Finite Group Databasebeta

← #6629797

Subgroups — #6629797

Order 512.

2016 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35253C211yesmaximal, characteristic
2C2×#2274C211yesmaximal, characteristic
3#33156C214yesmaximal, complemented
4#33155C214yesmaximal, complemented
5#33151C212yesmaximal, complemented
6#33150C212yesmaximal, complemented
7#33149C212yesmaximal, complemented
8#33146C212yesmaximal, complemented
9#21315C212yesmaximal, complemented
10#21173C212yesmaximal, complemented
11#21037C212yesmaximal, complemented
12#21035C212yesmaximal, complemented
13#20844C211yesmaximal, characteristic
14C2×#1852C211yesmaximal, characteristic
15#20702C211yesmaximal, characteristic
16#20699C211yesmaximal, characteristic
17#20628C211yesmaximal, characteristic
18C2×#544C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C2×#428C2212yes—
23C22×#107C2212yes—
24C2×#424C2212yes—
25C2×#407C2211yescharacteristic
26C22×C22⋊C8C2211yescharacteristic
27#2857C2212yes—
28#2851C2212yes—
29#2629C2212yes—
30#2628C2212yes—
31#2442—21no—
32#2441—21no—
33#2440—21no—
34#2439—21no—
35#2438—22no—
36#2437—22no—
37#2317C2211yescharacteristic
38#2317C2211yescharacteristic
39#2316C2211yescharacteristic
40#2280C2211yescharacteristic
41C2×#380C2211yescharacteristic
42#2274—22no—
43#2274—21no—
44#2274—21no—
45#2274C2212yes—
46#2274C2212yes—
47#2274C2214yes—
48C2×C4×C4⋊C4—21no—
49C2×C4×C4⋊C4—21no—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C22⋊C4—22no—
53#1989C2214yescomplemented
54#1989C2214yescomplemented
55#1988C2214yescomplemented
56#1987C2214yescomplemented
57#1949C2214yescomplemented
58#1949C2214yescomplemented
59#1947C2214yescomplemented
60#1947C2214yescomplemented
61#1921C2214yescomplemented
62#1921C2214yescomplemented
63#1921C2214yescomplemented
64#1920C2214yescomplemented
65#1919C2214yescomplemented
66#1918C2214yescomplemented
67#1918C2214yescomplemented
68#1918C2214yescomplemented
69#1853C2212yes—
70#1853C2212yes—
71#1852C2212yes—
72#1852C2212yes—
73#1852C2214yes—
74#1852C2212yes—
75#1852C2212yes—
76#1825C2212yes—
77#1825C2214yes—
78#1825C2212yes—
79#1824C2211yescharacteristic
80#1824C2212yes—
81#1824C2211yescharacteristic
82#1823C2211yescharacteristic
83#1823C2212yes—
84#1823C2211yescharacteristic
85#1798C2214yes—
86#1797C2212yes—
87#1797C2212yes—
88#1736C2212yes—
89#1736C2212yes—
90#1736C2212yes—
91#1735C2212yes—
92#1735C2212yes—
93C2×#340C2212yes—
94#1729C2212yes—
95C24×C4—21no—
96C24×C4C2311yescharacteristic
97C22×M16C2312yes—
98C23×C8C2312yes—
99#544—28no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2312yes—
102C2×#125—21no—
103C2×#125C2312yes—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—22no—
106C2×#117—21no—
107C2×#117—21no—
108C2×#117—24no—
109C2×#117—28no—
110C2×#117D412yes—
111C2×#117D411yescharacteristic
112C2×#117D411yescharacteristic
113C2×#117Q818yescomplemented
114C2×#117C2311yescharacteristic
115C2×#117C2311yescharacteristic
116C2×#117C2312yes—
117C2×#117C2312yes—
118C2×#117C2312yes—
119C2×#117C2311yescharacteristic
120C2×#117C2311yescharacteristic
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—28no—
125C22×C4⋊C4D411yescharacteristic
126C22×C4⋊C4D411yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2312yes—
134C22×C22⋊C4—21no—
135C22×C22⋊C4—21no—
136C22×C22⋊C4D412yes—
137C22×C42—21no—
138C22×C42—22no—
139C22×C42—22no—
140C22×C42C2312yes—
141#428—28no—
142C2×#107—28no—
143C2×#107C2314yes—
144C2×#107C2314yes—
145C2×#107C2314yes—
146C2×#106C2314yes—
147#424—28no—
148C2×C4⋊C8C2314yes—
149C2×C4⋊C8C2314yes—
150#407—24no—
151C2×C22⋊C8—24no—
152C2×C22⋊C8C2311yescharacteristic
153C2×C22⋊C8C2311yescharacteristic
154C2×C22⋊C8C2312yes—
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2312yes—
157#400—22no—
158#399—22no—
159#398—22no—
160#398—22no—
161#396—22no—
162#396—22no—
163#395—22no—
164#393—22no—
165#391—22no—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—22no—
171#389C2312yes—
172#389C2312yes—
173#389C2312yes—
174#389C2312yes—
175#387—22no—
176#387—22no—
177#385—22no—
178#385—22no—
179#385—22no—
180#385—22no—
181#384—21no—
182#384C2312yes—
183#382—21no—
184#382C2312yes—
185#381—21no—
186#380—22no—
187#380—21no—
188#380—21no—
189#380—22no—
190#380C2311yescharacteristic
191#380C2312yes—
192#380C2311yescharacteristic
193#380C2312yes—
194#380C2311yescharacteristic
195#380C2311yescharacteristic
196#379—21no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—24no—
200C4×C4⋊C4—24no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4C2314yes—
208C4×C4⋊C4C2314yes—
209C4×C4⋊C4C2314yes—
210C4×C4⋊C4C2314yes—
211C4×C22⋊C4—21no—
212C4×C22⋊C4—21no—
213C4×C22⋊C4—24no—
214C4×C22⋊C4—24no—
215C4×C22⋊C4—24no—
216C4×C22⋊C4—24no—
217C2×#95—21no—
218C2×#95—22no—
219C2×#95—22no—
220#340C2312yes—
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#336C2314yes—
237#336C2314yes—
238C25C2×D411yescharacteristic
239C22×Q8—22no—
240C23×C4—24no—
241C23×C4—24no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—22no—
245C23×C4—28no—
246C23×C4—22no—
247C23×C4—22no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—82no—
251C23×C4—82no—
252C23×C4C2×D412yes—
253C23×C4C2×D412yes—
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×Q814yes—
258C23×C4#4114yes—
259C23×C4C2411yescharacteristic
260C23×C4C2411yescharacteristic
261C23×C4C2411yescharacteristic
262C2×M16—24no—
263C2×M16—24no—
264C2×M16—24no—
265C22×C8—24no—
266C22×C8—24no—
267C22×C8—24no—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C22×C8C2414yes—
271C4⋊Q8—28no—
272C4⋊Q8—44no—
273#125—28no—
274#125—44no—
275C22⋊Q8—48no—
276C22⋊Q8—84no—
277#117—28no—
278#117—28no—
279#117—24no—
280#117—24no—
281#117—24no—
282#117—24no—
283#117—24no—
284#117—28no—
285#117—28no—
286#117—24no—
287#117—24no—
288#117C2×Q818yes—
289#117C2×Q818yes—
290#117#4118yes—
291C2×C4⋊C4—28no—
292C2×C4⋊C4—28no—
293C2×C4⋊C4—28no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—28no—
305C2×C4⋊C4—28no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×Q814yes—
326C2×C4⋊C4C2×Q814yes—
327C2×C4⋊C4#4118yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2414yes—
337C2×C22⋊C4—28no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—28no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—82no—
350C2×C22⋊C4C2×D412yes—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C22⋊C4C2×D412yes—
354C2×C42—28no—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—28no—
358C2×C42—21no—
359C2×C42—21no—
360C2×C42—24no—
361C2×C42—24no—
362C2×C42—24no—
363C2×C42—24no—
364C2×C42—84no—
365C2×C42—84no—
366C2×C42C2×D412yes—
367C2×C42C2×D412yes—
368C2×C42C2412yes—
369C2×C42C2412yes—
370#107—28no—
371#107—28no—
372#107—28no—
373#106—28no—
374C4⋊C8—28no—
375C4⋊C8—28no—
376C22⋊C8—22no—
377C22⋊C8—22no—
378C22⋊C8—24no—
379C22⋊C8—24no—
380C22⋊C8—24no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—22no—
386#95—22no—
387#95—21no—
388#95—21no—
389#95—21no—
390#95—21no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397C24—22no—
398C24—22no—
399C24—22no—
400C24—22no—
401C24—41no—
402C24—41no—
403C24C22⋊Q814yes—
404C24C22×D411yescharacteristic
405C24C22×D411yescharacteristic
406C24C22×D411yescharacteristic
407C2×Q8—44no—
408C2×Q8—82no—
409C2×Q8—82no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—216no—
417C22×C4—216no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—28no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—28no—
431C22×C4—28no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—44no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—44no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—82no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4—82no—
454C22×C4—816no—
455C22×C4—81no—
456C22×C4—81no—
457C22×C4—81no—
458C22×C4—81no—
459C22×C4C22⋊Q814yes—
460C22×C4C22⋊Q814yes—
461C22×C4C22⋊Q814yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×Q812yes—
470C22×C4C22×Q812yes—
471C22×C4C22×Q812yes—
472C22×C4C2×#4112yes—
473C22×C4C2×#4112yes—
474C22×C4C2×#4112yes—
475C22×C4C2×#4114yes—
476C22×C4C2×#4114yes—
477C22×C4C2511yescharacteristic
478M16—48no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—24no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—48no—
489C4⋊C4—28no—
490C4⋊C4—28no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—28no—
498C4⋊C4—24no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—24no—
503C4⋊C4—22no—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—42no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—42no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C4⋊C4—82no—
517C4⋊C4—82no—
518C4⋊C4#12314yes—
519C4⋊C4#12314yes—
520C4⋊C4C22×Q814yes—
521C4⋊C4C22×Q814yes—
522C4⋊C4C2×#4118yes—
523C22⋊C4—28no—
524C22⋊C4—28no—
525C22⋊C4—24no—
526C22⋊C4—24no—
527C22⋊C4—24no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—44no—
531C22⋊C4—42no—
532C22⋊C4—42no—
533C22⋊C4—81no—
534C22⋊C4—81no—
535C22⋊C4—82no—
536C22⋊C4#12318yes—
537C42—28no—
538C42—28no—
539C42—24no—
540C42—24no—
541C42—24no—
542C42—42no—
543C42—42no—
544C42—84no—
545C42—84no—
546C42—84no—
547C42—84no—
548C42C2×#4118yes—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—28no—
560C23—28no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23C2×C22⋊Q812yes—
571C23C2×C22⋊Q812yes—
572C23C2×C22⋊Q812yes—
573C23C2×#12312yes—
574C23#52512yes—
575C23#57612yes—
576C23#57812yes—
577C23C23×D411yescharacteristic
578Q8—164no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—28no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—28no—
589C2×C4—28no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—28no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—48no—
604C2×C4—48no—
605C2×C4—48no—
606C2×C4—416no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—48no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—88no—
633C2×C4—88no—
634C2×C4—162no—
635C2×C4—162no—
636C2×C4C2×C22⋊Q812yes—
637C2×C4C2×C22⋊Q812yes—
638C2×C4C2×C22⋊Q812yes—
639C2×C4C2×#12314yes—
640C2×C4C2×#12314yes—
641C2×C4C2×#12312yes—
642C2×C4#52512yes—
643C2×C4#52918yes—
644C2×C4#53318yes—
645C2×C4C23×Q811yescharacteristic
646C2×C4C22×#4111yescharacteristic
647C2×C4C22×#4112yes—
648C2×C4C22×#4112yes—
649C8—42no—
650C8—42no—
651C8—44no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—416no—
668C22—41no—
669C22—41no—
670C22—41no—
671C22—41no—
672C22#285514yes—
673C22#308314yes—
674C22C22×C22⋊Q811yescharacteristic
675C22C22×#12311yescharacteristic
676C22C2×#52511yescharacteristic
677C22C2×#57611yescharacteristic
678C22C2×#57811yescharacteristic
679C22#358511yescharacteristic
680C22#358611yescharacteristic
681C4—22no—
682C4—22no—
683C4—24no—
684C4—24no—
685C4—44no—
686C4—44no—
687C4—42no—
688C4—42no—
689C4—48no—
690C4—82no—
691C4—81no—
692C4—81no—
693C4—81no—
694C4—82no—
695C4—81no—
696C4#285514yes—
697C4C2×#52912yes—
698C4C2×#53312yes—
699C2—24no—
700C2—24no—
701C2—42no—
702C2C2×#285511yescharacteristic
703C2C2×#308311yescharacteristic
704C2#6049211yescharacteristic
705C2#6052111yescharacteristic
706C2#6052611yescharacteristic
707C2#6057811yescharacteristic
708C2#6058311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33156C2semidirect
4. #33155C2semidirect
5. #33151C2semidirect
6. #33150C2semidirect
7. #33149C2semidirect
8. #33146C2semidirect
9. #21315C2semidirect
10. #21173C2semidirect
11. #21037C2semidirect
12. #21035C2semidirect
53. #1989C22semidirect
54. #1989C22semidirect
55. #1988C22semidirect
56. #1987C22semidirect
57. #1949C22semidirect
58. #1949C22semidirect
59. #1947C22semidirect
60. #1947C22semidirect
61. #1921C22semidirect
62. #1921C22semidirect
63. #1921C22semidirect
64. #1920C22semidirect
65. #1919C22semidirect
66. #1918C22semidirect
67. #1918C22semidirect
68. #1918C22semidirect
113. C2×#117Q8semidirect

Maximal subgroups

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

subgroupindex
#352532
C2×#22742
#331562
#331552
#331512
#331502
#331492
#331462
#213152
#211732
#210372
#210352
#208442
C2×#18522
#207022
#206992
#206282