Finite Group Databasebeta

← #6477187

Subgroups — #6477187

Order 512.

2004 conjugacy classes of subgroups.

The subgroup lattice has 717 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C211yesmaximal, characteristic
2C2×#2857C211yesmaximal, characteristic
3#33207C212yesmaximal, complemented
4#33043C212yesmaximal, complemented
5#30359C214yesmaximal, complemented
6#29159C214yesmaximal, complemented
7#27960C211yesmaximal, characteristic, complemented
8#27905C211yesmaximal, characteristic, complemented
9#27876C212yesmaximal, complemented
10#25902C211yesmaximal, characteristic, complemented
11#25811C211yesmaximal, characteristic, complemented
12#25672C212yesmaximal, complemented
13#24551C214yesmaximal
14#22078C214yesmaximal, complemented
15C2×C4×C22⋊C8C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×C22⋊Q8—21no—
19C23×C42C2211yescharacteristic
20C2×#498C2212yescomplemented
21C2×#487C2212yescomplemented
22C2×#483C2214yes—
23C22×C22⋊C8C2212yes—
24C22×C4×C8C2212yes—
25#2857—21no—
26#2857—21no—
27#2857—21no—
28#2857—21no—
29#2857C2212yes—
30#2857C2212yes—
31#2857C2212yes—
32#2857C2212yes—
33#2857C2211yescharacteristic
34#2857C2211yescharacteristic
35#2857C2211yescharacteristic
36#2857C2211yescharacteristic
37#2855—21no—
38#2855C2212yes—
39#2851—21no—
40#2851—21no—
41#2848—21no—
42#2848C2212yes—
43C4×C22⋊Q8—21no—
44C4×C22⋊Q8—21no—
45C4×C22⋊Q8C2212yes—
46C4×C22⋊Q8C2212yes—
47#2296—21no—
48C2×#391—21no—
49C2×#389—21no—
50C2×#389C2212yes—
51#2285C2211yescharacteristic
52C2×#384C2211yescharacteristic
53#2280—21no—
54C2×#380—21no—
55C2×#380C2212yes—
56#2099C2214yescomplemented
57#2096C2214yescomplemented
58#2077C2218yes—
59#2058C2214yescomplemented
60#2035C2218yes—
61#2024C2214yescomplemented
62#1990C2212yescomplemented
63#1990C2212yescomplemented
64#1990C2212yescomplemented
65#1956C2212yescomplemented
66#1941C2218yescomplemented
67#1933C2218yescomplemented
68#1919C2218yescomplemented
69#1892C2218yes—
70#1872C2214yes—
71#1865C2218yescomplemented
72#1853C2214yes—
73#1844C2212yescomplemented
74#1834C2212yescomplemented
75#1823C2214yes—
76#1807C2218yes—
77C4×C22⋊C8C2211yescharacteristic
78C4×C22⋊C8C2211yescharacteristic
79C4×C22⋊C8C2212yes—
80C4×C22⋊C8C2212yes—
81C4×C22⋊C8C2212yes—
82C2×#336C2212yes—
83C23×Q8—21no—
84C24×C4D411yescharacteristic
85C24×C4D411yescharacteristic
86C24×C4C2311yescharacteristic
87C23×C8C2314yes—
88#533—44no—
89C2×C4⋊Q8—21no—
90C2×C4⋊Q8—21no—
91C2×C4⋊Q8—44no—
92C2×C4⋊Q8C2311yescharacteristic
93C2×C4⋊Q8C2312yes—
94C2×C4⋊Q8C2311yescharacteristic
95C2×C4⋊Q8C2312yes—
96C2×#125—21no—
97C2×C22⋊Q8—21no—
98C2×C22⋊Q8—21no—
99C2×C22⋊Q8—21no—
100C2×C22⋊Q8—22no—
101C2×C22⋊Q8—22no—
102C2×C22⋊Q8—22no—
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8—48no—
105C2×C22⋊Q8C2312yes—
106C2×C22⋊Q8C2312yes—
107C2×C22⋊Q8C2312yes—
108C2×C4×Q8—21no—
109C2×C4×Q8C2312yes—
110C2×#117—21no—
111C2×#117—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—22no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C22⋊C4—22no—
119C22×C42—44no—
120C22×C42D411yescharacteristic
121C22×C42D411yescharacteristic
122C22×C42D411yescharacteristic
123C22×C42D411yescharacteristic
124C22×C42D411yescharacteristic
125C22×C42D411yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132#498—44no—
133#487—44no—
134#483—216no—
135C2×#106C2318yes—
136C2×#106C2318yes—
137C2×#103C2318yes—
138C2×#103C2318yes—
139C2×C22⋊C8—28no—
140C2×C22⋊C8C2314yes—
141C2×C22⋊C8C2312yes—
142C2×C22⋊C8C2312yes—
143C2×C4×C8—44no—
144C2×C4×C8C2312yes—
145C2×C4×C8C2312yes—
146C2×C4×C8C2314yes—
147#391—21no—
148#391—21no—
149#391—22no—
150#391—22no—
151#391—22no—
152#391—22no—
153#391C2312yes—
154#391C2312yes—
155#389—21no—
156#389—21no—
157#389—21no—
158#389—21no—
159#389—21no—
160#389—21no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—22no—
166#389—22no—
167#389C2312yes—
168#389C2312yes—
169#389C2314yes—
170#389C2314yes—
171#387—22no—
172#387—22no—
173#387—22no—
174#387—22no—
175#385—22no—
176#385—22no—
177#385—22no—
178#385—22no—
179#385—22no—
180#385—22no—
181#384—21no—
182#384—21no—
183#384C2311yescharacteristic
184#384C2312yes—
185#384C2312yes—
186#384C2311yescharacteristic
187#382—21no—
188#381—22no—
189#380—21no—
190#380—21no—
191#380—21no—
192#380—21no—
193#380—21no—
194#380—21no—
195#380—21no—
196#380—22no—
197#380—22no—
198#380—22no—
199#380—22no—
200#380—22no—
201#380—22no—
202#380C2312yes—
203#380C2314yes—
204#380C2312yes—
205#380C2314yes—
206#380C2312yes—
207#379—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4C2312yes—
212C4×C22⋊C4—22no—
213C4×C22⋊C4—22no—
214C2×#95—22no—
215C2×#95—22no—
216#340C2318yes—
217#340C2318yes—
218#340C2318yes—
219#340C2318yes—
220#336C2312yes—
221#336C2312yes—
222#336C2314yes—
223C25C2×D411yescharacteristic
224C22×Q8—21no—
225C22×Q8—22no—
226C22×Q8—22no—
227C22×Q8—44no—
228C22×Q8—44no—
229C22×Q8—44no—
230C22×Q8C2412yes—
231C23×C4—24no—
232C23×C4—24no—
233C23×C4—24no—
234C23×C4—24no—
235C23×C4—22no—
236C23×C4—22no—
237C23×C4—42no—
238C23×C4—42no—
239C23×C4—42no—
240C23×C4—42no—
241C23×C4SD1618yes—
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2411yescharacteristic
258C23×C4C2411yescharacteristic
259C23×C4C2411yescharacteristic
260C22×C8—28no—
261C22×C8—28no—
262C22×C8—44no—
263C22×C8C2414yes—
264C22×C8C2414yes—
265C4⋊Q8—24no—
266C4⋊Q8—24no—
267C4⋊Q8—24no—
268C4⋊Q8—24no—
269C4⋊Q8—44no—
270C4⋊Q8—44no—
271C4⋊Q8—44no—
272#125—44no—
273C22⋊Q8—28no—
274C22⋊Q8—28no—
275C22⋊Q8—44no—
276C22⋊Q8—44no—
277C22⋊Q8—48no—
278C22⋊Q8—48no—
279C22⋊Q8—48no—
280C22⋊Q8—48no—
281C22⋊Q8—44no—
282C22⋊Q8—44no—
283C4×Q8—44no—
284C4×Q8—44no—
285#117—44no—
286#117—44no—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—48no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4C2414yes—
317C2×C4⋊C4C2414yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—48no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—42no—
336C2×C42—42no—
337C2×C42—42no—
338C2×C42—42no—
339C2×C42—42no—
340C2×C42—42no—
341C2×C42—42no—
342C2×C42—42no—
343C2×C42—42no—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360#106—216no—
361#106—48no—
362#103—216no—
363#103—48no—
364C22⋊C8—24no—
365C22⋊C8—28no—
366C22⋊C8#4118yes—
367C4×C8—24no—
368C4×C8—24no—
369C4×C8—44no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384C24—22no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—41no—
389C24—41no—
390C24C2×SD1614yes—
391C24C22×D411yescharacteristic
392C24C22×D411yescharacteristic
393C24C22×D411yescharacteristic
394C2×Q8—24no—
395C2×Q8—24no—
396C2×Q8—24no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—42no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8—44no—
406C2×Q8—816no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—416no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—416no—
459C22×C4—416no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4C2×SD1614yes—
467C22×C4C2×SD1614yes—
468C22×C4C2×SD1614yes—
469C22×C4C2×SD1614yes—
470C22×C4C2×SD1614yes—
471C22×C4C2×SD1614yes—
472C22×C4#13514yes—
473C22×C4#13714yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C2×#4114yes—
486C22×C42−1+414yes—
487C22×C4C2511yescharacteristic
488C2×C8—24no—
489C2×C8—24no—
490C2×C8—24no—
491C2×C8—24no—
492C2×C8—24no—
493C2×C8—416no—
494C2×C8C2×#4118yes—
495C4⋊C4—28no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—48no—
515C4⋊C4—48no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—42no—
519C4⋊C4—44no—
520C4⋊C4—42no—
521C4⋊C4—44no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—48no—
529C22⋊C4—48no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—41no—
542C42—41no—
543C42—41no—
544C42—41no—
545C42—41no—
546C42—41no—
547C42—44no—
548C42—44no—
549C42H(5,𝔽2)14yes—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—28no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—28no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23#53212yes—
572C23#53612yes—
573C23C22×SD1612yes—
574C23C22×SD1612yes—
575C23C22×SD1612yes—
576C23C2×#13512yes—
577C23C2×#13712yes—
578C23C23×D411yescharacteristic
579Q8—48no—
580Q8—88no—
581Q8—88no—
582Q8—84no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—28no—
598C2×C4—28no—
599C2×C4—28no—
600C2×C4—28no—
601C2×C4—28no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—88no—
640C2×C4—88no—
641C2×C4#53212yes—
642C2×C4#53412yes—
643C2×C4#53412yes—
644C2×C4#53612yes—
645C2×C4C22×SD1612yes—
646C2×C4C22×SD1612yes—
647C2×C4C22×SD1612yes—
648C2×C4C22×SD1612yes—
649C2×C4C2×#13512yes—
650C2×C4C2×#13512yes—
651C2×C4C2×#13712yes—
652C2×C4C2×#13712yes—
653C2×C4#57512yes—
654C2×C4C22×#4112yes—
655C2×C4C2×H(5,𝔽2)12yes—
656C2×C4C2×2−1+412yes—
657C8—28no—
658C8—44no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—416no—
675C22—41no—
676C22—41no—
677C22—41no—
678C22—41no—
679C22#316714yes—
680C22#322914yes—
681C22C2×#53211yescharacteristic
682C22C2×#53411yescharacteristic
683C22C2×#53611yescharacteristic
684C22C23×SD1611yescharacteristic
685C22C22×#13511yescharacteristic
686C22C22×#13711yescharacteristic
687C22C2×#57511yescharacteristic
688C4—22no—
689C4—22no—
690C4—22no—
691C4—22no—
692C4—22no—
693C4—22no—
694C4—22no—
695C4—41no—
696C4—41no—
697C4—82no—
698C4—82no—
699C4—84no—
700C4—82no—
701C4—82no—
702C4—84no—
703C4#318714yes—
704C4#319014yes—
705C4#352014yes—
706C2—24no—
707C2—24no—
708C2—42no—
709C2#3508511yescharacteristic
710C2C2×#316711yescharacteristic
711C2#6070111yescharacteristic
712C2C2×#318711yescharacteristic
713C2C2×#319011yescharacteristic
714C2C2×#322911yescharacteristic
715C2#6084611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33207C2semidirect
4. #33043C2semidirect
5. #30359C2semidirect
6. #29159C2semidirect
7. #27960C2semidirect
8. #27905C2semidirect
9. #27876C2semidirect
10. #25902C2semidirect
11. #25811C2semidirect
12. #25672C2semidirect
14. #22078C2semidirect
20. C2×#498C22semidirect
21. C2×#487C22semidirect
56. #2099C22semidirect
57. #2096C22semidirect
59. #2058C22semidirect
61. #2024C22semidirect
62. #1990C22semidirect
63. #1990C22semidirect
64. #1990C22semidirect
65. #1956C22semidirect
66. #1941C22semidirect
67. #1933C22semidirect
68. #1919C22semidirect
71. #1865C22semidirect
73. #1844C22semidirect
74. #1834C22semidirect

Maximal subgroups

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

subgroupindex
#366252
C2×#28572
#332072
#330432
#303592
#291592
#279602
#279052
#278762
#259022
#258112
#256722
#245512
#220782
C2×C4×C22⋊C82