Finite Group Databasebeta

← #6461713

Subgroups — #6461713

Order 512.

1756 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34074C211yesmaximal, characteristic, complemented
2C2×#2319C211yesmaximal, characteristic
3#29406C214yesmaximal, complemented
4#29404C214yesmaximal, complemented
5#29401C214yesmaximal, complemented
6#29400C214yesmaximal, complemented
7#23657C211yesmaximal, characteristic, complemented
8#23628C211yesmaximal, characteristic
9#23554C211yesmaximal, characteristic, complemented
10#23454C211yesmaximal, characteristic
11#21060C212yesmaximal
12#21034C212yesmaximal, complemented
13#21033C212yesmaximal, complemented
14#20873C212yesmaximal
15#20773C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C2×#428C2212yes—
19C2×#428C2212yes—
20C2×#418C2212yes—
21C2×#418C2212yes—
22#2331C2212yescomplemented
23#2331C2212yescomplemented
24#2330C2212yescomplemented
25#2330C2212yescomplemented
26#2329—21no—
27#2329—21no—
28#2329—21no—
29#2329—21no—
30#2319—22no—
31#2319—22no—
32#2319C2214yes—
33#2319C2214yes—
34#2296C2212yes—
35#2295—22no—
36#2295—22no—
37C2×#390C2211yescharacteristic
38C2×#390C2211yescharacteristic
39C2×#389C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#388—21no—
42C2×#388—21no—
43#2285C2211yescharacteristic
44C2×#381—21no—
45C2×#380—21no—
46C4×#117C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×#376C2211yescharacteristic
49#2085C2214yescomplemented
50#2084C2212yescomplemented
51#2083C2212yescomplemented
52#2073C2214yes—
53#2071C2212yes—
54#2071C2212yes—
55#2059C2214yescomplemented
56#2058C2212yescomplemented
57#2057C2212yescomplemented
58#2025C2214yes—
59#2023C2212yes—
60#2023C2212yes—
61#1926C2214yes—
62#1926C2214yes—
63#1925C2214yes—
64#1925C2214yes—
65#1921C2214yescomplemented
66#1921C2214yescomplemented
67#1920C2214yescomplemented
68#1920C2214yescomplemented
69#1919C2214yescomplemented
70#1919C2214yescomplemented
71#1918C2214yescomplemented
72#1918C2214yescomplemented
73#1864C2214yes—
74#1864C2214yes—
75#1862C2214yes—
76#1862C2214yes—
77#1840C2214yescomplemented
78#1840C2214yescomplemented
79#1736C2212yes—
80#1736C2212yes—
81#1736C2212yes—
82#1736C2212yes—
83#1728C2211yescharacteristic
84#1728C2212yes—
85#1728C2212yes—
86#1728C2211yescharacteristic
87C23×D4C2311yescharacteristic
88C24×C4—21no—
89C22×M16C2312yes—
90C22×M16C2312yes—
91#533—28no—
92C2×C4⋊Q8C2311yescharacteristic
93C2×C4⋊Q8C2311yescharacteristic
94C2×#125C2312yes—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8—21no—
97C2×C4×Q8—21no—
98C2×C4×Q8—21no—
99C2×C4×D4—24no—
100C2×C4×D4—24no—
101C2×C4×D4—22no—
102C2×C4×D4C2314yes—
103C2×#117C2312yes—
104C2×#117C2312yes—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C22⋊C4—21no—
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C42—21no—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119#428—28no—
120#428—28no—
121C2×#107C2314yes—
122C2×#107C2314yes—
123C2×#106C2314yes—
124C2×#106C2314yes—
125#418—28no—
126#418—28no—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131#391—22no—
132#391—22no—
133#391—22no—
134#391—22no—
135#390—21no—
136#390—21no—
137#390—22no—
138#390C2312yes—
139#390C2312yes—
140#390C2314yes—
141#389—21no—
142#389—21no—
143#389—22no—
144#389—22no—
145#389—22no—
146#389—22no—
147#389—22no—
148#389C2312yes—
149#389C2312yes—
150#389C2314yes—
151#389C2312yes—
152#389C2312yes—
153#389C2312yes—
154#389C2312yes—
155#388—24no—
156#388—24no—
157#388—24no—
158#388—24no—
159#387—22no—
160#387—22no—
161#387—22no—
162#387—22no—
163#387—22no—
164#387—22no—
165#387—22no—
166#387—22no—
167#385—22no—
168#385—22no—
169#384C2311yescharacteristic
170#384C2311yescharacteristic
171#382C2312yes—
172#381—24no—
173#381—24no—
174#380—21no—
175#380—21no—
176#380—24no—
177#380—24no—
178C4×C4⋊C4—22no—
179C4×C4⋊C4—21no—
180C4×C4⋊C4—21no—
181C4×C4⋊C4—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4C2314yes—
184C4×C4⋊C4C2312yes—
185C4×C4⋊C4C2312yes—
186C4×C4⋊C4C2312yes—
187C4×C4⋊C4C2312yes—
188C4×C22⋊C4—22no—
189C4×C22⋊C4—22no—
190#376—22no—
191#376D412yes—
192#376D412yes—
193#376C2312yes—
194#376C2312yes—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95D411yescharacteristic
199C2×#95D411yescharacteristic
200C2×#95D411yescharacteristic
201C2×#95D411yescharacteristic
202C43—22no—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#336C2312yes—
212#336C2312yes—
213#336C2312yes—
214#336C2312yes—
215#336C2314yes—
216#336C2314yes—
217#336C2314yes—
218#336C2314yes—
219C25—21no—
220C22×Q8—21no—
221C22×Q8—21no—
222C22×D4—21no—
223C22×D4—24no—
224C22×D4—22no—
225C22×D4—24no—
226C22×D4#4114yes—
227C22×D4C2412yes—
228C23×C4—21no—
229C23×C4—28no—
230C23×C4—24no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—82no—
234C23×C4—82no—
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2411yescharacteristic
242C2×M16—24no—
243C2×M16—24no—
244C2×M16—24no—
245C2×M16—24no—
246C2×M16—24no—
247C2×M16—24no—
248C22×C8C2414yes—
249C22×C8C2414yes—
250C4⋊Q8—24no—
251C4⋊Q8—24no—
252#125—24no—
253#125—24no—
254C22⋊Q8—28no—
255C22⋊Q8—28no—
256C4×Q8—28no—
257C4×Q8—28no—
258C4×D4—216no—
259C4×D4—28no—
260C4×D4—28no—
261#117—24no—
262#117—24no—
263#117—24no—
264#117#4118yes—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—82no—
287C2×C4⋊C4—82no—
288C2×C4⋊C4#4114yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—82no—
307C2×C22⋊C4—82no—
308C2×C22⋊C4—82no—
309C2×C22⋊C4—82no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—24no—
321C2×C42—82no—
322C2×C42C2×D412yes—
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D412yes—
326C2×C42C2412yes—
327C2×C42C2411yescharacteristic
328C2×C42C2411yescharacteristic
329C2×C42C2412yes—
330#107—28no—
331#107—28no—
332#106—28no—
333#106—28no—
334#102—28no—
335#102—28no—
336#102—28no—
337#102—28no—
338#95—24no—
339#95—24no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95C2×D412yes—
353#95C2×D412yes—
354#95C2×D412yes—
355#95C2×D412yes—
356#95C2×D412yes—
357#95C2×D412yes—
358#95C2×D412yes—
359#95C2×D412yes—
360C24—24no—
361C24—22no—
362C24—42no—
363C24—42no—
364C24—42no—
365C24—42no—
366C24—42no—
367C24—42no—
368C24C22×D411yescharacteristic
369C2×Q8—24no—
370C2×Q8—24no—
371C2×Q8—81no—
372C2×Q8—81no—
373C2×Q8—81no—
374C2×Q8—81no—
375C2×D4—24no—
376C2×D4—216no—
377C2×D4—24no—
378C2×D4—22no—
379C2×D4—24no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4—48no—
383C2×D4C2×#4114yes—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—216no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—84no—
420C22×C4—84no—
421C22×C4—816no—
422C22×C4C22×D412yes—
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C2×#4112yes—
432C22×C4C2×#4114yes—
433C22×C4C2×#4112yes—
434C22×C42−1+414yes—
435C22×C4C2511yescharacteristic
436M16—48no—
437M16—48no—
438C2×C8—24no—
439C2×C8—24no—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—24no—
444C4⋊C4—28no—
445C4⋊C4—28no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—24no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—82no—
461C4⋊C4—81no—
462C4⋊C4—81no—
463C4⋊C4—81no—
464C4⋊C4—82no—
465C4⋊C4—82no—
466C4⋊C4—82no—
467C4⋊C4—81no—
468C4⋊C4—84no—
469C4⋊C4—82no—
470C4⋊C4—82no—
471C4⋊C4C2×#4114yes—
472C4⋊C4C2×#4114yes—
473C4⋊C4C2×#4114yes—
474C22⋊C4—216no—
475C22⋊C4—28no—
476C22⋊C4—44no—
477C22⋊C4—84no—
478C22⋊C4—84no—
479C22⋊C4—82no—
480C22⋊C4—82no—
481C22⋊C4—82no—
482C22⋊C4—82no—
483C42—24no—
484C42—22no—
485C42—22no—
486C42—28no—
487C42—24no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—24no—
493C42—44no—
494C42—82no—
495C42—82no—
496C42—82no—
497C42—82no—
498C42—84no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—24no—
504C23—48no—
505C23—48no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—48no—
511C23—42no—
512C23—42no—
513C23—84no—
514C23—84no—
515C23—84no—
516C23—84no—
517C23#52512yes—
518C23#52512yes—
519C23#53612yes—
520C23C23×D411yescharacteristic
521Q8—84no—
522Q8—84no—
523D4—28no—
524D4—48no—
525D4—44no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—44no—
554C2×C4—48no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—816no—
579C2×C4—84no—
580C2×C4—84no—
581C2×C4#52512yes—
582C2×C4#52512yes—
583C2×C4#53612yes—
584C2×C4#55218yes—
585C2×C4#55618yes—
586C2×C4C22×#4111yescharacteristic
587C2×C4C22×#4112yes—
588C2×C4C22×#4111yescharacteristic
589C2×C4C2×2−1+412yes—
590C8—44no—
591C8—44no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—816no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22#259914yes—
607C22C2×#52511yescharacteristic
608C22C2×#52511yescharacteristic
609C22C2×#53611yescharacteristic
610C22#358511yescharacteristic
611C22#358511yescharacteristic
612C22#358511yescharacteristic
613C22#358611yescharacteristic
614C4—42no—
615C4—44no—
616C4—42no—
617C4—41no—
618C4—41no—
619C4—44no—
620C4—41no—
621C4—41no—
622C4—44no—
623C4—42no—
624C4—42no—
625C4—82no—
626C4—82no—
627C4—82no—
628C4—82no—
629C4#259914yes—
630C4C2×#55212yes—
631C4C2×#55612yes—
632C2—24no—
633C2—84no—
634C2C2×#259911yescharacteristic
635C2#6056411yescharacteristic
636C2#6056511yescharacteristic
637C2#6056711yescharacteristic
638C2#6056811yescharacteristic
639C2#6086811yescharacteristic
640C2#6087011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #34074C2semidirect
3. #29406C2semidirect
4. #29404C2semidirect
5. #29401C2semidirect
6. #29400C2semidirect
7. #23657C2semidirect
9. #23554C2semidirect
12. #21034C2semidirect
13. #21033C2semidirect
15. #20773C2semidirect
22. #2331C22semidirect
23. #2331C22semidirect
24. #2330C22semidirect
25. #2330C22semidirect
49. #2085C22semidirect
50. #2084C22semidirect
51. #2083C22semidirect
55. #2059C22semidirect
56. #2058C22semidirect
57. #2057C22semidirect
65. #1921C22semidirect
66. #1921C22semidirect
67. #1920C22semidirect
68. #1920C22semidirect
69. #1919C22semidirect
70. #1919C22semidirect
71. #1918C22semidirect
72. #1918C22semidirect
77. #1840C22semidirect
78. #1840C22semidirect

Maximal subgroups

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

subgroupindex
#340742
C2×#23192
#294062
#294042
#294012
#294002
#236572
#236282
#235542
#234542
#210602
#210342
#210332
#208732
#207732