Finite Group Databasebeta

← #473839

Subgroups — #473839

Order 512.

1300 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14702C211yesmaximal, characteristic
2#14686C211yesmaximal, characteristic
3#11315C212yesmaximal, complemented
4#11267C211yesmaximal, characteristic, complemented
5#11267C211yesmaximal, characteristic, complemented
6#11199C212yesmaximal
7#11190C212yesmaximal, complemented
8#11050C211yesmaximal, characteristic, complemented
9#11050C211yesmaximal, characteristic, complemented
10#10970C212yesmaximal
11#10712C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C23×C42C2211yescharacteristic
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×C22⋊C8C2212yes—
17C2×#391C2211yescharacteristic
18C2×#391C2211yescharacteristic
19C2×#384C2211yescharacteristic
20C2×#384C2211yescharacteristic
21C2×#380—21no—
22C2×#380—21no—
23C2×#380C2211yescharacteristic
24C2×#380C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#336C2211yescharacteristic
32C2×#336C2211yescharacteristic
33C2×#336C2212yes—
34#1436—21no—
35#1436—21no—
36C4⋊#95—21no—
37C4⋊#95—21no—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2212yes—
41C22⋊#95—21no—
42C22⋊#95—21no—
43C23×Q8C2311yescharacteristic
44C24×C4D411yescharacteristic
45C24×C4D411yescharacteristic
46C24×C4C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2312yes—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8—24no—
52C2×C22⋊Q8—42no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—22no—
66C22×C42—22no—
67C22×C42—22no—
68C22×C42—42no—
69C22×C42—42no—
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C2×#103—24no—
75C2×#103—42no—
76C2×#103—42no—
77C2×#103—42no—
78C2×#103—42no—
79C2×#103—42no—
80C2×C22⋊C8—24no—
81C2×C22⋊C8—42no—
82C2×C22⋊C8—42no—
83#391—22no—
84#391—22no—
85#391—42no—
86#384—42no—
87#384—42no—
88#380—24no—
89#380—42no—
90#380—42no—
91#380—44no—
92#380—44no—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—21no—
105#340—24no—
106#340—42no—
107#340—42no—
108#340—42no—
109#340—42no—
110#340—42no—
111#336—22no—
112#336—22no—
113#336—42no—
114C25C2×D411yescharacteristic
115C22×Q8—21no—
116C22×Q8—22no—
117C22×Q8—21no—
118C22×Q8—21no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8#4112yes—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—41no—
151C23×C4—41no—
152C23×C4—41no—
153C23×C4—41no—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4#4114yes—
160C23×C4C2411yescharacteristic
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22⋊Q8—48no—
176C22⋊Q8—48no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4#4112yes—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—42no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—48no—
234C2×C42—48no—
235C2×C42—41no—
236C2×C42—41no—
237C2×C42—41no—
238C2×C42—41no—
239C2×C42#4112yes—
240C2×C42#4112yes—
241#103—48no—
242#103—48no—
243C22⋊C8—48no—
244#95—44no—
245#95—44no—
246#95—42no—
247#95—42no—
248#95—42no—
249#95—42no—
250#95—42no—
251#95—42no—
252#95—42no—
253#95—42no—
254#95—44no—
255#95—42no—
256C24—21no—
257C24—21no—
258C24—21no—
259C24—21no—
260C24—21no—
261C24—21no—
262C24—21no—
263C24—21no—
264C24—21no—
265C24—21no—
266C24—21no—
267C24—21no—
268C24#12412yes—
269C24AGL(1,ℤ/8)12yes—
270C24#13712yes—
271C24C22×D411yescharacteristic
272C2×Q8—24no—
273C2×Q8—24no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—28no—
297C22×C4—28no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4#12312yes—
387C22×C4#12312yes—
388C22×C4#12412yes—
389C22×C4#13512yes—
390C22×C4#13512yes—
391C22×C4AGL(1,ℤ/8)12yes—
392C22×C4#13712yes—
393C22×C4#13712yes—
394C22×C4#13712yes—
395C22×C4C2×#4111yescharacteristic
396C22×C4C2×#4111yescharacteristic
397C22×C4C2×#4111yescharacteristic
398C22×C4C2×#4112yes—
399C22×C4C2×#4111yescharacteristic
400C22×C4H(5,𝔽2)12yes—
401C22×C4H(5,𝔽2)12yes—
402C22×C4H(5,𝔽2)11yescharacteristic
403C22×C4H(5,𝔽2)12yes—
404C22×C42−1+411yescharacteristic
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C4⋊C4—24no—
420C4⋊C4—48no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—84no—
435C4⋊C4—84no—
436C4⋊C4—84no—
437C4⋊C4—84no—
438C22⋊C4—48no—
439C22⋊C4—88no—
440C22⋊C4—84no—
441C42—24no—
442C42—24no—
443C42—44no—
444C42—44no—
445C42—42no—
446C42—42no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23#48914yes—
482C23C2×#12311yescharacteristic
483C23#52511yescharacteristic
484C23C2×#12411yescharacteristic
485C23#53411yescharacteristic
486C23#53411yescharacteristic
487C23#53511yescharacteristic
488C23#53711yescharacteristic
489C23C2×#13511yescharacteristic
490C23C2×#13511yescharacteristic
491C23C2×AGL(1,ℤ/8)11yescharacteristic
492C23C2×AGL(1,ℤ/8)11yescharacteristic
493C23C2×#13711yescharacteristic
494C23C2×#13711yescharacteristic
495C23C2×#13711yescharacteristic
496C23C2×#13711yescharacteristic
497Q8—44no—
498Q8—44no—
499Q8—82no—
500Q8—82no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—416no—
523C2×C4—416no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—81no—
547C2×C4—81no—
548C2×C4—81no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4—81no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—82no—
563C2×C4—81no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—82no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—82no—
580C2×C4—81no—
581C2×C4—82no—
582C2×C4#48514yes—
583C2×C4#48514yes—
584C2×C4#53812yes—
585C2×C4#53812yes—
586C2×C4#53812yes—
587C2×C4#53812yes—
588C2×C4#53912yes—
589C2×C4#54012yes—
590C2×C4#55112yes—
591C2×C4#55112yes—
592C2×C4#55211yescharacteristic
593C2×C4#55312yes—
594C2×C4#55311yescharacteristic
595C2×C4#55312yes—
596C2×C4#55511yescharacteristic
597C2×C4#55611yescharacteristic
598C2×C4#56012yes—
599C2×C4#56212yes—
600C8—82no—
601C8—82no—
602C22—24no—
603C22—24no—
604C22—24no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22#248511yescharacteristic
618C22#250411yescharacteristic
619C22#256111yescharacteristic
620C22#262211yescharacteristic
621C22#263511yescharacteristic
622C22#263911yescharacteristic
623C22#281311yescharacteristic
624C22C2×#48511yescharacteristic
625C22C2×#48511yescharacteristic
626C22#309211yescharacteristic
627C22#309211yescharacteristic
628C22#310611yescharacteristic
629C22#310711yescharacteristic
630C22#311211yescharacteristic
631C22#311311yescharacteristic
632C22#313111yescharacteristic
633C22C2×#48911yescharacteristic
634C22#313911yescharacteristic
635C22#314111yescharacteristic
636C22#318811yescharacteristic
637C22#318811yescharacteristic
638C22#319011yescharacteristic
639C22#319011yescharacteristic
640C22#319311yescharacteristic
641C22#319311yescharacteristic
642C22#319511yescharacteristic
643C22#319511yescharacteristic
644C22#321011yescharacteristic
645C22#321011yescharacteristic
646C22#321711yescharacteristic
647C22#321811yescharacteristic
648C22#324711yescharacteristic
649C22#324811yescharacteristic
650C22#325411yescharacteristic
651C22#325511yescharacteristic
652C4—24no—
653C4—24no—
654C4—42no—
655C4—42no—
656C4—42no—
657C4—82no—
658C4—82no—
659C4—161no—
660C4—162no—
661C4—162no—
662C4—161no—
663C4#275012yes—
664C4#275612yes—
665C4#277812yes—
666C4#278612yes—
667C2—44no—
668C2#1688411yescharacteristic
669C2#2763011yescharacteristic
670C2#2763111yescharacteristic
671C2#2797511yescharacteristic
672C2#2797811yescharacteristic
673C2#2910711yescharacteristic
674C2#2911111yescharacteristic
675C2#2989411yescharacteristic
676C2#2989411yescharacteristic
677C2#3011511yescharacteristic
678C2#3011611yescharacteristic
679C2#3018811yescharacteristic
680C2#3019911yescharacteristic
681C2#3272111yescharacteristic
682C2#3272511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11315C2semidirect
4. #11267C2semidirect
5. #11267C2semidirect
7. #11190C2semidirect
8. #11050C2semidirect
9. #11050C2semidirect

Maximal subgroups

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

subgroupindex
#147022
#146862
#113152
#112672
#111992
#111902
#110502
#109702
#107122