Finite Group Databasebeta

← #357690

Subgroups — #357690

Order 512.

2734 conjugacy classes of subgroups.

The subgroup lattice has 794 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#430C211yesmaximal, characteristic, complemented
2C2×#2896C211yesmaximal, characteristic
3C22×#417C211yesmaximal, characteristic
4C2×#2879C211yesmaximal, characteristic
5C2×#1742C211yesmaximal, characteristic, complemented
6C22×#343C211yesmaximal, characteristic, complemented
7C2×#1736C211yesmaximal, characteristic, complemented
8#11105C218yesmaximal, complemented
9C23×#41C2211yescharacteristic
10C23×M16C2211yescharacteristic
11C23×M16C2211yescharacteristic
12C23×M16C2211yescharacteristic
13C23×M16C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C2×#430—24no—
17C2×#430—24no—
18C2×#430—24no—
19C22×#108C2214yescomplemented
20#2896—24no—
21#2896—24no—
22C2×#417—22no—
23C2×#417—24no—
24C2×#417—24no—
25C2×#417C2214yes—
26C2×#417C2214yes—
27C2×#419—21no—
28C2×#418—21no—
29C2×#416—22no—
30C22×#103—21no—
31C22×#102—21no—
32#2879—24no—
33#2879C418yescomplemented
34C2×#408—22no—
35C2×#408C2214yes—
36C2×#407—22no—
37#1742—24no—
38C2×#343—24no—
39C2×#343C2214yescomplemented
40C2×#343C2214yescomplemented
41#1736—22no—
42#1736—22no—
43C2×#340C2214yescomplemented
44C22×#41—21no—
45C22×#41—22no—
46C22×#41—21no—
47C22×#41—22no—
48C22×#41—22no—
49C22×#41—22no—
50C22×#41—22no—
51C22×#41—22no—
52C22×#41—22no—
53C22×#41—22no—
54C22×#41—22no—
55C22×#41—22no—
56C22×#41C2×C414yes—
57C22×#41D412yes—
58C22×#41Q814yes—
59C22×#41C2312yes—
60C23×Q8D411yescharacteristic
61C23×D4—21no—
62C23×D4D411yescharacteristic
63C24×C4—21no—
64C24×C4—21no—
65C24×C4C2311yescharacteristic
66C22×M16—22no—
67C22×M16—22no—
68C22×M16—22no—
69C22×M16—22no—
70C22×M16—21no—
71C22×M16—22no—
72C22×M16—21no—
73C22×M16—22no—
74C22×M16—21no—
75C22×M16—22no—
76C22×M16—22no—
77C22×M16—22no—
78C22×M16—22no—
79C22×M16—22no—
80C22×M16—22no—
81C22×M16—22no—
82C22×M16—22no—
83C22×M16—22no—
84C22×M16—22no—
85C22×M16—22no—
86C22×M16—22no—
87C22×M16—22no—
88C22×M16C2×C414yes—
89C22×M16C2×C414yes—
90C22×M16D414yes—
91C22×M16D414yes—
92C22×M16D412yes—
93C22×M16D412yes—
94C22×M16D414yes—
95C22×M16C2312yes—
96C22×M16C2312yes—
97C22×M16C2312yes—
98C23×C8—21no—
99C23×C8—21no—
100C23×C8—21no—
101C23×C8C2312yes—
102C2×#117—22no—
103C2×#117—21no—
104C2×#117—21no—
105C2×#117—22no—
106C2×#117—21no—
107C2×#117—21no—
108C2×#117—22no—
109C2×#117D414yes—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—22no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—22no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4D414yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C22⋊C4—21no—
122C22×C42—21no—
123#430—232no—
124C2×#108—28no—
125C2×#108—28no—
126C2×#108—28no—
127#419—28no—
128#418—28no—
129#417—216no—
130#417—28no—
131#417—28no—
132#417—28no—
133#417—28no—
134#416—216no—
135C2×#103—28no—
136C2×#103—24no—
137C2×#103—24no—
138C2×#102—28no—
139C2×#102—24no—
140C2×#102—24no—
141#408—216no—
142#408—28no—
143#408C2×C4116yescomplemented
144#407—216no—
145C2×C22⋊C8—28no—
146#343—28no—
147#343—28no—
148#340—28no—
149C25—21no—
150C25#4111yescharacteristic
151C2×#41—24no—
152C2×#41—24no—
153C2×#41—24no—
154C2×#41—24no—
155C2×#41—24no—
156C2×#41—22no—
157C2×#41—22no—
158C2×#41—28no—
159C2×#41—28no—
160C2×#41—28no—
161C2×#41—28no—
162C2×#41—24no—
163C2×#41—24no—
164C2×#41—28no—
165C2×#41—28no—
166C2×#41—28no—
167C2×#41—28no—
168C2×#41—24no—
169C2×#41—24no—
170C2×#41—22no—
171C2×#41—22no—
172C2×#41—22no—
173C2×#41—22no—
174C2×#41—22no—
175C2×#41—22no—
176C2×#41—22no—
177C2×#41—22no—
178C2×#41—44no—
179C2×#41—42no—
180C2×#41—42no—
181C2×#41—42no—
182C2×#41—42no—
183C2×#41—44no—
184C2×#41—44no—
185C2×#41—44no—
186C2×#41—42no—
187C2×#41—42no—
188C2×#41—44no—
189C2×#41—42no—
190C2×#41—42no—
191C2×#41—44no—
192C2×#41—44no—
193C2×#41—44no—
194C2×#41C4⋊C418yes—
195C2×#41C22×C414yes—
196C2×#41C2×Q814yes—
197C22×Q8—21no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—22no—
203C22×Q8C4⋊C414yes—
204C22×Q8C2×D412yes—
205C22×D4—24no—
206C22×D4—22no—
207C22×D4—24no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—21no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—42no—
216C22×D4—42no—
217C22×D4—42no—
218C22×D4—42no—
219C22×D4C4⋊C414yes—
220C22×D4C2×D412yes—
221C23×C4—21no—
222C23×C4—24no—
223C23×C4—22no—
224C23×C4—24no—
225C23×C4—22no—
226C23×C4—24no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4—42no—
232C23×C4—42no—
233C23×C4—41no—
234C23×C4—41no—
235C23×C4—42no—
236C23×C4—84no—
237C23×C4C22×C412yes—
238C23×C4C2×D412yes—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D412yes—
244C23×C4C2×D412yes—
245C23×C4C2×D412yes—
246C23×C4C2×D412yes—
247C23×C4C2×Q812yes—
248C23×C4#4111yescharacteristic
249C23×C4#4112yes—
250C23×C4#4112yes—
251C23×C4#4112yes—
252C23×C4#4112yes—
253C23×C4#4112yes—
254C23×C4C2411yescharacteristic
255C2×M16—28no—
256C2×M16—28no—
257C2×M16—28no—
258C2×M16—24no—
259C2×M16—24no—
260C2×M16—24no—
261C2×M16—24no—
262C2×M16—28no—
263C2×M16—24no—
264C2×M16—24no—
265C2×M16—24no—
266C2×M16—22no—
267C2×M16—22no—
268C2×M16—22no—
269C2×M16—28no—
270C2×M16—28no—
271C2×M16—28no—
272C2×M16—28no—
273C2×M16—28no—
274C2×M16—28no—
275C2×M16—22no—
276C2×M16—22no—
277C2×M16—22no—
278C2×M16—22no—
279C2×M16—22no—
280C2×M16—22no—
281C2×M16—22no—
282C2×M16—22no—
283C2×M16—22no—
284C2×M16—22no—
285C2×M16—22no—
286C2×M16—22no—
287C2×M16—42no—
288C2×M16—42no—
289C2×M16—42no—
290C2×M16—42no—
291C2×M16—42no—
292C2×M16—42no—
293C2×M16—44no—
294C2×M16—44no—
295C2×M16—44no—
296C2×M16—44no—
297C2×M16—44no—
298C2×M16—44no—
299C2×M16C22⋊C418yes—
300C2×M16C22⋊C418yes—
301C2×M16C22×C414yes—
302C2×M16C2×D414yes—
303C2×M16C2×D414yes—
304C22×C8—22no—
305C22×C8—22no—
306C22×C8—22no—
307C22×C8—22no—
308C22×C8—24no—
309C22×C8—24no—
310C22×C8—24no—
311C22×C8—22no—
312C22×C8—22no—
313C22×C8—22no—
314C22×C8—42no—
315C22×C8—42no—
316C22×C8—42no—
317C22×C8—42no—
318C22×C8—42no—
319C22×C8—42no—
320C22×C8C22×C414yes—
321C22×C8C2×D414yes—
322C22×C8C2×D414yes—
323#117—24no—
324#117—24no—
325#117—24no—
326#117—24no—
327#117—42no—
328#117—42no—
329#117—42no—
330#117—42no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—42no—
347C2×C4⋊C4—42no—
348C2×C4⋊C4—41no—
349C2×C4⋊C4—41no—
350C2×C4⋊C4—42no—
351C2×C4⋊C4—42no—
352C2×C4⋊C4—42no—
353C2×C4⋊C4C2×D414yes—
354C2×C4⋊C4C2×D414yes—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—41no—
357C2×C22⋊C4—41no—
358C2×C22⋊C4—42no—
359C2×C42—24no—
360C2×C42—41no—
361C2×C42—41no—
362C2×C42—42no—
363#108—232no—
364#103—216no—
365#103—216no—
366#102—216no—
367#102—216no—
368C22⋊C8—232no—
369C24—22no—
370C24—24no—
371C24—42no—
372C24—42no—
373C24—42no—
374C24—42no—
375C24—82no—
376C24—82no—
377C24#11712yes—
378C24#12111yescharacteristic
379C24#12111yescharacteristic
380C24C22⋊Q812yes—
381C24#12312yes—
382C24#12312yes—
383C24#12412yes—
384C24#12412yes—
385C24C2×#4111yescharacteristic
386#41—24no—
387#41—24no—
388#41—24no—
389#41—24no—
390#41—24no—
391#41—24no—
392#41—24no—
393#41—24no—
394#41—216no—
395#41—216no—
396#41—42no—
397#41—42no—
398#41—42no—
399#41—42no—
400#41—42no—
401#41—42no—
402#41—42no—
403#41—42no—
404#41—48no—
405#41—48no—
406#41—48no—
407#41—48no—
408#41—48no—
409#41—48no—
410#41—48no—
411#41—48no—
412#41—48no—
413#41—48no—
414C2×Q8—22no—
415C2×Q8—24no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—28no—
421C2×Q8—28no—
422C2×Q8—44no—
423C2×Q8—42no—
424C2×Q8—42no—
425C2×Q8—44no—
426C2×Q8C2×C4⋊C414yes—
427C2×D4—24no—
428C2×D4—24no—
429C2×D4—28no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—24no—
436C2×D4—28no—
437C2×D4—28no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4—42no—
441C2×D4—42no—
442C2×D4—48no—
443C2×D4—48no—
444C2×D4—48no—
445C2×D4—44no—
446C2×D4—44no—
447C2×D4—48no—
448C2×D4—44no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4—44no—
452C2×D4C2×C4⋊C414yes—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—28no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—84no—
479C22×C4—84no—
480C22×C4—88no—
481C22×C4—82no—
482C22×C4—82no—
483C22×C4—84no—
484C22×C4—88no—
485C22×C4C2×C22⋊C412yes—
486C22×C4C2×C22⋊C412yes—
487C22×C4C2×C4⋊C412yes—
488C22×C4C2×C4⋊C412yes—
489C22×C4#11712yes—
490C22×C4C4×D414yes—
491C22×C4C4×D414yes—
492C22×C4C4×D414yes—
493C22×C4C4×D414yes—
494C22×C4#12012yes—
495C22×C4#12012yes—
496C22×C4#12012yes—
497C22×C4#12012yes—
498C22×C4#12014yes—
499C22×C4#12112yes—
500C22×C4#12112yes—
501C22×C4#12112yes—
502C22×C4#12112yes—
503C22×C4#12112yes—
504C22×C4#12112yes—
505C22×C4#12114yes—
506C22×C4#12114yes—
507C22×C4#12112yes—
508C22×C4#12112yes—
509C22×C4#12112yes—
510C22×C4#12112yes—
511C22×C4#12111yescharacteristic
512C22×C4#12111yescharacteristic
513C22×C4C22⋊Q812yes—
514C22×C4C22⋊Q812yes—
515C22×C4C22⋊Q814yes—
516C22×C4C22⋊Q814yes—
517C22×C4C22⋊Q814yes—
518C22×C4C22⋊Q814yes—
519C22×C4C22⋊Q812yes—
520C22×C4#12312yes—
521C22×C4#12312yes—
522C22×C4#12312yes—
523C22×C4#12312yes—
524C22×C4#12314yes—
525C22×C4#12312yes—
526C22×C4#12312yes—
527C22×C4#12414yes—
528C22×C4#12412yes—
529C22×C4#12412yes—
530C22×C4#12614yes—
531C22×C4#12614yes—
532C22×C4#12714yes—
533C22×C4C23×C411yescharacteristic
534C22×C4C22×D411yescharacteristic
535C22×C4C22×D411yescharacteristic
536C22×C4C22×D411yescharacteristic
537C22×C4C22×D411yescharacteristic
538C22×C4C22×D411yescharacteristic
539C22×C4C22×D411yescharacteristic
540C22×C4C22×D411yescharacteristic
541C22×C4C22×Q811yescharacteristic
542C22×C4C2×#4111yescharacteristic
543C22×C4C2×#4111yescharacteristic
544C22×C4C2×#4111yescharacteristic
545C22×C4C2×#4111yescharacteristic
546C22×C4C2×#4111yescharacteristic
547M16—216no—
548M16—216no—
549M16—24no—
550M16—24no—
551M16—24no—
552M16—24no—
553M16—24no—
554M16—24no—
555M16—42no—
556M16—42no—
557M16—42no—
558M16—42no—
559M16—42no—
560M16—42no—
561M16—48no—
562M16—48no—
563M16—48no—
564C2×C8—24no—
565C2×C8—24no—
566C2×C8—24no—
567C2×C8—24no—
568C2×C8—24no—
569C2×C8—24no—
570C2×C8—24no—
571C2×C8—24no—
572C2×C8—24no—
573C2×C8—42no—
574C2×C8—42no—
575C2×C8—42no—
576C2×C8—42no—
577C2×C8—42no—
578C2×C8—42no—
579C2×C8—48no—
580C2×C8—48no—
581C2×C8—48no—
582C2×C8C2×C22⋊C418yes—
583C4⋊C4—24no—
584C4⋊C4—24no—
585C4⋊C4—24no—
586C4⋊C4—24no—
587C4⋊C4—42no—
588C4⋊C4—42no—
589C4⋊C4—42no—
590C4⋊C4—42no—
591C4⋊C4—44no—
592C4⋊C4—44no—
593C22⋊C4—44no—
594C22⋊C4—44no—
595C42—44no—
596C42—44no—
597C23—22no—
598C23—22no—
599C23—22no—
600C23—22no—
601C23—22no—
602C23—22no—
603C23—24no—
604C23—42no—
605C23—42no—
606C23—44no—
607C23—44no—
608C23—42no—
609C23—44no—
610C23—42no—
611C23—44no—
612C23—42no—
613C23—42no—
614C23—48no—
615C23—44no—
616C23—44no—
617C23—88no—
618C23—84no—
619C23—84no—
620C23—88no—
621C23#38012yes—
622C23#38012yes—
623C23#38814yes—
624C23#38814yes—
625C23#39314yes—
626C23#39412yes—
627C23#39412yes—
628C23#39412yes—
629C23#39412yes—
630C23#39914yes—
631C23C2×#11711yescharacteristic
632C23C2×#12111yescharacteristic
633C23C2×C22⋊Q811yescharacteristic
634C23C2×#12311yescharacteristic
635C23C2×#12311yescharacteristic
636C23C2×#12411yescharacteristic
637C23C2×#12411yescharacteristic
638Q8—24no—
639Q8—24no—
640Q8—42no—
641Q8—42no—
642Q8—48no—
643D4—24no—
644D4—24no—
645D4—44no—
646D4—44no—
647D4—416no—
648D4—44no—
649D4—44no—
650D4—42no—
651D4—42no—
652D4—48no—
653C2×C4—24no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—48no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—416no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—84no—
665C2×C4—88no—
666C2×C4—88no—
667C2×C4—84no—
668C2×C4—82no—
669C2×C4—84no—
670C2×C4—82no—
671C2×C4—84no—
672C2×C4—84no—
673C2×C4#38012yes—
674C2×C4#38012yes—
675C2×C4#38514yes—
676C2×C4#38514yes—
677C2×C4#38514yes—
678C2×C4#38514yes—
679C2×C4#38614yes—
680C2×C4#38614yes—
681C2×C4#38814yes—
682C2×C4#38814yes—
683C2×C4#39014yes—
684C2×C4#39014yes—
685C2×C4#39214yes—
686C2×C4#39214yes—
687C2×C4#39314yes—
688C2×C4#39412yes—
689C2×C4#39412yes—
690C2×C4#39412yes—
691C2×C4#39412yes—
692C2×C4#39614yes—
693C2×C4#39614yes—
694C2×C4#39614yes—
695C2×C4#39614yes—
696C2×C4#39714yes—
697C2×C4#39714yes—
698C2×C4#39914yes—
699C2×C4C22×C22⋊C411yescharacteristic
700C2×C4C22×C4⋊C411yescharacteristic
701C2×C4C2×C4×D411yescharacteristic
702C2×C4C2×C4×D411yescharacteristic
703C2×C4C2×C4×D411yescharacteristic
704C2×C4C2×C4×D411yescharacteristic
705C2×C4C2×#12011yescharacteristic
706C2×C4C2×#12011yescharacteristic
707C2×C4C2×#12011yescharacteristic
708C2×C4C2×#12111yescharacteristic
709C2×C4C2×#12111yescharacteristic
710C2×C4C2×#12111yescharacteristic
711C2×C4C2×#12111yescharacteristic
712C2×C4C2×#12111yescharacteristic
713C2×C4C2×#12111yescharacteristic
714C2×C4C2×#12111yescharacteristic
715C2×C4C2×C22⋊Q811yescharacteristic
716C2×C4C2×C22⋊Q811yescharacteristic
717C2×C4C2×C22⋊Q811yescharacteristic
718C2×C4C2×C22⋊Q811yescharacteristic
719C2×C4C2×C22⋊Q811yescharacteristic
720C2×C4C2×#12311yescharacteristic
721C2×C4C2×#12311yescharacteristic
722C2×C4C2×#12311yescharacteristic
723C2×C4C2×#12411yescharacteristic
724C2×C4C2×#12611yescharacteristic
725C2×C4C2×#12611yescharacteristic
726C2×C4C2×#12711yescharacteristic
727C8—28no—
728C8—44no—
729C8—44no—
730C8—44no—
731C22—24no—
732C22—22no—
733C22—22no—
734C22—24no—
735C22—22no—
736C22—21no—
737C22—21no—
738C22—22no—
739C22—22no—
740C22—21no—
741C22—21no—
742C22—22no—
743C22—22no—
744C22—22no—
745C22—24no—
746C22—24no—
747C22—44no—
748C22—44no—
749C22—44no—
750C22—44no—
751C22—44no—
752C22—48no—
753C22—816no—
754C22—82no—
755C22—82no—
756C22—84no—
757C22#143914yes—
758C22#143914yes—
759C22C2×#38011yescharacteristic
760C22C2×#38811yescharacteristic
761C22C2×#38811yescharacteristic
762C22C2×#39311yescharacteristic
763C22C2×#39411yescharacteristic
764C22C2×#39411yescharacteristic
765C22C2×#39911yescharacteristic
766C22#306812yes—
767C22#306912yes—
768C4—44no—
769C4—84no—
770C4—84no—
771C4#143914yes—
772C4#143914yes—
773C4C2×#38511yescharacteristic
774C4C2×#38511yescharacteristic
775C4C2×#38611yescharacteristic
776C4C2×#39011yescharacteristic
777C4C2×#39211yescharacteristic
778C4C2×#39611yescharacteristic
779C4C2×#39611yescharacteristic
780C4C2×#39711yescharacteristic
781C2—24no—
782C2—21no—
783C2—21no—
784C2—24no—
785C2—21no—
786C2—21no—
787C2—84no—
788C2C2×#143911yescharacteristic
789C2#2067912yes—
790C2#2353111yescharacteristic
791C2#2353211yescharacteristic
792C2#2353312yes—
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#430C2semidirect
5. C2×#1742C2semidirect
6. C22×#343C2semidirect
7. C2×#1736C2semidirect
8. #11105C2semidirect
19. C22×#108C22semidirect
33. #2879C4semidirect
39. C2×#343C22semidirect
40. C2×#343C22semidirect
43. C2×#340C22semidirect
143. #408C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4302
C2×#28962
C22×#4172
C2×#28792
C2×#17422
C22×#3432
C2×#17362
#111052