Finite Group Databasebeta

← #6986815

Subgroups — #6986815

Order 512.

1649 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36505C211yesmaximal, characteristic
2#35316C211yesmaximal, characteristic
3#32787C212yesmaximal, complemented
4#32264C214yesmaximal, complemented
5#32006C214yesmaximal, complemented
6#30294C212yesmaximal, complemented
7#30176C212yesmaximal, complemented
8#29937C211yesmaximal, characteristic, complemented
9#29160C212yesmaximal, complemented
10#28704C212yesmaximal, complemented
11#26283C212yesmaximal, complemented
12#25840C212yesmaximal, complemented
13#25813C211yesmaximal, characteristic, complemented
14#25341C212yesmaximal, complemented
15#24301C211yesmaximal, characteristic, complemented
16#23757C211yesmaximal, characteristic, complemented
17#22469C211yesmaximal, characteristic
18C2×#557C2211yescharacteristic
19C2×#536—21no—
20C22×C4⋊Q8C2211yescharacteristic
21C2×C4×#41C2211yescharacteristic
22C2×#500C2212yes—
23C2×#475C2212yes—
24C2×C8×D4C2212yes—
25#2857C2212yes—
26#2857C2212yes—
27#2857C2211yescharacteristic
28#2851C2212yes—
29#2621—22no—
30#2562C2212yes—
31#2547—21no—
32#2547C2211yescharacteristic
33#2547C2212yes—
34#2546—21no—
35#2546C2212yes—
36#2538C2211yescharacteristic
37#2536C2212yes—
38#2486—22no—
39#2486C2214yes—
40#2475—22no—
41#2460—22no—
42#2446—21no—
43#2432—21no—
44#2367—21no—
45#2362—21no—
46#2355C2211yescharacteristic
47#2310—21no—
48C4×C22⋊Q8C2211yescharacteristic
49C4×C22⋊Q8C2212yes—
50#2285C2211yescharacteristic
51#2101C2214yes—
52#2100C2211yescharacteristic
53#2100C2212yes—
54#2095C2211yescharacteristic
55#2085C2214yes—
56#2084C2212yes—
57#2083C2211yescharacteristic
58#2077C2214yes—
59#2077C2214yes—
60#2076C2212yes—
61#2075C2214yes—
62#2075C2214yes—
63#2072C2211yescharacteristic
64#2059C2214yes—
65#2059C2214yes—
66#2058C2212yes—
67#2057C2212yes—
68#2057C2212yes—
69#2036C2212yes—
70#2035C2214yes—
71#2024C2212yes—
72#1990C2212yes—
73#1988C2212yes—
74#1988C2212yes—
75#1984C2212yes—
76#1948C2212yes—
77#1921C2214yes—
78#1920C2214yes—
79#1892C2214yes—
80#1872C2212yes—
81#1869C2214yes—
82#1863C2214yes—
83#1863C2214yes—
84#1853C2212yes—
85#1852C2214yes—
86#1817C2212yes—
87#1808C2214yes—
88#1808C2214yes—
89#1807C2214yes—
90#1797C2211yescharacteristic
91C4×#107C2212yes—
92C4×#103C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94C2×#336C2212yes—
95C22×#41D411yescharacteristic
96C23×Q8—21no—
97C23×C8C2314yes—
98#557—44no—
99#536—48no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—44no—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2312yes—
108C2×#125—22no—
109C2×#125C2312yes—
110C2×#124—21no—
111C2×#124—22no—
112C2×#123—24no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8C2311yescharacteristic
117C2×C22⋊Q8C2312yes—
118C4×#41—44no—
119C2×C4×Q8—21no—
120C2×C4×Q8D411yescharacteristic
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2312yes—
123C2×C4×Q8C2311yescharacteristic
124C2×C4×D4D411yescharacteristic
125C2×C4×D4D412yes—
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2312yes—
128C2×#117—21no—
129C2×#117D411yescharacteristic
130C2×#117D412yes—
131C22×C4⋊C4—21no—
132C22×C4⋊C4C2312yes—
133C22×C42C2311yescharacteristic
134C22×C42C2312yes—
135#500—44no—
136#475—44no—
137C8×D4—44no—
138C2×#107C2312yes—
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×#107C2312yes—
142C2×#106C2314yes—
143C2×#106C2314yes—
144C2×C4⋊C8C2312yes—
145C2×#103C2312yes—
146C2×#103C2312yes—
147C2×#103C2314yes—
148C2×C22⋊C8C2312yes—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2314yes—
151C2×C4×C8C2312yes—
152#400—22no—
153#398—24no—
154#398—24no—
155#398—22no—
156#397—24no—
157#397—24no—
158#397—22no—
159#396—22no—
160#394—22no—
161#394—22no—
162#391—21no—
163#391—22no—
164#391—22no—
165#391—21no—
166#391—22no—
167#391—22no—
168#391—21no—
169#391C2311yescharacteristic
170#390—21no—
171#389—21no—
172#389—21no—
173#389—22no—
174#389—22no—
175#389C2312yes—
176#389C2311yescharacteristic
177#389C2314yes—
178#389C2314yes—
179#388—22no—
180#388—22no—
181#388—22no—
182#388—22no—
183#387—22no—
184#387—22no—
185#387—22no—
186#387—22no—
187#387—21no—
188#385—22no—
189#385—22no—
190#385—22no—
191#385—21no—
192#384C2311yescharacteristic
193#384C2311yescharacteristic
194#384C2311yescharacteristic
195#382—21no—
196#382C2312yes—
197#381—24no—
198#380—21no—
199#380—22no—
200#380—22no—
201#380—21no—
202#380C2311yescharacteristic
203#380C2312yes—
204#380C2312yes—
205#380C2312yes—
206#380C2314yes—
207#380C2312yes—
208#380C2314yes—
209#380C2312yes—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—22no—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2312yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2314yes—
230#336C2312yes—
231#336C2311yescharacteristic
232#336C2311yescharacteristic
233#336C2314yes—
234#336C2314yes—
235C2×#41—24no—
236C2×#41—24no—
237C2×#41—42no—
238C22×Q8—21no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8—44no—
242C22×Q8—44no—
243C22×Q8—44no—
244C22×Q8C2×D411yescharacteristic
245C22×Q8C2411yescharacteristic
246C22×D4C2×D412yes—
247C22×D4C2×D411yescharacteristic
248C23×C4—22no—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D412yes—
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D412yes—
253C23×C4C2411yescharacteristic
254C23×C4C2412yes—
255C22×C8—28no—
256C22×C8—44no—
257C22×C8—44no—
258C22×C8C2414yes—
259C22×C8C2412yes—
260C22×C8C2412yes—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8—24no—
264C4⋊Q8—44no—
265C4⋊Q8—44no—
266C4⋊Q8—44no—
267C4⋊Q8—48no—
268C4⋊Q8—44no—
269#125—44no—
270#125—48no—
271#124—44no—
272#124—48no—
273#123—416no—
274C22⋊Q8—22no—
275C22⋊Q8—24no—
276C22⋊Q8—42no—
277C22⋊Q8—41no—
278C22⋊Q8—48no—
279C22⋊Q8—48no—
280C22⋊Q8—44no—
281C4×Q8—24no—
282C4×Q8—24no—
283C4×Q8—42no—
284C4×Q8—44no—
285C4×Q8—44no—
286C4×D4—24no—
287C4×D4—42no—
288C4×D4—44no—
289C4×D4—44no—
290#117—44no—
291#117—42no—
292#117—44no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2414yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C42—24no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42—22no—
346C2×C42—44no—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2412yes—
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354#107—24no—
355#107—42no—
356#107—44no—
357#107—44no—
358#106—44no—
359#106—44no—
360C4⋊C8—24no—
361#103—24no—
362#103—24no—
363#103—44no—
364C22⋊C8—24no—
365C22⋊C8—42no—
366C22⋊C8—44no—
367C4×C8—24no—
368#95—21no—
369#95—21no—
370#95—22no—
371#95—21no—
372#95—22no—
373#95—21no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—21no—
378#95—22no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387C24C22×D411yescharacteristic
388C24C22×D412yes—
389#41—416no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—21no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—42no—
399C2×Q8—44no—
400C2×Q8—42no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—42no—
405C2×Q8—816no—
406C2×Q82−1+412yes—
407C2×D4—24no—
408C2×D4—24no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—42no—
412C2×D4AGL(1,ℤ/8)14yes—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—42no—
438C22×C4—41no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4#13514yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C2511yescharacteristic
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—24no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—88no—
463C2×C82−1+414yes—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—48no—
491C4⋊C4—48no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—81no—
501C22⋊C4—24no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—81no—
513C22⋊C4—82no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—24no—
518C42—42no—
519C42—42no—
520C42—44no—
521C42—42no—
522C42—44no—
523C42—42no—
524C42—82no—
525C42H(5,𝔽2)14yes—
526C23—22no—
527C23—21no—
528C23—22no—
529C23—21no—
530C23—21no—
531C23—22no—
532C23—21no—
533C23—22no—
534C23—22no—
535C23—21no—
536C23—21no—
537C23—22no—
538C23C2×#13512yes—
539C23C2×AGL(1,ℤ/8)14yes—
540C23C23×D411yescharacteristic
541Q8—24no—
542Q8—44no—
543Q8—42no—
544Q8—88no—
545Q8—84no—
546Q8—88no—
547D4—28no—
548D4—48no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—28no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—44no—
570C2×C4—48no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—84no—
602C2×C4—88no—
603C2×C4#53412yes—
604C2×C4#53412yes—
605C2×C4C2×#13512yes—
606C2×C4C2×#13512yes—
607C2×C4C2×AGL(1,ℤ/8)12yes—
608C2×C4#57812yes—
609C2×C4C2×H(5,𝔽2)12yes—
610C2×C4C2×2−1+411yescharacteristic
611C2×C4C2×2−1+412yes—
612C2×C4#58511yescharacteristic
613C8—24no—
614C8—42no—
615C8—44no—
616C22—22no—
617C22—24no—
618C22—22no—
619C22—24no—
620C22—22no—
621C22—24no—
622C22—42no—
623C22—41no—
624C22—41no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—41no—
629C22—41no—
630C22C2×#53411yescharacteristic
631C22#348911yescharacteristic
632C22#349911yescharacteristic
633C22C22×#13511yescharacteristic
634C22C22×AGL(1,ℤ/8)11yescharacteristic
635C22C2×#57811yescharacteristic
636C22#358611yescharacteristic
637C4—22no—
638C4—22no—
639C4—22no—
640C4—42no—
641C4—42no—
642C4—41no—
643C4—42no—
644C4—42no—
645C4—81no—
646C4—82no—
647C4—84no—
648C4—82no—
649C4—81no—
650C4—82no—
651C4—81no—
652C4—82no—
653C4#318814yes—
654C4#355212yes—
655C4#356112yes—
656C2—42no—
657C2—44no—
658C2#4291211yescharacteristic
659C2C2×#318811yescharacteristic
660C2#6075911yescharacteristic
661C2#6210111yescharacteristic
662C2#6213211yescharacteristic
663C2#6247311yescharacteristic
664C2#6250311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32787C2semidirect
4. #32264C2semidirect
5. #32006C2semidirect
6. #30294C2semidirect
7. #30176C2semidirect
8. #29937C2semidirect
9. #29160C2semidirect
10. #28704C2semidirect
11. #26283C2semidirect
12. #25840C2semidirect
13. #25813C2semidirect
14. #25341C2semidirect
15. #24301C2semidirect
16. #23757C2semidirect

Maximal subgroups

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

subgroupindex
#365052
#353162
#327872
#322642
#320062
#302942
#301762
#299372
#291602
#287042
#262832
#258402
#258132
#253412
#243012
#237572
#224692