Finite Group Databasebeta

← #6627608

Subgroups — #6627608

Order 512.

1605 conjugacy classes of subgroups.

The subgroup lattice has 655 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36594C211yesmaximal, characteristic
2C4×#557C211yesmaximal, characteristic, complemented
3#33196C211yesmaximal, characteristic, complemented
4#33025C212yesmaximal
5#29356C212yesmaximal
6#29274C212yesmaximal, complemented
7#29219C211yesmaximal, characteristic, complemented
8#28279C211yesmaximal, characteristic
9#27578C214yesmaximal, complemented
10#27150C214yesmaximal, complemented
11#24991C212yesmaximal, complemented
12#24328C212yesmaximal, complemented
13#23066C212yesmaximal, complemented
14#22825C211yesmaximal, characteristic, complemented
15#22759C212yesmaximal, complemented
16#21615C212yesmaximal, complemented
17C4×#460C211yesmaximal, characteristic
18C2×#557C2211yescharacteristic
19C2×#556—21no—
20C2×#475C2212yes—
21C2×#474C2212yes—
22C2×#460C2211yescharacteristic
23C2×C4×SD16C2212yes—
24#2868C2211yescharacteristic
25#2858C2212yes—
26#2547—21no—
27#2545—22no—
28#2538—21no—
29#2538C2212yes—
30#2512—22no—
31#2512C2214yes—
32#2370—22no—
33#2370C2212yes—
34#2365—21no—
35#2365C2211yescharacteristic
36#2355—21no—
37#2355C2211yescharacteristic
38#2354—21no—
39#2354C2212yes—
40#2351C2212yes—
41Q8×C4⋊C4C2211yescharacteristic
42C4×C4⋊Q8—21no—
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2212yes—
45C4×#127C2211yescharacteristic
46C4×#125—22no—
47C4×#125C2212yes—
48C42×Q8—21no—
49C42×Q8C2212yes—
50C42×Q8C2211yescharacteristic
51C42×Q8C2211yescharacteristic
52#2101C2214yes—
53#2099C2214yes—
54#2073C2214yes—
55#2067C2214yes—
56#2059C2214yes—
57#2025C2214yes—
58#1990C2212yes—
59#1988C2212yes—
60#1988C2212yes—
61#1988C2212yes—
62#1984C2212yes—
63#1982C2212yes—
64#1982C2211yescharacteristic
65#1948C2212yes—
66#1948C2212yes—
67#1948C2211yescharacteristic
68#1947C2212yes—
69#1936C2214yes—
70#1936C2214yes—
71#1926C2214yes—
72#1926C2214yes—
73#1926C2212yes—
74#1921C2214yes—
75#1920C2214yes—
76#1919C2214yes—
77#1917C2214yes—
78#1880C2214yes—
79#1880C2214yes—
80#1880C2212yes—
81#1863C2214yes—
82#1842C2211yescharacteristic
83#1834C2212yes—
84#1831C2212yes—
85#1803C2212yes—
86#1803C2211yescharacteristic
87#1803C2212yes—
88#1802C2212yes—
89#1776C2214yes—
90#1770C2212yes—
91#1767C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93#1763C2214yes—
94C4×#102C2212yes—
95C22×SD16C2312yes—
96#557—24no—
97#557—24no—
98#556—48no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×#127C2311yescharacteristic
105C2×#126—24no—
106C2×#125—22no—
107C2×#125C2312yes—
108C2×#124—22no—
109C2×#124—22no—
110C2×C22⋊Q8—22no—
111C2×C4×Q8—21no—
112C2×C4×Q8—21no—
113C2×C4×Q8—21no—
114C2×C4×Q8—22no—
115C2×C4×Q8D411yescharacteristic
116C2×C4×Q8D411yescharacteristic
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2311yescharacteristic
123C2×C4×D4—21no—
124C2×C4×D4—21no—
125C2×C4×D4C2312yes—
126C2×#117—22no—
127#475—28no—
128#474—28no—
129#460—24no—
130C4×SD16—28no—
131C2×#107C2314yes—
132C2×#106C2314yes—
133C2×#106C2312yes—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2311yescharacteristic
137C2×C4⋊C8C2311yescharacteristic
138C2×C4⋊C8C2312yes—
139C2×#103C2314yes—
140C2×#103C2312yes—
141C2×#102C2314yes—
142C2×#102C2312yes—
143C2×#102C2312yes—
144C2×#102C2314yes—
145C2×#102C2312yes—
146C2×#97C2314yes—
147C2×C4×C8C2312yes—
148#397—24no—
149#396—24no—
150#396—24no—
151#391—21no—
152#391—21no—
153#391—21no—
154#391—22no—
155#390—21no—
156#390—21no—
157#390—22no—
158#390—22no—
159#390C2312yes—
160#390C2314yes—
161#390C2314yes—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—21no—
166#389—22no—
167#389—22no—
168#389C2312yes—
169#389C2312yes—
170#389C2314yes—
171#389C2312yes—
172#388—24no—
173#388—24no—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—22no—
180#384—21no—
181#384—21no—
182#384—21no—
183#384C2311yescharacteristic
184#384C2311yescharacteristic
185#384C2312yes—
186#384C2312yes—
187#382—22no—
188#382C2312yes—
189#380—22no—
190#380—22no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4D412yes—
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D412yes—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C43—22no—
219C43—21no—
220C43C2312yes—
221C43C2311yescharacteristic
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#336C2314yes—
232#336C2314yes—
233#336C2314yes—
234#336C2312yes—
235C22×Q8—21no—
236C22×Q8—22no—
237C22×Q8C2×D411yescharacteristic
238C22×Q8C2411yescharacteristic
239C22×D4—21no—
240C22×D4—21no—
241C22×D4C2412yes—
242C23×C4—22no—
243C2×SD16—24no—
244C2×SD16—24no—
245C2×SD16—24no—
246C22×C8C2412yes—
247C22×C8C2412yes—
248C22×C8C2414yes—
249C4⋊Q8—28no—
250C4⋊Q8—22no—
251C4⋊Q8—24no—
252C4⋊Q8—24no—
253C4⋊Q8—44no—
254C4⋊Q8—82no—
255C4⋊Q8#4114yes—
256#127—22no—
257#127—22no—
258#126—416no—
259#125—216no—
260#125—24no—
261#125—24no—
262#124—48no—
263#124—84no—
264C22⋊Q8—48no—
265C4×Q8—28no—
266C4×Q8—24no—
267C4×Q8—24no—
268C4×Q8—24no—
269C4×Q8—22no—
270C4×Q8—22no—
271C4×Q8—22no—
272C4×Q8—22no—
273C4×Q8—22no—
274C4×Q8—24no—
275C4×Q8—42no—
276C4×Q8—42no—
277C4×Q8—41no—
278C4×Q8—48no—
279C4×Q8—44no—
280C4×Q8—44no—
281C4×Q8#4114yes—
282C4×Q8#4114yes—
283C4×D4—24no—
284C4×D4—24no—
285C4×D4—44no—
286C4×D4—44no—
287#117—48no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2414yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C22⋊C4—24no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C42—22no—
330C2×C42—21no—
331C2×C42—21no—
332C2×C42—24no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42—22no—
338C2×C42—21no—
339C2×C42—22no—
340C2×C42—21no—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2412yes—
350C2×C42C2412yes—
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354#107—28no—
355#106—24no—
356#106—28no—
357C4⋊C8—22no—
358C4⋊C8—24no—
359C4⋊C8—24no—
360C4⋊C8—22no—
361C4⋊C8—42no—
362#103—24no—
363#103—28no—
364#102—24no—
365#102—24no—
366#102—28no—
367#102—42no—
368#102—44no—
369#97—44no—
370C4×C8—24no—
371#95—22no—
372#95—22no—
373#95—21no—
374#95—21no—
375#95—21no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—24no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—24no—
384#95C2×D412yes—
385#95C2×D412yes—
386#95C2×D412yes—
387C24—22no—
388C2×Q8—22no—
389C2×Q8—24no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—44no—
396C2×Q8—42no—
397C2×Q8—44no—
398C2×Q8—42no—
399C2×Q8—44no—
400C2×Q8C2×#4112yes—
401C2×Q8C2×#4112yes—
402C2×Q82−1+412yes—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—42no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C2511yescharacteristic
437SD16—48no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—24no—
441C2×C8—22no—
442C2×C8—24no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—24no—
447C4⋊C4—28no—
448C4⋊C4—28no—
449C4⋊C4—28no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—42no—
470C4⋊C4—42no—
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—81no—
482C4⋊C4—82no—
483C4⋊C4—82no—
484C4⋊C4—82no—
485C4⋊C4C2×#4112yes—
486C4⋊C4C2×#4112yes—
487C4⋊C4C2×#4114yes—
488C22⋊C4—48no—
489C22⋊C4—44no—
490C22⋊C4—48no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—84no—
494C22⋊C4—84no—
495C42—28no—
496C42—24no—
497C42—24no—
498C42—24no—
499C42—22no—
500C42—24no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—21no—
505C42—22no—
506C42—22no—
507C42—21no—
508C42—41no—
509C42—42no—
510C42—42no—
511C42—42no—
512C42—44no—
513C42—48no—
514C42—44no—
515C42—44no—
516C42—42no—
517C42—42no—
518C42—81no—
519C42—82no—
520C42AGL(1,ℤ/8)14yes—
521C42C2×#4112yes—
522C42C2×#4112yes—
523C42C2×#4112yes—
524C422−1+412yes—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23C23×D411yescharacteristic
533Q8—22no—
534Q8—22no—
535Q8—22no—
536Q8—44no—
537Q8—48no—
538Q8—88no—
539Q8#55614yescomplemented
540D4—42no—
541D4—42no—
542D4—42no—
543D4—42no—
544D4—84no—
545D4—84no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—44no—
591C2×C4—88no—
592C2×C4#52512yes—
593C2×C4#52512yes—
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×C4#57512yes—
600C2×C4#57712yes—
601C2×C4C22×#4111yescharacteristic
602C2×C4C22×#4111yescharacteristic
603C2×C4C22×#4111yescharacteristic
604C2×C4C2×2−1+411yescharacteristic
605C2×C4C2×2−1+411yescharacteristic
606C2×C4#58511yescharacteristic
607C8—42no—
608C8—44no—
609C8—42no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22C2×#52511yescharacteristic
618C22C2×#54811yescharacteristic
619C22#348911yescharacteristic
620C22C22×AGL(1,ℤ/8)11yescharacteristic
621C22C2×#57511yescharacteristic
622C22C2×#57711yescharacteristic
623C22#358611yescharacteristic
624C4—22no—
625C4—22no—
626C4—22no—
627C4—24no—
628C4—42no—
629C4—44no—
630C4—42no—
631C4—42no—
632C4—42no—
633C4—42no—
634C4—42no—
635C4—41no—
636C4—82no—
637C4—84no—
638C4—82no—
639C4—81no—
640C4#311514yes—
641C4#336814yes—
642C4C2×#55611yescharacteristic
643C4#348011yescharacteristic
644C4#350711yescharacteristic
645C4#352311yescharacteristic
646C2—162no—
647C2#3657911yescharacteristic
648C2C2×#311511yescharacteristic
649C2#6059911yescharacteristic
650C2C2×#336811yescharacteristic
651C2#6132311yescharacteristic
652C2#6211111yescharacteristic
653C2#6214511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
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
2. C4×#557C2semidirect
3. #33196C2semidirect
6. #29274C2semidirect
7. #29219C2semidirect
9. #27578C2semidirect
10. #27150C2semidirect
11. #24991C2semidirect
12. #24328C2semidirect
13. #23066C2semidirect
14. #22825C2semidirect
15. #22759C2semidirect
16. #21615C2semidirect
539. Q8#556semidirect

Maximal subgroups

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

subgroupindex
#365942
C4×#5572
#331962
#330252
#293562
#292742
#292192
#282792
#275782
#271502
#249912
#243282
#230662
#228252
#227592
#216152
C4×#4602