Finite Group Databasebeta

← #6848560

Subgroups — #6848560

Order 512.

1817 conjugacy classes of subgroups.

The subgroup lattice has 711 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36193C211yesmaximal, characteristic, complemented
2C2×#2547C211yesmaximal, characteristic
3#32956C211yesmaximal, characteristic
4#32187C214yesmaximal, complemented
5#32105C214yesmaximal, complemented
6#29626C212yesmaximal
7#28419C212yesmaximal, complemented
8#27756C212yesmaximal, complemented
9#27583C212yesmaximal, complemented
10#25856C211yesmaximal, characteristic, complemented
11#25273C212yesmaximal, complemented
12#24629C212yesmaximal, complemented
13#24508C212yesmaximal, complemented
14#24212C211yesmaximal, characteristic, complemented
15#22603C211yesmaximal, characteristic, complemented
16#21808C212yesmaximal
17C4×#450C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×#124C2212yes—
20C22×C4×D4C2211yescharacteristic
21C2×#487C2212yescomplemented
22C2×#476C2212yes—
23C2×#462C2212yescomplemented
24C2×#451C2212yes—
25C2×#450C2211yescharacteristic
26C2×C4×SD16C2212yes—
27C2×#432C2212yes—
28#2857C2211yescharacteristic
29#2855C2212yes—
30#2567C2211yescharacteristic, complemented
31#2552C2211yescharacteristic, complemented
32#2547—21no—
33#2547—21no—
34#2547—22no—
35#2547C2212yes—
36#2547C2212yes—
37#2547C2214yes—
38#2520—22no—
39#2511C2212yescomplemented
40#2508—21no—
41#2507C2212yescomplemented
42#2442—21no—
43#2438—22no—
44C2×#397—22no—
45#2370—21no—
46#2369C2211yescharacteristic, complemented
47#2355C2211yescharacteristic, complemented
48#2344—21no—
49C4×C22⋊Q8C2211yescharacteristic
50C2×#391C2212yes—
51#2285C2212yes—
52C2×#380—21no—
53C2×#380—21no—
54C2×C4×C4⋊C4C2211yescharacteristic
55#2096C2214yescomplemented
56#2072C2214yescomplemented
57#2066C2214yescomplemented
58#2056C2214yescomplemented
59#2043C2214yescomplemented
60#2043C2212yescomplemented
61#2042C2214yes—
62#2042C2214yes—
63#2016C2212yescomplemented
64#2015C2214yes—
65#2015C2214yes—
66#1983C2214yes—
67#1983C2214yes—
68#1981C2212yescomplemented
69#1971C2214yescomplemented
70#1959C2214yes—
71#1959C2214yes—
72#1956C2212yescomplemented
73#1939C2212yescomplemented
74#1935C2214yescomplemented
75#1927C2214yescomplemented
76#1927C2214yescomplemented
77#1917C2214yescomplemented
78#1877C2212yes—
79#1872C2211yescharacteristic
80#1864C2212yes—
81#1864C2214yes—
82#1853C2212yes—
83#1853C2211yescharacteristic
84#1817C2212yes—
85#1802C2212yes—
86#1797C2211yescharacteristic
87#1787C2212yes—
88#1787C2212yes—
89#1787C2212yes—
90#1786C2211yescharacteristic
91#1786C2212yes—
92#1770C2212yescomplemented
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2312yes—
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C22×Q16C2314yes—
99C22×SD16C2312yes—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—44no—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2311yescharacteristic
106C2×#125—21no—
107C2×#124—28no—
108C2×#124—22no—
109C2×#124—22no—
110C2×#124—44no—
111C2×#124C2314yes—
112C2×#124C2314yes—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8C2311yescharacteristic
115C2×C4×Q8C2311yescharacteristic
116C2×C4×D4—21no—
117C2×C4×D4—21no—
118C2×C4×D4—44no—
119C2×C4×D4C2312yes—
120C2×C4×D4C2312yes—
121C2×#117—22no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4D411yescharacteristic
125C22×C4⋊C4D411yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C22⋊C4—21no—
128C22×C22⋊C4—22no—
129C22×C22⋊C4—22no—
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132#487—44no—
133#476—44no—
134#462—44no—
135#451—44no—
136#450—24no—
137C4×SD16—44no—
138#432—44no—
139C2×#106C2314yes—
140C2×#106C2312yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2314yes—
143C2×#103C2312yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#102C2312yes—
152C2×#102C2314yes—
153C2×#102C2312yes—
154C2×#102C2314yes—
155C2×#102C2312yes—
156C2×C22⋊C8C2312yes—
157C2×C22⋊C8C2311yescharacteristic
158C2×C22⋊C8C2311yescharacteristic
159C2×#97C2314yes—
160C2×C4×C8C2312yes—
161#400—22no—
162#400—22no—
163#398—22no—
164#398—22no—
165#398—22no—
166#397—24no—
167#397—24no—
168#397—24no—
169#397—24no—
170#395—22no—
171#391—21no—
172#391—22no—
173#391—22no—
174#391—22no—
175#391C2312yes—
176#391C2312yes—
177#391C2314yes—
178#391C2314yes—
179#391C2311yescharacteristic
180#389—21no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—21no—
185#389—22no—
186#389C2312yes—
187#389C2312yes—
188#389C2312yes—
189#389C2311yescharacteristic
190#387—22no—
191#387—22no—
192#387—22no—
193#387—22no—
194#387—21no—
195#387—22no—
196#387—22no—
197#385—22no—
198#385—21no—
199#384C2312yes—
200#384C2311yescharacteristic
201#384C2312yes—
202#382—21no—
203#382—22no—
204#381—22no—
205#380—21no—
206#380—22no—
207#380—22no—
208#380—22no—
209#380—22no—
210#380—24no—
211#380—22no—
212#380—24no—
213#380C2312yes—
214#380C2311yescharacteristic
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—22no—
219C4×C4⋊C4D411yescharacteristic
220C4×C4⋊C4D411yescharacteristic
221C4×C4⋊C4D412yes—
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2311yescharacteristic
225C4×C4⋊C4C2311yescharacteristic
226C4×C4⋊C4C2311yescharacteristic
227C4×C4⋊C4C2312yes—
228C4×C22⋊C4—21no—
229C2×#95—22no—
230C2×#95—22no—
231C2×#95D412yes—
232#340C2314yes—
233#340C2312yes—
234#340C2312yes—
235#336C2312yes—
236#336C2314yes—
237#336C2314yes—
238#336C2312yes—
239C25—21no—
240C22×Q8—22no—
241C22×Q8—21no—
242C22×Q8—24no—
243C22×Q8—24no—
244C22×Q8—44no—
245C22×Q8—44no—
246C22×Q8C2411yescharacteristic
247C22×Q8C2414yes—
248C22×D4—21no—
249C22×D4—22no—
250C22×D4—42no—
251C22×D4—42no—
252C22×D4#4114yes—
253C22×D4C2412yes—
254C23×C4—22no—
255C23×C4—21no—
256C23×C4—22no—
257C23×C4—22no—
258C23×C4—44no—
259C23×C4—44no—
260C23×C4—44no—
261C23×C4C2×D412yes—
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D412yes—
265C23×C4C2411yescharacteristic
266C2×Q16—44no—
267C2×Q16—44no—
268C2×Q16—44no—
269C2×SD16—24no—
270C2×SD16—42no—
271C2×SD16—42no—
272C22×C8C2412yes—
273C22×C8C2414yes—
274C22×C8C2412yes—
275C4⋊Q8—24no—
276C4⋊Q8—24no—
277C4⋊Q8—24no—
278C4⋊Q8—44no—
279C4⋊Q8—44no—
280C4⋊Q8—44no—
281#125—44no—
282#124—28no—
283#124—28no—
284#124—44no—
285#124—44no—
286#124—48no—
287#124—48no—
288C22⋊Q8—22no—
289C22⋊Q8—22no—
290C22⋊Q8—82no—
291C4×Q8—24no—
292C4×D4—24no—
293C4×D4—24no—
294C4×D4—42no—
295C4×D4—42no—
296C4×D4—44no—
297C4×D4—44no—
298#117—84no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—42no—
319C2×C4⋊C4—42no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2412yes—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—48no—
344C2×C22⋊C4—48no—
345C2×C22⋊C4—44no—
346C2×C42—24no—
347C2×C42—24no—
348C2×C42—24no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D412yes—
356C2×C42C2×D412yes—
357C2×C42C2411yescharacteristic
358C2×C42C2412yes—
359C2×C42C2412yes—
360C2×C42C2411yescharacteristic
361#106—42no—
362#106—44no—
363C4⋊C8—24no—
364C4⋊C8—44no—
365#103—44no—
366#103—44no—
367#103—44no—
368#103—42no—
369#103—44no—
370#103—44no—
371#103—44no—
372#103—44no—
373#102—24no—
374#102—42no—
375#102—44no—
376#102—42no—
377#102—44no—
378C22⋊C8—22no—
379C22⋊C8—22no—
380C22⋊C8—24no—
381#97—44no—
382C4×C8—24no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—21no—
388#95—21no—
389#95—21no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—21no—
394#95—22no—
395#95—22no—
396#95—21no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95C2×D412yes—
403#95C2×D412yes—
404#95C2×D412yes—
405#95C2×D412yes—
406C24—24no—
407C24—22no—
408C24—42no—
409C24—42no—
410C24—42no—
411C24—42no—
412C24—42no—
413C24—42no—
414C24C22×D411yescharacteristic
415C2×Q8—21no—
416C2×Q8—24no—
417C2×Q8—24no—
418C2×Q8—22no—
419C2×Q8—24no—
420C2×Q8—24no—
421C2×Q8—21no—
422C2×Q8—24no—
423C2×Q8—24no—
424C2×Q8—22no—
425C2×Q8—44no—
426C2×Q8—44no—
427C2×Q8—42no—
428C2×Q8—44no—
429C2×Q8—81no—
430C2×Q8—81no—
431C2×Q8—88no—
432C2×Q8—88no—
433C2×D4—24no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—48no—
440C2×D4—42no—
441C2×D4—42no—
442C2×D4—48no—
443C2×D4C2×#4114yes—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—24no—
462C22×C4—41no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—42no—
468C22×C4—44no—
469C22×C4—42no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—42no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—816no—
478C22×C4#13714yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C2×#4112yes—
489C22×C4H(5,𝔽2)12yes—
490C22×C4C2511yescharacteristic
491Q16—88no—
492SD16—48no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—24no—
498C2×C8—24no—
499C2×C8—22no—
500C2×C8—24no—
501C2×C8—22no—
502C4⋊C4—24no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—21no—
508C4⋊C4—22no—
509C4⋊C4—24no—
510C4⋊C4—21no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—24no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—81no—
528C4⋊C4—82no—
529C4⋊C4—82no—
530C4⋊C4—82no—
531C4⋊C4—82no—
532C4⋊C4—81no—
533C4⋊C4—82no—
534C22⋊C4—48no—
535C22⋊C4—48no—
536C22⋊C4—48no—
537C22⋊C4—48no—
538C22⋊C4—44no—
539C22⋊C4—42no—
540C22⋊C4—42no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C42—22no—
544C42—22no—
545C42—24no—
546C42—24no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—42no—
553C42—42no—
554C42—82no—
555C42—82no—
556C42H(5,𝔽2)14yes—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—24no—
562C23—48no—
563C23—48no—
564C23—42no—
565C23—48no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—84no—
572C23—84no—
573C23—84no—
574C23—84no—
575C23#54612yes—
576C23C2×#13712yes—
577C23#57712yes—
578C23C23×D411yescharacteristic
579Q8—42no—
580Q8—44no—
581Q8—44no—
582Q8—42no—
583Q8—44no—
584Q8—44no—
585Q8—88no—
586Q8—162no—
587D4—24no—
588D4—24no—
589D4—48no—
590D4—42no—
591D4—42no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—28no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—21no—
615C2×C4—22no—
616C2×C4—44no—
617C2×C4—48no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—41no—
633C2×C4—42no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—88no—
643C2×C4—84no—
644C2×C4—84no—
645C2×C4—88no—
646C2×C4—88no—
647C2×C4#53512yes—
648C2×C4#53512yes—
649C2×C4#54612yes—
650C2×C4C2×#13712yes—
651C2×C4C2×#13712yes—
652C2×C4#57512yes—
653C2×C4C22×#4111yescharacteristic
654C2×C4C2×H(5,𝔽2)12yes—
655C2×C4C2×H(5,𝔽2)11yescharacteristic
656C2×C4#58512yes—
657C8—42no—
658C8—42no—
659C8—44no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—44no—
667C22—44no—
668C22—44no—
669C22—816no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22#333114yes—
675C22C2×#53511yescharacteristic
676C22C2×#54611yescharacteristic
677C22#348711yescharacteristic
678C22C22×#13711yescharacteristic
679C22C2×#57511yescharacteristic
680C22C2×#57711yescharacteristic
681C22#358611yescharacteristic
682C4—22no—
683C4—22no—
684C4—22no—
685C4—41no—
686C4—42no—
687C4—42no—
688C4—42no—
689C4—42no—
690C4—84no—
691C4—81no—
692C4—82no—
693C4—81no—
694C4—82no—
695C4—81no—
696C4—82no—
697C4—81no—
698C4#321514yes—
699C4#350512yes—
700C4#354212yes—
701C2—24no—
702C2—84no—
703C2#3946711yescharacteristic
704C2C2×#321511yescharacteristic
705C2#6081311yescharacteristic
706C2C2×#333111yescharacteristic
707C2#6115911yescharacteristic
708C2#6188911yescharacteristic
709C2#6189911yescharacteristic
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
4. #32187C2semidirect
5. #32105C2semidirect
7. #28419C2semidirect
8. #27756C2semidirect
9. #27583C2semidirect
10. #25856C2semidirect
11. #25273C2semidirect
12. #24629C2semidirect
13. #24508C2semidirect
14. #24212C2semidirect
15. #22603C2semidirect
21. C2×#487C22semidirect
23. C2×#462C22semidirect
30. #2567C22semidirect
31. #2552C22semidirect
39. #2511C22semidirect
41. #2507C22semidirect
46. #2369C22semidirect
47. #2355C22semidirect
55. #2096C22semidirect
56. #2072C22semidirect
57. #2066C22semidirect
58. #2056C22semidirect
59. #2043C22semidirect
60. #2043C22semidirect
63. #2016C22semidirect
68. #1981C22semidirect
69. #1971C22semidirect
72. #1956C22semidirect
73. #1939C22semidirect
74. #1935C22semidirect
75. #1927C22semidirect
76. #1927C22semidirect
77. #1917C22semidirect
92. #1770C22semidirect

Maximal subgroups

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

subgroupindex
#361932
C2×#25472
#329562
#321872
#321052
#296262
#284192
#277562
#275832
#258562
#252732
#246292
#245082
#242122
#226032
#218082
C4×#4502