Finite Group Databasebeta

← #6849211

Subgroups — #6849211

Order 512.

1611 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36188C211yesmaximal, characteristic
2#34929C211yesmaximal, characteristic
3#33196C211yesmaximal, characteristic, complemented
4#32065C214yesmaximal, complemented
5#31856C214yesmaximal, complemented
6#29664C212yesmaximal, complemented
7#28717C211yesmaximal, characteristic, complemented
8#28417C212yesmaximal, complemented
9#28255C211yesmaximal, characteristic, complemented
10#27823C212yesmaximal, complemented
11#27682C212yesmaximal, complemented
12#25570C212yesmaximal
13#24660C212yesmaximal, complemented
14#24123C212yesmaximal
15#22799C211yesmaximal, characteristic
16#21870C212yesmaximal, complemented
17C4×#462C211yesmaximal, characteristic, complemented
18C2×#556C2212yes—
19C2×#533C2212yes—
20C2×#487C2212yes—
21C2×#475C2212yes—
22C2×#462C2211yescharacteristic
23C2×#462C2212yes—
24C2×#462C2212yes—
25C2×C4×Q16C2212yes—
26C2×#432C2212yes—
27#2868C2211yescharacteristic
28#2857C2212yes—
29#2568C2212yes—
30#2556—22no—
31#2551C2211yescharacteristic
32#2547—21no—
33#2547—21no—
34#2546C2211yescharacteristic
35#2542C2211yescharacteristic
36#2538C2211yescharacteristic
37#2506—21no—
38#2499C2214yes—
39#2496—22no—
40#2496—22no—
41#2492C2212yes—
42#2452—21no—
43#2429C2212yes—
44#2396—22no—
45#2365—21no—
46#2365—21no—
47C4×C4⋊Q8—21no—
48C4×C4⋊Q8C2211yescharacteristic
49C4×C4⋊Q8C2211yescharacteristic
50C4×C22⋊Q8C2212yes—
51#2293—21no—
52#2285C2212yes—
53C42×Q8C2211yescharacteristic
54C42×D4C2211yescharacteristic
55#2101C2214yes—
56#2097C2214yes—
57#2072C2214yes—
58#2056C2214yes—
59#2035C2214yes—
60#2024C2214yes—
61#2016C2214yes—
62#2001C2214yes—
63#2001C2214yes—
64#2001C2214yes—
65#1993C2214yes—
66#1990C2212yes—
67#1988C2212yes—
68#1981C2211yescharacteristic
69#1971C2214yes—
70#1960C2214yes—
71#1954C2212yes—
72#1948C2212yes—
73#1948C2212yes—
74#1947C2211yescharacteristic
75#1935C2214yes—
76#1928C2212yes—
77#1924C2214yes—
78#1878C2212yes—
79#1873C2214yes—
80#1873C2214yes—
81#1873C2214yes—
82#1854C2214yes—
83#1842C2211yescharacteristic
84#1803C2212yes—
85#1803C2212yes—
86#1803C2212yes—
87#1803C2212yes—
88#1803C2211yescharacteristic
89#1803C2212yes—
90#1803C2212yes—
91#1797C2212yes—
92#1770C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C22×Q16C2312yes—
96C22×Q16C2314yes—
97#556—44no—
98#556—44no—
99#533—28no—
100#533—44no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2312yes—
109C2×#126—24no—
110C2×#125—22no—
111C2×#124—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8C2312yes—
114C2×C22⋊Q8C2312yes—
115C2×C22⋊Q8C2312yes—
116C2×C22⋊Q8C2314yes—
117C2×C4×Q8—22no—
118C2×C4×Q8—22no—
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2311yescharacteristic
124C2×C4×D4D411yescharacteristic
125C2×C4×D4D411yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×#117—22no—
128C2×#117C2314yes—
129C22×C42C2312yes—
130#487—44no—
131#475—44no—
132#462—24no—
133#462—44no—
134#462—44no—
135C4×Q16—44no—
136#432—44no—
137C2×#107C2312yes—
138C2×#106C2314yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2311yescharacteristic
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2311yescharacteristic
143C2×C4⋊C8C2312yes—
144C2×#103C2312yes—
145C2×#103C2312yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2312yes—
152C2×#103C2312yes—
153C2×C22⋊C8C2314yes—
154C2×C22⋊C8C2314yes—
155C2×#97C2314yes—
156C2×C4×C8C2312yes—
157#400—24no—
158#400—22no—
159#399—22no—
160#398—22no—
161#398—22no—
162#397—24no—
163#397—24no—
164#396—22no—
165#396—22no—
166#395—22no—
167#393—22no—
168#391—22no—
169#391—22no—
170#391—22no—
171#391C2312yes—
172#391C2312yes—
173#391C2314yes—
174#391C2312yes—
175#390—21no—
176#390—21no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—21no—
185#389—22no—
186#389—22no—
187#389—22no—
188#389C2312yes—
189#388—22no—
190#388—22no—
191#387—22no—
192#387—24no—
193#387—24no—
194#385—22no—
195#384—21no—
196#384—21no—
197#384—21no—
198#384C2311yescharacteristic
199#384C2311yescharacteristic
200#384C2312yes—
201#384C2312yes—
202#382C2314yes—
203#380—22no—
204#380—22no—
205#380—22no—
206#380—22no—
207#380C2312yes—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4D411yescharacteristic
219C4×C4⋊C4D411yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2311yescharacteristic
224C4×C22⋊C4—22no—
225C4×C22⋊C4D412yes—
226C4×C22⋊C4D412yes—
227C4×C22⋊C4C2312yes—
228C43—21no—
229C43C2311yescharacteristic
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2312yes—
235#340C2314yes—
236#340C2312yes—
237#336C2314yes—
238#336C2312yes—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8C2412yes—
242C22×Q8C2412yes—
243C22×Q8C2411yescharacteristic
244C22×D4C2×D411yescharacteristic
245C23×C4C2×D412yes—
246C23×C4C2×D412yes—
247C23×C4C2412yes—
248C2×Q16—24no—
249C2×Q16—42no—
250C2×Q16—42no—
251C2×Q16—44no—
252C2×Q16—44no—
253C2×Q16—44no—
254C22×C8C2412yes—
255C22×C8C2414yes—
256C22×C8C2412yes—
257C4⋊Q8—24no—
258C4⋊Q8—24no—
259C4⋊Q8—22no—
260C4⋊Q8—24no—
261C4⋊Q8—24no—
262C4⋊Q8—22no—
263C4⋊Q8—22no—
264C4⋊Q8—22no—
265C4⋊Q8—42no—
266C4⋊Q8—42no—
267C4⋊Q8—44no—
268C4⋊Q8—44no—
269C4⋊Q8—44no—
270#126—416no—
271#125—48no—
272#124—48no—
273C22⋊Q8—24no—
274C22⋊Q8—24no—
275C22⋊Q8—24no—
276C22⋊Q8—42no—
277C22⋊Q8—42no—
278C22⋊Q8—42no—
279C22⋊Q8—44no—
280C22⋊Q8—44no—
281C22⋊Q8—48no—
282C4×Q8—22no—
283C4×Q8—22no—
284C4×Q8—24no—
285C4×Q8—24no—
286C4×Q8—24no—
287C4×Q8—24no—
288C4×Q8—22no—
289C4×Q8—22no—
290C4×Q8—48no—
291C4×Q8—41no—
292C4×Q8—48no—
293C4×Q8#4114yes—
294C4×D4—24no—
295C4×D4—22no—
296C4×D4—22no—
297C4×D4—42no—
298#117—48no—
299#117—44no—
300#117—44no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2414yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2411yescharacteristic
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4C2×D412yes—
343C2×C22⋊C4C2×D412yes—
344C2×C22⋊C4C2×D412yes—
345C2×C42—21no—
346C2×C42—24no—
347C2×C42—24no—
348C2×C42—22no—
349C2×C42—24no—
350C2×C42—22no—
351C2×C42—21no—
352C2×C42—21no—
353C2×C42—21no—
354C2×C42—21no—
355C2×C42—21no—
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D412yes—
358C2×C42C2×D412yes—
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2411yescharacteristic
363C2×C42C2412yes—
364C2×C42C2411yescharacteristic
365C2×C42C2411yescharacteristic
366#107—42no—
367#106—44no—
368C4⋊C8—22no—
369C4⋊C8—22no—
370C4⋊C8—24no—
371C4⋊C8—24no—
372C4⋊C8—42no—
373#103—24no—
374#103—24no—
375#103—42no—
376#103—42no—
377#103—44no—
378#103—44no—
379#103—44no—
380#103—44no—
381#103—44no—
382C22⋊C8—44no—
383C22⋊C8—44no—
384#97—44no—
385C4×C8—24no—
386#95—22no—
387#95—22no—
388#95—21no—
389#95—21no—
390#95—21no—
391#95—24no—
392#95—24no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—24no—
400#95—24no—
401#95C2×D412yes—
402#95C2×D412yes—
403#95C2×D412yes—
404C24C22×D412yes—
405C2×Q8—21no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—21no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—21no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—44no—
423C2×Q8—44no—
424C2×Q8—44no—
425C2×Q8—44no—
426C2×Q8—44no—
427C2×Q8—82no—
428C2×Q8C2×#4112yes—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C2511yescharacteristic
463Q16—48no—
464Q16—88no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—24no—
470C2×C8—24no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—24no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—21no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—48no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—81no—
513C4⋊C4C2×#4112yes—
514C4⋊C42−1+412yes—
515C22⋊C4—24no—
516C22⋊C4—24no—
517C22⋊C4—42no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—48no—
522C22⋊C4—48no—
523C22⋊C4—82no—
524C42—24no—
525C42—24no—
526C42—24no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—21no—
532C42—21no—
533C42—41no—
534C42—42no—
535C42—48no—
536C42—42no—
537C42—44no—
538C42—44no—
539C42—42no—
540C42—42no—
541C42—81no—
542C42#13714yes—
543C42C2×#4112yes—
544C42H(5,𝔽2)12yes—
545C42H(5,𝔽2)12yes—
546C42H(5,𝔽2)12yes—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23C23×D411yescharacteristic
555Q8—22no—
556Q8—22no—
557Q8—41no—
558Q8—41no—
559Q8—42no—
560Q8—42no—
561Q8—42no—
562Q8—42no—
563Q8—42no—
564Q8—42no—
565Q8—42no—
566Q8—42no—
567Q8—88no—
568Q8—88no—
569D4—44no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—48no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—84no—
621C2×C4#53512yes—
622C2×C4#53512yes—
623C2×C4#54612yes—
624C2×C4#54612yes—
625C2×C4C2×#13712yes—
626C2×C4C2×#13712yes—
627C2×C4C2×#13712yes—
628C2×C4#57512yes—
629C2×C4#57812yes—
630C2×C4C22×#4111yescharacteristic
631C2×C4C2×H(5,𝔽2)11yescharacteristic
632C2×C4C2×H(5,𝔽2)11yescharacteristic
633C2×C4C2×H(5,𝔽2)11yescharacteristic
634C2×C4C2×2−1+411yescharacteristic
635C2×C4#58511yescharacteristic
636C8—42no—
637C8—44no—
638C8—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22C2×#53511yescharacteristic
647C22C2×#54611yescharacteristic
648C22#348911yescharacteristic
649C22C22×#13711yescharacteristic
650C22C2×#57511yescharacteristic
651C22C2×#57811yescharacteristic
652C22#358611yescharacteristic
653C4—22no—
654C4—22no—
655C4—22no—
656C4—42no—
657C4—42no—
658C4—41no—
659C4—42no—
660C4—42no—
661C4—42no—
662C4—42no—
663C4—82no—
664C4—82no—
665C4—84no—
666C4—81no—
667C4—82no—
668C4—82no—
669C4#321614yes—
670C4#332814yes—
671C4#350611yescharacteristic
672C4#351111yescharacteristic
673C4#354611yescharacteristic
674C4#354811yescharacteristic
675C2—82no—
676C2#3947111yescharacteristic
677C2C2×#321611yescharacteristic
678C2#6083711yescharacteristic
679C2C2×#332811yescharacteristic
680C2#6116511yescharacteristic
681C2#6211311yescharacteristic
682C2#6213311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33196C2semidirect
4. #32065C2semidirect
5. #31856C2semidirect
6. #29664C2semidirect
7. #28717C2semidirect
8. #28417C2semidirect
9. #28255C2semidirect
10. #27823C2semidirect
11. #27682C2semidirect
13. #24660C2semidirect
16. #21870C2semidirect
17. C4×#462C2semidirect

Maximal subgroups

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

subgroupindex
#361882
#349292
#331962
#320652
#318562
#296642
#287172
#284172
#282552
#278232
#276822
#255702
#246602
#241232
#227992
#218702
C4×#4622