Finite Group Databasebeta

← #6434297

Subgroups — #6434297

Order 512.

1726 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34637C211yesmaximal, characteristic
2C2×#2645C211yesmaximal, characteristic, complemented
3#30353C214yesmaximal, complemented
4#30352C214yesmaximal, complemented
5#30351C214yesmaximal, complemented
6#30350C214yesmaximal, complemented
7#23617C212yesmaximal, complemented
8#23554C211yesmaximal, characteristic, complemented
9#23547C212yesmaximal, complemented
10#23508C211yesmaximal, characteristic
11#23455C211yesmaximal, characteristic, complemented
12#23429C211yesmaximal, characteristic
13#21237C212yesmaximal, complemented
14#21093C212yesmaximal, complemented
15#20429C211yesmaximal, characteristic, complemented
16C22×#137C2212yes—
17C2×#533C2211yescharacteristic
18C2×#532C2211yescharacteristic
19C2×#428C2212yes—
20C2×#423C2212yes—
21C2×#417C2212yes—
22C2×#417C2212yes—
23C2×#417C2212yes—
24C2×#419C2212yes—
25C2×C4×M16C2212yes—
26#2645—22no—
27#2645—22no—
28#2645C2214yes—
29#2645C2214yes—
30#2642C2212yes—
31#2641C2212yes—
32#2640C2212yes—
33#2640C2212yes—
34#2639—21no—
35#2639—21no—
36#2639—21no—
37#2639—21no—
38C2×#401—21no—
39#2398—22no—
40#2398—22no—
41C2×#400—21no—
42C2×#400—21no—
43C2×#400—21no—
44#2297C2212yes—
45#2297C2212yes—
46C2×#391C2211yescharacteristic
47C2×#391C2211yescharacteristic
48C2×#391C2211yescharacteristic
49C2×#389C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#389C2211yescharacteristic
52C2×#376C2211yescharacteristic
53#2068C2214yes—
54#2067C2214yes—
55#2066C2214yes—
56#2065C2214yes—
57#2059C2214yes—
58#2058C2212yes—
59#2057C2212yes—
60#2056C2214yes—
61#2056C2214yes—
62#2055C2214yes—
63#2055C2214yes—
64#2043C2212yes—
65#2042C2214yes—
66#2041C2212yes—
67#2026C2214yes—
68#2024C2212yes—
69#2024C2212yes—
70#2016C2212yes—
71#2015C2214yes—
72#2014C2212yes—
73#1973C2214yes—
74#1971C2214yes—
75#1969C2214yes—
76#1968C2214yes—
77#1936C2214yes—
78#1935C2214yes—
79#1934C2214yes—
80#1933C2214yes—
81#1768C2214yes—
82#1768C2214yes—
83#1736C2212yes—
84#1736C2212yes—
85#1728C2211yescharacteristic
86#1728C2211yescharacteristic
87C22×#41C2312yes—
88C23×Q8C2311yescharacteristic
89C2×#137—28no—
90C2×#137—44no—
91C2×#137—44no—
92C22×Q16C2314yes—
93C22×SD16C2314yes—
94C22×M16C2312yes—
95C22×M16C2312yes—
96#533—44no—
97#532—44no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×#127C2311yescharacteristic
102C2×#125C2312yes—
103C2×#124C2312yes—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8—21no—
106C2×#121—21no—
107C2×#121—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8—21no—
110C2×C4×D4—21no—
111C2×C4×D4—21no—
112C2×#117C2312yes—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122#428—44no—
123C2×#107C2314yes—
124C2×#106C2314yes—
125#423—28no—
126C2×C4⋊C8C2314yes—
127C2×C4⋊C8C2314yes—
128#419—44no—
129#417—28no—
130#417—44no—
131#417—44no—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C4×M16—44no—
141C2×#97C2314yes—
142C2×C4×C8C2314yes—
143#401—24no—
144#401—24no—
145#400—22no—
146#400—22no—
147#400—22no—
148#400—22no—
149#400—24no—
150#400—24no—
151#400—24no—
152#400—24no—
153#400—24no—
154#400—24no—
155#397—22no—
156#397—22no—
157#397—22no—
158#397—22no—
159#394—22no—
160#394—22no—
161#394—22no—
162#394—22no—
163#393—22no—
164#393—22no—
165#393—22no—
166#393—22no—
167#391—21no—
168#391—21no—
169#391—22no—
170#391—22no—
171#391C2314yes—
172#391C2312yes—
173#391C2312yes—
174#391C2312yes—
175#391C2312yes—
176#391C2314yes—
177#391C2312yes—
178#391C2312yes—
179#390C2312yes—
180#390C2312yes—
181#390C2312yes—
182#390C2312yes—
183#389—21no—
184#389—21no—
185#389—22no—
186#389—22no—
187#389C2312yes—
188#389C2312yes—
189#389C2314yes—
190#389C2314yes—
191#386—22no—
192#386—22no—
193#386—22no—
194#386—22no—
195C4×C22⋊C4—22no—
196C4×C22⋊C4—22no—
197C4×C22⋊C4—22no—
198C4×C22⋊C4—22no—
199#376D412yes—
200#376D412yes—
201#376C2312yes—
202#376C2312yes—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95—21no—
210C2×#95—21no—
211C2×#95—21no—
212C2×#95D411yescharacteristic
213C2×#95D411yescharacteristic
214C2×#95D411yescharacteristic
215C2×#95D411yescharacteristic
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#336C2312yes—
221#336C2312yes—
222#336C2312yes—
223#336C2312yes—
224C2×#41—24no—
225C2×#41—24no—
226C2×#41—24no—
227C2×#41—24no—
228C2×#41—24no—
229C2×#41—24no—
230C22×Q8—21no—
231C22×Q8—21no—
232C22×Q8—21no—
233C22×Q8—24no—
234C22×Q8—22no—
235C22×Q8—24no—
236C22×Q8—22no—
237C22×Q8C2412yes—
238C22×Q8C2412yes—
239C22×D4—22no—
240C22×D4—21no—
241C22×D4—21no—
242C22×D4C2412yes—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—22no—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2411yescharacteristic
255#137—432no—
256C2×Q16—28no—
257C2×Q16—44no—
258C2×Q16—44no—
259C2×SD16—28no—
260C2×SD16—44no—
261C2×SD16—44no—
262C2×M16—24no—
263C2×M16—24no—
264C2×M16—24no—
265C2×M16—24no—
266C2×M16—24no—
267C2×M16#4118yes—
268C22×C8C2414yes—
269C22×C8C2414yes—
270C4⋊Q8—42no—
271C4⋊Q8—42no—
272C4⋊Q8—42no—
273#127—42no—
274#125—42no—
275#125—42no—
276#124—42no—
277#124—42no—
278C22⋊Q8—44no—
279C22⋊Q8—44no—
280#121—44no—
281#121—44no—
282C4×Q8—44no—
283C4×Q8—44no—
284C4×D4—44no—
285C4×D4—44no—
286#117—24no—
287#117—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—82no—
306C2×C4⋊C4—82no—
307C2×C4⋊C4—82no—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D412yes—
332C2×C42C2×D412yes—
333C2×C42C2411yescharacteristic
334C2×C42C2411yescharacteristic
335C2×C42C2412yes—
336C2×C42C2412yes—
337#107—44no—
338#106—44no—
339C4⋊C8—28no—
340C4⋊C8—28no—
341#103—28no—
342#103—44no—
343#103—44no—
344#103—44no—
345#103—44no—
346#102—28no—
347#102—44no—
348#102—44no—
349#97—44no—
350C4×C8—44no—
351#95—24no—
352#95—24no—
353#95—22no—
354#95—22no—
355#95—24no—
356#95—24no—
357#95—22no—
358#95—22no—
359#95—24no—
360#95—24no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95C2×D412yes—
372#95C2×D412yes—
373#95C2×D412yes—
374#95C2×D412yes—
375#95C2×D412yes—
376#95C2×D412yes—
377#95C2×D412yes—
378#95C2×D412yes—
379C24—22no—
380C24C22×D411yescharacteristic
381#41—48no—
382#41—48no—
383#41—48no—
384#41—48no—
385C2×Q8—22no—
386C2×Q8—24no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—24no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8—48no—
401C2×Q8—48no—
402C2×Q8—81no—
403C2×Q8—81no—
404C2×Q8—81no—
405C2×Q8—81no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—81no—
418C2×D4—81no—
419C2×D4—81no—
420C2×D4—81no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C2×#4114yes—
471C22×C4H(5,𝔽2)14yes—
472C22×C4H(5,𝔽2)12yes—
473C22×C42−1+412yes—
474C22×C4C2511yescharacteristic
475Q16—416no—
476SD16—416no—
477M16—216no—
478M16—48no—
479C2×C8—24no—
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—24no—
483C2×C8—24no—
484C2×C8C2×#4118yes—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—84no—
503C4⋊C4—84no—
504C4⋊C4—82no—
505C4⋊C4—82no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—82no—
510C22⋊C4—82no—
511C22⋊C4—82no—
512C22⋊C4—82no—
513C22⋊C4—82no—
514C22⋊C4—82no—
515C22⋊C4—82no—
516C22⋊C4—82no—
517C42—24no—
518C42—22no—
519C42—22no—
520C42—24no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—44no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23#53212yes—
540C23#53512yes—
541C23#53712yes—
542C23C23×D411yescharacteristic
543Q8—44no—
544Q8—42no—
545Q8—42no—
546Q8—44no—
547Q8—42no—
548Q8—42no—
549Q8—84no—
550Q8—84no—
551Q8—84no—
552D4—42no—
553D4—42no—
554D4—42no—
555D4—42no—
556D4—84no—
557D4—84no—
558D4—88no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—82no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4—82no—
609C2×C4—82no—
610C2×C4—84no—
611C2×C4—84no—
612C2×C4—84no—
613C2×C4—84no—
614C2×C4—88no—
615C2×C4#53212yes—
616C2×C4#53512yes—
617C2×C4#53712yes—
618C2×C4#53818yes—
619C2×C4#54218yes—
620C2×C4C22×#4112yes—
621C2×C4C2×H(5,𝔽2)11yescharacteristic
622C2×C4C2×H(5,𝔽2)12yes—
623C2×C4C2×2−1+411yescharacteristic
624C8—28no—
625C8—44no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22#266214yes—
640C22C2×#53211yescharacteristic
641C22C2×#53511yescharacteristic
642C22C2×#53711yescharacteristic
643C22#358511yescharacteristic
644C22#358611yescharacteristic
645C22#358611yescharacteristic
646C22#358611yescharacteristic
647C4—42no—
648C4—42no—
649C4—41no—
650C4—41no—
651C4—41no—
652C4—41no—
653C4—42no—
654C4—42no—
655C4—82no—
656C4—82no—
657C4—82no—
658C4—82no—
659C4—82no—
660C4—82no—
661C4—82no—
662C4#266214yes—
663C4C2×#53812yes—
664C4C2×#54212yes—
665C2—24no—
666C2—162no—
667C2C2×#266211yescharacteristic
668C2#6070311yescharacteristic
669C2#6070511yescharacteristic
670C2#6080911yescharacteristic
671C2#6081011yescharacteristic
672C2#6091911yescharacteristic
673C2#6092111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
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
2. C2×#2645C2semidirect
3. #30353C2semidirect
4. #30352C2semidirect
5. #30351C2semidirect
6. #30350C2semidirect
7. #23617C2semidirect
8. #23554C2semidirect
9. #23547C2semidirect
11. #23455C2semidirect
13. #21237C2semidirect
14. #21093C2semidirect
15. #20429C2semidirect

Maximal subgroups

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

subgroupindex
#346372
C2×#26452
#303532
#303522
#303512
#303502
#236172
#235542
#235472
#235082
#234552
#234292
#212372
#210932
#204292