Finite Group Databasebeta

← #6484538

Subgroups — #6484538

Order 512.

2160 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35239C211yesmaximal, characteristic, complemented
2C2×#2366C211yesmaximal, characteristic
3#32959C212yesmaximal, complemented
4#32956C212yesmaximal, complemented
5#29956C212yesmaximal, complemented
6#29950C212yesmaximal, complemented
7#26541C214yesmaximal, complemented
8#26529C212yesmaximal, complemented
9#26526C212yesmaximal, complemented
10#26472C214yesmaximal, complemented
11#26467C212yesmaximal, complemented
12#26464C212yesmaximal, complemented
13#21953C211yesmaximal, characteristic, complemented
14#21810C211yesmaximal, characteristic
15#21803C211yesmaximal, characteristic
16#21365C211yesmaximal, characteristic, complemented
17#21361C211yesmaximal, characteristic, complemented
18C22×#127C2211yescharacteristic
19C22×C4×D4—21no—
20C22×C4×D4C2212yes—
21C2×#478C2214yescomplemented
22C2×#476C2214yescomplemented
23C2×#432C2211yescharacteristic
24C2×#431C2211yescharacteristic
25#2857C2212yes—
26#2851C2212yes—
27#2666—22no—
28#2571C2211yescharacteristic, complemented
29#2570C2212yescomplemented
30#2552C2211yescharacteristic, complemented
31#2552C2211yescharacteristic, complemented
32#2551C2211yescharacteristic, complemented
33#2550C2212yescomplemented
34#2442—21no—
35#2440—21no—
36#2438—22no—
37#2366—21no—
38#2366—21no—
39#2366—22no—
40#2366C2212yes—
41#2366C2212yes—
42#2366C2214yes—
43#2331—21no—
44#2330—21no—
45C2×#390—21no—
46C2×#390C2212yes—
47C2×#386—22no—
48#2285C2211yescharacteristic
49#2283C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52#2093C2214yescomplemented
53#2092C2214yescomplemented
54#2067C2214yescomplemented
55#2067C2214yescomplemented
56#2067C2214yescomplemented
57#2065C2214yescomplemented
58#2025C2214yescomplemented
59#2023C2214yescomplemented
60#1982C2214yes—
61#1982C2212yes—
62#1982C2212yes—
63#1961C2212yes—
64#1961C2212yes—
65#1957C2212yes—
66#1957C2212yes—
67#1949C2212yescomplemented
68#1949C2212yescomplemented
69#1948C2212yescomplemented
70#1947C2212yescomplemented
71#1940C2214yescomplemented
72#1939C2212yescomplemented
73#1939C2212yescomplemented
74#1864C2214yescomplemented
75#1864C2214yescomplemented
76#1864C2214yescomplemented
77#1862C2214yescomplemented
78#1853C2211yescharacteristic
79#1853C2211yescharacteristic
80#1853C2212yes—
81#1852C2212yes—
82#1852C2211yescharacteristic
83#1852C2211yescharacteristic
84#1804C2214yescomplemented
85#1802C2214yescomplemented
86#1797C2212yes—
87#1797C2212yes—
88#1786C2212yes—
89#1786C2212yes—
90#1786C2212yes—
91#1786C2212yes—
92#1776C2214yescomplemented
93C4×#107C2212yescomplemented
94C4×#106C2212yescomplemented
95C23×D4—21no—
96C23×D4—21no—
97C23×D4C2312yes—
98C24×C4—22no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2312yes—
101C2×#127—21no—
102C2×#127—21no—
103C2×#127—24no—
104C2×#127Q818yescomplemented
105C2×#127C2312yes—
106C2×#127C2312yes—
107C2×#125—21no—
108C2×#125C2312yes—
109C2×C22⋊Q8—22no—
110C2×C4×D4—22no—
111C2×C4×D4—24no—
112C2×C4×D4—22no—
113C2×C4×D4—22no—
114C2×C4×D4—22no—
115C2×C4×D4—48no—
116C2×C4×D4—48no—
117C2×C4×D4C2314yes—
118C2×C4×D4C2314yes—
119C2×#117—21no—
120C22×C4⋊C4D411yescharacteristic
121C22×C4⋊C4D411yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C22⋊C4—22no—
125C22×C22⋊C4—22no—
126C22×C22⋊C4—22no—
127C22×C42—21no—
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130#478—48no—
131#476—48no—
132#432—24no—
133#431—24no—
134C2×#107C2312yes—
135C2×#107C2314yes—
136C2×#107C2312yes—
137C2×#106C2314yes—
138C2×#106C2312yes—
139C2×#106C2312yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2314yes—
143C2×C4⋊C8C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×C22⋊C8C2311yescharacteristic
153C2×C22⋊C8C2312yes—
154C2×C22⋊C8C2311yescharacteristic
155C2×#97C2312yes—
156C2×C4×C8C2312yes—
157#400—22no—
158#400—22no—
159#398—22no—
160#398—22no—
161#398—22no—
162#398—22no—
163#395—22no—
164#395—22no—
165#391—22no—
166#391—22no—
167#390—22no—
168#390—22no—
169#390—24no—
170#390—22no—
171#390—22no—
172#390C2314yes—
173#390C2314yes—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—21no—
178#389—22no—
179#389—22no—
180#389C2312yes—
181#389C2312yes—
182#389C2312yes—
183#389C2312yes—
184#387—22no—
185#387—22no—
186#387—22no—
187#387—22no—
188#386—24no—
189#386—24no—
190#386—24no—
191#386—24no—
192#385—22no—
193#385—22no—
194#384—21no—
195#384C2312yes—
196#384C2311yescharacteristic
197#384C2311yescharacteristic
198#383—21no—
199#382—21no—
200#382—21no—
201#382C2311yescharacteristic
202#382C2312yes—
203#382C2311yescharacteristic
204#381—21no—
205#380—22no—
206#380—21no—
207#380C2311yescharacteristic
208#380C2312yes—
209#380C2311yescharacteristic
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D412yes—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2311yescharacteristic
224C4×C22⋊C4—21no—
225#376—21no—
226C2×#95—21no—
227C2×#95D412yes—
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#340C2312yes—
233#340C2312yes—
234#336C2314yes—
235#336C2314yes—
236C25—22no—
237C22×Q8—22no—
238C22×D4—28no—
239C22×D4—28no—
240C22×D4—22no—
241C22×D4—22no—
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—24no—
246C22×D4—44no—
247C22×D4—44no—
248C22×D4—44no—
249C22×D4—44no—
250C22×D4—82no—
251C22×D4—82no—
252C22×D4C2×Q818yes—
253C22×D4C2414yes—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4—24no—
257C23×C4—24no—
258C23×C4—48no—
259C23×C4—48no—
260C23×C4—84no—
261C23×C4C2×D412yes—
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D412yes—
265C23×C4C2411yescharacteristic
266C22×C8C2412yes—
267C22×C8C2412yes—
268C22×C8C2414yes—
269C4⋊Q8—44no—
270C4⋊Q8—44no—
271#127—28no—
272#127—28no—
273#127—24no—
274#127—24no—
275#127C2×Q818yes—
276#127C2×Q818yes—
277#125—44no—
278#125—44no—
279C22⋊Q8—84no—
280C4×D4—28no—
281C4×D4—28no—
282C4×D4—44no—
283C4×D4—44no—
284C4×D4—48no—
285C4×D4—48no—
286C4×D4—416no—
287C4×D4—48no—
288C4×D4—48no—
289#117—82no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—42no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—48no—
333C2×C22⋊C4—48no—
334C2×C22⋊C4—84no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—21no—
338C2×C42—21no—
339C2×C42—22no—
340C2×C42—44no—
341C2×C42—44no—
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×Q814yes—
347C2×C42#4114yes—
348C2×C42C2412yes—
349C2×C42C2412yes—
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352#107—24no—
353#107—42no—
354#107—44no—
355#106—24no—
356#106—28no—
357#106—42no—
358C4⋊C8—24no—
359C4⋊C8—24no—
360C4⋊C8—44no—
361C4⋊C8—44no—
362#102—44no—
363#102—44no—
364#102—44no—
365#102—44no—
366#102—44no—
367#102—44no—
368#102—44no—
369#102—44no—
370C22⋊C8—22no—
371C22⋊C8—22no—
372C22⋊C8—24no—
373#97—24no—
374C4×C8—24no—
375#95—22no—
376#95—22no—
377#95—21no—
378#95—21no—
379#95—21no—
380#95—21no—
381#95—21no—
382#95—21no—
383#95—22no—
384#95—22no—
385#95—21no—
386#95—21no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95C2×D412yes—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397C24—28no—
398C24—24no—
399C24—44no—
400C24—44no—
401C24—44no—
402C24—44no—
403C24—44no—
404C24—44no—
405C24C22×D411yescharacteristic
406C2×Q8—44no—
407C2×Q8—82no—
408C2×Q8—82no—
409C2×D4—28no—
410C2×D4—28no—
411C2×D4—28no—
412C2×D4—28no—
413C2×D4—28no—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—24no—
419C2×D4—416no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—432no—
425C2×D4—416no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4—816no—
429C2×D4—82no—
430C2×D4—82no—
431C2×D4—82no—
432C2×D4—82no—
433C2×D4C22×Q818yes—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—48no—
464C22×C4—84no—
465C22×C4—84no—
466C22×C4—832no—
467C22×C4C22⋊Q814yes—
468C22×C4C22⋊Q814yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D412yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C22×Q812yes—
479C22×C4C2×#4112yes—
480C22×C4C2511yescharacteristic
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—24no—
484C2×C8—22no—
485C2×C8—24no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—24no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—81no—
512C4⋊C4—81no—
513C4⋊C4—82no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C4⋊C4—82no—
517C4⋊C4—82no—
518C4⋊C4—82no—
519C4⋊C4AGL(1,ℤ/8)14yes—
520C4⋊C4H(5,𝔽2)14yes—
521C22⋊C4—48no—
522C22⋊C4—48no—
523C22⋊C4—48no—
524C22⋊C4—44no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—48no—
528C22⋊C4—84no—
529C22⋊C4—84no—
530C42—24no—
531C42—24no—
532C42—42no—
533C42—42no—
534C42—44no—
535C42—44no—
536C42—42no—
537C42—42no—
538C42—82no—
539C42C22⋊Q814yes—
540C42C22⋊Q814yes—
541C42C22×Q812yes—
542C42C22×Q812yes—
543C42C2×#4112yes—
544C42C2×#4112yes—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—28no—
550C23—416no—
551C23—416no—
552C23—416no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—88no—
560C23—88no—
561C23—88no—
562C23—88no—
563C23C2×C22⋊Q812yes—
564C23#57612yes—
565C23#57712yes—
566C23C23×D411yescharacteristic
567Q8—164no—
568D4—28no—
569D4—28no—
570D4—416no—
571D4—416no—
572D4—416no—
573D4—44no—
574D4—44no—
575D4—88no—
576D4—88no—
577C2×C4—28no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—28no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—48no—
602C2×C4—48no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—816no—
627C2×C4—84no—
628C2×C4—816no—
629C2×C4—84no—
630C2×C4C2×C22⋊Q812yes—
631C2×C4C2×C22⋊Q812yes—
632C2×C4C2×C22⋊Q812yes—
633C2×C4C2×C22⋊Q812yes—
634C2×C4C2×C22⋊Q812yes—
635C2×C4C2×AGL(1,ℤ/8)14yes—
636C2×C4C2×AGL(1,ℤ/8)12yes—
637C2×C4#57512yes—
638C2×C4C23×Q811yescharacteristic
639C2×C4C22×#4111yescharacteristic
640C2×C4C2×H(5,𝔽2)12yes—
641C2×C4#58512yes—
642C8—42no—
643C8—42no—
644C8—44no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—48no—
652C22—48no—
653C22—48no—
654C22—832no—
655C22—84no—
656C22—84no—
657C22—84no—
658C22—84no—
659C22#308114yes—
660C22C22×C22⋊Q811yescharacteristic
661C22#348411yescharacteristic
662C22#348711yescharacteristic
663C22C22×AGL(1,ℤ/8)11yescharacteristic
664C22C2×#57511yescharacteristic
665C22C2×#57611yescharacteristic
666C22C2×#57711yescharacteristic
667C4—22no—
668C4—22no—
669C4—22no—
670C4—22no—
671C4—24no—
672C4—42no—
673C4—44no—
674C4—42no—
675C4—42no—
676C4—81no—
677C4—82no—
678C4—81no—
679C4—81no—
680C4—81no—
681C4—82no—
682C4—82no—
683C4#307714yes—
684C4#346012yes—
685C4#350112yes—
686C2—24no—
687C2—88no—
688C2#3517411yescharacteristic
689C2C2×#307711yescharacteristic
690C2C2×#308111yescharacteristic
691C2#6160011yescharacteristic
692C2#6160511yescharacteristic
693C2#6190411yescharacteristic
694C2#6190711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
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. #35239C2semidirect
3. #32959C2semidirect
4. #32956C2semidirect
5. #29956C2semidirect
6. #29950C2semidirect
7. #26541C2semidirect
8. #26529C2semidirect
9. #26526C2semidirect
10. #26472C2semidirect
11. #26467C2semidirect
12. #26464C2semidirect
13. #21953C2semidirect
16. #21365C2semidirect
17. #21361C2semidirect
21. C2×#478C22semidirect
22. C2×#476C22semidirect
28. #2571C22semidirect
29. #2570C22semidirect
30. #2552C22semidirect
31. #2552C22semidirect
32. #2551C22semidirect
33. #2550C22semidirect
52. #2093C22semidirect
53. #2092C22semidirect
54. #2067C22semidirect
55. #2067C22semidirect
56. #2067C22semidirect
57. #2065C22semidirect
58. #2025C22semidirect
59. #2023C22semidirect
67. #1949C22semidirect
68. #1949C22semidirect
69. #1948C22semidirect
70. #1947C22semidirect
71. #1940C22semidirect
72. #1939C22semidirect
73. #1939C22semidirect
74. #1864C22semidirect
75. #1864C22semidirect
76. #1864C22semidirect
77. #1862C22semidirect
84. #1804C22semidirect
85. #1802C22semidirect
92. #1776C22semidirect
93. C4×#107C22semidirect
94. C4×#106C22semidirect
104. C2×#127Q8semidirect

Maximal subgroups

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

subgroupindex
#352392
C2×#23662
#329592
#329562
#299562
#299502
#265412
#265292
#265262
#264722
#264672
#264642
#219532
#218102
#218032
#213652
#213612