Finite Group Databasebeta

← #6484397

Subgroups — #6484397

Order 512.

1974 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36645C211yesmaximal, characteristic, complemented
2C2×C4×#124C211yesmaximal, characteristic
3#32954C212yesmaximal, complemented
4#32953C212yesmaximal
5#29953C212yesmaximal, complemented
6#29952C212yesmaximal
7#29260C214yesmaximal, complemented
8#29197C214yesmaximal, complemented
9#29148C212yesmaximal, complemented
10#28520C212yesmaximal, complemented
11#28281C214yesmaximal, complemented
12#21986C212yesmaximal, complemented
13#21808C212yesmaximal
14#20810C211yesmaximal, characteristic, complemented
15C2×#533—21no—
16C22×#124C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C2×#475C2214yescomplemented
20C2×#474C2214yes—
21C2×#432C2212yes—
22#2868C2211yescharacteristic, complemented
23#2868C2211yescharacteristic, complemented
24#2862—21no—
25#2859—21no—
26#2859C2212yescomplemented
27#2857—21no—
28#2857—21no—
29#2857C2212yes—
30#2857C2212yes—
31C4×C4⋊Q8—21no—
32C4×C4⋊Q8—21no—
33C4×C4⋊Q8C2212yescomplemented
34C4×C4⋊Q8C2212yescomplemented
35C4×#124—21no—
36C4×#124—21no—
37C4×#124—21no—
38C4×#124—21no—
39C4×#124C2212yes—
40C4×#124C2212yes—
41C4×#124C2212yes—
42C4×#124C2212yes—
43C2×#391C2211yescharacteristic
44C2×#390C2211yescharacteristic
45C2×#388—21no—
46C2×#388—21no—
47#2285—21no—
48#2285C2212yes—
49C2×#382C2211yescharacteristic
50C2×C4×C22⋊C4—21no—
51C2×C4×C22⋊C4—21no—
52C2×C43C2211yescharacteristic
53#2095C2214yescomplemented
54#2094C2214yes—
55#2068C2218yescomplemented
56#2065C2218yes—
57#2024C2214yescomplemented
58#2023C2214yes—
59#1983C2218yes—
60#1959C2214yes—
61#1959C2214yes—
62#1949C2218yescomplemented
63#1865C2218yescomplemented
64#1862C2218yes—
65#1853C2214yes—
66#1852C2214yes—
67#1847C2214yescomplemented
68#1842C2212yescomplemented
69#1836C2214yescomplemented
70#1831C2212yescomplemented
71#1803C2214yescomplemented
72#1802C2214yes—
73#1797C2212yes—
74#1797C2212yes—
75#1787C2214yes—
76#1786C2214yes—
77#1767C2212yescomplemented
78#1767C2212yescomplemented
79C23×Q8C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C24×C4—21no—
82#533—48no—
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2312yes—
89C2×#125—22no—
90C2×#124—21no—
91C2×#124—21no—
92C2×#124—24no—
93C2×#124Q818yescomplemented
94C2×#124C2312yes—
95C2×#124C2312yes—
96C2×C22⋊Q8—22no—
97C2×C22⋊Q8—22no—
98C2×C4×Q8—22no—
99C2×C4×Q8—22no—
100C2×C4×Q8—21no—
101C2×C4×Q8—21no—
102C2×C4×Q8—44no—
103C2×C4×Q8C2312yes—
104C2×C4×Q8C2312yes—
105C2×C4×D4—21no—
106C2×C4×D4—21no—
107C2×C4×D4—44no—
108C2×C4×D4C2312yes—
109C2×C4×D4C2312yes—
110C2×#117—22no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C42—21no—
120C22×C42D411yescharacteristic
121C22×C42D411yescharacteristic
122C22×C42D411yescharacteristic
123C22×C42D411yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#475—48no—
128#474—48no—
129#432—28no—
130C2×#107C2318yes—
131C2×#106C2318yes—
132C2×C4⋊C8C2314yes—
133C2×C4⋊C8C2314yes—
134C2×C4⋊C8C2314yes—
135C2×#103C2318yes—
136C2×#103C2318yes—
137C2×#102C2318yes—
138C2×#102C2318yes—
139C2×C22⋊C8C2314yes—
140C2×#97C2314yes—
141#391—22no—
142#391—22no—
143#391—22no—
144#391—22no—
145#391—21no—
146#391—21no—
147#391C2312yes—
148#391C2312yes—
149#390—21no—
150#390—21no—
151#390C2312yes—
152#390C2312yes—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389C2314yes—
160#389C2314yes—
161#388—22no—
162#388—22no—
163#388—22no—
164#388—22no—
165#388—22no—
166#388—22no—
167#388—22no—
168#388—22no—
169#384—21no—
170#384—21no—
171#384—21no—
172#384—21no—
173#384C2312yes—
174#384C2312yes—
175#384C2312yes—
176#384C2312yes—
177#382—22no—
178#382—22no—
179#382—21no—
180#382—21no—
181#382C2312yes—
182#382C2312yes—
183#380—22no—
184#380—22no—
185#380—22no—
186#380C2314yes—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4C2314yes—
191C4×C22⋊C4—22no—
192C4×C22⋊C4—22no—
193C4×C22⋊C4—22no—
194C4×C22⋊C4—22no—
195C4×C22⋊C4—22no—
196C4×C22⋊C4—22no—
197C4×C22⋊C4—22no—
198C4×C22⋊C4—22no—
199C2×#95—21no—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95—21no—
203C43D411yescharacteristic
204C43D411yescharacteristic
205C43D411yescharacteristic
206C43D411yescharacteristic
207C43C2311yescharacteristic
208C43C2311yescharacteristic
209C43C2311yescharacteristic
210C43C2311yescharacteristic
211#340C2318yes—
212#340C2318yes—
213#336C2314yes—
214#336C2314yes—
215C25—21no—
216C22×Q8—21no—
217C22×Q8—22no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8—42no—
221C22×Q8—42no—
222C22×Q8C2×Q814yes—
223C22×Q8C2412yes—
224C22×D4—21no—
225C22×D4—22no—
226C22×D4—42no—
227C22×D4—42no—
228C22×D4C2×Q814yes—
229C22×D4C2412yes—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—22no—
234C23×C4—22no—
235C23×C4—44no—
236C23×C4—44no—
237C23×C4—44no—
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2411yescharacteristic
245C22×C8C2418yes—
246C4⋊Q8—28no—
247C4⋊Q8—44no—
248C4⋊Q8—44no—
249C4⋊Q8—44no—
250C4⋊Q8—44no—
251C4⋊Q8—44no—
252#125—48no—
253#124—28no—
254#124—28no—
255#124—24no—
256#124—24no—
257#124C2×Q818yes—
258#124C2×Q818yes—
259C22⋊Q8—48no—
260C22⋊Q8—48no—
261C4×Q8—28no—
262C4×Q8—44no—
263C4×Q8—44no—
264C4×Q8—44no—
265C4×Q8—48no—
266C4×Q8—48no—
267C4×D4—28no—
268C4×D4—44no—
269C4×D4—44no—
270C4×D4—44no—
271#117—48no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2414yes—
294C2×C4⋊C4C2414yes—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—42no—
318C2×C42—42no—
319C2×C42—42no—
320C2×C42—44no—
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×Q814yes—
344C2×C42#4114yes—
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351#107—48no—
352#106—48no—
353C4⋊C8—28no—
354C4⋊C8—28no—
355C4⋊C8—44no—
356#103—48no—
357#103—48no—
358#102—48no—
359#102—48no—
360C22⋊C8—28no—
361#97—28no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—24no—
374C24—24no—
375C24—22no—
376C24—42no—
377C24—42no—
378C24—42no—
379C24—42no—
380C24—42no—
381C24—42no—
382C24C22×D411yescharacteristic
383C2×Q8—24no—
384C2×Q8—22no—
385C2×Q8—24no—
386C2×Q8—24no—
387C2×Q8—48no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—48no—
391C2×Q8—44no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8C22×Q814yes—
398C2×D4—24no—
399C2×D4—22no—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—48no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—48no—
406C2×D4C22×Q814yes—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—816no—
444C22×C4C22⋊Q814yes—
445C22×C4C22⋊Q814yes—
446C22×C4AGL(1,ℤ/8)14yes—
447C22×C4#13714yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×Q812yes—
462C22×C4C2×#4112yes—
463C22×C4C2511yescharacteristic
464C2×C8—28no—
465C2×C8—28no—
466C2×C8—28no—
467C4⋊C4—28no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—28no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—48no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—48no—
486C4⋊C4—44no—
487C22⋊C4—28no—
488C22⋊C4—28no—
489C22⋊C4—28no—
490C22⋊C4—28no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—48no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—41no—
516C42—41no—
517C42—41no—
518C42—41no—
519C42—41no—
520C42—41no—
521C42—44no—
522C42—48no—
523C42—44no—
524C42C22⋊Q814yes—
525C42C22⋊Q814yes—
526C42C22×Q812yes—
527C42C22×Q812yes—
528C42C2×#4112yes—
529C42C2×#4112yes—
530C42H(5,𝔽2)14yes—
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—84no—
546C23—84no—
547C23—84no—
548C23—84no—
549C23C2×C22⋊Q812yes—
550C23C2×AGL(1,ℤ/8)12yes—
551C23C2×#13712yes—
552C23C23×D411yescharacteristic
553Q8—28no—
554Q8—48no—
555Q8—44no—
556Q8—88no—
557Q8—88no—
558D4—28no—
559D4—48no—
560D4—44no—
561C2×C4—28no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—28no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—48no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—88no—
612C2×C4—84no—
613C2×C4—84no—
614C2×C4—88no—
615C2×C4C2×C22⋊Q812yes—
616C2×C4C2×C22⋊Q812yes—
617C2×C4C2×C22⋊Q812yes—
618C2×C4C2×C22⋊Q812yes—
619C2×C4C2×C22⋊Q812yes—
620C2×C4#53512yes—
621C2×C4#53512yes—
622C2×C4C2×AGL(1,ℤ/8)12yes—
623C2×C4C2×AGL(1,ℤ/8)12yes—
624C2×C4C2×#13712yes—
625C2×C4C2×#13712yes—
626C2×C4#57512yes—
627C2×C4#57512yes—
628C2×C4C23×Q811yescharacteristic
629C2×C4C22×#4111yescharacteristic
630C2×C4C2×H(5,𝔽2)12yes—
631C2×C4#58512yes—
632C8—48no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—44no—
640C22—44no—
641C22—44no—
642C22—816no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22#307914yes—
648C22C22×C22⋊Q811yescharacteristic
649C22C2×#53511yescharacteristic
650C22#344911yescharacteristic
651C22C22×AGL(1,ℤ/8)11yescharacteristic
652C22C22×#13711yescharacteristic
653C22C2×#57511yescharacteristic
654C22C2×#57511yescharacteristic
655C4—22no—
656C4—22no—
657C4—22no—
658C4—22no—
659C4—22no—
660C4—22no—
661C4—22no—
662C4—22no—
663C4—22no—
664C4—22no—
665C4—41no—
666C4—44no—
667C4—41no—
668C4—84no—
669C4—82no—
670C4—84no—
671C4—82no—
672C4#321314yes—
673C4#350112yes—
674C4#350612yes—
675C2—24no—
676C2—84no—
677C2#3517211yescharacteristic
678C2C2×#307911yescharacteristic
679C2#6048411yescharacteristic
680C2C2×#321311yescharacteristic
681C2#6080411yescharacteristic
682C2#6094911yescharacteristic
683C2#6096011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36645C2semidirect
3. #32954C2semidirect
5. #29953C2semidirect
7. #29260C2semidirect
8. #29197C2semidirect
9. #29148C2semidirect
10. #28520C2semidirect
11. #28281C2semidirect
12. #21986C2semidirect
14. #20810C2semidirect
19. C2×#475C22semidirect
22. #2868C22semidirect
23. #2868C22semidirect
26. #2859C22semidirect
33. C4×C4⋊Q8C22semidirect
34. C4×C4⋊Q8C22semidirect
53. #2095C22semidirect
55. #2068C22semidirect
57. #2024C22semidirect
62. #1949C22semidirect
63. #1865C22semidirect
67. #1847C22semidirect
68. #1842C22semidirect
69. #1836C22semidirect
70. #1831C22semidirect
71. #1803C22semidirect
77. #1767C22semidirect
78. #1767C22semidirect
93. C2×#124Q8semidirect

Maximal subgroups

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

subgroupindex
#366452
C2×C4×#1242
#329542
#329532
#299532
#299522
#292602
#291972
#291482
#285202
#282812
#219862
#218082
#208102