Finite Group Databasebeta

← #6801229

Subgroups — #6801229

Order 512.

1627 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36560C211yesmaximal, characteristic, complemented
2#34934C211yesmaximal, characteristic
3#33195C211yesmaximal, characteristic, complemented
4#32490C214yesmaximal, complemented
5#31982C214yesmaximal, complemented
6#29995C212yesmaximal, complemented
7#29130C212yesmaximal, complemented
8#28716C211yesmaximal, characteristic, complemented
9#28648C212yesmaximal, complemented
10#28280C211yesmaximal, characteristic, complemented
11#28280C212yesmaximal, complemented
12#26510C212yesmaximal, complemented
13#25478C212yesmaximal, complemented
14#25272C212yesmaximal
15#22722C211yesmaximal, characteristic, complemented
16#22117C212yesmaximal, complemented
17C4×#476C211yesmaximal, characteristic, complemented
18C2×#557—21no—
19C2×#529—21no—
20C2×#486C2212yes—
21C2×#476C2212yes—
22C2×#476C2211yescharacteristic
23C2×#474C2212yes—
24C2×#446C2212yes—
25#2868C2211yescharacteristic
26#2859C2212yes—
27#2568—21no—
28#2565—22no—
29#2554—22no—
30#2552C2211yescharacteristic
31#2548C2211yescharacteristic
32#2546—21no—
33#2546C2212yes—
34#2541C2211yescharacteristic
35#2538C2211yescharacteristic
36#2512C2212yes—
37#2506—22no—
38#2506C2214yes—
39#2490—21no—
40#2445C2212yes—
41#2426—21no—
42#2392—22no—
43#2365—21no—
44#2365C2212yes—
45C4×C4⋊Q8C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×#127C2211yescharacteristic
48C4×#127C2212yes—
49C4×#121—21no—
50#2293C2212yes—
51C42×Q8C2211yescharacteristic
52C42×D4C2211yescharacteristic
53#2100C2214yes—
54#2088C2214yes—
55#2087C2214yes—
56#2071C2214yes—
57#2067C2214yes—
58#2065C2214yes—
59#2065C2214yes—
60#2065C2214yes—
61#2065C2214yes—
62#2041C2214yes—
63#1990C2212yes—
64#1988C2212yes—
65#1987C2211yescharacteristic
66#1982C2212yes—
67#1982C2214yes—
68#1948C2211yescharacteristic
69#1948C2212yes—
70#1948C2214yes—
71#1947C2212yes—
72#1947C2212yes—
73#1944C2214yes—
74#1940C2214yes—
75#1939C2212yes—
76#1936C2214yes—
77#1936C2214yes—
78#1936C2214yes—
79#1925C2214yes—
80#1920C2214yes—
81#1920C2212yes—
82#1864C2212yes—
83#1842C2212yes—
84#1842C2211yescharacteristic
85#1802C2212yes—
86#1802C2212yes—
87#1802C2212yes—
88#1802C2211yescharacteristic
89#1802C2212yes—
90#1802C2212yes—
91C4×#106C2212yes—
92#1767C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95#557—48no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2311yescharacteristic
101C2×#127C2311yescharacteristic
102C2×#127C2312yes—
103#529—48no—
104C2×#126—22no—
105C2×#125—21no—
106C2×#125—24no—
107C2×#125—22no—
108C2×#124—21no—
109C2×#123—22no—
110C2×C22⋊Q8—21no—
111C2×#121—21no—
112C2×#121—21no—
113C2×C4×Q8—22no—
114C2×C4×Q8—22no—
115C2×C4×Q8—21no—
116C2×C4×Q8D411yescharacteristic
117C2×C4×Q8D411yescharacteristic
118C2×C4×Q8C2311yescharacteristic
119C2×C4×D4—22no—
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4—22no—
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2312yes—
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×#117—22no—
128C22×C42—21no—
129#486—44no—
130#476—28no—
131#476—24no—
132#474—44no—
133#446—44no—
134C2×#107C2314yes—
135C2×#106C2314yes—
136C2×#106C2312yes—
137C2×#106C2312yes—
138C2×C4⋊C8C2311yescharacteristic
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2311yescharacteristic
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2314yes—
144C2×C4⋊C8C2314yes—
145C2×#102C2312yes—
146C2×#102C2312yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#97C2314yes—
153C2×C4×C8C2312yes—
154#400—24no—
155#398—24no—
156#398—24no—
157#398—22no—
158#397—22no—
159#396—22no—
160#396—22no—
161#394—22no—
162#394—22no—
163#394—22no—
164#392—22no—
165#391—22no—
166#391—22no—
167#391—21no—
168#391—22no—
169#390—21no—
170#390—21no—
171#390—22no—
172#390—22no—
173#390—21no—
174#390C2312yes—
175#390C2312yes—
176#390C2312yes—
177#390C2312yes—
178#390C2312yes—
179#389—21no—
180#389—21no—
181#389—22no—
182#389—22no—
183#389—22no—
184#389C2312yes—
185#389C2314yes—
186#389C2314yes—
187#389C2314yes—
188#389C2312yes—
189#389C2312yes—
190#388—22no—
191#388—22no—
192#388—22no—
193#387—24no—
194#387—24no—
195#386—22no—
196#384—21no—
197#384—21no—
198#384C2312yes—
199#384C2311yescharacteristic
200#384C2311yescharacteristic
201#384C2312yes—
202#384C2312yes—
203#382—22no—
204#380—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4D412yes—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D412yes—
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—21no—
221C4×C22⋊C4—21no—
222C43C2312yes—
223C43C2311yescharacteristic
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#340C2312yes—
232#340C2312yes—
233#336C2312yes—
234#336C2314yes—
235C22×Q8—21no—
236C22×Q8—22no—
237C22×Q8C2×D411yescharacteristic
238C22×D4—21no—
239C22×D4—22no—
240C22×D4—22no—
241C22×D4—21no—
242C22×D4C2412yes—
243C22×D4C2411yescharacteristic
244C23×C4—22no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—21no—
248C22×C8C2412yes—
249C22×C8C2412yes—
250C22×C8C2414yes—
251C4⋊Q8—24no—
252C4⋊Q8—22no—
253C4⋊Q8—24no—
254C4⋊Q8—44no—
255C4⋊Q8—48no—
256C4⋊Q8—44no—
257C4⋊Q8#4114yes—
258#127—24no—
259#127—22no—
260#127—22no—
261#127—42no—
262#126—48no—
263#125—48no—
264#125—44no—
265#125—416no—
266#124—44no—
267#123—48no—
268C22⋊Q8—44no—
269#121—44no—
270#121—44no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—22no—
274C4×Q8—22no—
275C4×Q8—48no—
276C4×Q8—44no—
277C4×Q8—48no—
278C4×D4—22no—
279C4×D4—22no—
280C4×D4—24no—
281C4×D4—22no—
282C4×D4—22no—
283C4×D4—22no—
284C4×D4—24no—
285C4×D4—41no—
286C4×D4—48no—
287C4×D4—44no—
288C4×D4—82no—
289C4×D4—84no—
290#117—48no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2414yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C42—21no—
337C2×C42—24no—
338C2×C42—22no—
339C2×C42—21no—
340C2×C42—21no—
341C2×C42—44no—
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D412yes—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D412yes—
349C2×C42C2×D411yescharacteristic
350C2×C42C2411yescharacteristic
351C2×C42C2412yes—
352C2×C42C2412yes—
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355#107—44no—
356#106—28no—
357#106—24no—
358#106—42no—
359C4⋊C8—24no—
360C4⋊C8—22no—
361C4⋊C8—24no—
362C4⋊C8—22no—
363C4⋊C8—42no—
364C4⋊C8—44no—
365C4⋊C8—44no—
366#102—24no—
367#102—24no—
368#102—28no—
369#102—44no—
370#102—44no—
371#102—44no—
372#102—44no—
373#97—44no—
374C4×C8—24no—
375#95—21no—
376#95—21no—
377#95—21no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—24no—
382#95—22no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95C2×D412yes—
387#95C2×D412yes—
388#95C2×D412yes—
389C24—21no—
390C24—22no—
391C2×Q8—22no—
392C2×Q8—24no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×D4—21no—
401C2×D4—22no—
402C2×D4—21no—
403C2×D4—22no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—21no—
410C2×D4—22no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—44no—
418C2×D4—42no—
419C2×D4—82no—
420C2×D4—81no—
421C2×D4—81no—
422C2×D4—82no—
423C2×D42−1+412yes—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—24no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—44no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—44no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—44no—
442C22×C4—81no—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D412yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C2511yescharacteristic
454C2×C8—22no—
455C2×C8—24no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—24no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—24no—
462C2×C8—22no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—21no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—48no—
489C4⋊C4—44no—
490C4⋊C4—48no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—48no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—42no—
497C4⋊C4—82no—
498C4⋊C4—81no—
499C4⋊C4AGL(1,ℤ/8)14yes—
500C4⋊C4C2×#4114yes—
501C22⋊C4—42no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—82no—
512C22⋊C4—82no—
513C22⋊C4—82no—
514C22⋊C4—82no—
515C42—24no—
516C42—24no—
517C42—22no—
518C42—24no—
519C42—22no—
520C42—22no—
521C42—21no—
522C42—22no—
523C42—22no—
524C42—21no—
525C42—22no—
526C42—41no—
527C42—42no—
528C42—42no—
529C42—42no—
530C42—42no—
531C42—48no—
532C42—44no—
533C42AGL(1,ℤ/8)14yes—
534C42C2×#4112yes—
535C42H(5,𝔽2)12yes—
536C422−1+412yes—
537C422−1+412yes—
538C23—22no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23C23×D411yescharacteristic
553Q8—48no—
554Q8—44no—
555Q8—88no—
556D4—22no—
557D4—22no—
558D4—41no—
559D4—42no—
560D4—42no—
561D4—42no—
562D4—41no—
563D4—42no—
564D4—88no—
565D4—84no—
566D4—88no—
567D4—84no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—44no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—44no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—84no—
610C2×C4—84no—
611C2×C4—88no—
612C2×C4—84no—
613C2×C4#53712yes—
614C2×C4#53712yes—
615C2×C4#54812yes—
616C2×C4#54812yes—
617C2×C4C2×AGL(1,ℤ/8)14yes—
618C2×C4C2×AGL(1,ℤ/8)12yes—
619C2×C4C2×AGL(1,ℤ/8)12yes—
620C2×C4C2×AGL(1,ℤ/8)12yes—
621C2×C4C2×AGL(1,ℤ/8)12yes—
622C2×C4#57712yes—
623C2×C4C22×#4111yescharacteristic
624C2×C4C2×H(5,𝔽2)11yescharacteristic
625C2×C4C2×2−1+411yescharacteristic
626C2×C4C2×2−1+411yescharacteristic
627C2×C4C2×2−1+411yescharacteristic
628C2×C4#58511yescharacteristic
629C8—42no—
630C8—44no—
631C8—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22C2×#53711yescharacteristic
647C22C2×#54811yescharacteristic
648C22#348911yescharacteristic
649C22C22×AGL(1,ℤ/8)11yescharacteristic
650C22C22×AGL(1,ℤ/8)11yescharacteristic
651C22C2×#57711yescharacteristic
652C22#358611yescharacteristic
653C4—22no—
654C4—22no—
655C4—24no—
656C4—22no—
657C4—44no—
658C4—42no—
659C4—41no—
660C4—42no—
661C4—42no—
662C4—81no—
663C4—82no—
664C4—81no—
665C4—82no—
666C4—81no—
667C4—84no—
668C4#325314yes—
669C4#336814yes—
670C4#351711yescharacteristic
671C4#352311yescharacteristic
672C4#355211yescharacteristic
673C4#355511yescharacteristic
674C2—82no—
675C2—162no—
676C2#3912711yescharacteristic
677C2C2×#325311yescharacteristic
678C2#6092311yescharacteristic
679C2C2×#336811yescharacteristic
680C2#6131711yescharacteristic
681C2#6210911yescharacteristic
682C2#6212811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2541C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36560C2semidirect
3. #33195C2semidirect
4. #32490C2semidirect
5. #31982C2semidirect
6. #29995C2semidirect
7. #29130C2semidirect
8. #28716C2semidirect
9. #28648C2semidirect
10. #28280C2semidirect
11. #28280C2semidirect
12. #26510C2semidirect
13. #25478C2semidirect
15. #22722C2semidirect
16. #22117C2semidirect
17. C4×#476C2semidirect

Maximal subgroups

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

subgroupindex
#365602
#349342
#331952
#324902
#319822
#299952
#291302
#287162
#286482
#282802
#265102
#254782
#252722
#227222
#221172
C4×#4762