Finite Group Databasebeta

← #6856087

Subgroups — #6856087

Order 512.

1681 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36577C211yesmaximal, characteristic
2C2×#2546C211yesmaximal, characteristic
3#33109C211yesmaximal, characteristic
4#32644C214yesmaximal, complemented
5#32591C214yesmaximal, complemented
6#29757C212yesmaximal
7#29335C212yesmaximal
8#29243C212yesmaximal
9#28646C212yesmaximal, complemented
10#26276C212yesmaximal, complemented
11#25902C211yesmaximal, characteristic
12#25647C212yesmaximal, complemented
13#25479C212yesmaximal, complemented
14#24301C211yesmaximal, characteristic
15#22510C211yesmaximal, characteristic
16#22085C212yesmaximal
17C4×#484C211yesmaximal, characteristic
18C2×#557C2211yescharacteristic
19C22×C4⋊Q8—21no—
20C22×C4⋊Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C2×#487C2212yes—
23C2×#484C2211yescharacteristic
24C2×#484C2212yes—
25C2×#475C2212yes—
26C2×#446C2212yes—
27#2857C2211yescharacteristic
28#2857C2212yes—
29#2582C2212yes—
30#2582C2211yescharacteristic
31#2568C2212yes—
32#2568C2211yescharacteristic
33#2546—21no—
34#2546—21no—
35#2546—22no—
36#2546C2212yes—
37#2546C2214yes—
38#2546C2212yes—
39#2453—21no—
40#2453—22no—
41#2442—22no—
42#2442—21no—
43C2×#398—22no—
44#2351—21no—
45Q8×C4⋊C4C2211yescharacteristic
46Q8×C22⋊C4—21no—
47C4×C22⋊Q8C2212yes—
48C4×C22⋊Q8C2211yescharacteristic
49C2×#391—21no—
50C2×#389C2212yes—
51C2×#389C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2099C2214yes—
54#2088C2214yes—
55#2083C2214yes—
56#2068C2214yes—
57#2068C2214yes—
58#2068C2214yes—
59#2058C2212yes—
60#2043C2214yes—
61#2043C2214yes—
62#2024C2214yes—
63#2024C2212yes—
64#1990C2212yes—
65#1988C2212yes—
66#1985C2214yes—
67#1984C2212yes—
68#1981C2212yes—
69#1947C2212yes—
70#1943C2214yes—
71#1935C2214yes—
72#1935C2214yes—
73#1935C2214yes—
74#1928C2212yes—
75#1928C2214yes—
76#1921C2212yes—
77#1918C2214yes—
78#1918C2214yes—
79#1872C2212yes—
80#1872C2212yes—
81#1872C2212yes—
82#1872C2212yes—
83#1872C2211yescharacteristic
84#1865C2214yes—
85#1865C2214yes—
86#1853C2212yes—
87#1852C2211yescharacteristic
88#1824C2211yescharacteristic
89#1797C2212yes—
90#1797C2211yescharacteristic
91C4×#107C2212yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×Q8D411yescharacteristic
97#557—44no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—24no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—44no—
104C2×C4⋊Q8—48no—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2312yes—
109C2×#125—21no—
110C2×#125—22no—
111C2×C22⋊Q8C2311yescharacteristic
112C2×C22⋊Q8C2312yes—
113C2×C4×Q8—21no—
114C2×C4×Q8—21no—
115C2×C4×Q8—44no—
116C2×C4×Q8D411yescharacteristic
117C2×C4×Q8D411yescharacteristic
118C2×C4×Q8D412yes—
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2312yes—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4D412yes—
128C22×C4⋊C4D411yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2312yes—
132C22×C42—21no—
133C22×C42C2311yescharacteristic
134C22×C42C2312yes—
135#487—44no—
136#484—24no—
137#484—28no—
138#475—44no—
139#446—44no—
140C2×#107C2312yes—
141C2×#107C2314yes—
142C2×#107C2312yes—
143C2×#106C2314yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2314yes—
146C2×#103C2312yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2312yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×C22⋊C8C2311yescharacteristic
154C2×C22⋊C8C2312yes—
155C2×C22⋊C8C2311yescharacteristic
156C2×#97C2314yes—
157C2×C4×C8C2312yes—
158#400—22no—
159#400—22no—
160#400—22no—
161#398—22no—
162#398—22no—
163#398—22no—
164#398—24no—
165#398—24no—
166#398—24no—
167#398—24no—
168#391—21no—
169#391—22no—
170#391—21no—
171#391—22no—
172#391—22no—
173#391—22no—
174#391—24no—
175#391C2312yes—
176#391C2311yescharacteristic
177#391C2312yes—
178#391C2312yes—
179#391C2312yes—
180#389—21no—
181#389—22no—
182#389—21no—
183#389—22no—
184#389—21no—
185#389—22no—
186#389—21no—
187#389—22no—
188#389—22no—
189#389—22no—
190#389—22no—
191#389—22no—
192#389C2311yescharacteristic
193#389C2314yes—
194#389C2314yes—
195#389C2312yes—
196#389C2314yes—
197#389C2314yes—
198#387—21no—
199#387—22no—
200#385—21no—
201#385—22no—
202#384C2312yes—
203#384C2311yescharacteristic
204#382—21no—
205#382—22no—
206#380—22no—
207#380—21no—
208#380—21no—
209#380—22no—
210#380C2311yescharacteristic
211C4×C4⋊C4—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2312yes—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—21no—
221C2×#95—21no—
222C2×#95—22no—
223C2×#95—22no—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#340C2314yes—
231#340C2312yes—
232#340C2314yes—
233#336C2312yes—
234#336C2314yes—
235#336C2314yes—
236#336C2312yes—
237C22×Q8—24no—
238C22×Q8—24no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8—42no—
242C22×Q8—44no—
243C22×Q8—44no—
244C22×Q8—44no—
245C22×Q8C2×D411yescharacteristic
246C22×Q8C2×D412yes—
247C22×Q8C2×D411yescharacteristic
248C22×Q8C2411yescharacteristic
249C22×Q8C2412yes—
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—22no—
253C23×C4C2×D412yes—
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D412yes—
257C23×C4C2411yescharacteristic
258C22×C8C2412yes—
259C22×C8C2412yes—
260C22×C8C2414yes—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8—24no—
264C4⋊Q8—44no—
265C4⋊Q8—44no—
266C4⋊Q8—44no—
267C4⋊Q8—44no—
268C4⋊Q8—416no—
269C4⋊Q8—48no—
270C4⋊Q8—48no—
271#125—48no—
272#125—44no—
273C22⋊Q8—22no—
274C22⋊Q8—24no—
275C22⋊Q8—22no—
276C22⋊Q8—24no—
277C4×Q8—24no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—44no—
281C4×Q8—44no—
282C4×Q8—42no—
283C4×Q8—42no—
284C4×Q8—44no—
285C4×Q8—44no—
286C4×Q8—44no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—42no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—48no—
317C2×C4⋊C4—48no—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4#4114yes—
326C2×C4⋊C4C2414yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C42—24no—
337C2×C42—22no—
338C2×C42—21no—
339C2×C42—22no—
340C2×C42—21no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—44no—
344C2×C42—44no—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2412yes—
350C2×C42C2411yescharacteristic
351C2×C42C2412yes—
352C2×C42C2411yescharacteristic
353#107—24no—
354#107—28no—
355#107—24no—
356#106—44no—
357C4⋊C8—24no—
358C4⋊C8—44no—
359#103—24no—
360#103—28no—
361#103—28no—
362#103—24no—
363#103—44no—
364#103—44no—
365#103—44no—
366C22⋊C8—22no—
367C22⋊C8—24no—
368C22⋊C8—22no—
369#97—44no—
370C4×C8—24no—
371#95—22no—
372#95—22no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—22no—
378#95—21no—
379#95—22no—
380#95—21no—
381#95—21no—
382#95—22no—
383#95—21no—
384#95—22no—
385#95—21no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392C24C22×D411yescharacteristic
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—24no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—24no—
399C2×Q8—22no—
400C2×Q8—21no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—24no—
404C2×Q8—22no—
405C2×Q8—24no—
406C2×Q8—21no—
407C2×Q8—22no—
408C2×Q8—42no—
409C2×Q8—416no—
410C2×Q8—44no—
411C2×Q8—44no—
412C2×Q8—44no—
413C2×Q8—44no—
414C2×Q8—44no—
415C2×Q8—44no—
416C2×Q8—816no—
417C2×Q8AGL(1,ℤ/8)14yes—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—24no—
438C22×C4—42no—
439C22×C4—41no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4#13714yes—
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C2×#4112yes—
460C22×C42−1+412yes—
461C22×C4C2511yescharacteristic
462C2×C8—22no—
463C2×C8—24no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8—22no—
468C2×C8—24no—
469C2×C8—22no—
470C2×C8—22no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—22no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—81no—
508C4⋊C4—81no—
509C4⋊C4—82no—
510C4⋊C4—82no—
511C4⋊C4C2×#4114yes—
512C4⋊C42−1+414yes—
513C22⋊C4—42no—
514C22⋊C4—44no—
515C22⋊C4—42no—
516C42—22no—
517C42—22no—
518C42—24no—
519C42—22no—
520C42—24no—
521C42—42no—
522C42—42no—
523C42—44no—
524C42—44no—
525C42—42no—
526C42—44no—
527C42—44no—
528C42—42no—
529C42H(5,𝔽2)14yes—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23#54812yes—
535C23C2×#13712yes—
536C23#57812yes—
537C23C23×D411yescharacteristic
538Q8—28no—
539Q8—48no—
540Q8—44no—
541Q8—42no—
542Q8—44no—
543Q8—42no—
544Q8—44no—
545Q8—88no—
546Q8—88no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—28no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—44no—
573C2×C4—48no—
574C2×C4—48no—
575C2×C4—48no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—44no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—84no—
602C2×C4—88no—
603C2×C4#53412yes—
604C2×C4#53412yes—
605C2×C4#54812yes—
606C2×C4C2×AGL(1,ℤ/8)14yes—
607C2×C4C2×AGL(1,ℤ/8)12yes—
608C2×C4C2×#13712yes—
609C2×C4C2×#13712yes—
610C2×C4C22×#4111yescharacteristic
611C2×C4C2×H(5,𝔽2)12yes—
612C2×C4C2×2−1+412yes—
613C2×C4C2×2−1+411yescharacteristic
614C8—42no—
615C8—44no—
616C8—42no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22#337414yes—
624C22C2×#53411yescharacteristic
625C22C2×#54811yescharacteristic
626C22#348611yescharacteristic
627C22C22×AGL(1,ℤ/8)11yescharacteristic
628C22C22×#13711yescharacteristic
629C22C2×#57811yescharacteristic
630C22#358611yescharacteristic
631C4—22no—
632C4—22no—
633C4—24no—
634C4—22no—
635C4—42no—
636C4—41no—
637C4—42no—
638C4—42no—
639C4—44no—
640C4—44no—
641C4—82no—
642C4—81no—
643C4—84no—
644C4—81no—
645C4—82no—
646C4—81no—
647C4—81no—
648C4#319314yes—
649C4#351112yes—
650C4#355212yes—
651C2—24no—
652C2#3972111yescharacteristic
653C2C2×#319311yescharacteristic
654C2#6075911yescharacteristic
655C2C2×#337411yescharacteristic
656C2#6131711yescharacteristic
657C2#6182911yescharacteristic
658C2#6183711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #32644C2semidirect
5. #32591C2semidirect
9. #28646C2semidirect
10. #26276C2semidirect
12. #25647C2semidirect
13. #25479C2semidirect

Maximal subgroups

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

subgroupindex
#365772
C2×#25462
#331092
#326442
#325912
#297572
#293352
#292432
#286462
#262762
#259022
#256472
#254792
#243012
#225102
#220852
C4×#4842