Finite Group Databasebeta

← #6568265

Subgroups — #6568265

Order 512.

1899 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36188C211yesmaximal, characteristic, complemented
2#34310C211yesmaximal, characteristic
3#30173C212yesmaximal, complemented
4#29864C211yesmaximal, characteristic, complemented
5#29258C211yesmaximal, characteristic, complemented
6#29109C212yesmaximal, complemented
7#28716C211yesmaximal, characteristic, complemented
8#27585C212yesmaximal, complemented
9#27539C214yesmaximal, complemented
10#27111C214yesmaximal, complemented
11#26298C212yesmaximal, complemented
12#25885C212yesmaximal
13#23775C211yesmaximal, characteristic, complemented
14#23760C212yesmaximal, complemented
15#23220C212yesmaximal, complemented
16#23066C212yesmaximal, complemented
17#22839C211yesmaximal, characteristic, complemented
18C2×#556—21no—
19C2×D42C2211yescharacteristic
20C2×#518—21no—
21C2×#486C2212yes—
22C2×#474C2212yes—
23#2868C2211yescharacteristic, complemented
24#2857C2212yescomplemented
25#2568—21no—
26#2547—21no—
27#2547C2212yes—
28#2499—22no—
29#2496—22no—
30#2496C2214yescomplemented
31#2366C2211yescharacteristic
32#2365—21no—
33#2365C2212yescomplemented
34#2363—21no—
35#2361C2212yes—
36#2358—22no—
37#2354C2211yescharacteristic
38D4×C4⋊C4C2211yescharacteristic
39D4×C22⋊C4—21no—
40D4×C22⋊C4C2212yes—
41#2324—21no—
42#2321—22no—
43C4×C4⋊Q8C2211yescharacteristic, complemented
44C4×#127C2211yescharacteristic
45C4×#121C2212yes—
46#2293C2211yescharacteristic
47C2×#390C2212yes—
48#2285C2212yescomplemented
49C42×D4C2211yescharacteristic
50#2100C2211yescharacteristic, complemented
51#2094C2211yescharacteristic
52#2094C2212yescomplemented
53#2088C2212yescomplemented
54#2083C2212yescomplemented
55#2083C2212yescomplemented
56#2077C2214yescomplemented
57#2077C2214yescomplemented
58#2071C2212yes—
59#2071C2212yes—
60#2071C2212yescomplemented
61#2071C2212yescomplemented
62#2065C2212yes—
63#2058C2211yescharacteristic, complemented
64#2033C2214yes—
65#2033C2214yes—
66#2023C2211yescharacteristic
67#2023C2212yescomplemented
68#1990C2212yescomplemented
69#1988C2212yescomplemented
70#1982C2212yes—
71#1954C2212yes—
72#1948C2212yescomplemented
73#1936C2214yes—
74#1936C2214yescomplemented
75#1926C2214yes—
76#1926C2214yescomplemented
77#1924C2214yescomplemented
78#1921C2214yescomplemented
79#1864C2214yescomplemented
80#1864C2214yescomplemented
81#1853C2214yescomplemented
82#1831C2211yescharacteristic, complemented
83#1824C2214yescomplemented
84#1817C2212yescomplemented
85#1802C2214yescomplemented
86#1786C2214yes—
87#1786C2214yes—
88#1776C2214yescomplemented
89C4×#106C2212yescomplemented
90#1770C2212yescomplemented
91C4×C4⋊C8C2211yescharacteristic, complemented
92#1762C2214yescomplemented
93C4×#102C2212yes—
94#1749C2212yescomplemented
95C23×D4—21no—
96C23×D4C2312yes—
97#556—48no—
98D42—24no—
99D42—24no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×#127C2311yescharacteristic
105C2×#126—24no—
106C2×#124—22no—
107C2×C22⋊Q8—22no—
108C2×#121—21no—
109C2×#121C2312yes—
110C2×#120—22no—
111#518—48no—
112C2×C4×Q8—22no—
113C2×C4×Q8—21no—
114C2×C4×D4—21no—
115C2×C4×D4—22no—
116C2×C4×D4—21no—
117C2×C4×D4—22no—
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4D411yescharacteristic
121C2×C4×D4D411yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2312yes—
124C2×C4×D4C2312yes—
125C2×C4×D4C2311yescharacteristic
126C2×#117—22no—
127C2×#117—21no—
128C2×#117—21no—
129C22×C4⋊C4C2312yes—
130C22×C22⋊C4—22no—
131C22×C22⋊C4—22no—
132C22×C22⋊C4—22no—
133C22×C42C2312yes—
134#486—28no—
135#474—28no—
136C2×#107C2314yes—
137C2×#106C2312yes—
138C2×#106C2314yes—
139C2×#106C2312yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2312yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2312yes—
146C2×#102C2312yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×C22⋊C8C2314yes—
150C2×C22⋊C8C2314yes—
151C2×#97C2314yes—
152C2×C4×C8C2312yes—
153#400—24no—
154#397—24no—
155#397—24no—
156#391—22no—
157#391—22no—
158#391—21no—
159#391—22no—
160#390—21no—
161#390—21no—
162#390—21no—
163#390—22no—
164#390—22no—
165#390C2312yes—
166#390C2311yescharacteristic
167#390C2311yescharacteristic
168#390C2312yes—
169#390C2312yes—
170#389—21no—
171#389—21no—
172#389—22no—
173#389C2312yes—
174#389C2311yescharacteristic
175#389C2311yescharacteristic
176#389C2314yes—
177#388—21no—
178#388—22no—
179#388—21no—
180#388—22no—
181#388—22no—
182#387—24no—
183#387—24no—
184#386—22no—
185#386—22no—
186#386—22no—
187#386—22no—
188#386—22no—
189#385—22no—
190#385—22no—
191#384—21no—
192#384—21no—
193#384C2311yescharacteristic
194#384C2311yescharacteristic
195#384C2312yes—
196#384C2312yes—
197#382—22no—
198#380—22no—
199#380—22no—
200#380—21no—
201#380C2312yes—
202#380C2314yes—
203#380C2314yes—
204#379—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2314yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C22⋊C4—21no—
215C4×C22⋊C4—22no—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4D412yes—
218C4×C22⋊C4C2312yes—
219C43C2311yescharacteristic
220#340C2314yes—
221#340C2314yes—
222#340C2312yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#336C2311yescharacteristic
230#336C2312yes—
231#336C2312yes—
232#336C2311yescharacteristic
233#336C2314yes—
234C25—22no—
235C22×Q8—21no—
236C22×Q8—22no—
237C22×D4—28no—
238C22×D4—22no—
239C22×D4—24no—
240C22×D4—24no—
241C22×D4—21no—
242C22×D4—21no—
243C22×D4—22no—
244C22×D4—44no—
245C22×D4—44no—
246C22×D4—44no—
247C22×D4—44no—
248C22×D4C2×D411yescharacteristic
249C22×D4C2412yes—
250C22×D4C2411yescharacteristic
251C23×C4—21no—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4—22no—
255C23×C4C2×D412yes—
256C23×C4C2×D412yes—
257C23×C4C2412yes—
258C22×C8C2412yes—
259C22×C8C2414yes—
260C22×C8C2412yes—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8—42no—
264C4⋊Q8—44no—
265C4⋊Q8—44no—
266#127—22no—
267#127#4114yes—
268#126—416no—
269#124—48no—
270C22⋊Q8—48no—
271#121—28no—
272#121—24no—
273#121—24no—
274#120—216no—
275C4×Q8—44no—
276C4×Q8—48no—
277C4×D4—24no—
278C4×D4—24no—
279C4×D4—22no—
280C4×D4—22no—
281C4×D4—22no—
282C4×D4—44no—
283C4×D4—44no—
284C4×D4—48no—
285C4×D4—44no—
286C4×D4—48no—
287C4×D4—44no—
288C4×D4—44no—
289C4×D4—82no—
290C4×D4#4114yes—
291#117—44no—
292#117—48no—
293#117—44no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2414yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C22⋊C4—21no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—22no—
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—21no—
337C2×C22⋊C4—48no—
338C2×C22⋊C4—48no—
339C2×C22⋊C4—84no—
340C2×C22⋊C4C2×D412yes—
341C2×C22⋊C4C2×D412yes—
342C2×C22⋊C4C2×D412yes—
343C2×C42—22no—
344C2×C42—21no—
345C2×C42—24no—
346C2×C42—22no—
347C2×C42—44no—
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2×D411yescharacteristic
354C2×C42C2411yescharacteristic
355C2×C42C2412yes—
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358#107—28no—
359#106—42no—
360#106—44no—
361#106—42no—
362C4⋊C8—24no—
363C4⋊C8—42no—
364#102—28no—
365#102—28no—
366#102—28no—
367#102—44no—
368#102—44no—
369#102—42no—
370#102—42no—
371C22⋊C8—44no—
372C22⋊C8—44no—
373#97—44no—
374C4×C8—24no—
375#95—21no—
376#95—22no—
377#95—21no—
378#95—24no—
379#95—22no—
380#95—24no—
381#95—22no—
382#95—21no—
383#95—22no—
384#95—24no—
385#95—24no—
386#95C2×D412yes—
387#95C2×D412yes—
388#95C2×D412yes—
389C24—28no—
390C24—21no—
391C24—22no—
392C24—44no—
393C24—44no—
394C24—82no—
395C24—82no—
396C24—82no—
397C24—82no—
398C24C22×D412yes—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—42no—
405C2×D4—28no—
406C2×D4—28no—
407C2×D4—22no—
408C2×D4—24no—
409C2×D4—24no—
410C2×D4—22no—
411C2×D4—21no—
412C2×D4—22no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—432no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—81no—
423C2×D4—81no—
424C2×D4—82no—
425C2×D4—82no—
426C2×D4—88no—
427C2×D4—88no—
428C2×D4AGL(1,ℤ/8)14yes—
429C2×D4C2×#4112yes—
430C2×D4C2×#4114yes—
431C22×C4—24no—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—44no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—82no—
455C22×C4—82no—
456C22×C4—82no—
457C22×C4—82no—
458C22×C4—82no—
459C22×C4—82no—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C2511yescharacteristic
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—24no—
473C2×C8—22no—
474C2×C8—24no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—24no—
478C2×C8—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—48no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—82no—
506C4⋊C4—81no—
507C4⋊C4C2×#4112yes—
508C22⋊C4—28no—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—24no—
512C22⋊C4—24no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—44no—
521C22⋊C4—42no—
522C22⋊C4—44no—
523C22⋊C4—81no—
524C22⋊C4—82no—
525C22⋊C4—82no—
526C22⋊C4—82no—
527C22⋊C4—82no—
528C42—22no—
529C42—22no—
530C42—24no—
531C42—22no—
532C42—22no—
533C42—42no—
534C42—42no—
535C42—41no—
536C42—44no—
537C42—48no—
538C42—42no—
539C42—44no—
540C42AGL(1,ℤ/8)14yes—
541C42C2×#4112yes—
542C42C2×#4112yes—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—24no—
551C23—416no—
552C23—41no—
553C23—42no—
554C23—42no—
555C23—41no—
556C23—42no—
557C23—41no—
558C23—41no—
559C23—42no—
560C23—41no—
561C23—42no—
562C23—41no—
563C23—42no—
564C23—88no—
565C23—88no—
566C23—164no—
567C23—164no—
568C23—164no—
569C23—164no—
570C23C2×AGL(1,ℤ/8)14yes—
571C23C23×D411yescharacteristic
572Q8—84no—
573Q8—88no—
574D4—28no—
575D4—24no—
576D4—28no—
577D4—416no—
578D4—48no—
579D4—44no—
580D4—42no—
581D4—84no—
582D4—88no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—48no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—48no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—84no—
629C2×C4—164no—
630C2×C4—164no—
631C2×C4—164no—
632C2×C4#52512yes—
633C2×C4#52512yes—
634C2×C4C2×AGL(1,ℤ/8)12yes—
635C2×C4C2×AGL(1,ℤ/8)12yes—
636C2×C4C2×AGL(1,ℤ/8)12yes—
637C2×C4C2×AGL(1,ℤ/8)12yes—
638C2×C4#57712yes—
639C2×C4C22×#4111yescharacteristic
640C2×C4C22×#4111yescharacteristic
641C2×C4#58512yes—
642C2×C4#58512yes—
643C8—44no—
644C8—42no—
645C8—42no—
646C22—24no—
647C22—24no—
648C22—24no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—44no—
655C22—44no—
656C22—42no—
657C22—42no—
658C22—44no—
659C22—82no—
660C22—81no—
661C22—82no—
662C22—81no—
663C22—82no—
664C22—81no—
665C22—82no—
666C22—81no—
667C22—1616no—
668C22C2×#52511yescharacteristic
669C22#348511yescharacteristic
670C22#350011yescharacteristic
671C22C22×AGL(1,ℤ/8)11yescharacteristic
672C22C22×AGL(1,ℤ/8)11yescharacteristic
673C22C2×#57711yescharacteristic
674C22#358611yescharacteristic
675C4—22no—
676C4—22no—
677C4—22no—
678C4—22no—
679C4—42no—
680C4—42no—
681C4—41no—
682C4—42no—
683C4—42no—
684C4—81no—
685C4—82no—
686C4—82no—
687C4—81no—
688C4—82no—
689C4—84no—
690C4#310914yes—
691C4#351012yes—
692C4#352112yes—
693C2—44no—
694C2—84no—
695C2—82no—
696C2#3603311yescharacteristic
697C2C2×#310911yescharacteristic
698C2#6059111yescharacteristic
699C2#6172211yescharacteristic
700C2#6172811yescharacteristic
701C2#6253911yescharacteristic
702C2#6258311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×D42C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36188C2semidirect
3. #30173C2semidirect
4. #29864C2semidirect
5. #29258C2semidirect
6. #29109C2semidirect
7. #28716C2semidirect
8. #27585C2semidirect
9. #27539C2semidirect
10. #27111C2semidirect
11. #26298C2semidirect
13. #23775C2semidirect
14. #23760C2semidirect
15. #23220C2semidirect
16. #23066C2semidirect
17. #22839C2semidirect
23. #2868C22semidirect
24. #2857C22semidirect
30. #2496C22semidirect
33. #2365C22semidirect
43. C4×C4⋊Q8C22semidirect
48. #2285C22semidirect
50. #2100C22semidirect
52. #2094C22semidirect
53. #2088C22semidirect
54. #2083C22semidirect
55. #2083C22semidirect
56. #2077C22semidirect
57. #2077C22semidirect
60. #2071C22semidirect
61. #2071C22semidirect
63. #2058C22semidirect
67. #2023C22semidirect
68. #1990C22semidirect
69. #1988C22semidirect
72. #1948C22semidirect
74. #1936C22semidirect
76. #1926C22semidirect
77. #1924C22semidirect
78. #1921C22semidirect
79. #1864C22semidirect
80. #1864C22semidirect
81. #1853C22semidirect
82. #1831C22semidirect
83. #1824C22semidirect
84. #1817C22semidirect
85. #1802C22semidirect
88. #1776C22semidirect
89. C4×#106C22semidirect
90. #1770C22semidirect
91. C4×C4⋊C8C22semidirect
92. #1762C22semidirect
94. #1749C22semidirect

Maximal subgroups

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

subgroupindex
#361882
#343102
#301732
#298642
#292582
#291092
#287162
#275852
#275392
#271112
#262982
#258852
#237752
#237602
#232202
#230662
#228392