Finite Group Databasebeta

← #6435206

Subgroups — #6435206

Order 512.

1656 conjugacy classes of subgroups.

The subgroup lattice has 701 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35275C211yesmaximal, characteristic, complemented
2#34060C211yesmaximal, characteristic
3#29993C214yesmaximal, complemented
4#29990C214yesmaximal, complemented
5#29988C212yesmaximal, complemented
6#29986C212yesmaximal, complemented
7#29983C212yesmaximal, complemented
8#29981C212yesmaximal, complemented
9#23748C212yesmaximal, complemented
10#23738C212yesmaximal
11#23556C212yesmaximal, complemented
12#23458C212yesmaximal
13#21170C211yesmaximal, characteristic, complemented
14C2×#1948C211yesmaximal, characteristic, complemented
15#20755C211yesmaximal, characteristic, complemented
16#20648C211yesmaximal, characteristic
17#20645C211yesmaximal, characteristic
18C2×#544C2212yes—
19C2×C4×#41C2211yescharacteristic
20C2×#428C2212yes—
21C22×#106C2212yes—
22C2×#423C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C2×#417C2212yes—
25C2×#416C2212yes—
26#2868C2212yes—
27#2859C2212yes—
28#2660C2214yes—
29#2552—21no—
30#2552—21no—
31#2551—21no—
32#2551—21no—
33#2440—22no—
34#2440—22no—
35#2320C2212yes—
36#2319C2212yes—
37#2318—22no—
38#2317C2211yescharacteristic
39#2317C2211yescharacteristic
40#2317C2212yes—
41#2314—21no—
42#2314—21no—
43#2311C2211yescharacteristic
44#2311C2211yescharacteristic
45#2295C2212yes—
46#2293C2212yes—
47#2290—22no—
48#2285—21no—
49#2285C2211yescharacteristic
50C2×#384C2211yescharacteristic
51#2283—21no—
52C4×#117C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54#2101C2214yes—
55#2101C2214yes—
56#2100C2214yes—
57#2099C2214yes—
58#2097C2214yes—
59#2096C2214yes—
60#2094C2214yes—
61#2092C2214yes—
62#2059C2214yes—
63#2059C2214yes—
64#2058C2214yes—
65#2058C2214yes—
66#2026C2214yes—
67#2025C2214yes—
68#2024C2214yes—
69#2023C2214yes—
70#1949C2214yes—
71#1948C2212yes—
72#1948C2212yes—
73#1948C2212yes—
74#1948C2214yes—
75#1947C2212yes—
76#1832C2214yes—
77#1831C2212yes—
78#1831C2212yes—
79#1805C2212yes—
80#1804C2212yes—
81#1803C2211yescharacteristic
82#1803C2212yes—
83#1803C2211yescharacteristic
84#1803C2212yes—
85#1802C2211yescharacteristic
86#1802C2212yes—
87#1802C2211yescharacteristic
88#1802C2212yes—
89#1736C2212yes—
90#1736C2212yes—
91#1735C2212yes—
92C2×#340C2212yes—
93#1729C2212yes—
94#1729C2212yes—
95C22×#41C2311yescharacteristic
96C22×M16C2312yes—
97C23×C8C2312yes—
98#544—44no—
99#544—44no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8C2314yes—
104C2×#125—22no—
105C2×#125C2314yes—
106C2×C22⋊Q8—24no—
107C4×#41—24no—
108C4×#41Q818yescomplemented
109C2×C4×Q8C2311yescharacteristic
110C2×C4×Q8C2311yescharacteristic
111C2×C4×D4—21no—
112C2×C4×D4—21no—
113C2×C4×D4C2311yescharacteristic
114C2×C4×D4C2311yescharacteristic
115C2×#117—21no—
116C2×#117—22no—
117C2×#117C2312yes—
118C2×#117C2311yescharacteristic
119C2×#117C2311yescharacteristic
120C2×#117C2311yescharacteristic
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2312yes—
126C22×C42—21no—
127C22×C42—21no—
128C22×C42C2311yescharacteristic
129#428—44no—
130C2×#107C2314yes—
131C2×#106—44no—
132C2×#106C2314yes—
133C2×#106C2314yes—
134C2×#106C2314yes—
135#423—24no—
136C2×C4⋊C8—24no—
137C2×C4⋊C8C2311yescharacteristic
138C2×C4⋊C8C2311yescharacteristic
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2312yes—
142#417—44no—
143#416—44no—
144C2×#103C2314yes—
145C2×#103C2312yes—
146C2×#103C2312yes—
147C2×#102C2314yes—
148C2×#102C2312yes—
149C2×#102C2312yes—
150#398—24no—
151#398—24no—
152#395—24no—
153#395—24no—
154#391C2312yes—
155#391C2312yes—
156#390C2312yes—
157#390C2312yes—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389C2312yes—
165#389C2312yes—
166#389C2312yes—
167#389C2312yes—
168#388—22no—
169#388—22no—
170#388—22no—
171#388—22no—
172#387—22no—
173#387—22no—
174#387—22no—
175#387—22no—
176#386—22no—
177#386—22no—
178#385—22no—
179#385—22no—
180#385—24no—
181#384—21no—
182#384—21no—
183#384—22no—
184#384—22no—
185#384—21no—
186#384—21no—
187#384C2311yescharacteristic
188#384C2312yes—
189#384C2311yescharacteristic
190#384C2312yes—
191#384C2314yes—
192#383—22no—
193#382—21no—
194#382—21no—
195#382—22no—
196#382C2312yes—
197#382C2314yes—
198#381—22no—
199#380—21no—
200#380—21no—
201#380—21no—
202#380—21no—
203#380—21no—
204#380—21no—
205#380—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C22⋊C4—22no—
218C4×C22⋊C4D412yes—
219C4×C22⋊C4D412yes—
220#376—22no—
221#376C2312yes—
222C2×#95—21no—
223C43C2312yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#336C2314yes—
235#336C2314yes—
236#336C2314yes—
237#336C2314yes—
238C2×#41—22no—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41C2×Q814yes—
244C22×Q8—24no—
245C22×Q8C2411yescharacteristic
246C22×D4—21no—
247C22×D4C2411yescharacteristic
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—22no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2411yescharacteristic
257C2×M16—24no—
258C2×M16—24no—
259C2×M16—24no—
260C22×C8—24no—
261C22×C8—24no—
262C22×C8—24no—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C22×C8C2414yes—
266C4⋊Q8—44no—
267C4⋊Q8—44no—
268C4⋊Q8—48no—
269C4⋊Q8—84no—
270C4⋊Q8—84no—
271#125—44no—
272#125—44no—
273#125—48no—
274C22⋊Q8—416no—
275C4×Q8—22no—
276C4×Q8—22no—
277C4×Q8C2×Q814yes—
278C4×Q8C2×Q814yes—
279C4×D4—28no—
280C4×D4—28no—
281C4×D4—22no—
282C4×D4—22no—
283C4×D4C2×Q814yes—
284C4×D4C2×Q814yes—
285#117—22no—
286#117—22no—
287#117—22no—
288#117—22no—
289#117—22no—
290#117—28no—
291#117—24no—
292#117—24no—
293#117—84no—
294#117C2×Q814yes—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—82no—
324C2×C4⋊C4—82no—
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2414yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2411yescharacteristic
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4C2×D412yes—
345C2×C22⋊C4C2×D412yes—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42—28no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—82no—
354C2×C42C2×D412yes—
355C2×C42C2×D412yes—
356C2×C42C2×D412yes—
357C2×C42C2×D412yes—
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×Q814yes—
361C2×C42#4114yes—
362C2×C42C2412yes—
363C2×C42C2412yes—
364C2×C42C2411yescharacteristic
365C2×C42C2411yescharacteristic
366#107—44no—
367#106—44no—
368#106—44no—
369#106—44no—
370C4⋊C8—22no—
371C4⋊C8—22no—
372C4⋊C8—24no—
373C4⋊C8—24no—
374C4⋊C8—24no—
375#103—42no—
376#103—42no—
377#103—44no—
378#102—42no—
379#102—42no—
380#102—44no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—24no—
394#95—24no—
395#95C2×D412yes—
396#95C2×D412yes—
397#95C2×D412yes—
398#95C2×D412yes—
399C24—21no—
400C24C22×D411yescharacteristic
401#41—28no—
402#41—28no—
403#41—44no—
404#41—44no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—48no—
411C2×Q8—84no—
412C2×Q8—84no—
413C2×Q8C22×Q812yes—
414C2×D4—24no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4C22×Q812yes—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—44no—
445C22×C4—81no—
446C22×C4—81no—
447C22×C4—81no—
448C22×C4—81no—
449C22×C4—81no—
450C22×C4—81no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4C22⋊Q814yes—
454C22×C4C22⋊Q814yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×Q812yes—
465C22×C4C2×#4112yes—
466C22×C4H(5,𝔽2)12yes—
467C22×C4H(5,𝔽2)12yes—
468C22×C42−1+412yes—
469C22×C42−1+412yes—
470C22×C4C2511yescharacteristic
471M16—48no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—24no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—24no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—24no—
481C2×C8—48no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—48no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—44no—
511C4⋊C4—42no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C4⋊C4—82no—
517C4⋊C4—82no—
518C4⋊C4—84no—
519C4⋊C4—84no—
520C4⋊C4C22×Q812yes—
521C4⋊C4C22×Q812yes—
522C22⋊C4—24no—
523C22⋊C4—24no—
524C22⋊C4—24no—
525C22⋊C4—24no—
526C22⋊C4—24no—
527C22⋊C4—44no—
528C22⋊C4—82no—
529C22⋊C4—82no—
530C22⋊C4—82no—
531C22⋊C4—82no—
532C42—24no—
533C42—24no—
534C42—24no—
535C42—24no—
536C42—24no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—42no—
542C42—42no—
543C42—44no—
544C42—44no—
545C42—82no—
546C42C22⋊Q814yes—
547C42C22⋊Q814yes—
548C42C22×Q812yes—
549C42C22×Q812yes—
550C42C2×#4112yes—
551C42C2×#4112yes—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—22no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23C2×C22⋊Q812yes—
564C23#53512yes—
565C23#53612yes—
566C23C23×D411yescharacteristic
567Q8—22no—
568Q8—22no—
569Q8—41no—
570Q8—41no—
571Q8—168no—
572D4—22no—
573D4—22no—
574D4—48no—
575D4—41no—
576D4—41no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—28no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
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—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—44no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—48no—
607C2×C4—48no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—84no—
624C2×C4—84no—
625C2×C4—84no—
626C2×C4—164no—
627C2×C4C2×C22⋊Q812yes—
628C2×C4C2×C22⋊Q812yes—
629C2×C4C2×C22⋊Q812yes—
630C2×C4C2×C22⋊Q812yes—
631C2×C4C2×C22⋊Q812yes—
632C2×C4#53512yes—
633C2×C4#53612yes—
634C2×C4#54214yes—
635C2×C4#54214yes—
636C2×C4#54414yes—
637C2×C4#54414yes—
638C2×C4#57712yes—
639C2×C4#57712yes—
640C2×C4C23×Q811yescharacteristic
641C2×C4C22×#4111yescharacteristic
642C2×C4C2×H(5,𝔽2)11yescharacteristic
643C2×C4C2×H(5,𝔽2)11yescharacteristic
644C2×C4C2×2−1+411yescharacteristic
645C2×C4C2×2−1+411yescharacteristic
646C8—42no—
647C8—42no—
648C8—44no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22#262814yes—
663C22C22×C22⋊Q811yescharacteristic
664C22C2×#53511yescharacteristic
665C22C2×#53611yescharacteristic
666C22C2×#57711yescharacteristic
667C22C2×#57711yescharacteristic
668C22#358511yescharacteristic
669C22#358611yescharacteristic
670C4—22no—
671C4—22no—
672C4—22no—
673C4—42no—
674C4—42no—
675C4—42no—
676C4—42no—
677C4—42no—
678C4—42no—
679C4—42no—
680C4—84no—
681C4—82no—
682C4—82no—
683C4—82no—
684C4—82no—
685C4#262814yes—
686C4#308414yes—
687C4C2×#54211yescharacteristic
688C4C2×#54211yescharacteristic
689C4C2×#54411yescharacteristic
690C4C2×#54411yescharacteristic
691C2—24no—
692C2—82no—
693C2C2×#262811yescharacteristic
694C2C2×#308411yescharacteristic
695C2#6049311yescharacteristic
696C2#6083311yescharacteristic
697C2#6083611yescharacteristic
698C2#6088911yescharacteristic
699C2#6089211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
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
1. #35275C2semidirect
3. #29993C2semidirect
4. #29990C2semidirect
5. #29988C2semidirect
6. #29986C2semidirect
7. #29983C2semidirect
8. #29981C2semidirect
9. #23748C2semidirect
11. #23556C2semidirect
13. #21170C2semidirect
14. C2×#1948C2semidirect
15. #20755C2semidirect
108. C4×#41Q8semidirect

Maximal subgroups

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

subgroupindex
#352752
#340602
#299932
#299902
#299882
#299862
#299832
#299812
#237482
#237382
#235562
#234582
#211702
C2×#19482
#207552
#206482
#206452