Finite Group Databasebeta

← #6513783

Subgroups — #6513783

Order 512.

1814 conjugacy classes of subgroups.

The subgroup lattice has 727 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34952C211yesmaximal, characteristic
2#34183C211yesmaximal, characteristic
3#27681C212yesmaximal, complemented
4#27648C212yesmaximal, complemented
5#27629C212yesmaximal, complemented
6#27629C212yesmaximal, complemented
7#25062C212yesmaximal, complemented
8#25016C212yesmaximal, complemented
9#24996C212yesmaximal, complemented
10#24971C212yesmaximal, complemented
11#24939C214yesmaximal, complemented
12#24867C214yesmaximal, complemented
13#21982C211yesmaximal, characteristic, complemented
14#21961C211yesmaximal, characteristic, complemented
15#21847C211yesmaximal, characteristic
16#20734C211yesmaximal, characteristic, complemented
17C2×#1803C211yesmaximal, characteristic, complemented
18C2×#520—21no—
19C2×C4×#41C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C2×#484C2214yes—
22C2×#431C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C22×#103C2212yes—
25#2582C2211yescharacteristic
26#2582C2211yescharacteristic
27#2581—21no—
28#2580—21no—
29#2579—22no—
30#2538C2211yescharacteristic
31#2538C2211yescharacteristic
32#2487C2212yes—
33#2409C2212yes—
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#2296—21no—
47#2285C2211yescharacteristic
48#2285C2212yes—
49#2285C2211yescharacteristic
50C2×#384C2212yes—
51C2×#384C2211yescharacteristic
52#2100C2214yes—
53#2095C2214yes—
54#2083C2214yes—
55#2076C2214yes—
56#2076C2214yes—
57#2072C2214yes—
58#2036C2214yes—
59#2036C2214yes—
60#1984C2212yes—
61#1954C2212yes—
62#1947C2212yes—
63#1947C2212yes—
64#1947C2212yes—
65#1947C2212yes—
66#1947C2212yes—
67#1947C2212yes—
68#1928C2214yes—
69#1928C2214yes—
70#1921C2214yes—
71#1921C2214yes—
72#1872C2214yes—
73#1869C2214yes—
74#1852C2214yes—
75#1852C2214yes—
76#1845C2212yes—
77#1834C2212yes—
78#1824C2214yes—
79#1824C2212yes—
80#1808C2214yes—
81#1803C2211yescharacteristic
82#1803C2212yes—
83#1803C2211yescharacteristic
84#1803C2212yes—
85#1803C2212yes—
86#1803C2212yes—
87#1803C2211yescharacteristic
88#1803C2211yescharacteristic
89#1797C2212yes—
90#1797C2212yes—
91#1787C2212yes—
92C4×#103C2212yes—
93C4×#103C2212yes—
94C2×#340C2212yes—
95C22×#41D411yescharacteristic
96C23×Q8C2311yescharacteristic
97C23×C8C2312yes—
98C2×C22⋊Q8C2312yes—
99C2×C22⋊Q8C2312yes—
100#520—48no—
101C4×#41—24no—
102C4×#41—24no—
103C2×C4×Q8—22no—
104C2×C4×Q8—21no—
105C2×C4×Q8—21no—
106C2×C4×Q8—24no—
107C2×C4×Q8—24no—
108C2×C4×Q8D411yescharacteristic
109C2×C4×Q8C2311yescharacteristic
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2311yescharacteristic
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2311yescharacteristic
114C2×C4×D4—24no—
115C2×C4×D4—22no—
116C2×C4×D4D412yes—
117C2×C4×D4D411yescharacteristic
118C2×C4×D4C2312yes—
119C2×C4×D4C2311yescharacteristic
120C2×#117—22no—
121C2×#117—21no—
122C2×#117—21no—
123C2×#117D412yes—
124C2×#117D411yescharacteristic
125C22×C4⋊C4—22no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—22no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2314yes—
132C22×C4⋊C4C2312yes—
133C22×C42—21no—
134C22×C42C2312yes—
135C22×C42C2311yescharacteristic
136#484—48no—
137#431—24no—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×#107C2314yes—
142C2×C4⋊C8—24no—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2311yescharacteristic
145C2×C4⋊C8C2311yescharacteristic
146C2×C4⋊C8C2312yes—
147C2×#103—44no—
148C2×#103C2314yes—
149C2×#103C2312yes—
150C2×#103C2312yes—
151C2×#103C2314yes—
152C2×#103C2312yes—
153C2×#103C2312yes—
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#389—21no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389C2312yes—
181#389C2312yes—
182#387—22no—
183#387—22no—
184#386—22no—
185#386—22no—
186#385—22no—
187#385—22no—
188#385—24no—
189#385—24no—
190#385—24no—
191#385—22no—
192#385—22no—
193#384—22no—
194#384—21no—
195#384—22no—
196#384—21no—
197#384C2312yes—
198#384C2311yescharacteristic
199#384C2312yes—
200#384C2311yescharacteristic
201#384C2311yescharacteristic
202#384C2311yescharacteristic
203#384C2314yes—
204#382—22no—
205#382—21no—
206#382—21no—
207#380—21no—
208#380—22no—
209#380—21no—
210#380C2312yes—
211#380C2312yes—
212#380C2312yes—
213#380C2314yes—
214#380C2312yes—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—22no—
219C4×C4⋊C4—22no—
220C4×C4⋊C4C2314yes—
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2312yes—
223C4×C22⋊C4—22no—
224C4×C22⋊C4—22no—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#340C2312yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#336C2314yes—
236#336C2314yes—
237C2×#41—24no—
238C2×#41—24no—
239C2×#41Q1618yes—
240C22×Q8—21no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—41no—
246C22×Q8—41no—
247C22×Q8C2×D411yescharacteristic
248C22×Q8C2411yescharacteristic
249C22×Q8C2411yescharacteristic
250C22×D4C2×D412yes—
251C22×D4C2×D411yescharacteristic
252C23×C4—21no—
253C23×C4—24no—
254C23×C4—22no—
255C23×C4—21no—
256C23×C4C2×D412yes—
257C23×C4C2×D412yes—
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2412yes—
261C23×C4C2411yescharacteristic
262C22×C8—24no—
263C22×C8—42no—
264C22×C8—42no—
265C22×C8C2414yes—
266C22×C8C2412yes—
267C22×C8C2412yes—
268C22⋊Q8—24no—
269C22⋊Q8—24no—
270C22⋊Q8—42no—
271C22⋊Q8—42no—
272C4×Q8—24no—
273C4×Q8—22no—
274C4×Q8—22no—
275C4×Q8—22no—
276C4×Q8—22no—
277C4×Q8—22no—
278C4×Q8—22no—
279C4×Q8—22no—
280C4×Q8—22no—
281C4×Q8—22no—
282C4×Q8—22no—
283C4×Q8—48no—
284C4×Q8—44no—
285C4×Q8—44no—
286C4×D4—28no—
287C4×D4—24no—
288C4×D4—24no—
289C4×D4—24no—
290C4×D4—22no—
291C4×D4—22no—
292C4×D4—416no—
293C4×D4—48no—
294#117—24no—
295#117—28no—
296#117—48no—
297#117—44no—
298#117—44no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—44no—
327C2×C4⋊C4—42no—
328C2×C4⋊C4—42no—
329C2×C4⋊C4—44no—
330C2×C4⋊C4—48no—
331C2×C4⋊C4—44no—
332C2×C4⋊C4—44no—
333C2×C4⋊C4—82no—
334C2×C4⋊C4—84no—
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2×D412yes—
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×D412yes—
339C2×C4⋊C4C2414yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2411yescharacteristic
342C2×C4⋊C4C2411yescharacteristic
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2412yes—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4C2×D412yes—
354C2×C22⋊C4C2×D412yes—
355C2×C22⋊C4C2×D412yes—
356C2×C42—24no—
357C2×C42—24no—
358C2×C42—24no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—21no—
362C2×C42—21no—
363C2×C42—21no—
364C2×C42—21no—
365C2×C42—44no—
366C2×C42C2×D412yes—
367C2×C42C2×D412yes—
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42#4114yes—
371C2×C42#4114yes—
372C2×C42C2412yes—
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375#107—44no—
376#107—44no—
377#107—44no—
378#107—44no—
379C4⋊C8—24no—
380C4⋊C8—24no—
381C4⋊C8—22no—
382C4⋊C8—22no—
383#103—28no—
384#103—24no—
385#103—24no—
386#103—44no—
387#103—42no—
388#103—42no—
389C22⋊C8—24no—
390C22⋊C8—44no—
391C22⋊C8—44no—
392C4×C8—24no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—24no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—24no—
406#95—22no—
407C24C22×D412yes—
408C24C22×D411yescharacteristic
409#41—232no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—28no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—42no—
424C2×Q8—42no—
425C2×Q8—44no—
426C2×Q8—44no—
427C2×Q8—44no—
428C2×Q8—42no—
429C2×Q8C2×Q1614yes—
430C2×Q8H(5,𝔽2)12yes—
431C2×Q82−1+412yes—
432C2×D4—24no—
433C2×D4—24no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4C2×Q1618yes—
437C2×D4C2×Q1614yes—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—41no—
461C22×C4—42no—
462C22×C4—41no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—48no—
469C22×C4—44no—
470C22×C4—81no—
471C22×C4—81no—
472C22×C4—82no—
473C22×C4—84no—
474C22×C4—82no—
475C22×C4—84no—
476C22×C4#12314yes—
477C22×C4#12314yes—
478C22×C4C2×Q1618yes—
479C22×C4C2×Q1614yes—
480C22×C4AGL(1,ℤ/8)14yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D412yes—
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C2×#4112yes—
490C22×C4C2×#4112yes—
491C22×C4C2511yescharacteristic
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—24no—
495C2×C8—22no—
496C2×C8—24no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—24no—
500C2×C8—22no—
501C2×C8—84no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—24no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—24no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—42no—
523C4⋊C4—48no—
524C4⋊C4—44no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—81no—
538C4⋊C4—81no—
539C4⋊C4—82no—
540C4⋊C4—82no—
541C4⋊C4—82no—
542C4⋊C4—84no—
543C4⋊C4—82no—
544C4⋊C4—84no—
545C22⋊C4—24no—
546C22⋊C4—24no—
547C22⋊C4—24no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—48no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—82no—
554C22⋊C4—82no—
555C42—24no—
556C42—24no—
557C42—22no—
558C42—22no—
559C42—48no—
560C42—42no—
561C42—42no—
562C42—42no—
563C42—42no—
564C42—44no—
565C42—44no—
566C42#12314yes—
567C42#12314yes—
568C42C2×#4112yes—
569C42C2×#4112yes—
570C42C2×#4112yes—
571C42C2×#4112yes—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—22no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—22no—
582C23C2×#12312yes—
583C23C22×Q1614yes—
584C23C22×Q1612yes—
585C23C2×AGL(1,ℤ/8)12yes—
586C23C23×D411yescharacteristic
587Q8—28no—
588Q8—22no—
589Q8—22no—
590Q8—22no—
591Q8—22no—
592Q8—48no—
593Q8—41no—
594Q8—41no—
595Q8—41no—
596Q8—41no—
597D4—216no—
598D4—28no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—216no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—44no—
621C2×C4—48no—
622C2×C4—48no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—84no—
653C2×C4—164no—
654C2×C4—168no—
655C2×C4#484116yescomplemented
656C2×C4C2×#12312yes—
657C2×C4C2×#12312yes—
658C2×C4C2×#12312yes—
659C2×C4C2×#12312yes—
660C2×C4C2×#12312yes—
661C2×C4C22×Q1614yes—
662C2×C4C22×Q1612yes—
663C2×C4C22×Q1612yes—
664C2×C4C2×AGL(1,ℤ/8)12yes—
665C2×C4C2×AGL(1,ℤ/8)12yes—
666C2×C4C22×#4111yescharacteristic
667C2×C4C22×#4111yescharacteristic
668C2×C4C2×H(5,𝔽2)11yescharacteristic
669C2×C4C2×2−1+411yescharacteristic
670C2×C4#58511yescharacteristic
671C2×C4#58511yescharacteristic
672C8—42no—
673C8—42no—
674C8—44no—
675C22—22no—
676C22—22no—
677C22—22no—
678C22—22no—
679C22—22no—
680C22—22no—
681C22—24no—
682C22—24no—
683C22—24no—
684C22—42no—
685C22—42no—
686C22—42no—
687C22—42no—
688C22C2×#48414yes—
689C22C22×#12311yescharacteristic
690C22#348811yescharacteristic
691C22#348911yescharacteristic
692C22C23×Q1611yescharacteristic
693C22C22×AGL(1,ℤ/8)11yescharacteristic
694C22#358511yescharacteristic
695C22#358611yescharacteristic
696C4—24no—
697C4—22no—
698C4—22no—
699C4—22no—
700C4—42no—
701C4—42no—
702C4—42no—
703C4—42no—
704C4—82no—
705C4—84no—
706C4—82no—
707C4—81no—
708C4—81no—
709C4—82no—
710C4—82no—
711C4C2×#48414yes—
712C4C2×#48414yes—
713C4#349211yescharacteristic
714C4#349311yescharacteristic
715C4#349411yescharacteristic
716C4#349511yescharacteristic
717C2—24no—
718C2—44no—
719C2#3551611yescharacteristic
720C2C22×#48411yescharacteristic
721C2#6051711yescharacteristic
722C2#6195211yescharacteristic
723C2#6196311yescharacteristic
724C2#6210011yescharacteristic
725C2#6211111yescharacteristic
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. #25062C2semidirect
8. #25016C2semidirect
9. #24996C2semidirect
10. #24971C2semidirect
11. #24939C2semidirect
12. #24867C2semidirect
13. #21982C2semidirect
14. #21961C2semidirect
16. #20734C2semidirect
17. C2×#1803C2semidirect
655. C2×C4#484semidirect

Maximal subgroups

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

subgroupindex
#349522
#341832
#276812
#276482
#276292
#250622
#250162
#249962
#249712
#249392
#248672
#219822
#219612
#218472
#207342
C2×#18032