Finite Group Databasebeta

← #6849045

Subgroups — #6849045

Order 512.

1657 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36180C211yesmaximal, characteristic, complemented
2#34932C211yesmaximal, characteristic
3#33197C211yesmaximal, characteristic, complemented
4#31758C214yesmaximal, complemented
5#31136C214yesmaximal, complemented
6#29692C212yesmaximal, complemented
7#29141C211yesmaximal, characteristic, complemented
8#28800C212yesmaximal, complemented
9#28284C211yesmaximal, characteristic, complemented
10#27756C212yesmaximal, complemented
11#27647C212yesmaximal, complemented
12#26234C212yesmaximal, complemented
13#24965C212yesmaximal
14#24711C212yesmaximal, complemented
15#22799C211yesmaximal, characteristic, complemented
16#21867C212yesmaximal, complemented
17C4×#476C211yesmaximal, characteristic, complemented
18C2×#554—21no—
19C2×#529C2212yes—
20C2×#483C2214yes—
21C2×#477C2212yes—
22C2×#476C2211yescharacteristic
23C2×#476C2212yes—
24C2×#432C2212yes—
25#2868C2211yescharacteristic
26#2857C2212yes—
27#2567—21no—
28#2553—22no—
29#2552—21no—
30#2552C2212yes—
31#2552C2211yescharacteristic
32#2547C2211yescharacteristic
33#2542C2211yescharacteristic
34#2537C2211yescharacteristic
35#2502—22no—
36#2496—21no—
37#2489C2212yes—
38#2440—22no—
39#2440C2214yes—
40#2427C2212yes—
41#2422—21no—
42#2395—22no—
43D4×C4⋊C4—21no—
44D4×C4⋊C4C2212yes—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×C22⋊Q8C2212yes—
48C4×#121—21no—
49#2285—21no—
50C2×#384C2212yes—
51C42×D4C2211yescharacteristic
52C42×D4C2211yescharacteristic
53#2084C2214yes—
54#2075C2214yes—
55#2072C2214yes—
56#2068C2214yes—
57#2059C2214yes—
58#2042C2214yes—
59#2035C2214yes—
60#2035C2214yes—
61#2033C2214yes—
62#2024C2214yes—
63#1992C2214yes—
64#1987C2212yes—
65#1987C2211yescharacteristic
66#1981C2212yes—
67#1959C2214yes—
68#1956C2212yes—
69#1954C2212yes—
70#1948C2211yescharacteristic
71#1948C2212yes—
72#1948C2212yes—
73#1947C2212yes—
74#1944C2214yes—
75#1935C2214yes—
76#1924C2214yes—
77#1920C2212yes—
78#1874C2214yes—
79#1872C2214yes—
80#1864C2212yes—
81#1842C2211yescharacteristic
82#1834C2212yes—
83#1823C2214yes—
84#1803C2212yes—
85#1803C2212yes—
86#1803C2212yes—
87#1802C2211yescharacteristic
88#1802C2212yes—
89#1802C2212yes—
90#1797C2212yes—
91C4×#106C2212yes—
92#1770C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95#554—48no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101#529—28no—
102#529—44no—
103C2×#126—22no—
104C2×#125—22no—
105C2×#125C2312yes—
106C2×#124C2312yes—
107C2×#123—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—24no—
110C2×C22⋊Q8C2312yes—
111C2×C22⋊Q8C2312yes—
112C2×#121—21no—
113C2×#121C2312yes—
114C2×C4×Q8—21no—
115C2×C4×Q8C2312yes—
116C2×C4×D4—22no—
117C2×C4×D4—22no—
118C2×C4×D4D411yescharacteristic
119C2×C4×D4D411yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×#117—22no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4C2314yes—
128C22×C42—21no—
129C22×C42C2312yes—
130#483—48no—
131#477—44no—
132#476—24no—
133#476—44no—
134#432—44no—
135C2×#107C2314yes—
136C2×#106C2312yes—
137C2×#106C2312yes—
138C2×#106C2314yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2311yescharacteristic
141C2×C4⋊C8C2311yescharacteristic
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2312yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#102C2312yes—
149C2×#102C2314yes—
150C2×#102C2312yes—
151C2×C22⋊C8C2314yes—
152C2×C22⋊C8C2314yes—
153C2×#97C2314yes—
154C2×C4×C8C2312yes—
155#400—22no—
156#399—22no—
157#398—22no—
158#398—24no—
159#398—24no—
160#397—22no—
161#397—22no—
162#397—22no—
163#395—24no—
164#394—22no—
165#394—22no—
166#391—21no—
167#391—22no—
168#391C2312yes—
169#391C2312yes—
170#390—21no—
171#389—22no—
172#389—22no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—22no—
178#389C2312yes—
179#389C2314yes—
180#389C2312yes—
181#388—22no—
182#387—22no—
183#386—22no—
184#385—22no—
185#385—24no—
186#385—24no—
187#384—21no—
188#384—21no—
189#384—22no—
190#384—21no—
191#384—21no—
192#384—22no—
193#384C2311yescharacteristic
194#384C2311yescharacteristic
195#384C2312yes—
196#384C2312yes—
197#384C2314yes—
198#382—22no—
199#380—22no—
200#380—21no—
201#380—21no—
202#380—21no—
203#380—22no—
204#380C2312yes—
205#380C2314yes—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C22⋊C4—22no—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—21no—
221C4×C22⋊C4D412yes—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4C2312yes—
224C43C2311yescharacteristic
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#340C2312yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8C2412yes—
239C22×D4—22no—
240C22×D4C2×D411yescharacteristic
241C22×D4C2411yescharacteristic
242C23×C4—21no—
243C23×C4—24no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4C2×D412yes—
247C23×C4C2×D412yes—
248C23×C4C2412yes—
249C22×C8C2412yes—
250C22×C8C2412yes—
251C22×C8C2414yes—
252C4⋊Q8—22no—
253C4⋊Q8—22no—
254C4⋊Q8—22no—
255C4⋊Q8—22no—
256C4⋊Q8—42no—
257C4⋊Q8—42no—
258C4⋊Q8—44no—
259C4⋊Q8—82no—
260#126—48no—
261#125—24no—
262#125—42no—
263#125—48no—
264#124—24no—
265#124—42no—
266#123—48no—
267C22⋊Q8—24no—
268C22⋊Q8—24no—
269C22⋊Q8—416no—
270C22⋊Q8—42no—
271C22⋊Q8—42no—
272C22⋊Q8—48no—
273#121—24no—
274#121—42no—
275#121—44no—
276C4×Q8—24no—
277C4×Q8—24no—
278C4×Q8—44no—
279C4×D4—24no—
280C4×D4—22no—
281C4×D4—24no—
282C4×D4—22no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—22no—
287C4×D4—41no—
288C4×D4—48no—
289C4×D4—48no—
290C4×D4#4114yes—
291#117—48no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—48no—
318C2×C4⋊C4—84no—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2414yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C22⋊C4—21no—
330C2×C22⋊C4—24no—
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—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C22⋊C4C2×D412yes—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42—44no—
350C2×C42C2×D412yes—
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D412yes—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2411yescharacteristic
357C2×C42C2412yes—
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360#107—44no—
361#106—24no—
362#106—28no—
363#106—24no—
364C4⋊C8—22no—
365C4⋊C8—22no—
366C4⋊C8—24no—
367C4⋊C8—24no—
368C4⋊C8—42no—
369#103—44no—
370#103—44no—
371#103—44no—
372#103—44no—
373#102—24no—
374#102—42no—
375#102—44no—
376C22⋊C8—44no—
377C22⋊C8—44no—
378#97—44no—
379C4×C8—24no—
380#95—21no—
381#95—21no—
382#95—21no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—24no—
392#95—24no—
393#95C2×D412yes—
394#95C2×D412yes—
395#95C2×D412yes—
396C24—21no—
397C24C22×D412yes—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—44no—
407C2×Q8—42no—
408C2×Q8—82no—
409C2×Q8—82no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—82no—
420C2×D4AGL(1,ℤ/8)14yes—
421C2×D4C2×#4112yes—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—24no—
426C22×C4—24no—
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—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—48no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—81no—
452C22×C4—84no—
453C22×C4—84no—
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C2511yescharacteristic
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8—22no—
468C2×C8—24no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—24no—
472C2×C8—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—48no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—48no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—81no—
506C4⋊C4—82no—
507C4⋊C4—84no—
508C4⋊C4—84no—
509C4⋊C4—82no—
510C4⋊C4—82no—
511C4⋊C4C2×#4112yes—
512C4⋊C4H(5,𝔽2)12yes—
513C22⋊C4—24no—
514C22⋊C4—24no—
515C22⋊C4—24no—
516C22⋊C4—24no—
517C22⋊C4—42no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—48no—
524C22⋊C4—44no—
525C22⋊C4—42no—
526C42—24no—
527C42—24no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—21no—
533C42—21no—
534C42—41no—
535C42—44no—
536C42—44no—
537C42—42no—
538C42—42no—
539C42—42no—
540C42#13714yes—
541C42C2×#4112yes—
542C42H(5,𝔽2)12yes—
543C422−1+412yes—
544C422−1+412yes—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23C2×AGL(1,ℤ/8)14yes—
559C23C23×D411yescharacteristic
560Q8—42no—
561Q8—42no—
562Q8—42no—
563Q8—42no—
564Q8—84no—
565Q8—164no—
566D4—28no—
567D4—22no—
568D4—22no—
569D4—41no—
570D4—41no—
571D4—88no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—48no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—48no—
598C2×C4—48no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—84no—
622C2×C4—84no—
623C2×C4—168no—
624C2×C4#53612yes—
625C2×C4#53612yes—
626C2×C4#54612yes—
627C2×C4#54612yes—
628C2×C4C2×AGL(1,ℤ/8)12yes—
629C2×C4C2×#13712yes—
630C2×C4C2×#13712yes—
631C2×C4C2×#13712yes—
632C2×C4#57712yes—
633C2×C4C22×#4111yescharacteristic
634C2×C4C2×H(5,𝔽2)11yescharacteristic
635C2×C4C2×H(5,𝔽2)11yescharacteristic
636C2×C4C2×2−1+411yescharacteristic
637C2×C4C2×2−1+411yescharacteristic
638C2×C4#58511yescharacteristic
639C8—42no—
640C8—42no—
641C8—44no—
642C22—24no—
643C22—24no—
644C22—24no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22C2×#53611yescharacteristic
657C22C2×#54611yescharacteristic
658C22#348811yescharacteristic
659C22C22×AGL(1,ℤ/8)11yescharacteristic
660C22C22×#13711yescharacteristic
661C22C2×#57711yescharacteristic
662C22#358611yescharacteristic
663C4—22no—
664C4—22no—
665C4—22no—
666C4—42no—
667C4—41no—
668C4—42no—
669C4—42no—
670C4—42no—
671C4—42no—
672C4—82no—
673C4—84no—
674C4—82no—
675C4—81no—
676C4—82no—
677C4—82no—
678C4#323814yes—
679C4#333114yes—
680C4#350711yescharacteristic
681C4#351111yescharacteristic
682C4#354711yescharacteristic
683C4#354811yescharacteristic
684C2—44no—
685C2—82no—
686C2#3947011yescharacteristic
687C2C2×#323811yescharacteristic
688C2#6088911yescharacteristic
689C2C2×#333111yescharacteristic
690C2#6116111yescharacteristic
691C2#6196011yescharacteristic
692C2#6198311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2537C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36180C2semidirect
3. #33197C2semidirect
4. #31758C2semidirect
5. #31136C2semidirect
6. #29692C2semidirect
7. #29141C2semidirect
8. #28800C2semidirect
9. #28284C2semidirect
10. #27756C2semidirect
11. #27647C2semidirect
12. #26234C2semidirect
14. #24711C2semidirect
15. #22799C2semidirect
16. #21867C2semidirect
17. C4×#476C2semidirect

Maximal subgroups

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

subgroupindex
#361802
#349322
#331972
#317582
#311362
#296922
#291412
#288002
#282842
#277562
#276472
#262342
#249652
#247112
#227992
#218672
C4×#4762