Finite Group Databasebeta

← #6606557

Subgroups — #6606557

Order 512.

2129 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3335C212yesmaximal
2#36180C211yesmaximal, characteristic, complemented
3C4×#547C211yesmaximal, characteristic
4#33197C211yesmaximal, characteristic, complemented
5#33108C212yesmaximal, complemented
6#28687C211yesmaximal, characteristic, complemented
7#28283C211yesmaximal, characteristic, complemented
8#27645C212yesmaximal, complemented
9#25453C212yesmaximal, complemented
10#25202C214yesmaximal, complemented
11#24751C212yesmaximal, complemented
12#24680C212yesmaximal, complemented
13#24580C214yesmaximal, complemented
14D4×C4⋊C8C211yesmaximal, characteristic, complemented
15#22487C212yesmaximal, complemented
16C4×#461C211yesmaximal, characteristic, complemented
17C2×#1948C212yesmaximal, complemented
18C2×#554C2212yes—
19C2×#547C2211yescharacteristic
20C2×C4×#41—21no—
21#3335—216no—
22C2×#498C2212yes—
23C2×#481C2214yes—
24C2×#483C2214yes—
25C2×#475C2212yes—
26C2×#476C2212yes—
27C2×#465C2214yes—
28C2×#461C2211yescharacteristic
29C2×C4×SD16C2212yes—
30C2×C8×D4C2212yes—
31C22×#106C2214yes—
32C22×C4⋊C8C2212yes—
33#2868C2211yescharacteristic
34#2857C2212yes—
35#2567C2212yes—
36#2552—21no—
37#2552—21no—
38#2502C2214yes—
39#2440—22no—
40#2440—22no—
41#2365C2211yescharacteristic
42#2362—21no—
43#2361C2212yes—
44#2359—22no—
45#2354C2211yescharacteristic
46D4×C4⋊C4—21no—
47D4×C4⋊C4—21no—
48D4×C4⋊C4C2211yescharacteristic
49#2344C2212yes—
50#2339—21no—
51C4×C4⋊Q8C2211yescharacteristic
52C4×C4⋊Q8C2211yescharacteristic
53C4×C22⋊Q8—21no—
54C4×#123—22no—
55C4×#121C2212yes—
56C2×#384C2212yes—
57C42×D4C2211yescharacteristic
58C42×D4C2211yescharacteristic
59C2×C4×C4⋊C4C2212yes—
60#2084C2214yes—
61#2084C2214yes—
62#2071C2214yes—
63#2066C2214yes—
64#2066C2214yes—
65#2055C2214yes—
66#1990C2212yes—
67#1987C2212yes—
68#1981C2211yescharacteristic
69#1981C2212yes—
70#1948C2212yes—
71#1948C2214yes—
72#1948C2212yes—
73#1948C2212yes—
74#1948C2212yes—
75#1948C2214yes—
76#1948C2212yes—
77#1948C2211yescharacteristic
78#1939C2212yes—
79#1927C2212yes—
80#1927C2214yes—
81#1927C2214yes—
82#1892C2214yes—
83#1877C2212yes—
84#1872C2214yes—
85#1850C2214yes—
86#1842C2211yescharacteristic
87#1824C2214yes—
88#1803C2212yes—
89#1802C2212yes—
90#1802C2212yes—
91#1802C2211yescharacteristic
92#1797C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C2×#340C2214yes—
96C22×#41—21no—
97C22×SD16C2312yes—
98C23×C8C2314yes—
99#554—28no—
100#554—28no—
101#547—24no—
102#547D418yes—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2311yescharacteristic
108C2×#125—22no—
109C2×#123—22no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—24no—
112C2×C22⋊Q8C2314yes—
113C2×#121C2312yes—
114C4×#41—48no—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8C2312yes—
118C2×C4×D4—21no—
119C2×C4×D4—22no—
120C2×C4×D4—22no—
121C2×C4×D4—22no—
122C2×C4×D4D411yescharacteristic
123C2×C4×D4D411yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×#117—21no—
129C2×#117—22no—
130C22×C4⋊C4—22no—
131C22×C4⋊C4—22no—
132C22×C4⋊C4C2314yes—
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2312yes—
135C22×C42—22no—
136C22×C42—21no—
137C22×C42C2312yes—
138#498—28no—
139#483—216no—
140#481—216no—
141#476—28no—
142#475—28no—
143#465—216no—
144#461—24no—
145C4×SD16—28no—
146C8×D4—28no—
147C2×#106—216no—
148C2×#106C2314yes—
149C2×#106C2314yes—
150C2×#106C2314yes—
151C2×#106C2314yes—
152C2×#106C2312yes—
153C2×C4⋊C8—28no—
154C2×C4⋊C8C2312yes—
155C2×C4⋊C8C2311yescharacteristic
156C2×C4⋊C8C2312yes—
157C2×C4⋊C8C2314yes—
158C2×C4⋊C8C2312yes—
159C2×C4⋊C8C2311yescharacteristic
160C2×#103C2314yes—
161C2×#103C2312yes—
162C2×#103C2312yes—
163C2×#103C2312yes—
164C2×#103C2314yes—
165C2×#102C2314yes—
166C2×#102C2312yes—
167C2×C22⋊C8C2314yes—
168C2×C4×C8C2312yes—
169#398—24no—
170#398—24no—
171#395—24no—
172#391—21no—
173#391—21no—
174#391C2312yes—
175#391C2314yes—
176#390C2312yes—
177#389—22no—
178#389—22no—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389—22no—
184#389—22no—
185#389C2312yes—
186#389C2312yes—
187#388—22no—
188#388—22no—
189#388—22no—
190#387—22no—
191#387—22no—
192#387—22no—
193#386—22no—
194#386—22no—
195#385—24no—
196#385—24no—
197#385—22no—
198#385—22no—
199#384—21no—
200#384—21no—
201#384—22no—
202#384—22no—
203#384C2311yescharacteristic
204#384C2311yescharacteristic
205#384C2314yes—
206#384C2312yes—
207#384C2312yes—
208#381—22no—
209#380—22no—
210#380—22no—
211#380—21no—
212#380C2312yes—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4D411yescharacteristic
219C4×C4⋊C4D411yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2311yescharacteristic
226C4×C4⋊C4C2311yescharacteristic
227C4×C22⋊C4—21no—
228C4×C22⋊C4—21no—
229C4×C22⋊C4—21no—
230C4×C22⋊C4—22no—
231C4×C22⋊C4D412yes—
232C4×C22⋊C4D412yes—
233C4×C22⋊C4C2312yes—
234C2×#95—22no—
235C43C2311yescharacteristic
236#340C2314yes—
237#340C2314yes—
238#340C2312yes—
239#340C2314yes—
240#340C2314yes—
241#340C2314yes—
242#340C2312yes—
243#336C2314yes—
244#336C2312yes—
245C2×#41—28no—
246C2×#41—44no—
247C2×#41—44no—
248C22×Q8—21no—
249C22×Q8—22no—
250C22×Q8C2412yes—
251C22×D4—22no—
252C22×D4C2×D411yescharacteristic
253C22×D4C2411yescharacteristic
254C23×C4—22no—
255C23×C4—21no—
256C23×C4—22no—
257C23×C4—24no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4C2×D412yes—
261C23×C4C2×D412yes—
262C23×C4C2412yes—
263C2×SD16—24no—
264C2×SD16—24no—
265C2×SD16—42no—
266C22×C8—28no—
267C22×C8—28no—
268C22×C8—44no—
269C22×C8C2414yes—
270C22×C8C2412yes—
271C22×C8C2412yes—
272C4⋊Q8—24no—
273C4⋊Q8—22no—
274C4⋊Q8—24no—
275C4⋊Q8—22no—
276C4⋊Q8—22no—
277C4⋊Q8—82no—
278C4⋊Q8—82no—
279C4⋊Q8C2×D414yes—
280#125—48no—
281#123—216no—
282C22⋊Q8—28no—
283C22⋊Q8—28no—
284C22⋊Q8—28no—
285C22⋊Q8—416no—
286#121—24no—
287#121C2×D418yes—
288C4×Q8—24no—
289C4×Q8—24no—
290C4×Q8—44no—
291C4×Q8—44no—
292C4×D4—24no—
293C4×D4—22no—
294C4×D4—22no—
295C4×D4—22no—
296C4×D4—22no—
297C4×D4—41no—
298C4×D4—48no—
299C4×D4—44no—
300C4×D4—48no—
301C4×D4—48no—
302C4×D4SD1618yescomplemented
303C4×D4C2×D414yes—
304C4×D4C2×D414yes—
305C4×D4#4114yes—
306#117—44no—
307#117—48no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—28no—
312C2×C4⋊C4—28no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—42no—
334C2×C4⋊C4—84no—
335C2×C4⋊C4—84no—
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2×D418yes—
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2414yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2411yescharacteristic
344C2×C4⋊C4C2411yescharacteristic
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2411yescharacteristic
347C2×C22⋊C4—21no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4C2×D412yes—
356C2×C22⋊C4C2×D412yes—
357C2×C22⋊C4C2×D412yes—
358C2×C42—24no—
359C2×C42—24no—
360C2×C42—21no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—21no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—21no—
367C2×C42—48no—
368C2×C42—44no—
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D412yes—
372C2×C42C2×D412yes—
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2411yescharacteristic
376C2×C42C2412yes—
377C2×C42C2411yescharacteristic
378C2×C42C2411yescharacteristic
379#106—24no—
380#106—28no—
381#106—28no—
382#106—28no—
383#106—28no—
384C4⋊C8—28no—
385C4⋊C8—24no—
386C4⋊C8—24no—
387C4⋊C8—22no—
388C4⋊C8—42no—
389C4⋊C8#4114yescomplemented
390#103—24no—
391#103—24no—
392#103—28no—
393#103—24no—
394#103—44no—
395#102—24no—
396#102—28no—
397C22⋊C8—28no—
398C4×C8—24no—
399#95—21no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—21no—
404#95—21no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—24no—
411#95—24no—
412#95C2×D412yes—
413#95C2×D412yes—
414#95C2×D412yes—
415C24—21no—
416C24C22×D412yes—
417#41—432no—
418C2×Q8—24no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—44no—
426C2×Q8—42no—
427C2×Q8—42no—
428C2×Q8—82no—
429C2×Q8—82no—
430C2×D4—28no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—44no—
437C2×D4—44no—
438C2×D4#12114yes—
439C2×D4C2×SD1614yes—
440C2×D4#13714yes—
441C2×D4C22×D412yes—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—28no—
449C22×C4—24no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—44no—
468C22×C4—48no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—81no—
474C22×C4—82no—
475C22×C4—84no—
476C22×C4—84no—
477C22×C4C2×SD1618yes—
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D412yes—
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D412yes—
482C22×C4C22×D414yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D412yes—
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C2×#4114yes—
489C22×C4C2511yescharacteristic
490SD16—48no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—24no—
495C2×C8—24no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—24no—
499C2×C8—416no—
500C2×C8C2×#4114yes—
501C4⋊C4—28no—
502C4⋊C4—24no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—22no—
506C4⋊C4—21no—
507C4⋊C4—22no—
508C4⋊C4—24no—
509C4⋊C4—21no—
510C4⋊C4—22no—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—22no—
515C4⋊C4—44no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—48no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—48no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—44no—
529C4⋊C4—81no—
530C4⋊C4—84no—
531C4⋊C4—84no—
532C4⋊C4—82no—
533C4⋊C4—82no—
534C4⋊C4#12114yes—
535C4⋊C4C2×SD1614yes—
536C4⋊C4C22×D412yes—
537C4⋊C4C22×D414yes—
538C4⋊C42−1+412yes—
539C22⋊C4—24no—
540C22⋊C4—24no—
541C22⋊C4—28no—
542C22⋊C4—42no—
543C22⋊C4—48no—
544C22⋊C4—44no—
545C22⋊C4—42no—
546C22⋊C4—44no—
547C22⋊C4—82no—
548C22⋊C4#12118yes—
549C22⋊C4C2×SD1618yes—
550C42—24no—
551C42—24no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—21no—
557C42—21no—
558C42—41no—
559C42—44no—
560C42—42no—
561C42—44no—
562C42—42no—
563C42—44no—
564C42—42no—
565C42—44no—
566C42C2×SD1614yes—
567C42C22×D412yes—
568C42C2×#4112yes—
569C42C2×#4112yes—
570C422−1+412yes—
571C23—22no—
572C23—22no—
573C23—22no—
574C23—22no—
575C23—22no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23C2×#12114yes—
583C23C22×SD1614yes—
584C23C2×#13714yes—
585C23C23×D411yescharacteristic
586Q8—48no—
587Q8—42no—
588Q8—42no—
589Q8—42no—
590Q8—42no—
591Q8—164no—
592D4—22no—
593D4—22no—
594D4—416no—
595D4—41no—
596D4—41no—
597D4#461116yescomplemented
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—216no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—24no—
616C2×C4—22no—
617C2×C4—21no—
618C2×C4—22no—
619C2×C4—21no—
620C2×C4—22no—
621C2×C4—21no—
622C2×C4—48no—
623C2×C4—41no—
624C2×C4—416no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—48no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—41no—
646C2×C4—84no—
647C2×C4—88no—
648C2×C4—84no—
649C2×C4—168no—
650C2×C4C2×#12112yes—
651C2×C4C2×#12114yes—
652C2×C4C2×#12112yes—
653C2×C4#54712yes—
654C2×C4#54712yes—
655C2×C4C22×SD1614yes—
656C2×C4C22×SD1612yes—
657C2×C4C22×SD1612yes—
658C2×C4C22×SD1612yes—
659C2×C4C2×#13712yes—
660C2×C4#57712yes—
661C2×C4C23×D411yescharacteristic
662C2×C4C22×#4111yescharacteristic
663C2×C4C22×#4111yescharacteristic
664C2×C4C2×2−1+411yescharacteristic
665C2×C4C2×2−1+411yescharacteristic
666C2×C4#58511yescharacteristic
667C8—24no—
668C8—44no—
669C8—42no—
670C22—24no—
671C22—24no—
672C22—24no—
673C22—24no—
674C22—24no—
675C22—24no—
676C22—42no—
677C22—42no—
678C22—42no—
679C22—81no—
680C22—81no—
681C22—81no—
682C22—81no—
683C22C2×#46118yes—
684C22C22×#12111yescharacteristic
685C22C2×#54711yescharacteristic
686C22#348911yescharacteristic
687C22C23×SD1611yescharacteristic
688C22C22×#13711yescharacteristic
689C22C2×#57711yescharacteristic
690C22#358511yescharacteristic
691C4—22no—
692C4—24no—
693C4—22no—
694C4—22no—
695C4—24no—
696C4—42no—
697C4—42no—
698C4—41no—
699C4—42no—
700C4—44no—
701C4—82no—
702C4—84no—
703C4—81no—
704C4—82no—
705C4—82no—
706C4C2×#46114yes—
707C4#333514yescomplemented
708C4D4×#4111yescharacteristic
709C4#348211yescharacteristic
710C4#349911yescharacteristic
711C4#351511yescharacteristic
712C2—28no—
713C2—82no—
714C2#3625911yescharacteristic
715C2C22×#46111yescharacteristic
716C2#6043111yescharacteristic
717C2C2×#333511yescharacteristic
718C2#6125411yescharacteristic
719C2#6209611yescharacteristic
720C2#6213111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #36180C2semidirect
4. #33197C2semidirect
5. #33108C2semidirect
6. #28687C2semidirect
7. #28283C2semidirect
8. #27645C2semidirect
9. #25453C2semidirect
10. #25202C2semidirect
11. #24751C2semidirect
12. #24680C2semidirect
13. #24580C2semidirect
14. D4×C4⋊C8C2semidirect
15. #22487C2semidirect
16. C4×#461C2semidirect
17. C2×#1948C2semidirect
302. C4×D4SD16semidirect
389. C4⋊C8#41semidirect
597. D4#461semidirect
707. C4#3335semidirect

Maximal subgroups

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

subgroupindex
C2×#33352
#361802
C4×#5472
#331972
#331082
#286872
#282832
#276452
#254532
#252022
#247512
#246802
#245802
D4×C4⋊C82
#224872
C4×#4612
C2×#19482