Finite Group Databasebeta

← #6540316

Subgroups — #6540316

Order 512.

1697 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36134C211yesmaximal, characteristic, complemented
2#34316C211yesmaximal, characteristic
3#30168C212yesmaximal, complemented
4#29981C211yesmaximal, characteristic, complemented
5#29141C211yesmaximal, characteristic, complemented
6#29113C212yesmaximal, complemented
7#28161C211yesmaximal, characteristic, complemented
8#26682C214yesmaximal, complemented
9#24972C212yesmaximal, complemented
10#24924C214yesmaximal, complemented
11#24241C212yesmaximal
12#23935C212yesmaximal, complemented
13#23677C211yesmaximal, characteristic, complemented
14#23577C212yesmaximal, complemented
15#23004C212yesmaximal, complemented
16#22839C211yesmaximal, characteristic, complemented
17#22759C212yesmaximal, complemented
18C2×#547C2211yescharacteristic
19C2×#533C2212yes—
20C2×#462C2212yes—
21C2×#452C2214yes—
22C2×#441C2214yes—
23C2×C4×SD16C2212yes—
24#2868C2211yescharacteristic
25#2857C2212yes—
26#2552—21no—
27#2542—21no—
28#2538C2212yes—
29#2441—22no—
30#2429—22no—
31#2423C2214yes—
32#2374—22no—
33#2365—21no—
34#2365C2211yescharacteristic
35#2363—21no—
36#2362C2212yes—
37#2354—21no—
38#2354C2211yescharacteristic
39#2354C2211yescharacteristic
40Q8×C4⋊C4C2212yes—
41#2345—21no—
42#2341C2212yes—
43#2335—22no—
44#2324—21no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×C22⋊Q8C2212yes—
48#2297C2212yes—
49#2293C2211yescharacteristic
50#2274—21no—
51C42×D4C2211yescharacteristic
52#2100C2212yes—
53#2094C2212yes—
54#2084C2211yescharacteristic
55#2077C2214yes—
56#2072C2212yes—
57#2071C2211yescharacteristic
58#2066C2212yes—
59#2058C2212yes—
60#2055C2212yes—
61#2043C2211yescharacteristic
62#2035C2214yes—
63#2035C2214yes—
64#2024C2212yes—
65#2024C2212yes—
66#2023C2212yes—
67#2016C2212yes—
68#2013C2211yescharacteristic
69#2001C2214yes—
70#1956C2212yes—
71#1954C2212yes—
72#1948C2212yes—
73#1936C2214yes—
74#1935C2214yes—
75#1924C2214yes—
76#1919C2214yes—
77#1880C2214yes—
78#1878C2214yes—
79#1878C2214yes—
80#1873C2214yes—
81#1865C2214yes—
82#1834C2212yes—
83#1831C2211yescharacteristic
84#1825C2214yes—
85#1817C2212yes—
86#1803C2212yes—
87#1803C2214yes—
88#1802C2212yes—
89#1788C2214yes—
90#1770C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92#1763C2214yes—
93C4×#103C2212yes—
94#1749C2212yes—
95C22×#41C2312yes—
96C22×Q16C2314yes—
97C22×SD16C2312yes—
98#547—24no—
99#547—24no—
100#533—48no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×#125—22no—
107C2×#123—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8C2312yes—
111C2×C22⋊Q8C2314yes—
112C2×#121—21no—
113C2×C4×Q8—22no—
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2314yes—
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2312yes—
118C2×C4×D4D411yescharacteristic
119C2×C4×D4D411yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2311yescharacteristic
122C2×#117—22no—
123C2×#117—21no—
124C2×#117—21no—
125C2×#117C2312yes—
126C22×C4⋊C4—21no—
127C22×C42C2312yes—
128#462—28no—
129#452—48no—
130#441—48no—
131C4×SD16—28no—
132C2×#106C2312yes—
133C2×#106C2314yes—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2312yes—
136C2×#103C2312yes—
137C2×#103C2312yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2312yes—
143C2×#102C2312yes—
144C2×#102C2314yes—
145C2×C22⋊C8C2314yes—
146C2×C22⋊C8C2314yes—
147C2×#97C2314yes—
148C2×C4×C8C2312yes—
149#399—24no—
150#398—24no—
151#396—24no—
152#391—22no—
153#391C2312yes—
154#391C2312yes—
155#391C2312yes—
156#390—21no—
157#390—22no—
158#390—22no—
159#390—22no—
160#390C2311yescharacteristic
161#390C2311yescharacteristic
162#390C2312yes—
163#389—21no—
164#389—22no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—22no—
170#389—24no—
171#389C2311yescharacteristic
172#389C2311yescharacteristic
173#389C2314yes—
174#388—22no—
175#388—22no—
176#388—22no—
177#388—24no—
178#388—21no—
179#388—21no—
180#388—22no—
181#387—22no—
182#387—22no—
183#387—22no—
184#387—24no—
185#386—22no—
186#386—22no—
187#385—22no—
188#384—22no—
189#384—21no—
190#384—21no—
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#384C2312yes—
194#384C2312yes—
195#380—22no—
196#380—21no—
197#380—22no—
198#380C2312yes—
199#380C2314yes—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C22⋊C4—22no—
215C4×C22⋊C4—22no—
216C4×C22⋊C4—21no—
217C4×C22⋊C4—22no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4D412yes—
220C4×C22⋊C4D412yes—
221C4×C22⋊C4C2312yes—
222C43C2311yescharacteristic
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#336C2312yes—
231#336C2311yescharacteristic
232#336C2311yescharacteristic
233#336C2314yes—
234#336C2312yes—
235C2×#41—24no—
236C2×#41—24no—
237C2×#41—44no—
238C2×#41—44no—
239C22×Q8—22no—
240C22×Q8C2412yes—
241C22×Q8C2412yes—
242C22×D4—22no—
243C22×D4C2×D411yescharacteristic
244C22×D4C2411yescharacteristic
245C23×C4—21no—
246C23×C4—22no—
247C23×C4C2×D412yes—
248C23×C4C2×D412yes—
249C23×C4C2412yes—
250C2×Q16—28no—
251C2×Q16—28no—
252C2×Q16—44no—
253C2×SD16—24no—
254C2×SD16—24no—
255C2×SD16—42no—
256C22×C8C2412yes—
257C22×C8C2414yes—
258C22×C8C2412yes—
259C4⋊Q8—24no—
260C4⋊Q8—22no—
261C4⋊Q8—44no—
262C4⋊Q8—44no—
263C4⋊Q8—82no—
264C4⋊Q8#4114yes—
265#125—48no—
266#123—216no—
267C22⋊Q8—24no—
268C22⋊Q8—24no—
269C22⋊Q8—28no—
270C22⋊Q8—44no—
271C22⋊Q8—48no—
272C22⋊Q8—84no—
273#121—28no—
274C4×Q8—28no—
275C4×Q8—24no—
276C4×Q8—24no—
277C4×Q8—44no—
278C4×Q8—44no—
279C4×Q8—44no—
280C4×Q8—48no—
281C4×D4—24no—
282C4×D4—22no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—42no—
286C4×D4#4114yes—
287#117—44no—
288#117—48no—
289#117—44no—
290#117—44no—
291C2×C4⋊C4—28no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2414yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4C2×D412yes—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D412yes—
338C2×C42—24no—
339C2×C42—22no—
340C2×C42—21no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—44no—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D412yes—
349C2×C42C2×D412yes—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2412yes—
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356#106—24no—
357#106—44no—
358C4⋊C8—24no—
359C4⋊C8—42no—
360#103—28no—
361#103—24no—
362#103—24no—
363#103—42no—
364#103—44no—
365#103—44no—
366#103—44no—
367#102—24no—
368#102—44no—
369C22⋊C8—44no—
370C22⋊C8—44no—
371#97—44no—
372C4×C8—24no—
373#95—21no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—21no—
380#95—24no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—21no—
385#95—24no—
386#95C2×D412yes—
387#95C2×D412yes—
388#95C2×D412yes—
389C24—21no—
390C24C22×D412yes—
391#41—88no—
392#41—88no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—24no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—24no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8C2×#4114yes—
405C2×Q8H(5,𝔽2)14yes—
406C2×D4—28no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—21no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—82no—
414C2×D4—82no—
415C2×D4C2×#4112yes—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—28no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—82no—
439C22×C4—82no—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D412yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C2511yescharacteristic
450Q16—416no—
451SD16—48no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—24no—
455C2×C8—22no—
456C2×C8—24no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—24no—
460C2×C8—22no—
461C4⋊C4—28no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—48no—
478C4⋊C4—48no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—81no—
489C4⋊C4—82no—
490C4⋊C4—82no—
491C4⋊C4—84no—
492C4⋊C4—82no—
493C4⋊C4—82no—
494C4⋊C4C2×#4112yes—
495C22⋊C4—28no—
496C22⋊C4—24no—
497C22⋊C4—24no—
498C22⋊C4—24no—
499C22⋊C4—42no—
500C22⋊C4—48no—
501C22⋊C4—44no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—81no—
506C22⋊C4—82no—
507C22⋊C4—82no—
508C22⋊C4—82no—
509C22⋊C4—84no—
510C42—22no—
511C42—22no—
512C42—24no—
513C42—22no—
514C42—22no—
515C42—42no—
516C42—42no—
517C42—41no—
518C42—48no—
519C42—44no—
520C42—44no—
521C42—42no—
522C42—82no—
523C42—82no—
524C42—82no—
525C42#13714yes—
526C42C2×#4112yes—
527C42C2×#4112yes—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23C23×D411yescharacteristic
543Q8—24no—
544Q8—24no—
545Q8—28no—
546Q8—42no—
547Q8—44no—
548Q8—42no—
549Q8—88no—
550D4—24no—
551D4—44no—
552D4—42no—
553D4—88no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—48no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—48no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—84no—
603C2×C4—88no—
604C2×C4#52512yes—
605C2×C4#52512yes—
606C2×C4C2×#13712yes—
607C2×C4C2×#13712yes—
608C2×C4C2×#13712yes—
609C2×C4#57512yes—
610C2×C4#57712yes—
611C2×C4C22×#4111yescharacteristic
612C2×C4C22×#4111yescharacteristic
613C2×C4C2×H(5,𝔽2)12yes—
614C2×C4#58512yes—
615C8—42no—
616C8—42no—
617C8—44no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22C2×#52511yescharacteristic
633C22#348411yescharacteristic
634C22#350011yescharacteristic
635C22C22×#13711yescharacteristic
636C22C2×#57511yescharacteristic
637C22C2×#57711yescharacteristic
638C22#358611yescharacteristic
639C4—22no—
640C4—22no—
641C4—22no—
642C4—22no—
643C4—42no—
644C4—41no—
645C4—42no—
646C4—42no—
647C4—44no—
648C4—44no—
649C4—42no—
650C4—81no—
651C4—82no—
652C4—81no—
653C4—84no—
654C4—82no—
655C4#311614yes—
656C4#349412yes—
657C4#351312yes—
658C2—82no—
659C2—82no—
660C2#3573011yescharacteristic
661C2C2×#311611yescharacteristic
662C2#6059911yescharacteristic
663C2#6158411yescharacteristic
664C2#6159611yescharacteristic
665C2#6254211yescharacteristic
666C2#6258711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36134C2semidirect
3. #30168C2semidirect
4. #29981C2semidirect
5. #29141C2semidirect
6. #29113C2semidirect
7. #28161C2semidirect
8. #26682C2semidirect
9. #24972C2semidirect
10. #24924C2semidirect
12. #23935C2semidirect
13. #23677C2semidirect
14. #23577C2semidirect
15. #23004C2semidirect
16. #22839C2semidirect
17. #22759C2semidirect

Maximal subgroups

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

subgroupindex
#361342
#343162
#301682
#299812
#291412
#291132
#281612
#266822
#249722
#249242
#242412
#239352
#236772
#235772
#230042
#228392
#227592