Finite Group Databasebeta

← #6802861

Subgroups — #6802861

Order 512.

1570 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36596C211yesmaximal, characteristic
2C2×#2571C211yesmaximal, characteristic, complemented
3#33107C211yesmaximal, characteristic, complemented
4#32405C214yesmaximal, complemented
5#31601C214yesmaximal, complemented
6#29754C212yesmaximal, complemented
7#28835C212yesmaximal, complemented
8#27918C212yesmaximal, complemented
9#27909C211yesmaximal, characteristic, complemented
10#27755C212yesmaximal, complemented
11#27651C212yesmaximal, complemented
12#27644C211yesmaximal, characteristic, complemented
13#26529C212yesmaximal, complemented
14#26087C212yesmaximal, complemented
15#22629C211yesmaximal, characteristic, complemented
16#22081C212yesmaximal, complemented
17C4×#481C211yesmaximal, characteristic, complemented
18C2×#553—21no—
19C2×#481C2211yescharacteristic
20C2×#481C2212yes—
21C2×#480C2212yes—
22C2×#480C2212yes—
23C2×#480C2212yes—
24#2856C2211yescharacteristic
25#2856C2212yes—
26#2571—21no—
27#2571—21no—
28#2571—22no—
29#2571C2212yes—
30#2571C2214yes—
31#2571C2212yes—
32#2554—22no—
33#2552C2211yescharacteristic
34#2543—22no—
35#2541C2211yescharacteristic
36#2512C2212yes—
37#2490—21no—
38#2445C2212yes—
39#2416—21no—
40C2×#400—22no—
41#2366C2211yescharacteristic
42#2363—21no—
43#2354C2211yescharacteristic
44C4×#121C2212yes—
45C4×#121C2211yescharacteristic
46C2×#389—21no—
47C2×#389C2212yes—
48C2×#384C2211yescharacteristic
49C2×#382—21no—
50C2×#382C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53#2093C2214yes—
54#2092C2214yes—
55#2092C2214yes—
56#2085C2214yes—
57#2083C2214yes—
58#2083C2212yes—
59#2073C2214yes—
60#2071C2212yes—
61#2067C2214yes—
62#2067C2214yes—
63#2041C2214yes—
64#1982C2212yes—
65#1961C2214yes—
66#1957C2212yes—
67#1949C2214yes—
68#1948C2212yes—
69#1947C2212yes—
70#1943C2214yes—
71#1939C2212yes—
72#1936C2214yes—
73#1926C2212yes—
74#1926C2214yes—
75#1921C2214yes—
76#1919C2212yes—
77#1917C2214yes—
78#1874C2211yescharacteristic
79#1874C2212yes—
80#1864C2214yes—
81#1853C2211yescharacteristic
82#1847C2214yes—
83#1845C2212yes—
84#1824C2211yescharacteristic
85#1823C2212yes—
86#1804C2214yes—
87#1802C2212yes—
88#1797C2212yes—
89#1797C2211yescharacteristic
90C4×#106C2212yes—
91#1770C2212yes—
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95#553—48no—
96C2×#127C2311yescharacteristic
97C2×#127C2312yes—
98C2×#126—22no—
99C2×#125—21no—
100C2×#123—22no—
101C2×#121—21no—
102C2×#121—22no—
103C2×#121—22no—
104C2×#121—21no—
105C2×#121—21no—
106C2×#121—21no—
107C2×#121C2312yes—
108C2×#121C2311yescharacteristic
109C2×C4×D4—21no—
110C2×C4×D4—22no—
111C2×C4×D4—21no—
112C2×C4×D4C2311yescharacteristic
113C2×C4×D4C2312yes—
114C2×#117—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4D411yescharacteristic
119C22×C4⋊C4D411yescharacteristic
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C42—21no—
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126#481—24no—
127#481—44no—
128#480—28no—
129#480—44no—
130#480—44no—
131C2×#107C2314yes—
132C2×#107C2314yes—
133C2×#107C2314yes—
134C2×#106C2312yes—
135C2×#106C2314yes—
136C2×#106C2312yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2314yes—
139C2×#102C2314yes—
140C2×#102C2312yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2312yes—
145C2×#102C2314yes—
146C2×C22⋊C8C2311yescharacteristic
147C2×C22⋊C8C2312yes—
148C2×C22⋊C8C2311yescharacteristic
149C2×#97C2314yes—
150C2×C4×C8C2312yes—
151#400—24no—
152#400—24no—
153#400—24no—
154#400—24no—
155#398—22no—
156#397—22no—
157#396—22no—
158#394—22no—
159#394—22no—
160#392—22no—
161#390—21no—
162#390—21no—
163#390—22no—
164#390—22no—
165#390—21no—
166#390C2312yes—
167#390C2312yes—
168#390C2311yescharacteristic
169#390C2312yes—
170#390C2312yes—
171#389—22no—
172#389—22no—
173#389—24no—
174#389—21no—
175#389—22no—
176#389—22no—
177#389C2314yes—
178#389C2314yes—
179#389C2311yescharacteristic
180#389C2312yes—
181#388—22no—
182#388—21no—
183#388—22no—
184#388—22no—
185#386—22no—
186#386—21no—
187#386—22no—
188#386—22no—
189#384—21no—
190#384—21no—
191#384C2312yes—
192#384C2312yes—
193#384C2312yes—
194#384C2311yescharacteristic
195#382—22no—
196#382—22no—
197#382—24no—
198#382—22no—
199#382—22no—
200#382C2314yes—
201#382C2314yes—
202#380—21no—
203#380C2311yescharacteristic
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D412yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C22⋊C4—21no—
217C4×C22⋊C4—22no—
218C2×#95—21no—
219C2×#95—22no—
220C2×#95—22no—
221C2×#95—22no—
222C2×#95D412yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#336C2312yes—
231#336C2314yes—
232#336C2314yes—
233#336C2312yes—
234C22×D4—21no—
235C22×D4—22no—
236C22×D4—21no—
237C22×D4—22no—
238C22×D4—22no—
239C22×D4—21no—
240C22×D4C2411yescharacteristic
241C22×D4C2412yes—
242C23×C4—22no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—22no—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D412yes—
250C23×C4C2×D412yes—
251C23×C4C2411yescharacteristic
252C22×C8C2412yes—
253C22×C8C2414yes—
254C22×C8C2412yes—
255#127—24no—
256#127—44no—
257#126—48no—
258#125—44no—
259#123—48no—
260#121—24no—
261#121—22no—
262#121—22no—
263#121—24no—
264#121—44no—
265#121—44no—
266#121—48no—
267#121—44no—
268#121—82no—
269#121—84no—
270C4×D4—24no—
271C4×D4—44no—
272C4×D4—44no—
273C4×D4—48no—
274C4×D4—44no—
275#117—44no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—82no—
300C2×C4⋊C4—84no—
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2414yes—
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329C2×C42—24no—
330C2×C42—22no—
331C2×C42—21no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—44no—
335C2×C42—44no—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D412yes—
339C2×C42C2×D412yes—
340C2×C42C2412yes—
341C2×C42C2412yes—
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344#107—28no—
345#107—44no—
346#107—44no—
347#106—24no—
348#106—42no—
349#106—44no—
350C4⋊C8—24no—
351C4⋊C8—44no—
352#102—28no—
353#102—24no—
354#102—44no—
355#102—42no—
356#102—44no—
357#102—44no—
358#102—44no—
359C22⋊C8—22no—
360C22⋊C8—24no—
361C22⋊C8#4114yes—
362#97—44no—
363C4×C8—24no—
364#95—22no—
365#95—22no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—21no—
373#95—22no—
374#95—21no—
375#95—22no—
376#95—21no—
377#95C2×D412yes—
378#95C2×D412yes—
379#95C2×D412yes—
380#95C2×D412yes—
381C24—21no—
382C24—22no—
383C24C22×D411yescharacteristic
384C2×D4—22no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—24no—
390C2×D4—24no—
391C2×D4—22no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—42no—
396C2×D4—44no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—44no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—81no—
405C2×D4—82no—
406C2×D4—82no—
407C2×D4—82no—
408C2×D4—81no—
409C2×D4—82no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—41no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—44no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—42no—
439C22×C4—44no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—82no—
444C22×C4—82no—
445C22×C4AGL(1,ℤ/8)14yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C2×#4112yes—
456C22×C4H(5,𝔽2)12yes—
457C22×C4C2511yescharacteristic
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—24no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—22no—
466C2×C8C2×#4114yes—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4—84no—
497C4⋊C4—84no—
498C4⋊C4—81no—
499C4⋊C4—82no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—44no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—44no—
510C22⋊C4—81no—
511C22⋊C4—82no—
512C22⋊C4—82no—
513C22⋊C4—82no—
514C22⋊C4—82no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—24no—
519C42—24no—
520C42—42no—
521C42—42no—
522C42—44no—
523C42—44no—
524C42—42no—
525C42—44no—
526C422−1+414yes—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—41no—
532C23—42no—
533C23—41no—
534C23—42no—
535C23—41no—
536C23—42no—
537C23—42no—
538C23—41no—
539C23—42no—
540C23—41no—
541C23—42no—
542C23—41no—
543C23—41no—
544C23—42no—
545C23#55012yes—
546C23C2×AGL(1,ℤ/8)12yes—
547C23#57612yes—
548C23C23×D411yescharacteristic
549D4—42no—
550D4—44no—
551D4—42no—
552D4—44no—
553D4—88no—
554D4—84no—
555D4—84no—
556D4—162no—
557D4—164no—
558D4—164no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—44no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—84no—
606C2×C4—84no—
607C2×C4—84no—
608C2×C4—84no—
609C2×C4—88no—
610C2×C4—84no—
611C2×C4—164no—
612C2×C4#53612yes—
613C2×C4#53612yes—
614C2×C4#55012yes—
615C2×C4C2×AGL(1,ℤ/8)12yes—
616C2×C4C2×AGL(1,ℤ/8)12yes—
617C2×C4#57512yes—
618C2×C4C22×#4111yescharacteristic
619C2×C4C2×H(5,𝔽2)11yescharacteristic
620C2×C4C2×2−1+412yes—
621C2×C4#58512yes—
622C8—24no—
623C8—44no—
624C8—42no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—82no—
632C22—81no—
633C22—81no—
634C22—82no—
635C22—81no—
636C22—82no—
637C22—82no—
638C22—81no—
639C22—82no—
640C22—81no—
641C22—81no—
642C22—82no—
643C22—81no—
644C22—82no—
645C22#339914yes—
646C22C2×#53611yescharacteristic
647C22C2×#55011yescharacteristic
648C22#348511yescharacteristic
649C22C22×AGL(1,ℤ/8)11yescharacteristic
650C22C2×#57511yescharacteristic
651C22C2×#57611yescharacteristic
652C22#358611yescharacteristic
653C4—22no—
654C4—22no—
655C4—22no—
656C4—42no—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—41no—
661C4—81no—
662C4—82no—
663C4—81no—
664C4—84no—
665C4—81no—
666C4—82no—
667C4—82no—
668C4#323414yes—
669C4#351512yes—
670C4#355912yes—
671C2—24no—
672C2—162no—
673C2—161no—
674C2#3913411yescharacteristic
675C2C2×#323411yescharacteristic
676C2#6087111yescharacteristic
677C2C2×#339911yescharacteristic
678C2#6140311yescharacteristic
679C2#6172211yescharacteristic
680C2#6174011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36596C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2571C2semidirect
3. #33107C2semidirect
4. #32405C2semidirect
5. #31601C2semidirect
6. #29754C2semidirect
7. #28835C2semidirect
8. #27918C2semidirect
9. #27909C2semidirect
10. #27755C2semidirect
11. #27651C2semidirect
12. #27644C2semidirect
13. #26529C2semidirect
14. #26087C2semidirect
15. #22629C2semidirect
16. #22081C2semidirect
17. C4×#481C2semidirect

Maximal subgroups

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

subgroupindex
#365962
C2×#25712
#331072
#324052
#316012
#297542
#288352
#279182
#279092
#277552
#276512
#276442
#265292
#260872
#226292
#220812
C4×#4812