Finite Group Databasebeta

← #6856910

Subgroups — #6856910

Order 512.

1515 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic, complemented
2#34929C211yesmaximal, characteristic
3#33162C211yesmaximal, characteristic
4#31984C214yesmaximal, complemented
5#31165C214yesmaximal
6#29829C212yesmaximal
7#29355C212yesmaximal, complemented
8#29286C211yesmaximal, characteristic, complemented
9#28748C211yesmaximal, characteristic, complemented
10#28646C212yesmaximal, complemented
11#28284C212yesmaximal, complemented
12#26466C212yesmaximal, complemented
13#25320C212yesmaximal
14#25301C212yesmaximal
15#22722C211yesmaximal, characteristic, complemented
16#22152C212yesmaximal
17C4×#474C211yesmaximal, characteristic
18C2×#533—21no—
19C2×#487C2212yes—
20C2×#477C2214yes—
21C2×#474C2211yescharacteristic
22C2×#461C2212yes—
23C2×#461C2212yes—
24C2×#461C2212yes—
25C2×#446C2212yes—
26#2868C2211yescharacteristic
27#2862C2212yes—
28#2582—21no—
29#2571—21no—
30#2570—22no—
31#2568C2212yes—
32#2556—22no—
33#2551C2211yescharacteristic
34#2546C2211yescharacteristic
35#2542C2211yescharacteristic
36#2538C2211yescharacteristic
37#2511—22no—
38#2506C2212yes—
39#2492—21no—
40#2452C2212yes—
41#2446C2214yes—
42#2429—21no—
43#2396—22no—
44#2365C2212yes—
45#2355—21no—
46#2355—21no—
47C4×C4⋊Q8C2211yescharacteristic
48C4×C4⋊Q8C2211yescharacteristic
49C4×C4⋊Q8C2212yes—
50C4×C4⋊Q8C2212yes—
51C4×C22⋊Q8—21no—
52#2293C2212yes—
53C42×Q8C2211yescharacteristic
54C42×D4C2211yescharacteristic
55#2086C2214yes—
56#2072C2214yes—
57#2068C2214yes—
58#2068C2214yes—
59#2068C2214yes—
60#2068C2214yes—
61#2055C2214yes—
62#2055C2214yes—
63#2043C2214yes—
64#2024C2214yes—
65#1988C2212yes—
66#1988C2212yes—
67#1988C2211yescharacteristic
68#1981C2212yes—
69#1981C2212yes—
70#1981C2212yes—
71#1981C2212yes—
72#1981C2212yes—
73#1973C2214yes—
74#1956C2212yes—
75#1947C2212yes—
76#1947C2212yes—
77#1947C2211yescharacteristic
78#1940C2214yes—
79#1939C2212yes—
80#1935C2214yes—
81#1935C2214yes—
82#1935C2214yes—
83#1918C2212yes—
84#1880C2214yes—
85#1863C2214yes—
86#1862C2212yes—
87#1842C2211yescharacteristic
88#1842C2212yes—
89#1803C2214yes—
90#1802C2211yescharacteristic
91C4×#107C2212yes—
92#1767C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95C22×SD16C2314yes—
96#533—48no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#125—22no—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—21no—
110C2×C4×Q8—21no—
111C2×C4×Q8—22no—
112C2×C4×Q8—21no—
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8C2314yes—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×D4C2311yescharacteristic
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×#117—22no—
121C22×C42—21no—
122#487—44no—
123#477—48no—
124#474—24no—
125#461—28no—
126#461—44no—
127#461—44no—
128#446—44no—
129C2×#107C2312yes—
130C2×#107C2312yes—
131C2×#107C2314yes—
132C2×C4⋊C8C2311yescharacteristic
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2314yes—
137C2×C4⋊C8C2314yes—
138C2×C4⋊C8C2312yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#102C2312yes—
147C2×#102C2312yes—
148C2×#97C2314yes—
149C2×C4×C8C2312yes—
150#400—24no—
151#400—24no—
152#400—22no—
153#399—22no—
154#398—24no—
155#398—22no—
156#398—22no—
157#396—22no—
158#396—22no—
159#395—22no—
160#393—22no—
161#391—22no—
162#391—21no—
163#391—21no—
164#391—22no—
165#391—21no—
166#391—22no—
167#391—22no—
168#391C2314yes—
169#391C2314yes—
170#391C2314yes—
171#390C2312yes—
172#390C2312yes—
173#389—22no—
174#389—21no—
175#389—21no—
176#389—22no—
177#389—22no—
178#389—22no—
179#389—21no—
180#389C2312yes—
181#389C2312yes—
182#389C2312yes—
183#389C2312yes—
184#389C2312yes—
185#389C2312yes—
186#388—22no—
187#388—22no—
188#387—24no—
189#387—24no—
190#387—24no—
191#387—22no—
192#385—22no—
193#384—21no—
194#384—21no—
195#384—21no—
196#384—21no—
197#384C2311yescharacteristic
198#384C2312yes—
199#384C2311yescharacteristic
200#382—22no—
201#382—22no—
202#382—22no—
203#380—21no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4D412yes—
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D412yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C22⋊C4—21no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—21no—
224C4×C22⋊C4—21no—
225C43C2312yes—
226C43C2311yescharacteristic
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#340C2314yes—
232#340C2312yes—
233#340C2312yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—21no—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8C2×D411yescharacteristic
240C22×Q8C2414yes—
241C22×D4C2411yescharacteristic
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C2×SD16—28no—
246C2×SD16—44no—
247C2×SD16—44no—
248C22×C8C2414yes—
249C22×C8C2412yes—
250C22×C8C2412yes—
251C4⋊Q8—24no—
252C4⋊Q8—22no—
253C4⋊Q8—24no—
254C4⋊Q8—22no—
255C4⋊Q8—24no—
256C4⋊Q8—22no—
257C4⋊Q8—22no—
258C4⋊Q8—24no—
259C4⋊Q8—24no—
260C4⋊Q8—42no—
261C4⋊Q8—42no—
262C4⋊Q8—42no—
263C4⋊Q8—44no—
264C4⋊Q8—44no—
265#125—48no—
266C22⋊Q8—48no—
267C22⋊Q8—44no—
268C22⋊Q8—44no—
269C22⋊Q8—44no—
270C4×Q8—24no—
271C4×Q8—24no—
272C4×Q8—22no—
273C4×Q8—22no—
274C4×Q8—48no—
275C4×Q8—44no—
276C4×Q8—44no—
277C4×Q8—44no—
278C4×Q8—44no—
279C4×D4—22no—
280C4×D4—22no—
281C4×D4—22no—
282C4×D4—22no—
283C4×D4—41no—
284C4×D4#4114yes—
285#117—48no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
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—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C22⋊C4—21no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C42—22no—
329C2×C42—21no—
330C2×C42—22no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—24no—
334C2×C42—44no—
335C2×C42C2×D412yes—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D412yes—
338C2×C42C2×D412yes—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D412yes—
342C2×C42C2×D411yescharacteristic
343C2×C42C2411yescharacteristic
344C2×C42C2412yes—
345C2×C42C2412yes—
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348#107—24no—
349#107—24no—
350#107—44no—
351C4⋊C8—24no—
352C4⋊C8—22no—
353C4⋊C8—24no—
354C4⋊C8—22no—
355C4⋊C8—44no—
356C4⋊C8—42no—
357C4⋊C8—44no—
358#103—28no—
359#103—28no—
360#103—44no—
361#103—44no—
362#103—44no—
363#103—44no—
364#103—44no—
365#102—24no—
366#102—42no—
367#97—44no—
368C4×C8—24no—
369#95—21no—
370#95—21no—
371#95—21no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—24no—
376#95—24no—
377#95—22no—
378#95—22no—
379#95—24no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95C2×D412yes—
387#95C2×D412yes—
388#95C2×D412yes—
389C24—21no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—24no—
394C2×Q8—24no—
395C2×Q8—24no—
396C2×Q8—24no—
397C2×Q8—24no—
398C2×Q8—24no—
399C2×Q8—44no—
400C2×Q8—44no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—82no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4C2×#4112yes—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—81no—
431C22×C4C22×D412yes—
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C2511yescharacteristic
442SD16—88no—
443C2×C8—22no—
444C2×C8—24no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C4⋊C4—22no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—24no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—44no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—42no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—81no—
488C4⋊C4—84no—
489C4⋊C4—82no—
490C4⋊C4—82no—
491C4⋊C4#13714yes—
492C4⋊C4C2×#4112yes—
493C4⋊C4H(5,𝔽2)12yes—
494C22⋊C4—24no—
495C22⋊C4—42no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—42no—
499C22⋊C4—44no—
500C22⋊C4—82no—
501C42—24no—
502C42—22no—
503C42—24no—
504C42—24no—
505C42—22no—
506C42—22no—
507C42—21no—
508C42—22no—
509C42—22no—
510C42—21no—
511C42—22no—
512C42—41no—
513C42—42no—
514C42—42no—
515C42—42no—
516C42—42no—
517C42—44no—
518C42—44no—
519C42—84no—
520C42AGL(1,ℤ/8)14yes—
521C42C2×#4112yes—
522C42H(5,𝔽2)12yes—
523C422−1+412yes—
524C422−1+412yes—
525C23—22no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23C23×D411yescharacteristic
533Q8—44no—
534Q8—44no—
535Q8—44no—
536Q8—44no—
537Q8—44no—
538Q8—88no—
539Q8—84no—
540Q8—84no—
541D4—22no—
542D4—22no—
543D4—41no—
544D4—41no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—48no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—84no—
591C2×C4—84no—
592C2×C4#53712yes—
593C2×C4#53712yes—
594C2×C4#54812yes—
595C2×C4#54812yes—
596C2×C4C2×AGL(1,ℤ/8)12yes—
597C2×C4C2×AGL(1,ℤ/8)12yes—
598C2×C4C2×AGL(1,ℤ/8)12yes—
599C2×C4C2×#13714yes—
600C2×C4C2×#13712yes—
601C2×C4#57812yes—
602C2×C4C22×#4111yescharacteristic
603C2×C4C2×H(5,𝔽2)11yescharacteristic
604C2×C4C2×H(5,𝔽2)11yescharacteristic
605C2×C4C2×2−1+411yescharacteristic
606C2×C4C2×2−1+411yescharacteristic
607C2×C4#58511yescharacteristic
608C8—42no—
609C8—44no—
610C8—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22C2×#53711yescharacteristic
619C22C2×#54811yescharacteristic
620C22#348811yescharacteristic
621C22C22×AGL(1,ℤ/8)11yescharacteristic
622C22C22×#13711yescharacteristic
623C22C2×#57811yescharacteristic
624C22#358611yescharacteristic
625C4—22no—
626C4—24no—
627C4—22no—
628C4—22no—
629C4—42no—
630C4—42no—
631C4—41no—
632C4—42no—
633C4—42no—
634C4—42no—
635C4—82no—
636C4—82no—
637C4—84no—
638C4—82no—
639C4—81no—
640C4—82no—
641C4#325414yes—
642C4#336714yes—
643C4#350711yescharacteristic
644C4#350811yescharacteristic
645C4#355011yescharacteristic
646C4#355411yescharacteristic
647C2—82no—
648C2#3972611yescharacteristic
649C2C2×#325411yescharacteristic
650C2#6092511yescharacteristic
651C2C2×#336711yescharacteristic
652C2#6132211yescharacteristic
653C2#6196211yescharacteristic
654C2#6198411yescharacteristic
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. #36598C2semidirect
4. #31984C2semidirect
7. #29355C2semidirect
8. #29286C2semidirect
9. #28748C2semidirect
10. #28646C2semidirect
11. #28284C2semidirect
12. #26466C2semidirect
15. #22722C2semidirect

Maximal subgroups

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

subgroupindex
#365982
#349292
#331622
#319842
#311652
#298292
#293552
#292862
#287482
#286462
#282842
#264662
#253202
#253012
#227222
#221522
C4×#4742