Finite Group Databasebeta

← #6457226

Subgroups — #6457226

Order 512.

2150 conjugacy classes of subgroups.

The subgroup lattice has 723 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3098C212yesmaximal
2#34267C211yesmaximal, characteristic
3#34183C211yesmaximal, characteristic, complemented
4#27656C214yesmaximal, complemented
5#27655C214yesmaximal, complemented
6#27652C212yesmaximal, complemented
7#27650C212yesmaximal, complemented
8#27648C212yesmaximal, complemented
9#27644C212yesmaximal, complemented
10#23657C211yesmaximal, characteristic, complemented
11C2×#2083C211yesmaximal, characteristic, complemented
12#23630C211yesmaximal, characteristic, complemented
13#23627C211yesmaximal, characteristic, complemented
14#21173C212yesmaximal, complemented
15#20703C212yesmaximal, complemented
16#20652C212yesmaximal, complemented
17#20630C211yesmaximal, characteristic, complemented
18C2×#537C2211yescharacteristic
19C2×#520C2211yescharacteristic
20C2×#520C2212yes—
21C2×C4×#41C2211yescharacteristic
22#3098—216no—
23C2×#484C2214yes—
24C2×#485C2214yes—
25C2×#482C2214yes—
26C2×#480C2214yes—
27C2×#428C2212yes—
28C22×#107C2212yes—
29C2×#424C2212yes—
30C2×#417C2212yes—
31C2×#416C2212yes—
32C2×#408C2212yes—
33#2364—22no—
34#2363—21no—
35#2362—21no—
36#2361—22no—
37#2361—21no—
38#2360—21no—
39Q8×C4⋊C4—21no—
40Q8×C4⋊C4—21no—
41D4×C4⋊C4—22no—
42D4×C4⋊C4—21no—
43D4×C4⋊C4—21no—
44D4×C4⋊C4—22no—
45#2324C2212yes—
46#2323C2212yes—
47#2317C2214yes—
48#2317C2212yes—
49#2311C2211yescharacteristic
50C4×C22⋊Q8C2212yes—
51C4×#121C2212yes—
52#2297C2211yescharacteristic
53C2×#389C2211yescharacteristic
54#2285C2212yes—
55#2285C2211yescharacteristic
56C2×#384C2212yes—
57C2×#384C2211yescharacteristic
58#2274C2211yescharacteristic
59#2085C2214yes—
60#2084C2212yes—
61#2083C2212yes—
62#2083C2212yes—
63#2083C2212yes—
64#2083C2214yes—
65#2074C2212yes—
66#2073C2212yes—
67#2072C2211yescharacteristic
68#2072C2212yes—
69#2072C2211yescharacteristic
70#2072C2212yes—
71#2071C2211yescharacteristic
72#2071C2212yes—
73#2071C2212yes—
74#2071C2211yescharacteristic
75#1949C2214yes—
76#1949C2214yes—
77#1947C2214yes—
78#1947C2214yes—
79#1825C2214yes—
80#1825C2214yes—
81#1824C2214yes—
82#1823C2214yes—
83#1805C2214yes—
84#1804C2214yes—
85#1803C2214yes—
86#1802C2214yes—
87#1798C2214yes—
88#1797C2212yes—
89#1797C2212yes—
90#1736C2212yes—
91#1736C2212yes—
92#1735C2212yes—
93C2×#340C2212yes—
94#1728C2211yescharacteristic
95C2×#336C2211yescharacteristic
96C22×#41C2311yescharacteristic
97C22×#41C2311yescharacteristic
98C22×M16C2312yes—
99C23×C8C2312yes—
100#537—24no—
101#537D418yes—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8C2312yes—
104C2×#121—21no—
105C2×#121—22no—
106C2×#121C2312yes—
107#520—28no—
108#520—28no—
109#520—28no—
110C4×#41—28no—
111C2×C4×Q8—22no—
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8C2312yes—
115C2×C4×D4—24no—
116C2×C4×D4—22no—
117C2×C4×D4—22no—
118C2×C4×D4—22no—
119C2×C4×D4D411yescharacteristic
120C2×C4×D4D412yes—
121C2×C4×D4D411yescharacteristic
122C2×C4×D4D412yes—
123C2×C4×D4C2312yes—
124C2×#117—21no—
125C2×#117C2311yescharacteristic
126C2×#117C2314yes—
127C2×#117C2312yes—
128C2×#117C2312yes—
129C2×#117C2311yescharacteristic
130C22×C4⋊C4—21no—
131C22×C4⋊C4—22no—
132C22×C4⋊C4—21no—
133C22×C4⋊C4—22no—
134C22×C4⋊C4—21no—
135C22×C4⋊C4—21no—
136C22×C4⋊C4C2311yescharacteristic
137C22×C4⋊C4C2314yes—
138C22×C4⋊C4C2312yes—
139C22×C4⋊C4C2311yescharacteristic
140C22×C42C2312yes—
141C22×C42C2311yescharacteristic
142#485—216no—
143#484—216no—
144#482—216no—
145#480—216no—
146#428—28no—
147C2×#107—28no—
148C2×#107C2314yes—
149C2×#107C2314yes—
150C2×#107C2314yes—
151C2×#106C2314yes—
152#424—28no—
153C2×C4⋊C8C2314yes—
154C2×C4⋊C8C2314yes—
155#417—28no—
156#416—28no—
157C2×#103C2312yes—
158C2×#103C2314yes—
159C2×#103C2312yes—
160C2×#102C2312yes—
161C2×#102C2314yes—
162C2×#102C2312yes—
163#408—28no—
164C2×C22⋊C8C2314yes—
165C2×C22⋊C8C2314yes—
166#391—21no—
167#391C2311yescharacteristic
168#391C2311yescharacteristic
169#390—21no—
170#390—22no—
171#390C2311yescharacteristic
172#390C2311yescharacteristic
173#389—24no—
174#389—24no—
175#389—22no—
176#389—22no—
177#389—22no—
178#389—21no—
179#389—21no—
180#389C2312yes—
181#389C2312yes—
182#388—22no—
183#388—22no—
184#388—22no—
185#388—22no—
186#387—22no—
187#387—22no—
188#386—22no—
189#386—21no—
190#386—22no—
191#386—21no—
192#386—22no—
193#385—22no—
194#385—22no—
195#385—24no—
196#385—24no—
197#385—24no—
198#384—22no—
199#384—21no—
200#384—21no—
201#384—22no—
202#384C2314yes—
203#384C2312yes—
204#384C2312yes—
205#384C2311yescharacteristic
206#384C2311yescharacteristic
207#382C2314yes—
208#382C2312yes—
209#380—21no—
210#380—22no—
211#380—21no—
212#380—22no—
213#380C2312yes—
214#380C2312yes—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—22no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2312yes—
224C4×C22⋊C4—22no—
225C4×C22⋊C4—22no—
226C4×C22⋊C4—21no—
227C4×C22⋊C4—22no—
228C4×C22⋊C4—21no—
229C2×#95—21no—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#340C2312yes—
237#340C2312yes—
238#340C2312yes—
239#340C2312yes—
240#336C2311yescharacteristic
241#336C2311yescharacteristic
242#336C2312yes—
243#336C2312yes—
244#336C2312yes—
245C2×#41—24no—
246C2×#41—22no—
247C2×#41—24no—
248C2×#41—24no—
249C2×#41—24no—
250C2×#41C2×D414yes—
251C2×#41C2×D414yes—
252C2×#41#4114yes—
253C22×Q8C2×D411yescharacteristic
254C22×Q8C2411yescharacteristic
255C22×D4—21no—
256C22×D4—22no—
257C22×D4C2×D412yes—
258C22×D4C2×D411yescharacteristic
259C22×D4C2411yescharacteristic
260C23×C4—21no—
261C23×C4—21no—
262C23×C4—24no—
263C23×C4—22no—
264C23×C4—21no—
265C23×C4—22no—
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D412yes—
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D412yes—
270C23×C4C2412yes—
271C23×C4C2411yescharacteristic
272C2×M16—24no—
273C2×M16—24no—
274C2×M16—24no—
275C22×C8—24no—
276C22×C8—24no—
277C22×C8—24no—
278C22×C8C2412yes—
279C22×C8C2412yes—
280C22×C8C2414yes—
281C22⋊Q8—28no—
282C22⋊Q8—24no—
283C22⋊Q8C2×D418yes—
284#121—28no—
285#121—216no—
286#121—24no—
287#121C2×D418yes—
288C4×Q8—24no—
289C4×Q8—24no—
290C4×Q8—28no—
291C4×Q8—48no—
292C4×D4—24no—
293C4×D4—24no—
294C4×D4—28no—
295C4×D4—28no—
296C4×D4—28no—
297C4×D4—416no—
298C4×D4—48no—
299C4×D4—48no—
300C4×D4—84no—
301#117—28no—
302#117—24no—
303#117—28no—
304#117—28no—
305#117—24no—
306#117—24no—
307#117—44no—
308#117#4118yes—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—28no—
314C2×C4⋊C4—28no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—44no—
334C2×C4⋊C4—82no—
335C2×C4⋊C4—84no—
336C2×C4⋊C4—82no—
337C2×C4⋊C4—84no—
338C2×C4⋊C4—82no—
339C2×C4⋊C4C2×D412yes—
340C2×C4⋊C4C2×D412yes—
341C2×C4⋊C4C2×D414yes—
342C2×C4⋊C4C2×D411yescharacteristic
343C2×C4⋊C4C2×D411yescharacteristic
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2414yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2411yescharacteristic
349C2×C4⋊C4C2411yescharacteristic
350C2×C4⋊C4C2412yes—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4C2×D412yes—
364C2×C22⋊C4C2×D412yes—
365C2×C22⋊C4C2×D412yes—
366C2×C42—24no—
367C2×C42—28no—
368C2×C42—22no—
369C2×C42—24no—
370C2×C42—22no—
371C2×C42—21no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42C2×D412yes—
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D412yes—
378C2×C42C2411yescharacteristic
379C2×C42C2411yescharacteristic
380C2×C42C2412yes—
381#107—28no—
382#107—28no—
383#107—28no—
384#106—28no—
385C4⋊C8—28no—
386C4⋊C8—28no—
387#103—24no—
388#103—24no—
389#103—28no—
390#102—24no—
391#102—24no—
392#102—28no—
393C22⋊C8—28no—
394C22⋊C8—28no—
395#95—22no—
396#95—22no—
397#95—21no—
398#95—21no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—21no—
403#95—21no—
404#95—22no—
405#95—22no—
406#95—24no—
407#95—24no—
408#95—22no—
409C24—21no—
410C24C22×D412yes—
411C24C22×D411yescharacteristic
412#41—216no—
413#41—216no—
414#41—216no—
415#41—48no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—21no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8#12114yes—
422C2×Q8C22×D412yes—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—28no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—21no—
431C2×D4—22no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—82no—
435C2×D4—82no—
436C2×D4#12118yes—
437C2×D4#12114yes—
438C2×D4C22×D412yes—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—42no—
460C22×C4—41no—
461C22×C4—42no—
462C22×C4—41no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—48no—
466C22×C4—44no—
467C22×C4—42no—
468C22×C4—44no—
469C22×C4—82no—
470C22×C4—82no—
471C22×C4—81no—
472C22×C4—81no—
473C22×C4—82no—
474C22×C4—84no—
475C22×C4—81no—
476C22×C4—82no—
477C22×C4—84no—
478C22×C4—81no—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D412yes—
482C22×C4C22×D414yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D412yes—
489C22×C4C2×#4114yes—
490C22×C4C2×#4112yes—
491C22×C4C2×#4114yes—
492C22×C42−1+414yes—
493C22×C4C2511yescharacteristic
494M16—48no—
495C2×C8—24no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—24no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—24no—
504C2×C8—48no—
505C4⋊C4—24no—
506C4⋊C4—28no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—24no—
513C4⋊C4—22no—
514C4⋊C4—24no—
515C4⋊C4—24no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—44no—
522C4⋊C4—48no—
523C4⋊C4—44no—
524C4⋊C4—42no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—42no—
528C4⋊C4—44no—
529C4⋊C4—42no—
530C4⋊C4—84no—
531C4⋊C4—84no—
532C4⋊C4—82no—
533C4⋊C4—82no—
534C4⋊C4—82no—
535C4⋊C4#12314yes—
536C4⋊C4#12314yes—
537C4⋊C4C22×D414yes—
538C22⋊C4—28no—
539C22⋊C4—28no—
540C22⋊C4—24no—
541C22⋊C4—24no—
542C22⋊C4—48no—
543C22⋊C4—44no—
544C22⋊C4—42no—
545C22⋊C4—44no—
546C22⋊C4—42no—
547C22⋊C4—42no—
548C22⋊C4—81no—
549C22⋊C4—81no—
550C22⋊C4—82no—
551C22⋊C4—82no—
552C22⋊C4#12318yes—
553C42—24no—
554C42—24no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—48no—
560C42—44no—
561C42—44no—
562C42—42no—
563C42—82no—
564C42—82no—
565C42C2×#4118yes—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—22no—
574C23—22no—
575C23—22no—
576C23—22no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23C2×#12114yes—
584C23C2×#12112yes—
585C23C2×#12312yes—
586C23#53612yes—
587C23C23×D411yescharacteristic
588Q8—28no—
589Q8—24no—
590Q8—42no—
591D4—216no—
592D4—24no—
593D4—28no—
594D4—48no—
595D4—42no—
596D4—88no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—28no—
608C2×C4—28no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—24no—
612C2×C4—22no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—48no—
618C2×C4—48no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—48no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—41no—
643C2×C4—42no—
644C2×C4—84no—
645C2×C4—164no—
646C2×C4—164no—
647C2×C4—168no—
648C2×C4—162no—
649C2×C4C2×#12114yes—
650C2×C4C2×#12112yes—
651C2×C4C2×#12314yes—
652C2×C4C2×#12314yes—
653C2×C4C2×#12312yes—
654C2×C4#53612yes—
655C2×C4#54718yes—
656C2×C4#55218yes—
657C2×C4#57612yes—
658C2×C4#57812yes—
659C2×C4C23×D411yescharacteristic
660C2×C4C22×#4112yes—
661C2×C4C22×#4111yescharacteristic
662C2×C4C2×2−1+412yes—
663C8—42no—
664C8—42no—
665C8—44no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—24no—
673C22—24no—
674C22—24no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22—42no—
681C22—42no—
682C22—81no—
683C22—81no—
684C22—81no—
685C22—81no—
686C22#248614yes—
687C22C22×#12111yescharacteristic
688C22C22×#12311yescharacteristic
689C22C2×#53611yescharacteristic
690C22C2×#57611yescharacteristic
691C22C2×#57811yescharacteristic
692C22#358511yescharacteristic
693C22#358611yescharacteristic
694C4—22no—
695C4—24no—
696C4—22no—
697C4—24no—
698C4—24no—
699C4—44no—
700C4—41no—
701C4—41no—
702C4—42no—
703C4—84no—
704C4—82no—
705C4—82no—
706C4—82no—
707C4—81no—
708C4—81no—
709C4#248614yes—
710C4C2×#54712yes—
711C4C2×#55212yes—
712C2—24no—
713C2—44no—
714C2—82no—
715C2C2×#248611yescharacteristic
716C2#6042811yescharacteristic
717C2#6043211yescharacteristic
718C2#6052111yescharacteristic
719C2#6052611yescharacteristic
720C2#6088011yescharacteristic
721C2#6088711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#537C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #34183C2semidirect
4. #27656C2semidirect
5. #27655C2semidirect
6. #27652C2semidirect
7. #27650C2semidirect
8. #27648C2semidirect
9. #27644C2semidirect
10. #23657C2semidirect
11. C2×#2083C2semidirect
12. #23630C2semidirect
13. #23627C2semidirect
14. #21173C2semidirect
15. #20703C2semidirect
16. #20652C2semidirect
17. #20630C2semidirect

Maximal subgroups

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

subgroupindex
C2×#30982
#342672
#341832
#276562
#276552
#276522
#276502
#276482
#276442
#236572
C2×#20832
#236302
#236272
#211732
#207032
#206522
#206302