Finite Group Databasebeta

← #6801067

Subgroups — #6801067

Order 512.

1451 conjugacy classes of subgroups.

The subgroup lattice has 657 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36563C211yesmaximal, characteristic, complemented
2#34954C211yesmaximal, characteristic
3#33198C211yesmaximal, characteristic, complemented
4#31833C214yesmaximal, complemented
5#31169C214yesmaximal, complemented
6#29994C212yesmaximal, complemented
7#29139C211yesmaximal, characteristic
8#29131C212yesmaximal, complemented
9#28254C212yesmaximal, complemented
10#28252C211yesmaximal, characteristic
11#28196C212yesmaximal, complemented
12#26109C212yesmaximal, complemented
13#25303C212yesmaximal
14#24733C212yesmaximal
15#22722C211yesmaximal, characteristic, complemented
16#22152C212yesmaximal, complemented
17C4×#462C211yesmaximal, characteristic, complemented
18C2×#477C2212yes—
19C2×#464C2214yes—
20C2×#462C2211yescharacteristic
21C2×#460C2212yes—
22C2×#461C2212yes—
23C2×#446C2212yes—
24C2×C4×Q16C2212yes—
25#2868C2211yescharacteristic
26#2859C2212yes—
27#2593—22no—
28#2582C2211yescharacteristic
29#2582C2212yes—
30#2571C2211yescharacteristic
31#2567—21no—
32#2552—21no—
33#2550—22no—
34#2548C2211yescharacteristic
35#2548C2211yescharacteristic
36#2513—21no—
37#2513C2212yes—
38#2507—22no—
39#2451C2214yes—
40#2448—21no—
41#2448C2212yes—
42#2395—22no—
43#2369—21no—
44#2355—21no—
45Q8×C4⋊C4C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2212yes—
48C4×#127C2211yescharacteristic
49C4×#124—21no—
50C4×#124C2212yes—
51#2293—21no—
52#2293C2212yes—
53C42×Q8C2211yescharacteristic
54C42×Q8C2211yescharacteristic
55#2101C2214yes—
56#2086C2214yes—
57#2086C2214yes—
58#2067C2214yes—
59#2067C2214yes—
60#2056C2214yes—
61#2055C2214yes—
62#2055C2214yes—
63#2042C2214yes—
64#2023C2214yes—
65#1987C2212yes—
66#1981C2211yescharacteristic
67#1973C2214yes—
68#1957C2212yes—
69#1947C2211yescharacteristic
70#1947C2212yes—
71#1947C2212yes—
72#1940C2214yes—
73#1939C2212yes—
74#1936C2214yes—
75#1935C2214yes—
76#1935C2214yes—
77#1928C2212yes—
78#1880C2214yes—
79#1878C2212yes—
80#1863C2214yes—
81#1842C2212yes—
82#1842C2211yescharacteristic
83#1834C2212yes—
84#1805C2214yes—
85#1803C2212yes—
86#1803C2212yes—
87#1803C2212yes—
88#1803C2211yescharacteristic
89#1802C2212yes—
90#1802C2212yes—
91#1802C2212yes—
92#1767C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C22×Q16C2312yes—
96C22×SD16C2314yes—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×#127C2311yescharacteristic
102C2×#125—22no—
103C2×#124—21no—
104C2×#124C2312yes—
105C2×C4×Q8D411yescharacteristic
106C2×C4×Q8D411yescharacteristic
107C2×C4×Q8C2312yes—
108C2×C4×Q8C2311yescharacteristic
109C2×C4×Q8C2311yescharacteristic
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2314yes—
112C2×C4×Q8C2311yescharacteristic
113C2×C4×D4—21no—
114C2×C4×D4—21no—
115C2×C4×D4C2312yes—
116C2×#117—22no—
117#477—44no—
118#464—48no—
119#462—24no—
120#461—28no—
121#460—44no—
122#446—44no—
123C4×Q16—44no—
124C2×#107C2314yes—
125C2×#107C2312yes—
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2311yescharacteristic
128C2×C4⋊C8C2312yes—
129C2×C4⋊C8C2312yes—
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2314yes—
132C2×C4⋊C8C2314yes—
133C2×#103C2314yes—
134C2×#103C2312yes—
135C2×#103C2314yes—
136C2×#103C2312yes—
137C2×#103C2312yes—
138C2×#103C2312yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#97C2314yes—
143C2×C4×C8C2312yes—
144#400—22no—
145#399—22no—
146#398—22no—
147#398—24no—
148#398—24no—
149#397—22no—
150#397—22no—
151#397—22no—
152#395—24no—
153#394—22no—
154#394—22no—
155#391—22no—
156#391—21no—
157#391—24no—
158#391C2312yes—
159#391C2314yes—
160#391C2312yes—
161#390—21no—
162#390—21no—
163#390—21no—
164#390—21no—
165#390—21no—
166#390C2312yes—
167#390C2312yes—
168#390C2312yes—
169#390C2312yes—
170#389—21no—
171#389—22no—
172#389—21no—
173#389—22no—
174#389—21no—
175#389—21no—
176#389—22no—
177#389—22no—
178#389—22no—
179#389—24no—
180#389C2312yes—
181#389C2314yes—
182#389C2314yes—
183#389C2312yes—
184#388—22no—
185#388—22no—
186#388—22no—
187#388—22no—
188#388—22no—
189#388—22no—
190#388—22no—
191#388—22no—
192#387—24no—
193#384—22no—
194#384—21no—
195#384—21no—
196#384C2311yescharacteristic
197#384C2311yescharacteristic
198#384C2312yes—
199#384C2312yes—
200#382—21no—
201#382—22no—
202#382C2312yes—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4D412yes—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D412yes—
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2312yes—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—22no—
223C43—21no—
224C43C2311yescharacteristic
225C43C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#336C2314yes—
234#336C2312yes—
235C22×Q8—22no—
236C22×Q8C2×D411yescharacteristic
237C22×Q8C2412yes—
238C22×Q8C2411yescharacteristic
239C22×D4—21no—
240C22×D4—21no—
241C22×D4C2412yes—
242C23×C4—22no—
243C2×Q16—24no—
244C2×Q16—24no—
245C2×Q16—42no—
246C2×SD16—28no—
247C2×SD16—44no—
248C2×SD16—44no—
249C22×C8C2414yes—
250C22×C8C2412yes—
251C22×C8C2412yes—
252C4⋊Q8—22no—
253C4⋊Q8—42no—
254C4⋊Q8—42no—
255C4⋊Q8—44no—
256C4⋊Q8—82no—
257C4⋊Q8#4114yes—
258#127—22no—
259#127—22no—
260#125—84no—
261#124—24no—
262#124—42no—
263#124—44no—
264C4×Q8—24no—
265C4×Q8—22no—
266C4×Q8—22no—
267C4×Q8—22no—
268C4×Q8—22no—
269C4×Q8—22no—
270C4×Q8—24no—
271C4×Q8—24no—
272C4×Q8—22no—
273C4×Q8—22no—
274C4×Q8—42no—
275C4×Q8—41no—
276C4×Q8—44no—
277C4×Q8—44no—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—44no—
281C4×D4—82no—
282#117—84no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2414yes—
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2411yescharacteristic
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×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C42—21no—
327C2×C42—21no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—22no—
331C2×C42—21no—
332C2×C42—24no—
333C2×C42—21no—
334C2×C42C2×D412yes—
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D412yes—
338C2×C42C2×D412yes—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D412yes—
341C2×C42C2×D411yescharacteristic
342C2×C42C2412yes—
343C2×C42C2411yescharacteristic
344C2×C42C2412yes—
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347#107—42no—
348#107—44no—
349C4⋊C8—24no—
350C4⋊C8—22no—
351C4⋊C8—24no—
352C4⋊C8—22no—
353C4⋊C8—44no—
354C4⋊C8—44no—
355C4⋊C8—42no—
356#103—28no—
357#103—24no—
358#103—28no—
359#103—42no—
360#103—42no—
361#103—42no—
362#102—44no—
363#102—44no—
364#102—44no—
365#97—44no—
366C4×C8—24no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—21no—
374#95—21no—
375#95—21no—
376#95—24no—
377#95—22no—
378#95—24no—
379#95—24no—
380#95—22no—
381#95—24no—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385C24—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—22no—
399C2×Q8—44no—
400C2×Q8—82no—
401C2×Q8—82no—
402C2×Q8C2×#4114yes—
403C2×Q8H(5,𝔽2)12yes—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—81no—
415C2×D4—81no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—24no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D412yes—
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C2511yescharacteristic
438Q16—48no—
439SD16—88no—
440C2×C8—24no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—24no—
445C2×C8—22no—
446C2×C8—24no—
447C2×C8—22no—
448C2×C8—22no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—24no—
467C4⋊C4—21no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—82no—
480C4⋊C4—82no—
481C4⋊C4—81no—
482C4⋊C4—84no—
483C4⋊C4—84no—
484C4⋊C4—84no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—82no—
494C22⋊C4—82no—
495C22⋊C4—82no—
496C22⋊C4—82no—
497C22⋊C4—82no—
498C42—24no—
499C42—22no—
500C42—24no—
501C42—22no—
502C42—22no—
503C42—21no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—21no—
508C42—41no—
509C42—42no—
510C42—42no—
511C42—42no—
512C42—42no—
513C42—44no—
514C42—42no—
515C42—42no—
516C42—42no—
517C42—81no—
518C42—84no—
519C42#13714yes—
520C42C2×#4112yes—
521C42H(5,𝔽2)12yes—
522C422−1+412yes—
523C422−1+412yes—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23C23×D411yescharacteristic
532Q8—24no—
533Q8—24no—
534Q8—22no—
535Q8—22no—
536Q8—44no—
537Q8—41no—
538Q8—42no—
539Q8—42no—
540Q8—41no—
541Q8—164no—
542D4—42no—
543D4—42no—
544D4—42no—
545D4—42no—
546D4—84no—
547D4—84no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—88no—
594C2×C4#53712yes—
595C2×C4#53712yes—
596C2×C4#54812yes—
597C2×C4#54812yes—
598C2×C4C2×#13712yes—
599C2×C4C2×#13712yes—
600C2×C4C2×#13712yes—
601C2×C4#57512yes—
602C2×C4#57612yes—
603C2×C4C22×#4111yescharacteristic
604C2×C4C2×H(5,𝔽2)11yescharacteristic
605C2×C4C2×H(5,𝔽2)11yescharacteristic
606C2×C4C2×2−1+411yescharacteristic
607C2×C4C2×2−1+411yescharacteristic
608C2×C4#58511yescharacteristic
609C8—44no—
610C8—42no—
611C8—42no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22C2×#53711yescharacteristic
620C22C2×#54811yescharacteristic
621C22#348811yescharacteristic
622C22C22×#13711yescharacteristic
623C22C2×#57511yescharacteristic
624C22C2×#57611yescharacteristic
625C22#358611yescharacteristic
626C4—22no—
627C4—22no—
628C4—22no—
629C4—42no—
630C4—42no—
631C4—42no—
632C4—41no—
633C4—44no—
634C4—42no—
635C4—42no—
636C4—81no—
637C4—82no—
638C4—81no—
639C4—84no—
640C4—82no—
641C4—81no—
642C4#325214yes—
643C4#337514yes—
644C4#351511yescharacteristic
645C4#351911yescharacteristic
646C4#354411yescharacteristic
647C4#354711yescharacteristic
648C2—162no—
649C2#3912611yescharacteristic
650C2C2×#325211yescharacteristic
651C2#6092811yescharacteristic
652C2C2×#337511yescharacteristic
653C2#6132011yescharacteristic
654C2#6196411yescharacteristic
655C2#6198811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36563C2semidirect
3. #33198C2semidirect
4. #31833C2semidirect
5. #31169C2semidirect
6. #29994C2semidirect
8. #29131C2semidirect
9. #28254C2semidirect
11. #28196C2semidirect
12. #26109C2semidirect
15. #22722C2semidirect
16. #22152C2semidirect
17. C4×#462C2semidirect

Maximal subgroups

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

subgroupindex
#365632
#349542
#331982
#318332
#311692
#299942
#291392
#291312
#282542
#282522
#281962
#261092
#253032
#247332
#227222
#221522
C4×#4622