Finite Group Databasebeta

← #6564154

Subgroups — #6564154

Order 512.

1812 conjugacy classes of subgroups.

The subgroup lattice has 691 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36504C211yesmaximal, characteristic
2#34183C211yesmaximal, characteristic, complemented
3#29928C212yesmaximal, complemented
4#29923C211yesmaximal, characteristic, complemented
5#27914C212yesmaximal, complemented
6#27910C211yesmaximal, characteristic, complemented
7#27681C212yesmaximal, complemented
8#27678C211yesmaximal, characteristic, complemented
9#26244C214yesmaximal, complemented
10#26234C212yesmaximal, complemented
11#25024C214yesmaximal, complemented
12#25010C212yesmaximal, complemented
13#23762C212yesmaximal, complemented
14#23755C211yesmaximal, characteristic, complemented
15#22728C212yesmaximal, complemented
16#22544C212yesmaximal, complemented
17#22532C211yesmaximal, characteristic, complemented
18C2×#556C2211yescharacteristic
19C2×#520—21no—
20C2×C4×#41C2211yescharacteristic
21C2×#484C2214yes—
22C2×#483C2214yes—
23C2×#480C2214yes—
24C2×#446C2212yes—
25#2857C2211yescharacteristic
26#2850C2212yes—
27#2542C2211yescharacteristic
28#2540C2212yes—
29#2537C2211yescharacteristic
30#2536C2212yes—
31#2479—22no—
32#2467—22no—
33#2423—21no—
34#2407—21no—
35#2363—21no—
36#2355C2211yescharacteristic
37#2354—21no—
38Q8×C4⋊C4—21no—
39Q8×C4⋊C4C2212yes—
40D4×C4⋊C4—21no—
41D4×C4⋊C4—22no—
42D4×C4⋊C4C2212yes—
43D4×C4⋊C4C2214yes—
44#2317—21no—
45#2317—22no—
46C4×C22⋊Q8C2212yes—
47C4×#121C2211yescharacteristic
48#2285C2211yescharacteristic
49#2285C2212yes—
50C2×#384C2211yescharacteristic
51C2×#384C2212yes—
52#2100C2212yes—
53#2100C2211yescharacteristic
54#2095C2212yes—
55#2095C2212yes—
56#2095C2212yes—
57#2094C2211yescharacteristic
58#2094C2212yes—
59#2084C2211yescharacteristic
60#2077C2212yes—
61#2075C2214yes—
62#2072C2212yes—
63#2071C2211yescharacteristic
64#2036C2214yes—
65#2035C2214yes—
66#2035C2214yes—
67#2034C2212yes—
68#2033C2214yes—
69#1983C2214yes—
70#1981C2212yes—
71#1959C2214yes—
72#1956C2212yes—
73#1955C2214yes—
74#1954C2212yes—
75#1949C2214yes—
76#1948C2212yes—
77#1947C2214yes—
78#1947C2212yes—
79#1874C2212yes—
80#1872C2214yes—
81#1853C2212yes—
82#1852C2214yes—
83#1847C2214yes—
84#1845C2212yes—
85#1803C2214yes—
86#1802C2212yes—
87#1797C2211yescharacteristic
88#1797C2212yes—
89#1776C2214yes—
90C4×#106C2212yes—
91#1763C2214yes—
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C22×#41D411yescharacteristic
96#556—28no—
97C2×C4⋊Q8C2311yescharacteristic
98C2×#126—22no—
99C2×#124C2312yes—
100C2×C22⋊Q8—21no—
101C2×C22⋊Q8C2312yes—
102C2×C22⋊Q8C2312yes—
103C2×C22⋊Q8C2312yes—
104C2×#121—21no—
105C2×#121C2311yescharacteristic
106C2×#121C2312yes—
107#520—48no—
108C4×#41—28no—
109C2×C4×Q8—22no—
110C2×C4×Q8D411yescharacteristic
111C2×C4×Q8C2312yes—
112C2×C4×Q8C2311yescharacteristic
113C2×C4×D4—24no—
114C2×C4×D4—22no—
115C2×C4×D4—21no—
116C2×C4×D4D411yescharacteristic
117C2×C4×D4D412yes—
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2312yes—
121C2×#117—22no—
122C2×#117—21no—
123C2×#117—21no—
124C2×#117D411yescharacteristic
125C2×#117D412yes—
126C22×C4⋊C4—22no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—22no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4C2314yes—
131C22×C4⋊C4C2312yes—
132C22×C42C2312yes—
133C22×C42C2311yescharacteristic
134#484—48no—
135#483—48no—
136#480—48no—
137#446—28no—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×#106C2312yes—
142C2×#106C2312yes—
143C2×#106C2314yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#102C2312yes—
151C2×#102C2314yes—
152C2×#102C2312yes—
153C2×C22⋊C8C2314yes—
154C2×C22⋊C8C2312yes—
155C2×C22⋊C8C2314yes—
156C2×C22⋊C8C2312yes—
157C2×#97C2314yes—
158C2×C4×C8C2312yes—
159#399—22no—
160#399—22no—
161#398—22no—
162#394—22no—
163#394—22no—
164#392—22no—
165#391—21no—
166#391C2312yes—
167#390—21no—
168#390—22no—
169#390—22no—
170#390C2311yescharacteristic
171#389—22no—
172#389—24no—
173#389—22no—
174#389—24no—
175#389—21no—
176#389—21no—
177#389C2311yescharacteristic
178#388—22no—
179#388—22no—
180#388—22no—
181#388—22no—
182#388—21no—
183#387—22no—
184#387—22no—
185#386—22no—
186#386—22no—
187#386—21no—
188#385—22no—
189#385—24no—
190#385—24no—
191#385—24no—
192#384—21no—
193#384—22no—
194#384—21no—
195#384—21no—
196#384—22no—
197#384C2312yes—
198#384C2314yes—
199#384C2311yescharacteristic
200#384C2311yescharacteristic
201#384C2311yescharacteristic
202#384C2312yes—
203#382—22no—
204#382—21no—
205#382C2312yes—
206#380—21no—
207#380—21no—
208#380—22no—
209#380—21no—
210#380—21no—
211#380C2312yes—
212#380C2314yes—
213#380C2311yescharacteristic
214#380C2312yes—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4—22no—
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2314yes—
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2311yescharacteristic
223C4×C22⋊C4—22no—
224C4×C22⋊C4—21no—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#340C2314yes—
232#336C2311yescharacteristic
233#336C2312yes—
234#336C2311yescharacteristic
235C2×#41—24no—
236C2×#41—24no—
237C2×#41—24no—
238C22×Q8C2×D411yescharacteristic
239C22×Q8C2412yes—
240C22×D4—21no—
241C22×D4—22no—
242C22×D4C2×D412yes—
243C22×D4C2×D411yescharacteristic
244C22×D4C2411yescharacteristic
245C23×C4—22no—
246C23×C4—24no—
247C23×C4—21no—
248C23×C4C2×D412yes—
249C23×C4C2×D412yes—
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2412yes—
253C23×C4C2411yescharacteristic
254C22×C8C2412yes—
255C22×C8C2414yes—
256C22×C8C2412yes—
257C4⋊Q8—24no—
258#126—48no—
259#124—28no—
260C22⋊Q8—24no—
261C22⋊Q8—42no—
262C22⋊Q8—44no—
263C22⋊Q8—44no—
264C22⋊Q8—44no—
265#121—22no—
266#121—41no—
267#121—44no—
268#121—44no—
269C4×Q8—24no—
270C4×Q8—28no—
271C4×Q8—48no—
272C4×Q8#4118yes—
273C4×D4—24no—
274C4×D4—28no—
275C4×D4—24no—
276C4×D4—28no—
277C4×D4—24no—
278C4×D4—416no—
279C4×D4—48no—
280C4×D4—44no—
281#117—24no—
282#117—28no—
283#117—48no—
284#117—44no—
285#117—44no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—48no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—84no—
313C2×C4⋊C4—82no—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2414yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C22⋊C4—22no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C22⋊C4C2×D412yes—
342C2×C42—24no—
343C2×C42—28no—
344C2×C42—24no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—21no—
348C2×C42—22no—
349C2×C42C2×D412yes—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D412yes—
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355C2×C42C2412yes—
356#107—28no—
357#107—44no—
358#107—44no—
359#106—24no—
360#106—24no—
361#106—44no—
362C4⋊C8—24no—
363C4⋊C8—28no—
364#103—28no—
365#103—44no—
366#103—44no—
367#103—44no—
368#102—24no—
369#102—42no—
370#102—44no—
371C22⋊C8—28no—
372C22⋊C8—24no—
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—22no—
382#95—24no—
383#95—22no—
384#95—22no—
385#95—24no—
386#95—22no—
387#95—22no—
388#95—21no—
389#95—22no—
390C24—21no—
391C24C22×D412yes—
392C24C22×D411yescharacteristic
393#41—232no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—24no—
397C2×Q8—24no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8#12314yes—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—24no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—44no—
413C2×D4—81no—
414C2×D4—82no—
415C2×D4—82no—
416C2×D4AGL(1,ℤ/8)14yes—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—21no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—42no—
443C22×C4—41no—
444C22×C4—44no—
445C22×C4—48no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—84no—
450C22×C4—82no—
451C22×C4—84no—
452C22×C4—82no—
453C22×C4#13714yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C2511yescharacteristic
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—24no—
471C2×C8H(5,𝔽2)14yes—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—48no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—81no—
499C4⋊C4—84no—
500C4⋊C4—82no—
501C4⋊C4—84no—
502C4⋊C4—82no—
503C4⋊C4—82no—
504C4⋊C4—82no—
505C4⋊C4#12314yes—
506C4⋊C4#12318yes—
507C4⋊C42−1+412yes—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—42no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—48no—
515C22⋊C4—44no—
516C22⋊C4—42no—
517C22⋊C4—44no—
518C22⋊C4—42no—
519C22⋊C4—44no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—82no—
523C22⋊C4—82no—
524C22⋊C4—82no—
525C42—24no—
526C42—22no—
527C42—24no—
528C42—22no—
529C42—22no—
530C42—48no—
531C42—44no—
532C42—44no—
533C42—42no—
534C42—82no—
535C42C2×#4114yes—
536C42C2×#4118yes—
537C23—21no—
538C23—22no—
539C23—22no—
540C23—21no—
541C23—22no—
542C23—21no—
543C23—21no—
544C23—22no—
545C23—21no—
546C23—22no—
547C23—22no—
548C23—21no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23C2×AGL(1,ℤ/8)14yes—
557C23C2×#13712yes—
558C23C23×D411yescharacteristic
559Q8—28no—
560Q8—44no—
561Q8—44no—
562D4—216no—
563D4—28no—
564D4—42no—
565D4—42no—
566D4—84no—
567D4—164no—
568C2×C4—216no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—28no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—48no—
588C2×C4—44no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—84no—
617C2×C4—168no—
618C2×C4—164no—
619C2×C4C2×#12314yes—
620C2×C4C2×#12312yes—
621C2×C4C2×#12314yes—
622C2×C4C2×#12312yes—
623C2×C4C2×AGL(1,ℤ/8)12yes—
624C2×C4C2×#13712yes—
625C2×C4C2×#13712yes—
626C2×C4#57812yes—
627C2×C4C22×#4112yes—
628C2×C4C2×H(5,𝔽2)12yes—
629C2×C4C2×2−1+411yescharacteristic
630C2×C4#58511yescharacteristic
631C8—24no—
632C8—42no—
633C8—44no—
634C22—24no—
635C22—22no—
636C22—22no—
637C22—24no—
638C22—22no—
639C22—24no—
640C22—42no—
641C22—41no—
642C22—42no—
643C22—42no—
644C22—41no—
645C22—41no—
646C22—42no—
647C22—41no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22C22×#12311yescharacteristic
656C22#348911yescharacteristic
657C22#349811yescharacteristic
658C22C22×AGL(1,ℤ/8)11yescharacteristic
659C22C22×#13711yescharacteristic
660C22C2×#57811yescharacteristic
661C22#358511yescharacteristic
662C4—22no—
663C4—24no—
664C4—22no—
665C4—24no—
666C4—22no—
667C4—42no—
668C4—41no—
669C4—42no—
670C4—81no—
671C4—82no—
672C4—82no—
673C4—81no—
674C4—82no—
675C4—84no—
676C4—82no—
677C4#309214yes—
678C4#351712yes—
679C4#352212yes—
680C2—44no—
681C2—42no—
682C2—161no—
683C2#3600511yescharacteristic
684C2C2×#309211yescharacteristic
685C2#6052611yescharacteristic
686C2#6210811yescharacteristic
687C2#6213211yescharacteristic
688C2#6242411yescharacteristic
689C2#6244711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36504C2
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. #29923C2semidirect
5. #27914C2semidirect
6. #27910C2semidirect
7. #27681C2semidirect
8. #27678C2semidirect
9. #26244C2semidirect
10. #26234C2semidirect
11. #25024C2semidirect
12. #25010C2semidirect
13. #23762C2semidirect
14. #23755C2semidirect
15. #22728C2semidirect
16. #22544C2semidirect
17. #22532C2semidirect

Maximal subgroups

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

subgroupindex
#365042
#341832
#299282
#299232
#279142
#279102
#276812
#276782
#262442
#262342
#250242
#250102
#237622
#237552
#227282
#225442
#225322