Finite Group Databasebeta

← #6555349

Subgroups — #6555349

Order 512.

1809 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36180C211yesmaximal, characteristic, complemented
2#34330C211yesmaximal, characteristic
3#29983C211yesmaximal, characteristic, complemented
4#29955C212yesmaximal, complemented
5#29254C211yesmaximal, characteristic, complemented
6#29163C212yesmaximal, complemented
7#28508C211yesmaximal, characteristic, complemented
8#27677C212yesmaximal, complemented
9#26530C212yesmaximal, complemented
10#26357C214yesmaximal, complemented
11#26092C212yesmaximal, complemented
12#25970C214yesmaximal, complemented
13#23776C211yesmaximal, characteristic, complemented
14#23734C212yesmaximal, complemented
15#22700C211yesmaximal, characteristic, complemented
16#22665C212yesmaximal, complemented
17#22432C212yesmaximal
18C2×#554—21no—
19C2×#550C2211yescharacteristic
20C2×#474C2214yes—
21C2×#436C2212yes—
22C2×#428C2214yes—
23C2×#417C2214yes—
24#2868C2211yescharacteristic
25#2857C2212yes—
26#2567—21no—
27#2552—21no—
28#2552C2212yes—
29#2502—22no—
30#2440—22no—
31#2440C2214yes—
32#2366—21no—
33#2366C2211yescharacteristic
34#2364—22no—
35#2363C2212yes—
36#2357—21no—
37#2355C2211yescharacteristic
38#2354C2211yescharacteristic
39D4×C4⋊C4—21no—
40D4×C4⋊C4C2212yes—
41#2339C2212yes—
42#2333—22no—
43#2328—21no—
44C4×C4⋊Q8C2211yescharacteristic
45C4×#127C2211yescharacteristic
46C4×#121C2212yes—
47#2297C2212yes—
48#2295C2211yescharacteristic
49C2×#384C2212yes—
50#2275—21no—
51C42×D4C2211yescharacteristic
52#2099C2211yescharacteristic
53#2099C2212yes—
54#2096C2212yes—
55#2096C2211yescharacteristic
56#2095C2214yes—
57#2094C2214yes—
58#2093C2212yes—
59#2093C2212yes—
60#2093C2212yes—
61#2088C2212yes—
62#2068C2214yes—
63#2066C2212yes—
64#2065C2214yes—
65#2058C2212yes—
66#2058C2211yescharacteristic
67#2024C2211yescharacteristic
68#2024C2212yes—
69#2023C2212yes—
70#1982C2212yes—
71#1957C2212yes—
72#1948C2212yes—
73#1947C2214yes—
74#1947C2212yes—
75#1939C2212yes—
76#1921C2214yes—
77#1895C2214yes—
78#1874C2214yes—
79#1864C2214yes—
80#1862C2214yes—
81#1854C2214yes—
82#1852C2214yes—
83#1851C2214yes—
84#1845C2212yes—
85#1831C2211yescharacteristic
86#1809C2214yes—
87#1803C2212yes—
88#1802C2214yes—
89#1797C2212yes—
90#1788C2214yes—
91C4×#106C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×#103C2212yes—
94#1728C2212yes—
95C22×#41C2312yes—
96C22×M16C2314yes—
97#554—48no—
98#550—24no—
99#550Q818yescomplemented
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×#127—21no—
105C2×#127C2311yescharacteristic
106C2×#125—22no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—24no—
109C2×#121—22no—
110C2×#121C2312yes—
111C2×C4×Q8—21no—
112C2×C4×Q8C2311yescharacteristic
113C2×C4×D4—22no—
114C2×C4×D4—22no—
115C2×C4×D4—21no—
116C2×C4×D4—21no—
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4D411yescharacteristic
120C2×C4×D4D411yescharacteristic
121C2×C4×D4C2312yes—
122C2×C4×D4C2312yes—
123C2×C4×D4C2311yescharacteristic
124C2×#117C2312yes—
125C22×C4⋊C4—22no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4C2314yes—
128C22×C42C2312yes—
129#474—48no—
130#436—28no—
131#428—48no—
132C2×#107C2314yes—
133C2×#107C2314yes—
134C2×#107C2314yes—
135C2×#106C2312yes—
136C2×#106C2312yes—
137C2×#106C2314yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2314yes—
141#417—48no—
142C2×#103C2312yes—
143C2×#103C2314yes—
144C2×#103C2312yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×C22⋊C8C2314yes—
150C2×C4×C8C2312yes—
151#398—24no—
152#398—24no—
153#395—24no—
154#391—21no—
155#391—22no—
156#391C2311yescharacteristic
157#391C2312yes—
158#391C2311yescharacteristic
159#390—21no—
160#390—21no—
161#390—22no—
162#390—21no—
163#390—21no—
164#390C2312yes—
165#390C2312yes—
166#389—22no—
167#389—22no—
168#389—21no—
169#389—22no—
170#389C2312yes—
171#389C2311yescharacteristic
172#389C2311yescharacteristic
173#389C2314yes—
174#388—22no—
175#388—22no—
176#388—22no—
177#387—21no—
178#387—21no—
179#387—22no—
180#386—22no—
181#386—22no—
182#386—22no—
183#386—22no—
184#386—22no—
185#386—22no—
186#386—22no—
187#385—24no—
188#385—24no—
189#384—22no—
190#384—21no—
191#384—21no—
192#384—22no—
193#384C2311yescharacteristic
194#384C2311yescharacteristic
195#384C2314yes—
196#384C2312yes—
197#384C2312yes—
198#382—21no—
199#381—22no—
200#380—22no—
201#380C2312yes—
202#380C2314yes—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4C2312yes—
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#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2312yes—
233#336C2311yescharacteristic
234#336C2314yes—
235C2×#41—24no—
236C2×#41—44no—
237C2×#41—44no—
238C2×#41C2×Q818yes—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8C2411yescharacteristic
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—21no—
245C22×D4—21no—
246C22×D4C2×D411yescharacteristic
247C22×D4C2412yes—
248C23×C4—22no—
249C23×C4—24no—
250C23×C4—22no—
251C23×C4—22no—
252C23×C4C2×D412yes—
253C23×C4C2×D412yes—
254C23×C4C2412yes—
255C2×M16—28no—
256C2×M16—28no—
257C2×M16—44no—
258C22×C8C2414yes—
259C22×C8C2412yes—
260C22×C8C2412yes—
261C4⋊Q8—42no—
262C4⋊Q8—44no—
263C4⋊Q8—44no—
264C4⋊Q8—82no—
265#127—28no—
266#127—22no—
267#127C2×Q814yes—
268#125—48no—
269C22⋊Q8—416no—
270C22⋊Q8—48no—
271#121—24no—
272#121—216no—
273#121C2×Q818yes—
274C4×Q8—22no—
275C4×Q8—44no—
276C4×Q8C2×Q814yes—
277C4×D4—28no—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—44no—
281C4×D4—44no—
282C4×D4—48no—
283C4×D4—44no—
284C4×D4—44no—
285C4×D4—82no—
286C4×D4—84no—
287C4×D4C2×Q814yes—
288C4×D4#4114yes—
289#117—44no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—48no—
311C2×C4⋊C4—84no—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2414yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C22⋊C4—22no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4C2×D412yes—
333C2×C22⋊C4C2×D412yes—
334C2×C22⋊C4C2×D412yes—
335C2×C42—21no—
336C2×C42—22no—
337C2×C42—21no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—44no—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2×D411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2412yes—
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351#107—28no—
352#107—44no—
353#107—44no—
354#106—24no—
355#106—42no—
356#106—44no—
357C4⋊C8—24no—
358C4⋊C8—24no—
359C4⋊C8—44no—
360#103—24no—
361#103—42no—
362#103—44no—
363#102—28no—
364#102—44no—
365#102—44no—
366#102—44no—
367C22⋊C8—28no—
368C4×C8—24no—
369#95—21no—
370#95—22no—
371#95—22no—
372#95—21no—
373#95—21no—
374#95—21no—
375#95—21no—
376#95—22no—
377#95—21no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—21no—
382#95—24no—
383#95—24no—
384#95C2×D412yes—
385#95C2×D412yes—
386#95C2×D412yes—
387C24—22no—
388C24C22×D412yes—
389#41—416no—
390#41—88no—
391C2×Q8—21no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8—44no—
398C2×Q8—82no—
399C2×Q8—82no—
400C2×Q8C22×Q812yes—
401C2×D4—28no—
402C2×D4—28no—
403C2×D4—24no—
404C2×D4—24no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—24no—
409C2×D4—24no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—81no—
413C2×D4—81no—
414C2×D4—82no—
415C2×D4—82no—
416C2×D4—82no—
417C2×D4—82no—
418C2×D4C22⋊Q814yes—
419C2×D4C22×Q814yes—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—28no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—48no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—82no—
441C22×C4—82no—
442C22×C4—82no—
443C22×C4—82no—
444C22×C4—82no—
445C22×C4—82no—
446C22×C4—84no—
447C22×C4—84no—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×Q814yes—
458C22×C4C2×#4114yes—
459C22×C4C2511yescharacteristic
460M16—416no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—24no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—22no—
469C2×C8H(5,𝔽2)14yes—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—21no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—81no—
494C4⋊C4—84no—
495C4⋊C4—84no—
496C4⋊C4—81no—
497C4⋊C4—82no—
498C4⋊C4—82no—
499C4⋊C4C22⋊Q814yes—
500C4⋊C4#13714yes—
501C4⋊C4C22×Q812yes—
502C22⋊C4—28no—
503C22⋊C4—24no—
504C22⋊C4—24no—
505C22⋊C4—48no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—82no—
510C22⋊C4—82no—
511C22⋊C4—82no—
512C22⋊C4—82no—
513C22⋊C4—82no—
514C22⋊C4C22⋊Q818yes—
515C42—24no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—24no—
520C42—22no—
521C42—42no—
522C42—42no—
523C42—41no—
524C42—44no—
525C42—42no—
526C42—82no—
527C42—82no—
528C42AGL(1,ℤ/8)14yes—
529C42C22×Q812yes—
530C42C2×#4112yes—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—24no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23C2×C22⋊Q814yes—
545C23C23×D411yescharacteristic
546Q8—24no—
547Q8—42no—
548Q8—84no—
549Q8—164no—
550D4—28no—
551D4—28no—
552D4—48no—
553D4—44no—
554D4—84no—
555D4—88no—
556D4—88no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—48no—
580C2×C4—48no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—88no—
604C2×C4—164no—
605C2×C4—164no—
606C2×C4—168no—
607C2×C4C2×C22⋊Q812yes—
608C2×C4C2×C22⋊Q814yes—
609C2×C4C2×C22⋊Q812yes—
610C2×C4C2×AGL(1,ℤ/8)12yes—
611C2×C4C2×AGL(1,ℤ/8)12yes—
612C2×C4C2×AGL(1,ℤ/8)12yes—
613C2×C4C2×#13714yes—
614C2×C4C2×#13712yes—
615C2×C4#57612yes—
616C2×C4C23×Q811yescharacteristic
617C2×C4C22×#4111yescharacteristic
618C2×C4C2×H(5,𝔽2)12yes—
619C2×C4#58512yes—
620C8—24no—
621C8—42no—
622C8—44no—
623C22—24no—
624C22—24no—
625C22—24no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—44no—
631C22—44no—
632C22—44no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22C22×C22⋊Q811yescharacteristic
638C22#348511yescharacteristic
639C22#349811yescharacteristic
640C22C22×AGL(1,ℤ/8)11yescharacteristic
641C22C22×#13711yescharacteristic
642C22C2×#57611yescharacteristic
643C22#358611yescharacteristic
644C4—22no—
645C4—22no—
646C4—24no—
647C4—22no—
648C4—22no—
649C4—24no—
650C4—42no—
651C4—42no—
652C4—41no—
653C4—42no—
654C4—82no—
655C4—81no—
656C4—82no—
657C4—82no—
658C4—84no—
659C4—81no—
660C4#307914yes—
661C4#350112yes—
662C4#351312yes—
663C2—44no—
664C2—84no—
665C2#3583611yescharacteristic
666C2C2×#307911yescharacteristic
667C2#6049411yescharacteristic
668C2#6171711yescharacteristic
669C2#6172611yescharacteristic
670C2#6242311yescharacteristic
671C2#6245411yescharacteristic
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. #36180C2semidirect
3. #29983C2semidirect
4. #29955C2semidirect
5. #29254C2semidirect
6. #29163C2semidirect
7. #28508C2semidirect
8. #27677C2semidirect
9. #26530C2semidirect
10. #26357C2semidirect
11. #26092C2semidirect
12. #25970C2semidirect
13. #23776C2semidirect
14. #23734C2semidirect
15. #22700C2semidirect
16. #22665C2semidirect
99. #550Q8semidirect

Maximal subgroups

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

subgroupindex
#361802
#343302
#299832
#299552
#292542
#291632
#285082
#276772
#265302
#263572
#260922
#259702
#237762
#237342
#227002
#226652
#224322