Finite Group Databasebeta

← #6593603

Subgroups — #6593603

Order 512.

1809 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36188C211yesmaximal, characteristic
2C4×D4×Q8C211yesmaximal, characteristic
3#33196C211yesmaximal, characteristic, complemented
4#32954C212yesmaximal, complemented
5#28717C211yesmaximal, characteristic, complemented
6#28283C211yesmaximal, characteristic, complemented
7#27628C212yesmaximal, complemented
8#27522C214yesmaximal, complemented
9#27034C214yesmaximal, complemented
10#26298C212yesmaximal
11#25884C212yesmaximal
12#24703C212yesmaximal, complemented
13#23214C212yesmaximal, complemented
14#22964C212yesmaximal, complemented
15#22795C211yesmaximal, characteristic
16C4×#475C211yesmaximal, characteristic, complemented
17#20734C212yesmaximal, complemented
18C2×#556—21no—
19C2×D4×Q8C2211yescharacteristic
20C22×C4×Q8C2212yes—
21C2×#487C2212yes—
22C2×#475C2212yes—
23C2×#475C2211yescharacteristic
24C2×#475C2212yes—
25C2×#431C2212yes—
26#2868C2211yescharacteristic
27#2857C2212yes—
28#2568—21no—
29#2547—21no—
30#2547C2212yes—
31#2499—22no—
32#2496—22no—
33#2496C2214yes—
34#2365—21no—
35#2365—21no—
36#2365C2212yes—
37#2365C2211yescharacteristic
38#2360—22no—
39#2360C2212yes—
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#2077C2214yes—
54#2077C2214yes—
55#2072C2214yes—
56#2066C2214yes—
57#2035C2214yes—
58#2035C2214yes—
59#1990C2212yes—
60#1988C2211yescharacteristic
61#1988C2212yes—
62#1981C2212yes—
63#1954C2212yes—
64#1948C2211yescharacteristic
65#1948C2212yes—
66#1948C2212yes—
67#1948C2212yes—
68#1941C2214yes—
69#1933C2214yes—
70#1927C2214yes—
71#1923C2214yes—
72#1920C2214yes—
73#1919C2212yes—
74#1865C2212yes—
75#1865C2214yes—
76#1865C2214yes—
77#1853C2214yes—
78#1842C2211yescharacteristic
79#1824C2214yes—
80#1803C2212yes—
81#1803C2212yes—
82#1803C2211yescharacteristic
83#1803C2212yes—
84#1803C2212yes—
85#1803C2212yes—
86#1797C2212yes—
87#1787C2214yes—
88#1787C2214yes—
89C4×#106C2212yes—
90C8×C4⋊C4C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92C4×#103C2212yes—
93C4×#103C2214yes—
94C4×#103C2214yes—
95C23×Q8C2312yes—
96#556—48no—
97D4×Q8—24no—
98D4×Q8—24no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#126—24no—
106C2×#124—22no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8C2312yes—
110C2×C4×Q8—22no—
111C2×C4×Q8—22no—
112C2×C4×Q8—21no—
113C2×C4×Q8—22no—
114C2×C4×Q8—28no—
115C2×C4×Q8—44no—
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2311yescharacteristic
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4D411yescharacteristic
125C2×C4×D4D411yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×#117—22no—
128C22×C4⋊C4—22no—
129C22×C4⋊C4C2312yes—
130C22×C42—22no—
131C22×C42C2312yes—
132#487—28no—
133#475—24no—
134#475—28no—
135#475—44no—
136#431—44no—
137C2×#106C2312yes—
138C2×#106C2314yes—
139C2×#106C2314yes—
140C2×#106C2312yes—
141C2×C4⋊C8C2311yescharacteristic
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2311yescharacteristic
145C2×C4⋊C8C2312yes—
146C2×#103C2314yes—
147C2×#103C2312yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2312yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×C22⋊C8C2314yes—
154C2×C22⋊C8C2314yes—
155C2×C4×C8C2312yes—
156C2×C4×C8C2314yes—
157#400—24no—
158#397—24no—
159#397—24no—
160#391—22no—
161#391—22no—
162#391—21no—
163#391—22no—
164#391—22no—
165#391—22no—
166#391C2312yes—
167#391C2312yes—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—22no—
174#389—22no—
175#389—22no—
176#389C2312yes—
177#389C2314yes—
178#389C2312yes—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—24no—
183#387—24no—
184#385—22no—
185#385—22no—
186#385—22no—
187#384—21no—
188#384—21no—
189#384—21no—
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#384C2312yes—
193#384C2312yes—
194#382—22no—
195#380—22no—
196#380—22no—
197#380—22no—
198#380C2312yes—
199#380C2314yes—
200#380C2314yes—
201C4×C4⋊C4—21no—
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—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2314yes—
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#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2314yes—
233#340C2312yes—
234#340C2312yes—
235#340C2312yes—
236#336C2314yes—
237#336C2312yes—
238C22×Q8—24no—
239C22×Q8—24no—
240C22×Q8—22no—
241C22×Q8—21no—
242C22×Q8—22no—
243C22×Q8—24no—
244C22×Q8—24no—
245C22×Q8—42no—
246C22×Q8—42no—
247C22×Q8C2412yes—
248C22×Q8C2411yescharacteristic
249C22×D4C2×D411yescharacteristic
250C23×C4—22no—
251C23×C4—22no—
252C23×C4C2×D412yes—
253C23×C4C2×D412yes—
254C23×C4C2412yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C22×C8C2414yes—
258C4⋊Q8—28no—
259C4⋊Q8—22no—
260C4⋊Q8—24no—
261C4⋊Q8—22no—
262C4⋊Q8—24no—
263C4⋊Q8—22no—
264C4⋊Q8—44no—
265C4⋊Q8—44no—
266C4⋊Q8#4114yes—
267#126—416no—
268#124—48no—
269C22⋊Q8—216no—
270C22⋊Q8—24no—
271C22⋊Q8—24no—
272C22⋊Q8—48no—
273C4×Q8—22no—
274C4×Q8—22no—
275C4×Q8—24no—
276C4×Q8—22no—
277C4×Q8—24no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—48no—
281C4×Q8—42no—
282C4×Q8—42no—
283C4×Q8—41no—
284C4×Q8—48no—
285C4×Q8—48no—
286C4×Q8—44no—
287C4×Q8#4114yes—
288C4×Q8#4114yes—
289C4×D4—28no—
290C4×D4—22no—
291C4×D4—22no—
292C4×D4—42no—
293C4×D4—44no—
294C4×D4SD1618yes—
295#117—48no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—42no—
321C2×C4⋊C4—48no—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2414yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2411yescharacteristic
332C2×C22⋊C4—22no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C42—21no—
340C2×C42—21no—
341C2×C42—24no—
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—21no—
352C2×C42—48no—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D412yes—
355C2×C42C2×D412yes—
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2412yes—
360C2×C42C2411yescharacteristic
361C2×C42C2411yescharacteristic
362C2×C42C2411yescharacteristic
363#106—28no—
364#106—24no—
365#106—42no—
366#106—44no—
367C4⋊C8—24no—
368C4⋊C8—22no—
369C4⋊C8—22no—
370C4⋊C8—24no—
371C4⋊C8—42no—
372#103—28no—
373#103—24no—
374#103—28no—
375#103—28no—
376#103—24no—
377#103—44no—
378#103—44no—
379C22⋊C8—44no—
380C22⋊C8—44no—
381C4×C8—24no—
382C4×C8—44no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—21no—
387#95—21no—
388#95—21no—
389#95—22no—
390#95—22no—
391#95—24no—
392#95—22no—
393#95—22no—
394#95—24no—
395#95—22no—
396#95—24no—
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—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—44no—
412C2×Q8—44no—
413C2×Q8—48no—
414C2×Q8—48no—
415C2×Q8—48no—
416C2×Q8—48no—
417C2×Q8—44no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q8—44no—
421C2×Q8—44no—
422C2×Q8—42no—
423C2×Q8C2×#4114yes—
424C2×Q8C2×#4112yes—
425C2×Q8C2×#4112yes—
426C2×Q82−1+412yes—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4C2×SD1614yes—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—28no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—82no—
456C22×C4C2×SD1618yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C2511yescharacteristic
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—24no—
471C2×C8—22no—
472C2×C8—24no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—24no—
476C4⋊C4—28no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—48no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—81no—
512C4⋊C4—82no—
513C4⋊C4C2×SD1614yes—
514C4⋊C4C2×#4112yes—
515C4⋊C4C2×#4112yes—
516C22⋊C4—28no—
517C22⋊C4—24no—
518C22⋊C4—42no—
519C22⋊C4—44no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C22⋊C4C2×SD1618yes—
523C42—24no—
524C42—24no—
525C42—24no—
526C42—24no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—21no—
532C42—21no—
533C42—41no—
534C42—44no—
535C42—44no—
536C42—42no—
537C42—44no—
538C42—48no—
539C42—44no—
540C42—42no—
541C42—42no—
542C42—81no—
543C42C2×SD1614yes—
544C42C2×#4112yes—
545C42C2×#4112yes—
546C42C2×#4112yes—
547C42H(5,𝔽2)12yes—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23C22×SD1614yes—
555C23C23×D411yescharacteristic
556Q8—22no—
557Q8—22no—
558Q8—22no—
559Q8—24no—
560Q8—24no—
561Q8—416no—
562Q8—42no—
563Q8—42no—
564Q8—88no—
565Q8—84no—
566Q8#55614yescomplemented
567D4—28no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—216no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—416no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—84no—
619C2×C4—88no—
620C2×C4#52512yes—
621C2×C4#52512yes—
622C2×C4#54612yes—
623C2×C4#54612yes—
624C2×C4C22×SD1614yes—
625C2×C4C22×SD1612yes—
626C2×C4C22×SD1612yes—
627C2×C4C22×SD1612yes—
628C2×C4#57512yes—
629C2×C4#57812yes—
630C2×C4C22×#4111yescharacteristic
631C2×C4C22×#4111yescharacteristic
632C2×C4C22×#4111yescharacteristic
633C2×C4C2×H(5,𝔽2)11yescharacteristic
634C2×C4C2×2−1+411yescharacteristic
635C2×C4#58511yescharacteristic
636C8—42no—
637C8—44no—
638C8—42no—
639C22—24no—
640C22—24no—
641C22—24no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22C2×#52511yescharacteristic
647C22C2×#54611yescharacteristic
648C22#348911yescharacteristic
649C22C23×SD1611yescharacteristic
650C22C2×#57511yescharacteristic
651C22C2×#57811yescharacteristic
652C22#358611yescharacteristic
653C4—24no—
654C4—22no—
655C4—22no—
656C4—24no—
657C4—22no—
658C4—42no—
659C4—44no—
660C4—42no—
661C4—41no—
662C4—42no—
663C4—42no—
664C4—84no—
665C4—82no—
666C4—82no—
667C4—81no—
668C4—82no—
669C4#310314yes—
670C4#329814yes—
671C4C2×#55611yescharacteristic
672C4#347511yescharacteristic
673C4#350611yescharacteristic
674C4#352011yescharacteristic
675C2—44no—
676C2#3616111yescharacteristic
677C2C2×#310311yescharacteristic
678C2#6059411yescharacteristic
679C2C2×#329811yescharacteristic
680C2#6116511yescharacteristic
681C2#6209911yescharacteristic
682C2#6214611yescharacteristic
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. #33196C2semidirect
4. #32954C2semidirect
5. #28717C2semidirect
6. #28283C2semidirect
7. #27628C2semidirect
8. #27522C2semidirect
9. #27034C2semidirect
12. #24703C2semidirect
13. #23214C2semidirect
14. #22964C2semidirect
16. C4×#475C2semidirect
17. #20734C2semidirect
566. Q8#556semidirect

Maximal subgroups

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

subgroupindex
#361882
C4×D4×Q82
#331962
#329542
#287172
#282832
#276282
#275222
#270342
#262982
#258842
#247032
#232142
#229642
#227952
C4×#4752
#207342