Finite Group Databasebeta

← #6451249

Subgroups — #6451249

Order 512.

1688 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35630C211yesmaximal, characteristic, complemented
2#34188C211yesmaximal, characteristic
3#27505C214yesmaximal, complemented
4#27502C214yesmaximal, complemented
5#27488C214yesmaximal, complemented
6#27487C214yesmaximal, complemented
7#23738C211yesmaximal, characteristic
8C2×#2096C211yesmaximal, characteristic, complemented
9#23658C211yesmaximal, characteristic, complemented
10#23545C211yesmaximal, characteristic
11#21234C211yesmaximal, characteristic
12#21174C211yesmaximal, characteristic, complemented
13#21068C211yesmaximal, characteristic
14#21066C211yesmaximal, characteristic, complemented
15#21064C211yesmaximal, characteristic, complemented
16#21026C211yesmaximal, characteristic, complemented
17#20927C211yesmaximal, characteristic
18#20872C211yesmaximal, characteristic
19#20454C211yesmaximal, characteristic, complemented
20C22×#137C2211yescharacteristic
21C2×C4×#41C2211yescharacteristic
22C2×#428C2211yescharacteristic
23C2×#428C2211yescharacteristic
24C2×#427C2211yescharacteristic
25C22×#106C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C2×#424C2211yescharacteristic
28C2×#417C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#416C2211yescharacteristic
31C2×#416C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C2×#405C2211yescharacteristic
35#2514—22no—
36#2514C2214yes—
37#2511—22no—
38#2511C2214yes—
39#2493—24no—
40#2355C2214yes—
41#2354C2214yes—
42#2352—24no—
43#2317C2211yescharacteristic
44#2314—22no—
45#2311C2211yescharacteristic
46#2311C2211yescharacteristic
47#2310C2211yescharacteristic
48#2310C2211yescharacteristic
49#2296C2211yescharacteristic
50C2×#391C2211yescharacteristic
51#2285—22no—
52#2285C2211yescharacteristic
53C2×#384C2211yescharacteristic
54#2274C2211yescharacteristic
55#2274C2211yescharacteristic
56#2274C2211yescharacteristic
57#2097C2212yes—
58#2096C2214yes—
59#2096C2214yes—
60#2096C2212yes—
61#2094C2212yes—
62#2092C2212yes—
63#2084C2214yes—
64#2083C2214yes—
65#2056C2214yes—
66#2055C2214yes—
67#1973C2212yes—
68#1971C2212yes—
69#1969C2212yes—
70#1968C2212yes—
71#1949C2214yes—
72#1948C2214yes—
73#1928C2214yes—
74#1928C2214yes—
75#1927C2214yes—
76#1927C2214yes—
77#1927C2214yes—
78#1926C2214yes—
79#1917C2214yes—
80#1916C2214yes—
81#1878C2214yes—
82#1877C2214yes—
83#1865C2212yes—
84#1864C2212yes—
85#1863C2212yes—
86#1862C2212yes—
87#1776C2214yes—
88C4×#106C2214yes—
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1736C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1728C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C2×#137—24no—
100C2×#137—24no—
101C2×#137—42no—
102C22×Q16C2312yes—
103C22×SD16C2312yes—
104C22×M16C2311yescharacteristic
105C22×M16C2311yescharacteristic
106C23×C8C2311yescharacteristic
107C23×C8C2311yescharacteristic
108C4×#41—24no—
109C4×#41—24no—
110C2×C4×Q8C2312yes—
111C2×C4×D4—22no—
112C2×C4×D4C2312yes—
113C2×#117—24no—
114C2×#117—21no—
115C2×#117C2311yescharacteristic
116C2×#117C2311yescharacteristic
117C2×#117C2311yescharacteristic
118C2×#117C2311yescharacteristic
119C2×#117C2311yescharacteristic
120C22×C4⋊C4—22no—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C42—21no—
125C22×C42C2311yescharacteristic
126#428—42no—
127#428—42no—
128#427—24no—
129C2×#107C2312yes—
130C2×#107C2312yes—
131C2×#107C2312yes—
132C2×#106—24no—
133C2×#106—42no—
134C2×#106C2312yes—
135C2×#106C2312yes—
136C2×#106C2312yes—
137C2×#106C2312yes—
138C2×#106C2312yes—
139C2×#106C2314yes—
140#424—24no—
141C2×C4⋊C8C2314yes—
142#419—24no—
143#417—42no—
144#416—24no—
145#416—24no—
146C2×#103—24no—
147C2×#103—42no—
148C2×#103C2312yes—
149C2×#103C2311yescharacteristic
150C2×#103C2312yes—
151C2×#103C2312yes—
152C2×#103C2311yescharacteristic
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2312yes—
156C2×#102C2312yes—
157C2×#102C2311yescharacteristic
158C2×#102C2311yescharacteristic
159C2×#102C2312yes—
160#405—24no—
161C2×#97C2312yes—
162C2×C4×C8C2312yes—
163#401—24no—
164#400—24no—
165#400—24no—
166#400—24no—
167#391—22no—
168#391—22no—
169#391—21no—
170#391—21no—
171#391C2314yes—
172#391C2312yes—
173#391C2312yes—
174#391C2312yes—
175#391C2312yes—
176#391C2314yes—
177#390C2312yes—
178#390C2312yes—
179#389—22no—
180#389C2312yes—
181#389C2312yes—
182#388—24no—
183#388—24no—
184#387—24no—
185#387—24no—
186#387—24no—
187#387—24no—
188#387—24no—
189#386—22no—
190#386—22no—
191#385—22no—
192#385—22no—
193#384—21no—
194#384—21no—
195#384D412yes—
196#384C2312yes—
197#384C2314yes—
198#383—24no—
199#382—24no—
200#382D412yes—
201#382C2312yes—
202#381—22no—
203#381—22no—
204#380—22no—
205#380—22no—
206#380—24no—
207#380D414yes—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—24no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4C2314yes—
215C4×C4⋊C4C2314yes—
216C4×C4⋊C4C2314yes—
217C4×C4⋊C4C2314yes—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—22no—
221C2×#95—22no—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#340C2314yes—
227#336C2314yes—
228#336C2312yes—
229#336C2312yes—
230#336C2312yes—
231#336C2312yes—
232#336C2314yes—
233C2×#41—22no—
234C2×#41—22no—
235C2×#41—22no—
236C2×#41—22no—
237C2×#41—22no—
238C2×#41#4114yes—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8C2412yes—
247C22×Q8C2411yescharacteristic
248C22×D4—21no—
249C22×D4C2411yescharacteristic
250C23×C4—22no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4C2×D412yes—
257C23×C4C2411yescharacteristic
258#137—416no—
259C2×Q16—24no—
260C2×Q16—24no—
261C2×Q16—42no—
262C2×SD16—24no—
263C2×SD16—24no—
264C2×SD16—42no—
265C2×M16—22no—
266C2×M16—22no—
267C2×M16—22no—
268C2×M16—22no—
269C2×M16—22no—
270C2×M16#4114yes—
271C22×C8—22no—
272C22×C8—22no—
273C22×C8—22no—
274C22×C8—22no—
275C22×C8—22no—
276C22×C8#4114yes—
277C22×C8C2412yes—
278C22×C8C2412yes—
279C22×C8C2411yescharacteristic
280C22×C8C2411yescharacteristic
281C22×C8C2412yes—
282C4×Q8—24no—
283C4×Q8—24no—
284C4×D4—216no—
285C4×D4—24no—
286C4×D4—24no—
287#117—28no—
288#117—22no—
289#117—22no—
290#117—22no—
291#117—22no—
292#117—22no—
293#117—22no—
294#117—22no—
295#117—22no—
296#117—22no—
297#117—88no—
298#117#4114yes—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—84no—
315C2×C4⋊C4C2×D414yes—
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D414yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4C2×D412yes—
333C2×C22⋊C4C2×D412yes—
334C2×C42—28no—
335C2×C42—21no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—21no—
340C2×C42—21no—
341C2×C42—24no—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2412yes—
347C2×C42C2412yes—
348C2×C42C2412yes—
349#107—24no—
350#107—42no—
351#107—42no—
352#106—24no—
353#106—24no—
354#106—24no—
355#106—44no—
356#106—42no—
357#106—42no—
358C4⋊C8—28no—
359#103—24no—
360#103—24no—
361#103—24no—
362#103—22no—
363#103—22no—
364#103—28no—
365#103—44no—
366#103—42no—
367#102—24no—
368#102—22no—
369#102—22no—
370#102—42no—
371#97—24no—
372C4×C8—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—22no—
381#95—22no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95C2×D414yes—
388#95C2×D414yes—
389C24—21no—
390C24C22×D411yescharacteristic
391#41—28no—
392#41—28no—
393#41—44no—
394#41—44no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—21no—
403C2×Q8—22no—
404C2×Q8—21no—
405C2×Q8—22no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×Q8—44no—
412C2×Q8—44no—
413C2×Q8C2×#4112yes—
414C2×D4—24no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4C2×#4112yes—
423C22×C4—28no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—42no—
437C22×C4—44no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—44no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—82no—
448C22×C4—82no—
449C22×C4—82no—
450C22×C4C22×D414yes—
451C22×C4C22×D414yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C2×#4112yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C4C2×#4112yes—
459C22×C4H(5,𝔽2)12yes—
460C22×C42−1+412yes—
461C22×C4C2511yescharacteristic
462Q16—48no—
463SD16—48no—
464M16—28no—
465M16—44no—
466C2×C8—28no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—44no—
480C2×C8C2×#4114yes—
481C2×C8C2×#4112yes—
482C2×C8C2×#4112yes—
483C4⋊C4—28no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—28no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—21no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—48no—
506C4⋊C4—84no—
507C4⋊C4—82no—
508C4⋊C4—84no—
509C4⋊C4C2×#4114yes—
510C22⋊C4—28no—
511C22⋊C4—24no—
512C22⋊C4—24no—
513C22⋊C4—24no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—84no—
517C22⋊C4—84no—
518C42—28no—
519C42—22no—
520C42—24no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—42no—
529C42—42no—
530C42—84no—
531C42—84no—
532C42—84no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—42no—
537C23—41no—
538C23—41no—
539C23—42no—
540C23#52512yes—
541C23#54712yes—
542C23#55012yes—
543C23C23×D411yescharacteristic
544Q8—22no—
545Q8—22no—
546Q8—41no—
547Q8—41no—
548Q8—42no—
549Q8—42no—
550Q8—42no—
551Q8—42no—
552Q8—84no—
553D4—22no—
554D4—22no—
555D4—48no—
556D4—41no—
557D4—41no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—48no—
572C2×C4—416no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—48no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—84no—
595C2×C4—162no—
596C2×C4—162no—
597C2×C4—162no—
598C2×C4#52512yes—
599C2×C4#52914yes—
600C2×C4#54214yes—
601C2×C4#54712yes—
602C2×C4#55012yes—
603C2×C4#55114yes—
604C2×C4#55214yes—
605C2×C4#57714yes—
606C2×C4C22×#4111yescharacteristic
607C2×C4C22×#4111yescharacteristic
608C2×C4C22×#4111yescharacteristic
609C2×C4C22×#4111yescharacteristic
610C2×C4C2×H(5,𝔽2)11yescharacteristic
611C2×C4C2×2−1+411yescharacteristic
612C8—22no—
613C8—22no—
614C8—24no—
615C8—42no—
616C8—42no—
617C22—24no—
618C22—24no—
619C22—22no—
620C22—22no—
621C22—42no—
622C22—44no—
623C22—82no—
624C22—81no—
625C22—81no—
626C22#247914yes—
627C22C2×#52511yescharacteristic
628C22C2×#54711yescharacteristic
629C22C2×#55011yescharacteristic
630C22C2×#57712yes—
631C22#358612yes—
632C4—24no—
633C4—44no—
634C4—44no—
635C4—42no—
636C4—42no—
637C4—44no—
638C4—82no—
639C4—82no—
640C4—81no—
641C4—82no—
642C4—82no—
643C4—81no—
644C4—82no—
645C4#247914yes—
646C4C2×#52911yescharacteristic
647C4C2×#54211yescharacteristic
648C4C2×#55111yescharacteristic
649C4C2×#55211yescharacteristic
650C2—24no—
651C2—82no—
652C2C2×#247911yescharacteristic
653C2#6058512yes—
654C2#6128112yes—
655C2#6141011yescharacteristic
656C2#6141811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35630C2semidirect
3. #27505C2semidirect
4. #27502C2semidirect
5. #27488C2semidirect
6. #27487C2semidirect
8. C2×#2096C2semidirect
9. #23658C2semidirect
12. #21174C2semidirect
14. #21066C2semidirect
15. #21064C2semidirect
16. #21026C2semidirect
19. #20454C2semidirect

Maximal subgroups

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

subgroupindex
#356302
#341882
#275052
#275022
#274882
#274872
#237382
C2×#20962
#236582
#235452
#212342
#211742
#210682
#210662
#210642
#210262
#209272
#208722
#204542