Finite Group Databasebeta

← #411985

Subgroups — #411985

Order 512.

1568 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#467C211yesmaximal, characteristic
2C2×#1971C211yesmaximal, characteristic, complemented
3C2×#1971C211yesmaximal, characteristic, complemented
4C2×#1944C211yesmaximal, characteristic, complemented
5C2×#1807C211yesmaximal, characteristic
6C2×#1752C211yesmaximal, characteristic
7#15179C212yesmaximal
8#11190C212yesmaximal, complemented
9#11067C212yesmaximal, complemented
10#10970C212yesmaximal
11#10863C211yesmaximal, characteristic, complemented
12C23×Q16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C22⋊Q8C2212yes—
15C2×#467—24no—
16C2×#467—24no—
17C2×#467—24no—
18C22×#107C2211yescharacteristic
19C22×#106C2211yescharacteristic
20C22×#106C2211yescharacteristic
21C22×#103C2212yes—
22C22×#103C2212yes—
23C22×#103C2212yes—
24C22×C22⋊C8C2211yescharacteristic
25C22×#97C2211yescharacteristic
26C22×#97C2211yescharacteristic
27C2×#391C2212yes—
28C2×#391C2212yes—
29C2×#389C2212yes—
30C2×#380C2212yes—
31C2×C4×C22⋊C4C2211yescharacteristic
32#1971—24no—
33#1971—24no—
34#1944—24no—
35#1807—24no—
36#1752—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×#107D414yes—
88C2×#106—22no—
89C2×#106—22no—
90C2×#106—22no—
91C2×#106—22no—
92C2×#106—22no—
93C2×#106D414yes—
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×#97—22no—
107C2×#97—22no—
108C2×#97—22no—
109C2×#97—22no—
110C2×#97—22no—
111C2×#97—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×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D412yes—
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—24no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—82no—
256C2×C42—21no—
257C2×C42—21no—
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—
268#107—28no—
269#106—28no—
270#106—44no—
271#103—216no—
272#103—48no—
273#103—48no—
274C22⋊C8—28no—
275#97—44no—
276#97—44no—
277#95—22no—
278#95—22no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95—44no—
286#95—82no—
287C24—21no—
288C24—21no—
289C24—21no—
290C24—21no—
291C24—21no—
292C24—21no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24#12112yes—
297C24#13714yes—
298C24C22×D411yescharacteristic
299C2×Q8—24no—
300C2×Q8—24no—
301C2×Q8—24no—
302C2×Q8—24no—
303C2×Q8—24no—
304C2×Q8—24no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—42no—
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—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
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—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
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—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#12112yes—
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C2×#4111yescharacteristic
415C22×C4C2×#4111yescharacteristic
416C22×C4C2×#4111yescharacteristic
417C22×C4C2×#4111yescharacteristic
418C22×C4C2×#4111yescharacteristic
419C22×C4C2×#4111yescharacteristic
420C22×C4H(5,𝔽2)11yescharacteristic
421C22×C4H(5,𝔽2)11yescharacteristic
422C22×C4H(5,𝔽2)11yescharacteristic
423C22×C42−1+411yescharacteristic
424Q16—416no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—416no—
447C2×C8#12014yes—
448C2×C8#12114yes—
449C2×C8#12114yes—
450C2×C8#12114yes—
451C2×C8#12314yes—
452C2×C8#12314yes—
453C2×C8#12414yes—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—416no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—88no—
468C22⋊C4—44no—
469C22⋊C4—416no—
470C22⋊C4—44no—
471C22⋊C4—88no—
472C42—41no—
473C42—41no—
474C42—41no—
475C42—41no—
476C42—41no—
477C42—41no—
478C42—41no—
479C42—41no—
480C23—24no—
481C23—22no—
482C23—22no—
483C23—24no—
484C23—24no—
485C23—22no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23#47014yes—
503C23C2×#12111yescharacteristic
504C23#53211yescharacteristic
505C23#53211yescharacteristic
506C23D4211yescharacteristic
507C23D4211yescharacteristic
508C23D4211yescharacteristic
509C23#54711yescharacteristic
510C23C2×#13712yes—
511C23#57512yes—
512C23#57712yes—
513C23#57812yes—
514Q8—44no—
515Q8—44no—
516Q8—44no—
517Q8—44no—
518Q8—82no—
519Q8—82no—
520Q8—82no—
521Q8—82no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4C2×#12011yescharacteristic
586C2×C4C2×#12111yescharacteristic
587C2×C4C2×#12111yescharacteristic
588C2×C4C2×#12111yescharacteristic
589C2×C4C2×#12311yescharacteristic
590C2×C4C2×#12311yescharacteristic
591C2×C4#52511yescharacteristic
592C2×C4C2×#12411yescharacteristic
593C2×C4#52911yescharacteristic
594C2×C4#52911yescharacteristic
595C2×C4#53511yescharacteristic
596C2×C4#53711yescharacteristic
597C2×C4#54011yescharacteristic
598C2×C4#54011yescharacteristic
599C2×C4#54211yescharacteristic
600C2×C4#54211yescharacteristic
601C2×C4#54611yescharacteristic
602C2×C4#54611yescharacteristic
603C2×C4#54611yescharacteristic
604C2×C4#54611yescharacteristic
605C2×C4#54711yescharacteristic
606C2×C4#54811yescharacteristic
607C2×C4#54811yescharacteristic
608C2×C4#55011yescharacteristic
609C2×C4#55011yescharacteristic
610C2×C4#55311yescharacteristic
611C2×C4#55311yescharacteristic
612C2×C4#55511yescharacteristic
613C8—42no—
614C8—42no—
615C8—42no—
616C8#39418yes—
617C22—24no—
618C22—24no—
619C22—24no—
620C22—42no—
621C22—44no—
622C22—44no—
623C22—42no—
624C22—44no—
625C22—42no—
626C22—42no—
627C22—44no—
628C22#240811yescharacteristic
629C22#241611yescharacteristic
630C22#242211yescharacteristic
631C22#242811yescharacteristic
632C22#242811yescharacteristic
633C22#249111yescharacteristic
634C22#285411yescharacteristic
635C22C2×#47011yescharacteristic
636C22#305411yescharacteristic
637C22#305911yescharacteristic
638C22#306011yescharacteristic
639C22#317812yes—
640C22#317812yes—
641C22#318112yes—
642C22#318112yes—
643C22#326612yes—
644C22#326712yes—
645C22#326912yes—
646C22#327012yes—
647C22#327012yes—
648C22#327112yes—
649C22#334412yes—
650C22#334912yes—
651C4—42no—
652C4—84no—
653C4—84no—
654C4—81no—
655C4—84no—
656C4—81no—
657C4—81no—
658C4—81no—
659C4—162no—
660C4C2×#39411yescharacteristic
661C4#240711yescharacteristic
662C4#241411yescharacteristic
663C4#242011yescharacteristic
664C4#242611yescharacteristic
665C4#242711yescharacteristic
666C4#249011yescharacteristic
667C4#262411yescharacteristic
668C2—44no—
669C2#1512611yescharacteristic
670C2#2427111yescharacteristic
671C2#2427311yescharacteristic
672C2#2478311yescharacteristic
673C2#2478511yescharacteristic
674C2#2504312yes—
675C2#2542012yes—
676C2#2542312yes—
677C2#2779712yes—
678C2#3313311yescharacteristic
679C2#3313611yescharacteristic
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×#1971C2semidirect
3. C2×#1971C2semidirect
4. C2×#1944C2semidirect
8. #11190C2semidirect
9. #11067C2semidirect
11. #10863C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4672
C2×#19712
C2×#19442
C2×#18072
C2×#17522
#151792
#111902
#110672
#109702
#108632