Finite Group Databasebeta

← #6475835

Subgroups — #6475835

Order 512.

2578 conjugacy classes of subgroups.

The subgroup lattice has 743 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C211yesmaximal, characteristic
2C22×#391C211yesmaximal, characteristic
3#32982C212yesmaximal
4#32882C212yesmaximal
5#29951C212yesmaximal, complemented
6#29715C212yesmaximal, complemented
7#29111C212yesmaximal, complemented
8#28313C212yesmaximal, complemented
9#28124C214yesmaximal, complemented
10#27583C212yesmaximal, complemented
11#24508C214yesmaximal, complemented
12#23934C212yesmaximal, complemented
13#22050C214yesmaximal, complemented
14#20715C211yesmaximal, characteristic
15C24×Q8C2211yescharacteristic
16C2×#533C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C42C2211yescharacteristic
19C2×#483C2214yes—
20C2×#451C2214yes—
21C2×#451C2214yes—
22C2×#451C2214yes—
23C2×#432C2212yes—
24#2857C2211yescharacteristic
25#2857C2211yescharacteristic
26#2857C2211yescharacteristic
27#2857C2211yescharacteristic
28#2855—21no—
29#2855C2212yes—
30#2851—21no—
31#2851—21no—
32#2848—21no—
33#2848C2212yes—
34C4×C22⋊Q8—21no—
35C4×C22⋊Q8—21no—
36C4×C22⋊Q8C2212yes—
37C4×C22⋊Q8C2212yes—
38#2296—21no—
39C2×#391—21no—
40C2×#391—21no—
41C2×#391—21no—
42C2×#391—21no—
43C2×#391—21no—
44C2×#391—21no—
45C2×#391C2212yes—
46C2×#391C2212yes—
47C2×#391C2212yes—
48C2×#391C2212yes—
49C2×#391C2212yes—
50C2×#391C2212yes—
51#2285C2211yescharacteristic
52#2280—21no—
53C22×#95—21no—
54C22×#95—21no—
55#2096C2214yescomplemented
56#2072C2214yescomplemented
57#2066C2218yescomplemented
58#2056C2218yescomplemented
59#2024C2214yescomplemented
60#2016C2214yescomplemented
61#1971C2218yescomplemented
62#1935C2218yescomplemented
63#1927C2218yescomplemented
64#1878C2218yescomplemented
65#1872C2214yes—
66#1872C2214yes—
67#1853C2214yes—
68#1853C2214yes—
69#1823C2214yes—
70#1817C2212yes—
71#1797C2212yes—
72#1797C2212yes—
73#1787C2214yes—
74#1787C2214yes—
75#1787C2214yes—
76#1779C2212yes—
77#1749C2212yes—
78#1749C2212yes—
79C23×Q8—28no—
80C23×Q8—21no—
81C23×Q8—21no—
82C23×Q8—21no—
83C23×Q8—44no—
84C23×Q8—44no—
85C23×Q8C2312yes—
86C23×Q8C2312yes—
87C23×Q8C2312yes—
88C24×C4—21no—
89C24×C4—21no—
90C24×C4D411yescharacteristic
91C24×C4D411yescharacteristic
92C24×C4C2311yescharacteristic
93C22×Q16C2318yes—
94#533—44no—
95C2×C4⋊Q8C2311yescharacteristic
96C2×C4⋊Q8C2311yescharacteristic
97C2×#125—21no—
98C2×C22⋊Q8—21no—
99C2×C22⋊Q8—21no—
100C2×C22⋊Q8—21no—
101C2×C22⋊Q8C2312yes—
102C2×C22⋊Q8C2312yes—
103C2×C22⋊Q8C2312yes—
104C2×C4×Q8—21no—
105C2×C4×Q8C2312yes—
106C2×#117—21no—
107C2×#117—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—44no—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C42—44no—
116C22×C42D411yescharacteristic
117C22×C42D411yescharacteristic
118C22×C42D411yescharacteristic
119C22×C42D411yescharacteristic
120C22×C42D411yescharacteristic
121C22×C42D411yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#483—48no—
129#451—216no—
130#451—48no—
131#451—48no—
132#432—44no—
133C2×#106C2318yes—
134C2×C4⋊C8C2314yes—
135C2×#103C2318yes—
136C2×#103C2318yes—
137C2×#103C2318yes—
138C2×#103C2318yes—
139C2×C22⋊C8C2314yes—
140C2×C22⋊C8C2314yes—
141C2×C22⋊C8C2314yes—
142C2×#97C2314yes—
143#391—21no—
144#391—21no—
145#391—22no—
146#391—22no—
147#391—22no—
148#391—22no—
149#391—22no—
150#391—22no—
151#391—22no—
152#391—22no—
153#391—22no—
154#391—22no—
155#391—22no—
156#391—22no—
157#391C2312yes—
158#391C2314yes—
159#391C2314yes—
160#391C2312yes—
161#391C2314yes—
162#391C2314yes—
163#389—21no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—21no—
169#389C2312yes—
170#389C2312yes—
171#387—22no—
172#387—22no—
173#387—22no—
174#387—22no—
175#385—22no—
176#385—22no—
177#385—22no—
178#385—22no—
179#385—22no—
180#385—22no—
181#384C2311yescharacteristic
182#384C2311yescharacteristic
183#382—21no—
184#381—22no—
185#380—21no—
186#380—21no—
187#380—21no—
188#380—21no—
189#380—21no—
190#380—21no—
191#380—21no—
192#380C2312yes—
193#380C2312yes—
194#380C2312yes—
195#379—22no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4C2312yes—
200C4×C22⋊C4—22no—
201C4×C22⋊C4—22no—
202C2×#95—22no—
203C2×#95—22no—
204C2×#95—22no—
205C2×#95—22no—
206C2×#95—22no—
207C2×#95—22no—
208C2×#95—22no—
209C2×#95—22no—
210C2×#95—22no—
211C2×#95—22no—
212C2×#95—22no—
213C2×#95—22no—
214#340C2318yes—
215#340C2318yes—
216#336C2314yes—
217#336C2314yes—
218C25C2×D411yescharacteristic
219C22×Q8—28no—
220C22×Q8—28no—
221C22×Q8—28no—
222C22×Q8—28no—
223C22×Q8—28no—
224C22×Q8—22no—
225C22×Q8—22no—
226C22×Q8—22no—
227C22×Q8—21no—
228C22×Q8—24no—
229C22×Q8—28no—
230C22×Q8—28no—
231C22×Q8—44no—
232C22×Q8—44no—
233C22×Q8—44no—
234C22×Q8—44no—
235C22×Q8—464no—
236C22×Q8—44no—
237C22×Q8—44no—
238C22×Q8—44no—
239C22×Q8—44no—
240C22×Q8—44no—
241C22×Q8—44no—
242C22×Q8#4118yes—
243C22×Q8C2412yes—
244C22×Q8C2414yes—
245C23×C4—24no—
246C23×C4—24no—
247C23×C4—24no—
248C23×C4—28no—
249C23×C4—24no—
250C23×C4—22no—
251C23×C4—22no—
252C23×C4—22no—
253C23×C4—24no—
254C23×C4—24no—
255C23×C4—22no—
256C23×C4—22no—
257C23×C4—22no—
258C23×C4—42no—
259C23×C4—42no—
260C23×C4—42no—
261C23×C4—44no—
262C23×C4—44no—
263C23×C4—44no—
264C23×C4—44no—
265C23×C4—44no—
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D411yescharacteristic
273C23×C4C2×D411yescharacteristic
274C23×C4C2×D411yescharacteristic
275C23×C4C2×D411yescharacteristic
276C23×C4C2×D411yescharacteristic
277C23×C4C2×D411yescharacteristic
278C23×C4C2×D411yescharacteristic
279C23×C4C2×D411yescharacteristic
280C23×C4C2×D411yescharacteristic
281C23×C4C2411yescharacteristic
282C23×C4C2411yescharacteristic
283C23×C4C2411yescharacteristic
284C2×Q16—216no—
285C2×Q16—48no—
286C2×Q16—48no—
287C22×C8C2418yes—
288C4⋊Q8—42no—
289C4⋊Q8—42no—
290#125—44no—
291C22⋊Q8—28no—
292C22⋊Q8—28no—
293C22⋊Q8—28no—
294C22⋊Q8—44no—
295C22⋊Q8—44no—
296C22⋊Q8—44no—
297C4×Q8—44no—
298C4×Q8—44no—
299#117—44no—
300#117—44no—
301C2×C4⋊C4—28no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4—44no—
323C2×C4⋊C4C2414yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
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×C42—24no—
334C2×C42—24no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—42no—
338C2×C42—42no—
339C2×C42—42no—
340C2×C42—42no—
341C2×C42—42no—
342C2×C42—42no—
343C2×C42—42no—
344C2×C42—42no—
345C2×C42—42no—
346C2×C42—42no—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361C2×C42C2411yescharacteristic
362C2×C42C2411yescharacteristic
363#106—48no—
364C4⋊C8—44no—
365#103—216no—
366#103—48no—
367#103—48no—
368#103—48no—
369C22⋊C8—28no—
370C22⋊C8—28no—
371C22⋊C8—28no—
372#97—44no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391C24—22no—
392C24—22no—
393C24—22no—
394C24—22no—
395C24—22no—
396C24—22no—
397C24#13714yes—
398C24C22×D411yescharacteristic
399C24C22×D411yescharacteristic
400C24C22×D411yescharacteristic
401C2×Q8—28no—
402C2×Q8—28no—
403C2×Q8—28no—
404C2×Q8—24no—
405C2×Q8—24no—
406C2×Q8—28no—
407C2×Q8—28no—
408C2×Q8—24no—
409C2×Q8—24no—
410C2×Q8—416no—
411C2×Q8—432no—
412C2×Q8—432no—
413C2×Q8—432no—
414C2×Q8—432no—
415C2×Q8—432no—
416C2×Q8—44no—
417C2×Q8—44no—
418C2×Q8—44no—
419C2×Q8—44no—
420C2×Q8—416no—
421C2×Q8—42no—
422C2×Q8—42no—
423C2×Q8—44no—
424C2×Q8—44no—
425C2×Q8—42no—
426C2×Q8C2×#4118yes—
427C22×C4—232no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—28no—
446C22×C4—28no—
447C22×C4—28no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—41no—
475C22×C4—41no—
476C22×C4—41no—
477C22×C4—41no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—41no—
482C22×C4—41no—
483C22×C4—41no—
484C22×C4—41no—
485C22×C4—41no—
486C22×C4—416no—
487C22×C4—416no—
488C22×C4—432no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4—44no—
500C22×C4—44no—
501C22×C4—44no—
502C22×C4—44no—
503C22×C4—44no—
504C22×C4—816no—
505C22×C4#13714yes—
506C22×C4#13714yes—
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D411yescharacteristic
517C22×C4C22×D411yescharacteristic
518C22×C4C2×#4114yes—
519C22×C4H(5,𝔽2)14yes—
520C22×C4H(5,𝔽2)14yes—
521C22×C4C2511yescharacteristic
522Q16—816no—
523C2×C8—28no—
524C2×C8—28no—
525C2×C8—28no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—24no—
529C4⋊C4—28no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—44no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—41no—
561C42—41no—
562C42—41no—
563C42—41no—
564C42—41no—
565C42—41no—
566C42—41no—
567C42—41no—
568C42—44no—
569C42—44no—
570C23—28no—
571C23—28no—
572C23—28no—
573C23—28no—
574C23—28no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—28no—
582C23—28no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—21no—
588C23—21no—
589C23#52512yes—
590C23#53412yes—
591C23#53512yes—
592C23C2×#13712yes—
593C23C2×#13712yes—
594C23C2×#13712yes—
595C23C2×#13712yes—
596C23C2×#13712yes—
597C23#57512yes—
598C23C23×D411yescharacteristic
599Q8—216no—
600Q8—416no—
601Q8—416no—
602Q8—416no—
603Q8—48no—
604Q8—48no—
605Q8—84no—
606C2×C4—28no—
607C2×C4—28no—
608C2×C4—28no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—28no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—28no—
625C2×C4—28no—
626C2×C4—28no—
627C2×C4—28no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—416no—
651C2×C4—416no—
652C2×C4—416no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—44no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—44no—
671C2×C4—88no—
672C2×C4—88no—
673C2×C4—88no—
674C2×C4#52512yes—
675C2×C4#53412yes—
676C2×C4#53512yes—
677C2×C4C2×#13712yes—
678C2×C4C2×#13712yes—
679C2×C4C2×#13712yes—
680C2×C4C2×#13712yes—
681C2×C4C22×#4112yes—
682C2×C4C2×H(5,𝔽2)12yes—
683C2×C4C2×H(5,𝔽2)12yes—
684C8—48no—
685C22—232no—
686C22—22no—
687C22—22no—
688C22—22no—
689C22—22no—
690C22—22no—
691C22—22no—
692C22—22no—
693C22—22no—
694C22—22no—
695C22—22no—
696C22—22no—
697C22—22no—
698C22—22no—
699C22—22no—
700C22—22no—
701C22—22no—
702C22—22no—
703C22—22no—
704C22#311314yes—
705C22#319514yes—
706C22#321514yes—
707C22C2×#52511yescharacteristic
708C22C2×#53411yescharacteristic
709C22C2×#53511yescharacteristic
710C22C22×#13711yescharacteristic
711C22C22×#13711yescharacteristic
712C22C22×#13711yescharacteristic
713C22C2×#57511yescharacteristic
714C4—22no—
715C4—22no—
716C4—22no—
717C4—22no—
718C4—22no—
719C4—22no—
720C4—22no—
721C4—41no—
722C4—41no—
723C4—41no—
724C4—41no—
725C4—48no—
726C4—82no—
727C4—84no—
728C4—82no—
729C4—82no—
730C4—82no—
731C4#349014yes—
732C2—24no—
733C2—24no—
734C2—24no—
735C2#3505011yescharacteristic
736C2C2×#311311yescharacteristic
737C2#6056011yescharacteristic
738C2C2×#319511yescharacteristic
739C2#6075111yescharacteristic
740C2C2×#321511yescharacteristic
741C2#6080011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #29951C2semidirect
6. #29715C2semidirect
7. #29111C2semidirect
8. #28313C2semidirect
9. #28124C2semidirect
10. #27583C2semidirect
11. #24508C2semidirect
12. #23934C2semidirect
13. #22050C2semidirect
55. #2096C22semidirect
56. #2072C22semidirect
57. #2066C22semidirect
58. #2056C22semidirect
59. #2024C22semidirect
60. #2016C22semidirect
61. #1971C22semidirect
62. #1935C22semidirect
63. #1927C22semidirect
64. #1878C22semidirect

Maximal subgroups

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

subgroupindex
#366252
C22×#3912
#329822
#328822
#299512
#297152
#291112
#283132
#281242
#275832
#245082
#239342
#220502
#207152