Finite Group Databasebeta

← #6593604

Subgroups — #6593604

Order 512.

1729 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic
2C4×D4×Q8C211yesmaximal, characteristic
3#33198C211yesmaximal, characteristic, complemented
4#32957C212yesmaximal, complemented
5#28717C211yesmaximal, characteristic, complemented
6#28284C211yesmaximal, characteristic, complemented
7#27629C212yesmaximal, complemented
8#27524C214yesmaximal, complemented
9#27036C214yesmaximal, complemented
10#26302C212yesmaximal
11#25888C212yesmaximal
12#24702C212yesmaximal, complemented
13#23217C212yesmaximal, complemented
14#22965C212yesmaximal, complemented
15#22795C211yesmaximal, characteristic
16C4×#477C211yesmaximal, characteristic, complemented
17#20734C212yesmaximal, complemented
18C2×D4×Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20C2×#487C2212yes—
21C2×#477C2212yes—
22C2×#477C2211yescharacteristic
23C2×#477C2212yes—
24C2×#431C2212yes—
25#2868C2211yescharacteristic
26#2857C2212yes—
27#2571—21no—
28#2563C2212yes—
29#2538—21no—
30#2497C2214yes—
31#2492—22no—
32#2439—22no—
33#2369C2212yes—
34#2365—21no—
35#2365C2211yescharacteristic
36#2360—22no—
37#2360C2212yes—
38#2355—21no—
39#2354—21no—
40Q8×C4⋊C4C2211yescharacteristic
41D4×C4⋊C4—21no—
42D4×C4⋊C4C2211yescharacteristic
43Q8×C22⋊C4C2212yes—
44C4×C4⋊Q8—21no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×C22⋊Q8—22no—
48C4×C22⋊Q8C2212yes—
49#2285C2212yes—
50C42×Q8C2211yescharacteristic
51C42×D4—21no—
52C42×D4C2211yescharacteristic
53#2076C2214yes—
54#2076C2214yes—
55#2072C2214yes—
56#2068C2214yes—
57#2036C2214yes—
58#2036C2214yes—
59#1990C2212yes—
60#1987C2211yescharacteristic
61#1987C2212yes—
62#1981C2212yes—
63#1954C2212yes—
64#1947C2211yescharacteristic
65#1947C2212yes—
66#1947C2212yes—
67#1947C2212yes—
68#1942C2214yes—
69#1933C2214yes—
70#1928C2214yes—
71#1923C2214yes—
72#1921C2212yes—
73#1921C2214yes—
74#1863C2212yes—
75#1863C2214yes—
76#1863C2214yes—
77#1852C2214yes—
78#1842C2211yescharacteristic
79#1824C2214yes—
80#1803C2212yes—
81#1803C2212yes—
82#1803C2211yescharacteristic
83#1803C2212yes—
84#1803C2212yes—
85#1803C2212yes—
86#1797C2212yes—
87#1787C2214yes—
88#1787C2214yes—
89C4×#107C2212yes—
90C8×C4⋊C4C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92C4×#103C2212yes—
93C4×#103C2214yes—
94C4×#103C2214yes—
95C23×Q8C2312yes—
96D4×Q8—24no—
97D4×Q8—24no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8C2312yes—
107C2×C4×Q8—22no—
108C2×C4×Q8—21no—
109C2×C4×Q8—22no—
110C2×C4×Q8—28no—
111C2×C4×Q8—44no—
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2311yescharacteristic
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4D411yescharacteristic
121C2×C4×D4D411yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×#117—22no—
124C22×C4⋊C4—22no—
125C22×C4⋊C4C2312yes—
126C22×C42—22no—
127C22×C42C2312yes—
128#487—28no—
129#477—24no—
130#477—28no—
131#477—44no—
132#431—44no—
133C2×#107C2312yes—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×#107C2312yes—
137C2×C4⋊C8C2311yescharacteristic
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2311yescharacteristic
141C2×C4⋊C8C2312yes—
142C2×#103C2314yes—
143C2×#103C2312yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2312yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×C22⋊C8C2314yes—
150C2×C22⋊C8C2314yes—
151C2×C4×C8C2312yes—
152C2×C4×C8C2314yes—
153#400—24no—
154#396—24no—
155#393—24no—
156#391—21no—
157#391—22no—
158#391—22no—
159#391—22no—
160#391—24no—
161#391C2312yes—
162#391C2312yes—
163#390—22no—
164#389—21no—
165#389—21no—
166#389—22no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—22no—
171#389—22no—
172#389—22no—
173#389C2312yes—
174#389C2314yes—
175#389C2312yes—
176#388—24no—
177#387—22no—
178#387—24no—
179#387—22no—
180#387—22no—
181#385—22no—
182#385—22no—
183#385—22no—
184#384—21no—
185#384—21no—
186#384—21no—
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#384C2312yes—
190#384C2312yes—
191#382—22no—
192#382—22no—
193#380—22no—
194#380—22no—
195#380—22no—
196#380C2314yes—
197#380C2314yes—
198#380C2312yes—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2314yes—
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—
224C43—21no—
225C43C2311yescharacteristic
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#340C2312yes—
233#340C2312yes—
234#340C2312yes—
235#336C2314yes—
236#336C2312yes—
237C22×Q8—24no—
238C22×Q8—24no—
239C22×Q8—22no—
240C22×Q8—21no—
241C22×Q8—22no—
242C22×Q8—24no—
243C22×Q8—24no—
244C22×Q8—42no—
245C22×Q8—42no—
246C22×Q8C2412yes—
247C22×Q8C2411yescharacteristic
248C22×D4C2×D411yescharacteristic
249C23×C4—22no—
250C23×C4—22no—
251C23×C4C2×D412yes—
252C23×C4C2×D412yes—
253C23×C4C2412yes—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C22×C8C2414yes—
257C4⋊Q8—28no—
258C4⋊Q8—22no—
259C4⋊Q8—24no—
260C4⋊Q8—22no—
261C4⋊Q8—24no—
262C4⋊Q8—22no—
263C4⋊Q8—82no—
264C4⋊Q8—82no—
265C4⋊Q8#4114yes—
266C22⋊Q8—216no—
267C22⋊Q8—24no—
268C22⋊Q8—24no—
269C22⋊Q8—84no—
270C4×Q8—22no—
271C4×Q8—22no—
272C4×Q8—24no—
273C4×Q8—22no—
274C4×Q8—24no—
275C4×Q8—24no—
276C4×Q8—24no—
277C4×Q8—48no—
278C4×Q8—42no—
279C4×Q8—42no—
280C4×Q8—41no—
281C4×Q8—48no—
282C4×Q8—44no—
283C4×Q8#4114yes—
284C4×Q8#4114yes—
285C4×D4—28no—
286C4×D4—22no—
287C4×D4—22no—
288C4×D4—42no—
289C4×D4—44no—
290C4×D4Q1618yes—
291#117—84no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—42no—
317C2×C4⋊C4—48no—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2414yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2411yescharacteristic
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4C2×D412yes—
333C2×C22⋊C4C2×D412yes—
334C2×C22⋊C4C2×D412yes—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42—21no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42—24no—
347C2×C42—21no—
348C2×C42—48no—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2412yes—
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359#107—28no—
360#107—24no—
361#107—42no—
362#107—44no—
363C4⋊C8—24no—
364C4⋊C8—22no—
365C4⋊C8—22no—
366C4⋊C8—24no—
367C4⋊C8—42no—
368#103—28no—
369#103—24no—
370#103—28no—
371#103—28no—
372#103—24no—
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—22no—
382#95—21no—
383#95—21no—
384#95—21no—
385#95—22no—
386#95—22no—
387#95—24no—
388#95—24no—
389#95—22no—
390#95—22no—
391#95—24no—
392#95—22no—
393#95—24no—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397C24C22×D412yes—
398C2×Q8—28no—
399C2×Q8—21no—
400C2×Q8—22no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—42no—
410C2×Q8—42no—
411C2×Q8—48no—
412C2×Q8—48no—
413C2×Q8—48no—
414C2×Q8—48no—
415C2×Q8—44no—
416C2×Q8—42no—
417C2×Q8—82no—
418C2×Q8—82no—
419C2×Q8C2×#4114yes—
420C2×Q8C2×#4112yes—
421C2×Q8C2×#4112yes—
422C2×Q82−1+412yes—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4C2×Q1614yes—
426C22×C4—21no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—28no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—82no—
452C22×C4C2×Q1618yes—
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C2511yescharacteristic
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8—22no—
468C2×C8—24no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—24no—
472C4⋊C4—28no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—21no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—81no—
505C4⋊C4—82no—
506C4⋊C4—84no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4C2×Q1614yes—
510C4⋊C4C2×#4112yes—
511C4⋊C4C2×#4112yes—
512C22⋊C4—28no—
513C22⋊C4—24no—
514C22⋊C4—42no—
515C22⋊C4—44no—
516C22⋊C4—84no—
517C22⋊C4—84no—
518C22⋊C4C2×Q1618yes—
519C42—24no—
520C42—24no—
521C42—24no—
522C42—24no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—21no—
528C42—21no—
529C42—41no—
530C42—44no—
531C42—44no—
532C42—42no—
533C42—44no—
534C42—42no—
535C42—44no—
536C42—42no—
537C42—81no—
538C42—84no—
539C42C2×Q1614yes—
540C42C2×#4112yes—
541C42C2×#4112yes—
542C42C2×#4112yes—
543C42H(5,𝔽2)12yes—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23C22×Q1614yes—
551C23C23×D411yescharacteristic
552Q8—22no—
553Q8—22no—
554Q8—22no—
555Q8—24no—
556Q8—24no—
557Q8—416no—
558Q8—42no—
559Q8—42no—
560Q8—84no—
561Q8—164no—
562Q8#55614yes—
563D4—28no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—216no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—416no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—84no—
615C2×C4—88no—
616C2×C4#52512yes—
617C2×C4#52512yes—
618C2×C4#54612yes—
619C2×C4#54612yes—
620C2×C4C22×Q1612yes—
621C2×C4C22×Q1614yes—
622C2×C4C22×Q1612yes—
623C2×C4C22×Q1612yes—
624C2×C4#57512yes—
625C2×C4#57712yes—
626C2×C4C22×#4111yescharacteristic
627C2×C4C22×#4111yescharacteristic
628C2×C4C22×#4111yescharacteristic
629C2×C4C2×H(5,𝔽2)11yescharacteristic
630C2×C4C2×2−1+411yescharacteristic
631C2×C4#58511yescharacteristic
632C8—42no—
633C8—44no—
634C8—42no—
635C22—24no—
636C22—24no—
637C22—24no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22C2×#52511yescharacteristic
643C22C2×#54611yescharacteristic
644C22#348911yescharacteristic
645C22C23×Q1611yescharacteristic
646C22C2×#57511yescharacteristic
647C22C2×#57711yescharacteristic
648C22#358511yescharacteristic
649C4—24no—
650C4—22no—
651C4—22no—
652C4—24no—
653C4—22no—
654C4—42no—
655C4—44no—
656C4—42no—
657C4—41no—
658C4—42no—
659C4—42no—
660C4—84no—
661C4—81no—
662C4—82no—
663C4—82no—
664C4—82no—
665C4#310414yes—
666C4#329914yes—
667C4C2×#55611yescharacteristic
668C4#347511yescharacteristic
669C4#350611yescharacteristic
670C4#352011yescharacteristic
671C2—44no—
672C2#3616111yescharacteristic
673C2C2×#310411yescharacteristic
674C2#6059311yescharacteristic
675C2C2×#329911yescharacteristic
676C2#6116611yescharacteristic
677C2#6210011yescharacteristic
678C2#6214511yescharacteristic
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. #32957C2semidirect
5. #28717C2semidirect
6. #28284C2semidirect
7. #27629C2semidirect
8. #27524C2semidirect
9. #27036C2semidirect
12. #24702C2semidirect
13. #23217C2semidirect
14. #22965C2semidirect
16. C4×#477C2semidirect
17. #20734C2semidirect

Maximal subgroups

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

subgroupindex
#361942
C4×D4×Q82
#331982
#329572
#287172
#282842
#276292
#275242
#270362
#263022
#258882
#247022
#232172
#229652
#227952
C4×#4772
#207342