Finite Group Databasebeta

← #6483198

Subgroups — #6483198

Order 512.

2230 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3215C212yesmaximal
2#36645C211yesmaximal, characteristic, complemented
3C2×C4×#124C211yesmaximal, characteristic
4#32906C212yesmaximal, complemented
5#32905C212yesmaximal
6#29715C212yesmaximal, complemented
7#29714C212yesmaximal
8#28726C212yesmaximal, complemented
9#28620C214yesmaximal, complemented
10#28449C212yesmaximal, complemented
11#28254C214yesmaximal, complemented
12#28164C214yesmaximal, complemented
13#21866C212yesmaximal, complemented
14#20810C211yesmaximal, characteristic, complemented
15C2×#533C2212yes—
16C22×#124C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19#3215—216no—
20C2×#497C2214yes—
21C2×#489C2214yes—
22C2×#464C2218yes—
23C2×#462C2214yescomplemented
24C2×#461C2214yes—
25C2×#451C2214yes—
26C2×#450C2214yes—
27C2×#432C2212yes—
28#2868C2211yescharacteristic, complemented
29#2868C2211yescharacteristic, complemented
30#2862C2212yescomplemented
31#2859—21no—
32#2859—21no—
33#2857—21no—
34#2857—21no—
35#2857C2212yes—
36#2857C2212yes—
37C4×C4⋊Q8—21no—
38C4×C4⋊Q8—21no—
39C4×C4⋊Q8C2212yescomplemented
40C4×C4⋊Q8C2212yescomplemented
41C4×#124—21no—
42C4×#124—21no—
43C4×#124—21no—
44C4×#124—21no—
45C4×#124C2212yes—
46C4×#124C2212yes—
47C4×#124C2212yes—
48C4×#124C2212yes—
49C2×#391C2211yescharacteristic
50C2×#390C2211yescharacteristic
51C2×#388—21no—
52C2×#388—21no—
53#2285—21no—
54#2285—21no—
55C2×#382C2211yescharacteristic
56C2×C4×C22⋊C4—21no—
57C2×C4×C22⋊C4—21no—
58C2×C43C2211yescharacteristic
59#2072C2214yescomplemented
60#2071C2214yes—
61#2056C2218yescomplemented
62#2055C2218yes—
63#2016C2214yescomplemented
64#2013C2214yes—
65#1993C2214yescomplemented
66#1960C2214yescomplemented
67#1878C2218yescomplemented
68#1877C2218yes—
69#1872C2214yes—
70#1872C2214yes—
71#1842C2212yescomplemented
72#1831C2212yescomplemented
73#1805C2218yescomplemented
74#1803C2214yescomplemented
75#1802C2214yes—
76#1797C2212yes—
77#1797C2212yes—
78#1767C2212yescomplemented
79#1767C2212yescomplemented
80C23×Q8C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C24×C4—21no—
83C22×Q16C2318yes—
84C22×SD16C2318yes—
85#533—28no—
86#533—28no—
87C2×C4⋊Q8—21no—
88C2×C4⋊Q8—21no—
89C2×C4⋊Q8C2312yes—
90C2×C4⋊Q8C2312yes—
91C2×C4⋊Q8C2312yes—
92C2×C4⋊Q8C2312yes—
93C2×#125—22no—
94C2×#124—21no—
95C2×#124—21no—
96C2×#124—24no—
97C2×#124D418yes—
98C2×#124C2312yes—
99C2×#124C2312yes—
100C2×C22⋊Q8—22no—
101C2×C22⋊Q8C2314yes—
102C2×C4×Q8—22no—
103C2×C4×Q8—21no—
104C2×C4×Q8—21no—
105C2×C4×Q8—44no—
106C2×C4×Q8C2314yes—
107C2×C4×Q8C2312yes—
108C2×C4×Q8C2312yes—
109C2×C4×D4—21no—
110C2×C4×D4—21no—
111C2×C4×D4—44no—
112C2×C4×D4C2312yes—
113C2×C4×D4C2312yes—
114C2×#117—22no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C42—21no—
124C22×C42D411yescharacteristic
125C22×C42D411yescharacteristic
126C22×C42D411yescharacteristic
127C22×C42D411yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131#497—216no—
132#489—216no—
133#464—232no—
134#462—48no—
135#461—48no—
136#451—216no—
137#450—216no—
138#432—28no—
139C2×C4⋊C8C2314yes—
140C2×C4⋊C8C2314yes—
141C2×C4⋊C8C2314yes—
142C2×#103C2318yes—
143C2×#103C2318yes—
144C2×#103C2318yes—
145C2×#102C2318yes—
146C2×C22⋊C8C2314yes—
147C2×#97C2314yes—
148#391—22no—
149#391—22no—
150#391—21no—
151#391—21no—
152#391C2314yes—
153#391C2314yes—
154#391C2312yes—
155#391C2312yes—
156#390—21no—
157#390—21no—
158#390C2312yes—
159#390C2312yes—
160#389—22no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—22no—
166#389—22no—
167#389—22no—
168#388—22no—
169#388—22no—
170#388—22no—
171#388—22no—
172#388—22no—
173#388—22no—
174#388—22no—
175#388—22no—
176#384—21no—
177#384—21no—
178#384—21no—
179#384—21no—
180#384—21no—
181#384—21no—
182#384C2312yes—
183#384C2312yes—
184#382—22no—
185#382—22no—
186#382—21no—
187#382—21no—
188#382C2312yes—
189#382C2312yes—
190#380—22no—
191#380—22no—
192#380—22no—
193#380—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C22⋊C4—22no—
199C4×C22⋊C4—22no—
200C4×C22⋊C4—22no—
201C4×C22⋊C4—22no—
202C4×C22⋊C4—22no—
203C4×C22⋊C4—22no—
204C4×C22⋊C4—22no—
205C4×C22⋊C4—22no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95—21no—
210C43D411yescharacteristic
211C43D411yescharacteristic
212C43D411yescharacteristic
213C43D411yescharacteristic
214C43C2311yescharacteristic
215C43C2311yescharacteristic
216C43C2311yescharacteristic
217C43C2311yescharacteristic
218#340C2318yes—
219#340C2318yes—
220#336C2314yes—
221#336C2314yes—
222C25—21no—
223C22×Q8—21no—
224C22×Q8—22no—
225C22×Q8—22no—
226C22×Q8—42no—
227C22×Q8—42no—
228C22×Q8C2×D414yes—
229C22×Q8C2414yes—
230C22×Q8C2412yes—
231C22×D4—21no—
232C22×D4—22no—
233C22×D4—42no—
234C22×D4—42no—
235C22×D4C2×D414yes—
236C22×D4C2412yes—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—22no—
241C23×C4—22no—
242C23×C4—44no—
243C23×C4—44no—
244C23×C4—44no—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2411yescharacteristic
252C2×Q16—216no—
253C2×Q16—48no—
254C2×Q16—48no—
255C2×SD16—216no—
256C2×SD16—48no—
257C2×SD16—48no—
258C22×C8C2418yes—
259C4⋊Q8—24no—
260C4⋊Q8—24no—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8—28no—
264C4⋊Q8—44no—
265C4⋊Q8—44no—
266C4⋊Q8—44no—
267#125—48no—
268#124—28no—
269#124—28no—
270#124—24no—
271#124—24no—
272#124C2×D418yes—
273#124C2×D418yes—
274C22⋊Q8—28no—
275C22⋊Q8—28no—
276C22⋊Q8—48no—
277C4×Q8—28no—
278C4×Q8—28no—
279C4×Q8—28no—
280C4×Q8—44no—
281C4×Q8—44no—
282C4×Q8—44no—
283C4×Q8—48no—
284C4×D4—28no—
285C4×D4—44no—
286C4×D4—44no—
287C4×D4—44no—
288#117—48no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—42no—
307C2×C4⋊C4—42no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2414yes—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—42no—
335C2×C42—42no—
336C2×C42—42no—
337C2×C42—44no—
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D414yes—
361C2×C42#4114yes—
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364C2×C42C2411yescharacteristic
365C2×C42C2411yescharacteristic
366C2×C42C2411yescharacteristic
367C2×C42C2411yescharacteristic
368C4⋊C8—28no—
369C4⋊C8—28no—
370C4⋊C8—44no—
371#103—216no—
372#103—48no—
373#103—48no—
374#102—216no—
375C22⋊C8—28no—
376#97—28no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—24no—
389C24—24no—
390C24—22no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24—42no—
396C24—42no—
397C24C22×D411yescharacteristic
398C2×Q8—24no—
399C2×Q8—22no—
400C2×Q8—24no—
401C2×Q8—24no—
402C2×Q8—24no—
403C2×Q8—28no—
404C2×Q8—24no—
405C2×Q8—28no—
406C2×Q8—48no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8—48no—
410C2×Q8—44no—
411C2×Q8—44no—
412C2×Q8—44no—
413C2×Q8C22×D414yes—
414C2×D4—24no—
415C2×D4—22no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—48no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—48no—
422C2×D4C22×D414yes—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—816no—
460C22×C4#12114yes—
461C22×C4#12114yes—
462C22×C4#13714yes—
463C22×C4#13714yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D412yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C2×#4112yes—
479C22×C4C2511yescharacteristic
480Q16—816no—
481SD16—816no—
482C2×C8—28no—
483C2×C8—28no—
484C2×C8—28no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C22⋊C4—28no—
507C22⋊C4—28no—
508C22⋊C4—28no—
509C22⋊C4—28no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—48no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—41no—
535C42—41no—
536C42—41no—
537C42—41no—
538C42—41no—
539C42—41no—
540C42—44no—
541C42—48no—
542C42—44no—
543C42#12114yes—
544C42#12114yes—
545C42C22×D412yes—
546C42C22×D412yes—
547C42C2×#4112yes—
548C42C2×#4112yes—
549C42H(5,𝔽2)14yes—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—24no—
555C23—48no—
556C23—48no—
557C23—42no—
558C23—42no—
559C23—48no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—84no—
565C23—84no—
566C23—84no—
567C23—84no—
568C23C2×#12112yes—
569C23C2×#13712yes—
570C23C2×#13712yes—
571C23C23×D411yescharacteristic
572Q8—28no—
573Q8—48no—
574Q8—44no—
575Q8—48no—
576Q8—48no—
577Q8—88no—
578D4—28no—
579D4—48no—
580D4—44no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—28no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—28no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—48no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—88no—
632C2×C4—84no—
633C2×C4—84no—
634C2×C4—88no—
635C2×C4C2×#12112yes—
636C2×C4C2×#12112yes—
637C2×C4C2×#12112yes—
638C2×C4C2×#12112yes—
639C2×C4C2×#12112yes—
640C2×C4#53512yes—
641C2×C4#53512yes—
642C2×C4C2×#13712yes—
643C2×C4C2×#13712yes—
644C2×C4C2×#13712yes—
645C2×C4C2×#13712yes—
646C2×C4#57512yes—
647C2×C4#57512yes—
648C2×C4C23×D411yescharacteristic
649C2×C4C22×#4111yescharacteristic
650C2×C4C2×H(5,𝔽2)12yes—
651C2×C4#58512yes—
652C8—48no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—44no—
660C22—44no—
661C22—44no—
662C22—816no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22#303914yes—
668C22C22×#12111yescharacteristic
669C22C2×#53511yescharacteristic
670C22#344911yescharacteristic
671C22C22×#13711yescharacteristic
672C22C22×#13711yescharacteristic
673C22C2×#57511yescharacteristic
674C22C2×#57511yescharacteristic
675C4—22no—
676C4—22no—
677C4—22no—
678C4—22no—
679C4—22no—
680C4—22no—
681C4—22no—
682C4—22no—
683C4—22no—
684C4—22no—
685C4—41no—
686C4—44no—
687C4—41no—
688C4—84no—
689C4—82no—
690C4—82no—
691C4—84no—
692C4#321514yescomplemented
693C4#349612yes—
694C4#350612yes—
695C2—24no—
696C2—84no—
697C2#3516511yescharacteristic
698C2C2×#303911yescharacteristic
699C2#6041511yescharacteristic
700C2C2×#321511yescharacteristic
701C2#6080411yescharacteristic
702C2#6095011yescharacteristic
703C2#6096011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #36645C2semidirect
4. #32906C2semidirect
6. #29715C2semidirect
8. #28726C2semidirect
9. #28620C2semidirect
10. #28449C2semidirect
11. #28254C2semidirect
12. #28164C2semidirect
13. #21866C2semidirect
14. #20810C2semidirect
23. C2×#462C22semidirect
28. #2868C22semidirect
29. #2868C22semidirect
30. #2862C22semidirect
39. C4×C4⋊Q8C22semidirect
40. C4×C4⋊Q8C22semidirect
59. #2072C22semidirect
61. #2056C22semidirect
63. #2016C22semidirect
65. #1993C22semidirect
66. #1960C22semidirect
67. #1878C22semidirect
71. #1842C22semidirect
72. #1831C22semidirect
73. #1805C22semidirect
74. #1803C22semidirect
78. #1767C22semidirect
79. #1767C22semidirect
692. C4#3215semidirect

Maximal subgroups

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

subgroupindex
C2×#32152
#366452
C2×C4×#1242
#329062
#329052
#297152
#297142
#287262
#286202
#284492
#282542
#281642
#218662
#208102