Finite Group Databasebeta

← #6435882

Subgroups — #6435882

Order 512.

1771 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35281C211yesmaximal, characteristic, complemented
2#35264C211yesmaximal, characteristic
3#30216C212yesmaximal, complemented
4#30213C212yesmaximal, complemented
5#30211C214yesmaximal, complemented
6#30209C214yesmaximal, complemented
7#30208C212yesmaximal, complemented
8#30203C212yesmaximal, complemented
9#23634C212yesmaximal, complemented
10#23632C212yesmaximal, complemented
11#23454C211yesmaximal, characteristic, complemented
12#23452C211yesmaximal, characteristic, complemented
13#23451C211yesmaximal, characteristic, complemented
14C2×#2023C211yesmaximal, characteristic, complemented
15#21326C212yesmaximal, complemented
16#21206C212yesmaximal
17#20368C211yesmaximal, characteristic, complemented
18C2×#544C2211yescharacteristic
19C2×#535C2211yescharacteristic
20C2×#533C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C2×#417C2212yes—
23C2×#418C2212yes—
24C2×#416C2212yes—
25C22×#102C2212yes—
26C2×#405C2212yes—
27#2868C2212yescomplemented
28#2862C2212yescomplemented
29#2861C2212yes—
30#2858C2212yes—
31#2660C2214yescomplemented
32#2641C2212yes—
33#2640C2212yes—
34#2567—21no—
35#2567—21no—
36#2546—21no—
37#2546—21no—
38#2502—22no—
39#2491—21no—
40#2491—21no—
41#2490—21no—
42#2490—21no—
43#2472—22no—
44#2472—22no—
45#2434—22no—
46#2317C2212yes—
47#2317C2212yes—
48#2311C2211yescharacteristic
49#2309C2211yescharacteristic
50#2297C2211yescharacteristic
51C2×#390C2211yescharacteristic
52C4×#117C2211yescharacteristic
53#2074C2214yescomplemented
54#2073C2214yes—
55#2073C2214yes—
56#2073C2214yescomplemented
57#2072C2214yescomplemented
58#2071C2214yes—
59#2071C2214yes—
60#2071C2214yescomplemented
61#2026C2212yescomplemented
62#2025C2214yes—
63#2025C2212yescomplemented
64#2024C2211yescharacteristic, complemented
65#2024C2212yescomplemented
66#2024C2212yescomplemented
67#2024C2211yescharacteristic, complemented
68#2023C2212yes—
69#2023C2212yes—
70#2023C2214yes—
71#2023C2212yes—
72#2023C2212yes—
73#2023C2212yescomplemented
74#2023C2211yescharacteristic, complemented
75#2023C2212yescomplemented
76#2023C2211yescharacteristic, complemented
77#1993C2214yescomplemented
78#1992C2214yescomplemented
79#1991C2214yescomplemented
80#1990C2214yescomplemented
81#1961C2214yes—
82#1959C2214yes—
83#1958C2214yes—
84#1957C2214yes—
85#1746C2214yescomplemented
86#1745C2212yescomplemented
87#1744C2212yescomplemented
88#1736C2212yes—
89#1736C2212yes—
90#1735C2212yes—
91C2×#340C2212yes—
92#1729C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C23×Q8—21no—
97C23×D4C2311yescharacteristic
98C22×M16C2312yes—
99C23×C8C2312yes—
100#544—28no—
101#535—28no—
102#533—44no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—22no—
107C2×C4⋊Q8—22no—
108C2×C4⋊Q8—44no—
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2311yescharacteristic
111C2×C4⋊Q8C2312yes—
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2312yes—
114C2×#127C2312yes—
115C2×#125—21no—
116C2×#125C2312yes—
117C2×#125C2312yes—
118C2×#124—22no—
119C2×#124—22no—
120C2×#124C2312yes—
121C2×C22⋊Q8—21no—
122C2×C22⋊Q8—21no—
123C2×C22⋊Q8—22no—
124C2×#121—21no—
125C2×#121—21no—
126C2×#120—22no—
127C2×C4×Q8—21no—
128C2×C4×Q8—21no—
129C2×#117C2311yescharacteristic
130C2×#117C2312yes—
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C22×C4⋊C4—21no—
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2312yes—
136C22×C22⋊C4—21no—
137C22×C22⋊C4—21no—
138C22×C42C2311yescharacteristic
139#418—44no—
140#417—44no—
141#416—44no—
142C2×#103C2312yes—
143C2×#103C2314yes—
144C2×#103C2312yes—
145C2×#102—44no—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2312yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2312yes—
153#405—28no—
154C2×#97C2314yes—
155C2×C4×C8C2314yes—
156#398—24no—
157#398—24no—
158#397—22no—
159#397—22no—
160#395—24no—
161#395—24no—
162#394—22no—
163#394—22no—
164#394—22no—
165#394—22no—
166#393—22no—
167#393—22no—
168#391—22no—
169#391—22no—
170#391—22no—
171#391—22no—
172#391C2312yes—
173#391C2311yescharacteristic
174#391C2311yescharacteristic
175#390—21no—
176#390—21no—
177#390—22no—
178#390—22no—
179#390—21no—
180#390—21no—
181#390C2312yes—
182#390C2312yes—
183#390C2312yes—
184#390C2311yescharacteristic
185#390C2312yes—
186#390C2312yes—
187#390C2311yescharacteristic
188#389—22no—
189#389—22no—
190#389—22no—
191#389—22no—
192#388—22no—
193#388—22no—
194#388—22no—
195#388—22no—
196#386—21no—
197#386—21no—
198#386—22no—
199#385—24no—
200#385—24no—
201#384—22no—
202#384—22no—
203#384C2314yes—
204#384C2314yes—
205#382C2314yes—
206#382C2314yes—
207#380—21no—
208#380—21no—
209#379—22no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C22⋊C4—21no—
215C4×C22⋊C4—21no—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4D412yes—
218#376C2312yes—
219C43C2312yes—
220#340C2312yes—
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2314yes—
230#336C2314yes—
231#336C2314yes—
232#336C2311yescharacteristic
233#336C2311yescharacteristic
234#336C2312yes—
235#336C2312yes—
236#336C2312yes—
237C25—21no—
238C2×#41—24no—
239C2×#41—24no—
240C2×#41—22no—
241C2×#41—22no—
242C22×Q8—21no—
243C22×Q8—21no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—44no—
248C22×Q8—82no—
249C22×Q8—82no—
250C22×Q8C2411yescharacteristic
251C22×D4—21no—
252C22×D4—21no—
253C22×D4—22no—
254C22×D4—22no—
255C22×D4—24no—
256C22×D4—24no—
257C22×D4—22no—
258C22×D4—22no—
259C22×D4C2412yes—
260C22×D4C2411yescharacteristic
261C23×C4—21no—
262C23×C4—22no—
263C23×C4—21no—
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2411yescharacteristic
267C2×M16—24no—
268C2×M16—24no—
269C2×M16—24no—
270C22×C8—24no—
271C22×C8—24no—
272C22×C8—24no—
273C22×C8C2412yes—
274C22×C8C2412yes—
275C22×C8C2414yes—
276C4⋊Q8—28no—
277C4⋊Q8—44no—
278C4⋊Q8—44no—
279C4⋊Q8—42no—
280C4⋊Q8—42no—
281C4⋊Q8—44no—
282C4⋊Q8—44no—
283C4⋊Q8—44no—
284C4⋊Q8—84no—
285C4⋊Q8—84no—
286#127—28no—
287#125—28no—
288#125—42no—
289#125—42no—
290#125—44no—
291#124—28no—
292#124—84no—
293#124—84no—
294C22⋊Q8—44no—
295C22⋊Q8—44no—
296C22⋊Q8—48no—
297#121—44no—
298#121—44no—
299#120—48no—
300C4×Q8—44no—
301C4×Q8—44no—
302#117—22no—
303#117—22no—
304#117—24no—
305#117—24no—
306#117—24no—
307#117#4118yes—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—44no—
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2414yes—
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2411yescharacteristic
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—82no—
356C2×C22⋊C4—82no—
357C2×C22⋊C4C2×D412yes—
358C2×C22⋊C4C2×D412yes—
359C2×C42—24no—
360C2×C42—22no—
361C2×C42—21no—
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D412yes—
365C2×C42C2×D412yes—
366C2×C42C2×D412yes—
367C2×C42C2×D412yes—
368C2×C42C2412yes—
369C2×C42C2411yescharacteristic
370C2×C42C2411yescharacteristic
371C2×C42C2412yes—
372#103—42no—
373#103—42no—
374#103—44no—
375#102—44no—
376#102—44no—
377#102—42no—
378#102—42no—
379#102—44no—
380#102—44no—
381#102—44no—
382#97—28no—
383C4×C8—28no—
384#95—21no—
385#95—21no—
386#95—21no—
387#95—21no—
388#95—22no—
389#95—22no—
390#95—24no—
391#95—24no—
392#95—24no—
393#95—24no—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397#95C2×D412yes—
398C24—22no—
399C24—21no—
400C24—42no—
401C24—42no—
402C24—42no—
403C24—81no—
404C24—81no—
405C24—81no—
406C24—81no—
407C24C22×D411yescharacteristic
408#41—48no—
409#41—48no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—42no—
415C2×Q8—42no—
416C2×Q8—44no—
417C2×Q8—44no—
418C2×Q8—42no—
419C2×Q8—44no—
420C2×Q8—42no—
421C2×Q8—44no—
422C2×Q8—42no—
423C2×Q8—44no—
424C2×Q8—42no—
425C2×Q8—82no—
426C2×Q8—82no—
427C2×Q8—82no—
428C2×Q8—82no—
429C2×Q8—816no—
430C2×D4—24no—
431C2×D4—22no—
432C2×D4—24no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4—42no—
441C2×D4—42no—
442C2×D4—44no—
443C2×D4—44no—
444C2×D4—42no—
445C2×D4—42no—
446C2×D4—48no—
447C2×D4—48no—
448C2×D4—82no—
449C2×D4—82no—
450C2×D4—82no—
451C2×D4—82no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—24no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—44no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—82no—
471C22×C4—82no—
472C22×C4—81no—
473C22×C4—81no—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C2×#4114yes—
484C22×C4H(5,𝔽2)14yes—
485C22×C42−1+412yes—
486C22×C42−1+412yes—
487C22×C4C2511yescharacteristic
488M16—48no—
489C2×C8—22no—
490C2×C8—24no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—24no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—24no—
498C2×C8—48no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—84no—
522C4⋊C4—84no—
523C4⋊C4—84no—
524C4⋊C4—84no—
525C4⋊C4#12414yes—
526C4⋊C4#12414yes—
527C22⋊C4—24no—
528C22⋊C4—42no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—42no—
534C22⋊C4—44no—
535C22⋊C4—42no—
536C22⋊C4—82no—
537C22⋊C4—81no—
538C22⋊C4—82no—
539C22⋊C4—81no—
540C22⋊C4—82no—
541C22⋊C4—82no—
542C22⋊C4—82no—
543C22⋊C4—82no—
544C22⋊C4—82no—
545C22⋊C4—82no—
546C22⋊C4#12418yes—
547C42—22no—
548C42—22no—
549C42—24no—
550C42—24no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—42no—
556C42—42no—
557C42—42no—
558C42—44no—
559C42C2×#4118yes—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—41no—
565C23—42no—
566C23—41no—
567C23—42no—
568C23—42no—
569C23—41no—
570C23—41no—
571C23—42no—
572C23—41no—
573C23—42no—
574C23—41no—
575C23—42no—
576C23—41no—
577C23—42no—
578C23—84no—
579C23—84no—
580C23—84no—
581C23—162no—
582C23—162no—
583C23—162no—
584C23—162no—
585C23C2×#12412yes—
586C23#53512yes—
587C23#53612yes—
588C23C23×D411yescharacteristic
589Q8—42no—
590Q8—42no—
591Q8—84no—
592Q8—84no—
593Q8—88no—
594Q8—88no—
595Q8—164no—
596D4—44no—
597D4—44no—
598D4—42no—
599D4—42no—
600D4—84no—
601D4—84no—
602D4—164no—
603D4—164no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—21no—
614C2×C4—22no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—21no—
622C2×C4—22no—
623C2×C4—21no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—41no—
646C2×C4—42no—
647C2×C4—88no—
648C2×C4—84no—
649C2×C4—162no—
650C2×C4C2×#12414yes—
651C2×C4C2×#12414yes—
652C2×C4C2×#12412yes—
653C2×C4#53512yes—
654C2×C4#53612yes—
655C2×C4#54218yes—
656C2×C4#54218yes—
657C2×C4#57612yes—
658C2×C4#57712yes—
659C2×C4C22×#4112yes—
660C2×C4C2×H(5,𝔽2)12yes—
661C2×C4C2×2−1+411yescharacteristic
662C2×C4C2×2−1+411yescharacteristic
663C8—42no—
664C8—42no—
665C8—44no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—81no—
673C22—82no—
674C22—81no—
675C22—82no—
676C22—81no—
677C22—82no—
678C22—81no—
679C22—82no—
680C22—81no—
681C22—82no—
682C22—81no—
683C22—82no—
684C22—81no—
685C22—82no—
686C22—168no—
687C22#264014yes—
688C22C22×#12411yescharacteristic
689C22C2×#53511yescharacteristic
690C22C2×#53611yescharacteristic
691C22C2×#57611yescharacteristic
692C22C2×#57711yescharacteristic
693C22#358511yescharacteristic
694C22#358611yescharacteristic
695C4—22no—
696C4—22no—
697C4—24no—
698C4—24no—
699C4—41no—
700C4—41no—
701C4—42no—
702C4—42no—
703C4—81no—
704C4—84no—
705C4—82no—
706C4—82no—
707C4—82no—
708C4—81no—
709C4—81no—
710C4#264014yes—
711C4C2×#54212yes—
712C4C2×#54212yes—
713C2—24no—
714C2—161no—
715C2—162no—
716C2C2×#264011yescharacteristic
717C2#6062211yescharacteristic
718C2#6062511yescharacteristic
719C2#6081111yescharacteristic
720C2#6081311yescharacteristic
721C2#6088811yescharacteristic
722C2#6089211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
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. #35281C2semidirect
3. #30216C2semidirect
4. #30213C2semidirect
5. #30211C2semidirect
6. #30209C2semidirect
7. #30208C2semidirect
8. #30203C2semidirect
9. #23634C2semidirect
10. #23632C2semidirect
11. #23454C2semidirect
12. #23452C2semidirect
13. #23451C2semidirect
14. C2×#2023C2semidirect
15. #21326C2semidirect
17. #20368C2semidirect
27. #2868C22semidirect
28. #2862C22semidirect
31. #2660C22semidirect
53. #2074C22semidirect
56. #2073C22semidirect
57. #2072C22semidirect
60. #2071C22semidirect
61. #2026C22semidirect
63. #2025C22semidirect
64. #2024C22semidirect
65. #2024C22semidirect
66. #2024C22semidirect
67. #2024C22semidirect
73. #2023C22semidirect
74. #2023C22semidirect
75. #2023C22semidirect
76. #2023C22semidirect
77. #1993C22semidirect
78. #1992C22semidirect
79. #1991C22semidirect
80. #1990C22semidirect
85. #1746C22semidirect
86. #1745C22semidirect
87. #1744C22semidirect

Maximal subgroups

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

subgroupindex
#352812
#352642
#302162
#302132
#302112
#302092
#302082
#302032
#236342
#236322
#234542
#234522
#234512
C2×#20232
#213262
#212062
#203682