Finite Group Databasebeta

← #6434773

Subgroups — #6434773

Order 512.

1852 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35285C211yesmaximal, characteristic
2#34272C211yesmaximal, characteristic, complemented
3#30071C214yesmaximal, complemented
4#30069C214yesmaximal, complemented
5#30066C212yesmaximal, complemented
6#30066C212yesmaximal, complemented
7#30062C212yesmaximal, complemented
8#30059C212yesmaximal, complemented
9#23735C212yesmaximal, complemented
10#23554C211yesmaximal, characteristic, complemented
11#23551C211yesmaximal, characteristic, complemented
12#23505C211yesmaximal, characteristic
13C2×#2041C211yesmaximal, characteristic
14#23456C212yesmaximal, complemented
15#21065C212yesmaximal, complemented
16#21063C212yesmaximal, complemented
17#20774C211yesmaximal, characteristic, complemented
18C2×#544C2211yescharacteristic
19C2×#533C2211yescharacteristic
20C22×#127C2211yescharacteristic
21C2×#428C2212yes—
22C2×#427C2212yes—
23C2×#418C2212yes—
24C2×#418C2212yes—
25C22×#102C2212yes—
26C22×#102C2212yes—
27#2858C2212yes—
28#2858C2212yes—
29#2641C2214yes—
30#2548—21no—
31#2548—21no—
32#2548—21no—
33#2548—21no—
34#2435—22no—
35#2435—22no—
36#2370—21no—
37#2370—21no—
38#2369—21no—
39#2368—22no—
40#2365—21no—
41#2360—22no—
42#2331C2212yescomplemented
43#2330C2212yescomplemented
44#2317C2211yescharacteristic
45#2309C2212yes—
46#2309C2212yes—
47C4×C4⋊Q8C2212yescomplemented
48C4×#125C2212yescomplemented
49#2296C2211yescharacteristic
50C2×#389C2211yescharacteristic
51#2274C2211yescharacteristic
52#2274C2211yescharacteristic
53C4×#117C2211yescharacteristic
54#2094C2214yescomplemented
55#2093C2214yescomplemented
56#2092C2214yescomplemented
57#2092C2214yescomplemented
58#2059C2214yescomplemented
59#2059C2212yescomplemented
60#2059C2212yescomplemented
61#2058C2212yescomplemented
62#2058C2211yescharacteristic, complemented
63#2058C2211yescharacteristic, complemented
64#2057C2212yescomplemented
65#2057C2211yescharacteristic, complemented
66#2057C2211yescharacteristic, complemented
67#2041C2212yes—
68#2041C2212yes—
69#2041C2212yes—
70#2041C2212yes—
71#2041C2212yes—
72#2041C2214yes—
73#2041C2212yes—
74#2025C2214yescomplemented
75#2025C2214yescomplemented
76#2023C2214yescomplemented
77#2023C2214yescomplemented
78#1926C2214yescomplemented
79#1926C2214yescomplemented
80#1926C2214yescomplemented
81#1926C2214yescomplemented
82#1925C2214yescomplemented
83#1925C2214yescomplemented
84#1925C2214yescomplemented
85#1925C2214yescomplemented
86#1840C2214yescomplemented
87#1839C2212yescomplemented
88#1839C2212yescomplemented
89#1736C2212yes—
90#1735C2212yes—
91#1729C2212yes—
92#1729C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C23×D4—21no—
96C23×D4—21no—
97C23×D4C2312yes—
98C22×M16C2312yes—
99C23×C8C2312yes—
100#544—24no—
101#544—24no—
102#533—44no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2311yescharacteristic
107C2×#127—21no—
108C2×#127—21no—
109C2×#127—44no—
110C2×#127C2312yes—
111C2×#127C2312yes—
112C2×#125—21no—
113C2×#125C2312yes—
114C2×#125C2312yes—
115C2×#124—22no—
116C2×#124—22no—
117C2×#124—22no—
118C2×#124—22no—
119C2×C22⋊Q8—21no—
120C2×C22⋊Q8—21no—
121C2×C22⋊Q8—22no—
122C2×C4×Q8—21no—
123C2×C4×Q8—21no—
124C2×C4×Q8—22no—
125C2×#117C2312yes—
126C2×#117C2311yescharacteristic
127C2×#117C2311yescharacteristic
128C2×#117C2311yescharacteristic
129C2×#117C2311yescharacteristic
130C22×C4⋊C4—21no—
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2311yescharacteristic
133C22×C42C2311yescharacteristic
134#428—44no—
135#427—44no—
136C2×#107C2312yes—
137C2×#107C2314yes—
138C2×#107C2312yes—
139C2×#106C2312yes—
140C2×#106C2314yes—
141C2×#106C2312yes—
142#418—28no—
143#418—44no—
144C2×#102—28no—
145C2×#102—44no—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154#394—24no—
155#394—24no—
156#394—24no—
157#394—24no—
158#391—22no—
159#391—22no—
160#391—22no—
161#390—22no—
162#390—22no—
163#390—22no—
164#390—22no—
165#390—22no—
166#390—22no—
167#390—22no—
168#390—22no—
169#390C2314yes—
170#390C2314yes—
171#390C2314yes—
172#390C2314yes—
173#389—21no—
174#389—21no—
175#389—22no—
176#389—21no—
177#389—21no—
178#389—22no—
179#389—22no—
180#389C2311yescharacteristic
181#389C2312yes—
182#389C2311yescharacteristic
183#389C2311yescharacteristic
184#389C2312yes—
185#389C2311yescharacteristic
186#387—22no—
187#387—22no—
188#387—22no—
189#387—22no—
190#387—22no—
191#387—22no—
192#385—21no—
193#385—21no—
194#385—22no—
195#384C2312yes—
196#382C2312yes—
197#379—24no—
198#379—24no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C22⋊C4—22no—
217C4×C22⋊C4—22no—
218C4×C22⋊C4D412yes—
219C4×C22⋊C4D412yes—
220#376C2312yes—
221C2×#95—21no—
222C43C2312yes—
223#340C2312yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#336C2311yescharacteristic
230#336C2312yes—
231#336C2311yescharacteristic
232#336C2312yes—
233#336C2312yes—
234#336C2314yes—
235#336C2314yes—
236#336C2314yes—
237#336C2314yes—
238C25—22no—
239C22×Q8—22no—
240C22×Q8—21no—
241C22×Q8—21no—
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—24no—
248C22×D4—24no—
249C22×D4—24no—
250C22×D4—24no—
251C22×D4—24no—
252C22×D4—24no—
253C22×D4—44no—
254C22×D4—44no—
255C22×D4—82no—
256C22×D4—82no—
257C22×D4—82no—
258C22×D4—82no—
259C22×D4C2414yes—
260C23×C4—21no—
261C23×C4—21no—
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2411yescharacteristic
265C2×M16—24no—
266C2×M16—24no—
267C2×M16—24no—
268C22×C8—24no—
269C22×C8—24no—
270C22×C8—24no—
271C22×C8C2412yes—
272C22×C8C2412yes—
273C22×C8C2414yes—
274C4⋊Q8—28no—
275C4⋊Q8—24no—
276C4⋊Q8—24no—
277C4⋊Q8—42no—
278C4⋊Q8—42no—
279#127—44no—
280#127—44no—
281#127—44no—
282#127—44no—
283#125—28no—
284#125—24no—
285#125—24no—
286#125—42no—
287#125—42no—
288#124—84no—
289#124—84no—
290#124—84no—
291#124—84no—
292C22⋊Q8—216no—
293C22⋊Q8—44no—
294C22⋊Q8—44no—
295C4×Q8—44no—
296C4×Q8—44no—
297C4×Q8—84no—
298#117—22no—
299#117—22no—
300#117—22no—
301#117—24no—
302#117—24no—
303#117—24no—
304#117—24no—
305#117#4114yes—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—82no—
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4#4114yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2411yescharacteristic
338C2×C4⋊C4C2411yescharacteristic
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4C2×D412yes—
351C2×C22⋊C4C2×D412yes—
352C2×C42—21no—
353C2×C42—24no—
354C2×C42—22no—
355C2×C42—21no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42C2×D412yes—
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D412yes—
364C2×C42C2×D412yes—
365C2×C42C2×D412yes—
366C2×C42C2412yes—
367C2×C42C2412yes—
368C2×C42C2411yescharacteristic
369C2×C42C2411yescharacteristic
370#107—42no—
371#107—42no—
372#107—44no—
373#106—42no—
374#106—42no—
375#106—44no—
376#102—28no—
377#102—28no—
378#102—28no—
379#102—28no—
380#102—44no—
381#102—44no—
382#102—44no—
383#102—44no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—21no—
391#95—21no—
392#95—22no—
393#95—22no—
394#95—21no—
395#95—21no—
396#95C2×D412yes—
397#95C2×D412yes—
398#95C2×D412yes—
399#95C2×D412yes—
400C24—24no—
401C24—44no—
402C24—44no—
403C24—44no—
404C24—82no—
405C24—82no—
406C24—82no—
407C24—82no—
408C24C22×D411yescharacteristic
409C2×Q8—28no—
410C2×Q8—42no—
411C2×Q8—42no—
412C2×Q8—42no—
413C2×Q8—42no—
414C2×Q8—42no—
415C2×Q8—42no—
416C2×Q8—82no—
417C2×Q8—82no—
418C2×D4—24no—
419C2×D4—24no—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—48no—
434C2×D4—48no—
435C2×D4—48no—
436C2×D4—48no—
437C2×D4—44no—
438C2×D4—44no—
439C2×D4—82no—
440C2×D4—82no—
441C2×D4—82no—
442C2×D4—82no—
443C2×D4—816no—
444C2×D4—816no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—42no—
461C22×C4—81no—
462C22×C4—81no—
463C22×C4—81no—
464C22×C4—81no—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C2×#4112yes—
475C22×C4C2×#4112yes—
476C22×C4H(5,𝔽2)14yes—
477C22×C42−1+414yes—
478C22×C4C2511yescharacteristic
479M16—48no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—24no—
483C2×C8—22no—
484C2×C8—24no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—24no—
489C2×C8—48no—
490C4⋊C4—28no—
491C4⋊C4—28no—
492C4⋊C4—28no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—82no—
517C4⋊C4—81no—
518C4⋊C4—81no—
519C4⋊C4—82no—
520C4⋊C4—82no—
521C4⋊C4C2×#4114yes—
522C4⋊C4C2×#4112yes—
523C4⋊C4C2×#4112yes—
524C22⋊C4—24no—
525C22⋊C4—24no—
526C22⋊C4—24no—
527C22⋊C4—42no—
528C22⋊C4—44no—
529C22⋊C4—84no—
530C22⋊C4—84no—
531C22⋊C4—84no—
532C22⋊C4—84no—
533C22⋊C4—84no—
534C22⋊C4—84no—
535C22⋊C4—84no—
536C22⋊C4—84no—
537C42—24no—
538C42—22no—
539C42—22no—
540C42—24no—
541C42—24no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—24no—
547C42—42no—
548C42—42no—
549C42—42no—
550C42—44no—
551C42—82no—
552C42—82no—
553C42—82no—
554C42—82no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—88no—
567C23—88no—
568C23—88no—
569C23—164no—
570C23—164no—
571C23—164no—
572C23—164no—
573C23#52512yes—
574C23#53512yes—
575C23#53612yes—
576C23C23×D411yescharacteristic
577Q8—84no—
578Q8—84no—
579Q8—88no—
580D4—44no—
581D4—44no—
582D4—44no—
583D4—44no—
584D4—88no—
585D4—88no—
586D4—88no—
587D4—88no—
588D4—88no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—21no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—21no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—41no—
637C2×C4—84no—
638C2×C4—162no—
639C2×C4#52512yes—
640C2×C4#53512yes—
641C2×C4#53612yes—
642C2×C4#54218yes—
643C2×C4#54218yes—
644C2×C4#57612yes—
645C2×C4#57712yes—
646C2×C4C22×#4111yescharacteristic
647C2×C4C22×#4111yescharacteristic
648C2×C4C2×H(5,𝔽2)12yes—
649C2×C4C2×2−1+412yes—
650C8—42no—
651C8—42no—
652C8—44no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—84no—
660C22—84no—
661C22—84no—
662C22—84no—
663C22—84no—
664C22—84no—
665C22—84no—
666C22—1616no—
667C22#263214yes—
668C22C2×#52511yescharacteristic
669C22C2×#53511yescharacteristic
670C22C2×#53611yescharacteristic
671C22C2×#57611yescharacteristic
672C22C2×#57711yescharacteristic
673C22#358511yescharacteristic
674C22#358511yescharacteristic
675C4—22no—
676C4—22no—
677C4—41no—
678C4—41no—
679C4—42no—
680C4—44no—
681C4—42no—
682C4—42no—
683C4—42no—
684C4—42no—
685C4—42no—
686C4—81no—
687C4—82no—
688C4—82no—
689C4—81no—
690C4—82no—
691C4#263214yes—
692C4C2×#54212yes—
693C4C2×#54212yes—
694C2—24no—
695C2—164no—
696C2C2×#263211yescharacteristic
697C2#6059011yescharacteristic
698C2#6059311yescharacteristic
699C2#6081111yescharacteristic
700C2#6081211yescharacteristic
701C2#6087711yescharacteristic
702C2#6088311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
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
2. #34272C2semidirect
3. #30071C2semidirect
4. #30069C2semidirect
5. #30066C2semidirect
6. #30066C2semidirect
7. #30062C2semidirect
8. #30059C2semidirect
9. #23735C2semidirect
10. #23554C2semidirect
11. #23551C2semidirect
14. #23456C2semidirect
15. #21065C2semidirect
16. #21063C2semidirect
17. #20774C2semidirect
42. #2331C22semidirect
43. #2330C22semidirect
47. C4×C4⋊Q8C22semidirect
48. C4×#125C22semidirect
54. #2094C22semidirect
55. #2093C22semidirect
56. #2092C22semidirect
57. #2092C22semidirect
58. #2059C22semidirect
59. #2059C22semidirect
60. #2059C22semidirect
61. #2058C22semidirect
62. #2058C22semidirect
63. #2058C22semidirect
64. #2057C22semidirect
65. #2057C22semidirect
66. #2057C22semidirect
74. #2025C22semidirect
75. #2025C22semidirect
76. #2023C22semidirect
77. #2023C22semidirect
78. #1926C22semidirect
79. #1926C22semidirect
80. #1926C22semidirect
81. #1926C22semidirect
82. #1925C22semidirect
83. #1925C22semidirect
84. #1925C22semidirect
85. #1925C22semidirect
86. #1840C22semidirect
87. #1839C22semidirect
88. #1839C22semidirect

Maximal subgroups

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

subgroupindex
#352852
#342722
#300712
#300692
#300662
#300622
#300592
#237352
#235542
#235512
#235052
C2×#20412
#234562
#210652
#210632
#207742