Finite Group Databasebeta

← #6388449

Subgroups — #6388449

Order 512.

2046 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3101C211yesmaximal, characteristic
2#34225C211yesmaximal, characteristic, complemented
3#34126C211yesmaximal, characteristic
4#22945C214yesmaximal, complemented
5#22864C214yesmaximal
6#22806C212yesmaximal, complemented
7#22768C212yesmaximal, complemented
8#22765C212yesmaximal, complemented
9#22673C212yesmaximal, complemented
10#22146C212yesmaximal, complemented
11#22117C212yesmaximal, complemented
12#21985C211yesmaximal, characteristic, complemented
13#21892C211yesmaximal, characteristic, complemented
14C4×#476C212yesmaximal, complemented
15#21504C212yesmaximal, complemented
16#20646C211yesmaximal, characteristic
17#20550C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20#3101C4116yescomplemented
21C2×#483C2212yes—
22C2×#476C2211yescharacteristic
23C2×#476C2211yescharacteristic
24C2×#461C2211yescharacteristic
25C2×#461C2211yescharacteristic
26C2×#450C2212yes—
27C2×#440C2214yes—
28C2×C4×SD16C2212yes—
29C2×C4×SD16C2212yes—
30C2×#431C2211yescharacteristic
31C2×#423C2211yescharacteristic
32C2×#418C2212yes—
33#2370—22no—
34#2362C2214yes—
35#2357C2211yescharacteristic
36#2357C2211yescharacteristic
37#2356C2212yescomplemented
38#2353—21no—
39#2353—21no—
40#2351C2212yescomplemented
41#2349C2211yescharacteristic
42#2349C2211yescharacteristic
43#2347—21no—
44#2346—21no—
45#2345C2212yes—
46#2342—22no—
47D4×C22⋊C4C2212yes—
48#2323—22no—
49#2320—21no—
50C4×C4⋊Q8C2212yescomplemented
51C4×C4⋊Q8C2212yescomplemented
52#2293C2212yes—
53#2290—22no—
54#2285C2211yescharacteristic
55#2283C2211yescharacteristic
56#2280—21no—
57#2276—21no—
58C42×D4C2212yes—
59C4×#117C2211yescharacteristic
60C2×C4×C22⋊C4—21no—
61#1987C2212yescomplemented
62#1981C2212yescomplemented
63#1968C2214yes—
64#1948C2212yescomplemented
65#1947C2212yescomplemented
66#1944C2214yescomplemented
67#1936C2214yes—
68#1933C2214yescomplemented
69#1928C2214yescomplemented
70#1925C2214yes—
71#1920C2214yescomplemented
72#1920C2214yescomplemented
73#1894C2214yes—
74#1880C2214yes—
75#1877C2214yes—
76#1865C2214yescomplemented
77#1864C2214yes—
78#1854C2214yes—
79#1846C2212yescomplemented
80#1835C2212yescomplemented
81#1805C2212yescomplemented
82#1805C2212yescomplemented
83#1804C2212yes—
84#1804C2212yes—
85#1802C2212yes—
86#1802C2212yes—
87C4×#106C2214yescomplemented
88#1770C2212yescomplemented
89C8×C4⋊C4C2212yescomplemented
90#1767C2212yescomplemented
91C4×C4⋊C8C2212yescomplemented
92#1763C2214yescomplemented
93C4×#102C2214yes—
94#1753C2214yes—
95#1736C2212yes—
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C22×SD16C2312yes—
99C22×SD16C2312yes—
100C22×M16C2312yes—
101#533—24no—
102#533C2×C418yescomplemented
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×#125—21no—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8C2312yes—
108C2×C4×Q8—21no—
109C2×C4×Q8—22no—
110C2×C4×Q8C2312yes—
111C2×C4×Q8C2314yes—
112C2×C4×D4—21no—
113C2×C4×D4—21no—
114C2×C4×D4—24no—
115C2×C4×D4C2×C418yes—
116C2×C4×D4C2312yes—
117C2×C4×D4C2312yes—
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2311yescharacteristic
122C2×#117—21no—
123C2×#117—22no—
124C2×#117D412yes—
125C2×#117C2312yes—
126C22×C4⋊C4C2311yescharacteristic
127C22×C22⋊C4—21no—
128C22×C22⋊C4—22no—
129C22×C42—22no—
130C22×C42C2311yescharacteristic
131#483C2×C4116yescomplemented
132#476C2×C418yes—
133#476C2×C418yescomplemented
134#461C2×C418yes—
135#461C2×C418yescomplemented
136#450C2×C4116yes—
137#440—48no—
138C4×SD16C2×C4116yes—
139C4×SD16C2×C4116yescomplemented
140#431C2×C418yescomplemented
141C2×#107C2314yes—
142C2×#106C2314yes—
143C2×#106C2312yes—
144C2×#106C2312yes—
145#423—24no—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2311yescharacteristic
149C2×C4⋊C8C2311yescharacteristic
150#418—44no—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2312yes—
154C2×#103C2312yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2312yes—
158C2×#102C2312yes—
159C2×C22⋊C8C2312yes—
160C2×#97C2314yes—
161C2×C4×C8C2314yes—
162C2×C4×C8C2312yes—
163#391—22no—
164#391C2314yes—
165#390C2312yes—
166#390C2312yes—
167#389—24no—
168#389—22no—
169#389—22no—
170#389—22no—
171#389C2314yes—
172#389C2312yes—
173#389C2312yes—
174#388—22no—
175#388—22no—
176#388—24no—
177#388—24no—
178#388—22no—
179#388—22no—
180#387—24no—
181#387—24no—
182#387—22no—
183#387—22no—
184#386—22no—
185#386—22no—
186#385—22no—
187#385—22no—
188#384C2312yes—
189#383—21no—
190#383—22no—
191#382—22no—
192#382C2312yes—
193#382C2311yescharacteristic
194#382C2311yescharacteristic
195#381—21no—
196#381—21no—
197#381—21no—
198#380—21no—
199#380—21no—
200#380—22no—
201#380—21no—
202#380—21no—
203#380C2312yes—
204#379—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—24no—
212C4×C4⋊C4D412yes—
213C4×C4⋊C4C2314yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C22⋊C4—21no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—22no—
224C4×C22⋊C4—22no—
225C4×C22⋊C4—21no—
226C4×C22⋊C4—21no—
227C4×C22⋊C4D411yescharacteristic
228C4×C22⋊C4D411yescharacteristic
229C4×C22⋊C4C2312yes—
230#376—21no—
231#376—22no—
232#376D411yescharacteristic
233#376D411yescharacteristic
234C2×#95—21no—
235C2×#95—21no—
236C43C2312yes—
237#340C2314yes—
238#340C2314yes—
239#340C2314yes—
240#340C2314yes—
241#340C2314yes—
242#340C2314yes—
243#336C2314yes—
244#336C2314yes—
245C25—21no—
246C22×Q8—21no—
247C22×Q8C2412yes—
248C22×D4—22no—
249C22×D4—22no—
250C22×D4—21no—
251C22×D4—22no—
252C22×D4—41no—
253C22×D4—41no—
254C22×D4C22×C414yes—
255C22×D4C2411yescharacteristic
256C22×D4C2411yescharacteristic
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—28no—
261C23×C4—21no—
262C23×C4—21no—
263C23×C4—22no—
264C23×C4—22no—
265C23×C4—84no—
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2411yescharacteristic
269C2×SD16—24no—
270C2×SD16—24no—
271C2×SD16—42no—
272C2×SD16—42no—
273C2×SD16C22×C418yes—
274C2×SD16C22×C418yes—
275C2×M16—24no—
276C2×M16—42no—
277C2×M16—42no—
278C22×C8C2414yes—
279C22×C8C2412yes—
280C22×C8C2412yes—
281C4⋊Q8—22no—
282C4⋊Q8—22no—
283C4⋊Q8C22×C414yes—
284C4⋊Q8C22×C414yes—
285#125—28no—
286C22⋊Q8—28no—
287C22⋊Q8—24no—
288C22⋊Q8C22×C418yes—
289C4×Q8—28no—
290C4×Q8—24no—
291C4×Q8—48no—
292C4×Q8—44no—
293C4×Q8—44no—
294C4×Q8C22×C418yes—
295C4×D4—24no—
296C4×D4—24no—
297C4×D4—28no—
298C4×D4—28no—
299C4×D4—22no—
300C4×D4—22no—
301C4×D4—22no—
302C4×D4—22no—
303C4×D4—42no—
304C4×D4—42no—
305C4×D4C22×C414yes—
306C4×D4C22×C414yes—
307C4×D4C22×C414yes—
308C4×D4C22×C414yes—
309#117—28no—
310#117—44no—
311#117—42no—
312#117—42no—
313#117—84no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4C22×C414yes—
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2414yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2412yes—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—28no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—48no—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—24no—
356C2×C42—22no—
357C2×C42—24no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—84no—
361C2×C42C22×C414yes—
362C2×C42C2×D412yes—
363C2×C42C2×D412yes—
364C2×C42C2×D412yes—
365C2×C42C2×D412yes—
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42#4114yes—
369C2×C42C2412yes—
370C2×C42C2412yes—
371C2×C42C2411yescharacteristic
372C2×C42C2411yescharacteristic
373#107—44no—
374#106—44no—
375#106C22×C418yes—
376#106C22×C418yes—
377C4⋊C8—24no—
378C4⋊C8—24no—
379C4⋊C8C22×C414yes—
380C4⋊C8C22×C414yes—
381#103—44no—
382#103—44no—
383#103C22×C418yes—
384#103C22×C418yes—
385#102—28no—
386#102—44no—
387#102C22×C418yes—
388#102C22×C418yes—
389C22⋊C8C22×C418yes—
390#97—44no—
391C4×C8—44no—
392C4×C8C22×C418yes—
393#95—21no—
394#95—21no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—24no—
399#95—22no—
400#95—21no—
401#95—21no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—24no—
407#95—24no—
408#95—24no—
409#95C2×D411yescharacteristic
410#95C2×D411yescharacteristic
411#95C2×D411yescharacteristic
412#95C2×D411yescharacteristic
413#95C2×D412yes—
414#95C2×D412yes—
415C24—21no—
416C24—21no—
417C24—24no—
418C24—42no—
419C24—42no—
420C24—44no—
421C24—82no—
422C24C22×D411yescharacteristic
423C2×Q8—24no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—44no—
430C2×Q8C23×C414yes—
431C2×D4—28no—
432C2×D4—24no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—44no—
442C2×D4—44no—
443C2×D4—44no—
444C2×D4—42no—
445C2×D4—42no—
446C2×D4C23×C412yes—
447C2×D4C23×C412yes—
448C2×D4H(5,𝔽2)12yes—
449C2×D42−1+412yes—
450C22×C4—24no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—22no—
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—82no—
476C22×C4—82no—
477C22×C4—82no—
478C22×C4—82no—
479C22×C4—82no—
480C22×C4C4×D414yes—
481C22×C4C4×D414yes—
482C22×C4C23×C412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D412yes—
491C22×C4C22×D411yescharacteristic
492C22×C4C2×#4112yes—
493C22×C4C2511yescharacteristic
494SD16—216no—
495SD16—48no—
496M16—84no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—24no—
502C2×C8—24no—
503C2×C8—24no—
504C2×C8C23×C414yes—
505C2×C8C23×C414yes—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—28no—
511C4⋊C4—28no—
512C4⋊C4—24no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—22no—
516C4⋊C4—24no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—24no—
522C4⋊C4—42no—
523C4⋊C4—44no—
524C4⋊C4—48no—
525C4⋊C4—44no—
526C4⋊C4—82no—
527C4⋊C4—82no—
528C4⋊C4—84no—
529C4⋊C4C23×C414yes—
530C4⋊C4C23×C412yes—
531C4⋊C4C23×C412yes—
532C22⋊C4—28no—
533C22⋊C4—24no—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—24no—
537C22⋊C4—24no—
538C22⋊C4—42no—
539C22⋊C4—44no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—82no—
545C22⋊C4—82no—
546C22⋊C4—82no—
547C42—28no—
548C42—24no—
549C42—24no—
550C42—24no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—42no—
556C42—48no—
557C42—44no—
558C42—82no—
559C42—82no—
560C42—82no—
561C42—84no—
562C42C4×D414yes—
563C42C4×D414yes—
564C42C23×C412yes—
565C42C23×C412yes—
566C42C2×#4112yes—
567C42C2×#4112yes—
568C23—22no—
569C23—22no—
570C23—21no—
571C23—22no—
572C23—21no—
573C23—22no—
574C23—22no—
575C23—22no—
576C23—48no—
577C23—48no—
578C23—42no—
579C23—42no—
580C23—41no—
581C23—41no—
582C23—42no—
583C23—42no—
584C23—84no—
585C23—88no—
586C23—88no—
587C23C2×C4×D412yes—
588C23C23×D411yescharacteristic
589Q8—24no—
590Q8—24no—
591Q8—48no—
592Q8—42no—
593Q8—42no—
594D4—22no—
595D4—22no—
596D4—22no—
597D4—22no—
598D4—22no—
599D4—48no—
600D4—41no—
601D4—41no—
602D4#52014yes—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—28no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—48no—
635C2×C4—48no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—84no—
646C2×C4—84no—
647C2×C4—88no—
648C2×C4—84no—
649C2×C4—88no—
650C2×C4C2×C4×D412yes—
651C2×C4C2×C4×D412yes—
652C2×C4C2×C4×D412yes—
653C2×C4C2×C4×D412yes—
654C2×C4C2×C4×D412yes—
655C2×C4#57514yes—
656C2×C4#57714yes—
657C2×C4C24×C411yescharacteristic
658C2×C4C22×#4111yescharacteristic
659C2×C4C2×H(5,𝔽2)11yescharacteristic
660C2×C4C2×2−1+411yescharacteristic
661C2×C4#58511yescharacteristic
662C2×C4#58511yescharacteristic
663C8—24no—
664C8—24no—
665C8—44no—
666C22—22no—
667C22—22no—
668C22—24no—
669C22—22no—
670C22—42no—
671C22—44no—
672C22—42no—
673C22—44no—
674C22—42no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22—42no—
679C22—816no—
680C22—82no—
681C22—82no—
682C22C22×C4×D411yescharacteristic
683C22#348811yescharacteristic
684C22#348911yescharacteristic
685C22C2×#57512yes—
686C22C2×#57712yes—
687C4—22no—
688C4—22no—
689C4—24no—
690C4—22no—
691C4—22no—
692C4—42no—
693C4—42no—
694C4—42no—
695C4—42no—
696C4—44no—
697C4—42no—
698C4—44no—
699C4—84no—
700C4—82no—
701C4—82no—
702C4#294914yes—
703C4C2×#52011yescharacteristic
704C4#342811yescharacteristic
705C4#349411yescharacteristic
706C4#349511yescharacteristic
707C2—42no—
708C2—44no—
709C2—82no—
710C2#3420211yescharacteristic
711C2C4×#57511yescharacteristic
712C2C2×#294911yescharacteristic
713C2#6199612yes—
714C2#6212612yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34225C2semidirect
4. #22945C2semidirect
6. #22806C2semidirect
7. #22768C2semidirect
8. #22765C2semidirect
9. #22673C2semidirect
10. #22146C2semidirect
11. #22117C2semidirect
12. #21985C2semidirect
13. #21892C2semidirect
14. C4×#476C2semidirect
15. #21504C2semidirect
17. #20550C2semidirect
20. #3101C4semidirect
37. #2356C22semidirect
40. #2351C22semidirect
50. C4×C4⋊Q8C22semidirect
51. C4×C4⋊Q8C22semidirect
61. #1987C22semidirect
62. #1981C22semidirect
64. #1948C22semidirect
65. #1947C22semidirect
66. #1944C22semidirect
68. #1933C22semidirect
69. #1928C22semidirect
71. #1920C22semidirect
72. #1920C22semidirect
76. #1865C22semidirect
79. #1846C22semidirect
80. #1835C22semidirect
81. #1805C22semidirect
82. #1805C22semidirect
87. C4×#106C22semidirect
88. #1770C22semidirect
89. C8×C4⋊C4C22semidirect
90. #1767C22semidirect
91. C4×C4⋊C8C22semidirect
92. #1763C22semidirect
102. #533C2×C4semidirect
131. #483C2×C4semidirect
133. #476C2×C4semidirect
135. #461C2×C4semidirect
139. C4×SD16C2×C4semidirect
140. #431C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31012
#342252
#341262
#229452
#228642
#228062
#227682
#227652
#226732
#221462
#221172
#219852
#218922
C4×#4762
#215042
#206462
#205502