Finite Group Databasebeta

← #6491857

Subgroups — #6491857

Order 512.

1804 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35286C211yesmaximal, characteristic, complemented
2D4×#117C211yesmaximal, characteristic
3#33199C212yesmaximal, complemented
4#33072C212yesmaximal, complemented
5#29995C212yesmaximal, complemented
6#29986C212yesmaximal, complemented
7#26111C212yesmaximal, complemented
8#26091C212yesmaximal, complemented
9#26045C212yesmaximal, complemented
10#26025C212yesmaximal, complemented
11#25996C214yesmaximal, complemented
12#25952C214yesmaximal
13#21981C211yesmaximal, characteristic, complemented
14#21972C211yesmaximal, characteristic, complemented
15#21961C211yesmaximal, characteristic, complemented
16#20732C211yesmaximal, characteristic
17C2×#1802C211yesmaximal, characteristic
18C22×C4×D4C2211yescharacteristic
19C22×#117—21no—
20C2×#478C2214yes—
21C2×#431C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C22×#102C2212yes—
24#2859C2212yescomplemented
25#2859C2212yescomplemented
26#2647—22no—
27#2571—21no—
28#2571C2212yescomplemented
29#2552—21no—
30#2552C2212yescomplemented
31#2442—22no—
32#2442C2214yes—
33#2354C2211yescharacteristic
34#2354C2211yescharacteristic
35#2352—21no—
36#2352—21no—
37D4×C4⋊C4C2211yescharacteristic
38D4×C4⋊C4C2211yescharacteristic
39#2344C2212yes—
40#2342—22no—
41D4×C22⋊C4C2212yes—
42#2317—21no—
43#2316—21no—
44#2293C2212yes—
45#2290—22no—
46#2285C2211yescharacteristic
47#2285C2211yescharacteristic
48C2×#384C2211yescharacteristic
49#2280—21no—
50C42×D4C2212yes—
51C4×#117C2211yescharacteristic
52#2101C2214yescomplemented
53#2092C2214yes—
54#2088C2214yescomplemented
55#2087C2214yescomplemented
56#2067C2214yes—
57#2065C2214yes—
58#2059C2214yescomplemented
59#2025C2214yes—
60#1989C2214yescomplemented
61#1949C2214yescomplemented
62#1948C2212yescomplemented
63#1948C2212yescomplemented
64#1947C2212yescomplemented
65#1947C2212yescomplemented
66#1947C2212yescomplemented
67#1947C2212yescomplemented
68#1939C2212yescomplemented
69#1939C2212yescomplemented
70#1921C2214yescomplemented
71#1919C2214yescomplemented
72#1864C2214yes—
73#1862C2214yes—
74#1853C2214yes—
75#1850C2214yes—
76#1844C2212yescomplemented
77#1842C2212yescomplemented
78#1834C2212yescomplemented
79#1831C2212yescomplemented
80#1823C2212yes—
81#1808C2214yes—
82#1804C2214yes—
83#1802C2211yescharacteristic
84#1802C2212yes—
85#1802C2211yescharacteristic
86#1802C2211yescharacteristic
87#1802C2212yes—
88#1802C2212yes—
89#1802C2212yes—
90#1802C2211yescharacteristic
91#1786C2212yes—
92C4×#102C2212yes—
93C4×#102C2212yes—
94C2×#340C2212yes—
95C23×D4C2311yescharacteristic
96C24×C4—21no—
97C23×C8C2312yes—
98C2×#125—22no—
99C2×#125C2314yes—
100C2×C4×D4—21no—
101C2×C4×D4—21no—
102C2×C4×D4—24no—
103C2×C4×D4Q818yes—
104C2×C4×D4C2312yes—
105C2×C4×D4C2312yes—
106C2×C4×D4C2311yescharacteristic
107C2×C4×D4C2311yescharacteristic
108C2×C4×D4C2311yescharacteristic
109C2×C4×D4C2311yescharacteristic
110C2×#117—21no—
111C2×#117—21no—
112C2×#117—21no—
113C2×#117—21no—
114C2×#117—22no—
115C2×#117—22no—
116C2×#117—48no—
117C2×#117D411yescharacteristic
118C2×#117D411yescharacteristic
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2312yes—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—22no—
127C22×C42—22no—
128C22×C42C2311yescharacteristic
129#478—48no—
130#431—24no—
131C2×#107C2314yes—
132C2×#107C2314yes—
133C2×#106C2314yes—
134C2×#106C2314yes—
135C2×C4⋊C8—24no—
136C2×C4⋊C8C2311yescharacteristic
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2311yescharacteristic
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2314yes—
141C2×C4⋊C8C2314yes—
142C2×#102—44no—
143C2×#102C2312yes—
144C2×#102C2314yes—
145C2×#102C2312yes—
146C2×#102C2314yes—
147C2×#102C2312yes—
148C2×#102C2312yes—
149C2×C22⋊C8C2312yes—
150C2×C4×C8C2312yes—
151#400—24no—
152#400—24no—
153#398—24no—
154#398—24no—
155#390C2312yes—
156#390C2312yes—
157#389—22no—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389C2312yes—
166#389C2314yes—
167#389C2314yes—
168#389C2312yes—
169#388—22no—
170#388—22no—
171#388—22no—
172#388—22no—
173#387—22no—
174#387—22no—
175#386—22no—
176#386—22no—
177#385—24no—
178#385—22no—
179#385—22no—
180#384—22no—
181#384—21no—
182#384—21no—
183#384C2312yes—
184#384C2311yescharacteristic
185#384C2311yescharacteristic
186#384C2312yes—
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#382—22no—
190#382—22no—
191#382—21no—
192#382—21no—
193#382C2314yes—
194#381—22no—
195#380—21no—
196#380—21no—
197#380—21no—
198#380—21no—
199#380—21no—
200#380—21no—
201#380—24no—
202#380C2312yes—
203#380C2312yes—
204#379—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C22⋊C4—22no—
216C4×C22⋊C4—22no—
217C4×C22⋊C4—22no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4D412yes—
220C4×C22⋊C4C2312yes—
221#376D412yes—
222C43C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#336C2314yes—
234#336C2314yes—
235C25—21no—
236C22×D4—22no—
237C22×D4—22no—
238C22×D4—21no—
239C22×D4—22no—
240C22×D4—41no—
241C22×D4—41no—
242C22×D4C2×Q814yes—
243C22×D4C2411yescharacteristic
244C22×D4C2411yescharacteristic
245C23×C4—21no—
246C23×C4—28no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—22no—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4—44no—
255C23×C4—44no—
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2411yescharacteristic
259C22×C8—24no—
260C22×C8—42no—
261C22×C8—42no—
262C22×C8C2414yes—
263C22×C8C2412yes—
264C22×C8C2412yes—
265#125—44no—
266#125—44no—
267#125—84no—
268C4×D4—24no—
269C4×D4—24no—
270C4×D4—28no—
271C4×D4—28no—
272C4×D4—22no—
273C4×D4—22no—
274C4×D4—22no—
275C4×D4—22no—
276C4×D4—42no—
277C4×D4—42no—
278C4×D4C2×Q814yes—
279C4×D4C2×Q814yes—
280C4×D4C2×Q814yes—
281C4×D4C2×Q814yes—
282#117—24no—
283#117—24no—
284#117—48no—
285#117—48no—
286#117—44no—
287#117—44no—
288#117—44no—
289#117—44no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—42no—
312C2×C4⋊C4—42no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—82no—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×Q814yes—
322C2×C4⋊C4C2414yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C22⋊C4—28no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—48no—
343C2×C22⋊C4C2×D412yes—
344C2×C22⋊C4C2×D412yes—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—48no—
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2×D412yes—
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×Q814yes—
357C2×C42#4114yes—
358C2×C42C2412yes—
359C2×C42C2412yes—
360C2×C42C2411yescharacteristic
361C2×C42C2411yescharacteristic
362#107—44no—
363#107—44no—
364#106—44no—
365#106—44no—
366C4⋊C8—24no—
367C4⋊C8—24no—
368C4⋊C8—22no—
369C4⋊C8—22no—
370C4⋊C8—44no—
371C4⋊C8—44no—
372#102—28no—
373#102—24no—
374#102—24no—
375#102—44no—
376#102—42no—
377#102—42no—
378C22⋊C8—24no—
379C4×C8—24no—
380#95—22no—
381#95—22no—
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—
392#95C2×D412yes—
393#95C2×D412yes—
394#95C2×D412yes—
395#95C2×D412yes—
396C24—21no—
397C24—21no—
398C24—24no—
399C24—42no—
400C24—42no—
401C24—42no—
402C24—42no—
403C24—81no—
404C24—81no—
405C24C22×D411yescharacteristic
406C2×D4—28no—
407C2×D4—24no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4C22×Q812yes—
422C2×D4C22×Q812yes—
423C2×D4H(5,𝔽2)12yes—
424C2×D42−1+412yes—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—432no—
458C22×C4—44no—
459C22×C4—82no—
460C22×C4—82no—
461C22×C4—82no—
462C22×C4—82no—
463C22×C4C22⋊Q814yes—
464C22×C4C22⋊Q814yes—
465C22×C4#13514yes—
466C22×C4AGL(1,ℤ/8)14yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×Q812yes—
477C22×C4C2×#4112yes—
478C22×C4C2511yescharacteristic
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—24no—
482C2×C8—24no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—24no—
488C2×C8—84no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—81no—
518C4⋊C4—81no—
519C4⋊C4—82no—
520C4⋊C4—82no—
521C4⋊C4—84no—
522C4⋊C4—84no—
523C4⋊C4—84no—
524C4⋊C4—84no—
525C4⋊C4C22×Q812yes—
526C4⋊C4C22×Q812yes—
527C22⋊C4—24no—
528C22⋊C4—24no—
529C22⋊C4—24no—
530C22⋊C4—24no—
531C22⋊C4—24no—
532C22⋊C4—24no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—82no—
540C22⋊C4—82no—
541C42—24no—
542C42—24no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—42no—
548C42—42no—
549C42—44no—
550C42—44no—
551C42—44no—
552C42—44no—
553C42C22⋊Q814yes—
554C42C22⋊Q814yes—
555C42C22×Q812yes—
556C42C22×Q812yes—
557C42C2×#4112yes—
558C42C2×#4112yes—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—48no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—48no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—84no—
580C23—84no—
581C23—84no—
582C23—84no—
583C23—84no—
584C23C2×C22⋊Q812yes—
585C23C2×#13512yes—
586C23C2×AGL(1,ℤ/8)12yes—
587C23C23×D411yescharacteristic
588D4—22no—
589D4—22no—
590D4—22no—
591D4—22no—
592D4—22no—
593D4—48no—
594D4—41no—
595D4—41no—
596D4#54414yes—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—44no—
622C2×C4—48no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—48no—
628C2×C4—48no—
629C2×C4—48no—
630C2×C4—48no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—88no—
647C2×C4—88no—
648C2×C4—164no—
649C2×C4#478116yescomplemented
650C2×C4C2×C22⋊Q812yes—
651C2×C4C2×C22⋊Q812yes—
652C2×C4C2×C22⋊Q812yes—
653C2×C4C2×C22⋊Q812yes—
654C2×C4C2×C22⋊Q812yes—
655C2×C4C2×#13512yes—
656C2×C4C2×#13512yes—
657C2×C4C2×AGL(1,ℤ/8)12yes—
658C2×C4C2×AGL(1,ℤ/8)12yes—
659C2×C4C23×Q811yescharacteristic
660C2×C4C22×#4111yescharacteristic
661C2×C4C2×H(5,𝔽2)11yescharacteristic
662C2×C4C2×2−1+411yescharacteristic
663C2×C4#58511yescharacteristic
664C2×C4#58511yescharacteristic
665C8—42no—
666C8—44no—
667C8—42no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—24no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22—42no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—816no—
685C22—81no—
686C22—81no—
687C22—81no—
688C22—81no—
689C22C2×#47814yes—
690C22C22×C22⋊Q811yescharacteristic
691C22#348811yescharacteristic
692C22#348911yescharacteristic
693C22C22×#13511yescharacteristic
694C22C22×AGL(1,ℤ/8)11yescharacteristic
695C22#358511yescharacteristic
696C22#358611yescharacteristic
697C4—22no—
698C4—22no—
699C4—22no—
700C4—42no—
701C4—42no—
702C4—42no—
703C4—42no—
704C4—42no—
705C4—42no—
706C4—82no—
707C4—82no—
708C4—82no—
709C4—82no—
710C4—84no—
711C4C2×#47814yes—
712C4C2×#47814yes—
713C4C2×#54411yescharacteristic
714C4#345811yescharacteristic
715C4#349411yescharacteristic
716C4#349511yescharacteristic
717C2—24no—
718C2—44no—
719C2—82no—
720C2#3525311yescharacteristic
721C2C22×#47811yescharacteristic
722C2#6049311yescharacteristic
723C2#6194911yescharacteristic
724C2#6196211yescharacteristic
725C2#6209711yescharacteristic
726C2#6211011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35286C2semidirect
3. #33199C2semidirect
4. #33072C2semidirect
5. #29995C2semidirect
6. #29986C2semidirect
7. #26111C2semidirect
8. #26091C2semidirect
9. #26045C2semidirect
10. #26025C2semidirect
11. #25996C2semidirect
13. #21981C2semidirect
14. #21972C2semidirect
15. #21961C2semidirect
24. #2859C22semidirect
25. #2859C22semidirect
28. #2571C22semidirect
30. #2552C22semidirect
52. #2101C22semidirect
54. #2088C22semidirect
55. #2087C22semidirect
58. #2059C22semidirect
60. #1989C22semidirect
61. #1949C22semidirect
62. #1948C22semidirect
63. #1948C22semidirect
64. #1947C22semidirect
65. #1947C22semidirect
66. #1947C22semidirect
67. #1947C22semidirect
68. #1939C22semidirect
69. #1939C22semidirect
70. #1921C22semidirect
71. #1919C22semidirect
76. #1844C22semidirect
77. #1842C22semidirect
78. #1834C22semidirect
79. #1831C22semidirect
649. C2×C4#478semidirect

Maximal subgroups

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

subgroupindex
#352862
D4×#1172
#331992
#330722
#299952
#299862
#261112
#260912
#260452
#260252
#259962
#259522
#219812
#219722
#219612
#207322
C2×#18022