Finite Group Databasebeta

← #6627954

Subgroups — #6627954

Order 512.

1833 conjugacy classes of subgroups.

The subgroup lattice has 687 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36587C211yesmaximal, characteristic
2C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
3#32909C212yesmaximal, complemented
4#32903C211yesmaximal, characteristic
5#29336C212yesmaximal, complemented
6#29234C212yesmaximal, complemented
7#27973C211yesmaximal, characteristic
8#27681C211yesmaximal, characteristic, complemented
9#27572C214yesmaximal, complemented
10#26177C214yesmaximal, complemented
11#26109C212yesmaximal, complemented
12#24733C212yesmaximal, complemented
13#23003C212yesmaximal, complemented
14#22763C212yesmaximal, complemented
15#22629C211yesmaximal, characteristic, complemented
16#21454C212yesmaximal, complemented
17C4×#451C211yesmaximal, characteristic, complemented
18C22×C4×Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20C2×#464C2214yes—
21C2×#460C2212yes—
22C2×#451C2211yescharacteristic
23C2×#448C2212yes—
24C2×#440C2214yes—
25C2×C4×Q16C2212yes—
26#2856C2211yescharacteristic
27#2850C2212yes—
28#2597—22no—
29#2571C2211yescharacteristic
30#2548C2211yescharacteristic
31#2513—21no—
32#2513C2212yes—
33#2504—21no—
34#2447C2212yes—
35#2444—22no—
36#2366C2211yescharacteristic
37#2361—21no—
38#2357—21no—
39#2355C2211yescharacteristic
40Q8×C4⋊C4—22no—
41Q8×C4⋊C4—21no—
42Q8×C4⋊C4—21no—
43Q8×C4⋊C4C2214yes—
44Q8×C4⋊C4C2212yes—
45Q8×C4⋊C4C2212yes—
46C4×C22⋊Q8C2211yescharacteristic
47C4×#121C2212yes—
48C2×#389—22no—
49C2×#389C2212yes—
50C2×#384—21no—
51C2×#384C2211yescharacteristic
52C2×C4×C4⋊C4—21no—
53C2×C4×C4⋊C4C2211yescharacteristic
54#2101C2214yes—
55#2100C2212yes—
56#2095C2212yes—
57#2086C2214yes—
58#2071C2214yes—
59#2067C2214yes—
60#2042C2214yes—
61#1957C2212yes—
62#1947C2212yes—
63#1939C2212yes—
64#1936C2214yes—
65#1935C2214yes—
66#1918C2214yes—
67#1917C2214yes—
68#1880C2214yes—
69#1880C2214yes—
70#1880C2214yes—
71#1878C2212yes—
72#1878C2214yes—
73#1874C2212yes—
74#1874C2212yes—
75#1874C2212yes—
76#1872C2212yes—
77#1872C2211yescharacteristic
78#1863C2214yes—
79#1863C2212yes—
80#1862C2214yes—
81#1852C2212yes—
82#1852C2211yescharacteristic
83#1834C2212yes—
84#1805C2214yes—
85#1803C2212yes—
86#1803C2212yes—
87#1797C2211yescharacteristic
88#1797C2212yes—
89#1787C2211yescharacteristic
90#1770C2212yes—
91#1763C2214yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2311yescharacteristic
96C22×Q16C2312yes—
97C22×SD16C2314yes—
98C2×#127C2311yescharacteristic
99C2×#124C2312yes—
100C2×C22⋊Q8C2311yescharacteristic
101C2×#121—21no—
102C2×#121—21no—
103C2×#121C2312yes—
104C2×C4×Q8—28no—
105C2×C4×Q8—21no—
106C2×C4×Q8—21no—
107C2×C4×Q8—22no—
108C2×C4×Q8—22no—
109C2×C4×Q8—48no—
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2312yes—
112C2×C4×Q8C2314yes—
113C2×C4×Q8C2314yes—
114C2×C4×Q8C2312yes—
115C2×C4×D4—22no—
116C2×C4×D4—21no—
117C2×C4×D4C2312yes—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—22no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—22no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4D411yescharacteristic
124C22×C4⋊C4D411yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2312yes—
127C22×C42—21no—
128C22×C42—22no—
129C22×C42C2311yescharacteristic
130C22×C42C2312yes—
131#464—48no—
132#460—28no—
133#451—24no—
134#448—28no—
135#440—48no—
136C4×Q16—28no—
137C2×#107C2312yes—
138C2×#107C2314yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2314yes—
141C2×#103C2314yes—
142C2×#103C2312yes—
143C2×#103C2312yes—
144C2×#103C2312yes—
145C2×#103C2312yes—
146C2×#103C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×C22⋊C8C2312yes—
151C2×C22⋊C8C2311yescharacteristic
152C2×C22⋊C8C2311yescharacteristic
153C2×#97C2314yes—
154C2×C4×C8C2312yes—
155#400—22no—
156#397—22no—
157#397—22no—
158#397—22no—
159#397—22no—
160#394—22no—
161#391C2312yes—
162#391C2311yescharacteristic
163#390—21no—
164#390—21no—
165#390—21no—
166#390—21no—
167#390—21no—
168#390C2312yes—
169#390C2312yes—
170#390C2312yes—
171#389—21no—
172#389—22no—
173#389—22no—
174#389—24no—
175#389—24no—
176#389—24no—
177#389—24no—
178#389C2312yes—
179#389C2314yes—
180#389C2314yes—
181#389C2311yescharacteristic
182#388—22no—
183#388—22no—
184#388—22no—
185#388—22no—
186#388—22no—
187#387—21no—
188#387—22no—
189#386—22no—
190#386—22no—
191#386—22no—
192#386—22no—
193#385—21no—
194#385—22no—
195#384—22no—
196#384—22no—
197#384—24no—
198#384—21no—
199#384—21no—
200#384C2311yescharacteristic
201#384C2312yes—
202#384C2312yes—
203#382—21no—
204#382—21no—
205#382C2312yes—
206#381—22no—
207#380C2311yescharacteristic
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—24no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4D412yes—
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C22⋊C4—21no—
222C4×C22⋊C4—22no—
223C2×#95—22no—
224C2×#95D412yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#336C2312yes—
233#336C2314yes—
234#336C2312yes—
235#336C2314yes—
236C22×Q8—24no—
237C22×Q8—24no—
238C22×Q8—21no—
239C22×Q8—22no—
240C22×Q8#4114yes—
241C22×Q8C2411yescharacteristic
242C22×Q8C2412yes—
243C22×D4—22no—
244C22×D4—21no—
245C22×D4—21no—
246C22×D4C2412yes—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—22no—
250C23×C4—22no—
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D412yes—
254C23×C4C2×D412yes—
255C23×C4C2411yescharacteristic
256C2×Q16—24no—
257C2×Q16—24no—
258C2×Q16—24no—
259C2×SD16—28no—
260C2×SD16—28no—
261C2×SD16—28no—
262C22×C8C2414yes—
263C22×C8C2412yes—
264C22×C8C2412yes—
265#127—24no—
266#124—44no—
267C22⋊Q8—22no—
268C22⋊Q8—22no—
269#121—24no—
270#121—24no—
271#121—44no—
272#121—82no—
273C4×Q8—28no—
274C4×Q8—28no—
275C4×Q8—24no—
276C4×Q8—24no—
277C4×Q8—24no—
278C4×Q8—28no—
279C4×Q8—28no—
280C4×Q8—24no—
281C4×Q8—44no—
282C4×Q8—44no—
283C4×Q8—48no—
284C4×Q8—48no—
285C4×Q8#4118yes—
286C4×D4—44no—
287C4×D4—44no—
288C4×D4—84no—
289C2×C4⋊C4—28no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—42no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—48no—
314C2×C4⋊C4—82no—
315C2×C4⋊C4—84no—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2414yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C42—28no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—24no—
344C2×C42—21no—
345C2×C42—24no—
346C2×C42—24no—
347C2×C42—21no—
348C2×C42—22no—
349C2×C42—44no—
350C2×C42—48no—
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2412yes—
356C2×C42C2412yes—
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359#107—24no—
360#107—44no—
361C4⋊C8—24no—
362C4⋊C8—44no—
363#103—24no—
364#103—28no—
365#103—24no—
366#103—24no—
367#103—24no—
368#103—44no—
369#102—28no—
370#102—44no—
371#102—44no—
372C22⋊C8—24no—
373C22⋊C8—22no—
374C22⋊C8—22no—
375#97—44no—
376C4×C8—24no—
377#95—24no—
378#95—24no—
379#95—21no—
380#95—22no—
381#95—21no—
382#95—21no—
383#95—21no—
384#95—22no—
385#95—22no—
386#95—21no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95C2×D412yes—
391#95C2×D412yes—
392#95C2×D412yes—
393#95C2×D412yes—
394C24—22no—
395C24C22×D411yescharacteristic
396C2×Q8—24no—
397C2×Q8—24no—
398C2×Q8—216no—
399C2×Q8—216no—
400C2×Q8—24no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—21no—
406C2×Q8—22no—
407C2×Q8—21no—
408C2×Q8C2×#4114yes—
409C2×Q8C2×#4114yes—
410C2×Q8H(5,𝔽2)14yes—
411C2×D4—24no—
412C2×D4—22no—
413C2×D4—24no—
414C2×D4—22no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—81no—
423C2×D4—81no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—24no—
444C22×C4—42no—
445C22×C4—41no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—44no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—82no—
456C22×C4—82no—
457C22×C4#13714yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C2×#4112yes—
468C22×C42−1+412yes—
469C22×C4C2511yescharacteristic
470Q16—48no—
471SD16—416no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—24no—
475C2×C8—22no—
476C2×C8—24no—
477C2×C8—22no—
478C2×C8—24no—
479C2×C8—22no—
480C2×C8—22no—
481C4⋊C4—28no—
482C4⋊C4—24no—
483C4⋊C4—28no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—21no—
498C4⋊C4—42no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—81no—
508C4⋊C4—82no—
509C4⋊C4—84no—
510C4⋊C4—82no—
511C4⋊C4—84no—
512C4⋊C4—82no—
513C4⋊C4—81no—
514C4⋊C4C2×#4114yes—
515C22⋊C4—44no—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—82no—
521C22⋊C4—82no—
522C22⋊C4—82no—
523C22⋊C4—82no—
524C22⋊C4—82no—
525C22⋊C4—82no—
526C22⋊C4—82no—
527C42—28no—
528C42—28no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—24no—
533C42—24no—
534C42—42no—
535C42—42no—
536C42—48no—
537C42—48no—
538C42—42no—
539C42—42no—
540C42—82no—
541C42C2×#4114yes—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23#54812yes—
554C23C2×#13712yes—
555C23#57612yes—
556C23C23×D411yescharacteristic
557Q8—216no—
558Q8—24no—
559Q8—24no—
560Q8—24no—
561Q8—42no—
562Q8—42no—
563Q8#55318yes—
564D4—44no—
565D4—44no—
566D4—88no—
567D4—84no—
568D4—162no—
569C2×C4—216no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—24no—
573C2×C4—21no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—28no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—44no—
599C2×C4—48no—
600C2×C4—48no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—44no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—44no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—44no—
621C2×C4—84no—
622C2×C4—88no—
623C2×C4—164no—
624C2×C4#52512yes—
625C2×C4#52512yes—
626C2×C4#54812yes—
627C2×C4C2×#13712yes—
628C2×C4C2×#13712yes—
629C2×C4#57512yes—
630C2×C4C22×#4111yescharacteristic
631C2×C4C22×#4112yes—
632C2×C4C2×H(5,𝔽2)12yes—
633C2×C4C2×2−1+411yescharacteristic
634C8—44no—
635C8—42no—
636C8—42no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22#337514yes—
651C22C2×#52511yescharacteristic
652C22C2×#54811yescharacteristic
653C22#348411yescharacteristic
654C22C22×#13711yescharacteristic
655C22C2×#57511yescharacteristic
656C22C2×#57611yescharacteristic
657C22#358611yescharacteristic
658C4—22no—
659C4—22no—
660C4—28no—
661C4—22no—
662C4—42no—
663C4—42no—
664C4—42no—
665C4—42no—
666C4—41no—
667C4—44no—
668C4—82no—
669C4—81no—
670C4—84no—
671C4—81no—
672C4—81no—
673C4—81no—
674C4#311714yes—
675C4C2×#55312yes—
676C4#349212yes—
677C2—24no—
678C2—162no—
679C2#3658111yescharacteristic
680C2C2×#311711yescharacteristic
681C2#6058411yescharacteristic
682C2C2×#337511yescharacteristic
683C2#6132011yescharacteristic
684C2#6158411yescharacteristic
685C2#6160511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×Q8×C4⋊C4C2semidirect
3. #32909C2semidirect
5. #29336C2semidirect
6. #29234C2semidirect
8. #27681C2semidirect
9. #27572C2semidirect
10. #26177C2semidirect
11. #26109C2semidirect
12. #24733C2semidirect
13. #23003C2semidirect
14. #22763C2semidirect
15. #22629C2semidirect
16. #21454C2semidirect
17. C4×#451C2semidirect

Maximal subgroups

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

subgroupindex
#365872
C2×Q8×C4⋊C42
#329092
#329032
#293362
#292342
#279732
#276812
#275722
#261772
#261092
#247332
#230032
#227632
#226292
#214542
C4×#4512