Finite Group Databasebeta

← #6450679

Subgroups — #6450679

Order 512.

1910 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34948C211yesmaximal, characteristic, complemented
2C2×#2354C211yesmaximal, characteristic
3#26099C214yesmaximal, complemented
4#26098C214yesmaximal, complemented
5#26091C212yesmaximal, complemented
6#26090C212yesmaximal, complemented
7#26087C212yesmaximal, complemented
8#26086C212yesmaximal, complemented
9#23731C212yesmaximal
10#23618C212yesmaximal
11#21171C211yesmaximal, characteristic, complemented
12#21170C212yesmaximal, complemented
13#21146C211yesmaximal, characteristic, complemented
14#21033C212yesmaximal, complemented
15#20800C211yesmaximal, characteristic, complemented
16#20646C211yesmaximal, characteristic
17C4×#418C211yesmaximal, characteristic
18C22×C4×D4C2211yescharacteristic
19C22×#117—21no—
20C2×#428C2212yes—
21C2×#428C2212yes—
22C2×#428C2212yes—
23C2×#423C2212yes—
24C2×#423C2211yescharacteristic
25C2×#418C2211yescharacteristic
26C2×#418C2211yescharacteristic
27C2×#418C2212yes—
28C2×C4×M16C2211yescharacteristic
29#2571C2211yescharacteristic, complemented
30#2571C2211yescharacteristic, complemented
31#2570C2212yescomplemented
32#2570C2212yescomplemented
33#2569—22no—
34#2552C2211yescharacteristic, complemented
35#2552C2211yescharacteristic, complemented
36#2442—21no—
37#2442—21no—
38#2402—22no—
39#2354—21no—
40#2354—21no—
41#2354—22no—
42#2354C2212yes—
43#2354C2212yes—
44#2354C2214yes—
45C4×#125—21no—
46C4×#125—21no—
47#2296C2212yes—
48C2×#390C2212yes—
49C2×#388—22no—
50#2285C2211yescharacteristic
51#2285C2212yes—
52C2×#384C2211yescharacteristic
53C2×#380—21no—
54C4×#117C2211yescharacteristic
55C2×C4×C4⋊C4C2212yes—
56C2×C4×C4⋊C4C2211yescharacteristic
57#2094C2214yes—
58#2093C2214yes—
59#2092C2214yes—
60#2092C2214yes—
61#2067C2214yes—
62#2067C2214yes—
63#2065C2214yes—
64#2065C2214yes—
65#1949C2214yescomplemented
66#1949C2214yescomplemented
67#1948C2214yescomplemented
68#1948C2212yescomplemented
69#1948C2212yescomplemented
70#1947C2214yescomplemented
71#1947C2212yescomplemented
72#1947C2212yescomplemented
73#1940C2214yescomplemented
74#1939C2212yescomplemented
75#1939C2212yescomplemented
76#1921C2214yescomplemented
77#1920C2214yescomplemented
78#1919C2214yescomplemented
79#1918C2214yescomplemented
80#1847C2214yescomplemented
81#1845C2212yescomplemented
82#1844C2212yescomplemented
83#1804C2214yes—
84#1802C2212yes—
85#1802C2212yes—
86#1762C2212yes—
87#1762C2212yes—
88C4×#102C2212yes—
89C4×#102C2212yes—
90#1736C2211yescharacteristic
91#1736C2212yes—
92#1736C2212yes—
93#1736C2211yescharacteristic
94#1728C2212yes—
95C23×D4C2311yescharacteristic
96C24×C4—21no—
97C22×M16C2312yes—
98C22×M16C2311yescharacteristic
99C22×M16C2311yescharacteristic
100C2×#125—21no—
101C2×#125—21no—
102C2×C4×D4—21no—
103C2×C4×D4—21no—
104C2×C4×D4—24no—
105C2×C4×D4Q818yes—
106C2×C4×D4C2312yes—
107C2×C4×D4C2312yes—
108C2×#117—22no—
109C2×#117—24no—
110C2×#117—22no—
111C2×#117—48no—
112C2×#117C2311yescharacteristic
113C2×#117C2312yes—
114C2×#117C2311yescharacteristic
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4D411yescharacteristic
118C22×C4⋊C4D411yescharacteristic
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C22⋊C4—21no—
122C22×C22⋊C4—22no—
123C22×C22⋊C4—22no—
124C22×C42—22no—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127#428—28no—
128#428—44no—
129#428—44no—
130C2×#107C2314yes—
131C2×#107C2314yes—
132C2×#107C2314yes—
133C2×#106C2314yes—
134C2×#106C2314yes—
135C2×#106C2314yes—
136#423—28no—
137#423—24no—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2314yes—
141C2×C4⋊C8C2314yes—
142#418—24no—
143#418—24no—
144#418—44no—
145C2×#102C2312yes—
146C2×#102C2312yes—
147C2×#102C2312yes—
148C2×#102C2312yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C4×M16—24no—
152C2×#97C2312yes—
153C2×C4×C8C2312yes—
154#400—22no—
155#400—22no—
156#400—22no—
157#400—22no—
158#398—22no—
159#398—22no—
160#398—22no—
161#398—22no—
162#390—22no—
163#390—22no—
164#390C2314yes—
165#390C2314yes—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389C2312yes—
171#389C2312yes—
172#389C2312yes—
173#389C2312yes—
174#388—24no—
175#388—24no—
176#388—24no—
177#388—24no—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#385—22no—
183#385—22no—
184#384—21no—
185#384—21no—
186#384C2311yescharacteristic
187#384C2311yescharacteristic
188#384C2312yes—
189#384C2312yes—
190#384C2312yes—
191#384C2312yes—
192#382—21no—
193#382—21no—
194#382C2312yes—
195#382C2312yes—
196#382C2312yes—
197#380—21no—
198#380—22no—
199#380—22no—
200#380—21no—
201#380—22no—
202#380—22no—
203#380—24no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D412yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2314yes—
220C4×C4⋊C4C2314yes—
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2311yescharacteristic
224C4×C22⋊C4—21no—
225C4×C22⋊C4—21no—
226#376—21no—
227C2×#95—22no—
228C2×#95D412yes—
229C43—21no—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2312yes—
235#340C2312yes—
236#340C2312yes—
237#340C2312yes—
238#336C2314yes—
239#336C2314yes—
240C25—21no—
241C22×D4—21no—
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4C2×Q814yes—
248C22×D4C2412yes—
249C23×C4—21no—
250C23×C4—28no—
251C23×C4—22no—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4—44no—
255C23×C4—44no—
256C23×C4C2×D412yes—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D412yes—
259C23×C4C2×D411yescharacteristic
260C23×C4C2411yescharacteristic
261C2×M16—22no—
262C2×M16—22no—
263C2×M16—22no—
264C2×M16—22no—
265C2×M16—24no—
266C2×M16—24no—
267C2×M16—24no—
268C2×M16#4114yes—
269C2×M16#4114yes—
270C22×C8C2414yes—
271C22×C8C2412yes—
272C22×C8C2412yes—
273#125—44no—
274#125—44no—
275C4×D4—28no—
276C4×D4—28no—
277C4×D4—24no—
278C4×D4—24no—
279C4×D4C2×Q818yes—
280C4×D4C2×Q818yes—
281#117—22no—
282#117—22no—
283#117—24no—
284#117—24no—
285#117—22no—
286#117—22no—
287#117—48no—
288#117—416no—
289#117—48no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—42no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—82no—
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×Q814yes—
322C2×C4⋊C4C2414yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C22⋊C4—28no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—48no—
341C2×C22⋊C4—84no—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42—24no—
345C2×C42—24no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—22no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—21no—
352C2×C42—21no—
353C2×C42—48no—
354C2×C42C2×D412yes—
355C2×C42C2×D412yes—
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×Q814yes—
359C2×C42#4114yes—
360C2×C42C2412yes—
361C2×C42C2412yes—
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364#107—28no—
365#107—44no—
366#107—44no—
367#106—28no—
368#106—44no—
369#106—44no—
370C4⋊C8—28no—
371C4⋊C8—28no—
372C4⋊C8—24no—
373C4⋊C8—24no—
374#102—24no—
375#102—24no—
376#102—24no—
377#102—24no—
378#102—44no—
379#102—44no—
380#97—24no—
381C4×C8—24no—
382#95—24no—
383#95—24no—
384#95—21no—
385#95—21no—
386#95—21no—
387#95—21no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95C2×D412yes—
399#95C2×D412yes—
400#95C2×D412yes—
401#95C2×D412yes—
402C24—24no—
403C24—22no—
404C24—42no—
405C24—42no—
406C24—42no—
407C24—42no—
408C24—42no—
409C24—42no—
410C24C22×D411yescharacteristic
411C2×D4—24no—
412C2×D4—28no—
413C2×D4—28no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4C22×Q814yes—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—28no—
431C22×C4—28no—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—432no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4C22⋊Q814yes—
456C22×C4C22⋊Q814yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D412yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×Q812yes—
467C22×C4C2×#4112yes—
468C22×C4C2×#4112yes—
469C22×C4C2×#4112yes—
470C22×C4H(5,𝔽2)12yes—
471C22×C42−1+412yes—
472C22×C4C2511yescharacteristic
473M16—28no—
474M16—28no—
475M16—48no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—24no—
483C2×C8C2×#4114yes—
484C2×C8C2×#4114yes—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—24no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—81no—
516C4⋊C4—82no—
517C4⋊C4—81no—
518C4⋊C4—82no—
519C4⋊C4—82no—
520C4⋊C4—82no—
521C4⋊C4—82no—
522C4⋊C4—82no—
523C4⋊C4AGL(1,ℤ/8)14yes—
524C4⋊C4C22×Q814yes—
525C22⋊C4—28no—
526C22⋊C4—28no—
527C22⋊C4—48no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—44no—
531C22⋊C4—48no—
532C22⋊C4—84no—
533C22⋊C4—84no—
534C42—24no—
535C42—24no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—48no—
541C42—42no—
542C42—42no—
543C42—44no—
544C42—48no—
545C42—42no—
546C42—42no—
547C42—81no—
548C42—81no—
549C42C22⋊Q814yes—
550C42C22⋊Q814yes—
551C42C22×Q812yes—
552C42C22×Q812yes—
553C42C2×#4112yes—
554C42C2×#4112yes—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—24no—
560C23—48no—
561C23—48no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—48no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—84no—
570C23—84no—
571C23—84no—
572C23—84no—
573C23C2×C22⋊Q812yes—
574C23#54712yes—
575C23#55012yes—
576C23C23×D411yescharacteristic
577D4—24no—
578D4—24no—
579D4—48no—
580D4—42no—
581D4—42no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—28no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—44no—
602C2×C4—48no—
603C2×C4—416no—
604C2×C4—48no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—416no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—84no—
630C2×C4—88no—
631C2×C4C2×C22⋊Q812yes—
632C2×C4C2×C22⋊Q812yes—
633C2×C4C2×C22⋊Q812yes—
634C2×C4C2×C22⋊Q812yes—
635C2×C4C2×C22⋊Q812yes—
636C2×C4#53314yes—
637C2×C4#54414yes—
638C2×C4#54712yes—
639C2×C4#55012yes—
640C2×C4#55214yes—
641C2×C4#55314yes—
642C2×C4C2×AGL(1,ℤ/8)14yes—
643C2×C4C2×AGL(1,ℤ/8)12yes—
644C2×C4#57512yes—
645C2×C4C23×Q811yescharacteristic
646C2×C4C22×#4111yescharacteristic
647C2×C4C22×#4111yescharacteristic
648C2×C4C22×#4111yescharacteristic
649C2×C4C2×H(5,𝔽2)11yescharacteristic
650C2×C4C2×2−1+411yescharacteristic
651C8—24no—
652C8—24no—
653C8—44no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—44no—
661C22—44no—
662C22—44no—
663C22—816no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22#244114yes—
669C22C22×C22⋊Q811yescharacteristic
670C22C2×#54711yescharacteristic
671C22C2×#55011yescharacteristic
672C22C22×AGL(1,ℤ/8)11yescharacteristic
673C22C2×#57511yescharacteristic
674C22#358511yescharacteristic
675C22#358611yescharacteristic
676C4—22no—
677C4—24no—
678C4—22no—
679C4—22no—
680C4—44no—
681C4—42no—
682C4—42no—
683C4—42no—
684C4—81no—
685C4—81no—
686C4—81no—
687C4—82no—
688C4—84no—
689C4—81no—
690C4—82no—
691C4#244114yes—
692C4#307714yes—
693C4C2×#53311yescharacteristic
694C4C2×#54411yescharacteristic
695C4C2×#55211yescharacteristic
696C4C2×#55311yescharacteristic
697C2—24no—
698C2—84no—
699C2C2×#244111yescharacteristic
700C2C2×#307711yescharacteristic
701C2#6049211yescharacteristic
702C2#6125211yescharacteristic
703C2#6125611yescharacteristic
704C2#6140211yescharacteristic
705C2#6140311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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. #34948C2semidirect
3. #26099C2semidirect
4. #26098C2semidirect
5. #26091C2semidirect
6. #26090C2semidirect
7. #26087C2semidirect
8. #26086C2semidirect
11. #21171C2semidirect
12. #21170C2semidirect
13. #21146C2semidirect
14. #21033C2semidirect
15. #20800C2semidirect
29. #2571C22semidirect
30. #2571C22semidirect
31. #2570C22semidirect
32. #2570C22semidirect
34. #2552C22semidirect
35. #2552C22semidirect
65. #1949C22semidirect
66. #1949C22semidirect
67. #1948C22semidirect
68. #1948C22semidirect
69. #1948C22semidirect
70. #1947C22semidirect
71. #1947C22semidirect
72. #1947C22semidirect
73. #1940C22semidirect
74. #1939C22semidirect
75. #1939C22semidirect
76. #1921C22semidirect
77. #1920C22semidirect
78. #1919C22semidirect
79. #1918C22semidirect
80. #1847C22semidirect
81. #1845C22semidirect
82. #1844C22semidirect

Maximal subgroups

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

subgroupindex
#349482
C2×#23542
#260992
#260982
#260912
#260902
#260872
#260862
#237312
#236182
#211712
#211702
#211462
#210332
#208002
#206462
C4×#4182