Finite Group Databasebeta

← #6626292

Subgroups — #6626292

Order 512.

2131 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36564C211yesmaximal, characteristic, complemented
2C2×D4×C4⋊C4C211yesmaximal, characteristic
3#32956C211yesmaximal, characteristic
4#32953C212yesmaximal
5#29302C212yesmaximal, complemented
6#29244C212yesmaximal, complemented
7#27950C211yesmaximal, characteristic, complemented
8#27681C211yesmaximal, characteristic, complemented
9#27317C214yesmaximal, complemented
10#26384C214yesmaximal, complemented
11#26040C212yesmaximal, complemented
12#24815C212yesmaximal, complemented
13#22969C212yesmaximal, complemented
14#22671C212yesmaximal, complemented
15#22629C211yesmaximal, characteristic, complemented
16#21453C212yesmaximal
17C4×#450C211yesmaximal, characteristic
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2212yes—
20C23×C4⋊C4—21no—
21C2×#476C2212yes—
22C2×#474C2214yes—
23C2×#450C2211yescharacteristic
24C2×#440C2214yes—
25C2×C4×SD16C2212yes—
26#2857C2211yescharacteristic
27#2857C2212yes—
28#2586—22no—
29#2567C2211yescharacteristic, complemented
30#2551C2211yescharacteristic, complemented
31#2549—22no—
32#2507—21no—
33#2507C2212yescomplemented
34#2451C2212yescomplemented
35#2440—21no—
36#2369—21no—
37#2369C2211yescharacteristic, complemented
38Q8×C4⋊C4C2211yescharacteristic, complemented
39D4×C4⋊C4—22no—
40D4×C4⋊C4—21no—
41D4×C4⋊C4—21no—
42D4×C4⋊C4C2214yes—
43D4×C4⋊C4C2212yes—
44D4×C4⋊C4C2212yes—
45#2345—21no—
46C4×C22⋊Q8C2211yescharacteristic
47C4×C22⋊Q8C2212yes—
48C2×#389C2212yes—
49C2×#385—22no—
50C2×#384C2211yescharacteristic
51C2×#380—21no—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2100C2212yescomplemented
54#2095C2212yescomplemented
55#2086C2214yescomplemented
56#2084C2214yescomplemented
57#2083C2214yescomplemented
58#2066C2214yescomplemented
59#2057C2214yescomplemented
60#1990C2212yescomplemented
61#1947C2212yescomplemented
62#1944C2214yescomplemented
63#1944C2214yescomplemented
64#1939C2212yescomplemented
65#1920C2214yescomplemented
66#1917C2214yescomplemented
67#1877C2212yes—
68#1877C2214yes—
69#1872C2212yes—
70#1872C2211yescharacteristic
71#1864C2214yes—
72#1864C2212yes—
73#1863C2214yescomplemented
74#1862C2214yes—
75#1862C2214yes—
76#1862C2214yes—
77#1853C2212yes—
78#1853C2211yescharacteristic
79#1852C2212yes—
80#1852C2212yes—
81#1852C2212yes—
82#1845C2212yescomplemented
83#1803C2212yescomplemented
84#1802C2212yes—
85#1802C2214yes—
86#1797C2211yescharacteristic
87#1797C2212yes—
88#1786C2211yescharacteristic
89#1786C2212yes—
90#1770C2212yescomplemented
91#1762C2214yes—
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×D4C2311yescharacteristic
96C24×C4—21no—
97C24×C4—22no—
98C22×SD16C2312yes—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×C22⋊Q8—22no—
103C2×C22⋊Q8—21no—
104C2×C22⋊Q8C2311yescharacteristic
105C2×C4×Q8C2311yescharacteristic
106C2×C4×Q8C2312yes—
107C2×C4×D4—28no—
108C2×C4×D4—21no—
109C2×C4×D4—21no—
110C2×C4×D4—22no—
111C2×C4×D4—22no—
112C2×C4×D4—48no—
113C2×C4×D4C2312yes—
114C2×C4×D4C2314yes—
115C2×C4×D4C2314yes—
116C2×C4×D4C2312yes—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—22no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—22no—
121C22×C4⋊C4—24no—
122C22×C4⋊C4—22no—
123C22×C4⋊C4—48no—
124C22×C4⋊C4D411yescharacteristic
125C22×C4⋊C4D411yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2312yes—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—22no—
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132#476—28no—
133#474—48no—
134#450Q818yescomplemented
135#440—48no—
136C4×SD16—28no—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#106C2314yes—
141C2×#106C2312yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2314yes—
144C2×#103C2312yes—
145C2×#103C2314yes—
146C2×#102C2312yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2312yes—
150C2×#102C2312yes—
151C2×C22⋊C8C2311yescharacteristic
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2311yescharacteristic
154C2×#97C2314yes—
155C2×C4×C8C2312yes—
156#398—22no—
157#398—22no—
158#398—22no—
159#395—22no—
160#395—22no—
161#395—22no—
162#391—21no—
163#391—22no—
164#391—22no—
165#391—22no—
166#391C2312yes—
167#391C2311yescharacteristic
168#389—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—22no—
173#389—21no—
174#389—22no—
175#389—22no—
176#389—22no—
177#389—22no—
178#389C2314yes—
179#389C2312yes—
180#389C2312yes—
181#389C2312yes—
182#389C2312yes—
183#389C2314yes—
184#389C2311yescharacteristic
185#387—22no—
186#387—22no—
187#387—21no—
188#387—22no—
189#385—22no—
190#385—21no—
191#385—24no—
192#385—24no—
193#385—24no—
194#385—24no—
195#384—21no—
196#384—21no—
197#384—21no—
198#384C2312yes—
199#384C2311yescharacteristic
200#384C2312yes—
201#384C2312yes—
202#380—21no—
203#380—22no—
204#380—22no—
205#380—24no—
206#380—22no—
207#380C2312yes—
208#380C2311yescharacteristic
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4D412yes—
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C22⋊C4—21no—
222C4×C22⋊C4—22no—
223C2×#95—22no—
224C2×#95D412yes—
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#336C2312yes—
233#336C2314yes—
234#336C2314yes—
235#336C2312yes—
236C25—21no—
237C22×Q8—22no—
238C22×Q8—21no—
239C22×Q8C2411yescharacteristic
240C22×D4—24no—
241C22×D4—24no—
242C22×D4—21no—
243C22×D4—22no—
244C22×D4C2×Q814yes—
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×D412yes—
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D412yes—
262C23×C4C2411yescharacteristic
263C2×SD16—24no—
264C2×SD16—24no—
265C2×SD16C2×Q818yes—
266C22×C8C2412yes—
267C22×C8C2414yes—
268C22×C8C2412yes—
269C4⋊Q8—24no—
270C4⋊Q8—44no—
271C4⋊Q8—44no—
272C22⋊Q8—22no—
273C22⋊Q8—82no—
274C22⋊Q8—84no—
275C22⋊Q8C2×Q814yes—
276C4×Q8—24no—
277C4×Q8—44no—
278C4×D4—28no—
279C4×D4—28no—
280C4×D4—24no—
281C4×D4—24no—
282C4×D4—28no—
283C4×D4—28no—
284C4×D4—24no—
285C4×D4—44no—
286C4×D4—44no—
287C4×D4—48no—
288C4×D4—48no—
289C4×D4#4118yes—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—416no—
317C2×C4⋊C4—48no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—48no—
320C2×C4⋊C4—48no—
321C2×C4⋊C4—82no—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2414yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2412yes—
335C2×C22⋊C4—28no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—48no—
344C2×C42—24no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—44no—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D412yes—
353C2×C42C2×D412yes—
354C2×C42C2412yes—
355C2×C42C2412yes—
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358#107—44no—
359#107—44no—
360#107—44no—
361#106—24no—
362#106—28no—
363C4⋊C8—24no—
364C4⋊C8—44no—
365#103—24no—
366#103—44no—
367#102—24no—
368#102—28no—
369#102—28no—
370#102—24no—
371#102C2×Q818yes—
372C22⋊C8—22no—
373C22⋊C8—24no—
374C22⋊C8C2×Q814yes—
375#97—44no—
376C4×C8—24no—
377#95—24no—
378#95—24no—
379#95—21no—
380#95—21no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—21no—
387#95—22no—
388#95—21no—
389#95—22no—
390#95—21no—
391#95—22no—
392#95C2×D412yes—
393#95C2×D412yes—
394#95C2×D412yes—
395#95C2×D412yes—
396C24—24no—
397C24—24no—
398C24—24no—
399C24—22no—
400C24—42no—
401C24—42no—
402C24—42no—
403C24—42no—
404C24C22×D411yescharacteristic
405C2×Q8—21no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—42no—
409C2×Q8—44no—
410C2×Q8—81no—
411C2×Q8—82no—
412C2×Q8—82no—
413C2×Q8—81no—
414C2×Q8C22×Q812yes—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—216no—
418C2×D4—216no—
419C2×D4—24no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4C22×Q814yes—
424C2×D4C2×#4114yes—
425C2×D4H(5,𝔽2)14yes—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—28no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—28no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—42no—
447C22×C4—41no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—44no—
451C22×C4—432no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—48no—
464C22×C4—84no—
465C22×C4—84no—
466C22×C4—832no—
467C22×C4#13714yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D412yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×Q812yes—
478C22×C42−1+412yes—
479C22×C4C2511yescharacteristic
480SD16—216no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—24no—
489C2×C8C22×Q814yes—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—44no—
506C4⋊C4—48no—
507C4⋊C4—44no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—48no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—81no—
515C4⋊C4—82no—
516C4⋊C4—82no—
517C4⋊C4—82no—
518C4⋊C4—82no—
519C4⋊C4—81no—
520C4⋊C4—82no—
521C4⋊C4—82no—
522C4⋊C4—82no—
523C4⋊C4—82no—
524C4⋊C4—82no—
525C4⋊C4AGL(1,ℤ/8)14yes—
526C4⋊C4C22×Q812yes—
527C4⋊C4C2×#4114yes—
528C22⋊C4—28no—
529C22⋊C4—24no—
530C22⋊C4—28no—
531C22⋊C4—48no—
532C22⋊C4—48no—
533C22⋊C4—42no—
534C22⋊C4—44no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—24no—
539C42—24no—
540C42—42no—
541C42—42no—
542C42—42no—
543C42—42no—
544C42—82no—
545C42—82no—
546C42C2×#4114yes—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—216no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—48no—
556C23—48no—
557C23—48no—
558C23—48no—
559C23—48no—
560C23—42no—
561C23—48no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23D4×Q812yes—
566C23C2×#13712yes—
567C23#57612yes—
568C23C23×D411yescharacteristic
569Q8—24no—
570Q8—42no—
571Q8—164no—
572Q8—162no—
573D4—216no—
574D4—24no—
575D4—24no—
576D4—24no—
577D4#55418yes—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—28no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—44no—
605C2×C4—48no—
606C2×C4—416no—
607C2×C4—48no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—416no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—44no—
630C2×C4—816no—
631C2×C4—84no—
632C2×C4—816no—
633C2×C4—88no—
634C2×C4#52512yes—
635C2×C4#52512yes—
636C2×C4D4×Q812yes—
637C2×C4C2×AGL(1,ℤ/8)14yes—
638C2×C4C2×AGL(1,ℤ/8)12yes—
639C2×C4C2×#13712yes—
640C2×C4C2×#13712yes—
641C2×C4C23×Q811yescharacteristic
642C2×C4C22×#4112yes—
643C2×C4C2×H(5,𝔽2)12yes—
644C2×C4C2×2−1+411yescharacteristic
645C8—24no—
646C8—42no—
647C8—44no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—28no—
655C22—432no—
656C22—44no—
657C22—44no—
658C22—44no—
659C22—44no—
660C22—44no—
661C22—44no—
662C22#338514yes—
663C22C2×#52511yescharacteristic
664C22C2×D4×Q811yescharacteristic
665C22#348411yescharacteristic
666C22C22×AGL(1,ℤ/8)11yescharacteristic
667C22C22×#13711yescharacteristic
668C22C2×#57611yescharacteristic
669C22#358611yescharacteristic
670C4—22no—
671C4—24no—
672C4—22no—
673C4—22no—
674C4—44no—
675C4—42no—
676C4—42no—
677C4—42no—
678C4—41no—
679C4—42no—
680C4—42no—
681C4—84no—
682C4—82no—
683C4—81no—
684C4—81no—
685C4—82no—
686C4#311014yes—
687C4C2×#55412yes—
688C4#349212yes—
689C2—24no—
690C2—48no—
691C2#3654611yescharacteristic
692C2C2×#311011yescharacteristic
693C2#6058411yescharacteristic
694C2C2×#338511yescharacteristic
695C2Q8×AGL(1,ℤ/8)11yescharacteristic
696C2#6158611yescharacteristic
697C2#6159111yescharacteristic
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. #36564C2semidirect
5. #29302C2semidirect
6. #29244C2semidirect
7. #27950C2semidirect
8. #27681C2semidirect
9. #27317C2semidirect
10. #26384C2semidirect
11. #26040C2semidirect
12. #24815C2semidirect
13. #22969C2semidirect
14. #22671C2semidirect
15. #22629C2semidirect
29. #2567C22semidirect
30. #2551C22semidirect
33. #2507C22semidirect
34. #2451C22semidirect
37. #2369C22semidirect
38. Q8×C4⋊C4C22semidirect
53. #2100C22semidirect
54. #2095C22semidirect
55. #2086C22semidirect
56. #2084C22semidirect
57. #2083C22semidirect
58. #2066C22semidirect
59. #2057C22semidirect
60. #1990C22semidirect
61. #1947C22semidirect
62. #1944C22semidirect
63. #1944C22semidirect
64. #1939C22semidirect
65. #1920C22semidirect
66. #1917C22semidirect
73. #1863C22semidirect
82. #1845C22semidirect
83. #1803C22semidirect
90. #1770C22semidirect
134. #450Q8semidirect

Maximal subgroups

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

subgroupindex
#365642
C2×D4×C4⋊C42
#329562
#329532
#293022
#292442
#279502
#276812
#273172
#263842
#260402
#248152
#229692
#226712
#226292
#214532
C4×#4502