Finite Group Databasebeta

← #6444015

Subgroups — #6444015

Order 512.

2294 conjugacy classes of subgroups.

The subgroup lattice has 733 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3095C212yesmaximal, complemented
2C2×#2536C211yesmaximal, characteristic
3D4×#117C211yesmaximal, characteristic
4#25025C214yesmaximal, complemented
5#25024C214yesmaximal, complemented
6#25021C214yesmaximal, complemented
7#25020C214yesmaximal, complemented
8#23642C212yesmaximal, complemented
9#23481C212yesmaximal, complemented
10#21190C211yesmaximal, characteristic
11#21172C211yesmaximal, characteristic, complemented
12#20846C212yesmaximal, complemented
13#20801C211yesmaximal, characteristic, complemented
14#20647C211yesmaximal, characteristic, complemented
15C4×#419C211yesmaximal, characteristic, complemented
16C22×C22⋊Q8C2212yes—
17C22×C4×Q8C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C22×#117C2212yes—
20#3095—216no—
21C2×#484C2214yescomplemented
22C2×#485C2214yescomplemented
23C2×#485C2214yescomplemented
24C2×#483C2214yescomplemented
25C2×#428C2212yes—
26C2×#428C2212yes—
27C2×#428C2212yes—
28C2×#423C2211yescharacteristic
29C2×#419C2212yes—
30C2×#419C2211yescharacteristic
31C2×#419C2211yescharacteristic
32C2×#407C2212yes—
33C2×#407C2212yes—
34C2×C4×M16C2211yescharacteristic
35#2536—21no—
36#2536—22no—
37#2536—21no—
38#2536C2214yes—
39#2536C2212yes—
40#2536C2212yes—
41C2×#398—21no—
42C2×#394—21no—
43#2354C2212yes—
44#2352C2212yes—
45#2352C2212yes—
46D4×C4⋊C4C2212yes—
47#2344—21no—
48#2342—22no—
49D4×C22⋊C4—21no—
50#2293—21no—
51#2290—22no—
52C2×#388—22no—
53#2285C2211yescharacteristic
54C2×#382C2211yescharacteristic
55#2280C2212yes—
56C2×#380C2212yes—
57C42×D4—21no—
58C4×#117C2211yescharacteristic
59#2077C2214yescomplemented
60#2077C2214yescomplemented
61#2076C2214yescomplemented
62#2075C2214yescomplemented
63#2036C2214yescomplemented
64#2036C2214yescomplemented
65#2035C2214yescomplemented
66#2035C2214yescomplemented
67#1955C2212yes—
68#1955C2212yes—
69#1954C2214yes—
70#1949C2214yescomplemented
71#1949C2214yescomplemented
72#1854C2214yescomplemented
73#1854C2214yescomplemented
74#1853C2214yescomplemented
75#1852C2214yescomplemented
76#1847C2214yescomplemented
77#1846C2212yescomplemented
78#1846C2212yescomplemented
79#1805C2212yescomplemented
80#1805C2212yescomplemented
81#1803C2214yescomplemented
82#1763C2214yescomplemented
83C4×#103C2214yescomplemented
84#1736C2211yescharacteristic
85#1736C2212yes—
86#1736C2212yes—
87#1736C2211yescharacteristic
88C23×Q8C2311yescharacteristic
89C23×D4D411yescharacteristic
90C24×C4C2312yes—
91C22×M16C2312yes—
92C22×M16C2311yescharacteristic
93C22×M16C2311yescharacteristic
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8—28no—
97C2×C22⋊Q8—44no—
98C2×C22⋊Q8C2314yes—
99C2×C22⋊Q8C2314yes—
100C2×C4×Q8—21no—
101C2×C4×Q8—21no—
102C2×C4×Q8—24no—
103C2×C4×Q8—24no—
104C2×C4×Q8C2312yes—
105C2×C4×Q8C2312yes—
106C2×C4×D4—22no—
107C2×C4×D4—24no—
108C2×C4×D4—24no—
109C2×C4×D4D411yescharacteristic
110C2×C4×D4D412yes—
111C2×C4×D4D411yescharacteristic
112C2×C4×D4C2311yescharacteristic
113C2×C4×D4C2312yes—
114C2×C4×D4C2311yescharacteristic
115C2×#117—22no—
116C2×#117—22no—
117C2×#117—28no—
118C2×#117—44no—
119C2×#117C2312yes—
120C2×#117C2311yescharacteristic
121C2×#117C2312yes—
122C2×#117C2312yes—
123C2×#117C2311yescharacteristic
124C22×C4⋊C4—21no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4D411yescharacteristic
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2312yes—
131C22×C22⋊C4—22no—
132C22×C22⋊C4—22no—
133C22×C22⋊C4—22no—
134C22×C22⋊C4D412yes—
135C22×C42—22no—
136C22×C42—21no—
137C22×C42C2311yescharacteristic
138#485—216no—
139#485—216no—
140#484—216no—
141#483—216no—
142#428—28no—
143#428—28no—
144#428—44no—
145C2×#107C2314yes—
146C2×#107C2314yes—
147C2×#107C2314yes—
148C2×#106C2314yes—
149C2×#106C2314yes—
150C2×#106C2314yes—
151#423—24no—
152C2×C4⋊C8C2312yes—
153C2×C4⋊C8C2312yes—
154#419—28no—
155#419—24no—
156#419D418yescomplemented
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161#407—28no—
162#407—28no—
163C2×C22⋊C8C2314yes—
164C2×C22⋊C8C2314yes—
165C2×C22⋊C8C2314yes—
166C2×C22⋊C8C2314yes—
167C4×M16—24no—
168C2×#97C2312yes—
169C2×C4×C8C2312yes—
170#398—22no—
171#398—24no—
172#398—22no—
173#394—22no—
174#394—24no—
175#394—22no—
176#390—22no—
177#389—22no—
178#388—22no—
179#388—22no—
180#388—22no—
181#388—22no—
182#388—24no—
183#388—24no—
184#388—24no—
185#388—24no—
186#387—22no—
187#387—22no—
188#386—22no—
189#386—22no—
190#385—22no—
191#385—22no—
192#384C2312yes—
193#382—21no—
194#382—21no—
195#382C2311yescharacteristic
196#382C2311yescharacteristic
197#382C2312yes—
198#382C2312yes—
199#381—22no—
200#380—21no—
201#380—21no—
202#380—22no—
203#380—22no—
204#380C2312yes—
205#380C2314yes—
206#380C2312yes—
207#380C2314yes—
208#380C2312yes—
209#380C2312yes—
210#379—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C22⋊C4—21no—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—22no—
222#376—21no—
223C2×#95—21no—
224C2×#95—22no—
225C2×#95—21no—
226C43—21no—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233C25C2×D412yes—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8—22no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8C2×D414yes—
241C22×Q8C2412yes—
242C22×D4—24no—
243C22×D4—44no—
244C22×D4C2×D411yescharacteristic
245C22×D4C2×D412yes—
246C22×D4C2×D411yescharacteristic
247C23×C4—21no—
248C23×C4—24no—
249C23×C4—24no—
250C23×C4—24no—
251C23×C4—24no—
252C23×C4—24no—
253C23×C4—24no—
254C23×C4—22no—
255C23×C4—21no—
256C23×C4—22no—
257C23×C4—22no—
258C23×C4C2×D412yes—
259C23×C4C2×D412yes—
260C23×C4C2×D412yes—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2412yes—
264C23×C4C2412yes—
265C23×C4C2411yescharacteristic
266C2×M16—22no—
267C2×M16—22no—
268C2×M16—22no—
269C2×M16—22no—
270C2×M16—24no—
271C2×M16—24no—
272C2×M16—24no—
273C2×M16C2×D414yes—
274C2×M16#4114yes—
275C22×C8C2414yes—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C22⋊Q8—28no—
279C22⋊Q8—28no—
280C22⋊Q8—44no—
281C22⋊Q8—44no—
282C22⋊Q8—48no—
283C22⋊Q8—48no—
284C4×Q8—24no—
285C4×Q8—24no—
286C4×Q8—24no—
287C4×Q8—24no—
288C4×Q8—44no—
289C4×Q8—44no—
290C4×D4—24no—
291C4×D4—28no—
292C4×D4—24no—
293C4×D4—24no—
294C4×D4—22no—
295C4×D4—22no—
296C4×D4—22no—
297C4×D4—24no—
298C4×D4—22no—
299C4×D4—48no—
300#117—22no—
301#117—24no—
302#117—22no—
303#117—24no—
304#117—22no—
305#117—24no—
306#117—24no—
307#117—48no—
308#117—48no—
309#117—42no—
310#117—42no—
311#117C2×D414yes—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—42no—
330C2×C4⋊C4—42no—
331C2×C4⋊C4—48no—
332C2×C4⋊C4—44no—
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D412yes—
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2414yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—28no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—48no—
357C2×C22⋊C4—48no—
358C2×C22⋊C4—48no—
359C2×C22⋊C4C2×D412yes—
360C2×C22⋊C4C2×D412yes—
361C2×C22⋊C4C2×D412yes—
362C2×C22⋊C4C2×D412yes—
363C2×C42—21no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—21no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—48no—
373C2×C42—44no—
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42#4114yes—
377C2×C42#4114yes—
378C2×C42C2411yescharacteristic
379C2×C42C2411yescharacteristic
380#107—28no—
381#107—28no—
382#107—44no—
383#106—28no—
384#106—28no—
385#106—44no—
386C4⋊C8—24no—
387C4⋊C8—24no—
388#103—28no—
389#103—28no—
390#103—28no—
391#103C2×D4116yescomplemented
392C22⋊C8—28no—
393C22⋊C8—28no—
394C22⋊C8—28no—
395C22⋊C8—28no—
396#97—24no—
397C4×C8—24no—
398#95—24no—
399#95—24no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—21no—
407#95—21no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—21no—
413#95—21no—
414C24—24no—
415C24—24no—
416C24—24no—
417C24—24no—
418C24—42no—
419C24—42no—
420C24—42no—
421C24C22×D412yes—
422C24C22×D412yes—
423C24C22×D411yescharacteristic
424C2×Q8—24no—
425C2×Q8—22no—
426C2×Q8—28no—
427C2×Q8—28no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—42no—
433C2×Q8—44no—
434C2×Q8—42no—
435C2×Q8—44no—
436C2×Q8C22×D414yes—
437C2×D4—24no—
438C2×D4—24no—
439C2×D4—22no—
440C2×D4—24no—
441C2×D4—22no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4—416no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—48no—
481C22×C4—48no—
482C22×C4—48no—
483C22×C4—48no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—42no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—82no—
492C22×C4—82no—
493C22×C4#12314yes—
494C22×C4#12314yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D412yes—
503C22×C4C2×#4112yes—
504C22×C4C2×#4112yes—
505C22×C4C2×#4112yes—
506C22×C4H(5,𝔽2)12yes—
507C22×C42−1+412yes—
508C22×C4C2511yescharacteristic
509M16—28no—
510M16—28no—
511M16—48no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—24no—
517C2×C8—24no—
518C2×C8—24no—
519C2×C8C22×D414yes—
520C2×C8C2×#4114yes—
521C4⋊C4—22no—
522C4⋊C4—24no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—24no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—24no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—48no—
538C4⋊C4—48no—
539C4⋊C4—44no—
540C4⋊C4—42no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—44no—
544C4⋊C4—82no—
545C4⋊C4C22×D414yes—
546C22⋊C4—24no—
547C22⋊C4—24no—
548C22⋊C4—24no—
549C22⋊C4—24no—
550C22⋊C4—24no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—48no—
554C22⋊C4—48no—
555C22⋊C4—48no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—42no—
559C22⋊C4—48no—
560C22⋊C4—44no—
561C22⋊C4—82no—
562C22⋊C4—82no—
563C42—24no—
564C42—44no—
565C42—44no—
566C42—44no—
567C42—44no—
568C42—44no—
569C42—44no—
570C42—42no—
571C42—82no—
572C42#12314yes—
573C42#12314yes—
574C42C2×#4112yes—
575C42C2×#4112yes—
576C42C2×#4112yes—
577C42C2×#4112yes—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—22no—
582C23—22no—
583C23—22no—
584C23—22no—
585C23—22no—
586C23—22no—
587C23—22no—
588C23—22no—
589C23—22no—
590C23—22no—
591C23—22no—
592C23—22no—
593C23—48no—
594C23—48no—
595C23—48no—
596C23—48no—
597C23—84no—
598C23—84no—
599C23—84no—
600C23C2×#12312yes—
601C23D4212yes—
602C23#54712yes—
603C23#57514yes—
604C23C23×D411yescharacteristic
605Q8—24no—
606Q8—24no—
607Q8—48no—
608Q8—42no—
609Q8—42no—
610D4—28no—
611D4—48no—
612D4—44no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—24no—
620C2×C4—22no—
621C2×C4—24no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—48no—
630C2×C4—48no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—48no—
636C2×C4—48no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—88no—
658C2×C4—88no—
659C2×C4—84no—
660C2×C4—88no—
661C2×C4C2×#12312yes—
662C2×C4C2×#12312yes—
663C2×C4C2×#12312yes—
664C2×C4C2×#12312yes—
665C2×C4C2×#12312yes—
666C2×C4#53214yes—
667C2×C4#54214yes—
668C2×C4D4212yes—
669C2×C4#54712yes—
670C2×C4#54714yes—
671C2×C4#55314yes—
672C2×C4C23×D411yescharacteristic
673C2×C4C22×#4111yescharacteristic
674C2×C4C22×#4111yescharacteristic
675C2×C4C22×#4111yescharacteristic
676C2×C4C2×H(5,𝔽2)11yescharacteristic
677C2×C4C2×2−1+411yescharacteristic
678C8—24no—
679C8—24no—
680C8—44no—
681C22—24no—
682C22—22no—
683C22—22no—
684C22—24no—
685C22—24no—
686C22—24no—
687C22—24no—
688C22—42no—
689C22—42no—
690C22—42no—
691C22—42no—
692C22—42no—
693C22—42no—
694C22—42no—
695C22—42no—
696C22—42no—
697C22—42no—
698C22—42no—
699C22—816no—
700C22#242214yes—
701C22C22×#12311yescharacteristic
702C22C2×D4211yescharacteristic
703C22C2×#54711yescharacteristic
704C22C2×#57512yes—
705C22#358612yes—
706C4—22no—
707C4—22no—
708C4—22no—
709C4—22no—
710C4—42no—
711C4—42no—
712C4—44no—
713C4—44no—
714C4—82no—
715C4—82no—
716C4—82no—
717C4—82no—
718C4—84no—
719C4#242214yes—
720C4C2×#53211yescharacteristic
721C4C2×#54211yescharacteristic
722C4C2×#54711yescharacteristic
723C4C2×#55311yescharacteristic
724C2—24no—
725C2—44no—
726C2—82no—
727C2C2×#242211yescharacteristic
728C2#6051111yescharacteristic
729C2#6051911yescharacteristic
730C2#6106812yes—
731C2#6125312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#3095C2semidirect
4. #25025C2semidirect
5. #25024C2semidirect
6. #25021C2semidirect
7. #25020C2semidirect
8. #23642C2semidirect
9. #23481C2semidirect
11. #21172C2semidirect
12. #20846C2semidirect
13. #20801C2semidirect
14. #20647C2semidirect
15. C4×#419C2semidirect
21. C2×#484C22semidirect
22. C2×#485C22semidirect
23. C2×#485C22semidirect
24. C2×#483C22semidirect
59. #2077C22semidirect
60. #2077C22semidirect
61. #2076C22semidirect
62. #2075C22semidirect
63. #2036C22semidirect
64. #2036C22semidirect
65. #2035C22semidirect
66. #2035C22semidirect
70. #1949C22semidirect
71. #1949C22semidirect
72. #1854C22semidirect
73. #1854C22semidirect
74. #1853C22semidirect
75. #1852C22semidirect
76. #1847C22semidirect
77. #1846C22semidirect
78. #1846C22semidirect
79. #1805C22semidirect
80. #1805C22semidirect
81. #1803C22semidirect
82. #1763C22semidirect
83. C4×#103C22semidirect
156. #419D4semidirect
391. #103C2×D4semidirect

Maximal subgroups

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

subgroupindex
C2×#30952
C2×#25362
D4×#1172
#250252
#250242
#250212
#250202
#236422
#234812
#211902
#211722
#208462
#208012
#206472
C4×#4192