Finite Group Databasebeta

← #6502122

Subgroups — #6502122

Order 512.

1948 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2429C211yesmaximal, characteristic
2#34212C211yesmaximal, characteristic
3#29240C212yesmaximal, complemented
4#27409C214yesmaximal, complemented
5#26949C214yesmaximal, complemented
6#26636C214yesmaximal, complemented
7#25173C214yesmaximal, complemented
8#24212C212yesmaximal, complemented
9#24003C211yesmaximal, characteristic
10#24003C211yesmaximal, characteristic
11#23921C211yesmaximal, characteristic
12#23921C211yesmaximal, characteristic
13#22050C212yesmaximal, complemented
14#21906C211yesmaximal, characteristic
15#21455C212yesmaximal, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C2×#483C2212yes—
22C2×#483C2212yes—
23C2×#462C2214yescomplemented
24C2×#451C2212yes—
25C2×#451C2212yes—
26C2×#451C2212yes—
27C2×#451C2212yes—
28C2×#441C2214yescomplemented
29#2429—22no—
30#2429—22no—
31#2429C2214yes—
32#2429C2214yes—
33C2×#398—21no—
34C2×#396—21no—
35#2368—22no—
36#2360C2212yes—
37#2360C2212yes—
38#2345—22no—
39#2345C2212yes—
40Q8×C22⋊C4C2212yes—
41#2329—21no—
42#2329—21no—
43#2315—21no—
44C4×C22⋊Q8C2212yes—
45C4×C22⋊Q8C2212yes—
46C2×#391C2211yescharacteristic
47C2×#389—21no—
48C2×#389C2211yescharacteristic
49C2×#388—21no—
50C2×#380C2211yescharacteristic
51#2273—21no—
52C2×C4×C22⋊C4C2211yescharacteristic
53#2088C2214yescomplemented
54#2066C2214yescomplemented
55#2043C2214yescomplemented
56#2035C2214yes—
57#2035C2214yes—
58#2016C2214yescomplemented
59#2001C2214yes—
60#2001C2214yes—
61#1981C2214yescomplemented
62#1971C2214yescomplemented
63#1935C2214yescomplemented
64#1927C2214yescomplemented
65#1924C2214yes—
66#1924C2214yes—
67#1919C2214yescomplemented
68#1878C2214yescomplemented
69#1878C2214yescomplemented
70#1872C2212yes—
71#1872C2212yes—
72#1872C2212yes—
73#1872C2212yes—
74#1872C2212yes—
75#1872C2212yes—
76#1865C2214yescomplemented
77#1865C2214yescomplemented
78#1853C2212yes—
79#1853C2212yes—
80#1797C2212yes—
81#1787C2212yes—
82#1787C2212yes—
83#1779C2211yescharacteristic
84#1779C2211yescharacteristic
85#1763C2214yescomplemented
86#1749C2212yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C23×Q8C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C22×Q16C2314yes—
92C22×Q16C2314yes—
93C2×C4⋊Q8—22no—
94C2×C4⋊Q8—44no—
95C2×C4⋊Q8C2314yes—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8—21no—
98C2×C22⋊Q8—28no—
99C2×C22⋊Q8—21no—
100C2×C22⋊Q8—21no—
101C2×C22⋊Q8—21no—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8—44no—
104C2×C22⋊Q8—44no—
105C2×C22⋊Q8C2312yes—
106C2×C22⋊Q8C2312yes—
107C2×C22⋊Q8C2312yes—
108C2×C22⋊Q8C2312yes—
109C2×C22⋊Q8C2312yes—
110C2×C22⋊Q8C2312yes—
111C2×C4×Q8—22no—
112C2×C4×Q8—44no—
113C2×C4×Q8C2314yes—
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2312yes—
116C22×C4⋊C4—21no—
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⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4D411yescharacteristic
133C22×C22⋊C4D411yescharacteristic
134C22×C42—21no—
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137#483—44no—
138#483—44no—
139#462—48no—
140#451—28no—
141#451—28no—
142#451—44no—
143#451—44no—
144#441—48no—
145C2×#106C2314yes—
146C2×#106C2314yes—
147C2×C4⋊C8C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×C22⋊C8C2312yes—
157C2×C22⋊C8C2312yes—
158C2×C22⋊C8C2312yes—
159C2×C22⋊C8C2312yes—
160C2×C22⋊C8C2312yes—
161C2×C22⋊C8C2312yes—
162C2×#97C2314yes—
163#398—24no—
164#398—24no—
165#396—24no—
166#396—24no—
167#391—22no—
168#391—22no—
169#391—22no—
170#391C2312yes—
171#391C2312yes—
172#391C2314yes—
173#389—22no—
174#389—22no—
175#389—22no—
176#389—24no—
177#389—24no—
178#389C2314yes—
179#389C2312yes—
180#389C2312yes—
181#388—24no—
182#388—24no—
183#387—22no—
184#387—22no—
185#387—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#385—22no—
190#385—22no—
191#385—22no—
192#385—22no—
193#380—21no—
194#380—21no—
195#380—22no—
196#380—22no—
197#380C2312yes—
198#380C2312yes—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4C2312yes—
202C4×C4⋊C4C2312yes—
203C4×C22⋊C4—22no—
204C4×C22⋊C4—22no—
205C4×C22⋊C4—22no—
206C4×C22⋊C4—22no—
207C4×C22⋊C4—22no—
208C4×C22⋊C4—22no—
209C4×C22⋊C4D412yes—
210C4×C22⋊C4D412yes—
211C4×C22⋊C4C2312yes—
212C4×C22⋊C4C2312yes—
213C2×#95—21no—
214C2×#95—21no—
215C2×#95—21no—
216C2×#95—21no—
217C2×#95D411yescharacteristic
218C2×#95D411yescharacteristic
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#336C2314yes—
224#336C2314yes—
225C25C2×D411yescharacteristic
226C22×Q8—24no—
227C22×Q8—24no—
228C22×Q8—21no—
229C22×Q8—21no—
230C22×Q8—21no—
231C22×Q8—24no—
232C22×Q8—24no—
233C22×Q8—42no—
234C22×Q8—42no—
235C22×Q8—42no—
236C22×Q8—42no—
237C22×Q8—42no—
238C22×Q8C2412yes—
239C22×Q8C2412yes—
240C22×Q8C2412yes—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—24no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—42no—
251C23×C4—42no—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2411yescharacteristic
260C23×C4C2411yescharacteristic
261C23×C4C2411yescharacteristic
262C2×Q16—28no—
263C2×Q16—28no—
264C2×Q16—28no—
265C2×Q16—44no—
266C2×Q16—44no—
267C2×Q16—44no—
268C22×C8C2414yes—
269C22×C8C2414yes—
270C4⋊Q8—28no—
271C4⋊Q8—44no—
272C4⋊Q8—48no—
273C22⋊Q8—28no—
274C22⋊Q8—28no—
275C22⋊Q8—24no—
276C22⋊Q8—24no—
277C22⋊Q8—24no—
278C22⋊Q8—24no—
279C22⋊Q8—24no—
280C22⋊Q8—24no—
281C22⋊Q8—24no—
282C22⋊Q8—42no—
283C22⋊Q8—42no—
284C22⋊Q8—42no—
285C22⋊Q8—42no—
286C22⋊Q8—44no—
287C22⋊Q8—44no—
288C22⋊Q8—44no—
289C22⋊Q8—44no—
290C22⋊Q8#4118yes—
291C4×Q8—28no—
292C4×Q8—44no—
293C4×Q8—44no—
294C4×Q8—44no—
295C4×Q8—48no—
296C2×C4⋊C4—28no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—21no—
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—44no—
322C2×C4⋊C4—44no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4—44no—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C22⋊C4—28no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—44no—
344C2×C22⋊C4—44no—
345C2×C22⋊C4—44no—
346C2×C22⋊C4C2×D412yes—
347C2×C22⋊C4C2×D412yes—
348C2×C22⋊C4C2×D412yes—
349C2×C22⋊C4C2×D412yes—
350C2×C42—24no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—42no—
356C2×C42—42no—
357C2×C42—44no—
358C2×C42C2×D412yes—
359C2×C42C2×D412yes—
360C2×C42C2412yes—
361C2×C42C2412yes—
362#106—28no—
363#106—44no—
364C4⋊C8—44no—
365#103—28no—
366#103—28no—
367#103—28no—
368#103—28no—
369#103—44no—
370#103—44no—
371#103—44no—
372#103—44no—
373C22⋊C8—24no—
374C22⋊C8—24no—
375C22⋊C8—42no—
376C22⋊C8—42no—
377C22⋊C8—42no—
378C22⋊C8—42no—
379#97—44no—
380#95—22no—
381#95—22no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95C2×D412yes—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397C24—22no—
398C24—22no—
399C24—41no—
400C24—41no—
401C24—41no—
402C24—41no—
403C24#13714yes—
404C24C22×D411yescharacteristic
405C24C22×D411yescharacteristic
406C24C22×D411yescharacteristic
407C2×Q8—24no—
408C2×Q8—24no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—24no—
412C2×Q8—24no—
413C2×Q8—24no—
414C2×Q8—24no—
415C2×Q8—24no—
416C2×Q8—22no—
417C2×Q8—24no—
418C2×Q8—22no—
419C2×Q8—416no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8—42no—
423C2×Q8—42no—
424C2×Q8—42no—
425C2×Q8—81no—
426C2×Q8—81no—
427C2×Q8—88no—
428C2×Q8—88no—
429C2×Q8C2×#4114yes—
430C2×Q8H(5,𝔽2)14yes—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
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—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—88no—
484C22×C4—81no—
485C22×C4—81no—
486C22×C4—81no—
487C22×C4—81no—
488C22×C4—81no—
489C22×C4—81no—
490C22×C4—81no—
491C22×C4—81no—
492C22×C4#13714yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D412yes—
500C22×C4C2×#4114yes—
501C22×C4C2511yescharacteristic
502Q16—416no—
503Q16—88no—
504C2×C8—24no—
505C2×C8—24no—
506C2×C8—24no—
507C2×C8—24no—
508C2×C8—24no—
509C2×C8—24no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—28no—
513C4⋊C4—28no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—48no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—48no—
526C4⋊C4—48no—
527C4⋊C4—42no—
528C4⋊C4—48no—
529C4⋊C4—82no—
530C4⋊C4—82no—
531C4⋊C4—81no—
532C4⋊C4—81no—
533C4⋊C4—82no—
534C4⋊C4—82no—
535C4⋊C4C2×#4114yes—
536C4⋊C4H(5,𝔽2)14yes—
537C22⋊C4—28no—
538C22⋊C4—28no—
539C22⋊C4—24no—
540C22⋊C4—24no—
541C22⋊C4—24no—
542C22⋊C4—24no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C42—24no—
550C42—24no—
551C42—42no—
552C42—42no—
553C42—48no—
554C42—82no—
555C42—82no—
556C42—82no—
557C42—82no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—28no—
575C23—44no—
576C23—44no—
577C23—82no—
578C23—82no—
579C23—82no—
580C23—82no—
581C23#52512yes—
582C23C2×#13712yes—
583C23C2×#13712yes—
584C23C2×#13712yes—
585C23C2×#13712yes—
586C23C23×D411yescharacteristic
587Q8—28no—
588Q8—28no—
589Q8—48no—
590Q8—44no—
591Q8—44no—
592Q8—44no—
593Q8—44no—
594Q8—84no—
595Q8—84no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—28no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—48no—
624C2×C4—48no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
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—42no—
647C2×C4—82no—
648C2×C4—82no—
649C2×C4—82no—
650C2×C4—82no—
651C2×C4—84no—
652C2×C4—84no—
653C2×C4—84no—
654C2×C4—84no—
655C2×C4—162no—
656C2×C4—162no—
657C2×C4#52512yes—
658C2×C4C2×#13714yes—
659C2×C4C22×#4112yes—
660C2×C4C2×H(5,𝔽2)12yes—
661C2×C4C2×H(5,𝔽2)12yes—
662C8—44no—
663C8—44no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—41no—
677C22—41no—
678C22—41no—
679C22—41no—
680C22—41no—
681C22—41no—
682C22—41no—
683C22—41no—
684C22—88no—
685C22#311314yes—
686C22C2×#52511yescharacteristic
687C22#348411yescharacteristic
688C22#348411yescharacteristic
689C22C22×#13711yescharacteristic
690C22C22×#13711yescharacteristic
691C22#358611yescharacteristic
692C22#358611yescharacteristic
693C4—22no—
694C4—24no—
695C4—41no—
696C4—41no—
697C4—44no—
698C4—44no—
699C4—42no—
700C4—42no—
701C4—42no—
702C4—44no—
703C4—44no—
704C4—82no—
705C4—82no—
706C4—82no—
707C4—82no—
708C4#351914yes—
709C2—24no—
710C2—42no—
711C2—42no—
712C2#3535511yescharacteristic
713C2C2×#311311yescharacteristic
714C2#6056911yescharacteristic
715C2#6157311yescharacteristic
716C2#6157411yescharacteristic
717C2#6157411yescharacteristic
718C2#6158611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29240C2semidirect
4. #27409C2semidirect
5. #26949C2semidirect
6. #26636C2semidirect
7. #25173C2semidirect
8. #24212C2semidirect
13. #22050C2semidirect
15. #21455C2semidirect
23. C2×#462C22semidirect
28. C2×#441C22semidirect
53. #2088C22semidirect
54. #2066C22semidirect
55. #2043C22semidirect
58. #2016C22semidirect
61. #1981C22semidirect
62. #1971C22semidirect
63. #1935C22semidirect
64. #1927C22semidirect
67. #1919C22semidirect
68. #1878C22semidirect
69. #1878C22semidirect
76. #1865C22semidirect
77. #1865C22semidirect
85. #1763C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24292
#342122
#292402
#274092
#269492
#266362
#251732
#242122
#240032
#239212
#220502
#219062
#214552