Finite Group Databasebeta

← #383854

Subgroups — #383854

Order 512.

2406 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1923C212yesmaximal, complemented
2C2×#1779C212yesmaximal
3C4×#379C211yesmaximal, characteristic
4#11147C212yesmaximal, complemented
5#11042C212yesmaximal, complemented
6#11042C212yesmaximal, complemented
7#10922C211yesmaximal, characteristic, complemented
8#10922C212yesmaximal, complemented
9#10921C211yesmaximal, characteristic, complemented
10C25×C4C2211yescharacteristic
11C22×C4⋊C8C2212yescomplemented
12C22×C22⋊C8C2212yes—
13C22×C22⋊C8C2212yes—
14C22×C22⋊C8C2212yes—
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2212yes—
18C22×#97C2211yescharacteristic, complemented
19C22×C4×C8C2211yescharacteristic, complemented
20C2×#379C412yes—
21C2×C4×C22⋊C4C2212yes—
22C2×C4×C22⋊C4C2211yescharacteristic
23C2×C4×C22⋊C4C2212yes—
24C2×C4×C22⋊C4C2211yescharacteristic
25#1923—28no—
26#1779—28no—
27C2×#336C2212yescomplemented
28C2×#336C2212yescomplemented
29C2×#336C2212yescomplemented
30C2×#336C2211yescharacteristic, complemented
31C2×#336C2212yescomplemented
32C2×#336C2212yescomplemented
33C2×#336C2211yescharacteristic, complemented
34C22⋊#95C412yes—
35C22⋊#95C412yes—
36C22⋊#95C412yes—
37C26C2×C411yescharacteristic
38C24×C4—24no—
39C24×C4—24no—
40C24×C4—24no—
41C24×C4—41no—
42C24×C4—41no—
43C24×C4—42no—
44C24×C4—42no—
45C24×C4—42no—
46C24×C4C2×C411yescharacteristic
47C24×C4C2×C412yes—
48C24×C4C2312yes—
49C24×C4C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2312yes—
52C23×C8C2312yes—
53C23×C8C2312yes—
54C22×C22⋊C4C2×C412yes—
55C22×C22⋊C4C2×C412yes—
56C22×C22⋊C4C2×C412yes—
57C22×C22⋊C4C2×C412yes—
58C22×C22⋊C4C2×C412yes—
59C22×C22⋊C4C2×C412yes—
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C22×C42C2312yes—
63C2×C4⋊C8—24no—
64C2×C4⋊C8—42no—
65C2×C4⋊C8—42no—
66C2×C22⋊C8—24no—
67C2×C22⋊C8—24no—
68C2×C22⋊C8—24no—
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—42no—
76C2×C22⋊C8—42no—
77C2×C22⋊C8—42no—
78C2×C22⋊C8—42no—
79C2×C22⋊C8—42no—
80C2×C22⋊C8—42no—
81C2×C22⋊C8—42no—
82C2×C22⋊C8—42no—
83C2×C22⋊C8D418yes—
84C2×#97—42no—
85C2×#97—41no—
86C2×C4×C8—24no—
87C2×C4×C8—41no—
88#379—28no—
89C4×C22⋊C4—24no—
90C4×C22⋊C4—24no—
91C4×C22⋊C4—42no—
92C4×C22⋊C4—41no—
93C4×C22⋊C4—41no—
94C4×C22⋊C4—42no—
95C4×C22⋊C4—42no—
96C2×#95C2×C412yes—
97C2×#95C2×C412yes—
98C2×#95C2×C411yescharacteristic
99C2×#95C2×C412yes—
100C2×#95C2×C412yes—
101C2×#95C2×C412yes—
102C2×#95C2×C411yescharacteristic
103#336—24no—
104#336—24no—
105#336—42no—
106#336—42no—
107#336—42no—
108#336—41no—
109#336—41no—
110C25—22no—
111C25—24no—
112C25—24no—
113C25—24no—
114C25—24no—
115C25—41no—
116C25—42no—
117C25—41no—
118C25—42no—
119C25C22×C411yescharacteristic
120C25C22×C412yes—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—24no—
144C23×C4—24no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4—41no—
150C23×C4—42no—
151C23×C4—41no—
152C23×C4—44no—
153C23×C4—48no—
154C23×C4—48no—
155C23×C4—48no—
156C23×C4—48no—
157C23×C4—48no—
158C23×C4—44no—
159C23×C4—48no—
160C23×C4—48no—
161C23×C4—48no—
162C23×C4—48no—
163C23×C4—48no—
164C23×C4—48no—
165C23×C4—48no—
166C23×C4—44no—
167C23×C4—44no—
168C23×C4C22×C412yes—
169C23×C4C22×C412yes—
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C412yes—
172C23×C4C22×C412yes—
173C23×C4C22×C412yes—
174C23×C4C2×D414yes—
175C23×C4#4114yes—
176C23×C4#4114yes—
177C23×C4C2411yescharacteristic
178C22×C8—22no—
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×C8C2×D414yes—
205C22×C8C2×D414yes—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4M1618yes—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—21no—
231C2×C42—22no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—22no—
235C2×C42—21no—
236C2×C42—22no—
237C2×C42—22no—
238C4⋊C8—48no—
239C22⋊C8—48no—
240C22⋊C8—48no—
241C22⋊C8—48no—
242C22⋊C8—48no—
243C22⋊C8—48no—
244C22⋊C8—48no—
245#97—44no—
246C4×C8—44no—
247#95—24no—
248#95—24no—
249#95—41no—
250#95—42no—
251#95—42no—
252#95—41no—
253#95—42no—
254C24—22no—
255C24—22no—
256C24—22no—
257C24—21no—
258C24—21no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—28no—
274C24—216no—
275C24—48no—
276C24—44no—
277C24—48no—
278C24—48no—
279C24—48no—
280C24—48no—
281C24—48no—
282C24—48no—
283C24—48no—
284C24—44no—
285C24—44no—
286C24—48no—
287C24—48no—
288C24—48no—
289C24—48no—
290C24—44no—
291C24—44no—
292C24—44no—
293C24—48no—
294C24#11712yes—
295C24C4×D414yes—
296C24C2×M1614yes—
297C24#13114yes—
298C24#13114yes—
299C24C23×C411yescharacteristic
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
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—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—48no—
355C22×C4—44no—
356C22×C4—48no—
357C22×C4—48no—
358C22×C4—48no—
359C22×C4—48no—
360C22×C4—48no—
361C22×C4—48no—
362C22×C4—44no—
363C22×C4—48no—
364C22×C4—48no—
365C22×C4—44no—
366C22×C4—432no—
367C22×C4—432no—
368C22×C4—432no—
369C22×C4—432no—
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—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—41no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—41no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—41no—
394C22×C4—42no—
395C22×C4—41no—
396C22×C4#11712yes—
397C22×C4C4×D414yes—
398C22×C4C2×M1614yes—
399C22×C4C2×M1614yes—
400C22×C4C22×D412yes—
401C22×C4C2×#4112yes—
402C22×C4C2×#4112yes—
403C22×C4H(5,𝔽2)12yes—
404C22×C4H(5,𝔽2)12yes—
405C22×C4H(5,𝔽2)12yes—
406C22×C4H(5,𝔽2)11yescharacteristic
407C22×C4H(5,𝔽2)11yescharacteristic
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—24no—
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—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C22⋊C4—216no—
437C22⋊C4—48no—
438C22⋊C4—48no—
439C22⋊C4—48no—
440C22⋊C4—48no—
441C22⋊C4—48no—
442C42—42no—
443C42—42no—
444C42—42no—
445C42—42no—
446C42—41no—
447C42—42no—
448C42—41no—
449C42—42no—
450C42—41no—
451C42—41no—
452C23—22no—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—22no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—432no—
481C23—432no—
482C23—432no—
483C23—432no—
484C23—416no—
485C23—416no—
486C23—432no—
487C23—432no—
488C23—416no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—41no—
494C23—41no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—41no—
513C23—41no—
514C23—42no—
515C23#43314yes—
516C23#43518yes—
517C23C2×#11711yescharacteristic
518C23C2×C4×D412yes—
519C23#51811yescharacteristic
520C23#51811yescharacteristic
521C23#51812yes—
522C23C22×M1612yes—
523C23C2×#13112yes—
524C23C2×#13112yes—
525C23#56812yes—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—432no—
538C2×C4—416no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
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—44no—
564C2×C4—44no—
565C2×C4—44no—
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—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
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—28no—
601C22—28no—
602C22—28no—
603C22—28no—
604C22—24no—
605C22—464no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—42no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—42no—
624C22—44no—
625C22#231211yescharacteristic
626C22#232112yes—
627C22#234212yes—
628C22#235811yescharacteristic
629C22#237511yescharacteristic
630C22C2×#43311yescharacteristic
631C22#292011yescharacteristic
632C22#292412yes—
633C22C2×#43512yes—
634C22#292912yes—
635C22#293312yes—
636C22D4×M1612yes—
637C22#297412yes—
638C22#297412yes—
639C22#297412yes—
640C22#297512yes—
641C22#297612yes—
642C22#297612yes—
643C22#297712yes—
644C22#297711yescharacteristic
645C22#297811yescharacteristic
646C4—44no—
647C4—42no—
648C4—44no—
649C4—44no—
650C4—82no—
651C4—82no—
652C4#267812yes—
653C4#268412yes—
654C4#268512yes—
655C4#273512yes—
656C2—48no—
657C2—44no—
658C2#1436511yescharacteristic
659C2#2190411yescharacteristic
660C2#2190811yescharacteristic
661C2#2203012yes—
662C2#2203812yes—
663C2#2255312yes—
664C2#2255812yes—
665C2#2284612yes—
666C2#2330812yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC26C2×C4
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×#1923C2semidirect
4. #11147C2semidirect
5. #11042C2semidirect
6. #11042C2semidirect
7. #10922C2semidirect
8. #10922C2semidirect
9. #10921C2semidirect
11. C22×C4⋊C8C22semidirect
18. C22×#97C22semidirect
19. C22×C4×C8C22semidirect
27. C2×#336C22semidirect
28. C2×#336C22semidirect
29. C2×#336C22semidirect
30. C2×#336C22semidirect
31. C2×#336C22semidirect
32. C2×#336C22semidirect
33. C2×#336C22semidirect

Maximal subgroups

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

subgroupindex
C2×#19232
C2×#17792
C4×#3792
#111472
#110422
#109222
#109212