Finite Group Databasebeta

← #6993605

Subgroups — #6993605

Order 512.

1555 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36272C211yesmaximal, characteristic
2#35320C211yesmaximal, characteristic, complemented
3#32821C212yesmaximal, complemented
4#32300C214yesmaximal, complemented
5#31953C214yesmaximal
6#30310C212yesmaximal, complemented
7#29982C212yesmaximal, complemented
8#29688C211yesmaximal, characteristic, complemented
9#29560C212yesmaximal, complemented
10#28540C212yesmaximal, complemented
11#27737C212yesmaximal, complemented
12#25813C211yesmaximal, characteristic, complemented
13#25456C212yesmaximal, complemented
14#24816C212yesmaximal, complemented
15#24298C211yesmaximal, characteristic
16#23568C211yesmaximal, characteristic
17#22636C211yesmaximal, characteristic, complemented
18C2×#552—21no—
19C2×#533C2211yescharacteristic
20C22×C4⋊Q8C2211yescharacteristic
21C2×#500C2212yes—
22C2×#475C2212yes—
23C2×#474C2212yes—
24C2×C4×SD16C2212yes—
25#2862C2212yes—
26#2862C2212yes—
27#2859C2212yes—
28#2857C2211yescharacteristic
29#2642—22no—
30#2611—22no—
31#2568—21no—
32#2568—21no—
33#2568C2212yes—
34#2568C2212yes—
35#2553—22no—
36#2552C2211yescharacteristic
37#2542C2211yescharacteristic
38#2503—22no—
39#2503C2214yes—
40#2496C2212yes—
41#2489—21no—
42#2422C2212yes—
43#2415—21no—
44#2365C2211yescharacteristic
45#2360—21no—
46D4×C4⋊C4C2211yescharacteristic
47#2310—21no—
48#2310—21no—
49C4×#121C2211yescharacteristic
50#2293C2212yes—
51C4×#117C2211yescharacteristic
52#2101C2214yes—
53#2095C2214yes—
54#2092C2214yes—
55#2085C2214yes—
56#2085C2214yes—
57#2072C2214yes—
58#2068C2214yes—
59#2068C2214yes—
60#2058C2211yescharacteristic
61#2058C2212yes—
62#2057C2212yes—
63#2057C2212yes—
64#2057C2212yes—
65#2043C2211yescharacteristic
66#2043C2212yes—
67#2035C2212yes—
68#2035C2214yes—
69#2023C2211yescharacteristic
70#2023C2212yes—
71#2013C2211yescharacteristic
72#2000C2212yes—
73#1990C2212yes—
74#1988C2212yes—
75#1973C2214yes—
76#1949C2214yes—
77#1939C2212yes—
78#1935C2214yes—
79#1928C2214yes—
80#1924C2214yes—
81#1877C2214yes—
82#1874C2212yes—
83#1852C2212yes—
84#1852C2214yes—
85#1847C2214yes—
86#1836C2214yes—
87#1831C2212yes—
88#1804C2214yes—
89#1803C2212yes—
90#1797C2211yescharacteristic
91#1770C2212yes—
92C4×#103C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94#1745C2212yes—
95C23×Q8—21no—
96C22×SD16C2312yes—
97#552—48no—
98#533—44no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—44no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×#126—22no—
108C2×#125—21no—
109C2×#125—22no—
110C2×#125—21no—
111C2×#123—22no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8C2312yes—
117C2×#121C2311yescharacteristic
118C2×C4×Q8—21no—
119C2×C4×Q8—21no—
120C2×C4×Q8C2312yes—
121C2×C4×D4—21no—
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2312yes—
124C2×#117—21no—
125C2×#117—22no—
126C2×#117D411yescharacteristic
127C2×#117D411yescharacteristic
128C22×C4⋊C4—21no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4C2312yes—
131C22×C42C2311yescharacteristic
132#500—44no—
133#475—44no—
134#474—44no—
135C4×SD16—44no—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×#106C2314yes—
140C2×#106C2312yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2314yes—
143C2×#103C2312yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×#102C2312yes—
149C2×#102C2314yes—
150C2×C22⋊C8C2312yes—
151C2×C22⋊C8C2312yes—
152C2×#97C2314yes—
153C2×C4×C8C2312yes—
154#400—24no—
155#400—24no—
156#400—24no—
157#400—24no—
158#400—22no—
159#399—22no—
160#398—22no—
161#397—22no—
162#397—22no—
163#394—22no—
164#391—22no—
165#391—22no—
166#391—22no—
167#391—22no—
168#391—21no—
169#391—22no—
170#391—22no—
171#391—22no—
172#391—22no—
173#391—22no—
174#391—22no—
175#391—21no—
176#391C2314yes—
177#391C2314yes—
178#390C2312yes—
179#390C2311yescharacteristic
180#389—21no—
181#389—21no—
182#389—22no—
183#389—22no—
184#389C2312yes—
185#389C2311yescharacteristic
186#389C2312yes—
187#389C2312yes—
188#389C2312yes—
189#388—21no—
190#388—22no—
191#388—22no—
192#387—22no—
193#387—22no—
194#386—21no—
195#385—22no—
196#385—22no—
197#384C2311yescharacteristic
198#384C2312yes—
199#384C2312yes—
200#382—22no—
201#382—22no—
202#382—22no—
203#382C2314yes—
204#382C2314yes—
205#381—24no—
206#381—24no—
207#380—21no—
208#380—21no—
209#380C2312yes—
210#380C2311yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C22⋊C4—21no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4D412yes—
220C4×C22⋊C4C2312yes—
221#376D412yes—
222C43C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2314yes—
230#336C2312yes—
231#336C2311yescharacteristic
232#336C2311yescharacteristic
233#336C2314yes—
234#336C2314yes—
235C22×Q8—21no—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—44no—
240C22×Q8—44no—
241C22×Q8—44no—
242C22×Q8C2412yes—
243C22×D4—21no—
244C22×D4C2411yescharacteristic
245C23×C4—22no—
246C23×C4—21no—
247C23×C4—22no—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2411yescharacteristic
251C2×SD16—24no—
252C2×SD16—24no—
253C2×SD16—42no—
254C22×C8C2412yes—
255C22×C8C2414yes—
256C22×C8C2412yes—
257C4⋊Q8—24no—
258C4⋊Q8—24no—
259C4⋊Q8—24no—
260C4⋊Q8—44no—
261C4⋊Q8—42no—
262C4⋊Q8—42no—
263C4⋊Q8—44no—
264C4⋊Q8—44no—
265#126—48no—
266#125—44no—
267#125—48no—
268#125—44no—
269#123—48no—
270C22⋊Q8—42no—
271C22⋊Q8—42no—
272C22⋊Q8—44no—
273C22⋊Q8—44no—
274C22⋊Q8—44no—
275C22⋊Q8—44no—
276#121—22no—
277#121—41no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—44no—
281C4×Q8—44no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—42no—
285C4×D4—44no—
286#117—24no—
287#117—48no—
288#117—42no—
289#117—44no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2414yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C22⋊C4—22no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4C2×D412yes—
335C2×C22⋊C4C2×D412yes—
336C2×C42—24no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—21no—
340C2×C42C2×D412yes—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2412yes—
347C2×C42C2411yescharacteristic
348C2×C42C2412yes—
349C2×C42C2411yescharacteristic
350#107—44no—
351#107—44no—
352#107—44no—
353#106—42no—
354#106—44no—
355C4⋊C8—24no—
356C4⋊C8—44no—
357#103—24no—
358#103—44no—
359#103—42no—
360#103—42no—
361#103—44no—
362#102—42no—
363#102—44no—
364C22⋊C8—24no—
365C22⋊C8—42no—
366#97—28no—
367C4×C8—24no—
368#95—22no—
369#95—21no—
370#95—21no—
371#95—22no—
372#95—21no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—22no—
385#95C2×D412yes—
386#95C2×D412yes—
387#95C2×D412yes—
388#95C2×D412yes—
389C24—21no—
390C24C22×D411yescharacteristic
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—42no—
398C2×Q8—44no—
399C2×Q8—42no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—42no—
407C2×Q8—44no—
408C2×Q8—42no—
409C2×Q8—816no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—21no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—81no—
416C2×D42−1+412yes—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—41no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—82no—
443C22×C4AGL(1,ℤ/8)14yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C2511yescharacteristic
454SD16—48no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—24no—
459C2×C8—24no—
460C2×C8—22no—
461C2×C8—24no—
462C2×C8—22no—
463C2×C82−1+414yes—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—42no—
497C4⋊C4—81no—
498C4⋊C4—82no—
499C22⋊C4—24no—
500C22⋊C4—42no—
501C22⋊C4—44no—
502C22⋊C4—42no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—42no—
507C22⋊C4—44no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C42—22no—
512C42—22no—
513C42—24no—
514C42—24no—
515C42—22no—
516C42—24no—
517C42—22no—
518C42—22no—
519C42—42no—
520C42—42no—
521C42—44no—
522C42—44no—
523C42—42no—
524C42H(5,𝔽2)14yes—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—22no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23C2×AGL(1,ℤ/8)12yes—
539C23C23×D411yescharacteristic
540Q8—42no—
541Q8—42no—
542Q8—42no—
543Q8—42no—
544Q8—84no—
545Q8—88no—
546Q8—84no—
547Q8—88no—
548D4—24no—
549D4—42no—
550D4—84no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—28no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—44no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—88no—
601C2×C4—88no—
602C2×C4—84no—
603C2×C4#53512yes—
604C2×C4#53512yes—
605C2×C4C2×AGL(1,ℤ/8)12yes—
606C2×C4C2×AGL(1,ℤ/8)12yes—
607C2×C4#57512yes—
608C2×C4#57812yes—
609C2×C4C2×H(5,𝔽2)12yes—
610C2×C4C2×2−1+411yescharacteristic
611C2×C4C2×2−1+412yes—
612C2×C4#58511yescharacteristic
613C8—24no—
614C8—42no—
615C8—44no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22C2×#53511yescharacteristic
631C22#348911yescharacteristic
632C22#349911yescharacteristic
633C22C22×AGL(1,ℤ/8)11yescharacteristic
634C22C2×#57511yescharacteristic
635C22C2×#57811yescharacteristic
636C22#358611yescharacteristic
637C4—22no—
638C4—22no—
639C4—22no—
640C4—42no—
641C4—42no—
642C4—42no—
643C4—42no—
644C4—41no—
645C4—81no—
646C4—81no—
647C4—84no—
648C4—82no—
649C4—82no—
650C4—82no—
651C4—81no—
652C4—82no—
653C4#321314yes—
654C4#353812yes—
655C4#354712yes—
656C2—42no—
657C2—82no—
658C2#4299411yescharacteristic
659C2C2×#321311yescharacteristic
660C2#6081411yescharacteristic
661C2#6210911yescharacteristic
662C2#6213311yescharacteristic
663C2#6247911yescharacteristic
664C2#6251111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #35320C2semidirect
3. #32821C2semidirect
4. #32300C2semidirect
6. #30310C2semidirect
7. #29982C2semidirect
8. #29688C2semidirect
9. #29560C2semidirect
10. #28540C2semidirect
11. #27737C2semidirect
12. #25813C2semidirect
13. #25456C2semidirect
14. #24816C2semidirect
17. #22636C2semidirect

Maximal subgroups

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

subgroupindex
#362722
#353202
#328212
#323002
#319532
#303102
#299822
#296882
#295602
#285402
#277372
#258132
#254562
#248162
#242982
#235682
#226362