Finite Group Databasebeta

← #155673

Subgroups — #155673

Order 512.

4254 conjugacy classes of subgroups.

The subgroup lattice has 825 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#126C211yesmaximal, characteristic
2C2×#1441C212yesmaximal, complemented
3C2×#1439C212yesmaximal
4C2×C4⋊#95C211yesmaximal, characteristic, complemented
5C2×#1434C211yesmaximal, characteristic, complemented
6#7557C212yesmaximal, complemented
7#7557C212yesmaximal, complemented
8#7555C212yesmaximal
9#7555C212yesmaximal
10C22×#126—24no—
11C22×#126—24no—
12C22×#126—28no—
13C22×#126—28no—
14C22×#126C418yescomplemented
15C23×C4⋊C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C4⋊C4C2212yes—
18C23×C4⋊C4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22C23×C22⋊C4C2214yes—
23C23×C42C2211yescharacteristic
24C22×#95C2212yes—
25C22×#95C2212yes—
26C22×#95C2214yes—
27C22×#95C2214yes—
28C22×#95C2214yes—
29C22×#95C2212yes—
30C22×#95C2212yes—
31C22×#95C2212yes—
32#1441—28no—
33#1439—28no—
34C4⋊#95—24no—
35#1434—24no—
36C26C2311yescharacteristic
37C24×C4C2314yes—
38C24×C4C2314yes—
39C24×C4C2311yescharacteristic
40C24×C4C2311yescharacteristic
41C24×C4C2311yescharacteristic
42C24×C4C2311yescharacteristic
43C24×C4C2311yescharacteristic
44C24×C4C2311yescharacteristic
45C2×#126—264no—
46C2×#126—264no—
47C2×#126—232no—
48C2×#126—232no—
49C2×#126—232no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—24no—
55C22×C4⋊C4—24no—
56C22×C4⋊C4—24no—
57C22×C4⋊C4—24no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—24no—
61C22×C4⋊C4—24no—
62C22×C4⋊C4—28no—
63C22×C4⋊C4—24no—
64C22×C4⋊C4—24no—
65C22×C4⋊C4—44no—
66C22×C4⋊C4C2×C414yes—
67C22×C4⋊C4C2×C414yes—
68C22×C4⋊C4C2×C414yes—
69C22×C4⋊C4D418yes—
70C22×C22⋊C4—24no—
71C22×C22⋊C4—24no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—22no—
76C22×C22⋊C4—24no—
77C22×C22⋊C4—24no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—24no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4—28no—
82C22×C22⋊C4—28no—
83C22×C22⋊C4—28no—
84C22×C22⋊C4—44no—
85C22×C22⋊C4—44no—
86C22×C22⋊C4—44no—
87C22×C22⋊C4—44no—
88C22×C22⋊C4C2×C414yes—
89C22×C22⋊C4C2×C414yes—
90C22×C22⋊C4C2×C414yes—
91C22×C22⋊C4D418yes—
92C22×C42—24no—
93C22×C42—24no—
94C22×C42C2×C414yes—
95C22×C42D414yes—
96C22×C42D414yes—
97C2×#95—24no—
98C2×#95—28no—
99C2×#95—24no—
100C2×#95—24no—
101C2×#95—24no—
102C2×#95—24no—
103C2×#95—28no—
104C2×#95—24no—
105C2×#95—28no—
106C2×#95—28no—
107C2×#95—24no—
108C2×#95—28no—
109C2×#95—28no—
110C2×#95—44no—
111C2×#95—44no—
112C2×#95—44no—
113C2×#95—44no—
114C2×#95—44no—
115C2×#95—44no—
116C2×#95—44no—
117C25—21no—
118C25—21no—
119C25—22no—
120C25—24no—
121C25—24no—
122C25—22no—
123C25—22no—
124C25—22no—
125C25—22no—
126C25—22no—
127C25—22no—
128C25—22no—
129C25—22no—
130C25C22×C412yes—
131C25C2×D414yes—
132C25C2411yescharacteristic
133C23×C4—24no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—24no—
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—24no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4C22×C412yes—
195C23×C4C22×C412yes—
196C23×C4C22×C412yes—
197C23×C4C22×C412yes—
198C23×C4C22×C412yes—
199C23×C4C22×C412yes—
200C23×C4C2×D414yes—
201C23×C4C2×D414yes—
202C23×C4C2×D412yes—
203C23×C4C2×D412yes—
204C23×C4C2×D412yes—
205C23×C4C2×D412yes—
206C23×C4C2×D414yes—
207C23×C4C2×D412yes—
208C23×C4C2×D414yes—
209C23×C4C2×D412yes—
210C23×C4C2×D414yes—
211C23×C4#4118yes—
212C23×C4#4118yes—
213#126—2256no—
214C2×C4⋊C4—216no—
215C2×C4⋊C4—216no—
216C2×C4⋊C4—216no—
217C2×C4⋊C4—216no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—216no—
227C2×C4⋊C4—216no—
228C2×C4⋊C4—216no—
229C2×C4⋊C4—416no—
230C2×C4⋊C4—416no—
231C2×C4⋊C4—48no—
232C2×C4⋊C4—48no—
233C2×C4⋊C4C22⋊C4116yescomplemented
234C2×C22⋊C4—216no—
235C2×C22⋊C4—216no—
236C2×C22⋊C4—216no—
237C2×C22⋊C4—216no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—216no—
247C2×C22⋊C4—216no—
248C2×C22⋊C4—416no—
249C2×C22⋊C4—416no—
250C2×C22⋊C4—416no—
251C2×C22⋊C4—416no—
252C2×C22⋊C4—416no—
253C2×C22⋊C4—416no—
254C2×C22⋊C4—416no—
255C2×C22⋊C4C22⋊C4116yes—
256C2×C42—28no—
257C2×C42—28no—
258C2×C42—216no—
259C2×C42—216no—
260C2×C42C22⋊C4116yescomplemented
261#95—216no—
262#95—416no—
263#95—416no—
264#95—48no—
265#95—48no—
266#95—48no—
267#95—416no—
268#95—48no—
269C24—22no—
270C24—22no—
271C24—24no—
272C24—24no—
273C24—24no—
274C24—24no—
275C24—24no—
276C24—24no—
277C24—24no—
278C24—24no—
279C24—24no—
280C24—24no—
281C24—24no—
282C24—24no—
283C24—28no—
284C24—28no—
285C24—28no—
286C24—28no—
287C24—42no—
288C24—42no—
289C24—42no—
290C24—42no—
291C24—41no—
292C24—41no—
293C24—42no—
294C24—41no—
295C24—41no—
296C24—41no—
297C24—41no—
298C24—42no—
299C24—41no—
300C24—41no—
301C24—42no—
302C24—42no—
303C24—42no—
304C24—42no—
305C24—42no—
306C24—42no—
307C24—44no—
308C24—44no—
309C24—44no—
310C24—44no—
311C24—44no—
312C24—44no—
313C24—44no—
314C24—44no—
315C24—44no—
316C24—44no—
317C24—44no—
318C24—44no—
319C24—44no—
320C24—44no—
321C24—44no—
322C24—44no—
323C24—44no—
324C24—44no—
325C24—44no—
326C24—44no—
327C24C2×C22⋊C414yes—
328C24C23×C411yescharacteristic
329C24C22×D411yescharacteristic
330C24C22×D411yescharacteristic
331C24C22×D412yes—
332C24C22×D412yes—
333C24C2×#4114yes—
334C24C2×#4114yes—
335C24H(5,𝔽2)12yes—
336C24H(5,𝔽2)12yes—
337C24H(5,𝔽2)12yes—
338C24H(5,𝔽2)12yes—
339C242−1+412yes—
340C242−1+412yes—
341C242−1+412yes—
342C242−1+412yes—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—28no—
351C22×C4—28no—
352C22×C4—28no—
353C22×C4—28no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
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—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4—44no—
500C22×C4—44no—
501C22×C4—44no—
502C22×C4—44no—
503C22×C4—44no—
504C22×C4—44no—
505C22×C4—44no—
506C22×C4—44no—
507C22×C4—44no—
508C22×C4—44no—
509C22×C4C2×C22⋊C414yes—
510C22×C4C2×C22⋊C414yes—
511C22×C4C2×C22⋊C414yes—
512C22×C4C2×C22⋊C414yes—
513C22×C4C2×C22⋊C414yes—
514C22×C4C2×C22⋊C414yes—
515C22×C4C2×C22⋊C414yes—
516C22×C4C2×C22⋊C414yes—
517C22×C4#12018yes—
518C22×C4#12018yes—
519C4⋊C4—232no—
520C4⋊C4—232no—
521C4⋊C4—232no—
522C4⋊C4—432no—
523C22⋊C4—232no—
524C22⋊C4—232no—
525C22⋊C4—232no—
526C22⋊C4—464no—
527C42—232no—
528C23—28no—
529C23—28no—
530C23—28no—
531C23—24no—
532C23—42no—
533C23—44no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—48no—
538C23—48no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—48no—
546C23—48no—
547C23—44no—
548C23—44no—
549C23—48no—
550C23—48no—
551C23—48no—
552C23—44no—
553C23—44no—
554C23—48no—
555C23—48no—
556C23—48no—
557C23—48no—
558C23—48no—
559C23—48no—
560C23—48no—
561C23—42no—
562C23—42no—
563C23—81no—
564C23—81no—
565C23—81no—
566C23—81no—
567C23—84no—
568C23—84no—
569C23—82no—
570C23—82no—
571C23—82no—
572C23—82no—
573C23—82no—
574C23—82no—
575C23—82no—
576C23—82no—
577C23—82no—
578C23—82no—
579C23—84no—
580C23—84no—
581C23—84no—
582C23—84no—
583C23—84no—
584C23—84no—
585C23—84no—
586C23—84no—
587C23C22×C22⋊C411yescharacteristic
588C23C22×C22⋊C411yescharacteristic
589C23C22×C22⋊C411yescharacteristic
590C23C4×#4112yes—
591C23C4×#4112yes—
592C23#52012yes—
593C23#52012yes—
594C23#52012yes—
595C23#52012yes—
596C23C2×#12012yes—
597C23C2×#12012yes—
598C23#52512yes—
599C23#52512yes—
600C23#53212yes—
601C23#53212yes—
602C23#53511yescharacteristic
603C23#53511yescharacteristic
604C23#53511yescharacteristic
605C23#53511yescharacteristic
606C23#53512yes—
607C23#53512yes—
608C23#53611yescharacteristic
609C23#53611yescharacteristic
610C23#53611yescharacteristic
611C23#53611yescharacteristic
612C23#53612yes—
613C23#53612yes—
614C23#53712yes—
615C23#53712yes—
616C23#53712yes—
617C23#53712yes—
618C23#53712yes—
619C23#53712yes—
620C23#53712yes—
621C23#53712yes—
622C23#53814yes—
623C23#53814yes—
624C23#53914yes—
625C23#53914yes—
626C23#54114yes—
627C23#54114yes—
628C23#54214yes—
629C23#54214yes—
630C23#54614yes—
631C23#54614yes—
632C23#54614yes—
633C23#54714yes—
634C23#54714yes—
635C23#54714yes—
636C23#54814yes—
637C23#55014yes—
638C23#55114yes—
639C23#55214yes—
640C23#55314yes—
641C23#55614yes—
642C23#56014yes—
643C23#56314yes—
644C23#56314yes—
645C23#56314yes—
646C2×C4—24no—
647C2×C4—28no—
648C2×C4—28no—
649C2×C4—28no—
650C2×C4—28no—
651C2×C4—216no—
652C2×C4—24no—
653C2×C4—24no—
654C2×C4—28no—
655C2×C4—24no—
656C2×C4—24no—
657C2×C4—28no—
658C2×C4—28no—
659C2×C4—24no—
660C2×C4—24no—
661C2×C4—24no—
662C2×C4—24no—
663C2×C4—24no—
664C2×C4—24no—
665C2×C4—42no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—44no—
679C2×C4—44no—
680C2×C4—44no—
681C2×C4—44no—
682C2×C4—44no—
683C2×C4—44no—
684C2×C4—44no—
685C2×C4—44no—
686C2×C4—48no—
687C2×C4—48no—
688C2×C4—48no—
689C2×C4—44no—
690C2×C4—44no—
691C2×C4—44no—
692C2×C4—44no—
693C2×C4—42no—
694C2×C4—42no—
695C2×C4—48no—
696C2×C4—48no—
697C2×C4—48no—
698C2×C4—48no—
699C2×C4—48no—
700C2×C4—48no—
701C2×C4—48no—
702C2×C4—48no—
703C2×C4—48no—
704C2×C4—48no—
705C2×C4—48no—
706C2×C4—48no—
707C2×C4—48no—
708C2×C4—48no—
709C2×C4—84no—
710C2×C4—84no—
711C2×C4—84no—
712C2×C4—84no—
713C2×C4—84no—
714C2×C4—84no—
715C2×C4—84no—
716C2×C4—84no—
717C2×C4—84no—
718C2×C4—84no—
719C2×C4—84no—
720C2×C4—84no—
721C2×C4—84no—
722C2×C4—84no—
723C2×C4—84no—
724C2×C4—84no—
725C2×C4#37918yes—
726C22—44no—
727C22—44no—
728C22—44no—
729C22—416no—
730C22—84no—
731C22—84no—
732C22—88no—
733C22—88no—
734C22—84no—
735C22—84no—
736C22—82no—
737C22—82no—
738C22—88no—
739C22—82no—
740C22—82no—
741C22C2×#37911yescharacteristic
742C22#229711yescharacteristic
743C22#229711yescharacteristic
744C22#231111yescharacteristic
745C22#231112yes—
746C22#231112yes—
747C22#231111yescharacteristic
748C22#231111yescharacteristic
749C22#231111yescharacteristic
750C22#233414yes—
751C22#233414yes—
752C22#233912yes—
753C22#233912yes—
754C22#233912yes—
755C22#234112yes—
756C22#237314yes—
757C22#238314yes—
758C22#240412yes—
759C22#240612yes—
760C22#240714yes—
761C22#243014yes—
762C22#243112yes—
763C22#243212yes—
764C22#260114yes—
765C22#260312yes—
766C22#260412yes—
767C22#261512yes—
768C22#261512yes—
769C22#261612yes—
770C22#261612yes—
771C22#263612yes—
772C22#263612yes—
773C22#263712yes—
774C22#263712yes—
775C22#266212yes—
776C22#266212yes—
777C22#266412yes—
778C22#266412yes—
779C22#268514yes—
780C22#268914yes—
781C22#269114yes—
782C22#269514yes—
783C22#269814yes—
784C22#270214yes—
785C22#270514yes—
786C22#271414yes—
787C22#271514yes—
788C22#271614yes—
789C22#271814yes—
790C22#272114yes—
791C22#272214yes—
792C22#272314yes—
793C22#272514yes—
794C22#272714yes—
795C22#281811yescharacteristic
796C22#282111yescharacteristic
797C22#282111yescharacteristic
798C22#282111yescharacteristic
799C22#282314yes—
800C4—28no—
801C4—44no—
802C4—44no—
803C4—44no—
804C4—44no—
805C4—44no—
806C4—88no—
807C4—88no—
808C2—84no—
809C2#1393711yescharacteristic
810C2#1395611yescharacteristic
811C2#1416611yescharacteristic
812C2#1429111yescharacteristic
813C2#1429111yescharacteristic
814C2#1433711yescharacteristic
815C2#1447514yes—
816C2#1449314yes—
817C2#1466511yescharacteristic
818C2#1665512yes—
819C2#1665612yes—
820C2#1665814yes—
821C2#1665914yes—
822C2#1667312yes—
823C2#1667412yes—
subgroupG/H
center, Frattini, socle, cosocleC25C24
commutatorC24C23×C4
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C25 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1441C2semidirect
4. C2×C4⋊#95C2semidirect
5. C2×#1434C2semidirect
6. #7557C2semidirect
7. #7557C2semidirect
14. C22×#126C4semidirect
233. C2×C4⋊C4C22⋊C4semidirect
260. C2×C42C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C23×#1262
C2×#14412
C2×#14392
C2×C4⋊#952
C2×#14342
#75572
#75552