Finite Group Databasebeta

← #6471601

Subgroups — #6471601

Order 512.

2060 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35015C211yesmaximal, characteristic, complemented
2C2×D4×C4⋊C4C211yesmaximal, characteristic
3#33007C212yesmaximal
4#32953C211yesmaximal, characteristic
5#32905C211yesmaximal, characteristic
6#29966C211yesmaximal, characteristic, complemented
7#29909C212yesmaximal, complemented
8#29756C211yesmaximal, characteristic, complemented
9#27573C214yesmaximal, complemented
10#27317C214yesmaximal, complemented
11#26384C214yesmaximal, complemented
12#26182C214yesmaximal, complemented
13#22084C212yesmaximal, complemented
14#21916C211yesmaximal, characteristic, complemented
15#21453C212yesmaximal
16C22×C4×D4C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C23×C4⋊C4—21no—
20C2×#483C2212yes—
21C2×#474C2212yes—
22C2×#461C2212yes—
23C2×#450C2212yes—
24C2×#440C2214yes—
25#2857C2211yescharacteristic
26#2857C2211yescharacteristic
27#2855C2212yes—
28#2646—21no—
29#2627—21no—
30#2590—22no—
31#2588—22no—
32#2511C2212yescomplemented
33#2507C2212yescomplemented
34#2451C2212yescomplemented
35#2446C2212yescomplemented
36D4×C4⋊C4—22no—
37D4×C4⋊C4—22no—
38D4×C4⋊C4C2214yes—
39D4×C4⋊C4C2214yes—
40#2329—21no—
41#2313—21no—
42C4×C22⋊Q8C2212yes—
43C2×#389C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#385—21no—
46C2×#385—21no—
47C2×#384C2211yescharacteristic
48C2×#380—21no—
49C2×C4×C4⋊C4C2211yescharacteristic
50#2100C2214yescomplemented
51#2099C2212yescomplemented
52#2095C2214yescomplemented
53#2086C2214yescomplemented
54#2084C2212yescomplemented
55#2083C2214yescomplemented
56#2072C2214yescomplemented
57#2066C2214yescomplemented
58#2057C2212yescomplemented
59#2043C2212yescomplemented
60#1944C2214yescomplemented
61#1935C2214yescomplemented
62#1920C2214yescomplemented
63#1917C2214yescomplemented
64#1917C2214yescomplemented
65#1877C2214yes—
66#1877C2214yes—
67#1877C2214yes—
68#1877C2214yes—
69#1872C2212yes—
70#1872C2212yes—
71#1864C2214yes—
72#1864C2214yes—
73#1863C2214yescomplemented
74#1862C2214yes—
75#1862C2214yes—
76#1852C2212yes—
77#1852C2212yes—
78#1823C2212yes—
79#1817C2212yes—
80#1802C2212yes—
81#1802C2212yes—
82#1797C2211yescharacteristic
83#1797C2211yescharacteristic
84#1786C2212yes—
85#1762C2214yes—
86#1749C2212yes—
87C23×D4C2311yescharacteristic
88C24×C4—21no—
89C24×C4—21no—
90C24×C4—21no—
91C22×SD16C2314yes—
92C2×C4⋊Q8C2311yescharacteristic
93C2×C4⋊Q8C2311yescharacteristic
94C2×C22⋊Q8—21no—
95C2×C22⋊Q8C2312yes—
96C2×C4×Q8C2312yes—
97C2×C4×D4—24no—
98C2×C4×D4—24no—
99C2×C4×D4—22no—
100C2×C4×D4—22no—
101C2×C4×D4—21no—
102C2×C4×D4—21no—
103C2×C4×D4—44no—
104C2×C4×D4—44no—
105C2×C4×D4C2314yes—
106C2×C4×D4C2312yes—
107C2×C4×D4C2312yes—
108C2×C4×D4C2314yes—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—24no—
114C22×C4⋊C4—24no—
115C22×C4⋊C4—48no—
116C22×C4⋊C4D411yescharacteristic
117C22×C4⋊C4D411yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#483—28no—
128#474—44no—
129#461—44no—
130#450—28no—
131#440—48no—
132C2×#107C2314yes—
133C2×#107C2314yes—
134C2×#106C2314yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2312yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2312yes—
145C2×#97C2314yes—
146#400—22no—
147#400—22no—
148#400—22no—
149#398—22no—
150#398—22no—
151#398—22no—
152#398—22no—
153#395—22no—
154#391—21no—
155#391—22no—
156#391C2312yes—
157#391C2312yes—
158#391C2312yes—
159#391C2312yes—
160#389—21no—
161#389—21no—
162#389—21no—
163#389—22no—
164#389—21no—
165#389—21no—
166#389—22no—
167#389C2312yes—
168#389C2312yes—
169#389C2312yes—
170#389C2312yes—
171#389C2312yes—
172#389C2314yes—
173#389C2312yes—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#385—22no—
184#385—22no—
185#385—24no—
186#385—24no—
187#385—24no—
188#385—24no—
189#384—22no—
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#384C2314yes—
193#381—22no—
194#380—21no—
195#380—21no—
196#380—21no—
197#380—24no—
198#380—24no—
199#380C2312yes—
200C4×C4⋊C4—22no—
201C4×C4⋊C4D412yes—
202C4×C4⋊C4D412yes—
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2312yes—
206C4×C22⋊C4—22no—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95D411yescharacteristic
210C2×#95D411yescharacteristic
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#336C2312yes—
218#336C2312yes—
219#336C2314yes—
220#336C2314yes—
221#336C2314yes—
222C25—21no—
223C22×Q8—21no—
224C22×Q8C2412yes—
225C22×D4—24no—
226C22×D4—24no—
227C22×D4—21no—
228C22×D4—22no—
229C22×D4#4114yes—
230C22×D4C2412yes—
231C23×C4—28no—
232C23×C4—21no—
233C23×C4—24no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—24no—
237C23×C4—22no—
238C23×C4—22no—
239C23×C4—44no—
240C23×C4—44no—
241C23×C4—44no—
242C23×C4—44no—
243C23×C4—44no—
244C23×C4—44no—
245C23×C4—82no—
246C23×C4—82no—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2411yescharacteristic
254C2×SD16—28no—
255C2×SD16—28no—
256C2×SD16—28no—
257C22×C8C2414yes—
258C22×C8C2414yes—
259C4⋊Q8—42no—
260C4⋊Q8—42no—
261C22⋊Q8—24no—
262C22⋊Q8—24no—
263C22⋊Q8—44no—
264C4×Q8—44no—
265C4×D4—216no—
266C4×D4—28no—
267C4×D4—28no—
268C4×D4—28no—
269C4×D4—24no—
270C4×D4—24no—
271C4×D4—44no—
272C4×D4—42no—
273C4×D4—42no—
274C4×D4—44no—
275C4×D4—44no—
276C4×D4—48no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
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—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4—42no—
303C2×C4⋊C4—416no—
304C2×C4⋊C4—416no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—82no—
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4#4114yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C22⋊C4—28no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—42no—
334C2×C42—42no—
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2412yes—
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342C2×C42C2412yes—
343#107—44no—
344#107—44no—
345#106—28no—
346C4⋊C8—42no—
347C4⋊C8—42no—
348#103—28no—
349#103—44no—
350#103—44no—
351#102—28no—
352#102—28no—
353#102—28no—
354C22⋊C8—24no—
355C22⋊C8—24no—
356#97—44no—
357#95—24no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95C2×D412yes—
375#95C2×D412yes—
376#95C2×D412yes—
377#95C2×D412yes—
378C24—24no—
379C24—24no—
380C24—24no—
381C24—22no—
382C24—42no—
383C24—42no—
384C24—42no—
385C24—42no—
386C24C22×D411yescharacteristic
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—24no—
390C2×Q8—24no—
391C2×Q8—42no—
392C2×Q8—81no—
393C2×Q8—81no—
394C2×D4—24no—
395C2×D4—24no—
396C2×D4—216no—
397C2×D4—216no—
398C2×D4—24no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4C2×#4114yes—
403C2×D4H(5,𝔽2)14yes—
404C2×D42−1+414yes—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—216no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—48no—
433C22×C4—432no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—48no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—82no—
450C22×C4—82no—
451C22×C4—84no—
452C22×C4—816no—
453C22×C4—816no—
454C22×C4AGL(1,ℤ/8)14yes—
455C22×C4#13714yes—
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D412yes—
465C22×C4C2×#4112yes—
466C22×C4C2×#4112yes—
467C22×C4C2511yescharacteristic
468SD16—416no—
469C2×C8—24no—
470C2×C8—24no—
471C2×C8—24no—
472C2×C8—24no—
473C2×C8—24no—
474C2×C8—24no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—81no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4—81no—
501C4⋊C4—81no—
502C4⋊C4—82no—
503C4⋊C4—84no—
504C4⋊C4—82no—
505C4⋊C4—81no—
506C4⋊C4—81no—
507C4⋊C4—81no—
508C4⋊C4C2×#4114yes—
509C22⋊C4—216no—
510C22⋊C4—48no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C42—24no—
516C42—22no—
517C42—22no—
518C42—24no—
519C42—22no—
520C42—22no—
521C42—41no—
522C42—41no—
523C42—42no—
524C42—82no—
525C42—82no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—216no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—48no—
535C23—48no—
536C23—48no—
537C23—48no—
538C23—48no—
539C23—42no—
540C23—42no—
541C23—48no—
542C23—42no—
543C23—42no—
544C23#52512yes—
545C23C2×AGL(1,ℤ/8)12yes—
546C23C2×#13712yes—
547C23C23×D411yescharacteristic
548Q8—44no—
549Q8—44no—
550Q8—162no—
551D4—216no—
552D4—24no—
553D4—24no—
554D4—24no—
555D4#53918yes—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—28no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—44no—
584C2×C4—432no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—48no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—88no—
609C2×C4—88no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4—816no—
613C2×C4—84no—
614C2×C4—84no—
615C2×C4#52512yes—
616C2×C4C2×AGL(1,ℤ/8)12yes—
617C2×C4C2×AGL(1,ℤ/8)12yes—
618C2×C4C2×#13714yes—
619C2×C4C22×#4111yescharacteristic
620C2×C4C22×#4111yescharacteristic
621C2×C4C2×H(5,𝔽2)12yes—
622C2×C4C2×2−1+412yes—
623C8—44no—
624C8—44no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—28no—
632C22—432no—
633C22—44no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—44no—
638C22—44no—
639C22#311014yes—
640C22C2×#52511yescharacteristic
641C22#348411yescharacteristic
642C22#348611yescharacteristic
643C22C22×AGL(1,ℤ/8)11yescharacteristic
644C22C22×#13711yescharacteristic
645C22#358511yescharacteristic
646C22#358611yescharacteristic
647C4—24no—
648C4—22no—
649C4—22no—
650C4—22no—
651C4—44no—
652C4—42no—
653C4—42no—
654C4—41no—
655C4—41no—
656C4—42no—
657C4—82no—
658C4—82no—
659C4—82no—
660C4—82no—
661C4—82no—
662C4—82no—
663C4C2×#53912yes—
664C4#349212yes—
665C2—24no—
666C2—48no—
667C2#3498911yescharacteristic
668C2C2×#311011yescharacteristic
669C2#6056511yescharacteristic
670C2#6157111yescharacteristic
671C2#6158611yescharacteristic
672C2#6182611yescharacteristic
673C2#6182711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35015C2semidirect
6. #29966C2semidirect
7. #29909C2semidirect
8. #29756C2semidirect
9. #27573C2semidirect
10. #27317C2semidirect
11. #26384C2semidirect
12. #26182C2semidirect
13. #22084C2semidirect
14. #21916C2semidirect
32. #2511C22semidirect
33. #2507C22semidirect
34. #2451C22semidirect
35. #2446C22semidirect
50. #2100C22semidirect
51. #2099C22semidirect
52. #2095C22semidirect
53. #2086C22semidirect
54. #2084C22semidirect
55. #2083C22semidirect
56. #2072C22semidirect
57. #2066C22semidirect
58. #2057C22semidirect
59. #2043C22semidirect
60. #1944C22semidirect
61. #1935C22semidirect
62. #1920C22semidirect
63. #1917C22semidirect
64. #1917C22semidirect
73. #1863C22semidirect

Maximal subgroups

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

subgroupindex
#350152
C2×D4×C4⋊C42
#330072
#329532
#329052
#299662
#299092
#297562
#275732
#273172
#263842
#261822
#220842
#219162
#214532