Finite Group Databasebeta

← #6856743

Subgroups — #6856743

Order 512.

1549 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36593C211yesmaximal, characteristic, complemented
2#34941C211yesmaximal, characteristic
3#33162C211yesmaximal, characteristic
4#32100C214yesmaximal, complemented
5#30894C214yesmaximal
6#29798C212yesmaximal
7#29257C212yesmaximal, complemented
8#28755C212yesmaximal, complemented
9#28748C211yesmaximal, characteristic, complemented
10#28416C212yesmaximal
11#28284C211yesmaximal, characteristic, complemented
12#27777C212yesmaximal, complemented
13#25651C212yesmaximal, complemented
14#24187C212yesmaximal
15#22825C211yesmaximal, characteristic, complemented
16#22152C212yesmaximal
17C4×#461C211yesmaximal, characteristic
18C2×#557C2212yes—
19C2×#491C2214yes—
20C2×#487C2212yes—
21C2×#477C2212yes—
22C2×#461C2211yescharacteristic
23C2×#461C2212yes—
24C2×#461C2212yes—
25C2×#461C2212yes—
26C2×C4×SD16C2212yes—
27#2868C2211yescharacteristic
28#2868C2212yes—
29#2574—22no—
30#2567C2211yescharacteristic
31#2563C2211yescharacteristic
32#2552—22no—
33#2552—22no—
34#2552—21no—
35#2552—21no—
36#2552C2211yescharacteristic
37#2546C2212yes—
38#2546C2214yes—
39#2538C2211yescharacteristic
40#2508C2212yes—
41#2497—21no—
42#2454C2212yes—
43#2423—21no—
44#2396—22no—
45Q8×C4⋊C4—21no—
46Q8×C4⋊C4—21no—
47C4×C4⋊Q8C2211yescharacteristic
48C4×C4⋊Q8C2212yes—
49C4×C4⋊Q8C2211yescharacteristic
50C4×#125C2212yes—
51C4×C22⋊Q8—21no—
52#2293C2212yes—
53#2285—21no—
54C42×Q8C2211yescharacteristic
55C42×D4C2211yescharacteristic
56#2097C2214yes—
57#2068C2214yes—
58#2043C2214yes—
59#2043C2214yes—
60#2043C2214yes—
61#2043C2214yes—
62#2043C2214yes—
63#2026C2214yes—
64#2013C2214yes—
65#2013C2214yes—
66#1984C2212yes—
67#1981C2212yes—
68#1981C2212yes—
69#1981C2212yes—
70#1981C2212yes—
71#1981C2211yescharacteristic
72#1981C2212yes—
73#1981C2212yes—
74#1973C2214yes—
75#1960C2214yes—
76#1956C2212yes—
77#1948C2211yescharacteristic
78#1947C2212yes—
79#1935C2214yes—
80#1927C2212yes—
81#1917C2214yes—
82#1880C2214yes—
83#1877C2212yes—
84#1865C2214yes—
85#1863C2214yes—
86#1863C2214yes—
87#1842C2211yescharacteristic
88#1831C2212yes—
89#1803C2212yes—
90#1802C2211yescharacteristic
91#1770C2212yes—
92#1767C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C22×SD16C2312yes—
96C22×SD16C2314yes—
97#557—44no—
98#557—44no—
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2314yes—
103C2×C4⋊Q8C2314yes—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2311yescharacteristic
106C2×#125—22no—
107C2×#125—24no—
108C2×#125C2312yes—
109C2×C22⋊Q8—21no—
110C2×C4×Q8—22no—
111C2×C4×Q8—21no—
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2314yes—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×D4C2311yescharacteristic
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×#117—22no—
121C22×C42—21no—
122#491—48no—
123#487—44no—
124#477—44no—
125#461—24no—
126#461—28no—
127#461—44no—
128#461—44no—
129C4×SD16—44no—
130C2×#107C2314yes—
131C2×#106C2312yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2312yes—
137C2×#103C2312yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2312yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×#102C2312yes—
150C2×#97C2314yes—
151C2×C4×C8C2312yes—
152#400—22no—
153#399—22no—
154#398—22no—
155#398—22no—
156#398—24no—
157#398—24no—
158#398—24no—
159#396—22no—
160#396—22no—
161#395—22no—
162#393—22no—
163#391—21no—
164#391—21no—
165#391C2312yes—
166#391C2314yes—
167#390C2312yes—
168#390C2312yes—
169#389—22no—
170#389—22no—
171#389—24no—
172#389—24no—
173#389—21no—
174#389—21no—
175#389—22no—
176#389—21no—
177#389—21no—
178#389—22no—
179#389C2312yes—
180#389C2312yes—
181#389C2312yes—
182#389C2312yes—
183#389C2312yes—
184#388—22no—
185#388—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#385—22no—
190#384—21no—
191#384—21no—
192#384—22no—
193#384—21no—
194#384—22no—
195#384—21no—
196#384—22no—
197#384—21no—
198#384—21no—
199#384—22no—
200#384C2311yescharacteristic
201#384C2311yescharacteristic
202#382—22no—
203#382C2312yes—
204#380—21no—
205#380—21no—
206#380—21no—
207#380—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4D412yes—
216C4×C4⋊C4D412yes—
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4D411yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2312yes—
225C4×C22⋊C4—21no—
226C4×C22⋊C4—21no—
227C4×C22⋊C4—22no—
228C4×C22⋊C4—21no—
229C43C2312yes—
230C43C2311yescharacteristic
231#340C2312yes—
232#340C2314yes—
233#340C2312yes—
234#336C2314yes—
235#336C2314yes—
236#336C2314yes—
237#336C2312yes—
238C22×Q8—21no—
239C22×Q8C2×D411yescharacteristic
240C22×Q8C2412yes—
241C22×Q8C2414yes—
242C22×D4C2411yescharacteristic
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C2×SD16—24no—
247C2×SD16—28no—
248C2×SD16—42no—
249C2×SD16—42no—
250C2×SD16—44no—
251C2×SD16—44no—
252C22×C8C2412yes—
253C22×C8C2414yes—
254C22×C8C2412yes—
255C4⋊Q8—22no—
256C4⋊Q8—24no—
257C4⋊Q8—22no—
258C4⋊Q8—22no—
259C4⋊Q8—24no—
260C4⋊Q8—24no—
261C4⋊Q8—24no—
262C4⋊Q8—22no—
263C4⋊Q8—44no—
264C4⋊Q8—44no—
265C4⋊Q8—42no—
266C4⋊Q8—44no—
267C4⋊Q8—42no—
268C4⋊Q8—44no—
269#125—24no—
270#125—42no—
271#125—416no—
272#125—48no—
273C22⋊Q8—44no—
274C4×Q8—24no—
275C4×Q8—24no—
276C4×Q8—22no—
277C4×Q8—22no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—44no—
281C4×Q8—44no—
282C4×Q8—48no—
283C4×Q8—82no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—22no—
287C4×D4—22no—
288C4×D4—41no—
289C4×D4#4114yes—
290#117—84no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C42—22no—
337C2×C42—24no—
338C2×C42—22no—
339C2×C42—21no—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42—44no—
343C2×C42C2×D412yes—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D412yes—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D412yes—
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D412yes—
350C2×C42C2×D411yescharacteristic
351C2×C42C2411yescharacteristic
352C2×C42C2412yes—
353C2×C42C2412yes—
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356#107—44no—
357#106—42no—
358C4⋊C8—22no—
359C4⋊C8—24no—
360C4⋊C8—24no—
361C4⋊C8—22no—
362C4⋊C8—42no—
363#103—28no—
364#103—24no—
365#103—28no—
366#103—24no—
367#103—44no—
368#103—42no—
369#103—44no—
370#103—44no—
371#103—44no—
372#103—44no—
373#103—44no—
374#103—44no—
375#102—42no—
376#97—44no—
377C4×C8—24no—
378#95—21no—
379#95—21no—
380#95—21no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—24no—
389#95—22no—
390#95—24no—
391#95—22no—
392#95—22no—
393#95C2×D412yes—
394#95C2×D412yes—
395#95C2×D412yes—
396C24—21no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—24no—
401C2×Q8—22no—
402C2×Q8—24no—
403C2×Q8—22no—
404C2×Q8—24no—
405C2×Q8—22no—
406C2×Q8—24no—
407C2×Q8—22no—
408C2×Q8—24no—
409C2×Q8—24no—
410C2×Q8—22no—
411C2×Q8—42no—
412C2×Q8—81no—
413C2×Q8—81no—
414C2×Q8#13714yes—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4C2×#4112yes—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—24no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—81no—
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C2511yescharacteristic
449SD16—48no—
450SD16—88no—
451C2×C8—22no—
452C2×C8—24no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—24no—
457C2×C8—24no—
458C2×C8—22no—
459C2×C8—22no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—48no—
480C4⋊C4—48no—
481C4⋊C4—48no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—82no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4—81no—
497C4⋊C4—82no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4—84no—
501C4⋊C4—84no—
502C4⋊C4C2×#4112yes—
503C4⋊C4H(5,𝔽2)12yes—
504C22⋊C4—24no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—44no—
508C22⋊C4—82no—
509C22⋊C4—82no—
510C22⋊C4—82no—
511C42—24no—
512C42—22no—
513C42—24no—
514C42—24no—
515C42—22no—
516C42—22no—
517C42—21no—
518C42—22no—
519C42—22no—
520C42—21no—
521C42—22no—
522C42—41no—
523C42—42no—
524C42—48no—
525C42—44no—
526C42—42no—
527C42—42no—
528C42—42no—
529C42—44no—
530C42#13714yes—
531C42C2×#4112yes—
532C42H(5,𝔽2)12yes—
533C42H(5,𝔽2)12yes—
534C422−1+412yes—
535C23—22no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23C23×D411yescharacteristic
543Q8—28no—
544Q8—42no—
545Q8—44no—
546Q8—44no—
547Q8—42no—
548Q8—42no—
549Q8—44no—
550Q8—42no—
551Q8—44no—
552Q8—84no—
553D4—22no—
554D4—22no—
555D4—41no—
556D4—41no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—48no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—84no—
603C2×C4—84no—
604C2×C4#53512yes—
605C2×C4#53512yes—
606C2×C4#54812yes—
607C2×C4#54812yes—
608C2×C4C2×#13714yes—
609C2×C4C2×#13712yes—
610C2×C4C2×#13712yes—
611C2×C4C2×#13712yes—
612C2×C4C2×#13712yes—
613C2×C4#57712yes—
614C2×C4C22×#4111yescharacteristic
615C2×C4C2×H(5,𝔽2)11yescharacteristic
616C2×C4C2×H(5,𝔽2)11yescharacteristic
617C2×C4C2×H(5,𝔽2)11yescharacteristic
618C2×C4C2×2−1+411yescharacteristic
619C2×C4#58511yescharacteristic
620C8—42no—
621C8—44no—
622C8—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22C2×#53511yescharacteristic
631C22C2×#54811yescharacteristic
632C22#348811yescharacteristic
633C22C22×#13711yescharacteristic
634C22C22×#13711yescharacteristic
635C22C2×#57711yescharacteristic
636C22#358611yescharacteristic
637C4—22no—
638C4—24no—
639C4—22no—
640C4—22no—
641C4—42no—
642C4—42no—
643C4—41no—
644C4—42no—
645C4—42no—
646C4—42no—
647C4—82no—
648C4—84no—
649C4—82no—
650C4—81no—
651C4—82no—
652C4—82no—
653C4#321814yes—
654C4#337214yes—
655C4#350611yescharacteristic
656C4#350811yescharacteristic
657C4#354811yescharacteristic
658C4#355411yescharacteristic
659C2—82no—
660C2#3972511yescharacteristic
661C2C2×#321811yescharacteristic
662C2#6083611yescharacteristic
663C2C2×#337211yescharacteristic
664C2#6132211yescharacteristic
665C2#6196011yescharacteristic
666C2#6200211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36593C2semidirect
4. #32100C2semidirect
7. #29257C2semidirect
8. #28755C2semidirect
9. #28748C2semidirect
11. #28284C2semidirect
12. #27777C2semidirect
13. #25651C2semidirect
15. #22825C2semidirect

Maximal subgroups

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

subgroupindex
#365932
#349412
#331622
#321002
#308942
#297982
#292572
#287552
#287482
#284162
#282842
#277772
#256512
#241872
#228252
#221522
C4×#4612