Finite Group Databasebeta

← #6469387

Subgroups — #6469387

Order 512.

1916 conjugacy classes of subgroups.

The subgroup lattice has 734 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#533C211yesmaximal, characteristic, complemented
2C4×#532C211yesmaximal, characteristic
3#29393C212yesmaximal, complemented
4#29389C212yesmaximal, complemented
5#29280C212yesmaximal, complemented
6#28207C212yesmaximal, complemented
7#27650C212yesmaximal, complemented
8#27615C212yesmaximal, complemented
9#24304C212yesmaximal, complemented
10#24062C212yesmaximal
11#23837C211yesmaximal, characteristic, complemented
12#23835C211yesmaximal, characteristic, complemented
13#23815C211yesmaximal, characteristic, complemented
14#23815C211yesmaximal, characteristic, complemented
15C4×#478C214yesmaximal, complemented
16C4×#482C214yesmaximal, complemented
17#20777C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#532C2211yescharacteristic
20C2×C4×#41—21no—
21C2×#482C2214yes—
22C2×#478C2214yes—
23C4×C4⋊Q8C2211yescharacteristic
24C4×C4⋊Q8C2211yescharacteristic
25C4×C4⋊Q8C2211yescharacteristic
26C4×C4⋊Q8C2211yescharacteristic
27C4×C4⋊Q8C2211yescharacteristic
28C4×C4⋊Q8C2211yescharacteristic
29C4×#127C2211yescharacteristic
30C4×#127C2211yescharacteristic
31C4×#125C2212yes—
32C4×#125C2212yes—
33C4×#124—21no—
34C4×#124—21no—
35C4×C22⋊Q8—21no—
36C4×C22⋊Q8—21no—
37C4×C22⋊Q8—21no—
38C4×C22⋊Q8—21no—
39C4×#121—21no—
40C4×#121—21no—
41C4×#121C2212yes—
42C4×#121C2212yes—
43#2297—21no—
44#2296C2212yes—
45#2295—21no—
46#2295—21no—
47#2293—21no—
48#2293C2212yes—
49#2285C2211yescharacteristic
50#2285C2211yescharacteristic
51C42×Q8—21no—
52C42×Q8—21no—
53C42×D4—21no—
54C42×D4C2212yes—
55C4×#117C2212yes—
56C2×C43C2211yescharacteristic
57#2099C2212yes—
58#2099C2212yes—
59#2084C2212yes—
60#2084C2212yes—
61#2083C2212yes—
62#2083C2212yes—
63#2071C2212yes—
64#2071C2212yes—
65#2058C2212yes—
66#2058C2212yes—
67#2057C2212yes—
68#2057C2212yes—
69#2041C2212yes—
70#2041C2212yes—
71#2023C2212yes—
72#2023C2212yes—
73#1989C2214yes—
74#1949C2214yes—
75#1926C2214yes—
76#1925C2214yes—
77#1921C2214yes—
78#1921C2214yes—
79#1920C2214yes—
80#1919C2214yes—
81#1875C2214yes—
82#1864C2214yes—
83#1862C2214yes—
84#1854C2214yes—
85#1839C2212yes—
86#1831C2212yes—
87#1825C2214yes—
88#1817C2212yes—
89#1804C2214yes—
90#1797C2212yes—
91C4×#107C2214yes—
92C4×#106C2214yes—
93C4×C4⋊C8C2212yes—
94C4×#102C2214yes—
95C4×#102C2214yes—
96C4×C22⋊C8C2212yes—
97#1744C2211yescharacteristic
98#1744C2211yescharacteristic
99C22×#41—21no—
100#533—28no—
101#532—28no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×#127C2311yescharacteristic
106C2×#125C2312yes—
107C2×#124—21no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—21no—
110C2×#121—21no—
111C2×#121C2312yes—
112C4×#41—48no—
113C2×C4×Q8—21no—
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8—21no—
119C2×C4×Q8—21no—
120C2×C4×Q8—21no—
121C2×C4×D4—22no—
122C2×C4×D4—22no—
123C2×C4×D4—21no—
124C2×C4×D4—21no—
125C2×C4×D4—21no—
126C2×C4×D4—21no—
127C2×C4×D4—21no—
128C2×C4×D4C2312yes—
129C2×C4×D4C2312yes—
130C2×C4×D4C2312yes—
131C2×#117—21no—
132C2×#117—22no—
133C2×#117C2312yes—
134C2×#117C2312yes—
135C22×C42—22no—
136C22×C42D411yescharacteristic
137C22×C42D411yescharacteristic
138C22×C42D411yescharacteristic
139C22×C42D411yescharacteristic
140C22×C42C2311yescharacteristic
141C22×C42C2311yescharacteristic
142C22×C42C2311yescharacteristic
143#482—216no—
144#478—216no—
145C2×#107C2314yes—
146C2×#107C2314yes—
147C2×#106C2314yes—
148C2×#106C2314yes—
149C2×C4⋊C8C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×C22⋊C8C2314yes—
155C2×C4×C8C2312yes—
156C2×C4×C8C2312yes—
157#391—21no—
158#391—21no—
159#391—21no—
160#391—21no—
161#391—21no—
162#391—21no—
163#390—21no—
164#390—21no—
165#390—21no—
166#390—21no—
167#390C2312yes—
168#390C2312yes—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389C2312yes—
176#389C2312yes—
177#389C2312yes—
178#389C2312yes—
179#389C2312yes—
180#389C2312yes—
181#388—22no—
182#388—22no—
183#388—22no—
184#388—22no—
185#387—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#386—22no—
190#386—22no—
191#385—22no—
192#385—22no—
193#384C2311yescharacteristic
194#384C2311yescharacteristic
195#384C2311yescharacteristic
196#384C2311yescharacteristic
197#382—21no—
198#382C2312yes—
199#380—21no—
200#380—21no—
201#380—21no—
202#380C2312yes—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2312yes—
225C4×C22⋊C4—22no—
226C4×C22⋊C4—22no—
227C4×C22⋊C4—22no—
228C4×C22⋊C4—22no—
229C4×C22⋊C4—22no—
230C4×C22⋊C4—22no—
231C4×C22⋊C4—22no—
232C4×C22⋊C4—22no—
233#376—22no—
234C43—22no—
235C43D411yescharacteristic
236C43D411yescharacteristic
237C43D411yescharacteristic
238C43D411yescharacteristic
239C43C2311yescharacteristic
240C43C2311yescharacteristic
241C43C2311yescharacteristic
242C43C2311yescharacteristic
243#340C2314yes—
244#340C2314yes—
245#340C2314yes—
246#340C2314yes—
247#340C2314yes—
248#340C2314yes—
249#340C2314yes—
250#340C2314yes—
251#336C2312yes—
252#336C2312yes—
253#336C2312yes—
254#336C2312yes—
255#336C2314yes—
256#336C2314yes—
257C2×#41—28no—
258C2×#41—44no—
259C2×#41—44no—
260C22×Q8—21no—
261C22×Q8—21no—
262C22×Q8—21no—
263C22×D4—22no—
264C22×D4—21no—
265C22×D4—21no—
266C22×D4C2412yes—
267C23×C4—22no—
268C23×C4—22no—
269C23×C4—22no—
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D411yescharacteristic
273C23×C4C2×D411yescharacteristic
274C23×C4C2×D411yescharacteristic
275C23×C4C2×D411yescharacteristic
276C23×C4C2411yescharacteristic
277C22×C8C2414yes—
278C22×C8C2414yes—
279C4⋊Q8—24no—
280C4⋊Q8—24no—
281C4⋊Q8—24no—
282#127—24no—
283#125—24no—
284#125#4118yescomplemented
285#124—28no—
286C22⋊Q8—28no—
287C22⋊Q8—28no—
288#121—28no—
289#121—24no—
290#121#4118yes—
291C4×Q8—28no—
292C4×Q8—28no—
293C4×Q8—44no—
294C4×Q8—44no—
295C4×Q8—44no—
296C4×Q8—44no—
297C4×Q8—44no—
298C4×Q8—44no—
299C4×D4—28no—
300C4×D4—24no—
301C4×D4—24no—
302C4×D4—28no—
303C4×D4—28no—
304C4×D4—44no—
305C4×D4—44no—
306C4×D4—48no—
307C4×D4—48no—
308C4×D4—44no—
309C4×D4—44no—
310#117—24no—
311#117—24no—
312#117—28no—
313#117—48no—
314#117—44no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2412yes—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C42—22no—
353C2×C42—24no—
354C2×C42—24no—
355C2×C42—24no—
356C2×C42—24no—
357C2×C42—24no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—42no—
367C2×C42—42no—
368C2×C42—48no—
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2411yescharacteristic
392C2×C42C2411yescharacteristic
393C2×C42C2411yescharacteristic
394C2×C42C2411yescharacteristic
395C2×C42C2411yescharacteristic
396C2×C42C2411yescharacteristic
397#107—28no—
398#107—28no—
399#106—28no—
400#106—28no—
401C4⋊C8—28no—
402#102—28no—
403#102—28no—
404#102—28no—
405#102—28no—
406C22⋊C8—28no—
407C4×C8—24no—
408C4×C8—24no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425C24—22no—
426C24C22×D411yescharacteristic
427#41—432no—
428C2×Q8—24no—
429C2×Q8—24no—
430C2×Q8—24no—
431C2×Q8—42no—
432C2×Q8—42no—
433C2×Q8—42no—
434C2×Q8—42no—
435C2×Q8—42no—
436C2×Q8—42no—
437C2×D4—24no—
438C2×D4—24no—
439C2×D4—28no—
440C2×D4—24no—
441C2×D4—22no—
442C2×D4—24no—
443C2×D4—42no—
444C2×D4—42no—
445C2×D4—44no—
446C2×D4—44no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4C2×#4114yes—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—28no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
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—22no—
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—44no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C2×#4114yes—
500C22×C4C2511yescharacteristic
501C2×C8—24no—
502C2×C8—24no—
503C2×C8—24no—
504C2×C8—24no—
505C2×C8—24no—
506C2×C8—24no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—28no—
511C4⋊C4—28no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—24no—
519C4⋊C4—44no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—44no—
524C4⋊C4—42no—
525C4⋊C4—44no—
526C4⋊C4—42no—
527C4⋊C4—44no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—44no—
531C4⋊C4—42no—
532C4⋊C4—44no—
533C4⋊C4C2×#4114yes—
534C4⋊C4C2×#4114yes—
535C4⋊C4C2×#4114yes—
536C22⋊C4—28no—
537C22⋊C4—28no—
538C22⋊C4—28no—
539C22⋊C4—44no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C42—28no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—41no—
569C42—41no—
570C42—41no—
571C42—41no—
572C42—44no—
573C42—44no—
574C42—44no—
575C42—44no—
576C42—44no—
577C42—44no—
578C42—44no—
579C42—44no—
580C42#13514yes—
581C42AGL(1,ℤ/8)14yes—
582C42C2×#4114yes—
583C422−1+414yes—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—21no—
588C23—21no—
589C23—21no—
590C23—24no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23—42no—
596C23—42no—
597C23#52512yes—
598C23C23×D411yescharacteristic
599Q8—48no—
600Q8—48no—
601Q8—48no—
602D4—28no—
603D4—48no—
604D4—48no—
605D4—416no—
606D4—44no—
607C2×C4—28no—
608C2×C4—21no—
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—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—28no—
629C2×C4—21no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—416no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—88no—
663C2×C4—88no—
664C2×C4#52512yes—
665C2×C4#52512yes—
666C2×C4#52512yes—
667C2×C4#53612yes—
668C2×C4#53612yes—
669C2×C4C2×#13512yes—
670C2×C4C2×#13512yes—
671C2×C4C2×#13512yes—
672C2×C4C2×AGL(1,ℤ/8)12yes—
673C2×C4C2×AGL(1,ℤ/8)12yes—
674C2×C4C2×AGL(1,ℤ/8)12yes—
675C2×C4#57512yes—
676C2×C4#57512yes—
677C2×C4C22×#4112yes—
678C2×C4C22×#4112yes—
679C2×C4C2×2−1+412yes—
680C8—44no—
681C8—44no—
682C22—22no—
683C22—22no—
684C22—22no—
685C22—41no—
686C22—41no—
687C22—41no—
688C22—41no—
689C22—44no—
690C22—44no—
691C22—44no—
692C22—82no—
693C22—82no—
694C22—82no—
695C22—82no—
696C22C2×#52511yescharacteristic
697C22C2×#52511yescharacteristic
698C22C2×#53611yescharacteristic
699C22C22×#13511yescharacteristic
700C22C22×AGL(1,ℤ/8)11yescharacteristic
701C22C2×#57511yescharacteristic
702C22C2×#57511yescharacteristic
703C4—22no—
704C4—22no—
705C4—22no—
706C4—22no—
707C4—22no—
708C4—22no—
709C4—22no—
710C4—22no—
711C4—22no—
712C4—22no—
713C4—41no—
714C4—44no—
715C4—44no—
716C4—44no—
717C4—41no—
718C4—82no—
719C4—82no—
720C4—82no—
721C4—82no—
722C4#322814yes—
723C4#348214yes—
724C2—42no—
725C2—84no—
726C2#3497111yescharacteristic
727C2#6053611yescharacteristic
728C2#6053611yescharacteristic
729C2#6055811yescharacteristic
730C2#6056111yescharacteristic
731C2C2×#322811yescharacteristic
732C2#6085411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutator, Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C4×#533C2semidirect
3. #29393C2semidirect
4. #29389C2semidirect
5. #29280C2semidirect
6. #28207C2semidirect
7. #27650C2semidirect
8. #27615C2semidirect
9. #24304C2semidirect
11. #23837C2semidirect
12. #23835C2semidirect
13. #23815C2semidirect
14. #23815C2semidirect
15. C4×#478C2semidirect
16. C4×#482C2semidirect
17. #20777C2semidirect
284. #125#41semidirect

Maximal subgroups

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

subgroupindex
C4×#5332
C4×#5322
#293932
#293892
#292802
#282072
#276502
#276152
#243042
#240622
#238372
#238352
#238152
C4×#4782
C4×#4822
#207772