Finite Group Databasebeta

← #6848666

Subgroups — #6848666

Order 512.

1541 conjugacy classes of subgroups.

The subgroup lattice has 665 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic
2#34929C211yesmaximal, characteristic
3#33198C211yesmaximal, characteristic, complemented
4#32200C214yesmaximal, complemented
5#31492C214yesmaximal, complemented
6#29691C212yesmaximal, complemented
7#28717C211yesmaximal, characteristic, complemented
8#28598C212yesmaximal
9#28284C211yesmaximal, characteristic, complemented
10#27629C212yesmaximal, complemented
11#26446C212yesmaximal, complemented
12#25340C212yesmaximal
13#25278C212yesmaximal, complemented
14#24837C212yesmaximal, complemented
15D4×C4⋊C8C211yesmaximal, characteristic
16#21885C212yesmaximal, complemented
17C4×#477C211yesmaximal, characteristic, complemented
18C2×#533C2212yes—
19C2×#487C2212yes—
20C2×#477C2211yescharacteristic
21C2×#477C2212yes—
22C2×#477C2212yes—
23C2×C8×D4C2212yes—
24C22×C4⋊C8C2212yes—
25#2868C2211yescharacteristic
26#2857C2212yes—
27#2571—21no—
28#2563C2212yes—
29#2556—22no—
30#2551C2211yescharacteristic
31#2546C2211yescharacteristic
32#2542C2211yescharacteristic
33#2538—21no—
34#2538C2211yescharacteristic
35#2506—21no—
36#2497C2214yes—
37#2492—22no—
38#2492C2212yes—
39#2452—21no—
40#2439—22no—
41#2429C2212yes—
42#2396—22no—
43#2369C2212yes—
44#2355—21no—
45#2354—21no—
46C4×C4⋊Q8—21no—
47C4×C4⋊Q8C2211yescharacteristic
48C4×C4⋊Q8C2211yescharacteristic
49C4×C22⋊Q8C2212yes—
50#2293—21no—
51#2285C2212yes—
52C42×Q8C2211yescharacteristic
53C42×D4C2211yescharacteristic
54#2087C2214yes—
55#2076C2214yes—
56#2072C2214yes—
57#2068C2214yes—
58#2068C2214yes—
59#2057C2214yes—
60#2043C2214yes—
61#2036C2214yes—
62#2036C2214yes—
63#2036C2214yes—
64#1990C2212yes—
65#1987C2211yescharacteristic
66#1987C2214yes—
67#1987C2212yes—
68#1981C2214yes—
69#1981C2212yes—
70#1947C2212yes—
71#1947C2212yes—
72#1947C2211yescharacteristic
73#1947C2212yes—
74#1939C2212yes—
75#1928C2214yes—
76#1921C2214yes—
77#1921C2212yes—
78#1892C2214yes—
79#1892C2214yes—
80#1892C2214yes—
81#1863C2212yes—
82#1850C2214yes—
83#1842C2211yescharacteristic
84#1808C2214yes—
85#1803C2212yes—
86#1803C2212yes—
87#1803C2212yes—
88#1803C2211yescharacteristic
89#1803C2212yes—
90#1803C2212yes—
91#1797C2212yes—
92C4×#107C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C23×C8C2314yes—
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×C22⋊Q8—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8C2312yes—
110C2×C22⋊Q8C2312yes—
111C2×C22⋊Q8C2312yes—
112C2×C4×Q8—22no—
113C2×C4×Q8—21no—
114C2×C4×Q8C2311yescharacteristic
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2311yescharacteristic
118C2×C4×D4D411yescharacteristic
119C2×C4×D4D411yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×#117—22no—
122C2×#117—22no—
123C22×C42C2312yes—
124#487—44no—
125#477—24no—
126#477—44no—
127#477—44no—
128C8×D4—44no—
129C2×#107C2312yes—
130C2×#107C2312yes—
131C2×#107C2314yes—
132C2×#107C2314yes—
133C2×C4⋊C8—44no—
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2311yescharacteristic
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2314yes—
140C2×#103C2312yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2312yes—
147C2×C22⋊C8C2314yes—
148C2×C4×C8C2312yes—
149#400—24no—
150#400—22no—
151#399—22no—
152#398—22no—
153#398—22no—
154#396—22no—
155#396—22no—
156#396—24no—
157#395—22no—
158#393—24no—
159#393—22no—
160#391—21no—
161#391—22no—
162#391—22no—
163#391—24no—
164#391C2312yes—
165#391C2312yes—
166#390—22no—
167#390—21no—
168#390—21no—
169#389—21no—
170#389—21no—
171#389—22no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—22no—
178#389—22no—
179#389C2312yes—
180#389C2314yes—
181#389C2312yes—
182#388—24no—
183#388—22no—
184#388—22no—
185#387—22no—
186#387—24no—
187#385—22no—
188#384—21no—
189#384—21no—
190#384—21no—
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#384C2312yes—
194#384C2312yes—
195#382—22no—
196#382—22no—
197#380—22no—
198#380—22no—
199#380C2314yes—
200#380C2314yes—
201#380C2312yes—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2314yes—
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×C4C2×D412yes—
243C23×C4C2×D412yes—
244C23×C4C2412yes—
245C22×C8—44no—
246C22×C8—44no—
247C22×C8—44no—
248C22×C8C2412yes—
249C22×C8C2414yes—
250C22×C8C2412yes—
251C4⋊Q8—24no—
252C4⋊Q8—24no—
253C4⋊Q8—22no—
254C4⋊Q8—22no—
255C4⋊Q8—24no—
256C4⋊Q8—22no—
257C4⋊Q8—24no—
258C4⋊Q8—22no—
259C4⋊Q8—42no—
260C4⋊Q8—42no—
261C4⋊Q8—44no—
262C4⋊Q8—82no—
263C4⋊Q8—82no—
264#125—48no—
265C22⋊Q8—24no—
266C22⋊Q8—24no—
267C22⋊Q8—24no—
268C22⋊Q8—42no—
269C22⋊Q8—42no—
270C22⋊Q8—42no—
271C22⋊Q8—48no—
272C22⋊Q8—84no—
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—22no—
284C4×D4—22no—
285C4×D4—42no—
286C4×D4Q1618yes—
287#117—48no—
288#117—84no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2414yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4C2×D412yes—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C42—21no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—21no—
339C2×C42—21no—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42—24no—
343C2×C42—21no—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2411yescharacteristic
351C2×C42C2412yes—
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354#107—24no—
355#107—24no—
356#107—44no—
357#107—44no—
358C4⋊C8—22no—
359C4⋊C8—22no—
360C4⋊C8—24no—
361C4⋊C8—24no—
362C4⋊C8—44no—
363C4⋊C8—42no—
364#103—24no—
365#103—44no—
366#103—42no—
367#103—44no—
368#103—44no—
369#103—44no—
370#103—44no—
371C22⋊C8—44no—
372C4×C8—24no—
373#95—22no—
374#95—22no—
375#95—21no—
376#95—21no—
377#95—21no—
378#95—24no—
379#95—24no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—24no—
386#95—22no—
387#95—24no—
388#95C2×D412yes—
389#95C2×D412yes—
390#95C2×D412yes—
391C24C22×D412yes—
392C2×Q8—21no—
393C2×Q8—22no—
394C2×Q8—21no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—21no—
402C2×Q8—22no—
403C2×Q8—44no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—42no—
409C2×Q8—82no—
410C2×Q8—82no—
411C2×Q8—82no—
412C2×Q8C2×#4112yes—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4C2×Q1614yes—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4C2×Q1618yes—
437C22×C4C22×D412yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D412yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D412yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C2511yescharacteristic
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—24no—
455C2×C8—22no—
456C2×C8—88no—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—81no—
493C4⋊C4—84no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4C2×Q1614yes—
497C4⋊C4C2×#4112yes—
498C4⋊C42−1+412yes—
499C22⋊C4—24no—
500C22⋊C4—42no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—82no—
505C22⋊C4—84no—
506C22⋊C4—84no—
507C22⋊C4C2×Q1618yes—
508C42—24no—
509C42—24no—
510C42—24no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—21no—
516C42—21no—
517C42—41no—
518C42—42no—
519C42—42no—
520C42—44no—
521C42—44no—
522C42—42no—
523C42—42no—
524C42—81no—
525C42—84no—
526C42C2×Q1614yes—
527C42C2×#4112yes—
528C42H(5,𝔽2)12yes—
529C42H(5,𝔽2)12yes—
530C422−1+412yes—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23C22×Q1614yes—
538C23C23×D411yescharacteristic
539Q8—22no—
540Q8—22no—
541Q8—41no—
542Q8—41no—
543Q8—42no—
544Q8—42no—
545Q8—42no—
546Q8—42no—
547Q8—84no—
548Q8—88no—
549Q8—164no—
550D4—28no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—216no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—84no—
601C2×C4#53712yes—
602C2×C4#53712yes—
603C2×C4#54612yes—
604C2×C4#54612yes—
605C2×C4C22×Q1612yes—
606C2×C4C22×Q1614yes—
607C2×C4C22×Q1612yes—
608C2×C4C22×Q1612yes—
609C2×C4#57512yes—
610C2×C4#57712yes—
611C2×C4C22×#4111yescharacteristic
612C2×C4C2×H(5,𝔽2)11yescharacteristic
613C2×C4C2×H(5,𝔽2)11yescharacteristic
614C2×C4C2×2−1+411yescharacteristic
615C2×C4C2×2−1+411yescharacteristic
616C2×C4#58511yescharacteristic
617C8—42no—
618C8—44no—
619C8—42no—
620C22—24no—
621C22—24no—
622C22—24no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22C2×#53711yescharacteristic
628C22C2×#54611yescharacteristic
629C22#348911yescharacteristic
630C22C23×Q1611yescharacteristic
631C22C2×#57511yescharacteristic
632C22C2×#57711yescharacteristic
633C22#358511yescharacteristic
634C4—22no—
635C4—24no—
636C4—22no—
637C4—22no—
638C4—42no—
639C4—42no—
640C4—41no—
641C4—42no—
642C4—42no—
643C4—42no—
644C4—82no—
645C4—82no—
646C4—82no—
647C4—82no—
648C4—81no—
649C4—84no—
650C4#324914yes—
651C4#329914yes—
652C4#350711yescharacteristic
653C4#351111yescharacteristic
654C4#354611yescharacteristic
655C4#355011yescharacteristic
656C2—44no—
657C2#3946811yescharacteristic
658C2C2×#324911yescharacteristic
659C2#6092411yescharacteristic
660C2C2×#329911yescharacteristic
661C2#6116611yescharacteristic
662C2#6210011yescharacteristic
663C2#6213411yescharacteristic
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. #32200C2semidirect
5. #31492C2semidirect
6. #29691C2semidirect
7. #28717C2semidirect
9. #28284C2semidirect
10. #27629C2semidirect
11. #26446C2semidirect
13. #25278C2semidirect
14. #24837C2semidirect
16. #21885C2semidirect
17. C4×#477C2semidirect

Maximal subgroups

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

subgroupindex
#361942
#349292
#331982
#322002
#314922
#296912
#287172
#285982
#282842
#276292
#264462
#253402
#252782
#248372
D4×C4⋊C82
#218852
C4×#4772