Finite Group Databasebeta

← #6849002

Subgroups — #6849002

Order 512.

1643 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36180C211yesmaximal, characteristic
2#34929C211yesmaximal, characteristic
3#33198C211yesmaximal, characteristic, complemented
4#31744C214yesmaximal, complemented
5#31122C214yesmaximal, complemented
6#29691C212yesmaximal, complemented
7#29141C211yesmaximal, characteristic, complemented
8#28789C212yesmaximal, complemented
9#28284C211yesmaximal, characteristic, complemented
10#27749C212yesmaximal, complemented
11#27648C212yesmaximal, complemented
12#26236C212yesmaximal
13#24971C212yesmaximal
14#24702C212yesmaximal, complemented
15#22795C211yesmaximal, characteristic
16#21860C212yesmaximal, complemented
17C4×#477C211yesmaximal, characteristic, complemented
18C2×#554—21no—
19C2×#533C2212yes—
20C2×#484C2214yes—
21C2×#477C2211yescharacteristic
22C2×#477C2212yes—
23C2×#477C2212yes—
24C2×#431C2212yes—
25#2868C2211yescharacteristic
26#2857C2212yes—
27#2567—21no—
28#2556—22no—
29#2552—21no—
30#2552C2212yes—
31#2551C2211yescharacteristic
32#2546C2211yescharacteristic
33#2542C2211yescharacteristic
34#2538C2211yescharacteristic
35#2506—21no—
36#2502—22no—
37#2492C2212yes—
38#2452—21no—
39#2440—22no—
40#2440C2214yes—
41#2429C2212yes—
42#2396—22no—
43D4×C4⋊C4—21no—
44D4×C4⋊C4C2212yes—
45C4×C4⋊Q8—21no—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C4×C22⋊Q8C2212yes—
49#2293—21no—
50C2×#384C2212yes—
51C42×Q8C2211yescharacteristic
52C42×D4C2211yescharacteristic
53#2083C2214yes—
54#2076C2214yes—
55#2072C2214yes—
56#2068C2214yes—
57#2057C2214yes—
58#2043C2214yes—
59#2036C2214yes—
60#2036C2214yes—
61#2036C2214yes—
62#2024C2214yes—
63#1990C2214yes—
64#1987C2212yes—
65#1987C2211yescharacteristic
66#1981C2212yes—
67#1956C2214yes—
68#1956C2212yes—
69#1954C2212yes—
70#1947C2211yescharacteristic
71#1947C2212yes—
72#1947C2212yes—
73#1947C2212yes—
74#1942C2214yes—
75#1933C2214yes—
76#1923C2214yes—
77#1921C2212yes—
78#1872C2214yes—
79#1872C2214yes—
80#1863C2212yes—
81#1842C2211yescharacteristic
82#1834C2212yes—
83#1824C2214yes—
84#1803C2212yes—
85#1803C2212yes—
86#1803C2211yescharacteristic
87#1803C2212yes—
88#1803C2212yes—
89#1803C2212yes—
90#1797C2212yes—
91C4×#107C2212yes—
92C8×C4⋊C4C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95#554—48no—
96#533—28no—
97#533—44no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×#125—22no—
107C2×#125—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—24no—
111C2×C22⋊Q8C2312yes—
112C2×C22⋊Q8C2312yes—
113C2×C22⋊Q8C2312yes—
114C2×C4×Q8—22no—
115C2×C4×Q8—21no—
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2311yescharacteristic
120C2×C4×D4—22no—
121C2×C4×D4D411yescharacteristic
122C2×C4×D4D411yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×#117—22no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4C2314yes—
128C22×C42C2312yes—
129#484—48no—
130#477—24no—
131#477—44no—
132#477—44no—
133#431—44no—
134C2×#107C2312yes—
135C2×#107C2314yes—
136C2×#107C2312yes—
137C2×#107C2314yes—
138C2×C4⋊C8C2311yescharacteristic
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2311yescharacteristic
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2312yes—
143C2×#103C2312yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×#103C2314yes—
150C2×C22⋊C8C2314yes—
151C2×C22⋊C8C2314yes—
152C2×C4×C8C2312yes—
153C2×C4×C8C2314yes—
154#400—22no—
155#399—22no—
156#398—22no—
157#398—22no—
158#398—24no—
159#398—24no—
160#396—22no—
161#396—22no—
162#395—24no—
163#395—22no—
164#393—22no—
165#391—21no—
166#391—22no—
167#391—22no—
168#391C2312yes—
169#391C2312yes—
170#390—21no—
171#390—21no—
172#389—21no—
173#389—21no—
174#389—22no—
175#389—22no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389—22no—
182#389—22no—
183#389C2312yes—
184#389C2314yes—
185#389C2312yes—
186#388—22no—
187#388—22no—
188#387—22no—
189#385—22no—
190#385—24no—
191#385—24no—
192#384—21no—
193#384—22no—
194#384—21no—
195#384—21no—
196#384—22no—
197#384C2311yescharacteristic
198#384C2311yescharacteristic
199#384C2312yes—
200#384C2312yes—
201#384C2314yes—
202#380—22no—
203#380C2312yes—
204#380C2314yes—
205C4×C4⋊C4—21no—
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⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C22⋊C4—22no—
220C4×C22⋊C4D412yes—
221C4×C22⋊C4D412yes—
222C4×C22⋊C4C2312yes—
223C43—21no—
224C43C2311yescharacteristic
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#340C2312yes—
233#340C2312yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8C2412yes—
240C22×Q8C2411yescharacteristic
241C22×D4C2×D411yescharacteristic
242C23×C4—24no—
243C23×C4C2×D412yes—
244C23×C4C2×D412yes—
245C23×C4C2412yes—
246C22×C8C2412yes—
247C22×C8C2412yes—
248C22×C8C2414yes—
249C4⋊Q8—24no—
250C4⋊Q8—24no—
251C4⋊Q8—22no—
252C4⋊Q8—22no—
253C4⋊Q8—22no—
254C4⋊Q8—22no—
255C4⋊Q8—42no—
256C4⋊Q8—42no—
257C4⋊Q8—42no—
258C4⋊Q8—42no—
259C4⋊Q8—44no—
260C4⋊Q8—44no—
261C4⋊Q8—82no—
262#125—48no—
263#125—48no—
264C22⋊Q8—24no—
265C22⋊Q8—24no—
266C22⋊Q8—24no—
267C22⋊Q8—416no—
268C22⋊Q8—42no—
269C22⋊Q8—42no—
270C22⋊Q8—42no—
271C22⋊Q8—48no—
272C22⋊Q8—48no—
273C4×Q8—22no—
274C4×Q8—22no—
275C4×Q8—24no—
276C4×Q8—24no—
277C4×Q8—22no—
278C4×Q8—22no—
279C4×Q8—48no—
280C4×Q8—41no—
281C4×Q8—44no—
282C4×Q8#4114yes—
283C4×D4—24no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—48no—
287C4×D4Q1618yes—
288#117—48no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—48no—
320C2×C4⋊C4—84no—
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2414yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C42—21no—
340C2×C42—24no—
341C2×C42—24no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2411yescharacteristic
356C2×C42C2412yes—
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359#107—24no—
360#107—28no—
361#107—24no—
362#107—44no—
363C4⋊C8—22no—
364C4⋊C8—22no—
365C4⋊C8—24no—
366C4⋊C8—24no—
367C4⋊C8—42no—
368#103—24no—
369#103—42no—
370#103—44no—
371#103—44no—
372#103—44no—
373#103—44no—
374#103—44no—
375C22⋊C8—44no—
376C22⋊C8—44no—
377C4×C8—24no—
378C4×C8—44no—
379#95—22no—
380#95—22no—
381#95—21no—
382#95—21no—
383#95—21no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—24no—
391#95—24no—
392#95C2×D412yes—
393#95C2×D412yes—
394#95C2×D412yes—
395C24C22×D412yes—
396C2×Q8—21no—
397C2×Q8—22no—
398C2×Q8—21no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—21no—
406C2×Q8—22no—
407C2×Q8—44no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q8—44no—
411C2×Q8—42no—
412C2×Q8—44no—
413C2×Q8—82no—
414C2×Q8—82no—
415C2×Q8—82no—
416C2×Q8C2×#4112yes—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4C2×Q1614yes—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—42no—
440C22×C4—48no—
441C22×C4—84no—
442C22×C4—84no—
443C22×C4C2×Q1618yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C2511yescharacteristic
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—24no—
457C2×C8—22no—
458C2×C8—24no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—24no—
462C2×C8—22no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—48no—
493C4⋊C4—44no—
494C4⋊C4—48no—
495C4⋊C4—44no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—44no—
499C4⋊C4—81no—
500C4⋊C4—84no—
501C4⋊C4—84no—
502C4⋊C4—82no—
503C4⋊C4—82no—
504C4⋊C4C2×Q1614yes—
505C4⋊C4AGL(1,ℤ/8)14yes—
506C4⋊C4C2×#4112yes—
507C4⋊C4H(5,𝔽2)12yes—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—48no—
514C22⋊C4—82no—
515C22⋊C4C2×Q1618yes—
516C42—24no—
517C42—24no—
518C42—24no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—21no—
524C42—21no—
525C42—41no—
526C42—42no—
527C42—42no—
528C42—44no—
529C42—44no—
530C42—42no—
531C42—42no—
532C42—81no—
533C42C2×Q1614yes—
534C42C2×#4112yes—
535C42H(5,𝔽2)12yes—
536C422−1+412yes—
537C422−1+412yes—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23C22×Q1614yes—
545C23C23×D411yescharacteristic
546Q8—22no—
547Q8—22no—
548Q8—41no—
549Q8—41no—
550Q8—42no—
551Q8—42no—
552Q8—42no—
553Q8—42no—
554Q8—88no—
555Q8—84no—
556Q8—164no—
557D4—28no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—216no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—48no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—168no—
608C2×C4#53612yes—
609C2×C4#53612yes—
610C2×C4#54612yes—
611C2×C4#54612yes—
612C2×C4C22×Q1612yes—
613C2×C4C22×Q1614yes—
614C2×C4C22×Q1612yes—
615C2×C4C22×Q1612yes—
616C2×C4C2×AGL(1,ℤ/8)14yes—
617C2×C4C2×AGL(1,ℤ/8)12yes—
618C2×C4#57712yes—
619C2×C4C22×#4111yescharacteristic
620C2×C4C2×H(5,𝔽2)11yescharacteristic
621C2×C4C2×H(5,𝔽2)11yescharacteristic
622C2×C4C2×2−1+411yescharacteristic
623C2×C4C2×2−1+411yescharacteristic
624C2×C4#58511yescharacteristic
625C8—42no—
626C8—42no—
627C8—44no—
628C22—24no—
629C22—24no—
630C22—24no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22C2×#53611yescharacteristic
636C22C2×#54611yescharacteristic
637C22#348811yescharacteristic
638C22C23×Q1611yescharacteristic
639C22C22×AGL(1,ℤ/8)11yescharacteristic
640C22C2×#57711yescharacteristic
641C22#358611yescharacteristic
642C4—22no—
643C4—24no—
644C4—22no—
645C4—22no—
646C4—24no—
647C4—42no—
648C4—42no—
649C4—41no—
650C4—42no—
651C4—42no—
652C4—82no—
653C4—82no—
654C4—84no—
655C4—81no—
656C4—82no—
657C4—82no—
658C4#323014yes—
659C4#329914yes—
660C4#350711yescharacteristic
661C4#351111yescharacteristic
662C4#354711yescharacteristic
663C4#354811yescharacteristic
664C2—44no—
665C2#3947011yescharacteristic
666C2C2×#323011yescharacteristic
667C2#6088911yescharacteristic
668C2C2×#329911yescharacteristic
669C2#6116111yescharacteristic
670C2#6195211yescharacteristic
671C2#6199911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33198C2semidirect
4. #31744C2semidirect
5. #31122C2semidirect
6. #29691C2semidirect
7. #29141C2semidirect
8. #28789C2semidirect
9. #28284C2semidirect
10. #27749C2semidirect
11. #27648C2semidirect
14. #24702C2semidirect
16. #21860C2semidirect
17. C4×#477C2semidirect

Maximal subgroups

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

subgroupindex
#361802
#349292
#331982
#317442
#311222
#296912
#291412
#287892
#282842
#277492
#276482
#262362
#249712
#247022
#227952
#218602
C4×#4772