Finite Group Databasebeta

← #383889

Subgroups — #383889

Order 512.

2426 conjugacy classes of subgroups.

The subgroup lattice has 734 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#11042C211yesmaximal, characteristic, complemented
7#11042C211yesmaximal, characteristic, complemented
8#10921C212yesmaximal, complemented
9#10921C212yesmaximal, complemented
10C25×C4C2211yescharacteristic
11C22×C4⋊C8C2211yescharacteristic, complemented
12C22×C4⋊C8C2211yescharacteristic, complemented
13C22×C22⋊C8C2212yes—
14C22×C22⋊C8C2212yes—
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2212yes—
19C22×C4×C8C2212yescomplemented
20C2×#379C411yescharacteristic
21C2×#379C411yescharacteristic
22C2×C4×C22⋊C4C2212yes—
23C2×C4×C22⋊C4C2211yescharacteristic
24C2×C4×C22⋊C4C2212yes—
25C2×C4×C22⋊C4C2211yescharacteristic
26#1923—28no—
27#1779—28no—
28C2×#336C2212yescomplemented
29C2×#336C2212yescomplemented
30C2×#336C2212yescomplemented
31C2×#336C2211yescharacteristic, complemented
32C2×#336C2212yescomplemented
33C2×#336C2211yescharacteristic, complemented
34C2×#336C2212yescomplemented
35C22⋊#95C412yes—
36C22⋊#95C412yes—
37C22⋊#95C411yescharacteristic
38C22⋊#95C411yescharacteristic
39C26C2×C411yescharacteristic
40C24×C4—24no—
41C24×C4—22no—
42C24×C4—22no—
43C24×C4—24no—
44C24×C4—41no—
45C24×C4—41no—
46C24×C4—42no—
47C24×C4—42no—
48C24×C4—41no—
49C24×C4—41no—
50C24×C4C2×C411yescharacteristic
51C24×C4C2×C412yes—
52C24×C4C2312yes—
53C24×C4C2311yescharacteristic
54C23×C8C2312yes—
55C23×C8C2312yes—
56C23×C8C2312yes—
57C23×C8C2312yes—
58C22×C22⋊C4C2×C412yes—
59C22×C22⋊C4C2×C412yes—
60C22×C22⋊C4C2×C412yes—
61C22×C22⋊C4C2×C411yescharacteristic
62C22×C22⋊C4C2×C411yescharacteristic
63C22×C22⋊C4C2×C411yescharacteristic
64C22×C22⋊C4C2×C412yes—
65C22×C22⋊C4C2×C411yescharacteristic
66C22×C42C2312yes—
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69C2×C4⋊C8—24no—
70C2×C4⋊C8—41no—
71C2×C4⋊C8—42no—
72C2×C4⋊C8—41no—
73C2×C22⋊C8—24no—
74C2×C22⋊C8—24no—
75C2×C22⋊C8—24no—
76C2×C22⋊C8—24no—
77C2×C22⋊C8—24no—
78C2×C22⋊C8—24no—
79C2×C22⋊C8—24no—
80C2×C22⋊C8—24no—
81C2×C22⋊C8—24no—
82C2×C22⋊C8—42no—
83C2×C22⋊C8—42no—
84C2×C22⋊C8—42no—
85C2×C22⋊C8—42no—
86C2×C22⋊C8—42no—
87C2×C22⋊C8—42no—
88C2×C22⋊C8—42no—
89C2×C22⋊C8—42no—
90C2×C22⋊C8D418yes—
91C2×C4×C8—24no—
92C2×C4×C8—42no—
93C2×C4×C8—42no—
94#379—24no—
95#379C818yescomplemented
96C4×C22⋊C4—24no—
97C4×C22⋊C4—24no—
98C4×C22⋊C4—41no—
99C4×C22⋊C4—41no—
100C4×C22⋊C4—42no—
101C4×C22⋊C4—42no—
102C4×C22⋊C4—41no—
103C4×C22⋊C4—41no—
104C2×#95C2×C412yes—
105C2×#95C2×C412yes—
106C2×#95C2×C411yescharacteristic
107C2×#95C2×C411yescharacteristic
108C2×#95C2×C412yes—
109C2×#95C2×C412yes—
110C2×#95C2×C412yes—
111#336—24no—
112#336—24no—
113#336—41no—
114#336—42no—
115#336—42no—
116#336—41no—
117#336—42no—
118C25—22no—
119C25—24no—
120C25—22no—
121C25—24no—
122C25—41no—
123C25—42no—
124C25—41no—
125C25—41no—
126C25—42no—
127C25—41no—
128C25C2×C814yes—
129C25C22×C411yescharacteristic
130C25C22×C412yes—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—24no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—41no—
161C23×C4—42no—
162C23×C4—41no—
163C23×C4—44no—
164C23×C4—48no—
165C23×C4—48no—
166C23×C4—48no—
167C23×C4—48no—
168C23×C4—48no—
169C23×C4—44no—
170C23×C4—48no—
171C23×C4—48no—
172C23×C4—44no—
173C23×C4—44no—
174C23×C4—44no—
175C23×C4—44no—
176C23×C4—48no—
177C23×C4—48no—
178C23×C4—44no—
179C23×C4—48no—
180C23×C4—44no—
181C23×C4C22×C411yescharacteristic
182C23×C4C22×C411yescharacteristic
183C23×C4C22×C411yescharacteristic
184C23×C4C22×C412yes—
185C23×C4C22×C411yescharacteristic
186C23×C4C22×C412yes—
187C23×C4C22×C411yescharacteristic
188C23×C4C22×C412yes—
189C23×C4C2×D414yes—
190C23×C4#4112yes—
191C23×C4#4112yes—
192C23×C4#4114yes—
193C23×C4C2411yescharacteristic
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×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8C2×D414yes—
221C22×C8C2×D414yes—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—41no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—41no—
238C2×C22⋊C4—41no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—41no—
241C2×C22⋊C4C2×C818yes—
242C2×C22⋊C4C2×C814yes—
243C2×C22⋊C4C2×C814yes—
244C2×C22⋊C4C2×C818yes—
245C2×C22⋊C4M1614yes—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—21no—
253C2×C42—22no—
254C2×C42—21no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—21no—
258C2×C42—22no—
259C2×C42—21no—
260C4⋊C8—44no—
261C4⋊C8—44no—
262C22⋊C8—48no—
263C22⋊C8—48no—
264C22⋊C8—48no—
265C22⋊C8—48no—
266C22⋊C8—48no—
267C22⋊C8—48no—
268C4×C8—48no—
269#95—24no—
270#95—24no—
271#95—42no—
272#95—42no—
273#95—41no—
274#95—41no—
275#95—42no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—21no—
280C24—21no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—21no—
290C24—21no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—28no—
297C24—216no—
298C24—48no—
299C24—44no—
300C24—44no—
301C24—48no—
302C24—44no—
303C24—48no—
304C24—48no—
305C24—48no—
306C24—48no—
307C24—44no—
308C24—44no—
309C24—48no—
310C24—44no—
311C24—48no—
312C24—44no—
313C24—44no—
314C24—48no—
315C24—48no—
316C24—44no—
317C24—48no—
318C24—44no—
319C24#11712yes—
320C24C4×D414yes—
321C24C22×C812yes—
322C24C22×C814yes—
323C24C2×M1612yes—
324C24#13114yes—
325C24C23×C411yescharacteristic
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
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—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—48no—
387C22×C4—44no—
388C22×C4—48no—
389C22×C4—48no—
390C22×C4—48no—
391C22×C4—48no—
392C22×C4—48no—
393C22×C4—48no—
394C22×C4—44no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—44no—
398C22×C4—416no—
399C22×C4—416no—
400C22×C4—432no—
401C22×C4—432no—
402C22×C4—416no—
403C22×C4—416no—
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—41no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—41no—
423C22×C4—42no—
424C22×C4—41no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—41no—
430C22×C4#11712yes—
431C22×C4C4×D414yes—
432C22×C4C22×C812yes—
433C22×C4C22×C814yes—
434C22×C4C22×C812yes—
435C22×C4C2×M1614yes—
436C22×C4C22×D412yes—
437C22×C4C2×#4111yescharacteristic
438C22×C4C2×#4111yescharacteristic
439C22×C4C2×#4112yes—
440C22×C4H(5,𝔽2)11yescharacteristic
441C22×C4H(5,𝔽2)12yes—
442C22×C4H(5,𝔽2)11yescharacteristic
443C22×C4H(5,𝔽2)12yes—
444C22×C4H(5,𝔽2)11yescharacteristic
445C22×C4H(5,𝔽2)11yescharacteristic
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—42no—
453C2×C8—42no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—42no—
458C2×C8—42no—
459C2×C8—42no—
460C2×C8—42no—
461C2×C8—42no—
462C2×C8—42no—
463C2×C8—42no—
464C2×C8—42no—
465C2×C8—42no—
466C2×C8—42no—
467C2×C8—42no—
468C2×C8—42no—
469C2×C8—42no—
470C2×C8—42no—
471C2×C8—42no—
472C2×C8—42no—
473C2×C8—42no—
474C22⋊C4—216no—
475C22⋊C4—48no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—48no—
480C22⋊C4—48no—
481C22⋊C4—44no—
482C42—42no—
483C42—42no—
484C42—42no—
485C42—42no—
486C42—41no—
487C42—42no—
488C42—41no—
489C42—41no—
490C42—42no—
491C42—41no—
492C23—22no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—22no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—22no—
510C23—22no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—22no—
516C23—24no—
517C23—24no—
518C23—22no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—432no—
523C23—416no—
524C23—432no—
525C23—416no—
526C23—416no—
527C23—416no—
528C23—416no—
529C23—432no—
530C23—432no—
531C23—416no—
532C23—416no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—41no—
538C23—42no—
539C23—41no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—41no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—41no—
558C23—42no—
559C23#43114yes—
560C23C8×D418yes—
561C23C2×#11711yescharacteristic
562C23C2×C4×D412yes—
563C23#51811yescharacteristic
564C23#51811yescharacteristic
565C23#51812yes—
566C23C23×C811yescharacteristic
567C23C22×M1611yescharacteristic
568C23C2×#13112yes—
569C23#56811yescharacteristic
570C23#56811yescharacteristic
571C23#56812yes—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—432no—
588C2×C4—416no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4#53412yes—
635C2×C4#53512yes—
636C2×C4#53811yescharacteristic
637C2×C4#53811yescharacteristic
638C2×C4#53811yescharacteristic
639C2×C4#53812yes—
640C2×C4#53811yescharacteristic
641C2×C4#53812yes—
642C2×C4D4211yescharacteristic
643C2×C4#54612yes—
644C2×C4#54612yes—
645C2×C4#54612yes—
646C2×C4#55112yes—
647C2×C4#55112yes—
648C2×C4#55111yescharacteristic
649C2×C4#56011yescharacteristic
650C2×C4#56012yes—
651C2×C4H(3,𝔽4)11yescharacteristic
652C8—44no—
653C8—44no—
654C8—82no—
655C8—82no—
656C22—28no—
657C22—28no—
658C22—28no—
659C22—28no—
660C22—24no—
661C22—464no—
662C22—44no—
663C22—42no—
664C22—44no—
665C22—44no—
666C22—44no—
667C22—42no—
668C22—44no—
669C22—44no—
670C22—44no—
671C22—44no—
672C22—42no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—44no—
677C22—42no—
678C22—44no—
679C22—42no—
680C22—44no—
681C22—42no—
682C22—44no—
683C22#231211yescharacteristic
684C22#232112yes—
685C22#234212yes—
686C22#235811yescharacteristic
687C22#237511yescharacteristic
688C22C2×#43111yescharacteristic
689C22#292111yescharacteristic
690C22#292412yes—
691C22C2×C8×D412yes—
692C22#293012yes—
693C22#293312yes—
694C22D4×M1612yes—
695C22#297212yes—
696C22#297211yescharacteristic
697C22#297211yescharacteristic
698C22#297311yescharacteristic
699C22#297412yes—
700C22#297511yescharacteristic
701C22#297612yes—
702C22#297612yes—
703C22#297711yescharacteristic
704C22#297812yes—
705C22#297811yescharacteristic
706C4—42no—
707C4—44no—
708C4—44no—
709C4—42no—
710C4—44no—
711C4—42no—
712C4—81no—
713C4—81no—
714C4#266811yescharacteristic
715C4#267011yescharacteristic
716C4#268512yes—
717C4#269712yes—
718C4#273511yescharacteristic
719C4#273611yescharacteristic
720C2—48no—
721C2—44no—
722C2#1436611yescharacteristic
723C2#2190211yescharacteristic
724C2#2190811yescharacteristic
725C2#2202812yes—
726C2#2203612yes—
727C2#2255212yes—
728C2#2255912yes—
729C2#2284211yescharacteristic
730C2#2285211yescharacteristic
731C2#2330611yescharacteristic
732C2#2330811yescharacteristic
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutatorC23C23×C8
socleC24C23×C4
prime radicalC26C2×C4
LCSG ▷ C23 ▷ 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. #11042C2semidirect
8. #10921C2semidirect
9. #10921C2semidirect
11. C22×C4⋊C8C22semidirect
12. C22×C4⋊C8C22semidirect
19. C22×C4×C8C22semidirect
28. C2×#336C22semidirect
29. C2×#336C22semidirect
30. C2×#336C22semidirect
31. C2×#336C22semidirect
32. C2×#336C22semidirect
33. C2×#336C22semidirect
34. C2×#336C22semidirect
95. #379C8semidirect

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
#109212