Finite Group Databasebeta

← #6592146

Subgroups — #6592146

Order 512.

1917 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36154C211yesmaximal, characteristic
2C2×#2354C211yesmaximal, characteristic
3#33148C211yesmaximal, characteristic, complemented
4#32984C212yesmaximal, complemented
5#27961C211yesmaximal, characteristic, complemented
6#27648C211yesmaximal, characteristic, complemented
7#27601C212yesmaximal, complemented
8#26299C212yesmaximal, complemented
9#26223C214yesmaximal, complemented
10#24972C212yesmaximal, complemented
11#24960C214yesmaximal, complemented
12#24556C212yesmaximal, complemented
13#22715C212yesmaximal, complemented
14#22502C212yesmaximal, complemented
15#22401C211yesmaximal, characteristic
16C4×#483C211yesmaximal, characteristic, complemented
17#20702C212yesmaximal, complemented
18C2×#554C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×#117C2212yes—
21C2×#484C2212yes—
22C2×#483C2211yescharacteristic
23C2×#436C2212yes—
24C2×#407C2212yes—
25#2857C2211yescharacteristic
26#2848C2212yes—
27#2563C2211yescharacteristic
28#2538C2211yescharacteristic
29#2487C2212yes—
30#2454—21no—
31#2450—22no—
32#2442—21no—
33#2437—22no—
34#2417C2212yes—
35#2354—21no—
36#2354—22no—
37#2354—21no—
38#2354C2214yes—
39#2354C2212yes—
40#2354C2212yes—
41#2349—21no—
42D4×C4⋊C4C2211yescharacteristic
43D4×C22⋊C4—21no—
44C4×C22⋊Q8C2211yescharacteristic
45C2×#390—21no—
46C2×#388—22no—
47C2×#384C2211yescharacteristic
48C2×#380C2212yes—
49C2×#380C2212yes—
50C2×C4×C4⋊C4—21no—
51C2×C4×C4⋊C4C2211yescharacteristic
52#2100C2214yescomplemented
53#2083C2212yescomplemented
54#2077C2214yescomplemented
55#2077C2214yescomplemented
56#2075C2214yescomplemented
57#2072C2212yescomplemented
58#2035C2214yescomplemented
59#1990C2212yescomplemented
60#1987C2212yescomplemented
61#1954C2212yes—
62#1948C2212yescomplemented
63#1947C2212yescomplemented
64#1927C2212yescomplemented
65#1920C2212yescomplemented
66#1920C2214yescomplemented
67#1894C2214yescomplemented
68#1873C2214yescomplemented
69#1873C2214yescomplemented
70#1872C2211yescharacteristic
71#1870C2214yes—
72#1854C2214yescomplemented
73#1854C2214yescomplemented
74#1853C2211yescharacteristic
75#1853C2212yes—
76#1853C2214yescomplemented
77#1834C2212yescomplemented
78#1825C2214yescomplemented
79#1825C2214yescomplemented
80#1824C2212yes—
81#1824C2212yes—
82#1824C2212yes—
83#1823C2211yescharacteristic
84#1823C2212yes—
85#1823C2214yescomplemented
86#1809C2214yescomplemented
87#1803C2212yescomplemented
88#1797C2211yescharacteristic
89#1779C2212yes—
90C4×#106C2212yescomplemented
91C4×#103C2212yescomplemented
92C4×C22⋊C8C2211yescharacteristic
93#1736C2214yescomplemented
94#1736C2214yescomplemented
95C23×D4D411yescharacteristic
96C24×C4C2312yes—
97C22×M16C2314yes—
98#554—28no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×#125—21no—
101C2×C22⋊Q8—21no—
102C2×C22⋊Q8—22no—
103C2×C22⋊Q8C2311yescharacteristic
104C2×C22⋊Q8C2312yes—
105C2×C22⋊Q8C2312yes—
106C2×C4×Q8C2311yescharacteristic
107C2×C4×D4—28no—
108C2×C4×D4—21no—
109C2×C4×D4—21no—
110C2×C4×D4D411yescharacteristic
111C2×C4×D4D412yes—
112C2×C4×D4D411yescharacteristic
113C2×C4×D4C2311yescharacteristic
114C2×C4×D4C2312yes—
115C2×C4×D4C2311yescharacteristic
116C2×#117—22no—
117C2×#117—22no—
118C2×#117—28no—
119C2×#117—44no—
120C2×#117C2314yes—
121C2×#117C2314yes—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4D411yescharacteristic
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2311yescharacteristic
130C22×C22⋊C4—22no—
131C22×C22⋊C4—22no—
132C22×C22⋊C4—22no—
133C22×C22⋊C4D412yes—
134C22×C42—22no—
135C22×C42—21no—
136C22×C42C2311yescharacteristic
137#484—44no—
138#483—24no—
139#436—28no—
140C2×#107C2314yes—
141C2×#106C2312yes—
142C2×#106C2314yes—
143C2×#106C2312yes—
144C2×C4⋊C8C2312yes—
145C2×#103C2312yes—
146C2×#103C2314yes—
147C2×#103C2312yes—
148#407—44no—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2311yescharacteristic
151C2×C22⋊C8C2311yescharacteristic
152C2×C22⋊C8C2314yes—
153C2×C22⋊C8C2314yes—
154C2×C22⋊C8C2314yes—
155C2×C4×C8C2312yes—
156#400—22no—
157#399—22no—
158#398—22no—
159#396—22no—
160#396—22no—
161#393—22no—
162#391—21no—
163#391C2311yescharacteristic
164#390—21no—
165#390—21no—
166#390—22no—
167#390—24no—
168#390—22no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389C2311yescharacteristic
173#389C2312yes—
174#388—24no—
175#388—24no—
176#388—24no—
177#388—24no—
178#387—21no—
179#386—22no—
180#386—22no—
181#385—22no—
182#385—22no—
183#385—22no—
184#385—21no—
185#385—22no—
186#385—22no—
187#384—21no—
188#384—21no—
189#384C2312yes—
190#384C2312yes—
191#384C2311yescharacteristic
192#382—21no—
193#380—21no—
194#380—22no—
195#380—22no—
196#380—22no—
197#380—22no—
198#380—22no—
199#380C2312yes—
200#380C2312yes—
201#380C2312yes—
202#380C2314yes—
203#380C2314yes—
204#380C2312yes—
205#380C2312yes—
206#380C2311yescharacteristic
207#379—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—24no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C22⋊C4—21no—
218C2×#95—22no—
219C2×#95—22no—
220C2×#95—21no—
221#340C2314yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#340C2314yes—
233#340C2314yes—
234#336C2312yes—
235#336C2312yes—
236C25C2×D412yes—
237C22×Q8—21no—
238C22×Q8C2411yescharacteristic
239C22×D4—24no—
240C22×D4—24no—
241C22×D4—42no—
242C22×D4C2×D411yescharacteristic
243C22×D4C2×D412yes—
244C22×D4C2×D411yescharacteristic
245C23×C4—21no—
246C23×C4—24no—
247C23×C4—28no—
248C23×C4—24no—
249C23×C4—22no—
250C23×C4—21no—
251C23×C4—22no—
252C23×C4—22no—
253C23×C4—44no—
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D412yes—
256C23×C4C2×D412yes—
257C23×C4C2×D412yes—
258C23×C4C2×D411yescharacteristic
259C23×C4C2412yes—
260C23×C4C2412yes—
261C23×C4C2411yescharacteristic
262C2×M16—28no—
263C2×M16—44no—
264C2×M16—44no—
265C22×C8C2412yes—
266C22×C8C2414yes—
267C22×C8C2412yes—
268C4⋊Q8—24no—
269#125—44no—
270C22⋊Q8—28no—
271C22⋊Q8—22no—
272C22⋊Q8—22no—
273C22⋊Q8—48no—
274C22⋊Q8—44no—
275C22⋊Q8—82no—
276C4×Q8—24no—
277C4×D4—24no—
278C4×D4—28no—
279C4×D4—24no—
280C4×D4—28no—
281C4×D4—24no—
282C4×D4—44no—
283C4×D4—44no—
284C4×D4#4118yes—
285#117—28no—
286#117—28no—
287#117—44no—
288#117—44no—
289#117—48no—
290#117—48no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—28no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—42no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—82no—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4#4114yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2414yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C22⋊C4—28no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—48no—
345C2×C22⋊C4—84no—
346C2×C22⋊C4—84no—
347C2×C22⋊C4C2×D412yes—
348C2×C22⋊C4C2×D412yes—
349C2×C22⋊C4C2×D412yes—
350C2×C22⋊C4C2×D412yes—
351C2×C42—24no—
352C2×C42—24no—
353C2×C42—24no—
354C2×C42—21no—
355C2×C42—21no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—48no—
359C2×C42—44no—
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364#107—44no—
365#106—24no—
366#106—42no—
367#106—44no—
368C4⋊C8—24no—
369#103—24no—
370#103—42no—
371#103—44no—
372C22⋊C8—28no—
373C22⋊C8—22no—
374C22⋊C8—22no—
375C22⋊C8—24no—
376C22⋊C8—44no—
377C22⋊C8—44no—
378C4×C8—24no—
379#95—21no—
380#95—24no—
381#95—24no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—21no—
388#95—21no—
389#95—21no—
390#95—21no—
391#95—22no—
392#95—22no—
393#95—22no—
394C24—24no—
395C24—24no—
396C24—24no—
397C24—24no—
398C24—42no—
399C24—42no—
400C24—42no—
401C24C22×D412yes—
402C24C22×D412yes—
403C24C22×D411yescharacteristic
404C2×Q8—21no—
405C2×Q8—22no—
406C2×Q8—21no—
407C2×Q8—22no—
408C2×Q8—42no—
409C2×Q8—81no—
410C2×Q8—81no—
411C2×D4—24no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—24no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—416no—
419C2×D4—42no—
420C2×D4#12314yes—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—21no—
450C22×C4—41no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—416no—
455C22×C4—416no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—44no—
460C22×C4—42no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—42no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—82no—
467C22×C4—82no—
468C22×C4—82no—
469C22×C4—82no—
470C22×C4#13714yes—
471C22×C4C22×D412yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D412yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C2×#4118yes—
479C22×C4C2×#4112yes—
480C22×C4H(5,𝔽2)12yes—
481C22×C4C2511yescharacteristic
482M16—88no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—24no—
487C2×C8—22no—
488C2×C8—24no—
489C2×C8—22no—
490C2×C8—24no—
491C2×C8—22no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—21no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—24no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—44no—
515C4⋊C4—42no—
516C4⋊C4—81no—
517C4⋊C4—82no—
518C4⋊C4—82no—
519C4⋊C4—82no—
520C4⋊C4—82no—
521C4⋊C4—81no—
522C4⋊C4—82no—
523C4⋊C4#12314yes—
524C4⋊C4C2×#4114yes—
525C22⋊C4—24no—
526C22⋊C4—24no—
527C22⋊C4—24no—
528C22⋊C4—48no—
529C22⋊C4—48no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—42no—
533C22⋊C4—42no—
534C22⋊C4—44no—
535C22⋊C4—82no—
536C22⋊C4—84no—
537C22⋊C4—84no—
538C22⋊C4—82no—
539C22⋊C4—82no—
540C22⋊C4—82no—
541C22⋊C4#12318yes—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—48no—
546C42—48no—
547C42—42no—
548C42—44no—
549C42—44no—
550C42—42no—
551C42C2×#4114yes—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—48no—
570C23—48no—
571C23—48no—
572C23—48no—
573C23—84no—
574C23—84no—
575C23—84no—
576C23C2×#12314yes—
577C23#54612yes—
578C23C2×#13712yes—
579C23#57712yes—
580C23C23×D411yescharacteristic
581Q8—42no—
582Q8—42no—
583Q8—162no—
584D4—28no—
585D4—48no—
586D4—44no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—28no—
596C2×C4—24no—
597C2×C4—28no—
598C2×C4—28no—
599C2×C4—24no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—48no—
608C2×C4—44no—
609C2×C4—48no—
610C2×C4—48no—
611C2×C4—48no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—84no—
641C2×C4—88no—
642C2×C4—164no—
643C2×C4—164no—
644C2×C4C2×#12312yes—
645C2×C4C2×#12314yes—
646C2×C4C2×#12312yes—
647C2×C4#54612yes—
648C2×C4C2×#13712yes—
649C2×C4C2×#13712yes—
650C2×C4#57512yes—
651C2×C4C22×#4111yescharacteristic
652C2×C4C22×#4112yes—
653C2×C4C2×H(5,𝔽2)11yescharacteristic
654C2×C4#58512yes—
655C8—42no—
656C8—44no—
657C8—42no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—24no—
665C22—24no—
666C22—24no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22—42no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22—816no—
679C22#333114yes—
680C22C22×#12311yescharacteristic
681C22C2×#54611yescharacteristic
682C22#348511yescharacteristic
683C22C22×#13711yescharacteristic
684C22C2×#57511yescharacteristic
685C22C2×#57711yescharacteristic
686C22#358511yescharacteristic
687C4—22no—
688C4—24no—
689C4—22no—
690C4—22no—
691C4—42no—
692C4—41no—
693C4—44no—
694C4—82no—
695C4—82no—
696C4—81no—
697C4—81no—
698C4—81no—
699C4—82no—
700C4—81no—
701C4—84no—
702C4#309514yes—
703C4#347512yes—
704C4#351312yes—
705C2—24no—
706C2—44no—
707C2—82no—
708C2#3615511yescharacteristic
709C2C2×#309511yescharacteristic
710C2#6052511yescharacteristic
711C2C2×#333111yescharacteristic
712C2#6116011yescharacteristic
713C2#6172111yescharacteristic
714C2#6173911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33148C2semidirect
4. #32984C2semidirect
5. #27961C2semidirect
6. #27648C2semidirect
7. #27601C2semidirect
8. #26299C2semidirect
9. #26223C2semidirect
10. #24972C2semidirect
11. #24960C2semidirect
12. #24556C2semidirect
13. #22715C2semidirect
14. #22502C2semidirect
16. C4×#483C2semidirect
17. #20702C2semidirect
52. #2100C22semidirect
53. #2083C22semidirect
54. #2077C22semidirect
55. #2077C22semidirect
56. #2075C22semidirect
57. #2072C22semidirect
58. #2035C22semidirect
59. #1990C22semidirect
60. #1987C22semidirect
62. #1948C22semidirect
63. #1947C22semidirect
64. #1927C22semidirect
65. #1920C22semidirect
66. #1920C22semidirect
67. #1894C22semidirect
68. #1873C22semidirect
69. #1873C22semidirect
72. #1854C22semidirect
73. #1854C22semidirect
76. #1853C22semidirect
77. #1834C22semidirect
78. #1825C22semidirect
79. #1825C22semidirect
85. #1823C22semidirect
86. #1809C22semidirect
87. #1803C22semidirect
90. C4×#106C22semidirect
91. C4×#103C22semidirect
93. #1736C22semidirect
94. #1736C22semidirect

Maximal subgroups

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

subgroupindex
#361542
C2×#23542
#331482
#329842
#279612
#276482
#276012
#262992
#262232
#249722
#249602
#245562
#227152
#225022
#224012
C4×#4832
#207022