Finite Group Databasebeta

← #6432492

Subgroups — #6432492

Order 512.

1956 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#34269C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic, complemented
3#23771C214yesmaximal, complemented
4#23767C214yesmaximal, complemented
5#23761C212yesmaximal, complemented
6#23760C212yesmaximal, complemented
7#23755C212yesmaximal, complemented
8#23754C212yesmaximal, complemented
9#23744C211yesmaximal, characteristic, complemented
10#23731C211yesmaximal, characteristic, complemented
11#23658C211yesmaximal, characteristic, complemented
12#23643C212yesmaximal, complemented
13C2×#2071C211yesmaximal, characteristic, complemented
14#23483C212yesmaximal
15#20453C212yesmaximal, complemented
16#20416C212yesmaximal, complemented
17#20378C211yesmaximal, characteristic, complemented
18C2×#535C2211yescharacteristic
19C2×#518C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C2×#428C2212yes—
22C2×#428C2212yes—
23C2×#418C2212yes—
24C22×#102C2212yes—
25C2×#408C2212yes—
26C2×#405C2212yes—
27#2367—21no—
28#2367—21no—
29#2366—21no—
30#2366—21no—
31#2358—22no—
32#2358—22no—
33#2355—21no—
34#2355—21no—
35#2354—21no—
36#2354—21no—
37#2352—22no—
38#2352—22no—
39#2324C2212yes—
40#2324C2212yes—
41#2314C2214yescomplemented
42#2311C2211yescharacteristic
43#2311C2212yes—
44C4×#121C2212yes—
45C4×#121C2212yes—
46#2296C2211yescharacteristic
47C2×#390C2211yescharacteristic
48#2285C2211yescharacteristic
49#2285C2212yescomplemented
50#2285C2212yescomplemented
51C2×#384C2211yescharacteristic
52#2274C2212yes—
53#2274C2211yescharacteristic
54#2101C2212yescomplemented
55#2101C2212yescomplemented
56#2100C2211yescharacteristic, complemented
57#2100C2211yescharacteristic, complemented
58#2099C2211yescharacteristic, complemented
59#2099C2211yescharacteristic, complemented
60#2094C2212yes—
61#2093C2212yes—
62#2092C2214yes—
63#2084C2212yescomplemented
64#2084C2212yescomplemented
65#2083C2212yescomplemented
66#2083C2212yescomplemented
67#2077C2214yescomplemented
68#2077C2214yescomplemented
69#2076C2214yescomplemented
70#2075C2214yescomplemented
71#2071C2212yes—
72#2071C2212yes—
73#2071C2214yes—
74#2034C2214yes—
75#2034C2214yes—
76#2033C2214yes—
77#2033C2214yes—
78#1776C2214yescomplemented
79#1776C2214yescomplemented
80C4×#107C2214yescomplemented
81C4×#106C2214yescomplemented
82#1762C2214yes—
83#1762C2214yes—
84C4×#102C2214yes—
85C4×#102C2214yes—
86#1750C2214yescomplemented
87#1749C2212yescomplemented
88C4×C22⋊C8C2212yescomplemented
89#1736C2212yes—
90#1735C2212yes—
91#1735C2212yes—
92C2×#340C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C23×D4C2311yescharacteristic
97C22×M16C2312yes—
98C23×C8C2312yes—
99#535—24no—
100#535—24no—
101C2×#127—21no—
102C2×#124—21no—
103C2×#121C2312yes—
104C2×#121C2312yes—
105C2×#120—22no—
106#518—28no—
107C4×#41—28no—
108C2×C4×Q8D411yescharacteristic
109C2×C4×Q8D411yescharacteristic
110C2×C4×D4—22no—
111C2×C4×D4—22no—
112C2×C4×D4—21no—
113C2×C4×D4—21no—
114C2×C4×D4D412yes—
115C2×C4×D4D412yes—
116C2×C4×D4D411yescharacteristic
117C2×C4×D4D411yescharacteristic
118C2×C4×D4C2312yes—
119C2×C4×D4C2312yes—
120C2×#117—22no—
121C2×#117—22no—
122C2×#117C2312yes—
123C2×#117C2311yescharacteristic
124C2×#117C2311yescharacteristic
125C2×#117C2312yes—
126C2×#117C2311yescharacteristic
127C22×C4⋊C4—21no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2311yescharacteristic
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C22⋊C4—22no—
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136#428—28no—
137#428—28no—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#106C2314yes—
141C2×#106C2314yes—
142#418—28no—
143C2×#102—28no—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148#408—28no—
149C2×C22⋊C8C2314yes—
150C2×C22⋊C8C2314yes—
151#405—28no—
152C2×#97C2314yes—
153C2×C4×C8C2314yes—
154#391—22no—
155#391—21no—
156#390—22no—
157#390—21no—
158#390—22no—
159#390—22no—
160#390—21no—
161#390—21no—
162#390C2312yes—
163#390C2312yes—
164#389C2311yescharacteristic
165#389C2311yescharacteristic
166#389C2311yescharacteristic
167#389C2311yescharacteristic
168#388—22no—
169#388—22no—
170#388—24no—
171#388—24no—
172#388—22no—
173#388—22no—
174#387—24no—
175#387—24no—
176#386—24no—
177#386—22no—
178#386—22no—
179#386—22no—
180#386—22no—
181#386—22no—
182#385—21no—
183#385—21no—
184#385—22no—
185#385—22no—
186#384—21no—
187#384—21no—
188#384C2312yes—
189#384C2312yes—
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#382—22no—
193#382—22no—
194#382C2312yes—
195#380—21no—
196#380—21no—
197#380—22no—
198#380—22no—
199#380C2314yes—
200#380C2314yes—
201#380C2312yes—
202#380C2312yes—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4C2314yes—
214C4×C4⋊C4C2314yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C22⋊C4—24no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—22no—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#340C2312yes—
231#340C2312yes—
232#340C2312yes—
233#336C2311yescharacteristic
234#336C2311yescharacteristic
235#336C2312yes—
236#336C2312yes—
237#336C2312yes—
238C25—21no—
239C2×#41—24no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—24no—
243C22×Q8C2×D411yescharacteristic
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—21no—
247C22×D4—24no—
248C22×D4—24no—
249C22×D4—22no—
250C22×D4C2×D412yes—
251C22×D4C2×D411yescharacteristic
252C22×D4#4114yes—
253C22×D4C2412yes—
254C23×C4—21no—
255C23×C4—22no—
256C23×C4—22no—
257C23×C4—22no—
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D412yes—
261C23×C4C2×D412yes—
262C23×C4C2412yes—
263C23×C4C2411yescharacteristic
264C2×M16—24no—
265C2×M16—24no—
266C2×M16#4118yes—
267C22×C8—24no—
268C22×C8—24no—
269C22×C8—24no—
270C22×C8C2412yes—
271C22×C8C2412yes—
272C22×C8C2414yes—
273#127—28no—
274#124—28no—
275#121—24no—
276#121—24no—
277#121—24no—
278#121—24no—
279#120—216no—
280C4×Q8—24no—
281C4×Q8—24no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—28no—
285C4×D4—28no—
286C4×D4—28no—
287C4×D4—28no—
288C4×D4—44no—
289C4×D4—44no—
290C4×D4—84no—
291C4×D4—84no—
292#117—22no—
293#117—28no—
294#117—24no—
295#117—24no—
296#117—24no—
297#117—84no—
298#117—84no—
299#117#4114yes—
300#117#4118yes—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—82no—
319C2×C4⋊C4—82no—
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2414yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2411yescharacteristic
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—48no—
348C2×C22⋊C4—82no—
349C2×C22⋊C4—82no—
350C2×C22⋊C4C2×D412yes—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C42—21no—
354C2×C42—24no—
355C2×C42—28no—
356C2×C42—21no—
357C2×C42—22no—
358C2×C42—24no—
359C2×C42—24no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42C2×D412yes—
363C2×C42C2×D412yes—
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2411yescharacteristic
367C2×C42C2411yescharacteristic
368C2×C42C2412yes—
369#107—28no—
370#107—28no—
371#106—28no—
372#106—28no—
373#102—28no—
374#102—28no—
375#102—28no—
376#102—28no—
377C22⋊C8—28no—
378C22⋊C8—28no—
379#97—28no—
380C4×C8—28no—
381#95—21no—
382#95—21no—
383#95—24no—
384#95—24no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—24no—
392#95—24no—
393#95—21no—
394#95—21no—
395C24—22no—
396C24—24no—
397C24—42no—
398C24—42no—
399C24—81no—
400C24—81no—
401C24—81no—
402C24—81no—
403C24C22×D412yes—
404C24C22×D411yescharacteristic
405#41—48no—
406#41—48no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—28no—
414C2×D4—28no—
415C2×D4—22no—
416C2×D4—24no—
417C2×D4—22no—
418C2×D4—216no—
419C2×D4—22no—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—22no—
423C2×D4—42no—
424C2×D4—48no—
425C2×D4—42no—
426C2×D4—82no—
427C2×D4—82no—
428C2×D4—82no—
429C2×D4—82no—
430C2×D4C2×#4114yes—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—41no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—41no—
455C22×C4—44no—
456C22×C4—42no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—82no—
460C22×C4—82no—
461C22×C4—82no—
462C22×C4—82no—
463C22×C4—81no—
464C22×C4—81no—
465C22×C4—82no—
466C22×C4—82no—
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C2×#4114yes—
476C22×C4C2×#4112yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4114yes—
479C22×C4C2511yescharacteristic
480M16—216no—
481C2×C8—24no—
482C2×C8—24no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—48no—
490C2×C8C2×#4118yes—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—42no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—81no—
515C4⋊C4—81no—
516C4⋊C4—82no—
517C4⋊C4—82no—
518C4⋊C4#12314yes—
519C4⋊C4#12314yes—
520C4⋊C4C2×#4112yes—
521C4⋊C4C2×#4112yes—
522C22⋊C4—28no—
523C22⋊C4—28no—
524C22⋊C4—24no—
525C22⋊C4—24no—
526C22⋊C4—24no—
527C22⋊C4—42no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—82no—
534C22⋊C4—82no—
535C22⋊C4—84no—
536C22⋊C4—84no—
537C22⋊C4—82no—
538C22⋊C4—82no—
539C22⋊C4#12318yes—
540C42—22no—
541C42—22no—
542C42—24no—
543C42—24no—
544C42—24no—
545C42—22no—
546C42—22no—
547C42—42no—
548C42—44no—
549C42—84no—
550C42—84no—
551C42—82no—
552C42—82no—
553C42C2×#4118yes—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—24no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—48no—
573C23—84no—
574C23—84no—
575C23—162no—
576C23—162no—
577C23—162no—
578C23—162no—
579C23C2×#12312yes—
580C23#52512yes—
581C23#53212yes—
582C23C23×D411yescharacteristic
583Q8—44no—
584D4—28no—
585D4—48no—
586D4—48no—
587D4—44no—
588D4—44no—
589D4—88no—
590D4—88no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—48no—
611C2×C4—48no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—164no—
639C2×C4—164no—
640C2×C4—162no—
641C2×C4—164no—
642C2×C4C2×#12314yes—
643C2×C4C2×#12314yes—
644C2×C4C2×#12312yes—
645C2×C4#52512yes—
646C2×C4#52918yes—
647C2×C4#52918yes—
648C2×C4#53212yes—
649C2×C4#57612yes—
650C2×C4#57712yes—
651C2×C4C22×#4112yes—
652C2×C4C22×#4111yescharacteristic
653C2×C4C22×#4111yescharacteristic
654C2×C4C22×#4112yes—
655C8—28no—
656C8—42no—
657C8—42no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—44no—
670C22—42no—
671C22—44no—
672C22—42no—
673C22—44no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22—168no—
679C22#239514yes—
680C22C22×#12311yescharacteristic
681C22C2×#52511yescharacteristic
682C22C2×#53211yescharacteristic
683C22C2×#57611yescharacteristic
684C22C2×#57711yescharacteristic
685C22#358511yescharacteristic
686C22#358611yescharacteristic
687C4—22no—
688C4—24no—
689C4—22no—
690C4—24no—
691C4—41no—
692C4—41no—
693C4—42no—
694C4—42no—
695C4—42no—
696C4—82no—
697C4—82no—
698C4—82no—
699C4—81no—
700C4—81no—
701C4—84no—
702C4#239514yes—
703C4C2×#52912yes—
704C4C2×#52912yes—
705C2—24no—
706C2—82no—
707C2—84no—
708C2C2×#239511yescharacteristic
709C2#6052111yescharacteristic
710C2#6052511yescharacteristic
711C2#6059011yescharacteristic
712C2#6059111yescharacteristic
713C2#6072211yescharacteristic
714C2#6072711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
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
2. #34188C2semidirect
3. #23771C2semidirect
4. #23767C2semidirect
5. #23761C2semidirect
6. #23760C2semidirect
7. #23755C2semidirect
8. #23754C2semidirect
9. #23744C2semidirect
10. #23731C2semidirect
11. #23658C2semidirect
12. #23643C2semidirect
13. C2×#2071C2semidirect
15. #20453C2semidirect
16. #20416C2semidirect
17. #20378C2semidirect
41. #2314C22semidirect
49. #2285C22semidirect
50. #2285C22semidirect
54. #2101C22semidirect
55. #2101C22semidirect
56. #2100C22semidirect
57. #2100C22semidirect
58. #2099C22semidirect
59. #2099C22semidirect
63. #2084C22semidirect
64. #2084C22semidirect
65. #2083C22semidirect
66. #2083C22semidirect
67. #2077C22semidirect
68. #2077C22semidirect
69. #2076C22semidirect
70. #2075C22semidirect
78. #1776C22semidirect
79. #1776C22semidirect
80. C4×#107C22semidirect
81. C4×#106C22semidirect
86. #1750C22semidirect
87. #1749C22semidirect
88. C4×C22⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#342692
#341882
#237712
#237672
#237612
#237602
#237552
#237542
#237442
#237312
#236582
#236432
C2×#20712
#234832
#204532
#204162
#203782