Finite Group Databasebeta

← #6632669

Subgroups — #6632669

Order 512.

1896 conjugacy classes of subgroups.

The subgroup lattice has 726 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35014C211yesmaximal, characteristic, complemented
2C2×#2426C211yesmaximal, characteristic
3#32957C211yesmaximal, characteristic, complemented
4#32906C211yesmaximal, characteristic, complemented
5#31504C214yesmaximal, complemented
6#30933C214yesmaximal, complemented
7#30862C214yesmaximal, complemented
8#30614C214yesmaximal, complemented
9#29952C211yesmaximal, characteristic, complemented
10#29714C211yesmaximal, characteristic, complemented
11#29243C212yesmaximal, complemented
12#24386C212yesmaximal, complemented
13#24385C212yesmaximal
14#24217C212yesmaximal, complemented
15#21934C211yesmaximal, characteristic, complemented
16C22×#124C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×#121C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C2×#477C2212yescomplemented
21C2×#470C2214yescomplemented
22C2×#468C2214yes—
23C2×#464C2214yescomplemented
24C2×#462C2212yescomplemented
25C2×#451C2212yes—
26C2×#450C2212yes—
27C2×#407C2212yes—
28#2857C2211yescharacteristic
29#2857C2211yescharacteristic
30#2848C2212yes—
31#2646—21no—
32#2626—21no—
33#2589—22no—
34#2587—22no—
35#2497C2212yescomplemented
36#2492C2212yescomplemented
37#2429C2212yescomplemented
38#2426—22no—
39#2426—22no—
40#2426C2214yes—
41#2426C2214yes—
42#2423C2212yescomplemented
43C2×#396—21no—
44C2×#394—21no—
45#2329—21no—
46#2312—21no—
47C4×C22⋊Q8C2212yes—
48C2×#390—21no—
49C2×#390C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#388—21no—
52C2×C4×C22⋊C4C2211yescharacteristic
53#2094C2212yescomplemented
54#2088C2214yescomplemented
55#2071C2212yescomplemented
56#2068C2214yescomplemented
57#2065C2214yescomplemented
58#2055C2214yescomplemented
59#2042C2214yescomplemented
60#2035C2214yescomplemented
61#2033C2214yes—
62#2023C2212yescomplemented
63#2015C2214yescomplemented
64#2013C2212yescomplemented
65#2001C2214yescomplemented
66#2000C2214yes—
67#1983C2214yescomplemented
68#1918C2214yescomplemented
69#1895C2214yes—
70#1894C2214yescomplemented
71#1878C2214yescomplemented
72#1872C2212yes—
73#1872C2212yes—
74#1865C2214yescomplemented
75#1863C2214yescomplemented
76#1853C2212yes—
77#1852C2212yes—
78#1851C2214yescomplemented
79#1851C2214yescomplemented
80#1803C2212yescomplemented
81#1803C2212yescomplemented
82#1797C2211yescharacteristic
83#1797C2211yescharacteristic
84#1797C2212yes—
85#1787C2212yes—
86#1786C2212yes—
87C23×Q8C2311yescharacteristic
88C23×D4—21no—
89C23×D4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C22×Q16C2314yes—
92C22×SD16C2314yes—
93C22×M16C2314yes—
94C2×C4⋊Q8C2311yescharacteristic
95C2×C4⋊Q8C2311yescharacteristic
96C2×#124—22no—
97C2×#124—44no—
98C2×#124C2314yes—
99C2×C22⋊Q8—21no—
100C2×C22⋊Q8—22no—
101C2×C22⋊Q8—44no—
102C2×C22⋊Q8C2312yes—
103C2×C22⋊Q8C2314yes—
104C2×C22⋊Q8C2312yes—
105C2×C22⋊Q8C2312yes—
106C2×#121—22no—
107C2×#121—44no—
108C2×#121C2314yes—
109C2×C4×Q8—21no—
110C2×C4×Q8—21no—
111C2×C4×Q8—44no—
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2312yes—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4D411yescharacteristic
127C22×C22⋊C4D411yescharacteristic
128C22×C42—21no—
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131#477—44no—
132#470—48no—
133#468—48no—
134#464—48no—
135#462—44no—
136#451—44no—
137#450—44no—
138C2×#107C2314yes—
139C2×#106C2314yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151#407—44no—
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2312yes—
154C2×C22⋊C8C2314yes—
155#400—22no—
156#400—22no—
157#399—22no—
158#398—22no—
159#398—22no—
160#396—22no—
161#396—22no—
162#396—24no—
163#396—24no—
164#394—24no—
165#394—24no—
166#393—22no—
167#391—21no—
168#391—22no—
169#391C2312yes—
170#391C2312yes—
171#390—24no—
172#390—24no—
173#390—21no—
174#390—21no—
175#390C2312yes—
176#390C2312yes—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—22no—
181#389—22no—
182#389C2312yes—
183#389C2314yes—
184#389C2312yes—
185#388—22no—
186#388—22no—
187#388—22no—
188#388—22no—
189#388—24no—
190#388—24no—
191#387—22no—
192#387—22no—
193#387—22no—
194#385—22no—
195#385—22no—
196#385—22no—
197#385—22no—
198#384C2311yescharacteristic
199#384C2311yescharacteristic
200#380—21no—
201#380—21no—
202#380—21no—
203#380C2312yes—
204#380C2312yes—
205#380C2312yes—
206#379—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4C2312yes—
209C4×C22⋊C4—22no—
210C4×C22⋊C4D412yes—
211C4×C22⋊C4D412yes—
212C4×C22⋊C4C2312yes—
213C4×C22⋊C4C2312yes—
214C2×#95—21no—
215C2×#95D411yescharacteristic
216C2×#95D411yescharacteristic
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#336C2312yes—
222#336C2312yes—
223#336C2314yes—
224C25—21no—
225C25C2×D411yescharacteristic
226C22×Q8—24no—
227C22×Q8—21no—
228C22×Q8—21no—
229C22×Q8—22no—
230C22×Q8—22no—
231C22×Q8—42no—
232C22×Q8C2412yes—
233C22×Q8C2412yes—
234C22×D4—24no—
235C22×D4—21no—
236C22×D4—22no—
237C22×D4—22no—
238C22×D4—42no—
239C22×D4—42no—
240C22×D4—44no—
241C22×D4—44no—
242C22×D4—82no—
243C22×D4C2412yes—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—22no—
247C23×C4—22no—
248C23×C4—21no—
249C23×C4—42no—
250C23×C4—42no—
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D412yes—
256C23×C4C2×D411yescharacteristic
257C23×C4C2412yes—
258C23×C4C2411yescharacteristic
259C2×Q16—28no—
260C2×Q16—44no—
261C2×Q16—44no—
262C2×SD16—44no—
263C2×SD16—44no—
264C2×SD16—44no—
265C2×M16—44no—
266C2×M16—44no—
267C2×M16—44no—
268C22×C8C2414yes—
269C22×C8C2414yes—
270C4⋊Q8—42no—
271C4⋊Q8—42no—
272#124—28no—
273#124—44no—
274#124—48no—
275C22⋊Q8—28no—
276C22⋊Q8—24no—
277C22⋊Q8—24no—
278C22⋊Q8—44no—
279C22⋊Q8—44no—
280C22⋊Q8—48no—
281C22⋊Q8—44no—
282C22⋊Q8—44no—
283#121—28no—
284#121—44no—
285#121—48no—
286C4×Q8—24no—
287C4×Q8—24no—
288C4×Q8—42no—
289C4×Q8—42no—
290C4×Q8—44no—
291C4×Q8—44no—
292C4×Q8—44no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—42no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—82no—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—42no—
338C2×C22⋊C4—44no—
339C2×C22⋊C4—44no—
340C2×C22⋊C4—44no—
341C2×C22⋊C4—44no—
342C2×C22⋊C4—44no—
343C2×C22⋊C4—82no—
344C2×C22⋊C4C2×D412yes—
345C2×C22⋊C4C2×D412yes—
346C2×C22⋊C4C2×D412yes—
347C2×C22⋊C4C2×D412yes—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—42no—
353C2×C42—42no—
354C2×C42—44no—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D412yes—
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360C2×C42C2412yes—
361#107—28no—
362#106—44no—
363C4⋊C8—42no—
364C4⋊C8—44no—
365C4⋊C8—42no—
366#103—28no—
367#103—28no—
368#103—44no—
369#103—44no—
370#103—44no—
371#102—44no—
372#102—44no—
373#102—44no—
374C22⋊C8—24no—
375C22⋊C8—24no—
376C22⋊C8—44no—
377#95—24no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95C2×D412yes—
389#95C2×D412yes—
390#95C2×D412yes—
391#95C2×D412yes—
392C24—22no—
393C24—22no—
394C24—22no—
395C24—22no—
396C24—41no—
397C24—41no—
398C24—41no—
399C24—41no—
400C24—42no—
401C24—42no—
402C24—42no—
403C24—42no—
404C24—42no—
405C24—42no—
406C24—42no—
407C24C22×D412yes—
408C24C22×D411yescharacteristic
409C2×Q8—24no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—24no—
415C2×Q8—24no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—48no—
420C2×Q8—48no—
421C2×Q8—42no—
422C2×Q8—42no—
423C2×Q8—42no—
424C2×Q8—81no—
425C2×Q8—81no—
426C2×Q8H(5,𝔽2)14yes—
427C2×D4—22no—
428C2×D4—24no—
429C2×D4—24no—
430C2×D4—22no—
431C2×D4—48no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—48no—
435C2×D4—42no—
436C2×D4—84no—
437C2×D4—84no—
438C2×D4—84no—
439C2×D4—816no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—41no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—84no—
480C22×C4—84no—
481C22×C4#13714yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4H(5,𝔽2)12yes—
491C22×C42−1+412yes—
492C22×C4C2511yescharacteristic
493Q16—88no—
494SD16—88no—
495M16—88no—
496C2×C8—24no—
497C2×C8—24no—
498C2×C8—24no—
499C2×C8—24no—
500C2×C8—24no—
501C2×C8—24no—
502C4⋊C4—24no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—48no—
512C4⋊C4—42no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—44no—
516C4⋊C4—81no—
517C4⋊C4—81no—
518C4⋊C4—81no—
519C4⋊C4—82no—
520C4⋊C4—84no—
521C4⋊C4—82no—
522C4⋊C4—82no—
523C4⋊C4—82no—
524C4⋊C4—81no—
525C4⋊C4—81no—
526C4⋊C4—81no—
527C4⋊C4H(5,𝔽2)14yes—
528C22⋊C4—24no—
529C22⋊C4—24no—
530C22⋊C4—42no—
531C22⋊C4—42no—
532C22⋊C4—48no—
533C22⋊C4—48no—
534C22⋊C4—48no—
535C22⋊C4—48no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C22⋊C4—82no—
541C22⋊C4—82no—
542C22⋊C4—82no—
543C22⋊C4—82no—
544C22⋊C4—82no—
545C22⋊C4—82no—
546C42—22no—
547C42—22no—
548C42—24no—
549C42—41no—
550C42—41no—
551C42—42no—
552C42—44no—
553C42—44no—
554C42—82no—
555C42—82no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—82no—
578C23—82no—
579C23—82no—
580C23—82no—
581C23—84no—
582C23—84no—
583C23—84no—
584C23—84no—
585C23—84no—
586C23—84no—
587C23—84no—
588C23#53712yes—
589C23C2×#13712yes—
590C23#57512yes—
591C23C23×D411yescharacteristic
592Q8—24no—
593Q8—24no—
594Q8—48no—
595Q8—42no—
596Q8—42no—
597Q8—44no—
598Q8—44no—
599Q8—162no—
600D4—44no—
601D4—44no—
602D4—84no—
603D4—816no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—22no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—22no—
629C2×C4—44no—
630C2×C4—48no—
631C2×C4—48no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—84no—
654C2×C4—82no—
655C2×C4—82no—
656C2×C4—82no—
657C2×C4—82no—
658C2×C4—84no—
659C2×C4#53712yes—
660C2×C4C2×#13712yes—
661C2×C4C2×#13712yes—
662C2×C4C2×H(5,𝔽2)12yes—
663C2×C4C2×H(5,𝔽2)12yes—
664C2×C4C2×H(5,𝔽2)11yescharacteristic
665C2×C4C2×2−1+411yescharacteristic
666C8—44no—
667C8—44no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—42no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22—42no—
681C22—88no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22—168no—
690C22#325214yes—
691C22C2×#53711yescharacteristic
692C22#348411yescharacteristic
693C22#348411yescharacteristic
694C22C22×#13711yescharacteristic
695C22C2×#57511yescharacteristic
696C22#358511yescharacteristic
697C22#358611yescharacteristic
698C4—22no—
699C4—22no—
700C4—22no—
701C4—42no—
702C4—44no—
703C4—42no—
704C4—41no—
705C4—41no—
706C4—44no—
707C4—42no—
708C4—82no—
709C4—82no—
710C4—82no—
711C4—82no—
712C4—82no—
713C4#353612yes—
714C4#353712yes—
715C2—24no—
716C2—82no—
717C2—162no—
718C2#3670611yescharacteristic
719C2C2×#325211yescharacteristic
720C2#6091911yescharacteristic
721C2#6157311yescharacteristic
722C2#6157811yescharacteristic
723C2#6158111yescharacteristic
724C2#6158811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35014C2semidirect
3. #32957C2semidirect
4. #32906C2semidirect
5. #31504C2semidirect
6. #30933C2semidirect
7. #30862C2semidirect
8. #30614C2semidirect
9. #29952C2semidirect
10. #29714C2semidirect
11. #29243C2semidirect
12. #24386C2semidirect
14. #24217C2semidirect
15. #21934C2semidirect
20. C2×#477C22semidirect
21. C2×#470C22semidirect
23. C2×#464C22semidirect
24. C2×#462C22semidirect
35. #2497C22semidirect
36. #2492C22semidirect
37. #2429C22semidirect
42. #2423C22semidirect
53. #2094C22semidirect
54. #2088C22semidirect
55. #2071C22semidirect
56. #2068C22semidirect
57. #2065C22semidirect
58. #2055C22semidirect
59. #2042C22semidirect
60. #2035C22semidirect
62. #2023C22semidirect
63. #2015C22semidirect
64. #2013C22semidirect
65. #2001C22semidirect
67. #1983C22semidirect
68. #1918C22semidirect
70. #1894C22semidirect
71. #1878C22semidirect
74. #1865C22semidirect
75. #1863C22semidirect
78. #1851C22semidirect
79. #1851C22semidirect
80. #1803C22semidirect
81. #1803C22semidirect

Maximal subgroups

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

subgroupindex
#350142
C2×#24262
#329572
#329062
#315042
#309332
#308622
#306142
#299522
#297142
#292432
#243862
#243852
#242172
#219342