309711yescharacteristic
688C2#6198511yescharacteristic
689C2#6199711yescharacteristic
690C2#6213411yescharacteristic
691C2#6214611yescharacteristic
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. #27629C2semidirect
6. #27628C2semidirect
7. #26319C2semidirect
8. #26301C2semidirect
9. #26283C2semidirect
10. #26231C2semidirect
11. #26226C2semidirect
12. #26154C2semidirect
13. #21894C2semidirect
14. #21860C2semidirect
16. #21362C2semidirect
17. #21171C2semidirect

Maximal subgroups

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

subgroupindex
#349792
#341882
#276822
#276492
#276292
#276282
#263192
#263012
#262832
#262312
#262262
#261542
#218942
#218602
#218492
#213622
#211712