Finite Group Databasebeta

← #383853

Subgroups — #383853

Order 512.

1856 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1924C212yesmaximal, complemented
2C2×#1780C212yesmaximal
3#13851C211yesmaximal, characteristic
4#11147C212yesmaximal, complemented
5#11042C212yesmaximal, complemented
6#11042C212yesmaximal, complemented
7#10922C211yesmaximal, characteristic, complemented
8#10921C211yesmaximal, characteristic, complemented
9#10921C212yesmaximal, complemented
10C23×#41C2211yescharacteristic
11C22×C4⋊C8C2212yes—
12C22×C22⋊C8C2212yes—
13C22×C22⋊C8C2212yes—
14C22×C22⋊C8C2212yes—
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2212yes—
18C22×#97C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20C2×C4×C22⋊C4C2212yes—
21C2×C4×C22⋊C4C2211yescharacteristic
22C2×C4×C22⋊C4C2212yes—
23C2×C4×C22⋊C4C2211yescharacteristic
24#1924—28no—
25#1780—28no—
26C2×#336C2212yes—
27C2×#336C2212yes—
28C2×#336C2212yes—
29C2×#336C2211yescharacteristic
30C2×#336C2212yes—
31C2×#336C2212yes—
32C2×#336C2211yescharacteristic
33#1446C411yescharacteristic
34#1446C411yescharacteristic
35#1438C411yescharacteristic
36#1438C412yes—
37#1438C412yes—
38#1438C411yescharacteristic
39C22×#41—24no—
40C22×#41—24no—
41C22×#41—24no—
42C22×#41—41no—
43C22×#41—41no—
44C22×#41—42no—
45C22×#41—42no—
46C22×#41—42no—
47C23×Q8C2×C411yescharacteristic
48C23×D4C2×C411yescharacteristic
49C23×D4C2×C412yes—
50C24×C4C2312yes—
51C24×C4C2311yescharacteristic
52C23×C8C2312yes—
53C23×C8C2312yes—
54C23×C8C2312yes—
55C23×C8C2312yes—
56C22×C22⋊C4C2×C412yes—
57C22×C22⋊C4C2×C412yes—
58C22×C22⋊C4C2×C412yes—
59C22×C22⋊C4C2×C412yes—
60C22×C22⋊C4C2×C411yescharacteristic
61C22×C22⋊C4C2×C411yescharacteristic
62C22×C22⋊C4C2×C412yes—
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C22×C42C2312yes—
66C2×C4⋊C8—24no—
67C2×C4⋊C8—42no—
68C2×C4⋊C8—42no—
69C2×C22⋊C8—24no—
70C2×C22⋊C8—24no—
71C2×C22⋊C8—24no—
72C2×C22⋊C8—24no—
73C2×C22⋊C8—24no—
74C2×C22⋊C8—24no—
75C2×C22⋊C8—24no—
76C2×C22⋊C8—24no—
77C2×C22⋊C8—24no—
78C2×C22⋊C8—42no—
79C2×C22⋊C8—42no—
80C2×C22⋊C8—42no—
81C2×C22⋊C8—42no—
82C2×C22⋊C8—42no—
83C2×C22⋊C8—42no—
84C2×C22⋊C8—42no—
85C2×C22⋊C8—42no—
86C2×C22⋊C8D418yes—
87C2×#97—24no—
88C2×#97—41no—
89C2×C4×C8—42no—
90C2×C4×C8—41no—
91C4×C22⋊C4—24no—
92C4×C22⋊C4—24no—
93C4×C22⋊C4—42no—
94C4×C22⋊C4—41no—
95C4×C22⋊C4—41no—
96C4×C22⋊C4—42no—
97C4×C22⋊C4—42no—
98C2×#95C2×C412yes—
99C2×#95C2×C412yes—
100C2×#95C2×C411yescharacteristic
101C2×#95C2×C411yescharacteristic
102C2×#95C2×C411yescharacteristic
103C2×#95C2×C412yes—
104C2×#95C2×C411yescharacteristic
105C2×#95C2×C412yes—
106#336—24no—
107#336—24no—
108#336—42no—
109#336—42no—
110#336—42no—
111#336—41no—
112#336—41no—
113C25C22×C411yescharacteristic
114C25C22×C412yes—
115C2×#41—44no—
116C2×#41—48no—
117C2×#41—48no—
118C2×#41—48no—
119C2×#41—48no—
120C2×#41—48no—
121C2×#41—44no—
122C2×#41—48no—
123C2×#41—48no—
124C2×#41—48no—
125C2×#41—48no—
126C2×#41—48no—
127C2×#41—48no—
128C2×#41—48no—
129C2×#41—44no—
130C2×#41—44no—
131C22×Q8—22no—
132C22×Q8—24no—
133C22×Q8—41no—
134C22×Q8—42no—
135C22×Q8—41no—
136C22×D4—24no—
137C22×D4—24no—
138C22×D4—24no—
139C22×D4—22no—
140C22×D4—24no—
141C22×D4—42no—
142C22×D4—41no—
143C22×D4—41no—
144C22×D4—42no—
145C22×D4—42no—
146C22×D4—42no—
147C22×D4—42no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4C22×C411yescharacteristic
168C23×C4C22×C411yescharacteristic
169C23×C4C22×C412yes—
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C412yes—
172C23×C4C22×C412yes—
173C23×C4C22×C411yescharacteristic
174C23×C4C22×C411yescharacteristic
175C23×C4C2×D414yes—
176C23×C4#4114yes—
177C23×C4#4114yes—
178C23×C4C2411yescharacteristic
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8C2×D414yes—
206C22×C8C2×D414yes—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—41no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—42no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—21no—
235C2×C42—22no—
236C2×C42—21no—
237C2×C42—22no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—22no—
241C2×C42—22no—
242C4⋊C8—48no—
243C22⋊C8—48no—
244C22⋊C8—48no—
245C22⋊C8—48no—
246C22⋊C8—48no—
247C22⋊C8—48no—
248C22⋊C8—48no—
249#97—44no—
250C4×C8—44no—
251#95—24no—
252#95—24no—
253#95—41no—
254#95—42no—
255#95—41no—
256#95—41no—
257#95—42no—
258#95—41no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—21no—
263C24—21no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—21no—
270C24—21no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—21no—
277C24—21no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24#11712yes—
284C24C4×D414yes—
285C24C23×C411yescharacteristic
286#41—432no—
287#41—432no—
288#41—432no—
289#41—432no—
290C2×Q8—48no—
291C2×Q8—44no—
292C2×Q8—44no—
293C2×Q8—48no—
294C2×Q8—44no—
295C2×D4—48no—
296C2×D4—48no—
297C2×D4—48no—
298C2×D4—44no—
299C2×D4—48no—
300C2×D4—48no—
301C2×D4—44no—
302C2×D4—48no—
303C2×D4—48no—
304C2×D4—48no—
305C2×D4—48no—
306C2×D4—44no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—41no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—41no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—41no—
390C22×C4—42no—
391C22×C4—41no—
392C22×C4#11712yes—
393C22×C4C4×D414yes—
394C22×C4#13114yes—
395C22×C4#13112yes—
396C22×C4#13112yes—
397C22×C4C22×D412yes—
398C22×C4C2×#4112yes—
399C22×C4C2×#4112yes—
400C22×C4H(5,𝔽2)12yes—
401C22×C4H(5,𝔽2)12yes—
402C22×C4H(5,𝔽2)12yes—
403C22×C4H(5,𝔽2)11yescharacteristic
404C22×C4H(5,𝔽2)11yescharacteristic
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
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—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C22⋊C4—48no—
434C22⋊C4—48no—
435C22⋊C4—44no—
436C22⋊C4—48no—
437C22⋊C4—44no—
438C22⋊C4—48no—
439C22⋊C4—48no—
440C42—42no—
441C42—42no—
442C42—42no—
443C42—42no—
444C42—41no—
445C42—42no—
446C42—41no—
447C42—42no—
448C42—41no—
449C42—41no—
450C23—22no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—22no—
456C23—22no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—41no—
467C23—41no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—41no—
483C23—41no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—41no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—41no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—41no—
507C23—41no—
508C23C2×#11711yescharacteristic
509C23C2×C4×D412yes—
510C23#51811yescharacteristic
511C23#51811yescharacteristic
512C23#51812yes—
513C23C2×#13111yescharacteristic
514C23C2×#13111yescharacteristic
515C23C2×#13112yes—
516C23#56811yescharacteristic
517C23#56811yescharacteristic
518C23#56812yes—
519Q8—416no—
520D4—416no—
521D4—432no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4#53412yes—
581C2×C4#53512yes—
582C2×C4#53812yes—
583C2×C4#53812yes—
584C2×C4#53812yes—
585C2×C4#53812yes—
586C2×C4D4211yescharacteristic
587C2×C4#54612yes—
588C2×C4#54612yes—
589C2×C4#54612yes—
590C2×C4#55112yes—
591C2×C4#55112yes—
592C2×C4#55111yescharacteristic
593C2×C4#56012yes—
594C2×C4#56011yescharacteristic
595C2×C4H(3,𝔽4)11yescharacteristic
596C8—44no—
597C8—44no—
598C8—82no—
599C8—82no—
600C22—44no—
601C22—44no—
602C22—42no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—44no—
610C22—42no—
611C22—42no—
612C22—81no—
613C22—82no—
614C22—82no—
615C22—81no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—81no—
623C22—81no—
624C22—82no—
625C22—82no—
626C22#231211yescharacteristic
627C22#232112yes—
628C22#234212yes—
629C22#235811yescharacteristic
630C22#237511yescharacteristic
631C22#292211yescharacteristic
632C22#292311yescharacteristic
633C22#292512yes—
634C22#293012yes—
635C22#293112yes—
636C22#293312yes—
637C22#293512yes—
638C22#297611yescharacteristic
639C22#297612yes—
640C22#297611yescharacteristic
641C22#297612yes—
642C22#297712yes—
643C22#297911yescharacteristic
644C22#298012yes—
645C22#298011yescharacteristic
646C22#298112yes—
647C22#298111yescharacteristic
648C22#298111yescharacteristic
649C4—42no—
650C4—42no—
651C4—44no—
652C4—42no—
653C4—44no—
654C4—82no—
655C4—81no—
656C4—81no—
657C4#267812yes—
658C4#268412yes—
659C4#268512yes—
660C4#273512yes—
661C2—84no—
662C2—82no—
663C2#1436511yescharacteristic
664C2#2190511yescharacteristic
665C2#2190911yescharacteristic
666C2#2203312yes—
667C2#2203712yes—
668C2#2255712yes—
669C2#2256012yes—
670C2#2284511yescharacteristic
671C2#2285311yescharacteristic
672C2#2331011yescharacteristic
673C2#2331111yescharacteristic
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC23×#41C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1924C2semidirect
4. #11147C2semidirect
5. #11042C2semidirect
6. #11042C2semidirect
7. #10922C2semidirect
8. #10921C2semidirect
9. #10921C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19242
C2×#17802
#138512
#111472
#110422
#109222
#109212