Finite Group Databasebeta

← #6477850

Subgroups — #6477850

Order 512.

1906 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35005C211yesmaximal, characteristic
2C2×#2362C211yesmaximal, characteristic, complemented
3#32981C212yesmaximal, complemented
4#32957C211yesmaximal, characteristic, complemented
5#32903C211yesmaximal, characteristic, complemented
6#29968C211yesmaximal, characteristic, complemented
7#29754C211yesmaximal, characteristic, complemented
8#29595C212yesmaximal, complemented
9#27103C214yesmaximal, complemented
10#26935C214yesmaximal, complemented
11#26793C214yesmaximal, complemented
12#26665C214yesmaximal, complemented
13#22041C212yesmaximal, complemented
14#21902C211yesmaximal, characteristic, complemented
15C4×#480C212yesmaximal, complemented
16C2×#538—21no—
17C22×C22⋊Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C2×#482C2214yes—
21C2×#480C2212yes—
22C2×#477C2212yescomplemented
23C2×#460C2212yescomplemented
24C2×#448C2212yes—
25#2857C2211yescharacteristic
26#2856C2211yescharacteristic
27#2847C2212yes—
28#2639—21no—
29#2617—21no—
30#2559—22no—
31#2557—22no—
32#2496C2212yes—
33#2491C2212yes—
34#2428C2212yes—
35#2422C2212yes—
36#2362—22no—
37#2362—22no—
38#2362C2214yes—
39#2362C2214yes—
40#2325—21no—
41C4×#121C2212yes—
42C2×#391C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#388—21no—
45C2×#388—21no—
46C2×#387—21no—
47C2×C4×C4⋊C4—21no—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2211yescharacteristic
50#2100C2212yescomplemented
51#2088C2214yescomplemented
52#2083C2212yescomplemented
53#2067C2214yescomplemented
54#2058C2212yescomplemented
55#2041C2212yescomplemented
56#2036C2214yes—
57#2036C2214yes—
58#2000C2214yes—
59#2000C2214yes—
60#1968C2214yescomplemented
61#1933C2214yescomplemented
62#1927C2214yescomplemented
63#1925C2214yescomplemented
64#1923C2214yes—
65#1923C2214yes—
66#1918C2214yescomplemented
67#1880C2214yescomplemented
68#1877C2214yescomplemented
69#1875C2214yes—
70#1874C2212yes—
71#1874C2212yes—
72#1863C2214yescomplemented
73#1854C2214yes—
74#1853C2212yes—
75#1852C2212yes—
76#1825C2214yes—
77#1823C2212yes—
78#1803C2212yescomplemented
79#1803C2212yescomplemented
80#1797C2211yescharacteristic
81#1797C2211yescharacteristic
82#1787C2212yes—
83#1779C2212yes—
84C4×#107C2214yescomplemented
85C4×#102C2214yescomplemented
86C4×C22⋊C8C2212yes—
87C23×Q8C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C22×SD16C2314yes—
90#538—48no—
91C2×C4⋊Q8C2311yescharacteristic
92C2×#127C2311yescharacteristic
93C2×#126—22no—
94C2×#124—22no—
95C2×#123—22no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—22no—
98C2×C22⋊Q8—21no—
99C2×C22⋊Q8—24no—
100C2×C22⋊Q8C2314yes—
101C2×#121—21no—
102C2×#121—21no—
103C2×#121—22no—
104C2×#121C2312yes—
105C2×#121C2312yes—
106C2×#121C2312yes—
107C2×#120—22no—
108C2×C4×Q8—21no—
109C2×C4×Q8—21no—
110C2×C4×Q8—44no—
111C2×C4×Q8C2312yes—
112C2×C4×Q8C2312yes—
113C2×C4×D4—22no—
114C2×C4×D4C2312yes—
115C2×#117—22no—
116C2×#117—44no—
117C2×#117C2314yes—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4D411yescharacteristic
127C22×C22⋊C4D411yescharacteristic
128C22×C42—21no—
129C22×C42—21no—
130C22×C42—21no—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133#482—48no—
134#480—28no—
135#477—44no—
136#460—44no—
137#448—28no—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#106C2314yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2312yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2312yes—
151C2×C22⋊C8C2314yes—
152C2×C22⋊C8C2314yes—
153C2×C4×C8C2314yes—
154#400—22no—
155#397—22no—
156#397—22no—
157#394—22no—
158#394—22no—
159#394—22no—
160#393—22no—
161#392—22no—
162#391—21no—
163#391—22no—
164#391C2314yes—
165#390—21no—
166#390—21no—
167#390C2312yes—
168#390C2312yes—
169#389—21no—
170#389—21no—
171#389—21no—
172#389C2312yes—
173#389C2312yes—
174#389C2312yes—
175#389C2312yes—
176#388—24no—
177#388—24no—
178#388—24no—
179#388—24no—
180#388—22no—
181#388—22no—
182#387—22no—
183#387—24no—
184#387—24no—
185#386—22no—
186#386—22no—
187#386—22no—
188#386—22no—
189#386—22no—
190#385—22no—
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#380—21no—
194#380C2312yes—
195#380C2312yes—
196#380C2312yes—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—24no—
199C4×C4⋊C4—24no—
200C4×C4⋊C4C2314yes—
201C4×C4⋊C4C2312yes—
202C4×C22⋊C4—22no—
203C4×C22⋊C4D412yes—
204C4×C22⋊C4D412yes—
205C4×C22⋊C4C2312yes—
206C4×C22⋊C4C2312yes—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95—21no—
210C2×#95—21no—
211C2×#95D411yescharacteristic
212C2×#95D411yescharacteristic
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#336C2312yes—
220#336C2312yes—
221#336C2314yes—
222C25C2×D411yescharacteristic
223C22×Q8—21no—
224C22×Q8—21no—
225C22×Q8—22no—
226C22×Q8—42no—
227C22×Q8—42no—
228C22×Q8#4114yes—
229C22×Q8C2412yes—
230C22×D4—22no—
231C22×D4—22no—
232C22×D4—22no—
233C22×D4—21no—
234C22×D4—21no—
235C22×D4C2412yes—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—22no—
240C23×C4—22no—
241C23×C4—21no—
242C23×C4—22no—
243C23×C4—42no—
244C23×C4—42no—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D412yes—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2412yes—
252C23×C4C2411yescharacteristic
253C2×SD16—28no—
254C2×SD16—44no—
255C2×SD16—44no—
256C22×C8C2414yes—
257C22×C8C2414yes—
258C4⋊Q8—42no—
259#127—42no—
260#126—48no—
261#124—48no—
262#123—48no—
263C22⋊Q8—216no—
264C22⋊Q8—28no—
265C22⋊Q8—28no—
266C22⋊Q8—44no—
267#121—24no—
268#121—24no—
269#121—48no—
270#121—44no—
271#121—44no—
272#121—44no—
273#121—44no—
274#120—48no—
275C4×Q8—24no—
276C4×Q8—24no—
277C4×Q8—42no—
278C4×Q8—42no—
279C4×Q8—44no—
280C4×Q8—44no—
281C4×D4—44no—
282C4×D4—48no—
283#117—28no—
284#117—44no—
285#117—48no—
286C2×C4⋊C4—28no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—42no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4#4114yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
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—22no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C22⋊C4C2×D412yes—
334C2×C22⋊C4C2×D412yes—
335C2×C42—24no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—24no—
340C2×C42—42no—
341C2×C42—42no—
342C2×C42—44no—
343C2×C42—44no—
344C2×C42—44no—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D412yes—
348C2×C42C2412yes—
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351#107—28no—
352#107—28no—
353#106—44no—
354C4⋊C8—42no—
355C4⋊C8—42no—
356#103—28no—
357#103—44no—
358#103—44no—
359#102—28no—
360#102—28no—
361#102—44no—
362C22⋊C8—24no—
363C22⋊C8—24no—
364C22⋊C8—44no—
365C22⋊C8—44no—
366C4×C8—44no—
367#95—24no—
368#95—24no—
369#95—22no—
370#95—22no—
371#95—24no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95C2×D412yes—
379#95C2×D412yes—
380#95C2×D412yes—
381#95C2×D412yes—
382C24—22no—
383C24—22no—
384C24—22no—
385C24—22no—
386C24—41no—
387C24—41no—
388C24—41no—
389C24—41no—
390C24C22×D412yes—
391C24C22×D411yescharacteristic
392C2×Q8—24no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—48no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8—48no—
402C2×Q8—42no—
403C2×Q8—81no—
404C2×Q8—81no—
405C2×Q8C2×#4114yes—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—24no—
409C2×D4—24no—
410C2×D4—42no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—42no—
417C2×D4—81no—
418C2×D4—81no—
419C2×D4—81no—
420C2×D4—81no—
421C2×D4—82no—
422C2×D4—82no—
423C2×D4—82no—
424C2×D4—82no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—44no—
469C22×C4—82no—
470C22×C4—82no—
471C22×C4#13514yes—
472C22×C4AGL(1,ℤ/8)14yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D412yes—
481C22×C4C2×#4112yes—
482C22×C4C2×#4112yes—
483C22×C4C2511yescharacteristic
484SD16—88no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—24no—
490C2×C8—24no—
491C4⋊C4—24no—
492C4⋊C4—216no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—44no—
511C4⋊C4—81no—
512C4⋊C4—81no—
513C4⋊C4C2×#4114yes—
514C4⋊C4H(5,𝔽2)14yes—
515C22⋊C4—216no—
516C22⋊C4—24no—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—48no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—82no—
529C22⋊C4—82no—
530C22⋊C4—82no—
531C22⋊C4—82no—
532C42—22no—
533C42—22no—
534C42—24no—
535C42—42no—
536C42—41no—
537C42—41no—
538C42—48no—
539C42—44no—
540C42—44no—
541C42—48no—
542C42—44no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—82no—
565C23—82no—
566C23—82no—
567C23—82no—
568C23#52512yes—
569C23C2×#13512yes—
570C23C2×AGL(1,ℤ/8)12yes—
571C23C23×D411yescharacteristic
572Q8—24no—
573Q8—24no—
574Q8—48no—
575Q8—42no—
576Q8—42no—
577Q8—162no—
578D4—44no—
579D4—44no—
580D4—88no—
581D4—162no—
582D4—162no—
583D4—164no—
584D4—164no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—48no—
613C2×C4—48no—
614C2×C4—44no—
615C2×C4—48no—
616C2×C4—42no—
617C2×C4—48no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—82no—
638C2×C4—82no—
639C2×C4—84no—
640C2×C4—164no—
641C2×C4#52512yes—
642C2×C4C2×#13512yes—
643C2×C4C2×#13512yes—
644C2×C4C2×AGL(1,ℤ/8)14yes—
645C2×C4C22×#4111yescharacteristic
646C2×C4C22×#4111yescharacteristic
647C2×C4C2×H(5,𝔽2)12yes—
648C2×C4#58512yes—
649C8—44no—
650C8—44no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—88no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22#310614yes—
673C22C2×#52511yescharacteristic
674C22#348411yescharacteristic
675C22#348511yescharacteristic
676C22C22×#13511yescharacteristic
677C22C22×AGL(1,ℤ/8)11yescharacteristic
678C22#358611yescharacteristic
679C22#358611yescharacteristic
680C4—24no—
681C4—22no—
682C4—22no—
683C4—22no—
684C4—42no—
685C4—42no—
686C4—44no—
687C4—44no—
688C4—41no—
689C4—41no—
690C4—44no—
691C4—82no—
692C4—82no—
693C4—82no—
694C4—82no—
695C4#346012yes—
696C4#350512yes—
697C2—24no—
698C2—82no—
699C2—162no—
700C2#3509111yescharacteristic
701C2C2×#310611yescharacteristic
702C2#6056611yescharacteristic
703C2#6156411yescharacteristic
704C2#6158511yescharacteristic
705C2#6170111yescharacteristic
706C2#6170411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35005C2
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
2. C2×#2362C2semidirect
3. #32981C2semidirect
4. #32957C2semidirect
5. #32903C2semidirect
6. #29968C2semidirect
7. #29754C2semidirect
8. #29595C2semidirect
9. #27103C2semidirect
10. #26935C2semidirect
11. #26793C2semidirect
12. #26665C2semidirect
13. #22041C2semidirect
14. #21902C2semidirect
15. C4×#480C2semidirect
22. C2×#477C22semidirect
23. C2×#460C22semidirect
50. #2100C22semidirect
51. #2088C22semidirect
52. #2083C22semidirect
53. #2067C22semidirect
54. #2058C22semidirect
55. #2041C22semidirect
60. #1968C22semidirect
61. #1933C22semidirect
62. #1927C22semidirect
63. #1925C22semidirect
66. #1918C22semidirect
67. #1880C22semidirect
68. #1877C22semidirect
72. #1863C22semidirect
78. #1803C22semidirect
79. #1803C22semidirect
84. C4×#107C22semidirect
85. C4×#102C22semidirect

Maximal subgroups

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

subgroupindex
#350052
C2×#23622
#329812
#329572
#329032
#299682
#297542
#295952
#271032
#269352
#267932
#266652
#220412
#219022
C4×#4802