Finite Group Databasebeta

← #6433947

Subgroups — #6433947

Order 512.

1703 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34635C211yesmaximal, characteristic
2C2×#2628C211yesmaximal, characteristic, complemented
3#30374C214yesmaximal, complemented
4#30372C214yesmaximal, complemented
5#30370C214yesmaximal, complemented
6#30368C214yesmaximal, complemented
7#23746C211yesmaximal, characteristic, complemented
8#23733C211yesmaximal, characteristic, complemented
9#23642C212yesmaximal, complemented
10#23552C211yesmaximal, characteristic, complemented
11#23482C212yesmaximal
12#23452C211yesmaximal, characteristic, complemented
13#21159C212yesmaximal, complemented
14#21093C212yesmaximal, complemented
15#20377C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C2×#525C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C2×#428C2212yes—
20C2×#428C2212yes—
21C2×#417C2212yes—
22C2×#417C2212yes—
23C2×#407C2212yes—
24C2×C4×M16C2212yes—
25#2628—22no—
26#2628—22no—
27#2628C2214yes—
28#2628C2214yes—
29#2625C2212yes—
30#2625C2212yes—
31#2624C2212yes—
32#2624C2212yes—
33#2623—21no—
34#2623—21no—
35#2622—21no—
36#2622—21no—
37#2395—22no—
38#2395—22no—
39C2×#400—21no—
40C2×#399—21no—
41C2×#395—21no—
42C2×#393—21no—
43#2297C2211yescharacteristic
44#2297C2211yescharacteristic
45#2296C2211yescharacteristic
46#2296C2211yescharacteristic
47C2×#389C2212yes—
48C2×#382C2211yescharacteristic
49#2280C2211yescharacteristic
50C2×#380C2212yes—
51#2101C2214yescomplemented
52#2101C2214yescomplemented
53#2097C2214yescomplemented
54#2092C2214yescomplemented
55#2077C2214yes—
56#2077C2214yes—
57#2076C2214yes—
58#2075C2214yes—
59#2058C2214yescomplemented
60#2057C2214yescomplemented
61#2036C2214yes—
62#2035C2214yes—
63#2034C2214yes—
64#2033C2214yes—
65#2024C2214yescomplemented
66#2023C2214yescomplemented
67#1944C2214yescomplemented
68#1943C2214yescomplemented
69#1942C2214yescomplemented
70#1941C2214yescomplemented
71#1936C2214yescomplemented
72#1935C2214yescomplemented
73#1934C2214yescomplemented
74#1933C2214yescomplemented
75#1750C2214yes—
76#1750C2214yes—
77#1736C2212yes—
78#1736C2212yes—
79#1736C2212yes—
80#1736C2212yes—
81#1728C2211yescharacteristic
82#1728C2211yescharacteristic
83C22×#41C2311yescharacteristic
84C23×Q8—21no—
85C24×C4C2311yescharacteristic
86C22×M16C2312yes—
87C22×M16C2312yes—
88C2×C4⋊Q8—22no—
89C2×C4⋊Q8—44no—
90C2×C4⋊Q8C2314yes—
91#525—44no—
92C2×#123—21no—
93C2×C22⋊Q8C2311yescharacteristic
94C2×C22⋊Q8C2311yescharacteristic
95C2×#121C2311yescharacteristic
96C2×#121C2311yescharacteristic
97C2×#120—21no—
98C2×C4×D4—21no—
99C2×C4×D4—21no—
100C2×#117—44no—
101C2×#117D412yes—
102C2×#117D412yes—
103C2×#117C2311yescharacteristic
104C2×#117C2311yescharacteristic
105C2×#117C2311yescharacteristic
106C2×#117C2312yes—
107C2×#117C2312yes—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—22no—
110C22×C4⋊C4D411yescharacteristic
111C22×C4⋊C4D411yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C22⋊C4—22no—
115C22×C22⋊C4D411yescharacteristic
116C22×C22⋊C4D411yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119#428—28no—
120#428—44no—
121C2×#107C2314yes—
122C2×#107C2314yes—
123C2×#106C2314yes—
124C2×#106C2314yes—
125#417—28no—
126#417—44no—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131#407—28no—
132C2×C22⋊C8C2314yes—
133C2×C22⋊C8C2314yes—
134C4×M16—44no—
135C2×#97C2314yes—
136C2×C4×C8C2314yes—
137#400—22no—
138#400—22no—
139#400—24no—
140#400—24no—
141#399—22no—
142#399—22no—
143#399—24no—
144#399—24no—
145#398—22no—
146#398—22no—
147#397—22no—
148#397—22no—
149#397—22no—
150#397—22no—
151#397—22no—
152#397—22no—
153#395—24no—
154#395—24no—
155#394—22no—
156#394—22no—
157#394—22no—
158#394—22no—
159#393—24no—
160#393—24no—
161#391C2312yes—
162#391C2312yes—
163#390C2312yes—
164#390C2312yes—
165#389—22no—
166#389—22no—
167#389C2312yes—
168#389C2312yes—
169#389C2312yes—
170#389C2312yes—
171#389C2314yes—
172#389C2314yes—
173#386—21no—
174#386—21no—
175#386—21no—
176#386—21no—
177#385—21no—
178#385—21no—
179#385—21no—
180#385—21no—
181#382—22no—
182#382C2314yes—
183#381—21no—
184#380—22no—
185#380—22no—
186#380C2312yes—
187#380C2314yes—
188#380C2312yes—
189#380C2314yes—
190#379—21no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—22no—
193C4×C22⋊C4—21no—
194C4×C22⋊C4—21no—
195C4×C22⋊C4—22no—
196C4×C22⋊C4—22no—
197C2×#95—21no—
198C2×#95—22no—
199C2×#95—22no—
200C2×#95—21no—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#336C2314yes—
210#336C2314yes—
211C25C2×D411yescharacteristic
212C2×#41—24no—
213C2×#41—24no—
214C2×#41—22no—
215C2×#41—22no—
216C22×Q8—24no—
217C22×Q8—44no—
218C22×Q8—84no—
219C22×Q8C2411yescharacteristic
220C22×D4—21no—
221C22×D4—21no—
222C22×D4—21no—
223C22×D4C2411yescharacteristic
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—21no—
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C2×M16—24no—
242C2×M16—24no—
243C2×M16—24no—
244C2×M16—24no—
245C2×M16—24no—
246C2×M16#4118yes—
247C22×C8C2414yes—
248C22×C8C2414yes—
249C4⋊Q8—44no—
250C4⋊Q8—44no—
251C4⋊Q8—48no—
252#123—44no—
253C22⋊Q8—42no—
254C22⋊Q8—42no—
255#121—42no—
256#121—42no—
257#120—44no—
258C4×D4—44no—
259C4×D4—44no—
260#117—24no—
261#117—24no—
262#117—22no—
263#117—22no—
264#117—44no—
265#117—44no—
266#117—42no—
267#117—42no—
268#117—42no—
269#117—42no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—84no—
289C2×C4⋊C4C2×D412yes—
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2414yes—
294C2×C4⋊C4C2414yes—
295C2×C4⋊C4C2411yescharacteristic
296C2×C4⋊C4C2411yescharacteristic
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—42no—
315C2×C22⋊C4—42no—
316C2×C22⋊C4—84no—
317C2×C22⋊C4C2×D412yes—
318C2×C22⋊C4C2×D412yes—
319C2×C22⋊C4C2×D412yes—
320C2×C22⋊C4C2×D412yes—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—21no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42C2×D412yes—
327C2×C42C2×D412yes—
328C2×C42C2412yes—
329C2×C42C2412yes—
330#107—28no—
331#107—44no—
332#106—28no—
333#106—44no—
334#103—28no—
335#103—44no—
336#102—28no—
337#102—44no—
338C22⋊C8—28no—
339C22⋊C8—28no—
340#97—44no—
341C4×C8—44no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—21no—
362C24—42no—
363C24—42no—
364C24C22×D411yescharacteristic
365C24C22×D411yescharacteristic
366C24C22×D411yescharacteristic
367#41—48no—
368#41—48no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8—48no—
374C2×Q8—84no—
375C2×Q8—84no—
376C2×Q8—816no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—82no—
387C2×D4—82no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
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—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—24no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4C22×D412yes—
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C2×#4114yes—
439C22×C4H(5,𝔽2)14yes—
440C22×C42−1+412yes—
441C22×C42−1+412yes—
442C22×C4C2511yescharacteristic
443M16—216no—
444M16—48no—
445C2×C8—24no—
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8C2×#4118yes—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—48no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—82no—
471C4⋊C4—82no—
472C4⋊C4—82no—
473C4⋊C4—82no—
474C4⋊C4—84no—
475C4⋊C4—84no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—82no—
488C22⋊C4—82no—
489C22⋊C4—82no—
490C22⋊C4—82no—
491C22⋊C4—82no—
492C22⋊C4—82no—
493C22⋊C4—84no—
494C22⋊C4—84no—
495C42—24no—
496C42—24no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—42no—
502C42—42no—
503C42—42no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—48no—
517C23—44no—
518C23—42no—
519C23—42no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—84no—
524C23—82no—
525C23—82no—
526C23#53212yes—
527C23#53412yes—
528C23#53612yes—
529C23C23×D411yescharacteristic
530Q8—42no—
531Q8—42no—
532Q8—816no—
533D4—42no—
534D4—42no—
535D4—84no—
536D4—84no—
537D4—84no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—88no—
587C2×C4—88no—
588C2×C4—84no—
589C2×C4#53212yes—
590C2×C4#53412yes—
591C2×C4#53612yes—
592C2×C4#53918yes—
593C2×C4#54218yes—
594C2×C4C22×#4112yes—
595C2×C4C2×H(5,𝔽2)12yes—
596C2×C4C2×2−1+411yescharacteristic
597C2×C4C2×2−1+411yescharacteristic
598C8—28no—
599C8—44no—
600C22—24no—
601C22—22no—
602C22—22no—
603C22—24no—
604C22—41no—
605C22—41no—
606C22—42no—
607C22—42no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—42no—
613C22—42no—
614C22—88no—
615C22—81no—
616C22—82no—
617C22—81no—
618C22—81no—
619C22—82no—
620C22#266314yes—
621C22C2×#53211yescharacteristic
622C22C2×#53411yescharacteristic
623C22C2×#53611yescharacteristic
624C22#358512yes—
625C22#358612yes—
626C4—41no—
627C4—41no—
628C4—42no—
629C4—42no—
630C4—42no—
631C4—42no—
632C4—81no—
633C4—82no—
634C4—84no—
635C4—84no—
636C4—81no—
637C4—81no—
638C4—82no—
639C4#266314yes—
640C4C2×#53912yes—
641C4C2×#54212yes—
642C2—24no—
643C2—81no—
644C2—81no—
645C2—161no—
646C2C2×#266311yescharacteristic
647C2#6070312yes—
648C2#6075311yescharacteristic
649C2#6075411yescharacteristic
650C2#6086712yes—
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×#2628C2semidirect
3. #30374C2semidirect
4. #30372C2semidirect
5. #30370C2semidirect
6. #30368C2semidirect
7. #23746C2semidirect
8. #23733C2semidirect
9. #23642C2semidirect
10. #23552C2semidirect
12. #23452C2semidirect
13. #21159C2semidirect
14. #21093C2semidirect
15. #20377C2semidirect
51. #2101C22semidirect
52. #2101C22semidirect
53. #2097C22semidirect
54. #2092C22semidirect
59. #2058C22semidirect
60. #2057C22semidirect
65. #2024C22semidirect
66. #2023C22semidirect
67. #1944C22semidirect
68. #1943C22semidirect
69. #1942C22semidirect
70. #1941C22semidirect
71. #1936C22semidirect
72. #1935C22semidirect
73. #1934C22semidirect
74. #1933C22semidirect

Maximal subgroups

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

subgroupindex
#346352
C2×#26282
#303742
#303722
#303702
#303682
#237462
#237332
#236422
#235522
#234822
#234522
#211592
#210932
#203772