Finite Group Databasebeta

← #6489380

Subgroups — #6489380

Order 512.

1656 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#35283C211yesmaximal, characteristic, complemented
2#34208C211yesmaximal, characteristic
3#32965C212yesmaximal, complemented
4#32963C212yesmaximal, complemented
5#30086C212yesmaximal, complemented
6#30084C212yesmaximal, complemented
7#28727C211yesmaximal, characteristic, complemented
8#28727C211yesmaximal, characteristic, complemented
9#28520C211yesmaximal, characteristic, complemented
10#28519C211yesmaximal, characteristic, complemented
11#28213C212yesmaximal, complemented
12#28210C212yesmaximal, complemented
13#27956C212yesmaximal, complemented
14#27597C212yesmaximal, complemented
15#27595C212yesmaximal, complemented
16#25863C212yesmaximal
17#22224C212yesmaximal, complemented
18#22223C212yesmaximal, complemented
19#20783C211yesmaximal, characteristic, complemented
20C2×#532C2211yescharacteristic
21C2×#520C2211yescharacteristic
22C2×#520C2211yescharacteristic
23C2×#489C2212yes—
24C2×#489C2212yes—
25C2×#477C2212yes—
26C2×#475C2212yes—
27C2×#476C2212yes—
28C2×#474C2212yes—
29#2868C2211yescharacteristic
30#2862C2211yescharacteristic
31#2857C2212yes—
32#2647C2212yes—
33#2628C2212yes—
34#2511—22no—
35#2492—22no—
36#2451—22no—
37#2423—22no—
38#2364—22no—
39#2364—22no—
40#2352—22no—
41#2352—22no—
42#2331C2212yes—
43#2328C2211yescharacteristic
44#2328C2211yescharacteristic
45#2327C2211yescharacteristic
46#2327C2211yescharacteristic
47#2314C2212yes—
48#2311C2211yescharacteristic
49#2311C2211yescharacteristic
50C4×C4⋊Q8C2212yes—
51C4×#124C2211yescharacteristic
52C4×#124C2211yescharacteristic
53C4×#124C2211yescharacteristic
54C4×#124C2211yescharacteristic
55C2×#382C2211yescharacteristic
56C4×#117C2211yescharacteristic
57#2096C2212yes—
58#2095C2212yes—
59#2094C2212yes—
60#2093C2212yes—
61#2085C2214yes—
62#2085C2214yes—
63#2074C2212yes—
64#2074C2212yes—
65#2073C2212yes—
66#2073C2212yes—
67#2043C2214yes—
68#2041C2214yes—
69#1993C2212yes—
70#1993C2212yes—
71#1959C2212yes—
72#1959C2212yes—
73#1928C2214yes—
74#1927C2214yes—
75#1926C2214yes—
76#1925C2214yes—
77#1865C2214yes—
78#1864C2214yes—
79#1863C2214yes—
80#1862C2214yes—
81#1853C2214yes—
82#1852C2214yes—
83#1847C2212yes—
84#1847C2212yes—
85#1840C2212yes—
86#1832C2212yes—
87#1818C2212yes—
88#1805C2212yes—
89#1805C2212yes—
90#1804C2212yes—
91#1804C2212yes—
92#1798C2212yes—
93#1788C2214yes—
94#1788C2214yes—
95#1767C2211yescharacteristic
96#1767C2211yescharacteristic
97#1749C2212yes—
98#1745C2212yes—
99C22×#41C2311yescharacteristic
100C22×#41C2311yescharacteristic
101#532—24no—
102#532—24no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×#127C2311yescharacteristic
108C2×#124C2311yescharacteristic
109C2×#124C2311yescharacteristic
110C2×C22⋊Q8—22no—
111C2×#121—22no—
112#520—44no—
113#520—44no—
114C2×C4×Q8—22no—
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2311yescharacteristic
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4—22no—
124C2×C4×D4C2311yescharacteristic
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×#117—21no—
129C2×#117—21no—
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C22×C4⋊C4—21no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2311yescharacteristic
136C22×C42C2311yescharacteristic
137#489—28no—
138#489—28no—
139#477—44no—
140#476—44no—
141#475—44no—
142#474—44no—
143C2×#107C2314yes—
144C2×#107C2314yes—
145C2×#106C2314yes—
146C2×#106C2314yes—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2312yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×C22⋊C8C2314yes—
158C2×#97C2312yes—
159C2×#97C2312yes—
160#400—22no—
161#400—24no—
162#399—22no—
163#398—24no—
164#395—22no—
165#393—22no—
166#391—22no—
167#391—22no—
168#391—22no—
169#391—22no—
170#391—24no—
171#391C2311yescharacteristic
172#391C2311yescharacteristic
173#391C2311yescharacteristic
174#391C2311yescharacteristic
175#390C2311yescharacteristic
176#390C2311yescharacteristic
177#390C2311yescharacteristic
178#390C2311yescharacteristic
179#389—22no—
180#389—22no—
181#389C2314yes—
182#389C2314yes—
183#388—24no—
184#388—22no—
185#388—22no—
186#388—22no—
187#388—22no—
188#387—24no—
189#387—22no—
190#387—22no—
191#387—22no—
192#387—22no—
193#387—24no—
194#386—22no—
195#386—22no—
196#386—22no—
197#386—22no—
198#386—22no—
199#386—22no—
200#384—22no—
201#384C2312yes—
202#384C2312yes—
203#382—22no—
204#382C2312yes—
205#382C2312yes—
206#382C2311yescharacteristic
207#382C2311yescharacteristic
208#382C2311yescharacteristic
209#382C2311yescharacteristic
210#380—22no—
211#380—22no—
212#380—22no—
213#380C2314yes—
214C4×C4⋊C4—22no—
215C4×C4⋊C4D412yes—
216C4×C4⋊C4D412yes—
217C4×C4⋊C4C2314yes—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4D412yes—
224#376C2311yescharacteristic
225#376C2311yescharacteristic
226C2×#95—22no—
227C43C2311yescharacteristic
228C43C2311yescharacteristic
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#336C2314yes—
234#336C2312yes—
235#336C2312yes—
236#336C2314yes—
237#336C2314yes—
238C2×#41—22no—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—24no—
243C2×#41—24no—
244C2×#41—42no—
245C2×#41—42no—
246C22×Q8—22no—
247C22×Q8—22no—
248C22×Q8C2411yescharacteristic
249C22×Q8C2411yescharacteristic
250C22×D4—21no—
251C22×D4—21no—
252C22×D4—22no—
253C22×D4C2411yescharacteristic
254C22×D4C2411yescharacteristic
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—21no—
261C23×C4—21no—
262C23×C4—21no—
263C23×C4C2×D412yes—
264C23×C4C2411yescharacteristic
265C22×C8C2414yes—
266C22×C8C2414yes—
267C4⋊Q8—22no—
268C4⋊Q8—22no—
269C4⋊Q8—24no—
270C4⋊Q8—24no—
271C4⋊Q8—24no—
272C4⋊Q8—24no—
273C4⋊Q8—84no—
274#127—22no—
275#127—22no—
276#124—22no—
277#124—22no—
278#124#4114yes—
279#124#4114yes—
280C22⋊Q8—84no—
281#121—216no—
282C4×Q8—24no—
283C4×Q8—24no—
284C4×Q8—42no—
285C4×Q8—42no—
286C4×Q8—84no—
287C4×D4—216no—
288C4×D4—24no—
289C4×D4—24no—
290C4×D4—42no—
291C4×D4—42no—
292C4×D4—44no—
293C4×D4—44no—
294C4×D4—44no—
295C4×D4—44no—
296#117—24no—
297#117—24no—
298#117—44no—
299#117—44no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—42no—
315C2×C4⋊C4—42no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2414yes—
326C2×C4⋊C4C2414yes—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4C2×D411yescharacteristic
339C2×C22⋊C4C2×D411yescharacteristic
340C2×C22⋊C4C2×D412yes—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—24no—
348C2×C42C2×D412yes—
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359#107—44no—
360#107—44no—
361#106—44no—
362#106—44no—
363C4⋊C8—24no—
364C4⋊C8—24no—
365#103—28no—
366#103—28no—
367#103—44no—
368#103—44no—
369#102—28no—
370#102—28no—
371#102—44no—
372#102—44no—
373C22⋊C8—28no—
374#97—24no—
375#97—24no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—22no—
389#95—24no—
390#95—22no—
391#95C2×D412yes—
392#95C2×D412yes—
393#95C2×D412yes—
394#95C2×D412yes—
395C24—21no—
396C24—21no—
397C24C22×D411yescharacteristic
398#41—48no—
399#41—48no—
400#41—48no—
401#41—48no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×Q8—44no—
412C2×Q8—84no—
413C2×Q8C2×#4112yes—
414C2×Q8C2×#4112yes—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—28no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—82no—
429C2×D4—82no—
430C2×D4C2×#4112yes—
431C2×D4C2×#4112yes—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—24no—
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—82no—
459C22×C4—82no—
460C22×C4—82no—
461C22×C4—82no—
462C22×C4—82no—
463C22×C4—82no—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C42−1+414yes—
474C22×C4C2511yescharacteristic
475C2×C8—24no—
476C2×C8—24no—
477C2×C8—24no—
478C2×C8—24no—
479C2×C8—24no—
480C2×C8—24no—
481C4⋊C4—28no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—84no—
504C4⋊C4—84no—
505C4⋊C4—84no—
506C22⋊C4—24no—
507C22⋊C4—28no—
508C22⋊C4—28no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—82no—
516C22⋊C4—82no—
517C22⋊C4—82no—
518C22⋊C4—82no—
519C22⋊C4—84no—
520C42—24no—
521C42—21no—
522C42—21no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—42no—
527C42—42no—
528C42—42no—
529C42—42no—
530C42—42no—
531C42—42no—
532C42—42no—
533C42—42no—
534C42—84no—
535C42C2×#4112yes—
536C42C2×#4112yes—
537C42H(5,𝔽2)14yes—
538C23—22no—
539C23—22no—
540C23—21no—
541C23—21no—
542C23—22no—
543C23—22no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23#53612yes—
553C23C23×D411yescharacteristic
554Q8—24no—
555Q8—24no—
556Q8—42no—
557Q8—42no—
558Q8—88no—
559Q8—164no—
560D4—24no—
561D4—24no—
562D4—48no—
563D4—48no—
564D4—42no—
565D4—42no—
566D4—88no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—48no—
591C2×C4—48no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—44no—
610C2×C4—84no—
611C2×C4—162no—
612C2×C4—162no—
613C2×C4—162no—
614C2×C4—162no—
615C2×C4—162no—
616C2×C4—162no—
617C2×C4#52514yes—
618C2×C4#53512yes—
619C2×C4#53512yes—
620C2×C4#53612yes—
621C2×C4C22×#4111yescharacteristic
622C2×C4C22×#4111yescharacteristic
623C2×C4C2×H(5,𝔽2)12yes—
624C2×C4C2×2−1+412yes—
625C8—44no—
626C8—44no—
627C22—22no—
628C22—24no—
629C22—42no—
630C22—41no—
631C22—41no—
632C22—44no—
633C22—44no—
634C22—42no—
635C22—42no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—82no—
641C22—82no—
642C22C2×#52511yescharacteristic
643C22C2×#53511yescharacteristic
644C22C2×#53611yescharacteristic
645C22#358512yes—
646C22#358612yes—
647C4—24no—
648C4—22no—
649C4—22no—
650C4—22no—
651C4—42no—
652C4—41no—
653C4—41no—
654C4—42no—
655C4—42no—
656C4—42no—
657C4—42no—
658C4—84no—
659C4—81no—
660C4—81no—
661C4—84no—
662C4—81no—
663C4—81no—
664C4#352012yes—
665C4#352012yes—
666C2—42no—
667C2—82no—
668C2—82no—
669C2#3524011yescharacteristic
670C2#6056712yes—
671C2#6082511yescharacteristic
672C2#6082611yescharacteristic
673C2#6088712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
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
1. #35283C2semidirect
3. #32965C2semidirect
4. #32963C2semidirect
5. #30086C2semidirect
6. #30084C2semidirect
7. #28727C2semidirect
8. #28727C2semidirect
9. #28520C2semidirect
10. #28519C2semidirect
11. #28213C2semidirect
12. #28210C2semidirect
13. #27956C2semidirect
14. #27597C2semidirect
15. #27595C2semidirect
17. #22224C2semidirect
18. #22223C2semidirect
19. #20783C2semidirect

Maximal subgroups

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

subgroupindex
#352832
#342082
#329652
#329632
#300862
#300842
#287272
#285202
#285192
#282132
#282102
#279562
#275972
#275952
#258632
#222242
#222232
#207832