Finite Group Databasebeta

← #6513959

Subgroups — #6513959

Order 512.

1786 conjugacy classes of subgroups.

The subgroup lattice has 709 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34947C211yesmaximal, characteristic
2#34183C211yesmaximal, characteristic
3#27681C212yesmaximal, complemented
4#27648C212yesmaximal, complemented
5#27629C212yesmaximal, complemented
6#27629C212yesmaximal, complemented
7#26322C212yesmaximal, complemented
8#26302C212yesmaximal, complemented
9#26282C212yesmaximal, complemented
10#26236C212yesmaximal, complemented
11#26204C214yesmaximal, complemented
12#26132C214yesmaximal, complemented
13#21885C211yesmaximal, characteristic, complemented
14#21860C211yesmaximal, characteristic, complemented
15#21847C211yesmaximal, characteristic
16#21358C211yesmaximal, characteristic, complemented
17C2×#1947C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#520—21no—
20C2×C4×#41C2211yescharacteristic
21C2×#484C2214yes—
22C2×#431C2211yescharacteristic
23C22×#107C2212yes—
24C22×C4⋊C8C2211yescharacteristic
25#2568C2211yescharacteristic
26#2567C2211yescharacteristic
27#2566—21no—
28#2565—21no—
29#2564—22no—
30#2563C2211yescharacteristic
31#2542C2211yescharacteristic
32#2417C2212yes—
33#2417C2212yes—
34#2396—22no—
35Q8×C4⋊C4—21no—
36Q8×C4⋊C4C2212yes—
37D4×C4⋊C4—22no—
38D4×C4⋊C4—21no—
39D4×C4⋊C4C2214yes—
40D4×C4⋊C4C2212yes—
41#2317—22no—
42#2317—21no—
43C4×C22⋊Q8C2212yes—
44C4×C22⋊Q8C2212yes—
45#2297—21no—
46#2285—21no—
47#2285C2211yescharacteristic
48#2285C2212yes—
49C2×#384C2212yes—
50C2×#384C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52#2100C2214yes—
53#2095C2214yes—
54#2083C2214yes—
55#2076C2214yes—
56#2076C2214yes—
57#2072C2214yes—
58#2036C2214yes—
59#2036C2214yes—
60#1990C2212yes—
61#1987C2212yes—
62#1984C2212yes—
63#1981C2212yes—
64#1956C2212yes—
65#1954C2212yes—
66#1947C2211yescharacteristic
67#1947C2212yes—
68#1947C2211yescharacteristic
69#1947C2212yes—
70#1947C2211yescharacteristic
71#1947C2211yescharacteristic
72#1947C2212yes—
73#1947C2212yes—
74#1928C2214yes—
75#1928C2214yes—
76#1921C2214yes—
77#1921C2214yes—
78#1892C2214yes—
79#1872C2214yes—
80#1872C2214yes—
81#1869C2214yes—
82#1852C2212yes—
83#1852C2214yes—
84#1852C2212yes—
85#1824C2214yes—
86#1803C2212yes—
87#1803C2212yes—
88#1803C2212yes—
89#1803C2212yes—
90#1797C2212yes—
91#1797C2212yes—
92C4×#107C2212yes—
93C4×#107C2212yes—
94C2×#340C2212yes—
95C22×#41D411yescharacteristic
96C23×C8C2312yes—
97#533—24no—
98#533—24no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×#125—21no—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8C2312yes—
104C2×C22⋊Q8C2312yes—
105C2×C22⋊Q8C2312yes—
106#520—48no—
107C4×#41—24no—
108C4×#41—24no—
109C2×C4×Q8—22no—
110C2×C4×Q8—21no—
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2311yescharacteristic
114C2×C4×D4—24no—
115C2×C4×D4—22no—
116C2×C4×D4D412yes—
117C2×C4×D4D411yescharacteristic
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2312yes—
120C2×#117—22no—
121C2×#117—21no—
122C2×#117—21no—
123C2×#117D412yes—
124C2×#117D411yescharacteristic
125C22×C4⋊C4—22no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—22no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2314yes—
132C22×C4⋊C4C2312yes—
133C22×C42—21no—
134C22×C42C2312yes—
135C22×C42C2311yescharacteristic
136#484—48no—
137#431—24no—
138C2×#107—44no—
139C2×#107C2312yes—
140C2×#107C2314yes—
141C2×#107C2312yes—
142C2×#107C2312yes—
143C2×#107C2314yes—
144C2×#107C2312yes—
145C2×C4⋊C8—24no—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2311yescharacteristic
148C2×C4⋊C8C2311yescharacteristic
149C2×C4⋊C8C2312yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×C22⋊C8C2312yes—
155C2×C22⋊C8C2314yes—
156C2×C22⋊C8C2314yes—
157C2×C4×C8C2312yes—
158#400—22no—
159#399—22no—
160#398—22no—
161#398—22no—
162#396—22no—
163#396—22no—
164#395—22no—
165#393—22no—
166#391—21no—
167#391—21no—
168#391C2312yes—
169#391C2312yes—
170#390—21no—
171#390—21no—
172#389—22no—
173#389—24no—
174#389—22no—
175#389—24no—
176#389C2312yes—
177#389C2312yes—
178#387—22no—
179#387—22no—
180#386—22no—
181#386—22no—
182#385—22no—
183#385—22no—
184#385—24no—
185#385—24no—
186#385—24no—
187#384—21no—
188#384—21no—
189#384—22no—
190#384—21no—
191#384—22no—
192#384—21no—
193#384C2311yescharacteristic
194#384C2312yes—
195#384C2311yescharacteristic
196#384C2314yes—
197#384C2312yes—
198#382—21no—
199#382—21no—
200#382—22no—
201#382—21no—
202#380—22no—
203#380—21no—
204#380—22no—
205#380—22no—
206#380C2312yes—
207#380C2312yes—
208#380C2314yes—
209#380C2312yes—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2314yes—
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C22⋊C4—22no—
223C4×C22⋊C4—22no—
224C2×#95—21no—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#340C2314yes—
234#340C2312yes—
235#336C2314yes—
236#336C2314yes—
237C2×#41—24no—
238C2×#41—24no—
239C2×#41Q1618yes—
240C22×Q8—21no—
241C22×Q8C2×D411yescharacteristic
242C22×Q8C2412yes—
243C22×D4C2×D412yes—
244C22×D4C2×D411yescharacteristic
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—24no—
248C23×C4—22no—
249C23×C4C2×D412yes—
250C23×C4C2×D412yes—
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2412yes—
254C23×C4C2411yescharacteristic
255C22×C8—24no—
256C22×C8—42no—
257C22×C8—42no—
258C22×C8C2414yes—
259C22×C8C2412yes—
260C22×C8C2412yes—
261C4⋊Q8—22no—
262C4⋊Q8—22no—
263C4⋊Q8—22no—
264C4⋊Q8—22no—
265#125—44no—
266C22⋊Q8—24no—
267C22⋊Q8—24no—
268C22⋊Q8—24no—
269C22⋊Q8—24no—
270C22⋊Q8—42no—
271C22⋊Q8—42no—
272C22⋊Q8—44no—
273C4×Q8—24no—
274C4×Q8—22no—
275C4×Q8—22no—
276C4×Q8—24no—
277C4×Q8—24no—
278C4×Q8—48no—
279C4×Q8—44no—
280C4×D4—28no—
281C4×D4—24no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—416no—
287C4×D4—48no—
288#117—24no—
289#117—28no—
290#117—48no—
291#117—44no—
292#117—44no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—41no—
319C2×C4⋊C4—41no—
320C2×C4⋊C4—42no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4—42no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4—48no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4—82no—
327C2×C4⋊C4—84no—
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2414yes—
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2411yescharacteristic
340C2×C22⋊C4—24no—
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⋊C4C2×D412yes—
348C2×C22⋊C4C2×D412yes—
349C2×C22⋊C4C2×D412yes—
350C2×C42—24no—
351C2×C42—24no—
352C2×C42—24no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—21no—
356C2×C42—21no—
357C2×C42—21no—
358C2×C42—21no—
359C2×C42—44no—
360C2×C42C2×D412yes—
361C2×C42C2×D412yes—
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42#4114yes—
365C2×C42#4114yes—
366C2×C42C2412yes—
367C2×C42C2411yescharacteristic
368C2×C42C2411yescharacteristic
369#107—28no—
370#107—24no—
371#107—24no—
372#107—44no—
373#107—42no—
374#107—42no—
375C4⋊C8—24no—
376C4⋊C8—24no—
377C4⋊C8—22no—
378C4⋊C8—22no—
379#103—44no—
380#103—44no—
381#103—44no—
382#103—44no—
383C22⋊C8—24no—
384C22⋊C8—44no—
385C22⋊C8—44no—
386C4×C8—24no—
387#95—21no—
388#95—21no—
389#95—21no—
390#95—21no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—24no—
400#95—22no—
401#95—24no—
402#95—22no—
403C24C22×D412yes—
404C24C22×D411yescharacteristic
405#41—232no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—42no—
415C2×Q8—81no—
416C2×Q8—81no—
417C2×Q8C2×Q1614yes—
418C2×D4—24no—
419C2×D4—24no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4C2×Q1618yes—
423C2×D4C2×Q1614yes—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—41no—
447C22×C4—42no—
448C22×C4—41no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—48no—
455C22×C4—44no—
456C22×C4—81no—
457C22×C4—81no—
458C22×C4—82no—
459C22×C4—84no—
460C22×C4—82no—
461C22×C4—84no—
462C22×C4#12314yes—
463C22×C4#12314yes—
464C22×C4C2×Q1618yes—
465C22×C4C2×Q1614yes—
466C22×C4AGL(1,ℤ/8)14yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D412yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C2×#4112yes—
476C22×C4C2×#4112yes—
477C22×C4C2511yescharacteristic
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—24no—
481C2×C8—22no—
482C2×C8—24no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—24no—
486C2×C8—22no—
487C2×C8—48no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—24no—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—24no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—22no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—48no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—42no—
523C4⋊C4—82no—
524C4⋊C4—82no—
525C4⋊C4—82no—
526C4⋊C4—84no—
527C4⋊C4—82no—
528C4⋊C4—84no—
529C4⋊C4H(5,𝔽2)12yes—
530C4⋊C42−1+412yes—
531C22⋊C4—24no—
532C22⋊C4—24no—
533C22⋊C4—24no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—48no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—82no—
540C22⋊C4—82no—
541C42—24no—
542C42—24no—
543C42—22no—
544C42—22no—
545C42—48no—
546C42—44no—
547C42—42no—
548C42—42no—
549C42—42no—
550C42—42no—
551C42—44no—
552C42#12314yes—
553C42#12314yes—
554C42C2×#4112yes—
555C42C2×#4112yes—
556C42C2×#4112yes—
557C42C2×#4112yes—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23C2×#12312yes—
569C23C22×Q1614yes—
570C23C22×Q1612yes—
571C23C2×AGL(1,ℤ/8)12yes—
572C23C23×D411yescharacteristic
573Q8—28no—
574Q8—42no—
575Q8—42no—
576Q8—42no—
577Q8—42no—
578Q8—84no—
579D4—216no—
580D4—28no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—216no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—44no—
603C2×C4—48no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—84no—
634C2×C4—84no—
635C2×C4—164no—
636C2×C4—168no—
637C2×C4#484116yescomplemented
638C2×C4C2×#12312yes—
639C2×C4C2×#12312yes—
640C2×C4C2×#12312yes—
641C2×C4C2×#12312yes—
642C2×C4C2×#12312yes—
643C2×C4C22×Q1614yes—
644C2×C4C22×Q1612yes—
645C2×C4C22×Q1612yes—
646C2×C4C2×AGL(1,ℤ/8)12yes—
647C2×C4C2×AGL(1,ℤ/8)12yes—
648C2×C4C22×#4111yescharacteristic
649C2×C4C22×#4111yescharacteristic
650C2×C4C2×H(5,𝔽2)11yescharacteristic
651C2×C4C2×2−1+411yescharacteristic
652C2×C4#58511yescharacteristic
653C2×C4#58511yescharacteristic
654C8—44no—
655C8—42no—
656C8—42no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—24no—
664C22—24no—
665C22—24no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22C2×#48414yes—
671C22C22×#12311yescharacteristic
672C22#348811yescharacteristic
673C22#348911yescharacteristic
674C22C23×Q1611yescharacteristic
675C22C22×AGL(1,ℤ/8)11yescharacteristic
676C22#358511yescharacteristic
677C22#358611yescharacteristic
678C4—24no—
679C4—22no—
680C4—22no—
681C4—22no—
682C4—42no—
683C4—42no—
684C4—42no—
685C4—42no—
686C4—81no—
687C4—84no—
688C4—82no—
689C4—82no—
690C4—82no—
691C4—81no—
692C4—82no—
693C4C2×#48414yes—
694C4C2×#48414yes—
695C4#350611yescharacteristic
696C4#350711yescharacteristic
697C4#351111yescharacteristic
698C4#351211yescharacteristic
699C2—24no—
700C2—44no—
701C2#3551711yescharacteristic
702C2C22×#48411yescharacteristic
703C2#6051711yescharacteristic
704C2#6195211yescharacteristic
705C2#6195911yescharacteristic
706C2#6210011yescharacteristic
707C2#6210711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #27681C2semidirect
4. #27648C2semidirect
5. #27629C2semidirect
6. #27629C2semidirect
7. #26322C2semidirect
8. #26302C2semidirect
9. #26282C2semidirect
10. #26236C2semidirect
11. #26204C2semidirect
12. #26132C2semidirect
13. #21885C2semidirect
14. #21860C2semidirect
16. #21358C2semidirect
17. C2×#1947C2semidirect
637. C2×C4#484semidirect

Maximal subgroups

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

subgroupindex
#349472
#341832
#276812
#276482
#276292
#263222
#263022
#262822
#262362
#262042
#261322
#218852
#218602
#218472
#213582
C2×#19472