Finite Group Databasebeta

← #406713

Subgroups — #406713

Order 512.

1644 conjugacy classes of subgroups.

The subgroup lattice has 687 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2C2×#1825C212yesmaximal
3#13756C211yesmaximal, characteristic
4#11384C211yesmaximal, characteristic, complemented
5#11384C211yesmaximal, characteristic, complemented
6#11293C212yesmaximal, complemented
7#11292C212yesmaximal, complemented
8#11267C211yesmaximal, characteristic, complemented
9#11267C211yesmaximal, characteristic, complemented
10#10731C212yesmaximal
11#10712C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8—21no—
14C22×#117C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C23×C42C2211yescharacteristic
17C22×C22⋊C8C2212yes—
18#2857—28no—
19C2×#391—21no—
20C2×#389—21no—
21C2×#389C2212yes—
22C2×#384C2211yescharacteristic
23C2×#380—21no—
24C2×#380C2212yes—
25#1825—28no—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36C2×#336C2212yes—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2211yescharacteristic
41C22⋊#95—21no—
42C22⋊#95—21no—
43C4×#95—21no—
44C4×#95—21no—
45C23×Q8—21no—
46C24×C4D411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—42no—
55C2×C22⋊Q8—28no—
56C2×C22⋊Q8—44no—
57C2×C22⋊Q8—44no—
58C2×#117—22no—
59C2×#117—22no—
60C2×#117—22no—
61C2×#117—24no—
62C2×#117—24no—
63C2×#117—24no—
64C2×#117—42no—
65C2×#117Q814yescomplemented
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—22no—
79C22×C42—24no—
80C22×C42—21no—
81C22×C42—21no—
82C22×C42—22no—
83C22×C42—22no—
84C22×C42—42no—
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C2×C22⋊C8—24no—
90C2×C22⋊C8—24no—
91C2×C22⋊C8—24no—
92#391—28no—
93#389—28no—
94#389—24no—
95#389—24no—
96#384—24no—
97#380—28no—
98#380—24no—
99#380D418yes—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110#340—24no—
111#340—24no—
112#340—24no—
113#340—24no—
114#340—24no—
115#340—24no—
116#340—42no—
117#340—42no—
118#336—22no—
119#336—22no—
120#336—42no—
121C25C2×D411yescharacteristic
122C22×Q8—24no—
123C22×Q8—42no—
124C22×Q8—42no—
125C22×Q8—42no—
126C22×Q8—42no—
127C22×Q8—42no—
128C22×Q8—42no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—41no—
152C23×C4—41no—
153C23×C4—41no—
154C23×C4—41no—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D414yes—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×Q812yes—
162C23×C4#4114yes—
163C23×C4#4112yes—
164C23×C4C2411yescharacteristic
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×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C4⋊Q8—48no—
180C4⋊Q8—48no—
181C22⋊Q8—432no—
182#117—216no—
183#117—48no—
184#117—48no—
185#117—48no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
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—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4C2×D414yes—
221C2×C4⋊C4C2×D414yes—
222C2×C4⋊C4C2×Q812yes—
223C2×C4⋊C4C2×Q812yes—
224C2×C4⋊C4#4112yes—
225C2×C4⋊C4#4112yes—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—24no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—24no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—22no—
247C2×C42—48no—
248C2×C42—48no—
249C2×C42—41no—
250C2×C42—41no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42#4112yes—
256C2×C42#4112yes—
257C22⋊C8—48no—
258#95—28no—
259#95—28no—
260#95—42no—
261#95—42no—
262#95—42no—
263#95—42no—
264#95—42no—
265#95—42no—
266#95—42no—
267#95—42no—
268C24—21no—
269C24—21no—
270C24—21no—
271C24—21no—
272C24—21no—
273C24—21no—
274C24—21no—
275C24—21no—
276C24—21no—
277C24—21no—
278C24—21no—
279C24—21no—
280C24#12012yes—
281C24C22⋊Q812yes—
282C24#12412yes—
283C24C22×D411yescharacteristic
284C2×Q8—48no—
285C2×Q8—48no—
286C2×Q8—48no—
287C2×Q8—84no—
288C2×Q8—84no—
289C2×Q8—84no—
290C2×Q8—84no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—28no—
318C22×C4—28no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
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—42no—
384C22×C4—42no—
385C22×C4#12012yes—
386C22×C4C22⋊Q818yes—
387C22×C4C22⋊Q812yes—
388C22×C4#12312yes—
389C22×C4#12312yes—
390C22×C4#12418yes—
391C22×C4#12412yes—
392C22×C4#13512yes—
393C22×C4#13512yes—
394C22×C4AGL(1,ℤ/8)12yes—
395C22×C4AGL(1,ℤ/8)12yes—
396C22×C4#13712yes—
397C22×C4#13712yes—
398C22×C4C22×D412yes—
399C22×C4C22×Q811yescharacteristic
400C22×C4C2×#4112yes—
401C22×C4C2×#4111yescharacteristic
402C22×C4C2×#4111yescharacteristic
403C22×C4C2×#4111yescharacteristic
404C22×C4C2×#4111yescharacteristic
405C22×C4C2×#4111yescharacteristic
406C22×C4H(5,𝔽2)12yes—
407C22×C42−1+412yes—
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—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—48no—
431C4⋊C4—48no—
432C4⋊C4—416no—
433C4⋊C4—48no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C22⋊C4—48no—
443C22⋊C4—416no—
444C22⋊C4—48no—
445C42—24no—
446C42—24no—
447C42—44no—
448C42—44no—
449C42—48no—
450C42—42no—
451C42—42no—
452C42—48no—
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—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23#39314yes—
488C23C2×#12011yescharacteristic
489C23C2×C22⋊Q811yescharacteristic
490C23C2×#12311yescharacteristic
491C23#52511yescharacteristic
492C23C2×#12411yescharacteristic
493C23#53411yescharacteristic
494C23#53611yescharacteristic
495C23C2×#13511yescharacteristic
496C23C2×#13511yescharacteristic
497C23C2×AGL(1,ℤ/8)11yescharacteristic
498C23C2×AGL(1,ℤ/8)11yescharacteristic
499C23C2×#13711yescharacteristic
500C23C2×#13711yescharacteristic
501C23#57511yescharacteristic
502C23#57511yescharacteristic
503Q8—816no—
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—28no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—28no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—416no—
533C2×C4—416no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—82no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—82no—
577C2×C4—81no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4#39314yes—
585C2×C4#48214yes—
586C2×C4#48514yes—
587C2×C4C2×C22⋊Q812yes—
588C2×C4C2×#12412yes—
589C2×C4#52911yescharacteristic
590C2×C4#52911yescharacteristic
591C2×C4#53311yescharacteristic
592C2×C4#53311yescharacteristic
593C2×C4#53912yes—
594C2×C4#54412yes—
595C2×C4#54612yes—
596C2×C4#54612yes—
597C2×C4#54712yes—
598C2×C4#54712yes—
599C2×C4#55112yes—
600C2×C4#55112yes—
601C2×C4#55212yes—
602C2×C4#55212yes—
603C8—82no—
604C8—82no—
605C22—24no—
606C22—24no—
607C22—24no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22C2×#39311yescharacteristic
621C22#240511yescharacteristic
622C22#240611yescharacteristic
623C22#250311yescharacteristic
624C22#256111yescharacteristic
625C22#262711yescharacteristic
626C22#285511yescharacteristic
627C22#300211yescharacteristic
628C22#300211yescharacteristic
629C22#300311yescharacteristic
630C22#300411yescharacteristic
631C22#307611yescharacteristic
632C22#307611yescharacteristic
633C22#307911yescharacteristic
634C22#307911yescharacteristic
635C22C2×#48211yescharacteristic
636C22C2×#48511yescharacteristic
637C22#309411yescharacteristic
638C22#309511yescharacteristic
639C22#310611yescharacteristic
640C22#310711yescharacteristic
641C22#311511yescharacteristic
642C22#311611yescharacteristic
643C22#313111yescharacteristic
644C22#313511yescharacteristic
645C22#314111yescharacteristic
646C22#314111yescharacteristic
647C22#318811yescharacteristic
648C22#319011yescharacteristic
649C22#319411yescharacteristic
650C22#319611yescharacteristic
651C22#322811yescharacteristic
652C22#323111yescharacteristic
653C22#323511yescharacteristic
654C22#323711yescharacteristic
655C4—24no—
656C4—24no—
657C4—24no—
658C4—42no—
659C4—42no—
660C4—82no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4—84no—
665C4—84no—
666C4#243812yes—
667C4#243812yes—
668C4#248812yes—
669C4#248812yes—
670C2—44no—
671C2#1493411yescharacteristic
672C2#2345111yescharacteristic
673C2#2345211yescharacteristic
674C2#2400911yescharacteristic
675C2#2401011yescharacteristic
676C2#2406211yescharacteristic
677C2#2406511yescharacteristic
678C2#2796211yescharacteristic
679C2#2796811yescharacteristic
680C2#2910711yescharacteristic
681C2#2911311yescharacteristic
682C2#2997011yescharacteristic
683C2#2997111yescharacteristic
684C2#3315311yescharacteristic
685C2#3315411yescharacteristic
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
4. #11384C2semidirect
5. #11384C2semidirect
6. #11293C2semidirect
7. #11292C2semidirect
8. #11267C2semidirect
9. #11267C2semidirect
65. C2×#117Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#18252
#137562
#113842
#112932
#112922
#112672
#107312
#107122