Finite Group Databasebeta

← #6470950

Subgroups — #6470950

Order 512.

1842 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35020C211yesmaximal, characteristic
2C2×#2360C211yesmaximal, characteristic
3#33150C212yesmaximal, complemented
4#32960C212yesmaximal, complemented
5#29973C212yesmaximal, complemented
6#29957C212yesmaximal, complemented
7#26418C214yesmaximal, complemented
8#26416C214yesmaximal, complemented
9#26346C214yesmaximal, complemented
10#26344C214yesmaximal, complemented
11#21930C211yesmaximal, characteristic
12#21759C211yesmaximal, characteristic
13#21759C211yesmaximal, characteristic
14C2×#1853C211yesmaximal, characteristic
15C2×#1852C211yesmaximal, characteristic
16C22×C22⋊Q8C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C2×#479C2214yescomplemented
20C22×#107C2212yes—
21C22×#106C2212yes—
22C22×C22⋊C8C2211yescharacteristic
23C22×C22⋊C8C2211yescharacteristic
24#2851C2212yes—
25#2851C2212yes—
26#2646—21no—
27#2646—21no—
28#2569—22no—
29#2569—22no—
30#2442C2212yes—
31#2442C2212yes—
32#2442C2212yes—
33#2442C2212yes—
34#2360—22no—
35#2360—22no—
36#2360C2214yes—
37#2360C2214yes—
38#2315—21no—
39#2315—21no—
40C2×#391C2211yescharacteristic
41C2×#389—21no—
42C2×#389C2211yescharacteristic
43C2×#387—21no—
44C2×#385—21no—
45C2×#380C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4—21no—
49C2×C4×C22⋊C4C2211yescharacteristic
50#2101C2214yescomplemented
51#2097C2214yescomplemented
52#2088C2214yescomplemented
53#2086C2214yescomplemented
54#2068C2214yescomplemented
55#2066C2214yescomplemented
56#2059C2214yescomplemented
57#2026C2214yescomplemented
58#1989C2214yescomplemented
59#1949C2214yescomplemented
60#1920C2214yescomplemented
61#1918C2214yescomplemented
62#1892C2214yes—
63#1892C2214yes—
64#1872C2212yes—
65#1872C2212yes—
66#1872C2212yes—
67#1872C2212yes—
68#1865C2214yescomplemented
69#1863C2214yescomplemented
70#1853C2212yes—
71#1853C2214yes—
72#1853C2212yes—
73#1852C2212yes—
74#1852C2214yes—
75#1852C2212yes—
76#1850C2214yes—
77#1850C2214yes—
78#1824C2212yes—
79#1824C2212yes—
80#1805C2214yescomplemented
81#1797C2212yes—
82#1797C2212yes—
83#1787C2212yes—
84#1787C2212yes—
85C2×#340C2212yes—
86C2×#340C2212yes—
87C23×Q8C2311yescharacteristic
88C24×C4—21no—
89C24×C4C2311yescharacteristic
90C23×C8C2312yes—
91C23×C8C2312yes—
92C2×#125C2312yes—
93C2×#125C2312yes—
94C2×C22⋊Q8—21no—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8Q818yes—
98C2×C22⋊Q8C2312yes—
99C2×C22⋊Q8C2312yes—
100C2×C4×Q8—22no—
101C2×C4×Q8—44no—
102C2×C4×Q8C2314yes—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—44no—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4D411yescharacteristic
123C22×C22⋊C4D411yescharacteristic
124C22×C42—21no—
125C22×C42—21no—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#479—48no—
129C2×#107—44no—
130C2×#107C2314yes—
131C2×#107C2314yes—
132C2×#106—44no—
133C2×#106C2314yes—
134C2×#106C2314yes—
135C2×C4⋊C8C2314yes—
136C2×C4⋊C8C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×C22⋊C8—24no—
142C2×C22⋊C8—24no—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2312yes—
147#400—22no—
148#400—22no—
149#400—22no—
150#400—22no—
151#398—22no—
152#398—22no—
153#398—22no—
154#398—22no—
155#391—22no—
156#391C2314yes—
157#389—22no—
158#389—24no—
159#389—24no—
160#389C2314yes—
161#389C2312yes—
162#389C2312yes—
163#389C2312yes—
164#389C2312yes—
165#387—22no—
166#387—22no—
167#387—22no—
168#387—22no—
169#387—24no—
170#387—24no—
171#385—22no—
172#385—22no—
173#385—22no—
174#385—22no—
175#385—22no—
176#385—22no—
177#385—22no—
178#385—22no—
179#385—24no—
180#385—24no—
181#382C2312yes—
182#382C2312yes—
183#380—21no—
184#380—21no—
185#380—21no—
186#380—21no—
187#380—22no—
188#380—22no—
189#380—22no—
190#380—22no—
191#380C2312yes—
192#380C2312yes—
193#380C2312yes—
194#380C2312yes—
195C4×C4⋊C4—22no—
196C4×C4⋊C4C2314yes—
197C4×C22⋊C4—24no—
198C4×C22⋊C4—24no—
199C4×C22⋊C4D412yes—
200C4×C22⋊C4D412yes—
201C4×C22⋊C4C2312yes—
202C4×C22⋊C4C2312yes—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95D411yescharacteristic
210C2×#95D411yescharacteristic
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#336C2314yes—
220#336C2314yes—
221C25C2×D411yescharacteristic
222C22×Q8—21no—
223C22×Q8—22no—
224C22×Q8—42no—
225C22×Q8—42no—
226C22×Q8C2×Q814yes—
227C22×Q8C2412yes—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—42no—
239C23×C4—42no—
240C23×C4—44no—
241C23×C4—44no—
242C23×C4—82no—
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×Q814yes—
251C23×C4#4114yes—
252C23×C4C2411yescharacteristic
253C23×C4C2411yescharacteristic
254C23×C4C2411yescharacteristic
255C22×C8—24no—
256C22×C8—24no—
257C22×C8—42no—
258C22×C8—42no—
259C22×C8—42no—
260C22×C8—42no—
261C22×C8C2414yes—
262C22×C8C2414yes—
263#125—44no—
264#125—44no—
265C22⋊Q8—28no—
266C22⋊Q8—28no—
267C22⋊Q8—24no—
268C22⋊Q8—24no—
269C22⋊Q8C2×Q818yes—
270C22⋊Q8C2×Q818yes—
271C4×Q8—28no—
272C4×Q8—44no—
273C4×Q8—48no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4—42no—
303C2×C4⋊C4—42no—
304C2×C4⋊C4—42no—
305C2×C4⋊C4—42no—
306C2×C4⋊C4—42no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—82no—
312C2×C4⋊C4C2×Q814yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C22⋊C4—28no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
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—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C22⋊C4C2×D412yes—
334C2×C22⋊C4C2×D412yes—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—42no—
338C2×C42—42no—
339C2×C42—44no—
340C2×C42—82no—
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42C2412yes—
344C2×C42C2412yes—
345#107—28no—
346#107—44no—
347#106—28no—
348#106—44no—
349C4⋊C8—44no—
350C4⋊C8—44no—
351#103—44no—
352#103—44no—
353#103—44no—
354#103—44no—
355C22⋊C8—24no—
356C22⋊C8—24no—
357C22⋊C8—24no—
358C22⋊C8—24no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—24no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95C2×D412yes—
376#95C2×D412yes—
377#95C2×D412yes—
378#95C2×D412yes—
379C24—41no—
380C24—41no—
381C24—41no—
382C24—41no—
383C24C22⋊Q814yes—
384C24#13514yes—
385C24#13714yes—
386C24C22×D411yescharacteristic
387C24C22×D411yescharacteristic
388C24C22×D411yescharacteristic
389C2×Q8—24no—
390C2×Q8—22no—
391C2×Q8—24no—
392C2×Q8—24no—
393C2×Q8—48no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—48no—
397C2×Q8C22×Q814yes—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
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—44no—
436C22×C4—44no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—82no—
453C22×C4—82no—
454C22×C4—81no—
455C22×C4—81no—
456C22×C4—81no—
457C22×C4—81no—
458C22×C4—81no—
459C22×C4—81no—
460C22×C4—81no—
461C22×C4—81no—
462C22×C4—816no—
463C22×C4C22⋊Q814yes—
464C22×C4C22⋊Q814yes—
465C22×C4C22⋊Q814yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×Q812yes—
474C22×C4C22×Q812yes—
475C22×C4C22×Q812yes—
476C22×C4C2×#4112yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4112yes—
479C22×C4C2511yescharacteristic
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—24no—
483C2×C8—24no—
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—48no—
487C2×C8—48no—
488C4⋊C4—24no—
489C4⋊C4—28no—
490C4⋊C4—28no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—48no—
503C4⋊C4—42no—
504C4⋊C4—82no—
505C4⋊C4—82no—
506C4⋊C4—82no—
507C4⋊C4—82no—
508C4⋊C4—81no—
509C4⋊C4—81no—
510C4⋊C4—81no—
511C4⋊C4—81no—
512C4⋊C4—84no—
513C4⋊C4—82no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C4⋊C4—82no—
517C4⋊C4C22×Q814yes—
518C4⋊C4H(5,𝔽2)14yes—
519C22⋊C4—28no—
520C22⋊C4—28no—
521C22⋊C4—24no—
522C22⋊C4—24no—
523C22⋊C4—24no—
524C22⋊C4—24no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C42—24no—
530C42—24no—
531C42—42no—
532C42—42no—
533C42—48no—
534C42—84no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23#479116yescomplemented
554C23C2×C22⋊Q812yes—
555C23C2×C22⋊Q812yes—
556C23C2×C22⋊Q812yes—
557C23C2×#13512yes—
558C23C2×#13512yes—
559C23C2×#13512yes—
560C23C2×#13712yes—
561C23C2×#13712yes—
562C23C2×#13712yes—
563C23C23×D411yescharacteristic
564Q8—28no—
565Q8—48no—
566Q8—44no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—48no—
599C2×C4—44no—
600C2×C4—48no—
601C2×C4—48no—
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—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—82no—
624C2×C4—82no—
625C2×C4—88no—
626C2×C4—88no—
627C2×C4—84no—
628C2×C4—162no—
629C2×C4—162no—
630C2×C4—162no—
631C2×C4—162no—
632C2×C4C2×C22⋊Q812yes—
633C2×C4C2×C22⋊Q812yes—
634C2×C4C2×C22⋊Q812yes—
635C2×C4C23×Q811yescharacteristic
636C2×C4C22×#4111yescharacteristic
637C2×C4C2×H(5,𝔽2)12yes—
638C2×C4#58512yes—
639C8—44no—
640C8—44no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—416no—
660C22C2×#47914yes—
661C22C2×#47914yes—
662C22C2×#47914yes—
663C22C22×C22⋊Q811yescharacteristic
664C22#348411yescharacteristic
665C22#348511yescharacteristic
666C22C22×#13511yescharacteristic
667C22C22×#13711yescharacteristic
668C22#358511yescharacteristic
669C22#358611yescharacteristic
670C4—22no—
671C4—22no—
672C4—22no—
673C4—42no—
674C4—42no—
675C4—44no—
676C4—42no—
677C4—44no—
678C4—42no—
679C4—44no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4—82no—
684C4—82no—
685C4#345912yes—
686C4#350112yes—
687C2—24no—
688C2—24no—
689C2—24no—
690C2#3498511yescharacteristic
691C2C22×#47911yescharacteristic
692C2#6049211yescharacteristic
693C2#6156111yescharacteristic
694C2#6157411yescharacteristic
695C2#6169611yescharacteristic
696C2#6170911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33150C2semidirect
4. #32960C2semidirect
5. #29973C2semidirect
6. #29957C2semidirect
7. #26418C2semidirect
8. #26416C2semidirect
9. #26346C2semidirect
10. #26344C2semidirect
19. C2×#479C22semidirect
50. #2101C22semidirect
51. #2097C22semidirect
52. #2088C22semidirect
53. #2086C22semidirect
54. #2068C22semidirect
55. #2066C22semidirect
56. #2059C22semidirect
57. #2026C22semidirect
58. #1989C22semidirect
59. #1949C22semidirect
60. #1920C22semidirect
61. #1918C22semidirect
68. #1865C22semidirect
69. #1863C22semidirect
80. #1805C22semidirect
553. C23#479semidirect

Maximal subgroups

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

subgroupindex
#350202
C2×#23602
#331502
#329602
#299732
#299572
#264182
#264162
#263462
#263442
#219302
#217592
C2×#18532
C2×#18522