Finite Group Databasebeta

← #6591967

Subgroups — #6591967

Order 512.

1767 conjugacy classes of subgroups.

The subgroup lattice has 685 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#32954C212yesmaximal, complemented
5#29141C211yesmaximal, characteristic, complemented
6#28283C211yesmaximal, characteristic, complemented
7#27682C212yesmaximal, complemented
8#26531C212yesmaximal, complemented
9#26430C214yesmaximal, complemented
10#26113C212yesmaximal, complemented
11#25979C214yesmaximal, complemented
12#24845C212yesmaximal, complemented
13#22715C212yesmaximal
14#22650C211yesmaximal, characteristic
15#22430C212yesmaximal
16C4×#475C211yesmaximal, characteristic, complemented
17#20647C212yesmaximal, complemented
18C2×D4×Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20C2×#477C2212yes—
21C2×#475C2211yescharacteristic
22C2×#475C2212yes—
23C2×#436C2212yes—
24C2×#423C2212yes—
25C2×#419C2214yes—
26#2868C2211yescharacteristic
27#2857C2212yes—
28#2571C2212yes—
29#2563—21no—
30#2538—21no—
31#2497—22no—
32#2492—22no—
33#2439C2214yes—
34#2369—21no—
35#2365—21no—
36#2365C2211yescharacteristic
37#2360—22no—
38#2360C2212yes—
39#2355—21no—
40#2354C2212yes—
41Q8×C4⋊C4C2211yescharacteristic
42D4×C4⋊C4—21no—
43D4×C4⋊C4C2211yescharacteristic
44Q8×C22⋊C4C2212yes—
45C4×C4⋊Q8—21no—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C4×C22⋊Q8—22no—
49C4×C22⋊Q8C2212yes—
50#2285C2212yes—
51C42×Q8C2211yescharacteristic
52C42×D4—21no—
53C42×D4C2211yescharacteristic
54#2101C2214yes—
55#2097C2214yes—
56#2088C2214yes—
57#2068C2214yes—
58#2066C2214yes—
59#2024C2214yes—
60#1988C2211yescharacteristic
61#1987C2212yes—
62#1981C2212yes—
63#1956C2212yes—
64#1948C2212yes—
65#1948C2211yescharacteristic
66#1947C2212yes—
67#1947C2212yes—
68#1939C2212yes—
69#1919C2212yes—
70#1894C2214yes—
71#1873C2214yes—
72#1865C2214yes—
73#1865C2214yes—
74#1865C2212yes—
75#1854C2214yes—
76#1853C2214yes—
77#1851C2214yes—
78#1842C2211yescharacteristic
79#1834C2212yes—
80#1809C2214yes—
81#1805C2214yes—
82#1805C2214yes—
83#1803C2212yes—
84#1803C2212yes—
85#1803C2212yes—
86#1803C2212yes—
87#1803C2211yescharacteristic
88#1803C2212yes—
89#1797C2212yes—
90#1787C2214yes—
91C4×#106C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×#103C2212yes—
94#1736C2214yes—
95C23×Q8C2312yes—
96C22×M16C2314yes—
97D4×Q8—24no—
98D4×Q8Q818yescomplemented
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8C2312yes—
108C2×C4×Q8—22no—
109C2×C4×Q8—21no—
110C2×C4×Q8—22no—
111C2×C4×Q8—28no—
112C2×C4×Q8—44no—
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2311yescharacteristic
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2311yescharacteristic
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4D411yescharacteristic
122C2×C4×D4D411yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×#117C2314yes—
125C22×C4⋊C4—22no—
126C22×C4⋊C4C2312yes—
127C22×C42—22no—
128C22×C42C2312yes—
129#477—44no—
130#475—24no—
131#475—44no—
132#436—28no—
133C2×#107C2314yes—
134C2×#106C2312yes—
135C2×#106C2314yes—
136C2×#106C2312yes—
137#423—44no—
138C2×C4⋊C8C2311yescharacteristic
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2311yescharacteristic
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2314yes—
144#419—48no—
145C2×#103C2312yes—
146C2×#103C2314yes—
147C2×#103C2312yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×C22⋊C8C2314yes—
153C2×C4×C8C2312yes—
154#400—24no—
155#396—24no—
156#393—24no—
157#391—21no—
158#391—22no—
159#391—22no—
160#391—22no—
161#391—24no—
162#391C2312yes—
163#391C2312yes—
164#390—22no—
165#389—21no—
166#389—21no—
167#389—22no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—22no—
172#389—22no—
173#389—22no—
174#389C2312yes—
175#389C2314yes—
176#389C2312yes—
177#388—24no—
178#387—22no—
179#387—24no—
180#387—22no—
181#387—22no—
182#385—22no—
183#385—22no—
184#385—22no—
185#384—21no—
186#384—21no—
187#384—21no—
188#384C2311yescharacteristic
189#384C2311yescharacteristic
190#384C2312yes—
191#384C2312yes—
192#382—22no—
193#382C2314yes—
194#380—22no—
195#380—22no—
196#380—22no—
197#380—22no—
198#380C2312yes—
199#380C2314yes—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
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⋊C4—22no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4D412yes—
224C4×C22⋊C4C2312yes—
225C43—21no—
226C43C2311yescharacteristic
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2312yes—
234#340C2312yes—
235#340C2312yes—
236#336C2314yes—
237#336C2312yes—
238C22×Q8—22no—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8—24no—
242C22×Q8—24no—
243C22×Q8—24no—
244C22×Q8—42no—
245C22×Q8—42no—
246C22×Q8C2×Q818yes—
247C22×Q8C2412yes—
248C22×Q8C2411yescharacteristic
249C22×D4C2×D411yescharacteristic
250C23×C4—22no—
251C23×C4—22no—
252C23×C4C2×D412yes—
253C23×C4C2×D412yes—
254C23×C4C2412yes—
255C2×M16—28no—
256C2×M16—44no—
257C2×M16—44no—
258C22×C8C2412yes—
259C22×C8C2414yes—
260C22×C8C2412yes—
261C4⋊Q8—28no—
262C4⋊Q8—22no—
263C4⋊Q8—22no—
264C4⋊Q8—22no—
265C4⋊Q8—42no—
266C4⋊Q8—42no—
267C4⋊Q8—44no—
268C4⋊Q8—82no—
269C4⋊Q8C2×Q814yes—
270C22⋊Q8—24no—
271C22⋊Q8—216no—
272C22⋊Q8—84no—
273C22⋊Q8C2×Q818yes—
274C4×Q8—22no—
275C4×Q8—22no—
276C4×Q8—24no—
277C4×Q8—22no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—24no—
281C4×Q8—48no—
282C4×Q8—42no—
283C4×Q8—42no—
284C4×Q8—41no—
285C4×Q8—48no—
286C4×Q8—44no—
287C4×Q8C2×Q814yes—
288C4×Q8#4114yes—
289C4×D4—28no—
290C4×D4—24no—
291C4×D4—42no—
292C4×D4—44no—
293C4×D4C2×Q814yes—
294C4×D4#4114yes—
295#117—44no—
296#117—44no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—42no—
322C2×C4⋊C4—48no—
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2414yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2411yescharacteristic
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42—21no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—24no—
352C2×C42—21no—
353C2×C42—48no—
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D412yes—
356C2×C42C2×D412yes—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2412yes—
361C2×C42C2411yescharacteristic
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364#107—44no—
365#106—24no—
366#106—42no—
367#106—44no—
368C4⋊C8—24no—
369C4⋊C8—22no—
370C4⋊C8—22no—
371C4⋊C8—24no—
372C4⋊C8—44no—
373C4⋊C8—42no—
374#103—24no—
375#103—24no—
376#103—28no—
377#103—44no—
378#103—44no—
379#103—44no—
380#103—44no—
381C22⋊C8—28no—
382C4×C8—24no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—21no—
387#95—21no—
388#95—21no—
389#95—24no—
390#95—24no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—24no—
395#95—22no—
396#95—22no—
397#95—24no—
398#95C2×D412yes—
399#95C2×D412yes—
400#95C2×D412yes—
401C24C22×D412yes—
402C2×Q8—28no—
403C2×Q8—21no—
404C2×Q8—22no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—216no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—44no—
413C2×Q8—44no—
414C2×Q8—48no—
415C2×Q8—48no—
416C2×Q8—48no—
417C2×Q8—44no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q8—42no—
421C2×Q8—82no—
422C2×Q8—82no—
423C2×Q8C22×Q814yes—
424C2×Q8C22×Q812yes—
425C2×Q8C2×#4112yes—
426C2×Q8H(5,𝔽2)12yes—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4C22⋊Q814yes—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—28no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—82no—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×Q814yes—
465C22×C4C2×#4114yes—
466C22×C4C2511yescharacteristic
467M16—88no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—24no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—24no—
474C2×C8—24no—
475C2×C8—22no—
476C2×C8—22no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—28no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—24no—
494C4⋊C4—21no—
495C4⋊C4—22no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—81no—
510C4⋊C4—82no—
511C4⋊C4—84no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C4C22⋊Q814yes—
515C4⋊C4C22×Q812yes—
516C4⋊C4C2×#4112yes—
517C22⋊C4—28no—
518C22⋊C4—24no—
519C22⋊C4—42no—
520C22⋊C4—44no—
521C22⋊C4—84no—
522C22⋊C4—84no—
523C22⋊C4C22⋊Q818yes—
524C42—24no—
525C42—24no—
526C42—24no—
527C42—24no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—21no—
533C42—21no—
534C42—41no—
535C42—44no—
536C42—44no—
537C42—42no—
538C42—44no—
539C42—44no—
540C42—42no—
541C42—42no—
542C42—81no—
543C42—84no—
544C42#13714yes—
545C42C22×Q812yes—
546C42C2×#4112yes—
547C42C2×#4112yes—
548C42H(5,𝔽2)12yes—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23C2×C22⋊Q814yes—
556C23C23×D411yescharacteristic
557Q8—22no—
558Q8—22no—
559Q8—22no—
560Q8—24no—
561Q8—24no—
562Q8—416no—
563Q8—42no—
564Q8—42no—
565Q8—84no—
566Q8—164no—
567Q8#55414yes—
568D4—28no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—416no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—48no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—84no—
620C2×C4—88no—
621C2×C4C2×C22⋊Q812yes—
622C2×C4C2×C22⋊Q814yes—
623C2×C4C2×C22⋊Q812yes—
624C2×C4#54612yes—
625C2×C4#54612yes—
626C2×C4C2×#13712yes—
627C2×C4C2×#13712yes—
628C2×C4C2×#13712yes—
629C2×C4#57512yes—
630C2×C4#57712yes—
631C2×C4C23×Q811yescharacteristic
632C2×C4C22×#4111yescharacteristic
633C2×C4C22×#4111yescharacteristic
634C2×C4C2×H(5,𝔽2)11yescharacteristic
635C2×C4C2×H(5,𝔽2)11yescharacteristic
636C2×C4#58511yescharacteristic
637C8—42no—
638C8—44no—
639C8—42no—
640C22—24no—
641C22—24no—
642C22—24no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22C22×C22⋊Q811yescharacteristic
648C22C2×#54611yescharacteristic
649C22#348811yescharacteristic
650C22C22×#13711yescharacteristic
651C22C2×#57511yescharacteristic
652C22C2×#57711yescharacteristic
653C22#358611yescharacteristic
654C4—22no—
655C4—24no—
656C4—24no—
657C4—22no—
658C4—22no—
659C4—41no—
660C4—44no—
661C4—42no—
662C4—42no—
663C4—42no—
664C4—42no—
665C4—82no—
666C4—81no—
667C4—84no—
668C4—82no—
669C4—82no—
670C4#307814yes—
671C4#332914yes—
672C4C2×#55411yescharacteristic
673C4#347511yescharacteristic
674C4#350111yescharacteristic
675C4#351911yescharacteristic
676C2—44no—
677C2#3615411yescharacteristic
678C2C2×#307811yescharacteristic
679C2#6049711yescharacteristic
680C2C2×#332911yescharacteristic
681C2#6116511yescharacteristic
682C2#6196611yescharacteristic
683C2#6199711yescharacteristic
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. #32954C2semidirect
5. #29141C2semidirect
6. #28283C2semidirect
7. #27682C2semidirect
8. #26531C2semidirect
9. #26430C2semidirect
10. #26113C2semidirect
11. #25979C2semidirect
12. #24845C2semidirect
16. C4×#475C2semidirect
17. #20647C2semidirect
98. D4×Q8Q8semidirect

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
#329542
#291412
#282832
#276822
#265312
#264302
#261132
#259792
#248452
#227152
#226502
#224302
C4×#4752
#206472