Finite Group Databasebeta

← #6593945

Subgroups — #6593945

Order 512.

2095 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36193C211yesmaximal, characteristic, complemented
2C2×D4×C4⋊C4C211yesmaximal, characteristic
3#33007C212yesmaximal
4#32956C211yesmaximal, characteristic
5#27960C211yesmaximal, characteristic, complemented
6#27647C211yesmaximal, characteristic, complemented
7#27613C212yesmaximal, complemented
8#27552C214yesmaximal, complemented
9#27296C214yesmaximal, complemented
10#26110C212yesmaximal, complemented
11#24815C212yesmaximal, complemented
12#24601C212yesmaximal, complemented
13#22963C212yesmaximal, complemented
14#22770C212yesmaximal, complemented
15#22599C211yesmaximal, characteristic, complemented
16C4×#450C211yesmaximal, characteristic
17#20732C212yesmaximal
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2212yes—
20C23×C4⋊C4—21no—
21C2×#483C2212yes—
22C2×#476C2212yes—
23C2×#450C2211yescharacteristic
24C2×C4×SD16C2212yes—
25C2×#431C2212yes—
26#2857C2211yescharacteristic
27#2855C2212yes—
28#2567C2211yescharacteristic, complemented
29#2552C2211yescharacteristic, complemented
30#2520—22no—
31#2511C2212yescomplemented
32#2508—21no—
33#2507C2212yescomplemented
34#2442—21no—
35#2438—22no—
36#2370—21no—
37#2369C2211yescharacteristic, complemented
38#2355C2211yescharacteristic, complemented
39D4×C4⋊C4—21no—
40D4×C4⋊C4—21no—
41D4×C4⋊C4—22no—
42D4×C4⋊C4C2214yes—
43D4×C4⋊C4C2212yes—
44D4×C4⋊C4C2212yes—
45#2344—21no—
46C4×C22⋊Q8C2211yescharacteristic
47C2×#389C2212yes—
48C2×#385—22no—
49#2285C2212yes—
50C2×#384C2211yescharacteristic
51C2×#380—21no—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2100C2214yescomplemented
54#2099C2214yescomplemented
55#2088C2214yescomplemented
56#2084C2212yescomplemented
57#2084C2214yescomplemented
58#2072C2212yescomplemented
59#2066C2214yescomplemented
60#1990C2212yescomplemented
61#1948C2212yescomplemented
62#1941C2214yescomplemented
63#1939C2212yescomplemented
64#1933C2214yescomplemented
65#1919C2214yescomplemented
66#1919C2214yescomplemented
67#1916C2214yescomplemented
68#1877C2212yes—
69#1877C2214yes—
70#1877C2214yes—
71#1872C2211yescharacteristic
72#1864C2212yes—
73#1864C2214yes—
74#1864C2214yes—
75#1864C2214yes—
76#1853C2212yes—
77#1853C2211yescharacteristic
78#1834C2212yescomplemented
79#1823C2212yes—
80#1823C2212yes—
81#1823C2212yes—
82#1817C2212yes—
83#1803C2212yescomplemented
84#1802C2214yes—
85#1802C2214yes—
86#1802C2212yes—
87#1797C2211yescharacteristic
88#1786C2212yes—
89#1786C2211yescharacteristic
90C8×C4⋊C4C2212yescomplemented
91C4×#102C2214yes—
92C4×#102C2214yes—
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×D4C2311yescharacteristic
96C24×C4—21no—
97C24×C4—22no—
98C22×SD16C2312yes—
99C2×C4⋊Q8C2311yescharacteristic
100C2×#125—21no—
101C2×C22⋊Q8—21no—
102C2×C22⋊Q8C2311yescharacteristic
103C2×C4×Q8C2311yescharacteristic
104C2×C4×D4—28no—
105C2×C4×D4—28no—
106C2×C4×D4—21no—
107C2×C4×D4—21no—
108C2×C4×D4—22no—
109C2×C4×D4—22no—
110C2×C4×D4—44no—
111C2×C4×D4C2312yes—
112C2×C4×D4C2314yes—
113C2×C4×D4C2314yes—
114C2×C4×D4C2312yes—
115C2×#117—22no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—22no—
120C22×C4⋊C4—24no—
121C22×C4⋊C4—22no—
122C22×C4⋊C4—48no—
123C22×C4⋊C4D411yescharacteristic
124C22×C4⋊C4D411yescharacteristic
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2311yescharacteristic
127C22×C22⋊C4—21no—
128C22×C22⋊C4—22no—
129C22×C42C2311yescharacteristic
130C22×C42C2312yes—
131#483—44no—
132#476—28no—
133#450—24no—
134C4×SD16—28no—
135#431—44no—
136C2×#106C2314yes—
137C2×#106C2312yes—
138C2×#106C2314yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2314yes—
141C2×#103C2312yes—
142C2×#103C2314yes—
143C2×#102C2314yes—
144C2×#102C2312yes—
145C2×#102C2312yes—
146C2×#102C2314yes—
147C2×#102C2312yes—
148C2×C22⋊C8C2312yes—
149C2×C22⋊C8C2311yescharacteristic
150C2×C22⋊C8C2311yescharacteristic
151C2×C4×C8C2312yes—
152C2×C4×C8C2314yes—
153#400—22no—
154#400—22no—
155#398—22no—
156#398—22no—
157#398—22no—
158#395—22no—
159#391—21no—
160#391—22no—
161#391C2312yes—
162#391C2311yescharacteristic
163#391C2312yes—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—22no—
170#389—22no—
171#389—22no—
172#389C2314yes—
173#389C2311yescharacteristic
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#389C2314yes—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—21no—
183#387—22no—
184#387—22no—
185#385—22no—
186#385—21no—
187#385—24no—
188#385—24no—
189#385—24no—
190#385—24no—
191#384—21no—
192#384—21no—
193#384C2312yes—
194#384C2311yescharacteristic
195#384C2312yes—
196#384C2312yes—
197#384C2312yes—
198#382—21no—
199#382—22no—
200#381—22no—
201#380—21no—
202#380—22no—
203#380—22no—
204#380—22no—
205#380—24no—
206#380C2312yes—
207#380C2311yescharacteristic
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D412yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C22⋊C4—21no—
222C2×#95—22no—
223C2×#95D412yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#336C2312yes—
234#336C2314yes—
235#336C2314yes—
236#336C2312yes—
237C25—21no—
238C22×Q8—21no—
239C22×Q8C2411yescharacteristic
240C22×D4—24no—
241C22×D4—24no—
242C22×D4—21no—
243C22×D4—22no—
244C22×D4#4114yes—
245C22×D4C2412yes—
246C23×C4—28no—
247C23×C4—21no—
248C23×C4—22no—
249C23×C4—22no—
250C23×C4—22no—
251C23×C4—24no—
252C23×C4—24no—
253C23×C4—44no—
254C23×C4—48no—
255C23×C4—44no—
256C23×C4—48no—
257C23×C4—84no—
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D412yes—
261C23×C4C2×D412yes—
262C23×C4C2411yescharacteristic
263C2×SD16—24no—
264C2×SD16—24no—
265C2×SD16—24no—
266C22×C8C2412yes—
267C22×C8C2414yes—
268C22×C8C2412yes—
269C4⋊Q8—24no—
270#125—44no—
271C22⋊Q8—22no—
272C22⋊Q8—22no—
273C22⋊Q8—82no—
274C4×Q8—24no—
275C4×D4—28no—
276C4×D4—28no—
277C4×D4—24no—
278C4×D4—24no—
279C4×D4—28no—
280C4×D4—28no—
281C4×D4—24no—
282C4×D4—44no—
283C4×D4—44no—
284C4×D4—48no—
285C4×D4—48no—
286C4×D4#4118yes—
287#117—84no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—42no—
313C2×C4⋊C4—48no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—416no—
316C2×C4⋊C4—48no—
317C2×C4⋊C4—48no—
318C2×C4⋊C4—82no—
319C2×C4⋊C4SD1618yescomplemented
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2414yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C22⋊C4—28no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—48no—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42—24no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2×D411yescharacteristic
353C2×C42C2411yescharacteristic
354C2×C42C2412yes—
355C2×C42C2412yes—
356C2×C42C2411yescharacteristic
357#106—24no—
358#106—28no—
359#106—44no—
360C4⋊C8—24no—
361C4⋊C8—44no—
362#103—24no—
363#103—44no—
364#102—24no—
365#102—24no—
366#102—28no—
367#102—28no—
368#102—24no—
369C22⋊C8—22no—
370C22⋊C8—22no—
371C22⋊C8—24no—
372C4×C8—24no—
373C4×C8—44no—
374#95—24no—
375#95—24no—
376#95—21no—
377#95—21no—
378#95—21no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—21no—
383#95—22no—
384#95—22no—
385#95—21no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95C2×D412yes—
392#95C2×D412yes—
393#95C2×D412yes—
394#95C2×D412yes—
395C24—24no—
396C24—24no—
397C24—24no—
398C24—22no—
399C24—42no—
400C24—42no—
401C24—42no—
402C24—42no—
403C24C22×D411yescharacteristic
404C2×Q8—22no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—22no—
408C2×Q8—42no—
409C2×Q8—81no—
410C2×Q8—81no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—216no—
414C2×D4—216no—
415C2×D4—24no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4C2×#4114yes—
420C2×D4C2×#4114yes—
421C2×D42−1+414yes—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—28no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—28no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—24no—
442C22×C4—41no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—44no—
446C22×C4—432no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—84no—
460C22×C4—84no—
461C22×C4—832no—
462C22×C4C2×SD1618yes—
463C22×C4C2×SD1614yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C2×#4112yes—
474C22×C4H(5,𝔽2)12yes—
475C22×C4C2511yescharacteristic
476SD16—48no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—24no—
481C2×C8—22no—
482C2×C8—24no—
483C2×C8—24no—
484C2×C8—22no—
485C2×C8—22no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—21no—
496C4⋊C4—22no—
497C4⋊C4—24no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—48no—
506C4⋊C4—48no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—81no—
511C4⋊C4—82no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C4⋊C4—82no—
517C4⋊C4—81no—
518C4⋊C4—82no—
519C4⋊C4C2×SD1618yes—
520C4⋊C4C2×SD1614yes—
521C4⋊C4C2×SD1614yes—
522C4⋊C4C2×#4114yes—
523C22⋊C4—28no—
524C22⋊C4—28no—
525C22⋊C4—48no—
526C22⋊C4—48no—
527C22⋊C4—42no—
528C22⋊C4—42no—
529C22⋊C4—44no—
530C42—22no—
531C42—22no—
532C42—24no—
533C42—24no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—42no—
540C42—42no—
541C42—82no—
542C42—82no—
543C42C2×#4114yes—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—216no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—48no—
553C23—48no—
554C23—48no—
555C23—48no—
556C23—48no—
557C23—42no—
558C23—48no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23#54612yes—
563C23C22×SD1612yes—
564C23#57612yes—
565C23C23×D411yescharacteristic
566Q8—42no—
567Q8—42no—
568Q8—162no—
569D4—216no—
570D4—24no—
571D4—24no—
572D4—24no—
573D4#55218yes—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—216no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—28no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—44no—
599C2×C4—416no—
600C2×C4—48no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—416no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—44no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—816no—
626C2×C4—816no—
627C2×C4—84no—
628C2×C4—88no—
629C2×C4#52512yes—
630C2×C4#52512yes—
631C2×C4#54612yes—
632C2×C4C22×SD1612yes—
633C2×C4C22×SD1614yes—
634C2×C4C22×SD1614yes—
635C2×C4C22×SD1612yes—
636C2×C4#57512yes—
637C2×C4C22×#4111yescharacteristic
638C2×C4C22×#4112yes—
639C2×C4C2×H(5,𝔽2)11yescharacteristic
640C2×C4C2×2−1+412yes—
641C8—42no—
642C8—42no—
643C8—44no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—28no—
651C22—432no—
652C22—44no—
653C22—44no—
654C22—44no—
655C22—44no—
656C22—44no—
657C22—44no—
658C22#329514yes—
659C22C2×#52511yescharacteristic
660C22C2×#54611yescharacteristic
661C22#348611yescharacteristic
662C22C23×SD1611yescharacteristic
663C22C2×#57511yescharacteristic
664C22C2×#57611yescharacteristic
665C22#358611yescharacteristic
666C4—22no—
667C4—22no—
668C4—22no—
669C4—24no—
670C4—24no—
671C4—44no—
672C4—41no—
673C4—42no—
674C4—42no—
675C4—81no—
676C4—81no—
677C4—82no—
678C4—81no—
679C4—82no—
680C4—84no—
681C4—81no—
682C4#310114yes—
683C4C2×#55212yes—
684C4#349212yes—
685C2—24no—
686C2—48no—
687C2#3616311yescharacteristic
688C2C2×#310111yescharacteristic
689C2#6058411yescharacteristic
690C2C2×#329511yescharacteristic
691C2#6115911yescharacteristic
692C2#6181811yescharacteristic
693C2#6183511yescharacteristic
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. #36193C2semidirect
5. #27960C2semidirect
6. #27647C2semidirect
7. #27613C2semidirect
8. #27552C2semidirect
9. #27296C2semidirect
10. #26110C2semidirect
11. #24815C2semidirect
12. #24601C2semidirect
13. #22963C2semidirect
14. #22770C2semidirect
15. #22599C2semidirect
28. #2567C22semidirect
29. #2552C22semidirect
31. #2511C22semidirect
33. #2507C22semidirect
37. #2369C22semidirect
38. #2355C22semidirect
53. #2100C22semidirect
54. #2099C22semidirect
55. #2088C22semidirect
56. #2084C22semidirect
57. #2084C22semidirect
58. #2072C22semidirect
59. #2066C22semidirect
60. #1990C22semidirect
61. #1948C22semidirect
62. #1941C22semidirect
63. #1939C22semidirect
64. #1933C22semidirect
65. #1919C22semidirect
66. #1919C22semidirect
67. #1916C22semidirect
78. #1834C22semidirect
83. #1803C22semidirect
90. C8×C4⋊C4C22semidirect
319. C2×C4⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
#361932
C2×D4×C4⋊C42
#330072
#329562
#279602
#276472
#276132
#275522
#272962
#261102
#248152
#246012
#229632
#227702
#225992
C4×#4502
#207322