Finite Group Databasebeta

← #6540319

Subgroups — #6540319

Order 512.

1753 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36134C211yesmaximal, characteristic, complemented
2#34330C211yesmaximal, characteristic
3#30165C212yesmaximal, complemented
4#29984C211yesmaximal, characteristic, complemented
5#29139C211yesmaximal, characteristic, complemented
6#29113C212yesmaximal, complemented
7#28160C211yesmaximal, characteristic, complemented
8#26686C214yesmaximal, complemented
9#24970C212yesmaximal, complemented
10#24919C214yesmaximal, complemented
11#24242C212yesmaximal
12#23935C212yesmaximal, complemented
13#23679C211yesmaximal, characteristic, complemented
14#23574C212yesmaximal, complemented
15#23003C212yesmaximal, complemented
16#22839C211yesmaximal, characteristic, complemented
17#22763C212yesmaximal, complemented
18C2×#550C2211yescharacteristic
19C2×#533C2212yes—
20C2×#460C2212yes—
21C2×#452C2214yes—
22C2×#440C2214yes—
23C2×C4×Q16C2212yes—
24#2868C2211yescharacteristic
25#2857C2212yes—
26#2552—21no—
27#2542—21no—
28#2538C2212yes—
29#2441—22no—
30#2429—22no—
31#2423C2214yes—
32#2366—21no—
33#2366C2211yescharacteristic
34#2365—21no—
35#2364—22no—
36#2363C2212yes—
37#2357—21no—
38#2355C2211yescharacteristic
39#2354—21no—
40#2354C2211yescharacteristic
41Q8×C4⋊C4C2212yes—
42#2339C2212yes—
43#2333—22no—
44#2328—21no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×#127C2211yescharacteristic
47C4×#121C2212yes—
48#2297C2212yes—
49#2295C2211yescharacteristic
50#2275—21no—
51C42×D4C2211yescharacteristic
52#2100C2212yes—
53#2095C2212yes—
54#2083C2211yescharacteristic
55#2075C2214yes—
56#2072C2211yescharacteristic
57#2071C2212yes—
58#2068C2212yes—
59#2057C2212yes—
60#2056C2212yes—
61#2043C2211yescharacteristic
62#2036C2214yes—
63#2033C2214yes—
64#2024C2212yes—
65#2023C2212yes—
66#2023C2212yes—
67#2016C2211yescharacteristic
68#2013C2212yes—
69#2000C2214yes—
70#1957C2212yes—
71#1954C2212yes—
72#1947C2212yes—
73#1936C2214yes—
74#1935C2214yes—
75#1924C2214yes—
76#1918C2214yes—
77#1880C2214yes—
78#1878C2214yes—
79#1877C2214yes—
80#1873C2214yes—
81#1862C2214yes—
82#1834C2212yes—
83#1831C2211yescharacteristic
84#1825C2214yes—
85#1817C2212yes—
86#1803C2212yes—
87#1803C2212yes—
88#1802C2214yes—
89#1788C2214yes—
90#1770C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92#1763C2214yes—
93C4×#103C2212yes—
94#1749C2212yes—
95C22×#41C2312yes—
96C22×Q16C2312yes—
97C22×SD16C2314yes—
98#550—24no—
99#550—24no—
100#533—48no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#127—21no—
106C2×#127C2311yescharacteristic
107C2×#125—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8C2314yes—
111C2×#121—22no—
112C2×#121C2312yes—
113C2×C4×Q8—22no—
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2314yes—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×D4—22no—
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4D411yescharacteristic
123C2×C4×D4D411yescharacteristic
124C2×C4×D4C2312yes—
125C2×C4×D4C2312yes—
126C2×C4×D4C2311yescharacteristic
127C2×#117—22no—
128C2×#117C2312yes—
129C22×C42C2312yes—
130#460—28no—
131#452—48no—
132#440—48no—
133C4×Q16—28no—
134C2×#107C2312yes—
135C2×#107C2314yes—
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2312yes—
138C2×#103C2312yes—
139C2×#103C2312yes—
140C2×#103C2312yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2312yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×C22⋊C8C2314yes—
148C2×C22⋊C8C2314yes—
149C2×#97C2314yes—
150C2×C4×C8C2312yes—
151#399—24no—
152#398—24no—
153#396—24no—
154#391—22no—
155#391C2312yes—
156#391C2311yescharacteristic
157#391C2311yescharacteristic
158#391C2312yes—
159#390—21no—
160#390—22no—
161#390—21no—
162#390—22no—
163#390—21no—
164#390—21no—
165#390C2312yes—
166#390C2312yes—
167#389—22no—
168#389—21no—
169#389—21no—
170#389—22no—
171#389—24no—
172#389C2311yescharacteristic
173#389C2311yescharacteristic
174#389C2314yes—
175#388—22no—
176#388—24no—
177#388—22no—
178#388—22no—
179#387—21no—
180#387—21no—
181#387—22no—
182#387—24no—
183#386—22no—
184#386—22no—
185#386—22no—
186#386—22no—
187#386—22no—
188#386—22no—
189#386—22no—
190#384—22no—
191#384—21no—
192#384—21no—
193#384C2311yescharacteristic
194#384C2311yescharacteristic
195#384C2312yes—
196#384C2312yes—
197#382—21no—
198#381—22no—
199#380—22no—
200#380C2312yes—
201#380C2314yes—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C22⋊C4—22no—
214C4×C22⋊C4—22no—
215C4×C22⋊C4—22no—
216C4×C22⋊C4—22no—
217C4×C22⋊C4D412yes—
218C4×C22⋊C4D412yes—
219C4×C22⋊C4C2312yes—
220#376—21no—
221C2×#95—22no—
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×Q8C2411yescharacteristic
241C22×Q8C2412yes—
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—21no—
245C22×D4—21no—
246C22×D4C2×D411yescharacteristic
247C22×D4C2412yes—
248C23×C4—22no—
249C23×C4—22no—
250C23×C4—22no—
251C23×C4C2×D412yes—
252C23×C4C2×D412yes—
253C23×C4C2412yes—
254C2×Q16—24no—
255C2×Q16—24no—
256C2×Q16—42no—
257C2×SD16—28no—
258C2×SD16—28no—
259C2×SD16—44no—
260C22×C8C2414yes—
261C22×C8C2412yes—
262C22×C8C2412yes—
263C4⋊Q8—24no—
264C4⋊Q8—44no—
265C4⋊Q8—44no—
266C4⋊Q8—82no—
267#127—28no—
268#127—22no—
269#127#4114yes—
270#125—48no—
271C22⋊Q8—28no—
272C22⋊Q8—44no—
273C22⋊Q8—48no—
274C22⋊Q8—84no—
275#121—216no—
276#121—24no—
277#121—24no—
278C4×Q8—22no—
279C4×Q8—28no—
280C4×Q8—24no—
281C4×Q8—24no—
282C4×Q8—44no—
283C4×Q8—48no—
284C4×Q8#4114yes—
285C4×D4—28no—
286C4×D4—24no—
287C4×D4—22no—
288C4×D4—22no—
289C4×D4—42no—
290C4×D4—44no—
291C4×D4—44no—
292C4×D4—44no—
293C4×D4—44no—
294C4×D4—82no—
295C4×D4—84no—
296#117—48no—
297#117—44no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2414yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C22⋊C4—22no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C42—21no—
340C2×C42—24no—
341C2×C42—21no—
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×C42C2411yescharacteristic
353C2×C42C2412yes—
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356#107—24no—
357#107—44no—
358C4⋊C8—24no—
359C4⋊C8—42no—
360#103—24no—
361#103—24no—
362#103—24no—
363#103—42no—
364#103—44no—
365#103—44no—
366#102—28no—
367#102—44no—
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—21no—
377#95—21no—
378#95—21no—
379#95—21no—
380#95—22no—
381#95—24no—
382#95—21no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—21no—
387#95—24no—
388#95C2×D412yes—
389#95C2×D412yes—
390#95C2×D412yes—
391C24—22no—
392C24C22×D412yes—
393#41—88no—
394#41—88no—
395C2×Q8—21no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8C2×#4112yes—
407C2×Q8H(5,𝔽2)14yes—
408C2×D4—28no—
409C2×D4—28no—
410C2×D4—22no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—81no—
421C2×D4—81no—
422C2×D4—82no—
423C2×D4—82no—
424C2×D4—82no—
425C2×D4—82no—
426C2×D4C2×#4114yes—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—28no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—82no—
449C22×C4—82no—
450C22×C4—82no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4—82no—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C2511yescharacteristic
464Q16—48no—
465SD16—416no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—24no—
469C2×C8—22no—
470C2×C8—24no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—24no—
474C2×C8—22no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—81no—
500C4⋊C4—81no—
501C4⋊C4—84no—
502C4⋊C4—82no—
503C4⋊C4—82no—
504C4⋊C4C2×#4112yes—
505C22⋊C4—28no—
506C22⋊C4—24no—
507C22⋊C4—24no—
508C22⋊C4—42no—
509C22⋊C4—48no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—82no—
514C22⋊C4—82no—
515C22⋊C4—82no—
516C22⋊C4—82no—
517C22⋊C4—82no—
518C22⋊C4—84no—
519C42—24no—
520C42—22no—
521C42—22no—
522C42—24no—
523C42—22no—
524C42—22no—
525C42—42no—
526C42—42no—
527C42—41no—
528C42—48no—
529C42—44no—
530C42—42no—
531C42—82no—
532C42—82no—
533C42#13714yes—
534C42C2×#4112yes—
535C42C2×#4112yes—
536C23—22no—
537C23—22no—
538C23—24no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23C23×D411yescharacteristic
551Q8—24no—
552Q8—24no—
553Q8—24no—
554Q8—42no—
555Q8—42no—
556Q8—42no—
557Q8—88no—
558D4—28no—
559D4—44no—
560D4—48no—
561D4—44no—
562D4—84no—
563D4—88no—
564D4—88no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—48no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—44no—
612C2×C4—84no—
613C2×C4—88no—
614C2×C4—164no—
615C2×C4—164no—
616C2×C4#52512yes—
617C2×C4#52512yes—
618C2×C4C2×#13712yes—
619C2×C4C2×#13712yes—
620C2×C4C2×#13712yes—
621C2×C4#57512yes—
622C2×C4#57612yes—
623C2×C4C22×#4111yescharacteristic
624C2×C4C22×#4111yescharacteristic
625C2×C4C2×H(5,𝔽2)12yes—
626C2×C4#58512yes—
627C8—42no—
628C8—42no—
629C8—44no—
630C22—42no—
631C22—42no—
632C22—44no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—44no—
637C22—42no—
638C22—42no—
639C22—44no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22C2×#52511yescharacteristic
645C22#348411yescharacteristic
646C22#350011yescharacteristic
647C22C22×#13711yescharacteristic
648C22C2×#57511yescharacteristic
649C22C2×#57611yescharacteristic
650C22#358611yescharacteristic
651C4—22no—
652C4—22no—
653C4—22no—
654C4—22no—
655C4—41no—
656C4—42no—
657C4—42no—
658C4—42no—
659C4—44no—
660C4—42no—
661C4—42no—
662C4—82no—
663C4—81no—
664C4—81no—
665C4—84no—
666C4—82no—
667C4#311614yes—
668C4#349412yes—
669C4#351312yes—
670C2—82no—
671C2—84no—
672C2#3573011yescharacteristic
673C2C2×#311611yescharacteristic
674C2#6059611yescharacteristic
675C2#6158411yescharacteristic
676C2#6160511yescharacteristic
677C2#6253711yescharacteristic
678C2#6258711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#550C22
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. #30165C2semidirect
4. #29984C2semidirect
5. #29139C2semidirect
6. #29113C2semidirect
7. #28160C2semidirect
8. #26686C2semidirect
9. #24970C2semidirect
10. #24919C2semidirect
12. #23935C2semidirect
13. #23679C2semidirect
14. #23574C2semidirect
15. #23003C2semidirect
16. #22839C2semidirect
17. #22763C2semidirect

Maximal subgroups

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

subgroupindex
#361342
#343302
#301652
#299842
#291392
#291132
#281602
#266862
#249702
#249192
#242422
#239352
#236792
#235742
#230032
#228392
#227632