Finite Group Databasebeta

← #6467752

Subgroups — #6467752

Order 512.

2220 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3182C212yesmaximal
2#36645C211yesmaximal, characteristic, complemented
3C4×#532C211yesmaximal, characteristic
4#28748C211yesmaximal, characteristic, complemented
5#28716C211yesmaximal, characteristic, complemented
6#28689C211yesmaximal, characteristic, complemented
7#28656C212yesmaximal, complemented
8#28613C212yesmaximal, complemented
9#28577C214yesmaximal, complemented
10#28438C211yesmaximal, characteristic, complemented
11#24759C214yesmaximal, complemented
12#23683C214yesmaximal, complemented
13#21504C214yesmaximal, complemented
14#21409C212yesmaximal, complemented
15C4×#423C211yesmaximal, characteristic, complemented
16C2×#533C2212yes—
17C2×#532C2211yescharacteristic
18C2×C4×#41—21no—
19#3182—216no—
20C2×#500C2212yes—
21C2×#492C2212yes—
22C2×#487C2212yes—
23C2×#486C2212yes—
24C2×#468C2218yes—
25C2×#461C2214yes—
26C2×#440C2218yes—
27C2×#423C2212yes—
28C2×C4×M16C2212yes—
29#2868C2211yescharacteristic
30#2868C2211yescharacteristic
31#2862C2212yes—
32#2859—21no—
33#2859—21no—
34#2857—21no—
35#2857—21no—
36#2857C2212yes—
37#2857C2212yes—
38C4×C4⋊Q8—21no—
39C4×C4⋊Q8—21no—
40C4×C4⋊Q8C2211yescharacteristic
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2212yes—
43C4×C4⋊Q8C2212yes—
44C4×#127C2211yescharacteristic
45C4×#127C2211yescharacteristic
46C4×#124—21no—
47C4×#124—21no—
48C4×#121—21no—
49C4×#121—21no—
50C4×#121C2212yes—
51C4×#121C2212yes—
52#2297—21no—
53#2293—21no—
54#2293C2212yes—
55#2285—21no—
56#2285—21no—
57#2285C2211yescharacteristic
58C42×D4—21no—
59C42×D4C2212yes—
60C2×C43C2211yescharacteristic
61#2083C2214yes—
62#2071C2214yes—
63#2068C2218yes—
64#2055C2218yes—
65#2043C2214yes—
66#2013C2214yes—
67#1990C2212yes—
68#1990C2212yes—
69#1981C2214yes—
70#1956C2212yes—
71#1956C2212yes—
72#1947C2214yes—
73#1928C2218yes—
74#1894C2218yes—
75#1877C2218yes—
76#1802C2214yes—
77#1767C2212yes—
78#1767C2212yes—
79C4×C4⋊C8C2211yescharacteristic
80C4×C4⋊C8C2211yescharacteristic
81C4×C4⋊C8C2212yes—
82#1763C2218yes—
83#1728C2212yes—
84C22×#41—21no—
85C22×SD16C2318yes—
86C22×M16C2314yes—
87#533—28no—
88#533—28no—
89#532—24no—
90#532D418yes—
91C2×C4⋊Q8—21no—
92C2×C4⋊Q8—21no—
93C2×C4⋊Q8C2312yes—
94C2×C4⋊Q8C2312yes—
95C2×C4⋊Q8C2312yes—
96C2×C4⋊Q8C2312yes—
97C2×C4⋊Q8C2311yescharacteristic
98C2×#127C2311yescharacteristic
99C2×#125—22no—
100C2×#124—21no—
101C2×C22⋊Q8—22no—
102C2×C22⋊Q8C2314yes—
103C2×#121—21no—
104C2×#121C2312yes—
105C4×#41—48no—
106C2×C4×Q8—22no—
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8C2314yes—
110C2×C4×D4—22no—
111C2×C4×D4—22no—
112C2×C4×D4—21no—
113C2×C4×D4—21no—
114C2×C4×D4—21no—
115C2×C4×D4—21no—
116C2×C4×D4—21no—
117C2×C4×D4C2312yes—
118C2×C4×D4C2312yes—
119C2×C4×D4C2312yes—
120C2×#117—21no—
121C2×#117—22no—
122C2×#117—22no—
123C22×C42—22no—
124C22×C42D411yescharacteristic
125C22×C42D411yescharacteristic
126C22×C42D411yescharacteristic
127C22×C42D411yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131#500—28no—
132#492—28no—
133#487—28no—
134#486—28no—
135#468—232no—
136#461—216no—
137#440—232no—
138C2×#107C2318yes—
139#423—28no—
140C2×C4⋊C8C2314yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2312yes—
143C2×#103C2318yes—
144C2×#103C2318yes—
145C2×#103C2318yes—
146C2×#102C2318yes—
147C4×M16—28no—
148C2×#97C2314yes—
149C2×C4×C8C2312yes—
150C2×C4×C8C2312yes—
151#391—22no—
152#391—22no—
153#391—21no—
154#391—21no—
155#391C2314yes—
156#391C2314yes—
157#390—21no—
158#390—21no—
159#390—21no—
160#390—21no—
161#390C2312yes—
162#390C2312yes—
163#389—22no—
164#389—22no—
165#389—22no—
166#389—22no—
167#389—22no—
168#389—22no—
169#389—22no—
170#389—22no—
171#389—21no—
172#389—21no—
173#389C2312yes—
174#389C2312yes—
175#388—22no—
176#388—22no—
177#388—22no—
178#388—22no—
179#386—22no—
180#386—22no—
181#384—21no—
182#384—21no—
183#384—21no—
184#384—21no—
185#384—21no—
186#384—21no—
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#384C2312yes—
190#384C2312yes—
191#382—22no—
192#382—22no—
193#382—21no—
194#380—22no—
195#380—22no—
196#380—22no—
197#380—22no—
198#380—21no—
199#380C2312yes—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C22⋊C4—22no—
211C4×C22⋊C4—22no—
212C4×C22⋊C4—22no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4—22no—
215C4×C22⋊C4—22no—
216C43D411yescharacteristic
217C43D411yescharacteristic
218C43D411yescharacteristic
219C43D411yescharacteristic
220C43C2311yescharacteristic
221C43C2311yescharacteristic
222C43C2311yescharacteristic
223C43C2311yescharacteristic
224#340C2318yes—
225#340C2318yes—
226#336C2312yes—
227#336C2312yes—
228#336C2314yes—
229C2×#41—28no—
230C2×#41—44no—
231C2×#41—44no—
232C22×Q8—21no—
233C22×Q8—22no—
234C22×Q8C2414yes—
235C22×D4—22no—
236C22×D4—21no—
237C22×D4—21no—
238C22×D4C2412yes—
239C23×C4—22no—
240C23×C4—22no—
241C23×C4—22no—
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2411yescharacteristic
249C2×SD16—216no—
250C2×SD16—216no—
251C2×SD16—48no—
252C2×M16—28no—
253C2×M16—28no—
254C2×M16—44no—
255C22×C8C2414yes—
256C22×C8C2414yes—
257C4⋊Q8—24no—
258C4⋊Q8—24no—
259C4⋊Q8—22no—
260C4⋊Q8—24no—
261C4⋊Q8—24no—
262C4⋊Q8—28no—
263C4⋊Q8—44no—
264C4⋊Q8—44no—
265C4⋊Q8—44no—
266C4⋊Q8C2×D414yes—
267#127—22no—
268#127C2×D414yes—
269#125—48no—
270#124—28no—
271C22⋊Q8—28no—
272C22⋊Q8—28no—
273C22⋊Q8—48no—
274#121—28no—
275#121—24no—
276#121C2×D418yes—
277C4×Q8—28no—
278C4×Q8—28no—
279C4×Q8—44no—
280C4×Q8—48no—
281C4×Q8—44no—
282C4×D4—28no—
283C4×D4—24no—
284C4×D4—28no—
285C4×D4—44no—
286C4×D4—44no—
287C4×D4—48no—
288C4×D4—48no—
289C4×D4—44no—
290C4×D4—44no—
291C4×D4—44no—
292C4×D4C2×D418yes—
293#117—48no—
294#117—44no—
295#117—48no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2414yes—
319C2×C4⋊C4C2412yes—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—22no—
335C2×C42—24no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—42no—
339C2×C42—42no—
340C2×C42—48no—
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×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42#4114yes—
365C2×C42C2411yescharacteristic
366C2×C42C2411yescharacteristic
367C2×C42C2411yescharacteristic
368C2×C42C2411yescharacteristic
369C2×C42C2411yescharacteristic
370C2×C42C2411yescharacteristic
371#107—216no—
372C4⋊C8—28no—
373C4⋊C8—24no—
374C4⋊C8#4118yescomplemented
375#103—216no—
376#103—216no—
377#103—48no—
378#102—216no—
379#97—28no—
380C4×C8—24no—
381C4×C8—24no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—24no—
389#95—24no—
390#95—24no—
391#95—24no—
392C24—22no—
393C24C22×D411yescharacteristic
394#41—432no—
395C2×Q8—24no—
396C2×Q8—24no—
397C2×Q8—28no—
398C2×Q8—24no—
399C2×Q8—28no—
400C2×Q8—44no—
401C2×Q8—42no—
402C2×Q8—44no—
403C2×Q8—42no—
404C2×Q8—44no—
405C2×D4—28no—
406C2×D4—24no—
407C2×D4—24no—
408C2×D4—24no—
409C2×D4—22no—
410C2×D4—24no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4C22×D414yes—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—28no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4#12114yes—
451C22×C4#12114yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C2×#4112yes—
467C22×C4C2511yescharacteristic
468SD16—432no—
469M16—416no—
470C2×C8—24no—
471C2×C8—24no—
472C2×C8—24no—
473C2×C8—24no—
474C2×C8—24no—
475C2×C8C2×#4118yes—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4C22×D414yes—
502C22⋊C4—28no—
503C22⋊C4—28no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—48no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—41no—
529C42—41no—
530C42—41no—
531C42—41no—
532C42—44no—
533C42—44no—
534C42—48no—
535C42—44no—
536C42#12114yes—
537C42#12114yes—
538C42AGL(1,ℤ/8)14yes—
539C42#13714yes—
540C42#13714yes—
541C42C22×D412yes—
542C42C22×D412yes—
543C42C2×#4112yes—
544C42C2×#4112yes—
545C42C2×#4114yes—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—24no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23C2×#12112yes—
560C23C23×D411yescharacteristic
561Q8—48no—
562Q8—48no—
563Q8—48no—
564Q8—88no—
565D4—28no—
566D4—416no—
567D4—44no—
568D4—48no—
569D4—48no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—28no—
589C2×C4—28no—
590C2×C4—24no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—416no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—88no—
617C2×C4—88no—
618C2×C4C2×#12112yes—
619C2×C4C2×#12112yes—
620C2×C4C2×#12112yes—
621C2×C4C2×#12112yes—
622C2×C4C2×#12112yes—
623C2×C4#53212yes—
624C2×C4#53212yes—
625C2×C4C2×AGL(1,ℤ/8)12yes—
626C2×C4C2×AGL(1,ℤ/8)12yes—
627C2×C4C2×AGL(1,ℤ/8)12yes—
628C2×C4C2×#13712yes—
629C2×C4C2×#13712yes—
630C2×C4C2×#13712yes—
631C2×C4C2×#13712yes—
632C2×C4C2×#13712yes—
633C2×C4C2×#13712yes—
634C2×C4#57512yes—
635C2×C4C23×D411yescharacteristic
636C2×C4C22×#4112yes—
637C2×C4C22×#4111yescharacteristic
638C2×C4#58512yes—
639C8—28no—
640C8—44no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22—44no—
649C22—44no—
650C22—44no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22C22×#12111yescharacteristic
656C22C2×#53211yescharacteristic
657C22#344911yescharacteristic
658C22C22×AGL(1,ℤ/8)11yescharacteristic
659C22C22×#13711yescharacteristic
660C22C22×#13711yescharacteristic
661C22C2×#57511yescharacteristic
662C4—22no—
663C4—22no—
664C4—22no—
665C4—22no—
666C4—22no—
667C4—22no—
668C4—22no—
669C4—22no—
670C4—41no—
671C4—41no—
672C4—44no—
673C4—82no—
674C4—82no—
675C4—84no—
676C4—84no—
677C4#303614yes—
678C4#303914yes—
679C4#318214yescomplemented
680C4D4×#4112yes—
681C4#348012yes—
682C2—42no—
683C2—84no—
684C2#3493711yescharacteristic
685C2C2×#303611yescharacteristic
686C2C2×#303911yescharacteristic
687C2C2×#318211yescharacteristic
688C2#6070111yescharacteristic
689C2#6096611yescharacteristic
690C2#6096911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
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. #28748C2semidirect
5. #28716C2semidirect
6. #28689C2semidirect
7. #28656C2semidirect
8. #28613C2semidirect
9. #28577C2semidirect
10. #28438C2semidirect
11. #24759C2semidirect
12. #23683C2semidirect
13. #21504C2semidirect
14. #21409C2semidirect
15. C4×#423C2semidirect
374. C4⋊C8#41semidirect
679. C4#3182semidirect

Maximal subgroups

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

subgroupindex
C2×#31822
#366452
C4×#5322
#287482
#287162
#286892
#286562
#286132
#285772
#284382
#247592
#236832
#215042
#214092
C4×#4232