Finite Group Databasebeta

← #6563283

Subgroups — #6563283

Order 512.

1739 conjugacy classes of subgroups.

The subgroup lattice has 689 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36444C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic, complemented
3#29939C212yesmaximal, complemented
4#29937C211yesmaximal, characteristic, complemented
5#27955C212yesmaximal, complemented
6#27950C211yesmaximal, characteristic, complemented
7#27650C212yesmaximal, complemented
8#27644C211yesmaximal, characteristic, complemented
9#26314C214yesmaximal, complemented
10#26301C212yesmaximal, complemented
11#25027C214yesmaximal, complemented
12#25011C212yesmaximal, complemented
13#23760C212yesmaximal, complemented
14#23755C211yesmaximal, characteristic, complemented
15#22728C212yesmaximal, complemented
16#22532C211yesmaximal, characteristic, complemented
17#22523C212yesmaximal, complemented
18C2×D4×Q8C2211yescharacteristic
19C2×#548—21no—
20C2×C4×#41C2211yescharacteristic
21C2×#482C2214yes—
22C2×#480C2212yes—
23C2×#446C2212yes—
24#2857C2211yescharacteristic
25#2857C2212yes—
26#2547C2211yescharacteristic
27#2547C2212yes—
28#2537C2211yescharacteristic
29#2537C2212yes—
30#2427—22no—
31#2427—21no—
32#2415—22no—
33#2415—21no—
34#2355—21no—
35#2355C2212yes—
36#2354—21no—
37#2354C2212yes—
38#2352—22no—
39#2352C2214yes—
40Q8×C4⋊C4C2211yescharacteristic
41Q8×C22⋊C4—21no—
42#2314—22no—
43#2311—21no—
44C4×#121C2212yes—
45C4×#121C2211yescharacteristic
46#2285C2211yescharacteristic
47#2285C2212yes—
48#2285C2212yes—
49C2×#384C2211yescharacteristic
50#2274—21no—
51#2100C2211yescharacteristic
52#2100C2212yes—
53#2094C2211yescharacteristic
54#2084C2212yes—
55#2084C2211yescharacteristic
56#2083C2212yes—
57#2083C2212yes—
58#2083C2212yes—
59#2077C2214yes—
60#2077C2214yes—
61#2077C2212yes—
62#2076C2214yes—
63#2076C2214yes—
64#2071C2211yescharacteristic
65#2071C2212yes—
66#2034C2212yes—
67#2033C2214yes—
68#1993C2214yes—
69#1990C2212yes—
70#1988C2212yes—
71#1987C2214yes—
72#1955C2214yes—
73#1954C2212yes—
74#1949C2214yes—
75#1948C2212yes—
76#1947C2212yes—
77#1947C2214yes—
78#1854C2214yes—
79#1854C2214yes—
80#1852C2214yes—
81#1847C2214yes—
82#1845C2212yes—
83#1825C2214yes—
84#1823C2212yes—
85#1804C2214yes—
86#1802C2212yes—
87#1797C2212yes—
88#1797C2211yescharacteristic
89#1776C2214yes—
90C4×#106C2212yes—
91#1762C2214yes—
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C22×#41D411yescharacteristic
96C23×Q8—21no—
97D4×Q8—28no—
98#548—48no—
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2311yescharacteristic
101C2×#124—21no—
102C2×#124—22no—
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8—22no—
107C2×#121—21no—
108C2×#121—22no—
109C2×#121C2311yescharacteristic
110C2×#121C2312yes—
111C4×#41—28no—
112C2×C4×Q8—21no—
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8C2311yescharacteristic
115C2×C4×D4—22no—
116C2×C4×D4—21no—
117C2×C4×D4D412yes—
118C2×C4×D4D411yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2312yes—
121C2×C4×D4C2312yes—
122C2×C4×D4C2311yescharacteristic
123C2×#117—21no—
124C2×#117—22no—
125C2×#117D412yes—
126C2×#117D411yescharacteristic
127C2×#117C2314yes—
128C22×C4⋊C4—21no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4C2312yes—
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133#482—48no—
134#480—44no—
135#446—28no—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×#106C2312yes—
140C2×#106C2314yes—
141C2×#106C2312yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2314yes—
144C2×#102C2314yes—
145C2×#102C2312yes—
146C2×#102C2312yes—
147C2×C22⋊C8C2312yes—
148C2×C22⋊C8C2314yes—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2314yes—
151C2×#97C2314yes—
152C2×C4×C8C2312yes—
153#397—22no—
154#397—22no—
155#397—22no—
156#394—22no—
157#394—22no—
158#394—22no—
159#391—22no—
160#391—21no—
161#391—22no—
162#391—22no—
163#391—21no—
164#390—22no—
165#390C2311yescharacteristic
166#389—21no—
167#389—22no—
168#389—22no—
169#389—21no—
170#389C2312yes—
171#389C2311yescharacteristic
172#388—24no—
173#388—24no—
174#388—22no—
175#388—21no—
176#387—24no—
177#387—24no—
178#386—24no—
179#386—21no—
180#386—22no—
181#384—21no—
182#384—21no—
183#384C2312yes—
184#384C2312yes—
185#384C2311yescharacteristic
186#384C2311yescharacteristic
187#384C2312yes—
188#384C2311yescharacteristic
189#382—21no—
190#382—22no—
191#382C2314yes—
192#380—22no—
193#380—22no—
194#380—21no—
195#380—22no—
196#380—21no—
197#380—22no—
198#380—21no—
199#380—22no—
200#380C2314yes—
201#380C2312yes—
202#380C2312yes—
203#380C2312yes—
204#380C2311yescharacteristic
205#380C2312yes—
206#380C2312yes—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2314yes—
216C4×C4⋊C4C2312yes—
217C4×C22⋊C4—24no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—21no—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#336C2311yescharacteristic
232#336C2312yes—
233#336C2311yescharacteristic
234C2×#41—24no—
235C2×#41—24no—
236C2×#41—42no—
237C22×Q8—22no—
238C22×Q8—21no—
239C22×Q8—44no—
240C22×Q8—44no—
241C22×Q8—44no—
242C22×Q8C2×D411yescharacteristic
243C22×D4—22no—
244C22×D4—21no—
245C22×D4C2×D411yescharacteristic
246C22×D4C2×D412yes—
247C22×D4C2411yescharacteristic
248C23×C4—22no—
249C23×C4—22no—
250C23×C4—21no—
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D412yes—
253C23×C4C2×D412yes—
254C23×C4C2×D411yescharacteristic
255C23×C4C2411yescharacteristic
256C23×C4C2412yes—
257C22×C8C2412yes—
258C22×C8C2414yes—
259C22×C8C2412yes—
260C4⋊Q8—24no—
261C4⋊Q8—28no—
262#124—48no—
263#124—44no—
264C22⋊Q8—48no—
265C22⋊Q8—44no—
266C22⋊Q8—48no—
267C22⋊Q8—44no—
268#121—22no—
269#121—24no—
270#121—41no—
271#121—42no—
272#121—44no—
273#121—48no—
274C4×Q8—24no—
275C4×Q8—44no—
276C4×Q8#4118yes—
277C4×D4—28no—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—28no—
281C4×D4—24no—
282C4×D4—28no—
283C4×D4—48no—
284C4×D4—44no—
285#117—24no—
286#117—28no—
287#117—44no—
288#117—44no—
289#117—44no—
290#117—84no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4—42no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—82no—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2414yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C42—24no—
342C2×C42—28no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—21no—
346C2×C42—24no—
347C2×C42—24no—
348C2×C42C2×D412yes—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D412yes—
351C2×C42C2×D411yescharacteristic
352C2×C42C2411yescharacteristic
353C2×C42C2412yes—
354C2×C42C2411yescharacteristic
355#107—44no—
356#107—44no—
357#107—44no—
358#106—24no—
359#106—42no—
360#106—44no—
361C4⋊C8—24no—
362C4⋊C8—28no—
363#102—28no—
364#102—24no—
365#102—24no—
366C22⋊C8—24no—
367C22⋊C8—28no—
368C22⋊C8—42no—
369C22⋊C8—44no—
370#97—28no—
371C4×C8—24no—
372#95—24no—
373#95—24no—
374#95—22no—
375#95—21no—
376#95—22no—
377#95—21no—
378#95—24no—
379#95—22no—
380#95—24no—
381#95—22no—
382#95—22no—
383#95—21no—
384#95—22no—
385#95—22no—
386#95—22no—
387C24—21no—
388C24C22×D412yes—
389C24C22×D411yescharacteristic
390#41—416no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—44no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8—816no—
400C2×Q8#12314yes—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—21no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—42no—
414C2×D4—81no—
415C2×D4—82no—
416C2×D4H(5,𝔽2)12yes—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—41no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4AGL(1,ℤ/8)14yes—
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
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×C82−1+414yes—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—82no—
501C4⋊C4—81no—
502C4⋊C4—82no—
503C4⋊C4—82no—
504C4⋊C4#12318yes—
505C4⋊C4#12314yes—
506C22⋊C4—24no—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C22⋊C4—44no—
510C22⋊C4—42no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—42no—
516C22⋊C4—44no—
517C22⋊C4—42no—
518C22⋊C4—44no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—84no—
522C22⋊C4—84no—
523C42—24no—
524C42—22no—
525C42—24no—
526C42—22no—
527C42—22no—
528C42—44no—
529C42—42no—
530C42—44no—
531C42—84no—
532C42—84no—
533C42C2×#4114yes—
534C42C2×#4118yes—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23C2×AGL(1,ℤ/8)12yes—
556C23C23×D411yescharacteristic
557Q8—28no—
558Q8—88no—
559Q8—84no—
560D4—24no—
561D4—48no—
562D4—44no—
563D4—42no—
564D4—84no—
565D4—88no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—28no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—44no—
584C2×C4—48no—
585C2×C4—48no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—88no—
615C2×C4—84no—
616C2×C4—164no—
617C2×C4C2×#12312yes—
618C2×C4C2×#12314yes—
619C2×C4C2×#12314yes—
620C2×C4C2×#12312yes—
621C2×C4C2×AGL(1,ℤ/8)12yes—
622C2×C4C2×AGL(1,ℤ/8)12yes—
623C2×C4#57512yes—
624C2×C4#57812yes—
625C2×C4C22×#4112yes—
626C2×C4C2×H(5,𝔽2)11yescharacteristic
627C2×C4C2×2−1+412yes—
628C2×C4#58511yescharacteristic
629C8—24no—
630C8—42no—
631C8—44no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22C22×#12311yescharacteristic
654C22#348811yescharacteristic
655C22#349911yescharacteristic
656C22C22×AGL(1,ℤ/8)11yescharacteristic
657C22C2×#57511yescharacteristic
658C22C2×#57811yescharacteristic
659C22#358511yescharacteristic
660C4—22no—
661C4—24no—
662C4—22no—
663C4—24no—
664C4—22no—
665C4—42no—
666C4—41no—
667C4—84no—
668C4—81no—
669C4—82no—
670C4—81no—
671C4—82no—
672C4—82no—
673C4—82no—
674C4—81no—
675C4#309414yes—
676C4#351612yes—
677C4#352212yes—
678C2—42no—
679C2—82no—
680C2—82no—
681C2#3599611yescharacteristic
682C2C2×#309411yescharacteristic
683C2#6052611yescharacteristic
684C2#6196111yescharacteristic
685C2#6198511yescharacteristic
686C2#6247911yescharacteristic
687C2#6250811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36444C2
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. #34188C2semidirect
3. #29939C2semidirect
4. #29937C2semidirect
5. #27955C2semidirect
6. #27950C2semidirect
7. #27650C2semidirect
8. #27644C2semidirect
9. #26314C2semidirect
10. #26301C2semidirect
11. #25027C2semidirect
12. #25011C2semidirect
13. #23760C2semidirect
14. #23755C2semidirect
15. #22728C2semidirect
16. #22532C2semidirect
17. #22523C2semidirect

Maximal subgroups

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

subgroupindex
#364442
#341882
#299392
#299372
#279552
#279502
#276502
#276442
#263142
#263012
#250272
#250112
#237602
#237552
#227282
#225322
#225232