Finite Group Databasebeta

← #6660695

Subgroups — #6660695

Order 512.

2020 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C211yesmaximal, characteristic
2C2×#2857C211yesmaximal, characteristic
3#32724C214yesmaximal, complemented
4#30242C212yesmaximal, complemented
5#30187C212yesmaximal, complemented
6#29953C212yesmaximal, complemented
7#29923C212yesmaximal, complemented
8#29715C212yesmaximal, complemented
9#29664C212yesmaximal, complemented
10#27681C212yesmaximal, complemented
11#24752C214yesmaximal, complemented
12#24380C214yesmaximal
13#24212C212yesmaximal, complemented
14C2×#1797C211yesmaximal, characteristic
15C2×#533C2211yescharacteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C23×C42C2211yescharacteristic
19C2×#467C2218yes—
20C2×#462C2214yescomplemented
21C2×#451C2214yes—
22C22×C4⋊C8C2212yes—
23C22×C22⋊C8C2212yes—
24#2857—21no—
25#2857—21no—
26#2857—21no—
27#2857—21no—
28#2857C2211yescharacteristic
29#2857C2211yescharacteristic
30#2857C2212yes—
31#2857C2212yes—
32#2857C2212yes—
33#2857C2212yes—
34#2857C2211yescharacteristic
35#2857C2211yescharacteristic
36#2855—21no—
37#2855C2212yes—
38#2851—21no—
39#2851—21no—
40#2848—21no—
41#2848C2212yes—
42C4×C22⋊Q8—21no—
43C4×C22⋊Q8—21no—
44C4×C22⋊Q8C2212yes—
45C4×C22⋊Q8C2212yes—
46#2296—21no—
47C2×#391C2212yes—
48C2×#389—21no—
49C2×#389—21no—
50#2285C2211yescharacteristic
51C2×#384C2211yescharacteristic
52#2280—21no—
53C2×#380—21no—
54C2×#380—21no—
55#2100C2214yescomplemented
56#2095C2214yescomplemented
57#2095C2214yescomplemented
58#2072C2214yescomplemented
59#2068C2218yescomplemented
60#2056C2218yescomplemented
61#2043C2214yescomplemented
62#2036C2218yes—
63#2024C2214yescomplemented
64#2016C2214yescomplemented
65#2016C2214yescomplemented
66#2001C2218yes—
67#1981C2214yescomplemented
68#1947C2214yescomplemented
69#1892C2218yes—
70#1872C2214yes—
71#1852C2214yes—
72#1803C2214yescomplemented
73#1797C2212yes—
74#1797C2212yes—
75#1797C2212yes—
76#1797C2212yes—
77#1787C2214yes—
78C2×#336C2212yes—
79C23×Q8C2312yes—
80C24×C4D411yescharacteristic
81C24×C4D411yescharacteristic
82C24×C4C2311yescharacteristic
83C22×Q16C2318yes—
84C23×C8C2314yes—
85#533—44no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8—21no—
88C2×C4⋊Q8—44no—
89C2×C4⋊Q8C2312yes—
90C2×C4⋊Q8C2312yes—
91C2×C4⋊Q8C2311yescharacteristic
92C2×C4⋊Q8C2311yescharacteristic
93C2×#125—21no—
94C2×C22⋊Q8—21no—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8—22no—
98C2×C22⋊Q8—22no—
99C2×C22⋊Q8—48no—
100C2×C22⋊Q8C2312yes—
101C2×C22⋊Q8C2312yes—
102C2×C22⋊Q8C2314yes—
103C2×C22⋊Q8C2312yes—
104C2×C22⋊Q8C2314yes—
105C2×C4×Q8—21no—
106C2×C4×Q8C2312yes—
107C2×#117—21no—
108C2×#117—21no—
109C22×C4⋊C4—22no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4C2312yes—
115C22×C22⋊C4—22no—
116C22×C42—44no—
117C22×C42D411yescharacteristic
118C22×C42D411yescharacteristic
119C22×C42D411yescharacteristic
120C22×C42D411yescharacteristic
121C22×C42D411yescharacteristic
122C22×C42D411yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129#467—416no—
130#462—48no—
131#451—48no—
132C2×#107C2318yes—
133C2×C4⋊C8—44no—
134C2×C4⋊C8C2314yes—
135C2×C4⋊C8C2314yes—
136C2×#103C2318yes—
137C2×#103C2318yes—
138C2×#103C2318yes—
139C2×#103C2318yes—
140C2×C22⋊C8—44no—
141C2×C22⋊C8C2314yes—
142C2×C22⋊C8C2314yes—
143#391—21no—
144#391—21no—
145#391—22no—
146#391—22no—
147#391C2312yes—
148#391C2314yes—
149#391C2314yes—
150#391C2312yes—
151#389—21no—
152#389—21no—
153#389—21no—
154#389—21no—
155#389—21no—
156#389—21no—
157#389—22no—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389C2312yes—
166#389C2312yes—
167#387—22no—
168#387—22no—
169#387—22no—
170#387—22no—
171#385—22no—
172#385—22no—
173#385—22no—
174#385—22no—
175#385—22no—
176#385—22no—
177#384—21no—
178#384—21no—
179#384C2312yes—
180#384C2311yescharacteristic
181#384C2311yescharacteristic
182#384C2312yes—
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#380—22no—
193#380—22no—
194#380—22no—
195#380—22no—
196#380—22no—
197#380—22no—
198#380—22no—
199#380—22no—
200#380C2312yes—
201#380C2312yes—
202#380C2312yes—
203#379—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4C2312yes—
208C4×C22⋊C4—22no—
209C4×C22⋊C4—22no—
210C2×#95—22no—
211C2×#95—22no—
212#340C2318yes—
213#340C2318yes—
214#336C2314yes—
215#336C2314yes—
216C25C2×D411yescharacteristic
217C22×Q8—21no—
218C22×Q8—22no—
219C22×Q8—24no—
220C22×Q8—24no—
221C22×Q8—44no—
222C22×Q8—44no—
223C22×Q8C2412yes—
224C22×Q8C2414yes—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—42no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4—42no—
235C23×C4—42no—
236C23×C4Q1618yescomplemented
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2411yescharacteristic
253C23×C4C2411yescharacteristic
254C23×C4C2411yescharacteristic
255C2×Q16—48no—
256C2×Q16—48no—
257C2×Q16—48no—
258C22×C8—28no—
259C22×C8—44no—
260C22×C8—44no—
261C22×C8C2418yes—
262C4⋊Q8—28no—
263C4⋊Q8—42no—
264C4⋊Q8—44no—
265C4⋊Q8—42no—
266C4⋊Q8—44no—
267C4⋊Q8—44no—
268#125—44no—
269C22⋊Q8—216no—
270C22⋊Q8—24no—
271C22⋊Q8—28no—
272C22⋊Q8—24no—
273C22⋊Q8—48no—
274C22⋊Q8—44no—
275C22⋊Q8—44no—
276C22⋊Q8—48no—
277C22⋊Q8—48no—
278C22⋊Q8—44no—
279C22⋊Q8—44no—
280C4×Q8—28no—
281C4×Q8—44no—
282#117—44no—
283#117—44no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—48no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4C2414yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—48no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—42no—
333C2×C42—42no—
334C2×C42—42no—
335C2×C42—42no—
336C2×C42—42no—
337C2×C42—42no—
338C2×C42—42no—
339C2×C42—42no—
340C2×C42—42no—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357#107—48no—
358C4⋊C8—28no—
359C4⋊C8—44no—
360#103—48no—
361#103—48no—
362#103—48no—
363#103—48no—
364C22⋊C8—28no—
365C22⋊C8—44no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380C24—22no—
381C24—22no—
382C24—22no—
383C24—22no—
384C24—41no—
385C24—41no—
386C24C2×Q1614yes—
387C24C22×D411yescharacteristic
388C24C22×D411yescharacteristic
389C24C22×D411yescharacteristic
390C2×Q8—24no—
391C2×Q8—24no—
392C2×Q8—28no—
393C2×Q8—28no—
394C2×Q8—24no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—24no—
398C2×Q8—42no—
399C2×Q8—44no—
400C2×Q8—42no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—42no—
404C2×Q8—88no—
405C2×Q8—88no—
406C22×C4—22no—
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—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—416no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—416no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4C2×Q1614yes—
470C22×C4C2×Q1614yes—
471C22×C4C2×Q1614yes—
472C22×C4C2×Q1614yes—
473C22×C4C2×Q1614yes—
474C22×C4C2×Q1614yes—
475C22×C4#13714yes—
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4H(5,𝔽2)14yes—
488C22×C4H(5,𝔽2)12yes—
489C22×C42−1+412yes—
490C22×C4C2511yescharacteristic
491Q16—816no—
492C2×C8—28no—
493C2×C8—28no—
494C2×C8—28no—
495C2×C8—88no—
496C4⋊C4—22no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—28no—
500C4⋊C4—22no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—48no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—48no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—42no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C22⋊C4—48no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—48no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—41no—
543C42—41no—
544C42—41no—
545C42—41no—
546C42—41no—
547C42—41no—
548C42—44no—
549C42—44no—
550C42H(5,𝔽2)14yes—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—28no—
563C23—28no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23#53512yes—
573C23#53712yes—
574C23C22×Q1612yes—
575C23C22×Q1612yes—
576C23C22×Q1612yes—
577C23C2×#13712yes—
578C23#57512yes—
579C23C23×D411yescharacteristic
580Q8—44no—
581Q8—48no—
582Q8—44no—
583Q8—48no—
584Q8—84no—
585Q8—88no—
586C2×C4—28no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—28no—
604C2×C4—28no—
605C2×C4—28no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—88no—
648C2×C4—88no—
649C2×C4#53412yes—
650C2×C4#53412yes—
651C2×C4#53512yes—
652C2×C4#53712yes—
653C2×C4C22×Q1612yes—
654C2×C4C22×Q1612yes—
655C2×C4C22×Q1612yes—
656C2×C4C22×Q1612yes—
657C2×C4C2×#13712yes—
658C2×C4C2×#13712yes—
659C2×C4#57512yes—
660C2×C4C2×H(5,𝔽2)12yes—
661C2×C4C2×H(5,𝔽2)12yes—
662C2×C4C2×H(5,𝔽2)11yescharacteristic
663C2×C4C2×2−1+411yescharacteristic
664C8—48no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—22no—
678C22—22no—
679C22—22no—
680C22—416no—
681C22—41no—
682C22—41no—
683C22—41no—
684C22—41no—
685C22#320914yes—
686C22#324914yes—
687C22C2×#53411yescharacteristic
688C22C2×#53511yescharacteristic
689C22C2×#53711yescharacteristic
690C22C23×Q1611yescharacteristic
691C22C22×#13711yescharacteristic
692C22C2×#57511yescharacteristic
693C22C2×#57511yescharacteristic
694C4—22no—
695C4—22no—
696C4—22no—
697C4—22no—
698C4—22no—
699C4—22no—
700C4—22no—
701C4—22no—
702C4—41no—
703C4—41no—
704C4—84no—
705C4—82no—
706C4—82no—
707C4—82no—
708C4—84no—
709C4—82no—
710C4#318914yes—
711C4#353612yes—
712C4#353712yes—
713C2—24no—
714C2—24no—
715C2—42no—
716C2#3743911yescharacteristic
717C2C2×#318911yescharacteristic
718C2#6075111yescharacteristic
719C2C2×#320911yescharacteristic
720C2#6079011yescharacteristic
721C2C2×#324911yescharacteristic
722C2#6091811yescharacteristic
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
3. #32724C2semidirect
4. #30242C2semidirect
5. #30187C2semidirect
6. #29953C2semidirect
7. #29923C2semidirect
8. #29715C2semidirect
9. #29664C2semidirect
10. #27681C2semidirect
11. #24752C2semidirect
13. #24212C2semidirect
20. C2×#462C22semidirect
55. #2100C22semidirect
56. #2095C22semidirect
57. #2095C22semidirect
58. #2072C22semidirect
59. #2068C22semidirect
60. #2056C22semidirect
61. #2043C22semidirect
63. #2024C22semidirect
64. #2016C22semidirect
65. #2016C22semidirect
67. #1981C22semidirect
68. #1947C22semidirect
72. #1803C22semidirect
236. C23×C4Q16semidirect

Maximal subgroups

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

subgroupindex
#366252
C2×#28572
#327242
#302422
#301872
#299532
#299232
#297152
#296642
#276812
#247522
#243802
#242122
C2×#17972