Finite Group Databasebeta

← #6644652

Subgroups — #6644652

Order 512.

2000 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35006C211yesmaximal, characteristic, complemented
2C2×#2542C211yesmaximal, characteristic, complemented
3#33002C212yesmaximal, complemented
4#32426C214yesmaximal, complemented
5#32342C214yesmaximal, complemented
6#31634C214yesmaximal, complemented
7#31518C214yesmaximal, complemented
8#29638C212yesmaximal, complemented
9#29103C211yesmaximal, characteristic, complemented
10#28301C211yesmaximal, characteristic
11#28114C212yesmaximal, complemented
12#27960C211yesmaximal, characteristic, complemented
13#27909C211yesmaximal, characteristic, complemented
14#27644C212yesmaximal, complemented
15#21912C211yesmaximal, characteristic, complemented
16C2×#539—21no—
17C22×C4⋊Q8C2211yescharacteristic
18C22×C22⋊Q8—21no—
19C22×C22⋊Q8—21no—
20C22×C4×D4C2211yescharacteristic
21C2×#480C2212yes—
22C2×#480C2212yes—
23C2×#480C2212yes—
24C2×#447C2212yes—
25#2857C2211yescharacteristic
26#2856C2211yescharacteristic
27#2854C2212yes—
28#2639—21no—
29#2618—21no—
30#2560—22no—
31#2558—22no—
32#2542—22no—
33#2542—22no—
34#2542C2214yes—
35#2542C2214yes—
36#2512C2212yescomplemented
37#2506C2212yescomplemented
38#2452C2212yescomplemented
39#2445C2212yescomplemented
40C2×#399—21no—
41C2×#399—21no—
42#2325—21no—
43C4×#121C2212yes—
44C2×#390C2211yescharacteristic
45C2×#390C2211yescharacteristic
46C2×#384C2211yescharacteristic
47C2×#384C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49#2099C2214yescomplemented
50#2093C2212yes—
51#2093C2214yescomplemented
52#2083C2214yescomplemented
53#2071C2212yes—
54#2071C2214yescomplemented
55#2067C2214yes—
56#2057C2214yescomplemented
57#2057C2214yescomplemented
58#2054C2214yes—
59#2023C2212yes—
60#2023C2214yescomplemented
61#2023C2214yescomplemented
62#2011C2212yes—
63#1990C2212yescomplemented
64#1970C2214yes—
65#1957C2212yescomplemented
66#1947C2214yescomplemented
67#1934C2214yes—
68#1925C2214yes—
69#1916C2214yescomplemented
70#1916C2214yescomplemented
71#1879C2214yes—
72#1879C2214yes—
73#1879C2214yes—
74#1862C2214yes—
75#1862C2214yes—
76#1845C2212yescomplemented
77#1834C2212yescomplemented
78#1823C2212yes—
79#1823C2212yes—
80#1823C2212yes—
81#1817C2212yes—
82#1802C2214yescomplemented
83#1797C2212yes—
84#1786C2212yes—
85C4×C22⋊C8C2211yescharacteristic
86C4×C22⋊C8C2211yescharacteristic
87C23×Q8—21no—
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C22×D8C2314yes—
91#539—48no—
92C2×C4⋊Q8—22no—
93C2×C4⋊Q8—44no—
94C2×C4⋊Q8C2314yes—
95C2×C4⋊Q8C2311yescharacteristic
96C2×#127C2311yescharacteristic
97C2×#126—22no—
98C2×#125—22no—
99C2×#123—22no—
100C2×#123—22no—
101C2×C22⋊Q8—21no—
102C2×C22⋊Q8—22no—
103C2×C22⋊Q8—24no—
104C2×C22⋊Q8—24no—
105C2×C22⋊Q8—24no—
106C2×C22⋊Q8—24no—
107C2×C22⋊Q8—48no—
108C2×C22⋊Q8—48no—
109C2×#121—21no—
110C2×#121—21no—
111C2×#121C2312yes—
112C2×C4×D4—22no—
113C2×C4×D4—22no—
114C2×C4×D4—44no—
115C2×C4×D4C2312yes—
116C2×C4×D4C2314yes—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4D411yescharacteristic
123C22×C4⋊C4D411yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#480—44no—
136#480—44no—
137#480—44no—
138#447—44no—
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×#107C2314yes—
142C2×C4⋊C8C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2312yes—
151C2×C4×C8C2312yes—
152C2×C4×C8C2312yes—
153#400—22no—
154#399—24no—
155#399—24no—
156#399—24no—
157#399—24no—
158#398—22no—
159#397—22no—
160#396—22no—
161#396—22no—
162#394—22no—
163#394—22no—
164#393—22no—
165#391—21no—
166#390—21no—
167#390—21no—
168#390—22no—
169#390—21no—
170#390—21no—
171#390C2312yes—
172#390C2314yes—
173#390C2312yes—
174#390C2312yes—
175#390C2312yes—
176#390C2312yes—
177#390C2312yes—
178#389—21no—
179#389C2312yes—
180#389C2312yes—
181#389C2312yes—
182#389C2312yes—
183#388—22no—
184#388—22no—
185#388—22no—
186#388—22no—
187#387—22no—
188#386—22no—
189#386—22no—
190#386—22no—
191#385—22no—
192#384—22no—
193#384—21no—
194#384—21no—
195#384C2314yes—
196#384C2311yescharacteristic
197#384C2311yescharacteristic
198#384C2312yes—
199#384C2312yes—
200#380—21no—
201#380C2312yes—
202C4×C4⋊C4D412yes—
203C4×C4⋊C4D412yes—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2312yes—
207C4×C22⋊C4—22no—
208C2×#95D411yescharacteristic
209C2×#95D411yescharacteristic
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#336C2314yes—
217#336C2314yes—
218#336C2314yes—
219#336C2312yes—
220#336C2312yes—
221C25—21no—
222C22×Q8—24no—
223C22×Q8—21no—
224C22×Q8—44no—
225C22×Q8—44no—
226C22×Q8—44no—
227C22×D4—24no—
228C22×D4—22no—
229C22×D4—21no—
230C22×D4—21no—
231C22×D4—21no—
232C22×D4—24no—
233C22×D4—42no—
234C22×D4C2412yes—
235C22×D4C2412yes—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—24no—
240C23×C4—22no—
241C23×C4—44no—
242C23×C4—44no—
243C23×C4—44no—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2411yescharacteristic
251C2×D8—28no—
252C2×D8—44no—
253C2×D8—44no—
254C22×C8C2414yes—
255C22×C8C2414yes—
256C4⋊Q8—48no—
257C4⋊Q8—42no—
258C4⋊Q8—48no—
259#127—42no—
260#126—48no—
261#125—48no—
262#123—48no—
263#123—48no—
264C22⋊Q8—416no—
265C22⋊Q8—416no—
266C22⋊Q8—44no—
267C22⋊Q8—416no—
268C22⋊Q8—416no—
269C22⋊Q8—48no—
270#121—24no—
271#121—24no—
272#121—44no—
273#121—44no—
274C4×D4—48no—
275C4×D4—48no—
276C4×D4—44no—
277C4×D4—48no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4D818yescomplemented
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
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⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4—44no—
339C2×C22⋊C4—44no—
340C2×C42—24no—
341C2×C42—22no—
342C2×C42—42no—
343C2×C42—42no—
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2412yes—
349C2×C42C2412yes—
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352#107—44no—
353#107—44no—
354#107—44no—
355C4⋊C8—44no—
356#102—28no—
357#102—44no—
358#102—44no—
359#102—44no—
360#102—44no—
361#102—44no—
362C22⋊C8—24no—
363C22⋊C8—24no—
364C4×C8—42no—
365C4×C8—42no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95C2×D412yes—
373#95C2×D412yes—
374#95C2×D412yes—
375#95C2×D412yes—
376C24—24no—
377C24—22no—
378C24—22no—
379C24—42no—
380C24—42no—
381C24—42no—
382C24—42no—
383C24—42no—
384C24—42no—
385C24C22×D411yescharacteristic
386C2×Q8—48no—
387C2×Q8—48no—
388C2×Q8—48no—
389C2×Q8—42no—
390C2×Q8—81no—
391C2×Q8—81no—
392C2×Q8—816no—
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—24no—
399C2×D4—24no—
400C2×D4—24no—
401C2×D4—416no—
402C2×D4—42no—
403C2×D4—44no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—81no—
410C2×D4—81no—
411C2×D4—81no—
412C2×D4—81no—
413C2×D42−1+414yes—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—48no—
448C22×C4—44no—
449C22×C4—82no—
450C22×C4—82no—
451C22×C4—816no—
452C22×C4C2×D814yes—
453C22×C4C2×D814yes—
454C22×C4C2×D814yes—
455C22×C4AGL(1,ℤ/8)14yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4H(5,𝔽2)14yes—
466C22×C4C2511yescharacteristic
467D8—88no—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—24no—
471C2×C8—24no—
472C2×C8—24no—
473C2×C8—24no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—48no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—81no—
501C4⋊C4—81no—
502C4⋊C4C2×D818yes—
503C4⋊C4C2×D818yes—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C22⋊C4—48no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—82no—
519C22⋊C4—82no—
520C22⋊C4—82no—
521C22⋊C4—82no—
522C42—24no—
523C42—24no—
524C42—22no—
525C42—22no—
526C42—24no—
527C42—41no—
528C42—42no—
529C42—41no—
530C42—44no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—24no—
536C23—48no—
537C23—48no—
538C23—48no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—84no—
553C23—84no—
554C23—84no—
555C23—84no—
556C23#53512yes—
557C23C22×D812yes—
558C23C2×AGL(1,ℤ/8)12yes—
559C23C23×D411yescharacteristic
560Q8—816no—
561Q8—162no—
562D4—28no—
563D4—48no—
564D4—44no—
565D4—44no—
566D4—44no—
567D4—88no—
568D4—162no—
569D4—162no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—28no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—28no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—816no—
616C2×C4—84no—
617C2×C4—84no—
618C2×C4—84no—
619C2×C4—164no—
620C2×C4#53512yes—
621C2×C4C22×D814yes—
622C2×C4C22×D812yes—
623C2×C4C22×D812yes—
624C2×C4C22×D814yes—
625C2×C4C2×AGL(1,ℤ/8)12yes—
626C2×C4C2×AGL(1,ℤ/8)12yes—
627C2×C4C2×H(5,𝔽2)12yes—
628C2×C4C2×2−1+412yes—
629C2×C4#58512yes—
630C8—44no—
631C8—44no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—44no—
639C22—44no—
640C22—44no—
641C22—816no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22#320514yes—
654C22C2×#53511yescharacteristic
655C22#348511yescharacteristic
656C22#348611yescharacteristic
657C22C23×D811yescharacteristic
658C22C22×AGL(1,ℤ/8)11yescharacteristic
659C22#358611yescharacteristic
660C22#358611yescharacteristic
661C4—22no—
662C4—22no—
663C4—24no—
664C4—22no—
665C4—24no—
666C4—22no—
667C4—22no—
668C4—42no—
669C4—42no—
670C4—82no—
671C4—82no—
672C4—82no—
673C4—82no—
674C4—82no—
675C4—82no—
676C4#353514yes—
677C2—24no—
678C2—84no—
679C2—162no—
680C2#3711811yescharacteristic
681C2C2×#320511yescharacteristic
682C2#6080711yescharacteristic
683C2#6169411yescharacteristic
684C2#6170611yescharacteristic
685C2#6181711yescharacteristic
686C2#6182711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#28301C2
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
1. #35006C2semidirect
2. C2×#2542C2semidirect
3. #33002C2semidirect
4. #32426C2semidirect
5. #32342C2semidirect
6. #31634C2semidirect
7. #31518C2semidirect
8. #29638C2semidirect
9. #29103C2semidirect
11. #28114C2semidirect
12. #27960C2semidirect
13. #27909C2semidirect
14. #27644C2semidirect
15. #21912C2semidirect
36. #2512C22semidirect
37. #2506C22semidirect
38. #2452C22semidirect
39. #2445C22semidirect
49. #2099C22semidirect
51. #2093C22semidirect
52. #2083C22semidirect
54. #2071C22semidirect
56. #2057C22semidirect
57. #2057C22semidirect
60. #2023C22semidirect
61. #2023C22semidirect
63. #1990C22semidirect
65. #1957C22semidirect
66. #1947C22semidirect
69. #1916C22semidirect
70. #1916C22semidirect
76. #1845C22semidirect
77. #1834C22semidirect
82. #1802C22semidirect
306. C2×C4⋊C4D8semidirect

Maximal subgroups

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

subgroupindex
#350062
C2×#25422
#330022
#324262
#323422
#316342
#315182
#296382
#291032
#283012
#281142
#279602
#279092
#276442
#219122