Finite Group Databasebeta

← #6434099

Subgroups — #6434099

Order 512.

1972 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
1C2×#2855C211yesmaximal, characteristic
2C2×#2285C211yesmaximal, characteristic
3#29915C214yesmaximal, complemented
4#29913C214yesmaximal, complemented
5#29911C214yesmaximal, complemented
6#29909C212yesmaximal, complemented
7#29908C212yesmaximal, complemented
8#23747C212yesmaximal, complemented
9#23732C212yesmaximal, complemented
10#23657C212yesmaximal, complemented
11#23629C212yesmaximal, complemented
12#20911C212yesmaximal
13#20846C212yesmaximal
14#20682C211yesmaximal, characteristic
15C22×C22⋊Q8C2211yescharacteristic
16C22×#117C2212yes—
17C23×C42C2211yescharacteristic
18C2×#428C2214yes—
19C2×#419C2214yes—
20C2×#407C2212yes—
21#2855—21no—
22#2855—21no—
23#2855—21no—
24#2855—21no—
25#2855C2212yes—
26#2855C2212yes—
27#2855C2212yes—
28#2855C2212yes—
29C2×#391C2211yescharacteristic
30C2×#391C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#387—21no—
34C2×#387—21no—
35C2×#385—21no—
36#2285—21no—
37#2285—21no—
38#2285—21no—
39#2285—21no—
40#2285C2212yes—
41#2285C2212yes—
42#2285C2212yes—
43#2285C2212yes—
44C2×#384C2211yescharacteristic
45C2×#384C2211yescharacteristic
46C2×#382C2212yes—
47C2×#381—21no—
48C2×#380—21no—
49C2×#380—21no—
50C2×#380C2211yescharacteristic
51C2×C4×C4⋊C4—21no—
52C2×C4×C4⋊C4—21no—
53#2101C2218yescomplemented
54#2100C2214yescomplemented
55#2099C2214yescomplemented
56#2097C2218yescomplemented
57#2096C2214yescomplemented
58#2095C2214yescomplemented
59#2085C2218yescomplemented
60#2084C2214yescomplemented
61#2083C2214yescomplemented
62#2074C2218yescomplemented
63#2072C2214yescomplemented
64#2072C2214yescomplemented
65#1873C2218yes—
66#1872C2214yes—
67#1872C2214yes—
68#1854C2218yes—
69#1853C2214yes—
70#1852C2214yes—
71#1817C2212yes—
72#1817C2212yes—
73#1817C2212yes—
74#1817C2212yes—
75#1736C2214yes—
76#1736C2214yes—
77#1728C2212yes—
78#1728C2212yes—
79C23×Q8C2311yescharacteristic
80C24×C4D411yescharacteristic
81C24×C4D411yescharacteristic
82C24×C4C2311yescharacteristic
83C22×M16C2314yes—
84C2×C22⋊Q8—21no—
85C2×C22⋊Q8—21no—
86C2×C22⋊Q8—28no—
87C2×C22⋊Q8C2312yes—
88C2×C22⋊Q8C2312yes—
89C2×#117—22no—
90C2×#117—22no—
91C2×#117—48no—
92C2×#117C2314yes—
93C2×#117C2314yes—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C22⋊C4—22no—
107C22×C22⋊C4—21no—
108C22×C22⋊C4—21no—
109C22×C42—28no—
110C22×C42—22no—
111C22×C42D411yescharacteristic
112C22×C42D411yescharacteristic
113C22×C42D411yescharacteristic
114C22×C42D411yescharacteristic
115C22×C42D411yescharacteristic
116C22×C42D411yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123#428—48no—
124C2×#107C2318yes—
125C2×#106C2318yes—
126#419—48no—
127C2×#103C2318yes—
128C2×#103C2318yes—
129#407—28no—
130C2×C22⋊C8C2314yes—
131C2×C22⋊C8C2314yes—
132#391—21no—
133#391—21no—
134#391—21no—
135#391—21no—
136#391C2312yes—
137#391C2312yes—
138#391C2312yes—
139#391C2312yes—
140#389—21no—
141#389—21no—
142#389—21no—
143#389—21no—
144#389C2312yes—
145#389C2312yes—
146#389C2312yes—
147#389C2312yes—
148#387—22no—
149#387—22no—
150#387—22no—
151#387—22no—
152#387—22no—
153#387—22no—
154#387—22no—
155#387—22no—
156#385—22no—
157#385—22no—
158#385—22no—
159#385—22no—
160#384—21no—
161#384—21no—
162#384—21no—
163#384—21no—
164#384C2312yes—
165#384C2312yes—
166#384C2312yes—
167#384C2312yes—
168#382—22no—
169#382—22no—
170#382C2314yes—
171#382C2314yes—
172#381—22no—
173#381—22no—
174#381—22no—
175#381—22no—
176#380—21no—
177#380—21no—
178#380—22no—
179#380—22no—
180#380—22no—
181#380—22no—
182#380—22no—
183#380—22no—
184#380—22no—
185#380—22no—
186#380C2312yes—
187#380C2312yes—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C2×#95—22no—
197C2×#95—22no—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95—21no—
203C2×#95—21no—
204C2×#95—21no—
205#340C2318yes—
206#340C2318yes—
207#340C2318yes—
208#340C2318yes—
209#336C2314yes—
210#336C2314yes—
211#336C2314yes—
212#336C2314yes—
213C25C2×D411yescharacteristic
214C22×Q8—21no—
215C22×Q8—24no—
216C22×Q8—24no—
217C22×Q8—24no—
218C22×Q8C2412yes—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—21no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—42no—
231C23×C4—42no—
232C23×C4—42no—
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
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×C4#4118yes—
249C23×C4C2411yescharacteristic
250C23×C4C2411yescharacteristic
251C23×C4C2411yescharacteristic
252C2×M16—28no—
253C2×M16—28no—
254C2×M16—28no—
255C22×C8C2418yes—
256C22⋊Q8—28no—
257C22⋊Q8—28no—
258C22⋊Q8—44no—
259C22⋊Q8—44no—
260#117—28no—
261#117—28no—
262#117—44no—
263#117—44no—
264#117—48no—
265#117—48no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4C2414yes—
298C2×C4⋊C4C2414yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—48no—
310C2×C22⋊C4—44no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—48no—
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×C42C2411yescharacteristic
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342#107—48no—
343#106—48no—
344#103—48no—
345#103—48no—
346C22⋊C8—28no—
347C22⋊C8—28no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—41no—
371C24—41no—
372C24#12314yes—
373C24C22×D411yescharacteristic
374C24C22×D411yescharacteristic
375C24C22×D411yescharacteristic
376C2×Q8—24no—
377C2×Q8—24no—
378C2×Q8—24no—
379C2×Q8—42no—
380C2×Q8—42no—
381C2×Q8—42no—
382C2×Q8—416no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—232no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
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—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
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—416no—
436C22×C4—416no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4#12314yes—
456C22×C4#12314yes—
457C22×C4#12314yes—
458C22×C4#13714yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C2×#4114yes—
471C22×C4C2×#4114yes—
472C22×C4C2×#4114yes—
473C22×C4H(5,𝔽2)14yes—
474C22×C42−1+414yes—
475C22×C4C2511yescharacteristic
476M16—416no—
477C2×C8—28no—
478C2×C8—28no—
479C2×C8—28no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—48no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—48no—
511C22⋊C4—44no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—48no—
529C42—48no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—28no—
539C23—28no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23C2×#12312yes—
552C23C2×#12312yes—
553C23C2×#12312yes—
554C23#53512yes—
555C23#53612yes—
556C23C2×#13712yes—
557C23#57512yes—
558C23C23×D411yescharacteristic
559Q8—48no—
560Q8—84no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—28no—
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—28no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
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—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—84no—
621C2×C4—88no—
622C2×C4—88no—
623C2×C4—84no—
624C2×C4—84no—
625C2×C4—84no—
626C2×C4C2×#12312yes—
627C2×C4C2×#12312yes—
628C2×C4C2×#12312yes—
629C2×C4#53512yes—
630C2×C4#53612yes—
631C2×C4#539116yes—
632C2×C4C2×#13712yes—
633C2×C4C2×#13712yes—
634C2×C4#57512yes—
635C2×C4#57512yes—
636C2×C4C22×#4112yes—
637C2×C4C2×H(5,𝔽2)12yes—
638C2×C4C2×2−1+412yes—
639C8—48no—
640C22—22no—
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—416no—
656C22—41no—
657C22—41no—
658C22—41no—
659C22—41no—
660C22#262314yes—
661C22#309514yes—
662C22C22×#12311yescharacteristic
663C22C2×#53511yescharacteristic
664C22C2×#53611yescharacteristic
665C22C22×#13711yescharacteristic
666C22C2×#57511yescharacteristic
667C22C2×#57511yescharacteristic
668C22C2×#57511yescharacteristic
669C4—22no—
670C4—22no—
671C4—22no—
672C4—22no—
673C4—22no—
674C4—22no—
675C4—41no—
676C4—41no—
677C4—41no—
678C4—41no—
679C4—82no—
680C4—82no—
681C4—84no—
682C4—84no—
683C4—82no—
684C4—82no—
685C4#262314yes—
686C4C2×#53914yes—
687C2—24no—
688C2—24no—
689C2—42no—
690C2C2×#262311yescharacteristic
691C2C2×#309511yescharacteristic
692C2#6051111yescharacteristic
693C2#6080411yescharacteristic
694C2#6080511yescharacteristic
695C2#6086511yescharacteristic
696C2#6086611yescharacteristic
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. #29915C2semidirect
4. #29913C2semidirect
5. #29911C2semidirect
6. #29909C2semidirect
7. #29908C2semidirect
8. #23747C2semidirect
9. #23732C2semidirect
10. #23657C2semidirect
11. #23629C2semidirect
53. #2101C22semidirect
54. #2100C22semidirect
55. #2099C22semidirect
56. #2097C22semidirect
57. #2096C22semidirect
58. #2095C22semidirect
59. #2085C22semidirect
60. #2084C22semidirect
61. #2083C22semidirect
62. #2074C22semidirect
63. #2072C22semidirect
64. #2072C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28552
C2×#22852
#299152
#299132
#299112
#299092
#299082
#237472
#237322
#236572
#236292
#209112
#208462
#206822