Finite Group Databasebeta

← #6571309

Subgroups — #6571309

Order 512.

1821 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36493C211yesmaximal, characteristic
2#34183C211yesmaximal, characteristic, complemented
3#29928C212yesmaximal, complemented
4#29921C211yesmaximal, characteristic, complemented
5#27978C212yesmaximal, complemented
6#27973C211yesmaximal, characteristic, complemented
7#27648C211yesmaximal, characteristic, complemented
8#27644C212yesmaximal, complemented
9#26244C214yesmaximal, complemented
10#26230C212yesmaximal, complemented
11#24977C214yesmaximal, complemented
12#24969C212yesmaximal, complemented
13#23760C212yesmaximal, complemented
14#23757C211yesmaximal, characteristic, complemented
15#22743C212yesmaximal, complemented
16#22546C212yesmaximal, complemented
17#22532C211yesmaximal, characteristic, complemented
18C2×#559C2211yescharacteristic
19C2×#520—21no—
20C2×C4×#41C2211yescharacteristic
21C2×#484C2212yes—
22C2×#480C2214yes—
23C2×#446C2212yes—
24#2856C2211yescharacteristic
25#2850C2212yes—
26#2541C2211yescharacteristic
27#2540C2212yes—
28#2538C2211yescharacteristic
29#2535C2212yes—
30#2471—22no—
31#2456—22no—
32#2448—21no—
33#2431—21no—
34#2366—21no—
35#2363—21no—
36#2355C2211yescharacteristic
37Q8×C4⋊C4—21no—
38Q8×C4⋊C4C2212yes—
39D4×C4⋊C4—21no—
40D4×C4⋊C4—22no—
41D4×C4⋊C4C2212yes—
42D4×C4⋊C4C2214yes—
43#2317—21no—
44#2317—22no—
45C4×C22⋊Q8C2211yescharacteristic
46C4×#121C2212yes—
47#2285C2211yescharacteristic
48#2285C2212yes—
49C2×#384C2211yescharacteristic
50C2×#384C2212yes—
51#2100C2211yescharacteristic
52#2095C2211yescharacteristic
53#2094C2212yes—
54#2083C2212yes—
55#2083C2211yescharacteristic
56#2077C2214yes—
57#2076C2212yes—
58#2072C2212yes—
59#2072C2211yescharacteristic
60#2071C2212yes—
61#2071C2212yes—
62#2071C2212yes—
63#2036C2212yes—
64#2035C2214yes—
65#2034C2214yes—
66#2033C2214yes—
67#2033C2214yes—
68#1983C2214yes—
69#1982C2212yes—
70#1959C2214yes—
71#1957C2212yes—
72#1955C2214yes—
73#1954C2212yes—
74#1949C2214yes—
75#1948C2212yes—
76#1947C2212yes—
77#1947C2214yes—
78#1874C2214yes—
79#1874C2214yes—
80#1872C2214yes—
81#1836C2214yes—
82#1834C2212yes—
83#1824C2212yes—
84#1823C2214yes—
85#1803C2212yes—
86#1802C2214yes—
87#1797C2211yescharacteristic
88#1797C2212yes—
89#1776C2214yes—
90C4×#107C2212yes—
91#1762C2214yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C22×#41D411yescharacteristic
96#559—28no—
97C2×#127—21no—
98C2×#127C2311yescharacteristic
99C2×#124—21no—
100C2×#124C2312yes—
101C2×#123—22no—
102C2×C22⋊Q8C2311yescharacteristic
103C2×C22⋊Q8C2312yes—
104C2×#121—21no—
105C2×#121—21no—
106C2×#121C2312yes—
107C2×#121C2312yes—
108C2×#121C2312yes—
109#520—48no—
110C4×#41—28no—
111C2×C4×Q8—22no—
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2311yescharacteristic
115C2×C4×D4—24no—
116C2×C4×D4—22no—
117C2×C4×D4—22no—
118C2×C4×D4—21no—
119C2×C4×D4D411yescharacteristic
120C2×C4×D4D412yes—
121C2×C4×D4C2312yes—
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2312yes—
124C2×#117—22no—
125C2×#117—21no—
126C2×#117D411yescharacteristic
127C2×#117D412yes—
128C22×C4⋊C4—22no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4—22no—
131C22×C4⋊C4—21no—
132C22×C4⋊C4C2314yes—
133C22×C4⋊C4C2312yes—
134C22×C42C2312yes—
135C22×C42C2311yescharacteristic
136#484—44no—
137#480—48no—
138#446—28no—
139C2×#107C2312yes—
140C2×#107C2314yes—
141C2×#107C2312yes—
142C2×#106C2314yes—
143C2×C4⋊C8C2314yes—
144C2×C4⋊C8C2312yes—
145C2×#103C2314yes—
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×C22⋊C8C2314yes—
153C2×C22⋊C8C2312yes—
154C2×C22⋊C8C2312yes—
155C2×C22⋊C8C2314yes—
156C2×#97C2314yes—
157C2×C4×C8C2312yes—
158#399—22no—
159#397—22no—
160#396—22no—
161#392—22no—
162#392—22no—
163#392—22no—
164#391—21no—
165#391C2311yescharacteristic
166#390—21no—
167#390—21no—
168#390—22no—
169#390—22no—
170#390—21no—
171#390—21no—
172#390C2312yes—
173#389—22no—
174#389—24no—
175#389—22no—
176#389—24no—
177#389—21no—
178#389C2311yescharacteristic
179#388—22no—
180#388—22no—
181#388—22no—
182#387—22no—
183#387—21no—
184#386—22no—
185#386—22no—
186#386—22no—
187#386—22no—
188#386—22no—
189#385—21no—
190#385—24no—
191#385—24no—
192#385—24no—
193#384—22no—
194#384—21no—
195#384—21no—
196#384—22no—
197#384C2314yes—
198#384C2312yes—
199#384C2312yes—
200#384C2311yescharacteristic
201#384C2311yescharacteristic
202#384C2311yescharacteristic
203#382—22no—
204#382—21no—
205#382C2312yes—
206#380—22no—
207#380—21no—
208#380C2311yescharacteristic
209#380C2312yes—
210#380C2312yes—
211#380C2314yes—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2314yes—
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C22⋊C4—21no—
221C4×C22⋊C4—22no—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#336C2311yescharacteristic
232#336C2312yes—
233#336C2311yescharacteristic
234C2×#41—24no—
235C2×#41—24no—
236C2×#41—24no—
237C22×Q8C2×D411yescharacteristic
238C22×Q8C2411yescharacteristic
239C22×D4—22no—
240C22×D4—22no—
241C22×D4—22no—
242C22×D4—21no—
243C22×D4—21no—
244C22×D4C2×D412yes—
245C22×D4C2×D411yescharacteristic
246C22×D4C2412yes—
247C23×C4—22no—
248C23×C4—24no—
249C23×C4—22no—
250C23×C4C2×D412yes—
251C23×C4C2×D412yes—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2412yes—
255C23×C4C2411yescharacteristic
256C22×C8C2412yes—
257C22×C8C2412yes—
258C22×C8C2414yes—
259#127—24no—
260#127—44no—
261#124—28no—
262#124—44no—
263#123—48no—
264C22⋊Q8—22no—
265C22⋊Q8—41no—
266C22⋊Q8—44no—
267#121—24no—
268#121—42no—
269#121—44no—
270#121—44no—
271#121—44no—
272#121—82no—
273C4×Q8—24no—
274C4×Q8—24no—
275C4×Q8—48no—
276C4×Q8#4118yes—
277C4×D4—24no—
278C4×D4—28no—
279C4×D4—24no—
280C4×D4—28no—
281C4×D4—28no—
282C4×D4—416no—
283C4×D4—48no—
284C4×D4—44no—
285C4×D4—84no—
286#117—24no—
287#117—28no—
288#117—48no—
289#117—44no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—42no—
312C2×C4⋊C4—48no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—84no—
315C2×C4⋊C4—82no—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2414yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C22⋊C4—22no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4C2×D412yes—
342C2×C22⋊C4C2×D412yes—
343C2×C22⋊C4C2×D412yes—
344C2×C42—24no—
345C2×C42—28no—
346C2×C42—24no—
347C2×C42—22no—
348C2×C42—21no—
349C2×C42—21no—
350C2×C42—22no—
351C2×C42C2×D412yes—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D412yes—
355C2×C42C2411yescharacteristic
356C2×C42C2412yes—
357C2×C42C2411yescharacteristic
358#107—24no—
359#107—28no—
360#107—42no—
361#106—44no—
362C4⋊C8—24no—
363C4⋊C8—28no—
364#103—42no—
365#103—42no—
366#103—44no—
367#102—44no—
368#102—44no—
369#102—44no—
370#102—44no—
371C22⋊C8—44no—
372C22⋊C8—42no—
373C22⋊C8—44no—
374C22⋊C8—42no—
375#97—28no—
376C4×C8—24no—
377#95—21no—
378#95—22no—
379#95—21no—
380#95—22no—
381#95—24no—
382#95—21no—
383#95—22no—
384#95—21no—
385#95—24no—
386#95—22no—
387#95—21no—
388#95—22no—
389#95—22no—
390C24—22no—
391C24C22×D412yes—
392C24C22×D411yescharacteristic
393#41—232no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—22no—
400C2×Q8#12314yes—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—24no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—24no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—81no—
419C2×D4—81no—
420C2×D4—82no—
421C2×D4—82no—
422C2×D4—82no—
423C2×D4—82no—
424C2×D4—82no—
425C2×D4AGL(1,ℤ/8)14yes—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—42no—
452C22×C4—41no—
453C22×C4—44no—
454C22×C4—48no—
455C22×C4—44no—
456C22×C4—82no—
457C22×C4—82no—
458C22×C4—84no—
459C22×C4—82no—
460C22×C4—84no—
461C22×C4—82no—
462C22×C4#13714yes—
463C22×C4C22×D412yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D412yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C2511yescharacteristic
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—24no—
475C2×C8—22no—
476C2×C8—24no—
477C2×C8—22no—
478C2×C8—24no—
479C2×C8—22no—
480C2×C8—22no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—42no—
497C4⋊C4—48no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—84no—
506C4⋊C4—82no—
507C4⋊C4—84no—
508C4⋊C4—82no—
509C4⋊C4—81no—
510C4⋊C4—82no—
511C4⋊C4#12314yes—
512C4⋊C4#12318yes—
513C4⋊C4H(5,𝔽2)12yes—
514C22⋊C4—24no—
515C22⋊C4—24no—
516C22⋊C4—24no—
517C22⋊C4—44no—
518C22⋊C4—42no—
519C22⋊C4—44no—
520C22⋊C4—48no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—82no—
527C22⋊C4—82no—
528C22⋊C4—81no—
529C22⋊C4—82no—
530C22⋊C4—82no—
531C42—24no—
532C42—22no—
533C42—24no—
534C42—22no—
535C42—22no—
536C42—48no—
537C42—42no—
538C42—44no—
539C42—42no—
540C42—82no—
541C42C2×#4114yes—
542C42C2×#4118yes—
543C23—21no—
544C23—22no—
545C23—22no—
546C23—21no—
547C23—21no—
548C23—22no—
549C23—22no—
550C23—21no—
551C23—21no—
552C23—22no—
553C23—22no—
554C23—21no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23C2×AGL(1,ℤ/8)14yes—
563C23C2×#13712yes—
564C23C23×D411yescharacteristic
565Q8—28no—
566Q8—42no—
567Q8—42no—
568D4—216no—
569D4—28no—
570D4—44no—
571D4—44no—
572D4—88no—
573D4—84no—
574D4—162no—
575D4—164no—
576D4—164no—
577C2×C4—216no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—28no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—48no—
597C2×C4—44no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—164no—
626C2×C4—168no—
627C2×C4—164no—
628C2×C4C2×#12314yes—
629C2×C4C2×#12312yes—
630C2×C4C2×#12314yes—
631C2×C4C2×#12312yes—
632C2×C4C2×AGL(1,ℤ/8)12yes—
633C2×C4C2×#13712yes—
634C2×C4C2×#13712yes—
635C2×C4#57612yes—
636C2×C4C22×#4112yes—
637C2×C4C2×H(5,𝔽2)11yescharacteristic
638C2×C4#58512yes—
639C2×C4#58511yescharacteristic
640C8—42no—
641C8—44no—
642C8—42no—
643C22—22no—
644C22—24no—
645C22—22no—
646C22—24no—
647C22—22no—
648C22—24no—
649C22—42no—
650C22—41no—
651C22—42no—
652C22—41no—
653C22—42no—
654C22—41no—
655C22—41no—
656C22—42no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22C22×#12311yescharacteristic
665C22#348811yescharacteristic
666C22#350011yescharacteristic
667C22C22×AGL(1,ℤ/8)11yescharacteristic
668C22C22×#13711yescharacteristic
669C22C2×#57611yescharacteristic
670C22#358611yescharacteristic
671C4—22no—
672C4—24no—
673C4—22no—
674C4—24no—
675C4—22no—
676C4—42no—
677C4—42no—
678C4—41no—
679C4—82no—
680C4—84no—
681C4—82no—
682C4—81no—
683C4—82no—
684C4—81no—
685C4—81no—
686C4#309214yes—
687C4#351712yes—
688C4#352412yes—
689C2—42no—
690C2—44no—
691C2—162no—
692C2#3607611yescharacteristic
693C2C2×#309211yescharacteristic
694C2#6052111yescharacteristic
695C2#6196411yescharacteristic
696C2#6197911yescharacteristic
697C2#6253711yescharacteristic
698C2#6258311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36493C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34183C2semidirect
3. #29928C2semidirect
4. #29921C2semidirect
5. #27978C2semidirect
6. #27973C2semidirect
7. #27648C2semidirect
8. #27644C2semidirect
9. #26244C2semidirect
10. #26230C2semidirect
11. #24977C2semidirect
12. #24969C2semidirect
13. #23760C2semidirect
14. #23757C2semidirect
15. #22743C2semidirect
16. #22546C2semidirect
17. #22532C2semidirect

Maximal subgroups

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

subgroupindex
#364932
#341832
#299282
#299212
#279782
#279732
#276482
#276442
#262442
#262302
#249772
#249692
#237602
#237572
#227432
#225462
#225322