Finite Group Databasebeta

← #401635

Subgroups — #401635

Order 512.

1636 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2296C211yesmaximal, characteristic
2C2×#2084C211yesmaximal, characteristic, complemented
3C2×#2083C211yesmaximal, characteristic, complemented
4C2×#1854C212yesmaximal
5#14686C211yesmaximal, characteristic
6#11220C212yesmaximal, complemented
7#11219C212yesmaximal, complemented
8#11207C211yesmaximal, characteristic, complemented
9#11207C211yesmaximal, characteristic, complemented
10#10731C212yesmaximal
11#10712C211yesmaximal, characteristic
12C22×#117C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C22×#107C2212yes—
16C22×#106C2212yes—
17C22×C22⋊C8C2212yes—
18#2296—24no—
19#2296—24no—
20C2×#389C2211yescharacteristic
21C2×#389C2211yescharacteristic
22C2×#389C2211yescharacteristic
23C2×#389C2211yescharacteristic
24C2×#385—21no—
25C2×#385—21no—
26C2×#384C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C2×#380—21no—
29C2×#380—21no—
30C2×#380C2212yes—
31C2×C4×C4⋊C4—21no—
32C2×C4×C4⋊C4—21no—
33#2084—24no—
34#2083—24no—
35#1854—28no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2211yescharacteristic
43C2×#336C2211yescharacteristic
44C2×#336C2212yes—
45C4⋊#95—21no—
46C4⋊#95—21no—
47C4⋊#95C2212yes—
48C24×C4D411yescharacteristic
49C24×C4D411yescharacteristic
50C24×C4C2311yescharacteristic
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×#117—22no—
54C2×#117—22no—
55C2×#117—22no—
56C2×#117—22no—
57C2×#117—22no—
58C2×#117—22no—
59C2×#117—22no—
60C2×#117—24no—
61C2×#117—42no—
62C2×#117D414yes—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—22no—
80C22×C42—21no—
81C22×C42—21no—
82C22×C42—24no—
83C22×C42—22no—
84C22×C42—22no—
85C22×C42—42no—
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C2×#107—24no—
91C2×#107—24no—
92C2×#107—24no—
93C2×#106—24no—
94C2×#106—24no—
95C2×#106—24no—
96C2×C22⋊C8—24no—
97C2×C22⋊C8—24no—
98C2×C22⋊C8Q818yes—
99#389—22no—
100#389—22no—
101#389—22no—
102#389—22no—
103#389—22no—
104#389—22no—
105#389D414yes—
106#389D414yes—
107#385—28no—
108#385—28no—
109#384—22no—
110#384—22no—
111#384—42no—
112#380—24no—
113#380—24no—
114#380—44no—
115#380—44no—
116C4×C4⋊C4—28no—
117C4×C4⋊C4—28no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—22no—
125#340—24no—
126#340—24no—
127#340—24no—
128#340—24no—
129#340—42no—
130#340—42no—
131#336—22no—
132#336—22no—
133#336—42no—
134C25C2×D411yescharacteristic
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—41no—
158C23×C4—41no—
159C23×C4—41no—
160C23×C4—41no—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D412yes—
167C23×C4C2×Q814yes—
168C23×C4#4112yes—
169C23×C4#4114yes—
170C23×C4C2411yescharacteristic
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8C2×Q814yes—
184C22×C8C2×Q814yes—
185#117—28no—
186#117—28no—
187#117—28no—
188#117—28no—
189#117—48no—
190#117—48no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4C2×D412yes—
240C2×C4⋊C4C2×D412yes—
241C2×C4⋊C4C2×D412yes—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4C2×D412yes—
245C2×C4⋊C4#4112yes—
246C2×C4⋊C4#4112yes—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—24no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—48no—
271C2×C42—48no—
272C2×C42—41no—
273C2×C42—41no—
274C2×C42—42no—
275C2×C42—42no—
276C2×C42C2×D412yes—
277C2×C42C2×D412yes—
278#107—48no—
279#106—48no—
280C22⋊C8—216no—
281#95—24no—
282#95—24no—
283#95—24no—
284#95—24no—
285#95—44no—
286#95—44no—
287#95—44no—
288C24—21no—
289C24—21no—
290C24—21no—
291C24—21no—
292C24—21no—
293C24—21no—
294C24—21no—
295C24—21no—
296C24—21no—
297C24—21no—
298C24—21no—
299C24—21no—
300C24#12112yes—
301C24#12312yes—
302C24C4⋊Q812yes—
303C24C22×D411yescharacteristic
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—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—28no—
331C22×C4—28no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—44no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
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—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4#12012yes—
399C22×C4#12012yes—
400C22×C4#12112yes—
401C22×C4C22⋊Q818yes—
402C22×C4#12312yes—
403C22×C4#12518yes—
404C22×C4C4⋊Q812yes—
405C22×C4#13512yes—
406C22×C4#13512yes—
407C22×C4AGL(1,ℤ/8)12yes—
408C22×C4AGL(1,ℤ/8)12yes—
409C22×C4#13712yes—
410C22×C4#13712yes—
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×Q812yes—
415C22×C4C2×#4111yescharacteristic
416C22×C4C2×#4111yescharacteristic
417C22×C4C2×#4112yes—
418C22×C4C2×#4111yescharacteristic
419C22×C4H(5,𝔽2)12yes—
420C22×C42−1+412yes—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—48no—
444C4⋊C4—48no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—48no—
448C4⋊C4—48no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—84no—
458C4⋊C4—84no—
459C4⋊C4—84no—
460C4⋊C4—84no—
461C4⋊C4#12114yes—
462C4⋊C4#12114yes—
463C22⋊C4—216no—
464C22⋊C4—48no—
465C22⋊C4—88no—
466C42—216no—
467C42—24no—
468C42—24no—
469C42—44no—
470C42—44no—
471C42—48no—
472C42—42no—
473C42—42no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23#39814yes—
509C23C2×#12011yescharacteristic
510C23C2×#12111yescharacteristic
511C23C2×#12311yescharacteristic
512C23#52511yescharacteristic
513C23C2×C4⋊Q811yescharacteristic
514C23#53411yescharacteristic
515C23#53611yescharacteristic
516C23C2×#13511yescharacteristic
517C23C2×#13511yescharacteristic
518C23C2×AGL(1,ℤ/8)11yescharacteristic
519C23C2×AGL(1,ℤ/8)11yescharacteristic
520C23C2×#13711yescharacteristic
521C23C2×#13711yescharacteristic
522C23#57511yescharacteristic
523C23#57511yescharacteristic
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—28no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—416no—
555C2×C4—416no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4—81no—
589C2×C4—81no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4#39814yes—
605C2×C4#44914yes—
606C2×C4#45214yes—
607C2×C4C2×#12111yescharacteristic
608C2×C4C2×#12111yescharacteristic
609C2×C4C2×C22⋊Q812yes—
610C2×C4#52511yescharacteristic
611C2×C4#52511yescharacteristic
612C2×C4C2×#12512yes—
613C2×C4#53712yes—
614C2×C4#53912yes—
615C2×C4#54612yes—
616C2×C4#54612yes—
617C2×C4D4×Q812yes—
618C2×C4D4×Q812yes—
619C2×C4#55212yes—
620C2×C4#55212yes—
621C2×C4#55412yes—
622C2×C4#55412yes—
623C8—44no—
624C8—44no—
625C22—24no—
626C22—24no—
627C22—24no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22C2×#39811yescharacteristic
641C22#240511yescharacteristic
642C22#243411yescharacteristic
643C22#248411yescharacteristic
644C22#258311yescharacteristic
645C22#261811yescharacteristic
646C22#284711yescharacteristic
647C22C2×#44911yescharacteristic
648C22C2×#45211yescharacteristic
649C22#300611yescharacteristic
650C22#300711yescharacteristic
651C22#303511yescharacteristic
652C22#303511yescharacteristic
653C22#303811yescharacteristic
654C22#303911yescharacteristic
655C22#309111yescharacteristic
656C22#309111yescharacteristic
657C22#309211yescharacteristic
658C22#309211yescharacteristic
659C22#310611yescharacteristic
660C22#310711yescharacteristic
661C22#311511yescharacteristic
662C22#311611yescharacteristic
663C22#316011yescharacteristic
664C22#316211yescharacteristic
665C22#316511yescharacteristic
666C22#316511yescharacteristic
667C22#318811yescharacteristic
668C22#319011yescharacteristic
669C22#319411yescharacteristic
670C22#319611yescharacteristic
671C22#322811yescharacteristic
672C22#323111yescharacteristic
673C22#323511yescharacteristic
674C22#323711yescharacteristic
675C4—24no—
676C4—24no—
677C4—24no—
678C4—44no—
679C4—44no—
680C4—42no—
681C4—42no—
682C4—82no—
683C4—82no—
684C4—162no—
685C4—162no—
686C4#241712yes—
687C4#241712yes—
688C4#247312yes—
689C4#247312yes—
690C2—44no—
691C2#1482111yescharacteristic
692C2#2365611yescharacteristic
693C2#2365811yescharacteristic
694C2#2400411yescharacteristic
695C2#2400711yescharacteristic
696C2#2581211yescharacteristic
697C2#2581811yescharacteristic
698C2#2761211yescharacteristic
699C2#2761511yescharacteristic
700C2#2929911yescharacteristic
701C2#2930411yescharacteristic
702C2#2975811yescharacteristic
703C2#2975911yescharacteristic
704C2#3290411yescharacteristic
705C2#3290711yescharacteristic
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
2. C2×#2084C2semidirect
3. C2×#2083C2semidirect
6. #11220C2semidirect
7. #11219C2semidirect
8. #11207C2semidirect
9. #11207C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22962
C2×#20842
C2×#20832
C2×#18542
#146862
#112202
#112192
#112072
#107312
#107122