Finite Group Databasebeta

← #411982

Subgroups — #411982

Order 512.

1600 conjugacy classes of subgroups.

The subgroup lattice has 739 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#467C211yesmaximal, characteristic
2C2×#1969C211yesmaximal, characteristic, complemented
3C2×#1969C211yesmaximal, characteristic, complemented
4C2×#1942C211yesmaximal, characteristic, complemented
5C2×#1808C211yesmaximal, characteristic
6C2×C8×C22⋊C4C211yesmaximal, characteristic
7#15179C212yesmaximal
8#11190C212yesmaximal, complemented
9#11067C212yesmaximal, complemented
10#10970C212yesmaximal
11#10862C211yesmaximal, characteristic, complemented
12C23×Q16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C22⋊Q8C2212yes—
15C2×#467—24no—
16C2×#467—24no—
17C2×#467—24no—
18C22×#107C2211yescharacteristic
19C22×#107C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#103C2212yes—
22C22×#103C2212yes—
23C22×#103C2212yes—
24C22×C22⋊C8C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27C2×#391C2212yes—
28C2×#391C2212yes—
29C2×#389C2212yes—
30C2×#380C2212yes—
31C2×C4×C22⋊C4C2211yescharacteristic
32#1969—24no—
33#1969—24no—
34#1942—24no—
35#1808—24no—
36C8×C22⋊C4—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2211yescharacteristic
42#1442—22no—
43#1441—22no—
44#1440—22no—
45#1439—22no—
46C4⋊#95C2212yes—
47C23×Q8C2312yes—
48C24×C4C2311yescharacteristic
49C22×Q16—22no—
50C22×Q16—22no—
51C22×Q16—22no—
52C22×Q16—22no—
53C22×Q16—22no—
54C22×Q16—22no—
55C22×Q16D414yes—
56C23×C8—22no—
57C23×C8—22no—
58C23×C8—22no—
59C23×C8—22no—
60C23×C8—22no—
61C23×C8—22no—
62C23×C8D414yes—
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C2×C22⋊Q8—24no—
68C2×C22⋊Q8—24no—
69C2×C22⋊Q8—24no—
70C2×C22⋊Q8—24no—
71C2×C22⋊Q8—42no—
72C2×C22⋊Q8—42no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4D411yescharacteristic
81C22×C22⋊C4D411yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84#467—232no—
85C2×#107—22no—
86C2×#107—22no—
87C2×#107—22no—
88C2×#107—22no—
89C2×#107—22no—
90C2×#107—22no—
91C2×#107—22no—
92C2×#107D414yes—
93C2×#107D414yes—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—24no—
101C2×#103—42no—
102C2×#103—42no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106C2×C4×C8—22no—
107C2×C4×C8—22no—
108C2×C4×C8—22no—
109C2×C4×C8—22no—
110C2×C4×C8—22no—
111C2×C4×C8—22no—
112#391—24no—
113#391—24no—
114#391—42no—
115#391—42no—
116#389—24no—
117#389—42no—
118#380—24no—
119#380—42no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4—22no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95D411yescharacteristic
131C2×#95D411yescharacteristic
132#340—24no—
133#340—24no—
134#340—42no—
135#336—22no—
136#336—22no—
137C25C2×D411yescharacteristic
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8C2×D414yes—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D412yes—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4#4112yes—
175C23×C4C2411yescharacteristic
176C2×Q16—28no—
177C2×Q16—28no—
178C2×Q16—28no—
179C2×Q16—44no—
180C2×Q16—44no—
181C2×Q16—44no—
182C2×Q16—44no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—28no—
202C22×C8—28no—
203C22×C8—28no—
204C22×C8—44no—
205C22×C8—44no—
206C22×C8—44no—
207C22×C8—44no—
208C22×C8C2×D412yes—
209C22×C8C2×D412yes—
210C22×C8C2×D412yes—
211C22×C8C2×D412yes—
212C22×C8C2×D412yes—
213C22×C8#4112yes—
214C22×C8#4112yes—
215C22×C8#4112yes—
216C22×C8#4112yes—
217C22×C8#4112yes—
218C22⋊Q8—216no—
219C22⋊Q8—48no—
220C22⋊Q8—48no—
221C22⋊Q8—48no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—82no—
244C2×C4⋊C4C2×D414yes—
245C2×C4⋊C4C2×D414yes—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—82no—
257C2×C22⋊C4Q1614yescomplemented
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270#107—28no—
271#107—28no—
272#107—44no—
273#103—216no—
274#103—48no—
275#103—48no—
276C22⋊C8—28no—
277C4×C8—44no—
278C4×C8—44no—
279#95—22no—
280#95—22no—
281#95—44no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—82no—
289C24—21no—
290C24—21no—
291C24—21no—
292C24—21no—
293C24—21no—
294C24—21no—
295C24—21no—
296C24—21no—
297C24—21no—
298C24—21no—
299C24—21no—
300C24—21no—
301C24#12112yes—
302C24C2×Q1612yes—
303C24#13512yes—
304C24C22×D411yescharacteristic
305C2×Q8—24no—
306C2×Q8—24no—
307C2×Q8—24no—
308C2×Q8—24no—
309C2×Q8—24no—
310C2×Q8—24no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8—42no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
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—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4#12112yes—
420C22×C4C2×Q1612yes—
421C22×C4C2×Q1612yes—
422C22×C4#13512yes—
423C22×C4#13512yes—
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C2×#4111yescharacteristic
429C22×C4C2×#4111yescharacteristic
430C22×C4C2×#4111yescharacteristic
431C22×C4C2×#4111yescharacteristic
432C22×C4C2×#4111yescharacteristic
433C22×C4C2×#4111yescharacteristic
434C22×C4H(5,𝔽2)11yescharacteristic
435C22×C4H(5,𝔽2)11yescharacteristic
436C22×C4H(5,𝔽2)11yescharacteristic
437C22×C42−1+411yescharacteristic
438Q16—416no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—416no—
461C2×C8#12014yes—
462C2×C8#12114yes—
463C2×C8#12114yes—
464C2×C8#12114yes—
465C2×C8#12314yes—
466C2×C8#12314yes—
467C2×C8#12414yes—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—416no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—88no—
482C22⋊C4—28no—
483C22⋊C4—28no—
484C22⋊C4—416no—
485C22⋊C4—88no—
486C42—41no—
487C42—41no—
488C42—41no—
489C42—41no—
490C42—41no—
491C42—41no—
492C42—41no—
493C42—41no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23#46714yes—
529C23C2×#12111yescharacteristic
530C23#53211yescharacteristic
531C23#53211yescharacteristic
532C23D4211yescharacteristic
533C23D4211yescharacteristic
534C23D4211yescharacteristic
535C23#54711yescharacteristic
536C23C22×Q1611yescharacteristic
537C23C2×#13511yescharacteristic
538C23C2×#13511yescharacteristic
539C23C2×#13511yescharacteristic
540C23#57611yescharacteristic
541C23#57811yescharacteristic
542C23#57811yescharacteristic
543C23#57811yescharacteristic
544Q8—44no—
545Q8—44no—
546Q8—44no—
547Q8—44no—
548Q8—82no—
549Q8—82no—
550Q8—82no—
551Q8—82no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—82no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4—82no—
609C2×C4—82no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4C2×#12011yescharacteristic
624C2×C4C2×#12111yescharacteristic
625C2×C4C2×#12111yescharacteristic
626C2×C4C2×#12111yescharacteristic
627C2×C4C2×#12311yescharacteristic
628C2×C4C2×#12311yescharacteristic
629C2×C4#52511yescharacteristic
630C2×C4C2×#12411yescharacteristic
631C2×C4#52911yescharacteristic
632C2×C4#52911yescharacteristic
633C2×C4#53511yescharacteristic
634C2×C4#53711yescharacteristic
635C2×C4#54011yescharacteristic
636C2×C4#54011yescharacteristic
637C2×C4#54211yescharacteristic
638C2×C4#54211yescharacteristic
639C2×C4#54611yescharacteristic
640C2×C4#54611yescharacteristic
641C2×C4#54611yescharacteristic
642C2×C4#54611yescharacteristic
643C2×C4#54711yescharacteristic
644C2×C4#54811yescharacteristic
645C2×C4#54811yescharacteristic
646C2×C4#55011yescharacteristic
647C2×C4#55011yescharacteristic
648C2×C4#55311yescharacteristic
649C2×C4#55311yescharacteristic
650C2×C4#55511yescharacteristic
651C8—42no—
652C8—42no—
653C8—42no—
654C8#39418yes—
655C22—24no—
656C22—24no—
657C22—24no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22#240811yescharacteristic
671C22#241611yescharacteristic
672C22#242211yescharacteristic
673C22#242811yescharacteristic
674C22#242811yescharacteristic
675C22#249111yescharacteristic
676C22#285411yescharacteristic
677C22C2×#46711yescharacteristic
678C22#305011yescharacteristic
679C22#305911yescharacteristic
680C22#306111yescharacteristic
681C22#316811yescharacteristic
682C22#316811yescharacteristic
683C22#317011yescharacteristic
684C22#317011yescharacteristic
685C22#317511yescharacteristic
686C22#317511yescharacteristic
687C22#317711yescharacteristic
688C22#317711yescharacteristic
689C22#328011yescharacteristic
690C22#328011yescharacteristic
691C22#328311yescharacteristic
692C22#328311yescharacteristic
693C22#328511yescharacteristic
694C22#328511yescharacteristic
695C22#328511yescharacteristic
696C22#328511yescharacteristic
697C22D4×Q1611yescharacteristic
698C22D4×Q1611yescharacteristic
699C22D4×Q1611yescharacteristic
700C22#328711yescharacteristic
701C22#333611yescharacteristic
702C22#333611yescharacteristic
703C22#333811yescharacteristic
704C22#334211yescharacteristic
705C4—42no—
706C4—42no—
707C4—42no—
708C4—42no—
709C4—42no—
710C4—84no—
711C4—84no—
712C4—84no—
713C4—162no—
714C4C2×#39411yescharacteristic
715C4#240711yescharacteristic
716C4#241411yescharacteristic
717C4#242011yescharacteristic
718C4#242611yescharacteristic
719C4#242711yescharacteristic
720C4#249011yescharacteristic
721C4#262411yescharacteristic
722C2—44no—
723C2#1512611yescharacteristic
724C2#2426611yescharacteristic
725C2#2426811yescharacteristic
726C2#2477911yescharacteristic
727C2#2478111yescharacteristic
728C2#2503311yescharacteristic
729C2#2503511yescharacteristic
730C2#2541211yescharacteristic
731C2#2541311yescharacteristic
732C2#2541511yescharacteristic
733C2#2541511yescharacteristic
734C2#2778911yescharacteristic
735C2#2779211yescharacteristic
736C2#3313011yescharacteristic
737C2#3313911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1969C2semidirect
3. C2×#1969C2semidirect
4. C2×#1942C2semidirect
8. #11190C2semidirect
9. #11067C2semidirect
11. #10862C2semidirect
257. C2×C22⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
C22×#4672
C2×#19692
C2×#19422
C2×#18082
C2×C8×C22⋊C42
#151792
#111902
#110672
#109702
#108622