Finite Group Databasebeta

← #6388291

Subgroups — #6388291

Order 512.

2114 conjugacy classes of subgroups.

The subgroup lattice has 711 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3100C211yesmaximal, characteristic
2#34236C211yesmaximal, characteristic, complemented
3C2×#2357C211yesmaximal, characteristic, complemented
4#23108C214yesmaximal, complemented
5#23079C214yesmaximal, complemented
6#22816C212yesmaximal, complemented
7#22810C212yesmaximal, complemented
8#22772C212yesmaximal, complemented
9#22764C212yesmaximal, complemented
10#22046C212yesmaximal, complemented
11#22040C212yesmaximal, complemented
12#21983C211yesmaximal, characteristic, complemented
13#21865C211yesmaximal, characteristic, complemented
14#21578C212yesmaximal, complemented
15C4×#480C212yesmaximal, complemented
16#20739C211yesmaximal, characteristic, complemented
17#20535C211yesmaximal, characteristic, complemented
18C2×#532C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20#3100C4116yescomplemented
21C2×#480C2211yescharacteristic
22C2×#480C2211yescharacteristic
23C2×#474C2212yes—
24C2×#459C2212yes—
25C2×#447C2211yescharacteristic
26C2×#447C2211yescharacteristic
27C2×C4×D8C2212yes—
28C2×C4×D8C2212yes—
29C2×#432C2211yescharacteristic
30C2×#431C2211yescharacteristic
31#2369C2212yescomplemented
32#2366C2212yescomplemented
33#2361—22no—
34#2357—21no—
35#2357—22no—
36#2357—21no—
37#2357C2212yes—
38#2357C2212yes—
39#2357C2214yes—
40#2357C2211yescharacteristic, complemented
41#2357C2211yescharacteristic, complemented
42#2349C2211yescharacteristic, complemented
43#2349C2211yescharacteristic, complemented
44#2339—21no—
45#2339—21no—
46#2337—22no—
47#2319—21no—
48C4×#121C2212yes—
49C4×#121C2212yes—
50C2×#390C2212yes—
51C2×#388—22no—
52#2283C2211yescharacteristic
53C2×#382C2211yescharacteristic
54C2×#381—21no—
55#2275—21no—
56C2×C4×C4⋊C4C2212yes—
57C2×C4×C4⋊C4C2211yescharacteristic
58C2×C4×C22⋊C4—21no—
59#1993C2212yescomplemented
60#1972C2214yes—
61#1970C2214yes—
62#1961C2212yescomplemented
63#1949C2214yescomplemented
64#1936C2214yes—
65#1934C2214yes—
66#1926C2214yes—
67#1926C2214yescomplemented
68#1925C2214yes—
69#1925C2214yescomplemented
70#1921C2214yescomplemented
71#1918C2214yescomplemented
72#1879C2214yes—
73#1879C2214yes—
74#1874C2212yes—
75#1874C2212yes—
76#1853C2212yes—
77#1852C2212yes—
78#1846C2212yescomplemented
79#1835C2212yescomplemented
80#1823C2212yes—
81#1804C2212yes—
82#1804C2212yes—
83#1804C2212yescomplemented
84#1804C2212yescomplemented
85#1786C2212yes—
86#1776C2214yescomplemented
87C4×#107C2214yescomplemented
88#1771C2214yescomplemented
89#1770C2212yescomplemented
90C8×C4⋊C4C2212yescomplemented
91#1762C2214yes—
92#1762C2214yescomplemented
93C4×#102C2214yescomplemented
94#1749C2212yes—
95C4×C22⋊C8C2212yes—
96C22×#41—21no—
97C23×D4C2311yescharacteristic
98C24×C4—21no—
99C22×D8C2312yes—
100C22×D8C2312yes—
101#532—24no—
102#532C2×C418yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×#127C2311yescharacteristic
105C2×#124—21no—
106C2×#121—21no—
107C2×#121C2311yescharacteristic
108C2×#121C2311yescharacteristic
109C2×C4×D4—22no—
110C2×C4×D4—22no—
111C2×C4×D4—22no—
112C2×C4×D4—21no—
113C2×C4×D4—21no—
114C2×C4×D4—21no—
115C2×C4×D4—24no—
116C2×C4×D4C2×C418yescomplemented
117C2×C4×D4C2312yes—
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2312yes—
121C2×C4×D4C2312yes—
122C2×C4×D4C2312yes—
123C22×C4⋊C4D412yes—
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2312yes—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—22no—
128C22×C22⋊C4—22no—
129C22×C42—22no—
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132#480C2×C418yescomplemented
133#480C2×C418yescomplemented
134#474C2×C4116yescomplemented
135#459C2×C4116yescomplemented
136#447C2×C418yescomplemented
137#447C2×C418yescomplemented
138C4×D8C2×C4116yescomplemented
139C4×D8C2×C4116yescomplemented
140#432—24no—
141#431C2×C418yescomplemented
142C2×#107C2314yes—
143C2×#107C2312yes—
144C2×#107C2312yes—
145C2×#106C2314yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2312yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2312yes—
153C2×#102C2312yes—
154C2×#102C2312yes—
155C2×#102C2312yes—
156C2×C22⋊C8C2312yes—
157C2×C22⋊C8C2311yescharacteristic
158C2×C22⋊C8C2311yescharacteristic
159C2×#97C2312yes—
160C2×#97C2314yes—
161C2×C4×C8C2314yes—
162C2×C4×C8C2312yes—
163#391—22no—
164#391—22no—
165#390—22no—
166#390—22no—
167#390—22no—
168#390C2314yes—
169#390C2314yes—
170#390C2312yes—
171#390C2312yes—
172#389—22no—
173#389C2312yes—
174#389C2312yes—
175#388—22no—
176#388—22no—
177#388—24no—
178#388—24no—
179#388—24no—
180#388—24no—
181#388—22no—
182#388—22no—
183#386—22no—
184#386—22no—
185#386—22no—
186#386—22no—
187#385—22no—
188#385—22no—
189#383—21no—
190#382—21no—
191#382—21no—
192#382C2311yescharacteristic
193#382C2311yescharacteristic
194#382C2312yes—
195#382C2312yes—
196#381—21no—
197#381—21no—
198#381—21no—
199#381—22no—
200#381—22no—
201#381—24no—
202#380—21no—
203#380—21no—
204#380C2312yes—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D412yes—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2314yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—21no—
223C4×C22⋊C4—21no—
224C4×C22⋊C4—22no—
225C4×C22⋊C4—22no—
226C4×C22⋊C4—24no—
227#376—21no—
228C2×#95—21no—
229C2×#95—22no—
230C2×#95—21no—
231C2×#95—21no—
232C2×#95—21no—
233C2×#95D411yescharacteristic
234C2×#95D411yescharacteristic
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#340C2314yes—
239#340C2314yes—
240#340C2314yes—
241#336C2314yes—
242#336C2314yes—
243C25—21no—
244C2×#41—28no—
245C2×#41—84no—
246C22×Q8—21no—
247C22×D4—21no—
248C22×D4—21no—
249C22×D4—21no—
250C22×D4—21no—
251C22×D4—21no—
252C22×D4—22no—
253C22×D4—42no—
254C22×D4—42no—
255C22×D4C22×C414yes—
256C22×D4C2411yescharacteristic
257C22×D4C2411yescharacteristic
258C22×D4C2412yes—
259C23×C4—21no—
260C23×C4—28no—
261C23×C4—22no—
262C23×C4—22no—
263C23×C4—21no—
264C23×C4—21no—
265C23×C4—22no—
266C23×C4—22no—
267C23×C4—22no—
268C23×C4—84no—
269C23×C4C2×D412yes—
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D412yes—
272C23×C4C2×D411yescharacteristic
273C23×C4C2411yescharacteristic
274C2×D8—42no—
275C2×D8—42no—
276C2×D8—42no—
277C2×D8—42no—
278C2×D8C22×C418yes—
279C2×D8C22×C418yes—
280C22×C8C2414yes—
281C22×C8C2412yes—
282C22×C8C2412yes—
283C4⋊Q8—22no—
284C4⋊Q8C22×C414yes—
285#127—22no—
286#127C22×C414yes—
287#124—28no—
288#121—28no—
289#121—22no—
290#121—22no—
291#121C22×C414yes—
292#121C22×C414yes—
293C4×D4—28no—
294C4×D4—28no—
295C4×D4—28no—
296C4×D4—24no—
297C4×D4—24no—
298C4×D4—22no—
299C4×D4—22no—
300C4×D4—44no—
301C4×D4—44no—
302C4×D4—48no—
303C4×D4—84no—
304C4×D4—84no—
305C4×D4C22×C418yes—
306C4×D4C22×C418yes—
307C4×D4C22×C414yes—
308C4×D4C22×C414yes—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4C22×C414yes—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2414yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2411yescharacteristic
328C2×C22⋊C4—28no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—48no—
347C2×C22⋊C4—84no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—24no—
352C2×C42—24no—
353C2×C42—44no—
354C2×C42—84no—
355C2×C42C22×C414yes—
356C2×C42C2×D412yes—
357C2×C42C2×D412yes—
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42#4114yes—
361C2×C42C2412yes—
362C2×C42C2412yes—
363C2×C42C2411yescharacteristic
364C2×C42C2411yescharacteristic
365#107—44no—
366#107C22×C418yes—
367#107C22×C418yes—
368#106—44no—
369C4⋊C8—24no—
370C4⋊C8C22×C418yes—
371#102—44no—
372#102—44no—
373#102—44no—
374#102—44no—
375#102C22×C418yes—
376#102C22×C418yes—
377#102C22×C418yes—
378#102C22×C418yes—
379C22⋊C8—24no—
380C22⋊C8C22×C414yes—
381C22⋊C8C22×C414yes—
382#97—24no—
383#97—44no—
384C4×C8—44no—
385C4×C8C22×C418yes—
386#95—21no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—24no—
392#95—24no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—21no—
397#95—21no—
398#95—21no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95C2×D411yescharacteristic
403#95C2×D411yescharacteristic
404#95C2×D411yescharacteristic
405#95C2×D412yes—
406#95C2×D411yescharacteristic
407#95C2×D412yes—
408C24—24no—
409C24—22no—
410C24—21no—
411C24—21no—
412C24—44no—
413C24—42no—
414C24—42no—
415C24—44no—
416C24C22×D411yescharacteristic
417#41—816no—
418C2×Q8—24no—
419C2×Q8—82no—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—22no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—48no—
437C2×D4—44no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4—48no—
441C2×D4—44no—
442C2×D4—44no—
443C2×D4—82no—
444C2×D4C23×C414yes—
445C2×D4C23×C412yes—
446C2×D4C23×C412yes—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—28no—
458C22×C4—28no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—816no—
475C22×C4—84no—
476C22×C4—84no—
477C22×C4—82no—
478C22×C4—82no—
479C22×C4—82no—
480C22×C4—82no—
481C22×C4—82no—
482C22×C4—82no—
483C22×C4C4×D414yes—
484C22×C4C4×D414yes—
485C22×C4C23×C412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D412yes—
495C22×C4C2×#4112yes—
496C22×C4C2511yescharacteristic
497D8—48no—
498D8—48no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—24no—
504C2×C8—24no—
505C2×C8—24no—
506C2×C8C23×C414yes—
507C2×C8C23×C414yes—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—24no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—44no—
522C4⋊C4—82no—
523C4⋊C4—82no—
524C4⋊C4C23×C414yes—
525C4⋊C4C23×C412yes—
526C4⋊C4C23×C412yes—
527C22⋊C4—28no—
528C22⋊C4—28no—
529C22⋊C4—24no—
530C22⋊C4—24no—
531C22⋊C4—24no—
532C22⋊C4—24no—
533C22⋊C4—48no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—48no—
537C22⋊C4—44no—
538C22⋊C4—84no—
539C22⋊C4—84no—
540C22⋊C4—82no—
541C22⋊C4—82no—
542C22⋊C4—82no—
543C22⋊C4—82no—
544C42—24no—
545C42—24no—
546C42—42no—
547C42—42no—
548C42—44no—
549C42—84no—
550C42—84no—
551C42—82no—
552C42—82no—
553C42C4×D414yes—
554C42C4×D414yes—
555C42C23×C412yes—
556C42C23×C412yes—
557C42C2×#4112yes—
558C42C2×#4112yes—
559C23—21no—
560C23—21no—
561C23—22no—
562C23—24no—
563C23—22no—
564C23—22no—
565C23—48no—
566C23—48no—
567C23—48no—
568C23—41no—
569C23—41no—
570C23—42no—
571C23—42no—
572C23—41no—
573C23—41no—
574C23—44no—
575C23—42no—
576C23—42no—
577C23—44no—
578C23—42no—
579C23—42no—
580C23—88no—
581C23—88no—
582C23C2×C4×D412yes—
583C23#57614yes—
584C23C23×D411yescharacteristic
585Q8—84no—
586D4—24no—
587D4—24no—
588D4—24no—
589D4—24no—
590D4—48no—
591D4—48no—
592D4—48no—
593D4—48no—
594D4—42no—
595D4—42no—
596D4—42no—
597D4—42no—
598D4—84no—
599C2×C4—28no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—44no—
623C2×C4—48no—
624C2×C4—48no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—88no—
638C2×C4—88no—
639C2×C4—84no—
640C2×C4—84no—
641C2×C4—84no—
642C2×C4—88no—
643C2×C4—164no—
644C2×C4—164no—
645C2×C4C2×C4×D412yes—
646C2×C4C2×C4×D412yes—
647C2×C4C2×C4×D412yes—
648C2×C4C2×C4×D412yes—
649C2×C4C2×C4×D412yes—
650C2×C4#57514yes—
651C2×C4C24×C411yescharacteristic
652C2×C4C22×#4111yescharacteristic
653C2×C4#58512yes—
654C2×C4#58512yes—
655C8—24no—
656C8—24no—
657C8—44no—
658C22—24no—
659C22—22no—
660C22—24no—
661C22—22no—
662C22—44no—
663C22—42no—
664C22—42no—
665C22—44no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—44no—
671C22—816no—
672C22—82no—
673C22—82no—
674C22—84no—
675C22—82no—
676C22—82no—
677C22—84no—
678C22#294814yes—
679C22C22×C4×D411yescharacteristic
680C22#348511yescharacteristic
681C22#348711yescharacteristic
682C22C2×#57512yes—
683C22C2×#57612yes—
684C4—22no—
685C4—22no—
686C4—24no—
687C4—22no—
688C4—22no—
689C4—22no—
690C4—44no—
691C4—42no—
692C4—42no—
693C4—42no—
694C4—42no—
695C4—42no—
696C4—82no—
697C4—82no—
698C4—84no—
699C4#342812yes—
700C4#349312yes—
701C2—24no—
702C2—82no—
703C2—82no—
704C2—84no—
705C2#3420111yescharacteristic
706C2C4×#57511yescharacteristic
707C2C2×#294811yescharacteristic
708C2#6172712yes—
709C2#6189812yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC2×#3100C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34236C2semidirect
3. C2×#2357C2semidirect
4. #23108C2semidirect
5. #23079C2semidirect
6. #22816C2semidirect
7. #22810C2semidirect
8. #22772C2semidirect
9. #22764C2semidirect
10. #22046C2semidirect
11. #22040C2semidirect
12. #21983C2semidirect
13. #21865C2semidirect
14. #21578C2semidirect
15. C4×#480C2semidirect
16. #20739C2semidirect
17. #20535C2semidirect
20. #3100C4semidirect
31. #2369C22semidirect
32. #2366C22semidirect
40. #2357C22semidirect
41. #2357C22semidirect
42. #2349C22semidirect
43. #2349C22semidirect
59. #1993C22semidirect
62. #1961C22semidirect
63. #1949C22semidirect
67. #1926C22semidirect
69. #1925C22semidirect
70. #1921C22semidirect
71. #1918C22semidirect
78. #1846C22semidirect
79. #1835C22semidirect
83. #1804C22semidirect
84. #1804C22semidirect
86. #1776C22semidirect
87. C4×#107C22semidirect
88. #1771C22semidirect
89. #1770C22semidirect
90. C8×C4⋊C4C22semidirect
92. #1762C22semidirect
93. C4×#102C22semidirect
116. C2×C4×D4C2×C4semidirect
132. #480C2×C4semidirect
133. #480C2×C4semidirect
134. #474C2×C4semidirect
135. #459C2×C4semidirect
136. #447C2×C4semidirect
137. #447C2×C4semidirect
138. C4×D8C2×C4semidirect
139. C4×D8C2×C4semidirect
141. #431C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31002
#342362
C2×#23572
#231082
#230792
#228162
#228102
#227722
#227642
#220462
#220402
#219832
#218652
#215782
C4×#4802
#207392
#205352