Finite Group Databasebeta

← #6556040

Subgroups — #6556040

Order 512.

1731 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#36188C211yesmaximal, characteristic, complemented
2#34308C211yesmaximal, characteristic
3#30173C212yesmaximal, complemented
4#29864C211yesmaximal, characteristic, complemented
5#29301C212yesmaximal, complemented
6#29276C211yesmaximal, characteristic, complemented
7#28689C211yesmaximal, characteristic, complemented
8#27614C212yesmaximal, complemented
9#27475C214yesmaximal, complemented
10#27047C214yesmaximal, complemented
11#26298C212yesmaximal, complemented
12#25703C212yesmaximal
13#23775C211yesmaximal, characteristic, complemented
14#23760C212yesmaximal, complemented
15#23161C212yesmaximal, complemented
16#22968C212yesmaximal, complemented
17#22839C211yesmaximal, characteristic, complemented
18C2×#556—21no—
19C2×#547C2211yescharacteristic
20C2×#520—21no—
21C2×#500C2212yes—
22C2×#474C2212yes—
23#2868C2211yescharacteristic
24#2857C2212yes—
25#2568—21no—
26#2547—21no—
27#2547C2212yes—
28#2499—22no—
29#2496—22no—
30#2496C2214yes—
31#2365—21no—
32#2365C2211yescharacteristic
33#2365C2212yes—
34#2361C2212yes—
35#2360—21no—
36#2359—22no—
37D4×C4⋊C4C2211yescharacteristic
38D4×C4⋊C4C2211yescharacteristic
39#2344C2212yes—
40#2339—21no—
41#2323—21no—
42#2322—22no—
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×#121C2212yes—
46#2293C2211yescharacteristic
47C2×#389C2212yes—
48#2285C2212yes—
49C42×D4C2211yescharacteristic
50#2100C2212yes—
51#2100C2211yescharacteristic
52#2094C2211yescharacteristic
53#2088C2212yes—
54#2083C2212yes—
55#2083C2212yes—
56#2083C2212yes—
57#2083C2212yes—
58#2077C2214yes—
59#2077C2214yes—
60#2071C2212yes—
61#2071C2212yes—
62#2065C2212yes—
63#2058C2212yes—
64#2058C2211yescharacteristic
65#2033C2214yes—
66#2033C2214yes—
67#2023C2211yescharacteristic
68#1990C2212yes—
69#1988C2212yes—
70#1988C2212yes—
71#1954C2212yes—
72#1948C2212yes—
73#1947C2214yes—
74#1943C2214yes—
75#1943C2214yes—
76#1924C2214yes—
77#1921C2214yes—
78#1921C2214yes—
79#1921C2214yes—
80#1920C2214yes—
81#1864C2214yes—
82#1853C2214yes—
83#1852C2214yes—
84#1852C2214yes—
85#1831C2211yescharacteristic
86#1824C2214yes—
87#1817C2212yes—
88#1776C2214yes—
89C4×#106C2212yes—
90#1770C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92#1762C2214yes—
93C4×#102C2212yes—
94#1749C2212yes—
95C22×#41—21no—
96#556—48no—
97#547—24no—
98#547—24no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×#126—24no—
105C2×#124—22no—
106C2×#123—22no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—21no—
109C2×#121C2312yes—
110#520—48no—
111C2×C4×Q8—21no—
112C2×C4×Q8—22no—
113C2×C4×Q8—21no—
114C2×C4×Q8—21no—
115C2×C4×D4—21no—
116C2×C4×D4—22no—
117C2×C4×D4—22no—
118C2×C4×D4D411yescharacteristic
119C2×C4×D4D411yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2312yes—
122C2×C4×D4C2311yescharacteristic
123C2×#117—22no—
124C2×#117—22no—
125C2×#117—21no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2312yes—
130C22×C42C2312yes—
131#500—28no—
132#474—28no—
133C2×#107C2314yes—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×#106C2312yes—
137C2×#106C2312yes—
138C2×#106C2314yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2312yes—
141C2×#102C2312yes—
142C2×#102C2314yes—
143C2×#102C2312yes—
144C2×C22⋊C8C2314yes—
145C2×C22⋊C8C2314yes—
146C2×#97C2314yes—
147C2×C4×C8C2312yes—
148#400—24no—
149#397—24no—
150#397—24no—
151#391—22no—
152#391—22no—
153#391—21no—
154#391—21no—
155#391—21no—
156#391—22no—
157#390C2311yescharacteristic
158#390C2311yescharacteristic
159#389—21no—
160#389—21no—
161#389—21no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389C2312yes—
166#389C2312yes—
167#389C2311yescharacteristic
168#389C2311yescharacteristic
169#389C2312yes—
170#389C2314yes—
171#389C2312yes—
172#388—21no—
173#388—22no—
174#388—21no—
175#388—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—24no—
180#387—24no—
181#386—22no—
182#386—22no—
183#385—22no—
184#385—22no—
185#385—22no—
186#384—21no—
187#384—21no—
188#384C2311yescharacteristic
189#384C2311yescharacteristic
190#384C2312yes—
191#384C2312yes—
192#382—22no—
193#381—22no—
194#380—22no—
195#380—22no—
196#380—21no—
197#380C2312yes—
198#380C2314yes—
199#380C2314yes—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2314yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C22⋊C4—21no—
212C4×C22⋊C4—22no—
213C4×C22⋊C4D412yes—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4C2312yes—
216C2×#95—22no—
217C43C2311yescharacteristic
218#340C2312yes—
219#340C2314yes—
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2311yescharacteristic
233#336C2314yes—
234C2×#41—28no—
235C2×#41—44no—
236C2×#41—44no—
237C22×Q8—21no—
238C22×Q8—21no—
239C22×Q8—22no—
240C22×D4—22no—
241C22×D4C2×D411yescharacteristic
242C22×D4C2411yescharacteristic
243C23×C4—22no—
244C23×C4—21no—
245C23×C4—22no—
246C23×C4—22no—
247C23×C4C2×D412yes—
248C23×C4C2×D412yes—
249C23×C4C2412yes—
250C22×C8C2412yes—
251C22×C8C2414yes—
252C22×C8C2412yes—
253C4⋊Q8—22no—
254C4⋊Q8—24no—
255C4⋊Q8—24no—
256C4⋊Q8—42no—
257C4⋊Q8—44no—
258C4⋊Q8—44no—
259C4⋊Q8#4114yes—
260#126—416no—
261#124—48no—
262#123—216no—
263C22⋊Q8—28no—
264C22⋊Q8—48no—
265#121—24no—
266#121—24no—
267C4×Q8—44no—
268C4×Q8—44no—
269C4×Q8—48no—
270C4×Q8—44no—
271C4×D4—24no—
272C4×D4—24no—
273C4×D4—22no—
274C4×D4—22no—
275C4×D4—22no—
276C4×D4—44no—
277C4×D4—48no—
278C4×D4—48no—
279C4×D4—44no—
280C4×D4#4114yes—
281#117—48no—
282#117—48no—
283#117—44no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—24no—
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—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—48no—
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2414yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4C2×D412yes—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C42—22no—
334C2×C42—21no—
335C2×C42—22no—
336C2×C42—24no—
337C2×C42—22no—
338C2×C42—44no—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2×D411yescharacteristic
345C2×C42C2412yes—
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349#107—28no—
350#107—28no—
351#107—28no—
352#106—24no—
353#106—42no—
354#106—44no—
355C4⋊C8—24no—
356C4⋊C8—24no—
357#102—28no—
358#102—24no—
359#102—42no—
360C22⋊C8—28no—
361C22⋊C8—44no—
362#97—44no—
363C4×C8—24no—
364#95—21no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—21no—
370#95—24no—
371#95—22no—
372#95—24no—
373#95—22no—
374#95—22no—
375#95—21no—
376#95—22no—
377#95—24no—
378#95—24no—
379#95C2×D412yes—
380#95C2×D412yes—
381#95C2×D412yes—
382C24—21no—
383C24C22×D412yes—
384#41—432no—
385C2×Q8—24no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×Q8—44no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8—44no—
392C2×Q8—44no—
393C2×Q8—42no—
394C2×D4—28no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—21no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4AGL(1,ℤ/8)14yes—
403C2×D4C2×#4112yes—
404C22×C4—28no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—44no—
430C22×C4—82no—
431C22×C4—82no—
432C22×C4—82no—
433C22×C4—82no—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C2511yescharacteristic
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—22no—
450C2×C8—24no—
451C2×C8—22no—
452C2×C82−1+414yes—
453C4⋊C4—28no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—48no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—81no—
486C4⋊C4—82no—
487C4⋊C4C2×#4112yes—
488C4⋊C4C2×#4114yes—
489C22⋊C4—28no—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—24no—
494C22⋊C4—44no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—81no—
501C22⋊C4—82no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—24no—
506C42—22no—
507C42—42no—
508C42—42no—
509C42—41no—
510C42—44no—
511C42—48no—
512C42—44no—
513C42—42no—
514C42—44no—
515C42AGL(1,ℤ/8)14yes—
516C42C2×#4112yes—
517C42C2×#4112yes—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23C2×AGL(1,ℤ/8)14yes—
532C23C23×D411yescharacteristic
533Q8—48no—
534Q8—84no—
535Q8—88no—
536D4—28no—
537D4—24no—
538D4—416no—
539D4—42no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—48no—
563C2×C4—416no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—48no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—44no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—84no—
590C2×C4—164no—
591C2×C4—164no—
592C2×C4#52512yes—
593C2×C4#52512yes—
594C2×C4C2×AGL(1,ℤ/8)12yes—
595C2×C4C2×AGL(1,ℤ/8)12yes—
596C2×C4C2×AGL(1,ℤ/8)12yes—
597C2×C4C2×AGL(1,ℤ/8)12yes—
598C2×C4#57812yes—
599C2×C4C22×#4111yescharacteristic
600C2×C4C22×#4111yescharacteristic
601C2×C4C2×2−1+412yes—
602C2×C4#58512yes—
603C8—24no—
604C8—44no—
605C8—42no—
606C22—24no—
607C22—24no—
608C22—24no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22C2×#52511yescharacteristic
621C22#348511yescharacteristic
622C22#349911yescharacteristic
623C22C22×AGL(1,ℤ/8)11yescharacteristic
624C22C22×AGL(1,ℤ/8)11yescharacteristic
625C22C2×#57811yescharacteristic
626C22#358611yescharacteristic
627C4—22no—
628C4—22no—
629C4—22no—
630C4—22no—
631C4—42no—
632C4—42no—
633C4—41no—
634C4—42no—
635C4—42no—
636C4—44no—
637C4—81no—
638C4—81no—
639C4—82no—
640C4—82no—
641C4—82no—
642C4—84no—
643C4#310914yes—
644C4#350712yes—
645C4#351612yes—
646C2—44no—
647C2—82no—
648C2#3584611yescharacteristic
649C2C2×#310911yescharacteristic
650C2#6059411yescharacteristic
651C2#6172211yescharacteristic
652C2#6173711yescharacteristic
653C2#6247911yescharacteristic
654C2#6250311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
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. #36188C2semidirect
3. #30173C2semidirect
4. #29864C2semidirect
5. #29301C2semidirect
6. #29276C2semidirect
7. #28689C2semidirect
8. #27614C2semidirect
9. #27475C2semidirect
10. #27047C2semidirect
11. #26298C2semidirect
13. #23775C2semidirect
14. #23760C2semidirect
15. #23161C2semidirect
16. #22968C2semidirect
17. #22839C2semidirect

Maximal subgroups

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

subgroupindex
#361882
#343082
#301732
#298642
#293012
#292762
#286892
#276142
#274752
#270472
#262982
#257032
#237752
#237602
#231612
#229682
#228392