Finite Group Databasebeta

← #6432491

Subgroups — #6432491

Order 512.

1890 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34065C211yesmaximal, characteristic
2C2×#2314C211yesmaximal, characteristic, complemented
3#23773C214yesmaximal, complemented
4#23771C214yesmaximal, complemented
5#23769C214yesmaximal, complemented
6#23767C214yesmaximal, complemented
7#23746C211yesmaximal, characteristic, complemented
8#23733C211yesmaximal, characteristic, complemented
9#23658C211yesmaximal, characteristic, complemented
10#23642C212yesmaximal, complemented
11#23630C211yesmaximal, characteristic, complemented
12#23482C212yesmaximal
13C4×#428C212yesmaximal, complemented
14C4×#417C212yesmaximal, complemented
15#20377C211yesmaximal, characteristic, complemented
16C2×#525C2211yescharacteristic
17C2×C4×#41C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C2×#428C2212yes—
20C2×#428C2212yes—
21C2×#417C2212yes—
22C2×#417C2212yes—
23C2×#407C2212yes—
24C2×C4×M16C2212yes—
25#2324C2212yes—
26#2324C2212yes—
27#2323C2212yes—
28#2323C2212yes—
29#2322—22no—
30#2321—22no—
31#2314—21no—
32#2314—21no—
33#2314—21no—
34#2314—21no—
35#2314C2214yes—
36#2314C2214yes—
37#2297C2211yescharacteristic
38#2296C2211yescharacteristic
39#2293—21no—
40#2293—21no—
41#2290—21no—
42#2290—21no—
43C2×#388—22no—
44C2×#387—22no—
45C2×#384C2211yescharacteristic
46C2×#382C2211yescharacteristic
47#2280C2211yescharacteristic
48C2×#380C2212yes—
49C4×#117C2211yescharacteristic
50C2×C4×C4⋊C4C2212yes—
51#2101C2214yescomplemented
52#2101C2214yescomplemented
53#2097C2214yescomplemented
54#2092C2214yescomplemented
55#2084C2214yescomplemented
56#2083C2214yescomplemented
57#2077C2214yes—
58#2077C2214yes—
59#2076C2214yes—
60#2075C2214yes—
61#2072C2214yescomplemented
62#2071C2214yescomplemented
63#2036C2214yes—
64#2035C2214yes—
65#2034C2214yes—
66#2033C2214yes—
67#1776C2214yescomplemented
68#1776C2214yescomplemented
69C4×#107C2214yescomplemented
70C4×#106C2214yescomplemented
71#1763C2214yescomplemented
72#1762C2214yescomplemented
73C4×#103C2214yescomplemented
74C4×#102C2214yescomplemented
75#1750C2214yes—
76#1750C2214yes—
77#1736C2212yes—
78#1736C2212yes—
79#1736C2212yes—
80#1736C2212yes—
81#1728C2211yescharacteristic
82#1728C2211yescharacteristic
83C22×#41C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C22×M16C2312yes—
86C22×M16C2312yes—
87#525—24no—
88#525—24no—
89C2×#123—21no—
90C2×C22⋊Q8C2311yescharacteristic
91C2×C22⋊Q8C2311yescharacteristic
92C2×#121C2311yescharacteristic
93C2×#121C2311yescharacteristic
94C2×#120—21no—
95C4×#41—28no—
96C2×C4×Q8C2312yes—
97C2×C4×D4—21no—
98C2×C4×D4—21no—
99C2×C4×D4—22no—
100C2×C4×D4C2312yes—
101C2×#117—21no—
102C2×#117—28no—
103C2×#117D411yescharacteristic
104C2×#117D411yescharacteristic
105C2×#117D411yescharacteristic
106C2×#117D411yescharacteristic
107C2×#117C2311yescharacteristic
108C2×#117C2311yescharacteristic
109C2×#117C2311yescharacteristic
110C2×#117C2312yes—
111C2×#117C2312yes—
112C22×C4⋊C4—22no—
113C22×C4⋊C4D412yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C22⋊C4—22no—
117C22×C22⋊C4D412yes—
118C22×C42—22no—
119C22×C42—21no—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122#428—28no—
123#428—28no—
124C2×#107C2314yes—
125C2×#107C2314yes—
126C2×#106C2314yes—
127C2×#106C2314yes—
128#417—28no—
129#417—28no—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134#407—28no—
135C2×C22⋊C8C2314yes—
136C2×C22⋊C8C2314yes—
137C4×M16—28no—
138C2×#97C2314yes—
139C2×C4×C8C2314yes—
140#391C2312yes—
141#390—22no—
142#390—22no—
143#390C2312yes—
144#389—22no—
145#389—22no—
146#389C2312yes—
147#389C2312yes—
148#388—22no—
149#388—22no—
150#388—24no—
151#388—24no—
152#388—24no—
153#388—24no—
154#388—22no—
155#388—22no—
156#388—22no—
157#388—22no—
158#387—24no—
159#387—24no—
160#387—24no—
161#387—24no—
162#387—22no—
163#387—22no—
164#386—22no—
165#386—22no—
166#386—22no—
167#385—22no—
168#385—22no—
169#385—22no—
170#384—21no—
171#384—21no—
172#384C2314yes—
173#382—21no—
174#382—21no—
175#382C2314yes—
176#381—21no—
177#380—22no—
178#380—22no—
179#380C2314yes—
180#380C2314yes—
181#380C2312yes—
182#380C2312yes—
183#379—21no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—21no—
187C4×C4⋊C4—21no—
188C4×C4⋊C4C2314yes—
189C4×C4⋊C4C2314yes—
190C4×C4⋊C4C2312yes—
191C4×C4⋊C4C2312yes—
192C4×C22⋊C4—22no—
193C4×C22⋊C4—21no—
194C4×C22⋊C4—21no—
195C4×C22⋊C4—21no—
196C4×C22⋊C4—21no—
197#376—21no—
198C2×#95—22no—
199C2×#95—22no—
200C2×#95—22no—
201C43—21no—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#336C2314yes—
211#336C2314yes—
212C25C2×D411yescharacteristic
213C2×#41—22no—
214C2×#41—24no—
215C2×#41—24no—
216C2×#41#4114yes—
217C22×Q8C2411yescharacteristic
218C22×D4—21no—
219C22×D4—21no—
220C22×D4—21no—
221C22×D4C2411yescharacteristic
222C23×C4—21no—
223C23×C4—24no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—24no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D412yes—
234C23×C4C2×D412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C2×M16—24no—
239C2×M16—24no—
240C2×M16—24no—
241C2×M16—24no—
242C2×M16—24no—
243C2×M16#4118yes—
244C22×C8C2414yes—
245C22×C8C2414yes—
246#123—28no—
247C22⋊Q8—22no—
248C22⋊Q8—22no—
249C22⋊Q8—22no—
250C22⋊Q8—22no—
251#121—22no—
252#121—22no—
253#121—22no—
254#121—22no—
255#120—28no—
256C4×Q8—24no—
257C4×Q8—24no—
258C4×D4—28no—
259C4×D4—28no—
260C4×D4—24no—
261C4×D4—24no—
262C4×D4—48no—
263#117—22no—
264#117—24no—
265#117—24no—
266#117—24no—
267#117—24no—
268#117—24no—
269#117—24no—
270#117—24no—
271#117—24no—
272#117—24no—
273#117—24no—
274#117—44no—
275#117#4114yes—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—84no—
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2414yes—
296C2×C4⋊C4C2414yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2411yescharacteristic
300C2×C4⋊C4C2411yescharacteristic
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—84no—
317C2×C22⋊C4C2×D412yes—
318C2×C22⋊C4C2×D412yes—
319C2×C22⋊C4C2×D412yes—
320C2×C22⋊C4C2×D412yes—
321C2×C42—21no—
322C2×C42—24no—
323C2×C42—21no—
324C2×C42—21no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—44no—
331C2×C42—84no—
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42#4118yes—
337C2×C42C2412yes—
338C2×C42C2412yes—
339#107—28no—
340#107—28no—
341#106—28no—
342#106—28no—
343#103—28no—
344#103—28no—
345#102—28no—
346#102—28no—
347C22⋊C8—28no—
348C22⋊C8—28no—
349#97—28no—
350C4×C8—28no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365C24—22no—
366C24—21no—
367C24—22no—
368C24—22no—
369C24—42no—
370C24—42no—
371C24C22×D411yescharacteristic
372C24C22×D411yescharacteristic
373C24C22×D411yescharacteristic
374#41—216no—
375#41—48no—
376C2×Q8—21no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8C2×#4112yes—
380C2×D4—24no—
381C2×D4—24no—
382C2×D4—24no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—21no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—82no—
389C2×D4—82no—
390C2×D4C2×#4112yes—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—42no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—82no—
433C22×C4—82no—
434C22×C4#12318yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C2×#4114yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4112yes—
445C22×C4C2×#4114yes—
446C22×C4C2511yescharacteristic
447M16—216no—
448M16—48no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—24no—
454C2×C8C2×#4118yes—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C4⋊C4—84no—
477C4⋊C4—84no—
478C4⋊C4C2×#4112yes—
479C4⋊C4C2×#4112yes—
480C22⋊C4—24no—
481C22⋊C4—24no—
482C22⋊C4—24no—
483C22⋊C4—24no—
484C22⋊C4—24no—
485C22⋊C4—24no—
486C22⋊C4—24no—
487C22⋊C4—24no—
488C22⋊C4—24no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—44no—
492C22⋊C4—82no—
493C22⋊C4—82no—
494C22⋊C4—82no—
495C22⋊C4—82no—
496C22⋊C4—84no—
497C22⋊C4—84no—
498C42—22no—
499C42—22no—
500C42—24no—
501C42—22no—
502C42—22no—
503C42—42no—
504C42—44no—
505C42—42no—
506C42—82no—
507C42—82no—
508C42—84no—
509C42—84no—
510C42#12314yes—
511C42#12314yes—
512C42C2×#4114yes—
513C42C2×#4114yes—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—22no—
525C23—21no—
526C23—21no—
527C23—48no—
528C23—42no—
529C23—42no—
530C23—44no—
531C23—41no—
532C23—41no—
533C23—84no—
534C23—82no—
535C23—82no—
536C23C2×#12312yes—
537C23#52512yes—
538C23#53212yes—
539C23C23×D411yescharacteristic
540Q8—24no—
541Q8—42no—
542D4—24no—
543D4—48no—
544D4—42no—
545D4—84no—
546D4—84no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—48no—
567C2×C4—48no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—48no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—88no—
593C2×C4—88no—
594C2×C4—88no—
595C2×C4C2×#12314yes—
596C2×C4C2×#12314yes—
597C2×C4C2×#12312yes—
598C2×C4#52512yes—
599C2×C4#52918yes—
600C2×C4#52918yes—
601C2×C4#53212yes—
602C2×C4C22×#4112yes—
603C2×C4C22×#4111yescharacteristic
604C2×C4C22×#4111yescharacteristic
605C2×C4C22×#4112yes—
606C8—28no—
607C8—44no—
608C22—24no—
609C22—24no—
610C22—22no—
611C22—22no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—41no—
617C22—42no—
618C22—41no—
619C22—41no—
620C22—42no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—42no—
625C22—88no—
626C22—82no—
627C22—82no—
628C22#239514yes—
629C22C22×#12311yescharacteristic
630C22C2×#52511yescharacteristic
631C22C2×#53211yescharacteristic
632C22#358512yes—
633C22#358612yes—
634C4—24no—
635C4—24no—
636C4—42no—
637C4—42no—
638C4—42no—
639C4—42no—
640C4—41no—
641C4—41no—
642C4—42no—
643C4—82no—
644C4—84no—
645C4—82no—
646C4—84no—
647C4#239514yes—
648C4C2×#52912yes—
649C4C2×#52912yes—
650C2—24no—
651C2—81no—
652C2—81no—
653C2—82no—
654C2C2×#239511yescharacteristic
655C2#6051712yes—
656C2#6056811yescharacteristic
657C2#6056911yescharacteristic
658C2#6070312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2314C2semidirect
3. #23773C2semidirect
4. #23771C2semidirect
5. #23769C2semidirect
6. #23767C2semidirect
7. #23746C2semidirect
8. #23733C2semidirect
9. #23658C2semidirect
10. #23642C2semidirect
11. #23630C2semidirect
13. C4×#428C2semidirect
14. C4×#417C2semidirect
15. #20377C2semidirect
51. #2101C22semidirect
52. #2101C22semidirect
53. #2097C22semidirect
54. #2092C22semidirect
55. #2084C22semidirect
56. #2083C22semidirect
61. #2072C22semidirect
62. #2071C22semidirect
67. #1776C22semidirect
68. #1776C22semidirect
69. C4×#107C22semidirect
70. C4×#106C22semidirect
71. #1763C22semidirect
72. #1762C22semidirect
73. C4×#103C22semidirect
74. C4×#102C22semidirect

Maximal subgroups

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

subgroupindex
#340652
C2×#23142
#237732
#237712
#237692
#237672
#237462
#237332
#236582
#236422
#236302
#234822
C4×#4282
C4×#4172
#203772