Finite Group Databasebeta

← #6986701

Subgroups — #6986701

Order 512.

1701 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36450C211yesmaximal, characteristic, complemented
2#35317C211yesmaximal, characteristic
3#32838C212yesmaximal, complemented
4#32304C214yesmaximal, complemented
5#32076C214yesmaximal, complemented
6#30343C212yesmaximal, complemented
7#30066C212yesmaximal, complemented
8#29687C211yesmaximal, characteristic
9#29213C212yesmaximal, complemented
10#28540C212yesmaximal, complemented
11#27822C212yesmaximal, complemented
12#25813C211yesmaximal, characteristic, complemented
13#25449C212yesmaximal, complemented
14#24678C212yesmaximal, complemented
15#24301C211yesmaximal, characteristic, complemented
16#23570C211yesmaximal, characteristic
17#22643C211yesmaximal, characteristic, complemented
18C2×#536—21no—
19C2×#536—21no—
20C2×#532C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C2×#500C2212yes—
23C2×#475C2212yes—
24C2×#474C2212yes—
25C2×C4×SD16C2212yes—
26#2868C2212yes—
27#2863C2212yes—
28#2858C2212yes—
29#2857C2211yescharacteristic
30#2658—22no—
31#2577—22no—
32#2568C2211yescharacteristic
33#2552C2211yescharacteristic
34#2542—21no—
35#2542—21no—
36#2542C2212yes—
37#2542C2212yes—
38#2528—22no—
39#2514—21no—
40#2501—22no—
41#2501C2214yes—
42#2499—21no—
43#2494C2212yes—
44#2489C2212yes—
45#2365C2211yescharacteristic
46#2362—21no—
47#2355—21no—
48#2355C2211yescharacteristic
49C4×#125C2212yes—
50C4×C22⋊Q8C2211yescharacteristic
51C4×#117C2211yescharacteristic
52#2100C2214yes—
53#2095C2214yes—
54#2092C2214yes—
55#2083C2214yes—
56#2072C2214yes—
57#2067C2214yes—
58#2058C2211yescharacteristic
59#2058C2212yes—
60#2057C2212yes—
61#2057C2212yes—
62#2057C2212yes—
63#2055C2214yes—
64#2041C2211yescharacteristic
65#2041C2212yes—
66#2034C2212yes—
67#2033C2214yes—
68#2025C2214yes—
69#2024C2211yescharacteristic
70#2024C2212yes—
71#2014C2211yescharacteristic
72#2000C2212yes—
73#1992C2214yes—
74#1990C2212yes—
75#1988C2212yes—
76#1973C2214yes—
77#1948C2214yes—
78#1947C2214yes—
79#1939C2212yes—
80#1936C2214yes—
81#1926C2214yes—
82#1925C2214yes—
83#1924C2214yes—
84#1872C2212yes—
85#1853C2212yes—
86#1852C2214yes—
87#1847C2214yes—
88#1839C2212yes—
89#1802C2212yes—
90#1797C2211yescharacteristic
91#1770C2212yes—
92#1767C2212yes—
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C22×#41—21no—
96C23×Q8—21no—
97C22×SD16C2312yes—
98#536—48no—
99#536—48no—
100#532—44no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—44no—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2311yescharacteristic
110C2×#127C2311yescharacteristic
111C2×#125C2312yes—
112C2×#124—22no—
113C2×#124—22no—
114C2×#124—21no—
115C2×#123—24no—
116C2×#123—24no—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8—22no—
119C2×C22⋊Q8—22no—
120C2×C22⋊Q8—22no—
121C2×C22⋊Q8—22no—
122C2×C22⋊Q8C2311yescharacteristic
123C2×#121—21no—
124C2×#121C2312yes—
125C2×C4×Q8—21no—
126C2×C4×Q8C2311yescharacteristic
127C2×C4×D4—21no—
128C2×C4×D4C2312yes—
129C2×#117D411yescharacteristic
130C2×#117D411yescharacteristic
131C22×C4⋊C4—21no—
132C22×C4⋊C4C2312yes—
133C22×C42C2311yescharacteristic
134#500—44no—
135#475—44no—
136#474—44no—
137C4×SD16—44no—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×#106C2314yes—
142C2×#106C2312yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2314yes—
145C2×#103C2314yes—
146C2×#103C2312yes—
147C2×#102C2312yes—
148C2×#102C2312yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2312yes—
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2312yes—
154C2×#97C2314yes—
155C2×C4×C8C2312yes—
156#401—22no—
157#400—22no—
158#400—22no—
159#399—24no—
160#399—24no—
161#399—24no—
162#399—24no—
163#398—22no—
164#397—22no—
165#397—22no—
166#391—21no—
167#391—21no—
168#391—21no—
169#391—21no—
170#391C2312yes—
171#391C2312yes—
172#391C2311yescharacteristic
173#390—22no—
174#390—22no—
175#390C2314yes—
176#390C2314yes—
177#389—21no—
178#389—21no—
179#389C2311yescharacteristic
180#389C2312yes—
181#389C2312yes—
182#389C2312yes—
183#388—22no—
184#388—22no—
185#387—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#387—22no—
190#387—21no—
191#387—22no—
192#387—22no—
193#387—22no—
194#385—21no—
195#384—21no—
196#384—22no—
197#384—22no—
198#384C2311yescharacteristic
199#384C2314yes—
200#384C2314yes—
201#383—22no—
202#382—21no—
203#382—22no—
204#382C2312yes—
205#380—21no—
206#380C2312yes—
207#380C2311yescharacteristic
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C22⋊C4—21no—
220C4×C22⋊C4D412yes—
221C4×C22⋊C4C2312yes—
222#376D412yes—
223C43C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#336C2312yes—
232#336C2311yescharacteristic
233#336C2311yescharacteristic
234#336C2314yes—
235#336C2314yes—
236C2×#41—44no—
237C2×#41—44no—
238C2×#41—44no—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—21no—
243C22×Q8—44no—
244C22×Q8—44no—
245C22×Q8—44no—
246C22×Q8C2411yescharacteristic
247C22×D4—22no—
248C22×D4—21no—
249C22×D4—21no—
250C22×D4C2412yes—
251C23×C4—22no—
252C23×C4—22no—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2411yescharacteristic
256C2×SD16—24no—
257C2×SD16—42no—
258C2×SD16—42no—
259C22×C8C2412yes—
260C22×C8C2414yes—
261C22×C8C2412yes—
262C4⋊Q8—24no—
263C4⋊Q8—24no—
264C4⋊Q8—24no—
265C4⋊Q8—44no—
266C4⋊Q8—42no—
267C4⋊Q8—44no—
268C4⋊Q8—44no—
269C4⋊Q8—48no—
270C4⋊Q8—48no—
271#127—42no—
272#125—24no—
273#125—42no—
274#124—44no—
275#124—48no—
276#124—48no—
277#123—416no—
278#123—416no—
279C22⋊Q8—22no—
280C22⋊Q8—41no—
281C22⋊Q8—48no—
282C22⋊Q8—48no—
283C22⋊Q8—48no—
284C22⋊Q8—44no—
285C22⋊Q8—48no—
286#121—42no—
287#121—42no—
288#121—44no—
289C4×Q8—24no—
290C4×Q8—44no—
291C4×D4—24no—
292C4×D4—24no—
293C4×D4—44no—
294#117—24no—
295#117—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2414yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2411yescharacteristic
328C2×C22⋊C4—22no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C42—24no—
340C2×C42—21no—
341C2×C42—22no—
342C2×C42—21no—
343C2×C42—22no—
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D412yes—
349C2×C42C2×D412yes—
350C2×C42C2412yes—
351C2×C42C2411yescharacteristic
352C2×C42C2412yes—
353C2×C42C2411yescharacteristic
354#107—44no—
355#107—44no—
356#107—44no—
357#106—42no—
358#106—44no—
359C4⋊C8—28no—
360C4⋊C8—24no—
361#103—42no—
362#103—44no—
363#102—24no—
364#102—24no—
365#102—44no—
366#102—42no—
367#102—44no—
368C22⋊C8—24no—
369C22⋊C8—42no—
370#97—44no—
371C4×C8—24no—
372#95—21no—
373#95—21no—
374#95—22no—
375#95—21no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—21no—
380#95—22no—
381#95—21no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95C2×D412yes—
390#95C2×D412yes—
391#95C2×D412yes—
392#95C2×D412yes—
393C24—22no—
394C24C22×D411yescharacteristic
395#41—816no—
396C2×Q8—22no—
397C2×Q8—21no—
398C2×Q8—22no—
399C2×Q8—21no—
400C2×Q8—44no—
401C2×Q8—42no—
402C2×Q8—44no—
403C2×Q8—42no—
404C2×Q8—44no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—42no—
410C2×Q8—44no—
411C2×Q8—42no—
412C2×Q8—816no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—42no—
427C2×D4—42no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—41no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4AGL(1,ℤ/8)14yes—
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C2511yescharacteristic
462SD16—48no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—24no—
470C2×C8—24no—
471C2×C82−1+414yes—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4—48no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—48no—
501C4⋊C4—42no—
502C4⋊C4—81no—
503C4⋊C4—82no—
504C4⋊C4AGL(1,ℤ/8)14yes—
505C4⋊C42−1+412yes—
506C22⋊C4—24no—
507C22⋊C4—24no—
508C22⋊C4—42no—
509C22⋊C4—44no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—48no—
516C22⋊C4—81no—
517C42—22no—
518C42—22no—
519C42—24no—
520C42—24no—
521C42—24no—
522C42—22no—
523C42—24no—
524C42—22no—
525C42—22no—
526C42—42no—
527C42—42no—
528C42—42no—
529C42—44no—
530C42—82no—
531C42H(5,𝔽2)14yes—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23C2×AGL(1,ℤ/8)12yes—
546C23C23×D411yescharacteristic
547Q8—42no—
548Q8—42no—
549Q8—88no—
550Q8—84no—
551Q8—84no—
552Q8—88no—
553D4—42no—
554D4—42no—
555D4—42no—
556D4—42no—
557D4—84no—
558D4—84no—
559D4—88no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—28no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—44no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—88no—
609C2×C4—88no—
610C2×C4#53512yes—
611C2×C4#53512yes—
612C2×C4C2×AGL(1,ℤ/8)12yes—
613C2×C4C2×AGL(1,ℤ/8)14yes—
614C2×C4C2×AGL(1,ℤ/8)12yes—
615C2×C4C2×AGL(1,ℤ/8)12yes—
616C2×C4#57812yes—
617C2×C4C2×H(5,𝔽2)12yes—
618C2×C4C2×2−1+412yes—
619C2×C4C2×2−1+411yescharacteristic
620C2×C4#58511yescharacteristic
621C8—24no—
622C8—44no—
623C8—42no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22C2×#53511yescharacteristic
639C22#348911yescharacteristic
640C22#349911yescharacteristic
641C22C22×AGL(1,ℤ/8)11yescharacteristic
642C22C22×AGL(1,ℤ/8)11yescharacteristic
643C22C2×#57811yescharacteristic
644C22#358611yescharacteristic
645C4—24no—
646C4—22no—
647C4—22no—
648C4—22no—
649C4—42no—
650C4—41no—
651C4—42no—
652C4—42no—
653C4—42no—
654C4—81no—
655C4—82no—
656C4—84no—
657C4—82no—
658C4—81no—
659C4—81no—
660C4—82no—
661C4#321714yes—
662C4#354112yes—
663C4#355012yes—
664C2—42no—
665C2—162no—
666C2#4291111yescharacteristic
667C2C2×#321711yescharacteristic
668C2#6081411yescharacteristic
669C2#6211111yescharacteristic
670C2#6214811yescharacteristic
671C2#6247911yescharacteristic
672C2#6250311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36450C2semidirect
3. #32838C2semidirect
4. #32304C2semidirect
5. #32076C2semidirect
6. #30343C2semidirect
7. #30066C2semidirect
9. #29213C2semidirect
10. #28540C2semidirect
11. #27822C2semidirect
12. #25813C2semidirect
13. #25449C2semidirect
14. #24678C2semidirect
15. #24301C2semidirect
17. #22643C2semidirect

Maximal subgroups

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

subgroupindex
#364502
#353172
#328382
#323042
#320762
#303432
#300662
#296872
#292132
#285402
#278222
#258132
#254492
#246782
#243012
#235702
#226432