Finite Group Databasebeta

← #6435649

Subgroups — #6435649

Order 512.

1914 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2#34012C211yesmaximal, characteristic
3#29942C214yesmaximal, complemented
4#29941C214yesmaximal, complemented
5#29939C212yesmaximal, complemented
6#29938C212yesmaximal, complemented
7#29937C212yesmaximal, complemented
8#29936C212yesmaximal, complemented
9#23747C212yesmaximal, complemented
10#23657C212yesmaximal, complemented
11#23642C214yesmaximal
12#21313C212yesmaximal, complemented
13#21171C212yesmaximal, complemented
14#20626C211yesmaximal, characteristic
15C22×C4⋊Q8C2211yescharacteristic
16C22×C22⋊Q8—21no—
17C23×C42C2211yescharacteristic
18C2×#428C2214yes—
19C2×#423C2212yes—
20C2×#407C2212yes—
21#2857—21no—
22#2857—21no—
23#2857—21no—
24#2857—21no—
25#2857C2212yes—
26#2857C2212yes—
27#2857C2212yes—
28#2857C2212yes—
29C2×#391—21no—
30C2×#389—21no—
31C2×#389C2212yes—
32#2285C2211yescharacteristic
33#2285C2211yescharacteristic
34#2285C2211yescharacteristic
35#2285C2211yescharacteristic
36#2285C2211yescharacteristic
37#2285C2211yescharacteristic
38C2×#384C2211yescharacteristic
39#2283—21no—
40#2283—21no—
41#2283—21no—
42#2283—21no—
43#2283C2212yes—
44#2283C2212yes—
45#2280—21no—
46#2280—21no—
47#2280C2212yes—
48C2×#380—21no—
49C2×#380C2212yes—
50C4×#117—21no—
51C4×#117—21no—
52C4×#117C2212yes—
53#2101C2218yescomplemented
54#2100C2214yescomplemented
55#2099C2214yescomplemented
56#2085C2218yescomplemented
57#2084C2214yescomplemented
58#2083C2214yescomplemented
59#2077C2218yes—
60#2077C2218yes—
61#2076C2218yes—
62#2075C2218yes—
63#1988C2214yescomplemented
64#1988C2214yescomplemented
65#1987C2214yescomplemented
66#1987C2214yescomplemented
67#1948C2214yescomplemented
68#1948C2214yescomplemented
69#1947C2214yescomplemented
70#1947C2214yescomplemented
71#1797C2212yes—
72#1797C2212yes—
73#1797C2212yes—
74#1797C2212yes—
75#1736C2214yes—
76#1736C2214yes—
77#1736C2214yes—
78#1728C2212yes—
79C23×Q8—21no—
80C24×C4D411yescharacteristic
81C24×C4D411yescharacteristic
82C24×C4C2311yescharacteristic
83C22×M16C2314yes—
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8—28no—
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2312yes—
89C2×C22⋊Q8—22no—
90C2×C22⋊Q8—22no—
91C2×C22⋊Q8—22no—
92C2×C22⋊Q8—22no—
93C2×C22⋊Q8—48no—
94C2×#117—21no—
95C2×#117—21no—
96C2×#117—21no—
97C2×#117—21no—
98C2×#117C2312yes—
99C2×#117C2312yes—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—22no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C22⋊C4—22no—
107C22×C42—28no—
108C22×C42D411yescharacteristic
109C22×C42D411yescharacteristic
110C22×C42D411yescharacteristic
111C22×C42D411yescharacteristic
112C22×C42D411yescharacteristic
113C22×C42D411yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120#428—48no—
121C2×#107C2318yes—
122C2×#106C2318yes—
123#423—28no—
124C2×C4⋊C8C2314yes—
125C2×C4⋊C8C2314yes—
126#407—44no—
127C2×C22⋊C8C2314yes—
128C2×C22⋊C8C2314yes—
129#391—22no—
130#391—22no—
131#391—22no—
132#391—22no—
133#389—22no—
134#389—22no—
135#389—22no—
136#389—22no—
137#389—22no—
138#389—22no—
139#389C2314yes—
140#389C2314yes—
141#384—21no—
142#384—21no—
143#384C2311yescharacteristic
144#384C2311yescharacteristic
145#384C2312yes—
146#384C2312yes—
147#384C2311yescharacteristic
148#384C2311yescharacteristic
149#383—22no—
150#383—22no—
151#382—21no—
152#382—21no—
153#382—21no—
154#382—21no—
155#382C2312yes—
156#382C2312yes—
157#381—22no—
158#381—22no—
159#381—22no—
160#380—22no—
161#380—22no—
162#380—22no—
163#380—22no—
164#380—22no—
165#380—22no—
166#380—21no—
167#380—21no—
168#380—21no—
169#380—21no—
170#380—21no—
171#380—21no—
172#380—21no—
173#380—21no—
174#380C2312yes—
175#380C2314yes—
176#380C2312yes—
177#380C2314yes—
178#380C2312yes—
179#380C2312yes—
180#379—22no—
181C4×C4⋊C4—21no—
182C4×C4⋊C4—21no—
183C4×C4⋊C4—21no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4—21no—
186C4×C4⋊C4—21no—
187C4×C4⋊C4—21no—
188C4×C4⋊C4—21no—
189C4×C4⋊C4C2312yes—
190C4×C4⋊C4C2312yes—
191C4×C4⋊C4C2312yes—
192C4×C4⋊C4C2312yes—
193C4×C22⋊C4—22no—
194C4×C22⋊C4—22no—
195C4×C22⋊C4—22no—
196C4×C22⋊C4—22no—
197C4×C22⋊C4—22no—
198C4×C22⋊C4—22no—
199#376—22no—
200#376—22no—
201#376—22no—
202C2×#95—22no—
203C2×#95—22no—
204C43—22no—
205#340C2318yes—
206#340C2318yes—
207#340C2318yes—
208#340C2318yes—
209#340C2318yes—
210#340C2318yes—
211#336C2314yes—
212#336C2314yes—
213C25C2×D411yescharacteristic
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—44no—
217C22×Q8—44no—
218C22×Q8—44no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—22no—
225C23×C4—24no—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
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×C4C2411yescharacteristic
247C23×C4C2411yescharacteristic
248C23×C4C2411yescharacteristic
249C2×M16—28no—
250C2×M16—28no—
251C2×M16—28no—
252C22×C8C2418yes—
253C4⋊Q8—28no—
254C4⋊Q8—28no—
255C4⋊Q8—44no—
256C4⋊Q8—44no—
257C22⋊Q8—48no—
258C22⋊Q8—48no—
259C22⋊Q8—48no—
260C22⋊Q8—48no—
261#117—24no—
262#117—28no—
263#117—24no—
264#117—44no—
265#117—44no—
266#117—44no—
267#117—44no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
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—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—48no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4C2414yes—
294C2×C4⋊C4C2414yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—48no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
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—22no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42C2×D411yescharacteristic
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×C42#4118yes—
338C2×C42C2411yescharacteristic
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342#107—48no—
343#106—48no—
344C4⋊C8—28no—
345C4⋊C8—28no—
346C22⋊C8—44no—
347C22⋊C8—44no—
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—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—41no—
370C24—41no—
371C24C22×D411yescharacteristic
372C24C22×D411yescharacteristic
373C24C22×D411yescharacteristic
374C2×Q8—44no—
375C2×Q8—44no—
376C2×Q8—44no—
377C2×Q8—44no—
378C2×Q8—44no—
379C2×Q8—44no—
380C2×Q8—816no—
381C22×C4—22no—
382C22×C4—22no—
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—22no—
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—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—416no—
437C22×C4—416no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4#12314yes—
447C22×C4#12314yes—
448C22×C4AGL(1,ℤ/8)14yes—
449C22×C4#13714yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C2×#4114yes—
462C22×C4H(5,𝔽2)12yes—
463C22×C42−1+414yes—
464C22×C42−1+412yes—
465C22×C4C2511yescharacteristic
466M16—416no—
467C2×C8—28no—
468C2×C8—28no—
469C2×C8—28no—
470C4⋊C4—28no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—48no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—48no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C22⋊C4—48no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—48no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—44no—
518C42—44no—
519C42—44no—
520C42—44no—
521C42—44no—
522C42—44no—
523C42#12314yes—
524C42#12314yes—
525C42C2×#4114yes—
526C42C2×#4114yes—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—28no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23C2×#12312yes—
551C23#53612yes—
552C23#53712yes—
553C23C2×AGL(1,ℤ/8)12yes—
554C23C2×#13712yes—
555C23C23×D411yescharacteristic
556Q8—88no—
557Q8—88no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—88no—
616C2×C4—88no—
617C2×C4C2×#12312yes—
618C2×C4C2×#12312yes—
619C2×C4C2×#12312yes—
620C2×C4C2×#12312yes—
621C2×C4C2×#12312yes—
622C2×C4#53612yes—
623C2×C4#53712yes—
624C2×C4#54118yes—
625C2×C4#54218yes—
626C2×C4C2×AGL(1,ℤ/8)12yes—
627C2×C4C2×AGL(1,ℤ/8)12yes—
628C2×C4C2×#13712yes—
629C2×C4C2×#13712yes—
630C2×C4#57512yes—
631C2×C4#57512yes—
632C2×C4C22×#4112yes—
633C2×C4C2×H(5,𝔽2)11yescharacteristic
634C2×C4C2×2−1+411yescharacteristic
635C2×C4C2×2−1+412yes—
636C8—48no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—416no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—41no—
654C22—41no—
655C22—41no—
656C22—41no—
657C22—41no—
658C22#262514yes—
659C22C22×#12311yescharacteristic
660C22C2×#53611yescharacteristic
661C22C2×#53711yescharacteristic
662C22C22×AGL(1,ℤ/8)11yescharacteristic
663C22C22×#13711yescharacteristic
664C22C2×#57511yescharacteristic
665C22C2×#57511yescharacteristic
666C4—22no—
667C4—22no—
668C4—22no—
669C4—22no—
670C4—41no—
671C4—41no—
672C4—41no—
673C4—41no—
674C4—84no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4—84no—
679C4—82no—
680C4#262514yes—
681C4#309414yes—
682C4#309514yes—
683C4C2×#54112yes—
684C4C2×#54212yes—
685C2—24no—
686C2—42no—
687C2—42no—
688C2C2×#262511yescharacteristic
689C2C2×#309411yescharacteristic
690C2C2×#309511yescharacteristic
691C2#6086011yescharacteristic
692C2#6086311yescharacteristic
693C2#6091611yescharacteristic
694C2#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. #29942C2semidirect
4. #29941C2semidirect
5. #29939C2semidirect
6. #29938C2semidirect
7. #29937C2semidirect
8. #29936C2semidirect
9. #23747C2semidirect
10. #23657C2semidirect
12. #21313C2semidirect
13. #21171C2semidirect
53. #2101C22semidirect
54. #2100C22semidirect
55. #2099C22semidirect
56. #2085C22semidirect
57. #2084C22semidirect
58. #2083C22semidirect
63. #1988C22semidirect
64. #1988C22semidirect
65. #1987C22semidirect
66. #1987C22semidirect
67. #1948C22semidirect
68. #1948C22semidirect
69. #1947C22semidirect
70. #1947C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
#340122
#299422
#299412
#299392
#299382
#299372
#299362
#237472
#236572
#236422
#213132
#211712
#206262