Finite Group Databasebeta

← #6852258

Subgroups — #6852258

Order 512.

1747 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#36551C211yesmaximal, characteristic, complemented
2C2×#2563C211yesmaximal, characteristic, complemented
3#32956C211yesmaximal, characteristic, complemented
4#32351C214yesmaximal, complemented
5#31527C214yesmaximal, complemented
6#29956C212yesmaximal, complemented
7#28528C212yesmaximal, complemented
8#27952C212yesmaximal, complemented
9#27814C212yesmaximal, complemented
10#27684C212yesmaximal, complemented
11#25813C211yesmaximal, characteristic, complemented
12#25450C212yesmaximal, complemented
13#24812C212yesmaximal, complemented
14#24299C211yesmaximal, characteristic, complemented
15#22625C211yesmaximal, characteristic, complemented
16#22040C212yesmaximal
17C4×#447C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C2×#500C2212yescomplemented
21C2×#476C2212yes—
22C2×#476C2212yescomplemented
23C2×#447C2211yescharacteristic
24C2×C4×D8C2212yes—
25#2857C2211yescharacteristic
26#2851C2212yes—
27#2589—22no—
28#2563—22no—
29#2563—21no—
30#2563—21no—
31#2563C2214yes—
32#2563C2212yes—
33#2563C2212yes—
34#2563C2211yescharacteristic, complemented
35#2552C2211yescharacteristic, complemented
36#2509C2212yescomplemented
37#2497—21no—
38#2487—21no—
39#2454C2212yescomplemented
40#2450—22no—
41C2×#393—22no—
42#2369C2211yescharacteristic, complemented
43D4×C4⋊C4—21no—
44D4×C4⋊C4C2211yescharacteristic, complemented
45#2345—21no—
46C4×#121C2212yes—
47C4×#121C2211yescharacteristic
48C2×#390C2212yes—
49C2×#382C2212yes—
50C2×#380—21no—
51C2×#380—21no—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2101C2214yescomplemented
54#2092C2214yes—
55#2092C2214yescomplemented
56#2085C2214yescomplemented
57#2085C2214yescomplemented
58#2067C2214yes—
59#2065C2214yes—
60#2058C2212yescomplemented
61#2057C2214yescomplemented
62#2025C2214yes—
63#2023C2212yescomplemented
64#1991C2214yescomplemented
65#1990C2212yescomplemented
66#1987C2212yescomplemented
67#1970C2214yes—
68#1949C2214yescomplemented
69#1939C2212yescomplemented
70#1934C2214yes—
71#1925C2214yes—
72#1916C2214yescomplemented
73#1879C2214yes—
74#1879C2212yes—
75#1874C2212yes—
76#1874C2211yescharacteristic
77#1864C2214yes—
78#1862C2212yes—
79#1853C2212yes—
80#1853C2212yes—
81#1852C2212yes—
82#1852C2212yes—
83#1852C2211yescharacteristic
84#1846C2214yescomplemented
85#1804C2214yescomplemented
86#1802C2212yes—
87#1797C2212yes—
88#1797C2211yescharacteristic
89#1786C2211yescharacteristic
90#1786C2212yes—
91C8×C4⋊C4C2212yescomplemented
92C4×#102C2212yes—
93C4×C22⋊C8C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C22×D8C2312yes—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—44no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#125C2312yes—
106C2×C22⋊Q8—21no—
107C2×#121C2311yescharacteristic
108C2×C4×D4—21no—
109C2×C4×D4—22no—
110C2×C4×D4—21no—
111C2×C4×D4—21no—
112C2×C4×D4—44no—
113C2×C4×D4C2312yes—
114C2×C4×D4C2311yescharacteristic
115C2×C4×D4C2312yes—
116C2×C4×D4C2312yes—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4D411yescharacteristic
120C22×C4⋊C4D411yescharacteristic
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2311yescharacteristic
123C22×C22⋊C4—21no—
124C22×C22⋊C4—22no—
125C22×C22⋊C4—22no—
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128#500—44no—
129#476—44no—
130#476—44no—
131#447Q818yescomplemented
132C4×D8—44no—
133C2×#107C2314yes—
134C2×#107C2314yes—
135C2×#107C2312yes—
136C2×#106C2314yes—
137C2×#106C2314yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2314yes—
140C2×#102C2314yes—
141C2×#102C2312yes—
142C2×#102C2314yes—
143C2×#102C2312yes—
144C2×#102C2312yes—
145C2×#102C2314yes—
146C2×#102C2312yes—
147C2×C22⋊C8C2311yescharacteristic
148C2×C22⋊C8C2312yes—
149C2×C22⋊C8C2311yescharacteristic
150C2×C4×C8C2312yes—
151C2×C4×C8C2314yes—
152#400—22no—
153#400—22no—
154#399—22no—
155#398—22no—
156#396—22no—
157#393—24no—
158#393—24no—
159#393—24no—
160#393—24no—
161#393—22no—
162#391—21no—
163#391—22no—
164#390—22no—
165#390—22no—
166#390C2314yes—
167#390C2314yes—
168#390C2312yes—
169#390C2311yescharacteristic
170#389—22no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389C2312yes—
175#389C2311yescharacteristic
176#389C2312yes—
177#389C2312yes—
178#389C2312yes—
179#388—21no—
180#388—22no—
181#387—22no—
182#387—22no—
183#387—22no—
184#386—21no—
185#386—22no—
186#385—22no—
187#385—22no—
188#385—22no—
189#385—22no—
190#384—21no—
191#384C2311yescharacteristic
192#382—22no—
193#382—22no—
194#382C2314yes—
195#382C2314yes—
196#382C2312yes—
197#380—21no—
198#380—22no—
199#380—22no—
200#380—22no—
201#380—22no—
202#380—24no—
203#380—24no—
204#380—21no—
205#380—21no—
206#380—21no—
207#380—22no—
208#380C2311yescharacteristic
209#380C2312yes—
210C4×C4⋊C4—21no—
211C4×C4⋊C4D412yes—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—21no—
221C2×#95—22no—
222C2×#95—22no—
223C2×#95D412yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#336C2312yes—
232#336C2314yes—
233#336C2312yes—
234#336C2314yes—
235C25—21no—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—44no—
240C22×Q8—82no—
241C22×Q8—82no—
242C22×D4—21no—
243C22×D4—21no—
244C22×D4—22no—
245C22×D4—42no—
246C22×D4—42no—
247C22×D4C2×Q814yes—
248C22×D4C2411yescharacteristic
249C22×D4C2412yes—
250C23×C4—22no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—22no—
254C23×C4—22no—
255C23×C4—22no—
256C23×C4—44no—
257C23×C4—44no—
258C23×C4—44no—
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D412yes—
262C23×C4C2×D412yes—
263C23×C4C2411yescharacteristic
264C2×D8—42no—
265C2×D8—42no—
266C2×D8C2×Q818yescomplemented
267C22×C8C2414yes—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C4⋊Q8—24no—
271C4⋊Q8—24no—
272C4⋊Q8—24no—
273C4⋊Q8—44no—
274C4⋊Q8—44no—
275C4⋊Q8—44no—
276#125—44no—
277C22⋊Q8—82no—
278#121—22no—
279#121C2×Q814yes—
280C4×D4—24no—
281C4×D4—24no—
282C4×D4—24no—
283C4×D4—42no—
284C4×D4—42no—
285C4×D4—44no—
286C4×D4—44no—
287C4×D4—44no—
288C4×D4—44no—
289C4×D4—84no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—42no—
306C2×C4⋊C4—42no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2414yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C22⋊C4—21no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—44no—
339C2×C22⋊C4—84no—
340C2×C22⋊C4—84no—
341C2×C42—24no—
342C2×C42—21no—
343C2×C42—22no—
344C2×C42—44no—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2411yescharacteristic
350C2×C42C2412yes—
351C2×C42C2412yes—
352C2×C42C2411yescharacteristic
353#107—42no—
354#107—44no—
355#107—44no—
356#106—44no—
357#106—44no—
358C4⋊C8—24no—
359C4⋊C8—44no—
360#102—44no—
361#102—42no—
362#102—42no—
363#102—42no—
364#102—44no—
365#102—44no—
366#102C2×Q818yescomplemented
367C22⋊C8—22no—
368C22⋊C8—24no—
369C22⋊C8C2×Q814yes—
370C4×C8—24no—
371C4×C8—44no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—21no—
377#95—22no—
378#95—21no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—21no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95C2×D412yes—
388#95C2×D412yes—
389#95C2×D412yes—
390#95C2×D412yes—
391C24—24no—
392C24—21no—
393C24—22no—
394C24—42no—
395C24—42no—
396C24—42no—
397C24—42no—
398C24—42no—
399C24—42no—
400C24C22×D411yescharacteristic
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—42no—
404C2×Q8—82no—
405C2×Q8—82no—
406C2×Q8—82no—
407C2×Q8—82no—
408C2×Q8—81no—
409C2×Q8—81no—
410C2×Q8—816no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—21no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—48no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—48no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4C22×Q814yes—
428C2×D4C22×Q812yes—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—41no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—42no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—42no—
464C22×C4—44no—
465C22×C4—82no—
466C22×C4—82no—
467C22×C4—816no—
468C22×C4#13514yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×Q812yes—
479C22×C42−1+412yes—
480C22×C4C2511yescharacteristic
481D8—48no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—22no—
488C2×C8—24no—
489C2×C8—22no—
490C2×C8C22×Q814yes—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C4⋊C4—81no—
517C4⋊C4—82no—
518C4⋊C4—82no—
519C4⋊C4—82no—
520C4⋊C4C22×Q812yes—
521C22⋊C4—24no—
522C22⋊C4—44no—
523C22⋊C4—42no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—42no—
527C22⋊C4—42no—
528C22⋊C4—82no—
529C22⋊C4—81no—
530C22⋊C4—84no—
531C22⋊C4—84no—
532C22⋊C4—84no—
533C22⋊C4—84no—
534C22⋊C4—82no—
535C22⋊C4—82no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—24no—
540C42—24no—
541C42—42no—
542C42—42no—
543C42—42no—
544C42—82no—
545C42H(5,𝔽2)14yes—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—22no—
551C23—24no—
552C23—48no—
553C23—48no—
554C23—42no—
555C23—41no—
556C23—48no—
557C23—42no—
558C23—41no—
559C23—41no—
560C23—42no—
561C23—41no—
562C23—42no—
563C23—41no—
564C23—42no—
565C23—42no—
566C23—41no—
567C23—84no—
568C23—84no—
569C23—84no—
570C23—84no—
571C23D4×Q812yes—
572C23C2×#13512yes—
573C23#57612yes—
574C23C23×D411yescharacteristic
575Q8—88no—
576Q8—88no—
577Q8—162no—
578D4—24no—
579D4—24no—
580D4—24no—
581D4—48no—
582D4—48no—
583D4—42no—
584D4—42no—
585D4—42no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—28no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—21no—
611C2×C4—44no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—44no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—88no—
633C2×C4—84no—
634C2×C4—84no—
635C2×C4—84no—
636C2×C4—88no—
637C2×C4—84no—
638C2×C4—88no—
639C2×C4—164no—
640C2×C4#53512yes—
641C2×C4#53512yes—
642C2×C4D4×Q812yes—
643C2×C4C2×#13512yes—
644C2×C4C2×#13512yes—
645C2×C4#57512yes—
646C2×C4C23×Q811yescharacteristic
647C2×C4C2×H(5,𝔽2)12yes—
648C2×C4C2×2−1+411yescharacteristic
649C2×C4#58512yes—
650C8—24no—
651C8—42no—
652C8—44no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—42no—
660C22—44no—
661C22—44no—
662C22—42no—
663C22—42no—
664C22—44no—
665C22—816no—
666C22—81no—
667C22—82no—
668C22—81no—
669C22—82no—
670C22—82no—
671C22—81no—
672C22—81no—
673C22—82no—
674C22#338014yes—
675C22C2×#53511yescharacteristic
676C22C2×D4×Q811yescharacteristic
677C22#348511yescharacteristic
678C22C22×#13511yescharacteristic
679C22C2×#57511yescharacteristic
680C22C2×#57611yescharacteristic
681C22#358511yescharacteristic
682C4—22no—
683C4—22no—
684C4—22no—
685C4—42no—
686C4—42no—
687C4—42no—
688C4—42no—
689C4—41no—
690C4—42no—
691C4—81no—
692C4—82no—
693C4—81no—
694C4—82no—
695C4—84no—
696C4—82no—
697C4#321114yes—
698C4#350112yes—
699C4#354112yes—
700C2—24no—
701C2—82no—
702C2—84no—
703C2#3959411yescharacteristic
704C2C2×#321111yescharacteristic
705C2#6081111yescharacteristic
706C2C2×#338011yescharacteristic
707C2#6136711yescharacteristic
708C2#6170011yescharacteristic
709C2#6174011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#447C22
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. #36551C2semidirect
2. C2×#2563C2semidirect
3. #32956C2semidirect
4. #32351C2semidirect
5. #31527C2semidirect
6. #29956C2semidirect
7. #28528C2semidirect
8. #27952C2semidirect
9. #27814C2semidirect
10. #27684C2semidirect
11. #25813C2semidirect
12. #25450C2semidirect
13. #24812C2semidirect
14. #24299C2semidirect
15. #22625C2semidirect
20. C2×#500C22semidirect
22. C2×#476C22semidirect
34. #2563C22semidirect
35. #2552C22semidirect
36. #2509C22semidirect
39. #2454C22semidirect
42. #2369C22semidirect
44. D4×C4⋊C4C22semidirect
53. #2101C22semidirect
55. #2092C22semidirect
56. #2085C22semidirect
57. #2085C22semidirect
60. #2058C22semidirect
61. #2057C22semidirect
63. #2023C22semidirect
64. #1991C22semidirect
65. #1990C22semidirect
66. #1987C22semidirect
68. #1949C22semidirect
69. #1939C22semidirect
72. #1916C22semidirect
84. #1846C22semidirect
85. #1804C22semidirect
91. C8×C4⋊C4C22semidirect
131. #447Q8semidirect
266. C2×D8C2×Q8semidirect
366. #102C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#365512
C2×#25632
#329562
#323512
#315272
#299562
#285282
#279522
#278142
#276842
#258132
#254502
#248122
#242992
#226252
#220402
C4×#4472