Finite Group Databasebeta

← #6644488

Subgroups — #6644488

Order 512.

1930 conjugacy classes of subgroups.

The subgroup lattice has 754 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35010C211yesmaximal, characteristic
2C2×#2538C211yesmaximal, characteristic
3#33010C212yesmaximal, complemented
4#32000C214yesmaximal, complemented
5#31388C214yesmaximal, complemented
6#30563C214yesmaximal, complemented
7#30407C214yesmaximal, complemented
8#29909C212yesmaximal, complemented
9#27603C211yesmaximal, characteristic, complemented
10#24552C212yesmaximal, complemented
11#24380C212yesmaximal, complemented
12#24006C211yesmaximal, characteristic, complemented
13#24006C211yesmaximal, characteristic, complemented
14#23921C211yesmaximal, characteristic, complemented
15#21738C211yesmaximal, characteristic
16C22×C22⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C2×#484C2212yes—
22C2×#484C2212yescomplemented
23C2×#467C2214yescomplemented
24C2×#451C2212yes—
25C2×#451C2212yescomplemented
26C22×C22⋊C8C2212yes—
27#2855C2212yes—
28#2848C2212yes—
29#2627—21no—
30#2626—21no—
31#2579—22no—
32#2564—22no—
33#2538—22no—
34#2538—22no—
35#2538C2214yes—
36#2538C2214yes—
37#2487C2212yes—
38#2417C2212yes—
39#2417C2212yes—
40#2409C2212yes—
41C2×#396—21no—
42C2×#396—21no—
43#2315—21no—
44#2308—21no—
45C2×#390—21no—
46C2×#389—21no—
47C2×#384C2211yescharacteristic
48C2×#380C2211yescharacteristic
49C2×#380C2211yescharacteristic
50C2×#380C2211yescharacteristic
51#2100C2214yescomplemented
52#2095C2214yescomplemented
53#2083C2214yescomplemented
54#2076C2214yescomplemented
55#2076C2214yescomplemented
56#2072C2214yescomplemented
57#2036C2214yescomplemented
58#2036C2214yescomplemented
59#2036C2214yescomplemented
60#2036C2214yescomplemented
61#2001C2214yescomplemented
62#2001C2214yescomplemented
63#1892C2214yescomplemented
64#1892C2214yescomplemented
65#1878C2214yescomplemented
66#1878C2214yescomplemented
67#1872C2212yes—
68#1872C2212yes—
69#1872C2212yescomplemented
70#1872C2212yescomplemented
71#1869C2214yes—
72#1869C2214yes—
73#1863C2214yescomplemented
74#1863C2214yescomplemented
75#1852C2212yes—
76#1852C2212yes—
77#1852C2212yescomplemented
78#1852C2212yescomplemented
79#1824C2212yes—
80#1824C2212yescomplemented
81#1817C2212yes—
82#1808C2214yescomplemented
83#1787C2212yes—
84#1787C2212yescomplemented
85#1779C2211yescharacteristic
86#1779C2211yescharacteristic
87C23×Q8C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C23×D4D411yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C22×Q16C2314yes—
93C23×C8C2314yes—
94C2×C22⋊Q8—21no—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8—21no—
98C2×C22⋊Q8—22no—
99C2×C22⋊Q8—44no—
100C2×C22⋊Q8—44no—
101C2×C22⋊Q8—44no—
102C2×C22⋊Q8C2312yes—
103C2×C22⋊Q8C2312yes—
104C2×C22⋊Q8C2312yes—
105C2×C22⋊Q8C2314yes—
106C2×C22⋊Q8C2312yes—
107C2×C22⋊Q8C2312yes—
108C2×C22⋊Q8C2312yes—
109C2×C4×Q8—22no—
110C2×C4×Q8—44no—
111C2×C4×Q8C2314yes—
112C2×C4×D4—44no—
113C2×C4×D4D412yes—
114C2×C4×D4D412yes—
115C2×C4×D4C2312yes—
116C2×C4×D4C2312yes—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4D411yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C22⋊C4—21no—
134C22×C22⋊C4—21no—
135C22×C22⋊C4—21no—
136C22×C22⋊C4D411yescharacteristic
137C22×C22⋊C4D411yescharacteristic
138C22×C42—21no—
139C22×C42C2311yescharacteristic
140#484—44no—
141#484—44no—
142#467—48no—
143#451—44no—
144#451—44no—
145C2×#107C2314yes—
146C2×#107C2314yes—
147C2×#107C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×C22⋊C8—44no—
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2312yes—
157C2×C22⋊C8C2312yes—
158C2×C22⋊C8C2312yes—
159C2×C22⋊C8C2314yes—
160C2×C22⋊C8C2312yes—
161C2×C22⋊C8C2312yes—
162#400—22no—
163#399—22no—
164#398—22no—
165#398—22no—
166#396—22no—
167#396—22no—
168#396—24no—
169#396—24no—
170#396—24no—
171#396—24no—
172#395—22no—
173#393—22no—
174#391C2312yes—
175#391C2312yes—
176#390—24no—
177#390—24no—
178#389—24no—
179#389—24no—
180#389C2312yes—
181#389C2312yes—
182#387—22no—
183#387—22no—
184#386—22no—
185#386—22no—
186#386—22no—
187#386—22no—
188#385—22no—
189#385—22no—
190#385—22no—
191#385—22no—
192#385—22no—
193#385—22no—
194#384—22no—
195#384C2314yes—
196#381—22no—
197#380—21no—
198#380—21no—
199#380—21no—
200#380—21no—
201#380—22no—
202#380—22no—
203#380—22no—
204#380C2312yes—
205#380C2314yes—
206#380C2312yes—
207#380C2312yes—
208#380C2312yes—
209#380C2312yes—
210#380C2312yes—
211#379—22no—
212C2×#95—21no—
213C2×#95—21no—
214C2×#95—21no—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#336C2314yes—
222#336C2314yes—
223C25C2×D411yescharacteristic
224C25C2×D411yescharacteristic
225C22×Q8—24no—
226C22×Q8—21no—
227C22×Q8—21no—
228C22×Q8—24no—
229C22×Q8—42no—
230C22×Q8—42no—
231C22×Q8—42no—
232C22×Q8—42no—
233C22×Q8C2412yes—
234C22×Q8C2412yes—
235C22×D4—42no—
236C22×D4—42no—
237C22×D4Q1618yescomplemented
238C22×D4C2×D412yes—
239C22×D4C2×D412yes—
240C23×C4—21no—
241C23×C4—24no—
242C23×C4—24no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—42no—
251C23×C4—42no—
252C23×C4—42no—
253C23×C4—42no—
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D412yes—
261C23×C4C2411yescharacteristic
262C23×C4C2411yescharacteristic
263C23×C4C2412yes—
264C23×C4C2411yescharacteristic
265C2×Q16—28no—
266C2×Q16—44no—
267C2×Q16—44no—
268C22×C8—44no—
269C22×C8—44no—
270C22×C8—44no—
271C22×C8C2414yes—
272C22×C8C2414yes—
273C22⋊Q8—24no—
274C22⋊Q8—28no—
275C22⋊Q8—24no—
276C22⋊Q8—24no—
277C22⋊Q8—24no—
278C22⋊Q8—42no—
279C22⋊Q8—44no—
280C22⋊Q8—42no—
281C22⋊Q8—42no—
282C22⋊Q8—42no—
283C22⋊Q8—44no—
284C22⋊Q8—44no—
285C22⋊Q8—48no—
286C22⋊Q8—44no—
287C22⋊Q8—44no—
288C22⋊Q8—44no—
289C22⋊Q8—44no—
290C4×Q8—44no—
291C4×Q8—44no—
292C4×Q8—48no—
293C4×D4—44no—
294C4×D4—44no—
295C4×D4—42no—
296C4×D4—42no—
297C4×D4—42no—
298C4×D4—42no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—42no—
319C2×C4⋊C4—42no—
320C2×C4⋊C4—42no—
321C2×C4⋊C4—42no—
322C2×C4⋊C4—42no—
323C2×C4⋊C4—42no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4—44no—
327C2×C4⋊C4—44no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4—82no—
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2×D412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—42no—
353C2×C22⋊C4—42no—
354C2×C22⋊C4—44no—
355C2×C22⋊C4—44no—
356C2×C22⋊C4—44no—
357C2×C22⋊C4—44no—
358C2×C22⋊C4—82no—
359C2×C22⋊C4—82no—
360C2×C22⋊C4C2×D412yes—
361C2×C22⋊C4C2×D412yes—
362C2×C22⋊C4C2×D412yes—
363C2×C22⋊C4C2×D412yes—
364C2×C42—24no—
365C2×C42—24no—
366C2×C42—44no—
367C2×C42C2×D412yes—
368C2×C42C2412yes—
369#107—44no—
370#107—44no—
371#107—44no—
372#103—28no—
373#103—44no—
374#103—44no—
375#103—44no—
376#103—44no—
377#103—44no—
378C22⋊C8—24no—
379C22⋊C8—24no—
380C22⋊C8—44no—
381C22⋊C8—42no—
382C22⋊C8—42no—
383C22⋊C8—42no—
384C22⋊C8—42no—
385#95—22no—
386#95—24no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398C24—22no—
399C24—22no—
400C24—22no—
401C24—22no—
402C24—41no—
403C24—41no—
404C24—41no—
405C24—41no—
406C24—41no—
407C24—41no—
408C24—41no—
409C24—41no—
410C24C2×Q1614yes—
411C24C2×Q1614yes—
412C24C22×D412yes—
413C24C22×D411yescharacteristic
414C24C22×D411yescharacteristic
415C24C22×D411yescharacteristic
416C2×Q8—24no—
417C2×Q8—22no—
418C2×Q8—24no—
419C2×Q8—22no—
420C2×Q8—24no—
421C2×Q8—22no—
422C2×Q8—24no—
423C2×Q8—24no—
424C2×Q8—416no—
425C2×Q8—42no—
426C2×Q8—42no—
427C2×Q8—42no—
428C2×Q8—81no—
429C2×Q8—81no—
430C2×Q8—88no—
431C2×Q8H(5,𝔽2)14yes—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—416no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4C2×Q1618yes—
438C2×D4C2×Q1618yes—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—41no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—42no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4—42no—
476C22×C4—41no—
477C22×C4—41no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—88no—
496C22×C4—88no—
497C22×C4—81no—
498C22×C4—81no—
499C22×C4—81no—
500C22×C4—81no—
501C22×C4—81no—
502C22×C4—81no—
503C22×C4—81no—
504C22×C4—81no—
505C22×C4C2×Q1614yes—
506C22×C4C22×D412yes—
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D412yes—
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D412yes—
512C22×C4H(5,𝔽2)14yes—
513C22×C4C2511yescharacteristic
514Q16—88no—
515C2×C8—24no—
516C2×C8—24no—
517C2×C8—24no—
518C2×C8—24no—
519C2×C8—24no—
520C2×C8—24no—
521C2×C8—88no—
522C4⋊C4—24no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—24no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—48no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—48no—
536C4⋊C4—48no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—44no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—44no—
543C4⋊C4—82no—
544C4⋊C4—82no—
545C4⋊C4—81no—
546C4⋊C4—81no—
547C4⋊C4—84no—
548C4⋊C4—82no—
549C4⋊C4—82no—
550C22⋊C4—42no—
551C22⋊C4—42no—
552C22⋊C4—42no—
553C22⋊C4—42no—
554C22⋊C4—48no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C22⋊C4—82no—
562C22⋊C4—82no—
563C22⋊C4—84no—
564C22⋊C4—82no—
565C22⋊C4—82no—
566C22⋊C4—82no—
567C22⋊C4—82no—
568C42—24no—
569C42—22no—
570C42—22no—
571C42—48no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—22no—
577C23—21no—
578C23—21no—
579C23—22no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—22no—
585C23—22no—
586C23—21no—
587C23—21no—
588C23—22no—
589C23—21no—
590C23—21no—
591C23—21no—
592C23—21no—
593C23—22no—
594C23—28no—
595C23—28no—
596C23—44no—
597C23—44no—
598C23—44no—
599C23—44no—
600C23—82no—
601C23—82no—
602C23—82no—
603C23—82no—
604C23—82no—
605C23—82no—
606C23—82no—
607C23—82no—
608C23#53412yes—
609C23C22×Q1614yes—
610C23C22×Q1612yes—
611C23C22×Q1612yes—
612C23C22×Q1612yes—
613C23#57512yes—
614C23C23×D411yescharacteristic
615Q8—28no—
616Q8—48no—
617Q8—44no—
618Q8—44no—
619Q8—44no—
620Q8—84no—
621D4—48no—
622D4—48no—
623C2×C4—28no—
624C2×C4—22no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—22no—
628C2×C4—28no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—22no—
635C2×C4—24no—
636C2×C4—21no—
637C2×C4—22no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—22no—
641C2×C4—21no—
642C2×C4—44no—
643C2×C4—48no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—82no—
672C2×C4—84no—
673C2×C4—82no—
674C2×C4—82no—
675C2×C4—82no—
676C2×C4—84no—
677C2×C4—84no—
678C2×C4—84no—
679C2×C4—84no—
680C2×C4—84no—
681C2×C4—162no—
682C2×C4—162no—
683C2×C4#53412yes—
684C2×C4C22×Q1614yes—
685C2×C4C2×H(5,𝔽2)12yes—
686C2×C4C2×H(5,𝔽2)12yes—
687C2×C4#58512yes—
688C8—44no—
689C8—44no—
690C22—22no—
691C22—22no—
692C22—22no—
693C22—22no—
694C22—22no—
695C22—22no—
696C22—24no—
697C22—22no—
698C22—22no—
699C22—22no—
700C22—22no—
701C22—24no—
702C22—24no—
703C22—22no—
704C22—22no—
705C22—41no—
706C22—42no—
707C22—41no—
708C22—41no—
709C22—42no—
710C22—41no—
711C22—41no—
712C22—42no—
713C22—41no—
714C22—41no—
715C22—42no—
716C22—41no—
717C22—88no—
718C22—88no—
719C22#318914yes—
720C22C2×#53411yescharacteristic
721C22#348411yescharacteristic
722C22#348511yescharacteristic
723C22C23×Q1611yescharacteristic
724C22C2×#57511yescharacteristic
725C22#358511yescharacteristic
726C22#358611yescharacteristic
727C4—24no—
728C4—22no—
729C4—42no—
730C4—41no—
731C4—42no—
732C4—41no—
733C4—44no—
734C4—82no—
735C4—82no—
736C4—82no—
737C4—82no—
738C4—82no—
739C4—82no—
740C4—82no—
741C4#353414yes—
742C2—24no—
743C2—42no—
744C2—42no—
745C2—44no—
746C2#3711711yescharacteristic
747C2C2×#318911yescharacteristic
748C2#6075511yescharacteristic
749C2#6156311yescharacteristic
750C2#6157711yescharacteristic
751C2#6169811yescharacteristic
752C2#6171211yescharacteristic
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
3. #33010C2semidirect
4. #32000C2semidirect
5. #31388C2semidirect
6. #30563C2semidirect
7. #30407C2semidirect
8. #29909C2semidirect
9. #27603C2semidirect
10. #24552C2semidirect
11. #24380C2semidirect
12. #24006C2semidirect
13. #24006C2semidirect
14. #23921C2semidirect
22. C2×#484C22semidirect
23. C2×#467C22semidirect
25. C2×#451C22semidirect
51. #2100C22semidirect
52. #2095C22semidirect
53. #2083C22semidirect
54. #2076C22semidirect
55. #2076C22semidirect
56. #2072C22semidirect
57. #2036C22semidirect
58. #2036C22semidirect
59. #2036C22semidirect
60. #2036C22semidirect
61. #2001C22semidirect
62. #2001C22semidirect
63. #1892C22semidirect
64. #1892C22semidirect
65. #1878C22semidirect
66. #1878C22semidirect
69. #1872C22semidirect
70. #1872C22semidirect
73. #1863C22semidirect
74. #1863C22semidirect
77. #1852C22semidirect
78. #1852C22semidirect
80. #1824C22semidirect
82. #1808C22semidirect
84. #1787C22semidirect
237. C22×D4Q16semidirect

Maximal subgroups

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

subgroupindex
#350102
C2×#25382
#330102
#320002
#313882
#305632
#304072
#299092
#276032
#245522
#243802
#240062
#239212
#217382