Finite Group Databasebeta

← #118282

Subgroups — #118282

Order 512.

1296 conjugacy classes of subgroups.

The subgroup lattice has 723 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1872C211yesmaximal, characteristic
2C2×#1852C211yesmaximal, characteristic
3C22×#328C211yesmaximal, characteristic
4#14281C211yesmaximal, characteristic
5C2×#1294C211yesmaximal, characteristic
6C2×#1280C211yesmaximal, characteristic
7#7557C211yesmaximal, characteristic
8C22×C22⋊Q8C2211yescharacteristic
9C23×C4⋊C4C2211yescharacteristic
10C22×#107—21no—
11C22×#103—21no—
12C22×C22⋊C8C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14C2×#380C2211yescharacteristic
15C22×#95—22no—
16C22×#95C2211yescharacteristic
17C22×#95C2211yescharacteristic
18#1872—22no—
19#1872—22no—
20#1852—22no—
21#1852—22no—
22C2×#340—21no—
23C2×#340—21no—
24C2×#328—22no—
25C2×#328—22no—
26C2×#328C414yescomplemented
27#1446—21no—
28#1445—21no—
29#1441—21no—
30#1441—21no—
31C22⋊#95—22no—
32#1294—22no—
33#1280—22no—
34C23×Q8D411yescharacteristic
35C24×C4—21no—
36C24×C4—21no—
37C24×C4D412yes—
38C24×C4C2311yescharacteristic
39C23×C8—21no—
40C23×C8—21no—
41C2×C22⋊Q8—21no—
42C2×C22⋊Q8—21no—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8C2×C412yes—
52C2×C22⋊Q8Q812yes—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—44no—
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4D412yes—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D411yescharacteristic
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C2×#107—24no—
67C2×#107—44no—
68C2×#103—24no—
69C2×#103—44no—
70C2×C22⋊C8—21no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8—21no—
78C2×C22⋊C8C2×C412yes—
79C2×C22⋊C8D412yes—
80C2×C22⋊C8D412yes—
81C2×C22⋊C8D412yes—
82#380—21no—
83#380—21no—
84#380—21no—
85#380—21no—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95—21no—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—44no—
103C2×#95—44no—
104C2×#95—44no—
105C2×#95C2×C412yes—
106#340—24no—
107#340—24no—
108#328—28no—
109C25C2×D411yescharacteristic
110C22×Q8—21no—
111C22×Q8—21no—
112C22×Q8—21no—
113C22×Q8—21no—
114C22×Q8—21no—
115C22×Q8—42no—
116C22×Q8—42no—
117C22×Q8C4⋊C412yes—
118C23×C4—24no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—22no—
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—42no—
134C23×C4—42no—
135C23×C4—42no—
136C23×C4—42no—
137C23×C4—42no—
138C23×C4—42no—
139C23×C4—42no—
140C23×C4—41no—
141C23×C4—41no—
142C23×C4—42no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4—42no—
146C23×C4—41no—
147C23×C4—41no—
148C23×C4—41no—
149C23×C4—41no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4Q1614yes—
155C23×C4C22×C411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×Q811yescharacteristic
163C23×C4#4111yescharacteristic
164C23×C4#4111yescharacteristic
165C23×C4#4111yescharacteristic
166C23×C4#4111yescharacteristic
167C23×C4#4111yescharacteristic
168C23×C4#4111yescharacteristic
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—42no—
174C22×C8—42no—
175C22×C8—42no—
176C22×C8—42no—
177C22×C8—41no—
178C22×C8—41no—
179C22⋊Q8—22no—
180C22⋊Q8—22no—
181C22⋊Q8—22no—
182C22⋊Q8—22no—
183C22⋊Q8—22no—
184C22⋊Q8—22no—
185C22⋊Q8—22no—
186C22⋊Q8—22no—
187C22⋊Q8—41no—
188C22⋊Q8—41no—
189C22⋊Q8—41no—
190C22⋊Q8—41no—
191C22⋊Q8—41no—
192C22⋊Q8—41no—
193C22⋊Q8—41no—
194C22⋊Q8—41no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4C4⋊C412yes—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—41no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235#107—48no—
236#103—48no—
237C22⋊C8—22no—
238C22⋊C8—22no—
239C22⋊C8—22no—
240C22⋊C8—22no—
241C22⋊C8—22no—
242C22⋊C8—41no—
243C22⋊C8—41no—
244C22⋊C8C22⋊C414yes—
245#95—22no—
246#95—22no—
247#95—416no—
248#95—41no—
249#95—41no—
250#95—41no—
251#95—41no—
252#95—41no—
253#95—41no—
254#95—41no—
255#95—41no—
256#95—82no—
257#95—82no—
258#95—82no—
259#95—82no—
260C24—21no—
261C24—21no—
262C24—21no—
263C24—21no—
264C24—22no—
265C24—22no—
266C24C4×D411yescharacteristic
267C24#12011yescharacteristic
268C24#12111yescharacteristic
269C24#12111yescharacteristic
270C24C22⋊Q811yescharacteristic
271C24#12311yescharacteristic
272C24#12711yescharacteristic
273C24C2×Q1612yes—
274C24#13512yes—
275C24AGL(1,ℤ/8)12yes—
276C24#13712yes—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—44no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—44no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—82no—
286C2×Q8—81no—
287C2×Q8—81no—
288C2×Q8—82no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—42no—
322C22×C4—44no—
323C22×C4—42no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—44no—
347C22×C4—42no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—44no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—44no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—82no—
373C22×C4—82no—
374C22×C4—82no—
375C22×C4—82no—
376C22×C4—82no—
377C22×C4—82no—
378C22×C4—81no—
379C22×C4—81no—
380C22×C4—82no—
381C22×C4—82no—
382C22×C4—82no—
383C22×C4—82no—
384C22×C4—82no—
385C22×C4—82no—
386C22×C4—82no—
387C22×C4—81no—
388C22×C4—81no—
389C22×C4—82no—
390C22×C4—81no—
391C22×C4—81no—
392C22×C4—81no—
393C22×C4—81no—
394C22×C4—82no—
395C22×C4—82no—
396C22×C4—82no—
397C22×C4—82no—
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C4⋊C411yescharacteristic
400C22×C4#11711yescharacteristic
401C22×C4C4×D411yescharacteristic
402C22×C4C4×D411yescharacteristic
403C22×C4C4×D411yescharacteristic
404C22×C4#12011yescharacteristic
405C22×C4#12011yescharacteristic
406C22×C4#12111yescharacteristic
407C22×C4#12111yescharacteristic
408C22×C4#12111yescharacteristic
409C22×C4#12111yescharacteristic
410C22×C4#12111yescharacteristic
411C22×C4#12111yescharacteristic
412C22×C4C22⋊Q811yescharacteristic
413C22×C4C22⋊Q811yescharacteristic
414C22×C4C22⋊Q811yescharacteristic
415C22×C4C22⋊Q811yescharacteristic
416C22×C4C22⋊Q811yescharacteristic
417C22×C4#12311yescharacteristic
418C22×C4#12311yescharacteristic
419C22×C4#12311yescharacteristic
420C22×C4#12311yescharacteristic
421C22×C4#12411yescharacteristic
422C22×C4#12411yescharacteristic
423C22×C4#12411yescharacteristic
424C22×C4#12611yescharacteristic
425C22×C4#12611yescharacteristic
426C2×C8—24no—
427C2×C8—42no—
428C2×C8—44no—
429C2×C8—42no—
430C2×C8—44no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—82no—
435C2×C8—82no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—41no—
450C4⋊C4—42no—
451C4⋊C4—41no—
452C4⋊C4—81no—
453C4⋊C4—81no—
454C22⋊C4—42no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C22⋊C4—42no—
458C22⋊C4—82no—
459C22⋊C4—81no—
460C22⋊C4—81no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—22no—
496C23—22no—
497C23#38511yescharacteristic
498C23#38811yescharacteristic
499C23#39011yescharacteristic
500C23#39211yescharacteristic
501C23#39411yescharacteristic
502C23#39611yescharacteristic
503C23#39911yescharacteristic
504C23C4×Q1612yes—
505C23#44012yes—
506C23#44912yes—
507C23#45112yes—
508C23#45311yescharacteristic
509C23#45411yescharacteristic
510C23#45511yescharacteristic
511C23#45611yescharacteristic
512C23B(4,𝔽2)12yes—
513C23#45812yes—
514C23#46212yes—
515C23#46312yes—
516C23#46712yes—
517C23#46812yes—
518C23#47712yes—
519C23#47912yes—
520C23#48412yes—
521C23#48512yes—
522C23#49411yescharacteristic
523C23#49511yescharacteristic
524C23#49611yescharacteristic
525C23#49711yescharacteristic
526Q8—44no—
527Q8—82no—
528Q8—82no—
529Q8—82no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—44no—
563C2×C4—48no—
564C2×C4—48no—
565C2×C4—48no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—82no—
576C2×C4—84no—
577C2×C4—84no—
578C2×C4—84no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—84no—
584C2×C4—84no—
585C2×C4—84no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4—84no—
589C2×C4—82no—
590C2×C4—84no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—81no—
594C2×C4—82no—
595C2×C4—81no—
596C2×C4—82no—
597C2×C4—81no—
598C2×C4—81no—
599C2×C4—81no—
600C2×C4—82no—
601C2×C4—81no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—84no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4—82no—
609C2×C4—81no—
610C2×C4—81no—
611C2×C4—81no—
612C2×C4—81no—
613C2×C4—81no—
614C2×C4—81no—
615C2×C4#38011yescharacteristic
616C2×C4#38511yescharacteristic
617C2×C4#38611yescharacteristic
618C2×C4#38811yescharacteristic
619C2×C4#39311yescharacteristic
620C2×C4#39411yescharacteristic
621C2×C4#39611yescharacteristic
622C2×C4#39711yescharacteristic
623C8—44no—
624C8—82no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—21no—
651C22—21no—
652C22—21no—
653C22—22no—
654C22—21no—
655C22—21no—
656C22—22no—
657C22—21no—
658C22—21no—
659C22—22no—
660C22—21no—
661C22#143911yescharacteristic
662C22#159712yes—
663C22#160012yes—
664C22#160212yes—
665C22#160512yes—
666C22#160912yes—
667C22#161212yes—
668C22#161412yes—
669C22#161712yes—
670C22#185911yescharacteristic
671C22#186011yescharacteristic
672C22#186312yes—
673C22#186712yes—
674C22#192812yes—
675C22#196911yescharacteristic
676C22#197311yescharacteristic
677C22#200611yescharacteristic
678C22#200711yescharacteristic
679C22#201211yescharacteristic
680C22#201611yescharacteristic
681C22#202111yescharacteristic
682C22#202211yescharacteristic
683C22#203612yes—
684C22#203711yescharacteristic
685C22#203811yescharacteristic
686C22#204412yes—
687C22#206812yes—
688C22#209511yescharacteristic
689C22#209711yescharacteristic
690C4—21no—
691C4—21no—
692C4—21no—
693C4—21no—
694C4—21no—
695C4—21no—
696C4—21no—
697C4—21no—
698C4—84no—
699C4—82no—
700C4—82no—
701C4—84no—
702C4—84no—
703C4—161no—
704C4—161no—
705C2—21no—
706C2—22no—
707C2—21no—
708C2—21no—
709C2—22no—
710C2—21no—
711C2#886712yes—
712C2#886812yes—
713C2#996311yescharacteristic
714C2#996711yescharacteristic
715C2#997111yescharacteristic
716C2#997211yescharacteristic
717C2#1039912yes—
718C2#1040012yes—
719C2#1108711yescharacteristic
720C2#1111311yescharacteristic
721C2#1112411yescharacteristic
subgroupG/H
center, socleC24#120
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
26. C2×#328C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18722
C2×#18522
C22×#3282
#142812
C2×#12942
C2×#12802
#75572