Finite Group Databasebeta

← #6615694

Subgroups — #6615694

Order 512.

1607 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
2C4×#554C211yesmaximal, characteristic
3#33198C211yesmaximal, characteristic, complemented
4#33109C212yesmaximal, complemented
5#29208C211yesmaximal, characteristic, complemented
6#28284C211yesmaximal, characteristic, complemented
7#27629C212yesmaximal, complemented
8#27332C214yesmaximal, complemented
9#26790C214yesmaximal, complemented
10#25016C212yesmaximal
11#24652C212yesmaximal, complemented
12#24122C212yesmaximal
13#23063C212yesmaximal, complemented
14#22808C212yesmaximal, complemented
15#22795C211yesmaximal, characteristic
16C4×#462C211yesmaximal, characteristic, complemented
17#21358C212yesmaximal, complemented
18C2×#554C2211yescharacteristic
19C2×#484C2214yes—
20C2×#477C2212yes—
21C2×#477C2212yes—
22C2×#462C2211yescharacteristic
23C2×C4×Q16C2212yes—
24C2×#431C2212yes—
25#2868C2211yescharacteristic
26#2857C2212yes—
27#2571—21no—
28#2563—21no—
29#2538C2212yes—
30#2497—22no—
31#2492C2214yes—
32#2439—22no—
33#2370—21no—
34#2369—21no—
35#2369C2211yescharacteristic
36#2368—22no—
37#2360C2212yes—
38#2355C2212yes—
39#2355C2211yescharacteristic
40#2354—21no—
41#2349—21no—
42D4×C4⋊C4C2211yescharacteristic
43#2345C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×#125—21no—
47C4×C22⋊Q8—22no—
48C4×C22⋊Q8C2212yes—
49#2285C2212yes—
50C42×Q8C2211yescharacteristic
51C42×D4—21no—
52C42×D4C2211yescharacteristic
53C2×C4×C4⋊C4C2212yes—
54#2076C2214yes—
55#2068C2214yes—
56#2057C2214yes—
57#2036C2214yes—
58#2036C2214yes—
59#2001C2214yes—
60#1990C2212yes—
61#1987C2212yes—
62#1981C2212yes—
63#1981C2211yescharacteristic
64#1969C2214yes—
65#1954C2212yes—
66#1947C2212yes—
67#1947C2212yes—
68#1947C2212yes—
69#1947C2212yes—
70#1947C2211yescharacteristic
71#1947C2212yes—
72#1933C2214yes—
73#1928C2214yes—
74#1928C2214yes—
75#1928C2214yes—
76#1928C2212yes—
77#1923C2214yes—
78#1921C2214yes—
79#1878C2212yes—
80#1872C2214yes—
81#1852C2214yes—
82#1852C2214yes—
83#1845C2212yes—
84#1842C2211yescharacteristic
85#1803C2212yes—
86#1803C2212yes—
87#1803C2212yes—
88#1803C2211yescharacteristic
89#1797C2212yes—
90C4×#107C2214yes—
91C4×#107C2214yes—
92C8×C4⋊C4C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C22×Q16C2312yes—
96#554—24no—
97#554—24no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×#125—21no—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8C2312yes—
106C2×C22⋊Q8C2314yes—
107C2×C4×Q8—21no—
108C2×C4×Q8C2312yes—
109C2×C4×Q8C2311yescharacteristic
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2311yescharacteristic
112C2×C4×D4—21no—
113C2×C4×D4—21no—
114C2×C4×D4D411yescharacteristic
115C2×C4×D4D411yescharacteristic
116C2×C4×D4C2311yescharacteristic
117C2×#117—22no—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C42—22no—
121C22×C42C2312yes—
122#484—48no—
123#477—28no—
124#477—44no—
125#462—24no—
126C4×Q16—28no—
127#431—28no—
128C2×#107C2314yes—
129C2×#107C2314yes—
130C2×#107C2312yes—
131C2×#107C2314yes—
132C2×#107C2314yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2311yescharacteristic
138C2×#103C2314yes—
139C2×#103C2312yes—
140C2×#103C2312yes—
141C2×#103C2312yes—
142C2×#103C2312yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×C22⋊C8C2314yes—
146C2×C22⋊C8C2314yes—
147C2×C4×C8C2312yes—
148C2×C4×C8C2314yes—
149#400—24no—
150#396—24no—
151#393—24no—
152#391—21no—
153#391—22no—
154#391—24no—
155#391C2312yes—
156#391C2314yes—
157#391C2312yes—
158#390—22no—
159#389—21no—
160#389—21no—
161#389—22no—
162#389—21no—
163#389—21no—
164#389—21no—
165#389—22no—
166#389—22no—
167#389—22no—
168#389C2312yes—
169#389C2312yes—
170#388—24no—
171#387—22no—
172#387—22no—
173#387—22no—
174#387—22no—
175#387—22no—
176#387—24no—
177#385—22no—
178#385—22no—
179#385—22no—
180#384—21no—
181#384—21no—
182#384C2311yescharacteristic
183#384C2311yescharacteristic
184#384C2312yes—
185#384C2312yes—
186#382—21no—
187#382—22no—
188#382—22no—
189#380—22no—
190#380—22no—
191#380—22no—
192#380—22no—
193#380C2312yes—
194#380C2314yes—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2314yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C22⋊C4—22no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4D412yes—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4C2312yes—
224C2×#95—22no—
225C43—21no—
226C43C2311yescharacteristic
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#340C2314yes—
233#340C2312yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8C2412yes—
239C22×Q8C2411yescharacteristic
240C22×D4C2×D411yescharacteristic
241C23×C4—22no—
242C23×C4—22no—
243C23×C4C2×D412yes—
244C23×C4C2×D412yes—
245C23×C4C2412yes—
246C2×Q16—24no—
247C2×Q16—24no—
248C2×Q16—42no—
249C22×C8C2412yes—
250C22×C8C2414yes—
251C22×C8C2412yes—
252C4⋊Q8—22no—
253C4⋊Q8—22no—
254C4⋊Q8—22no—
255C4⋊Q8—42no—
256C4⋊Q8—42no—
257C4⋊Q8—44no—
258C4⋊Q8—82no—
259C4⋊Q8#4114yes—
260#125—28no—
261C22⋊Q8—216no—
262C22⋊Q8—24no—
263C22⋊Q8—24no—
264C22⋊Q8—44no—
265C22⋊Q8—44no—
266C4×Q8—22no—
267C4×Q8—22no—
268C4×Q8—24no—
269C4×Q8—22no—
270C4×Q8—24no—
271C4×Q8—22no—
272C4×Q8—41no—
273C4×Q8—82no—
274C4×Q8#4114yes—
275C4×D4—28no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4—42no—
279C4×D4—44no—
280C4×D4Q1618yescomplemented
281#117—84no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—42no—
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2414yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4C2×D412yes—
323C2×C22⋊C4C2×D412yes—
324C2×C22⋊C4C2×D412yes—
325C2×C42—21no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—21no—
329C2×C42—22no—
330C2×C42—21no—
331C2×C42—21no—
332C2×C42—22no—
333C2×C42—21no—
334C2×C42—24no—
335C2×C42—21no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—84no—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2411yescharacteristic
346C2×C42C2412yes—
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349#107—24no—
350#107—28no—
351#107—28no—
352#107—44no—
353#107—44no—
354C4⋊C8—24no—
355C4⋊C8—24no—
356C4⋊C8—22no—
357C4⋊C8—42no—
358C4⋊C8#4114yes—
359#103—24no—
360#103—24no—
361#103—24no—
362#103—28no—
363#103—24no—
364#103—44no—
365#103—44no—
366C22⋊C8—28no—
367C22⋊C8—44no—
368C4×C8—28no—
369C4×C8—24no—
370#95—21no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—21no—
378#95—21no—
379#95—24no—
380#95—24no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—24no—
386#95—24no—
387#95C2×D412yes—
388#95C2×D412yes—
389#95C2×D412yes—
390C24C22×D412yes—
391C2×Q8—24no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—21no—
398C2×Q8—22no—
399C2×Q8—21no—
400C2×Q8—22no—
401C2×Q8—21no—
402C2×Q8—22no—
403C2×Q8—44no—
404C2×Q8—81no—
405C2×Q8—81no—
406C2×Q8—82no—
407C2×Q8—82no—
408C2×Q8C2×#4112yes—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4C2×Q1614yes—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—28no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—82no—
436C22×C4—82no—
437C22×C4—82no—
438C22×C4C2×Q1618yes—
439C22×C4C22×D412yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D412yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D412yes—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C2511yescharacteristic
449Q16—48no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—24no—
453C2×C8—22no—
454C2×C8—24no—
455C2×C8—22no—
456C2×C8—24no—
457C2×C8—22no—
458C2×C8C2×#4114yes—
459C4⋊C4—28no—
460C4⋊C4—28no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—21no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—82no—
489C4⋊C4—81no—
490C4⋊C4—82no—
491C4⋊C4—84no—
492C4⋊C4—82no—
493C4⋊C4—82no—
494C4⋊C4C2×Q1614yes—
495C4⋊C4C2×#4112yes—
496C4⋊C4C2×#4114yes—
497C4⋊C4H(5,𝔽2)12yes—
498C22⋊C4—28no—
499C22⋊C4—24no—
500C22⋊C4—42no—
501C22⋊C4—44no—
502C22⋊C4—84no—
503C22⋊C4—84no—
504C22⋊C4C2×Q1618yes—
505C42—24no—
506C42—24no—
507C42—24no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—21no—
512C42—22no—
513C42—21no—
514C42—41no—
515C42—42no—
516C42—44no—
517C42—42no—
518C42—42no—
519C42—42no—
520C42—44no—
521C42—81no—
522C42—82no—
523C42—82no—
524C42—84no—
525C42C2×Q1614yes—
526C42C2×#4112yes—
527C42C2×#4112yes—
528C42C2×#4112yes—
529C422−1+412yes—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23C22×Q1614yes—
537C23C23×D411yescharacteristic
538Q8—22no—
539Q8—22no—
540Q8—41no—
541Q8—41no—
542Q8—42no—
543Q8—42no—
544Q8—42no—
545Q8—42no—
546Q8—84no—
547Q8—164no—
548D4—28no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—216no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—41no—
572C2×C4—42no—
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—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—84no—
599C2×C4—88no—
600C2×C4—88no—
601C2×C4#52512yes—
602C2×C4#52512yes—
603C2×C4#54712yes—
604C2×C4#54712yes—
605C2×C4C22×Q1612yes—
606C2×C4C22×Q1614yes—
607C2×C4C22×Q1612yes—
608C2×C4C22×Q1612yes—
609C2×C4#57512yes—
610C2×C4#57612yes—
611C2×C4C22×#4111yescharacteristic
612C2×C4C22×#4111yescharacteristic
613C2×C4C22×#4111yescharacteristic
614C2×C4C2×H(5,𝔽2)11yescharacteristic
615C2×C4C2×2−1+411yescharacteristic
616C2×C4#58511yescharacteristic
617C8—24no—
618C8—44no—
619C8—42no—
620C22—24no—
621C22—24no—
622C22—24no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22C2×#52511yescharacteristic
628C22C2×#54711yescharacteristic
629C22#348811yescharacteristic
630C22C23×Q1611yescharacteristic
631C22C2×#57511yescharacteristic
632C22C2×#57611yescharacteristic
633C22#358611yescharacteristic
634C4—22no—
635C4—24no—
636C4—22no—
637C4—22no—
638C4—41no—
639C4—42no—
640C4—42no—
641C4—42no—
642C4—42no—
643C4—42no—
644C4—44no—
645C4—82no—
646C4—84no—
647C4—82no—
648C4—81no—
649C4—82no—
650C4#310414yes—
651C4#333814yes—
652C4#347511yescharacteristic
653C4#348311yescharacteristic
654C4#351111yescharacteristic
655C4#351811yescharacteristic
656C2—44no—
657C2#3641111yescharacteristic
658C2C2×#310411yescharacteristic
659C2#6059611yescharacteristic
660C2C2×#333811yescharacteristic
661C2#6125711yescharacteristic
662C2#6195211yescharacteristic
663C2#6199511yescharacteristic
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. #33109C2semidirect
5. #29208C2semidirect
6. #28284C2semidirect
7. #27629C2semidirect
8. #27332C2semidirect
9. #26790C2semidirect
11. #24652C2semidirect
13. #23063C2semidirect
14. #22808C2semidirect
16. C4×#462C2semidirect
17. #21358C2semidirect
280. C4×D4Q16semidirect

Maximal subgroups

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

subgroupindex
#361942
C4×#5542
#331982
#331092
#292082
#282842
#276292
#273322
#267902
#250162
#246522
#241222
#230632
#228082
#227952
C4×#4622
#213582