Finite Group Databasebeta

← #378991

Subgroups — #378991

Order 512.

1908 conjugacy classes of subgroups.

The subgroup lattice has 709 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1954C212yesmaximal, complemented
2C2×#1954C211yesmaximal, characteristic, complemented
3C2×#1923C212yesmaximal, complemented
4C2×#1869C212yesmaximal, complemented
5C2×C8×C22⋊C4C211yesmaximal, characteristic, complemented
6C4×#392C211yesmaximal, characteristic
7#11322C211yesmaximal, characteristic, complemented
8#11042C211yesmaximal, characteristic, complemented
9#10964C211yesmaximal, characteristic, complemented
10#10921C212yesmaximal, complemented
11C8×#95C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C22×C4×D4C2212yes—
14C22×C4×D4C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2212yes—
19C22×C22⋊C8C2212yes—
20C22×C22⋊C8C2212yes—
21C22×C22⋊C8C2212yes—
22C22×C22⋊C8C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24C22×C4×C8C2212yes—
25C22×C4×C8C2211yescharacteristic
26C2×#392C411yescharacteristic
27C2×C4×C22⋊C4C2211yescharacteristic
28C2×C4×C22⋊C4C2212yes—
29#1954—28no—
30#1954—24no—
31#1923—28no—
32#1869—28no—
33C8×C22⋊C4—24no—
34C2×#336C2212yes—
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37C2×#336C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40#1444C411yescharacteristic
41#1439C411yescharacteristic
42#1439C412yes—
43#1438C412yes—
44#1438C411yescharacteristic
45C4×#95C2211yescharacteristic
46C23×D4C2×C412yes—
47C23×D4C2×C411yescharacteristic
48C24×C4C2312yes—
49C24×C4C2311yescharacteristic
50C23×C8—22no—
51C23×C8—22no—
52C23×C8—22no—
53C23×C8—24no—
54C23×C8—42no—
55C23×C8C2312yes—
56C23×C8C2311yescharacteristic
57C23×C8C2312yes—
58C23×C8C2311yescharacteristic
59C23×C8C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C2×C4×D4—24no—
62C2×C4×D4—24no—
63C2×C4×D4—24no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—24no—
67C2×C4×D4—42no—
68C2×C4×D4—42no—
69C2×C4×D4—42no—
70C2×C4×D4—42no—
71C22×C4⋊C4C2×C412yes—
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C22⋊C4C2×C411yescharacteristic
74C22×C22⋊C4C2×C412yes—
75C22×C22⋊C4C2×C412yes—
76C22×C22⋊C4C2×C412yes—
77C22×C22⋊C4C2×C411yescharacteristic
78C22×C22⋊C4C2×C412yes—
79C22×C22⋊C4C2×C412yes—
80C22×C42C2311yescharacteristic
81C22×C42C2312yes—
82C22×C42C2311yescharacteristic
83C2×C4⋊C8—24no—
84C2×C4⋊C8—24no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—42no—
87C2×C4⋊C8—42no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—24no—
91C2×C22⋊C8—24no—
92C2×C22⋊C8—24no—
93C2×C22⋊C8—24no—
94C2×C22⋊C8—24no—
95C2×C22⋊C8—24no—
96C2×C22⋊C8—24no—
97C2×C22⋊C8—24no—
98C2×C22⋊C8—24no—
99C2×C22⋊C8—24no—
100C2×C22⋊C8—24no—
101C2×C22⋊C8—24no—
102C2×C22⋊C8—42no—
103C2×C22⋊C8—42no—
104C2×C22⋊C8—42no—
105C2×C22⋊C8D414yescomplemented
106C2×C4×C8—24no—
107C2×C4×C8—24no—
108C2×C4×C8—22no—
109C2×C4×C8—24no—
110C2×C4×C8—42no—
111C2×C4×C8—42no—
112C2×C4×C8D414yescomplemented
113#392C818yescomplemented
114C4×C22⋊C4—24no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4—24no—
118C2×#95C2×C411yescharacteristic
119C2×#95C2×C412yes—
120C2×#95C2×C412yes—
121C2×#95C2×C411yescharacteristic
122C2×#95C2×C412yes—
123C2×#95C2×C411yescharacteristic
124C2×#95C2×C411yescharacteristic
125#336—24no—
126#336—22no—
127#336—24no—
128#336—22no—
129#336—22no—
130#336D418yescomplemented
131C25C22×C412yes—
132C25C22×C411yescharacteristic
133C22×D4—24no—
134C22×D4—24no—
135C22×D4—24no—
136C22×D4—22no—
137C22×D4—24no—
138C22×D4—22no—
139C22×D4—42no—
140C22×D4—42no—
141C22×D4—42no—
142C22×D4—42no—
143C22×D4C2×C818yes—
144C22×D4C2×C814yescomplemented
145C23×C4—22no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
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×C412yes—
168C23×C4C22×C412yes—
169C23×C4C22×C411yescharacteristic
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C22×C411yescharacteristic
173C23×C4C22×C412yes—
174C23×C4C22×C411yescharacteristic
175C23×C4C2×D412yes—
176C23×C4#4112yes—
177C23×C4C2411yescharacteristic
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—28no—
184C22×C8—21no—
185C22×C8—22no—
186C22×C8—21no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—21no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—21no—
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×C8—48no—
206C22×C8—48no—
207C22×C8—44no—
208C22×C8—44no—
209C22×C8C2×D414yes—
210C22×C8C2×D412yes—
211C22×C8C2×D412yes—
212C22×C8C2×D412yes—
213C22×C8C2×D414yes—
214C22×C8C2×D412yes—
215C22×C8#4112yes—
216C22×C8#4112yes—
217C4×D4—48no—
218C4×D4—48no—
219C4×D4—48no—
220C4×D4—48no—
221C4×D4—44no—
222C4×D4—44no—
223C4×D4—48no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4C2×C814yes—
249C2×C22⋊C4C2×C818yescomplemented
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—21no—
257C2×C42—22no—
258C2×C42—21no—
259C2×C42—22no—
260C2×C42C2×D414yes—
261C2×C42C2×D412yes—
262C2×C42#4112yes—
263C2×C42#4114yes—
264C4⋊C8—48no—
265C4⋊C8—44no—
266C22⋊C8—28no—
267C22⋊C8—48no—
268C22⋊C8—48no—
269C22⋊C8—48no—
270C22⋊C8—48no—
271C22⋊C8—48no—
272C4×C8—44no—
273C4×C8—48no—
274C4×C8—44no—
275#95—22no—
276#95—22no—
277#95—24no—
278#95—24no—
279#95—22no—
280#95—24no—
281#95C2×C814yescomplemented
282C24—21no—
283C24—22no—
284C24—21no—
285C24—22no—
286C24—21no—
287C24—21no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24C4×D412yes—
305C24C22×C814yes—
306C24C22×C812yes—
307C24#13112yes—
308C24C23×C411yescharacteristic
309C2×D4—216no—
310C2×D4—216no—
311C2×D4—28no—
312C2×D4—48no—
313C2×D4—48no—
314C2×D4—48no—
315C2×D4—48no—
316C2×D4—48no—
317C2×D4—48no—
318C2×D4—44no—
319C2×D4—44no—
320C2×D4—48no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—41no—
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—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
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×C4C4×D412yes—
403C22×C4C4×D414yes—
404C22×C4C4×D412yes—
405C22×C4C4×D412yes—
406C22×C4C4×D414yes—
407C22×C4C22×C812yes—
408C22×C4C22×C812yes—
409C22×C4C22×C814yes—
410C22×C4#13112yes—
411C22×C4#13114yes—
412C22×C4#13112yes—
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D412yes—
416C22×C4C2×#4111yescharacteristic
417C22×C4C2×#4111yescharacteristic
418C22×C4C2×#4111yescharacteristic
419C22×C4C2×#4111yescharacteristic
420C22×C4C2×#4112yes—
421C22×C4H(5,𝔽2)12yes—
422C22×C4H(5,𝔽2)12yes—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—24no—
437C2×C8—24no—
438C2×C8—416no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—42no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—42no—
445C2×C8—42no—
446C2×C8—42no—
447C2×C8—42no—
448C2×C8—42no—
449C2×C8—42no—
450C2×C8—42no—
451C2×C8—42no—
452C2×C8—42no—
453C2×C8—42no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8#12114yes—
457C2×C8#12714yes—
458C4⋊C4—48no—
459C4⋊C4—44no—
460C22⋊C4—28no—
461C22⋊C4—28no—
462C22⋊C4—216no—
463C22⋊C4—48no—
464C22⋊C4—48no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C42—22no—
468C42—22no—
469C42—24no—
470C42—24no—
471C42—24no—
472C42—24no—
473C42—24no—
474C42—41no—
475C42—41no—
476C42—42no—
477C23—24no—
478C23—22no—
479C23—24no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—24no—
484C23—22no—
485C23—22no—
486C23—24no—
487C23—24no—
488C23—22no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—41no—
497C23—41no—
498C23—42no—
499C23—41no—
500C23—41no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23C8×D414yes—
535C23C2×C4×D411yescharacteristic
536C23C2×C4×D412yes—
537C23C2×C4×D411yescharacteristic
538C23C4×#4111yescharacteristic
539C23#51812yes—
540C23C23×C811yescharacteristic
541C23C2×#13111yescharacteristic
542C23C2×#13111yescharacteristic
543C23C2×#13112yes—
544C23C2×#13111yescharacteristic
545C23#56812yes—
546D4—432no—
547D4—416no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—82no—
606C2×C4C8×D414yes—
607C2×C4C8×D418yes—
608C2×C4C2×#12111yescharacteristic
609C2×C4#52511yescharacteristic
610C2×C4#52911yescharacteristic
611C2×C4C2×#12711yescharacteristic
612C2×C4#53211yescharacteristic
613C2×C4#53212yes—
614C2×C4#53211yescharacteristic
615C2×C4#53411yescharacteristic
616C2×C4#53511yescharacteristic
617C2×C4#53812yes—
618C2×C4D4212yes—
619C2×C4#54612yes—
620C2×C4#54612yes—
621C2×C4#54612yes—
622C2×C4#55012yes—
623C2×C4#55112yes—
624C2×C4#55312yes—
625C2×C4#55512yes—
626C8—24no—
627C8—24no—
628C8—42no—
629C8—42no—
630C8—44no—
631C8—82no—
632C22—28no—
633C22—24no—
634C22—42no—
635C22—42no—
636C22—44no—
637C22—44no—
638C22—44no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—44no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22—44no—
648C22—44no—
649C22—44no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22C4×#12111yescharacteristic
659C22C4×#12711yescharacteristic
660C22#232111yescharacteristic
661C22#235812yes—
662C22#236312yes—
663C22C2×C8×D411yescharacteristic
664C22C2×C8×D412yes—
665C22C2×C8×D411yescharacteristic
666C22#292911yescharacteristic
667C22#293011yescharacteristic
668C22#293111yescharacteristic
669C22#293112yes—
670C22#293111yescharacteristic
671C22#293312yes—
672C22#293312yes—
673C22#293312yes—
674C22C8×#4111yescharacteristic
675C22#296611yescharacteristic
676C22#296912yes—
677C22#297212yes—
678C22#297612yes—
679C22#297712yes—
680C22#297912yes—
681C4—24no—
682C4—28no—
683C4—42no—
684C4—44no—
685C4—42no—
686C4—42no—
687C4—42no—
688C4—82no—
689C4#253911yescharacteristic
690C4#254111yescharacteristic
691C4#254311yescharacteristic
692C4#255412yes—
693C4#255512yes—
694C4#261011yescharacteristic
695C2—44no—
696C2—84no—
697C2#1422811yescharacteristic
698C2C8×#12111yescharacteristic
699C2#2149411yescharacteristic
700C2C8×#12711yescharacteristic
701C2#2168511yescharacteristic
702C2#2202811yescharacteristic
703C2#2202911yescharacteristic
704C2#2284212yes—
705C2#2285312yes—
706C2#2308312yes—
707C2#2309012yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutatorC23C23×C8
socleC24C23×C4
prime radicalC2×#392C4
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×#1954C2semidirect
2. C2×#1954C2semidirect
3. C2×#1923C2semidirect
4. C2×#1869C2semidirect
5. C2×C8×C22⋊C4C2semidirect
7. #11322C2semidirect
8. #11042C2semidirect
9. #10964C2semidirect
10. #10921C2semidirect
11. C8×#95C2semidirect
105. C2×C22⋊C8D4semidirect
112. C2×C4×C8D4semidirect
113. #392C8semidirect
130. #336D4semidirect
144. C22×D4C2×C8semidirect
249. C2×C22⋊C4C2×C8semidirect
281. #95C2×C8semidirect

Maximal subgroups

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

subgroupindex
C2×#19542
C2×#19232
C2×#18692
C2×C8×C22⋊C42
C4×#3922
#113222
#110422
#109642
#109212
C8×#952