Finite Group Databasebeta

← #6440317

Subgroups — #6440317

Order 512.

4948 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊Q8C211yesmaximal, characteristic
2C2×#3095C212yesmaximal, complemented
3C2×#2280C211yesmaximal, characteristic
4#24010C218yesmaximal, complemented
5#24006C214yesmaximal, complemented
6#24003C214yesmaximal, complemented
7#23481C214yesmaximal, complemented
8#20911C212yesmaximal, complemented
9#20844C212yesmaximal, complemented
10#20599C212yesmaximal, complemented
11#20579C211yesmaximal, characteristic
12C24×Q8C2211yescharacteristic
13C25×C4C2211yescharacteristic
14C22×C22⋊Q8—22no—
15C22×C22⋊Q8—21no—
16C22×C22⋊Q8—21no—
17C22×C22⋊Q8—22no—
18C22×C22⋊Q8—216no—
19C22×C22⋊Q8C2212yes—
20C22×C22⋊Q8C2214yes—
21C22×C22⋊Q8C2214yes—
22C22×C22⋊Q8C2212yes—
23C22×#117C2211yescharacteristic
24C22×#117C2211yescharacteristic
25C23×C4⋊C4—21no—
26C23×C4⋊C4C2211yescharacteristic
27C23×C22⋊C4—21no—
28#3095—216no—
29C2×#484C2214yescomplemented
30C2×#485C2218yescomplemented
31C2×#483C2214yescomplemented
32C2×#428C2214yescomplemented
33C2×#419C2214yescomplemented
34C2×#419C2214yescomplemented
35C2×#407C2212yes—
36C2×#407C2212yes—
37C2×#407C2212yes—
38#2280—21no—
39#2280—21no—
40#2280—22no—
41#2280C2212yes—
42#2280C2212yes—
43#2280C2214yes—
44C2×#381—21no—
45C2×#380C2212yes—
46C2×#380C2212yes—
47C2×#379—21no—
48C2×C4×C22⋊C4—22no—
49#2036C22116yescomplemented
50#2035C22116yescomplemented
51#1873C2218yescomplemented
52#1872C2218yescomplemented
53#1853C2218yescomplemented
54#1852C2218yescomplemented
55#1787C2218yescomplemented
56#1787C2218yescomplemented
57#1779C2214yes—
58#1779C2214yes—
59#1736C2214yescomplemented
60C26D411yescharacteristic
61C23×Q8—28no—
62C23×Q8—22no—
63C23×Q8—21no—
64C23×Q8—28no—
65C23×Q8—44no—
66C23×Q8C2312yes—
67C23×Q8C2314yes—
68C24×C4—21no—
69C24×C4—21no—
70C24×C4—28no—
71C24×C4—28no—
72C24×C4—44no—
73C24×C4D411yescharacteristic
74C24×C4D412yes—
75C24×C4D411yescharacteristic
76C24×C4D412yes—
77C24×C4D411yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80C24×C4C2312yes—
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C22×M16C2314yes—
84C2×C22⋊Q8—232no—
85C2×C22⋊Q8—232no—
86C2×C22⋊Q8—216no—
87C2×C22⋊Q8—216no—
88C2×C22⋊Q8—28no—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8—22no—
97C2×C22⋊Q8—232no—
98C2×C22⋊Q8—232no—
99C2×C22⋊Q8—28no—
100C2×C22⋊Q8D4116yes—
101C2×C22⋊Q8C2318yes—
102C2×C22⋊Q8C2314yes—
103C2×C22⋊Q8C2314yes—
104C2×#117—21no—
105C2×#117—21no—
106C2×#117—21no—
107C2×#117—21no—
108C2×#117—28no—
109C2×#117—44no—
110C2×#117C2312yes—
111C2×#117C2312yes—
112C2×#117C2312yes—
113C2×#117C2312yes—
114C22×C4⋊C4—216no—
115C22×C4⋊C4—22no—
116C22×C4⋊C4—24no—
117C22×C4⋊C4—24no—
118C22×C4⋊C4—22no—
119C22×C4⋊C4—22no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—28no—
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2314yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C22⋊C4—216no—
128C22×C22⋊C4—22no—
129C22×C22⋊C4—24no—
130C22×C22⋊C4—24no—
131C22×C22⋊C4—22no—
132C22×C22⋊C4—21no—
133C22×C22⋊C4—21no—
134C22×C22⋊C4—22no—
135C22×C22⋊C4—22no—
136C22×C42—21no—
137C22×C42—21no—
138#485—232no—
139#484—216no—
140#483—216no—
141#428—216no—
142C2×#107C2318yescomplemented
143C2×#106C2318yescomplemented
144#419—216no—
145#419—48no—
146C2×#103C2318yescomplemented
147C2×#103C2318yescomplemented
148C2×#103C23116yescomplemented
149#407—28no—
150#407—28no—
151#407—44no—
152C2×C22⋊C8C2314yes—
153C2×C22⋊C8C2314yes—
154C2×C22⋊C8C2318yes—
155C2×C22⋊C8C2318yes—
156#381—22no—
157#381—24no—
158#381—22no—
159#380—22no—
160#380—22no—
161#380—22no—
162#380—22no—
163#380C2314yes—
164#380C2314yes—
165#380C2314yes—
166#380C2314yes—
167#379—22no—
168#379—22no—
169#379—24no—
170C4×C22⋊C4—24no—
171C4×C22⋊C4—24no—
172C4×C22⋊C4—24no—
173C4×C22⋊C4—24no—
174C2×#95—22no—
175C2×#95—22no—
176#340C23116yescomplemented
177C25—24no—
178C25—28no—
179C25—24no—
180C25—42no—
181C25—44no—
182C25C2×D411yescharacteristic
183C25C2×D411yescharacteristic
184C25C2×D412yes—
185C25C2×D411yescharacteristic
186C25C2×D412yes—
187C22×Q8—28no—
188C22×Q8—216no—
189C22×Q8—216no—
190C22×Q8—24no—
191C22×Q8—24no—
192C22×Q8—22no—
193C22×Q8—28no—
194C22×Q8—216no—
195C22×Q8—28no—
196C22×Q8—48no—
197C22×Q8—464no—
198C22×Q8—44no—
199C22×Q8—48no—
200C22×Q8—48no—
201C22×Q8—44no—
202C22×Q8C2×D418yes—
203C22×Q8C2414yes—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—28no—
209C23×C4—28no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—22no—
214C23×C4—24no—
215C23×C4—22no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—28no—
219C23×C4—28no—
220C23×C4—24no—
221C23×C4—28no—
222C23×C4—28no—
223C23×C4—24no—
224C23×C4—28no—
225C23×C4—28no—
226C23×C4—24no—
227C23×C4—28no—
228C23×C4—28no—
229C23×C4—24no—
230C23×C4—24no—
231C23×C4—22no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—44no—
235C23×C4—44no—
236C23×C4—44no—
237C23×C4—44no—
238C23×C4—42no—
239C23×C4—42no—
240C23×C4—42no—
241C23×C4—464no—
242C23×C4—48no—
243C23×C4—44no—
244C23×C4—44no—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D412yes—
247C23×C4C2×D412yes—
248C23×C4C2×D412yes—
249C23×C4C2×D412yes—
250C23×C4C2×D412yes—
251C23×C4C2×D412yes—
252C23×C4C2×D412yes—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D418yes—
257C23×C4C2×D412yes—
258C23×C4C2×D411yescharacteristic
259C23×C4#4118yes—
260C23×C4C2412yes—
261C23×C4C2411yescharacteristic
262C23×C4C2411yescharacteristic
263C23×C4C2412yes—
264C23×C4C2411yescharacteristic
265C2×M16—28no—
266C2×M16—28no—
267C2×M16—28no—
268C22×C8C2418yes—
269C22⋊Q8—232no—
270C22⋊Q8—232no—
271C22⋊Q8—232no—
272C22⋊Q8—232no—
273C22⋊Q8—232no—
274C22⋊Q8—216no—
275C22⋊Q8—216no—
276C22⋊Q8—216no—
277C22⋊Q8—28no—
278C22⋊Q8—28no—
279C22⋊Q8—216no—
280C22⋊Q8C2×D4116yes—
281C22⋊Q8C2×D4116yes—
282#117—28no—
283#117—28no—
284#117—28no—
285#117—44no—
286#117—44no—
287#117—44no—
288#117—44no—
289#117—44no—
290C2×C4⋊C4—28no—
291C2×C4⋊C4—216no—
292C2×C4⋊C4—232no—
293C2×C4⋊C4—232no—
294C2×C4⋊C4—216no—
295C2×C4⋊C4—216no—
296C2×C4⋊C4—28no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—216no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4C2×D418yes—
310C2×C4⋊C4C2414yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2414yes—
314C2×C22⋊C4—216no—
315C2×C22⋊C4—232no—
316C2×C22⋊C4—232no—
317C2×C22⋊C4—216no—
318C2×C22⋊C4—28no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—48no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—48no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—44no—
338C2×C42—44no—
339#107—216no—
340#106—216no—
341#103—216no—
342#103—216no—
343#103—416no—
344C22⋊C8—28no—
345C22⋊C8—28no—
346C22⋊C8—216no—
347C22⋊C8—48no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352C24—22no—
353C24—22no—
354C24—24no—
355C24—24no—
356C24—232no—
357C24—24no—
358C24—24no—
359C24—22no—
360C24—24no—
361C24—22no—
362C24—22no—
363C24—24no—
364C24—24no—
365C24—24no—
366C24—24no—
367C24—24no—
368C24—24no—
369C24—24no—
370C24—22no—
371C24—22no—
372C24—24no—
373C24—42no—
374C24—42no—
375C24—42no—
376C24—42no—
377C24—42no—
378C24—42no—
379C24—416no—
380C24—432no—
381C24—416no—
382C24—432no—
383C24#12014yes—
384C24#12314yes—
385C24#13718yes—
386C24C22×D411yescharacteristic
387C24C22×D412yes—
388C24C22×D411yescharacteristic
389C24C22×D411yescharacteristic
390C24C22×D412yes—
391C2×Q8—216no—
392C2×Q8—28no—
393C2×Q8—24no—
394C2×Q8—232no—
395C2×Q8—28no—
396C2×Q8—28no—
397C2×Q8—464no—
398C2×Q8—432no—
399C2×Q8—48no—
400C2×Q8—464no—
401C2×Q8—48no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—416no—
405C2×Q8C22×D418yes—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—232no—
417C22×C4—232no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—216no—
425C22×C4—216no—
426C22×C4—28no—
427C22×C4—28no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—416no—
461C22×C4—432no—
462C22×C4—432no—
463C22×C4—432no—
464C22×C4—432no—
465C22×C4—416no—
466C22×C4—416no—
467C22×C4—432no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—432no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—48no—
476C22×C4—416no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4#12014yes—
485C22×C4#12014yes—
486C22×C4#12014yes—
487C22×C4#12314yes—
488C22×C4#12318yes—
489C22×C4C22×D412yes—
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D414yes—
493C22×C4C22×D414yes—
494C22×C4C22×D412yes—
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D414yes—
497C22×C4C2×#4118yes—
498C22×C4C2×#4114yes—
499C22×C4H(5,𝔽2)14yes—
500C22×C4C2511yescharacteristic
501M16—416no—
502C2×C8—28no—
503C2×C8—28no—
504C2×C8—28no—
505C4⋊C4—216no—
506C4⋊C4—232no—
507C4⋊C4—216no—
508C4⋊C4—216no—
509C4⋊C4—28no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—28no—
514C4⋊C4—44no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4C22×D418yes—
518C22⋊C4—232no—
519C22⋊C4—216no—
520C22⋊C4—216no—
521C22⋊C4—48no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—48no—
527C22⋊C4—48no—
528C22⋊C4—48no—
529C42—44no—
530C42—44no—
531C42—44no—
532C42—44no—
533C23—28no—
534C23—28no—
535C23—216no—
536C23—216no—
537C23—28no—
538C23—22no—
539C23—21no—
540C23—21no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—28no—
548C23—216no—
549C23—216no—
550C23—28no—
551C23—22no—
552C23—22no—
553C23—21no—
554C23—21no—
555C23—22no—
556C23—28no—
557C23—216no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—44no—
564C23—48no—
565C23—48no—
566C23—48no—
567C23—48no—
568C23—48no—
569C23—48no—
570C23—48no—
571C23—48no—
572C23—48no—
573C23—48no—
574C23—48no—
575C23—48no—
576C23—48no—
577C23—4128no—
578C23—48no—
579C23—44no—
580C23—48no—
581C23—44no—
582C23C2×#12012yes—
583C23C2×#12012yes—
584C23C2×#12012yes—
585C23C2×#12314yes—
586C23C2×#12312yes—
587C23#53412yes—
588C23C2×#13714yes—
589C23C2×#13714yes—
590C23C2×#13714yes—
591C23#57514yes—
592C23C23×D411yescharacteristic
593Q8—216no—
594Q8—432no—
595Q8—416no—
596Q8—48no—
597C2×C4—28no—
598C2×C4—28no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—28no—
606C2×C4—28no—
607C2×C4—216no—
608C2×C4—28no—
609C2×C4—28no—
610C2×C4—216no—
611C2×C4—28no—
612C2×C4—28no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—48no—
617C2×C4—48no—
618C2×C4—48no—
619C2×C4—48no—
620C2×C4—48no—
621C2×C4—48no—
622C2×C4—416no—
623C2×C4—432no—
624C2×C4—432no—
625C2×C4—44no—
626C2×C4—416no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—48no—
630C2×C4—48no—
631C2×C4—48no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—88no—
640C2×C4—84no—
641C2×C4—84no—
642C2×C4C2×#12012yes—
643C2×C4C2×#12012yes—
644C2×C4C2×#12012yes—
645C2×C4C2×#12314yes—
646C2×C4C2×#12312yes—
647C2×C4#53412yes—
648C2×C4#546116yes—
649C2×C4C23×D412yes—
650C2×C4C22×#4112yes—
651C2×C4C2×H(5,𝔽2)12yes—
652C8—48no—
653C22—232no—
654C22—24no—
655C22—24no—
656C22—24no—
657C22—24no—
658C22—24no—
659C22—24no—
660C22—24no—
661C22—24no—
662C22—22no—
663C22—22no—
664C22—24no—
665C22—22no—
666C22—24no—
667C22—22no—
668C22—24no—
669C22—24no—
670C22—22no—
671C22—22no—
672C22—24no—
673C22—22no—
674C22—432no—
675C22—416no—
676C22—432no—
677C22—416no—
678C22—42no—
679C22—42no—
680C22—42no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22#240514yes—
685C22#300414yes—
686C22#309518yescomplemented
687C22C22×#12011yescharacteristic
688C22C22×#12311yescharacteristic
689C22C2×#53411yescharacteristic
690C22C22×#13712yes—
691C22C2×#57512yes—
692C4—22no—
693C4—22no—
694C4—24no—
695C4—42no—
696C4—48no—
697C4—48no—
698C4—84no—
699C4—82no—
700C4—82no—
701C4#240514yes—
702C4C2×#54614yes—
703C2—28no—
704C2—24no—
705C2—24no—
706C2—44no—
707C2—42no—
708C2C2×#240511yescharacteristic
709C2C2×#300411yescharacteristic
710C2#6035911yescharacteristic
711C2C2×#309512yes—
712C2#6075112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC26D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3095C2semidirect
4. #24010C2semidirect
5. #24006C2semidirect
6. #24003C2semidirect
7. #23481C2semidirect
8. #20911C2semidirect
9. #20844C2semidirect
10. #20599C2semidirect
29. C2×#484C22semidirect
30. C2×#485C22semidirect
31. C2×#483C22semidirect
32. C2×#428C22semidirect
33. C2×#419C22semidirect
34. C2×#419C22semidirect
49. #2036C22semidirect
50. #2035C22semidirect
51. #1873C22semidirect
52. #1872C22semidirect
53. #1853C22semidirect
54. #1852C22semidirect
55. #1787C22semidirect
56. #1787C22semidirect
59. #1736C22semidirect
142. C2×#107C23semidirect
143. C2×#106C23semidirect
146. C2×#103C23semidirect
147. C2×#103C23semidirect
148. C2×#103C23semidirect
176. #340C23semidirect
686. C22#3095semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊Q82
C2×#30952
C2×#22802
#240102
#240062
#240032
#234812
#209112
#208442
#205992
#205792