Finite Group Databasebeta

← #6621062

Subgroups — #6621062

Order 512.

2155 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×#3367C212yesmaximal, complemented
2Q8×#127C211yesmaximal, characteristic, complemented
3C4×#556C211yesmaximal, characteristic, complemented
4#33198C212yesmaximal, complemented
5#33195C211yesmaximal, characteristic, complemented
6#28726C212yesmaximal, complemented
7#28717C211yesmaximal, characteristic, complemented
8#28253C212yesmaximal
9#28252C211yesmaximal, characteristic, complemented
10#25334C214yesmaximal, complemented
11#25320C212yesmaximal, complemented
12#24694C214yesmaximal, complemented
13#24676C212yesmaximal, complemented
14#22722C211yesmaximal, characteristic, complemented
15#22523C212yesmaximal, complemented
16#21505C212yesmaximal, complemented
17C4×#459C211yesmaximal, characteristic
18C2×#556C2211yescharacteristic
19C2×D4×Q8—21no—
20C2×D4×Q8—21no—
21C2×D4×Q8C2212yes—
22#3367—216no—
23C2×#489C2214yes—
24C2×#487C2212yes—
25C2×#477C2214yes—
26C2×#474C2212yes—
27C2×#460C2212yes—
28C2×#463C2214yes—
29C2×#461C2212yes—
30C2×#459C2211yescharacteristic
31C2×#446C2212yes—
32C2×#440C2214yes—
33C2×C4×D8C2212yes—
34#2868C2211yescharacteristic
35#2868C2212yes—
36#2542—22no—
37#2542—21no—
38#2542—22no—
39#2542—21no—
40#2542C2212yes—
41#2542C2214yes—
42#2373—22no—
43#2367C2212yes—
44#2365C2211yescharacteristic
45#2362—21no—
46#2356C2212yes—
47#2355C2211yescharacteristic
48#2354C2211yescharacteristic
49#2347—21no—
50C4×C4⋊Q8C2211yescharacteristic
51C4×#127C2212yes—
52C4×#127C2211yescharacteristic
53C4×#126—22no—
54C4×#124C2212yes—
55C4×C22⋊Q8—21no—
56C42×Q8C2212yes—
57C42×Q8C2211yescharacteristic
58C42×D4C2211yescharacteristic
59C4×#117—21no—
60#2072C2214yes—
61#2072C2214yes—
62#2071C2214yes—
63#2055C2214yes—
64#2055C2214yes—
65#2054C2214yes—
66#1993C2214yes—
67#1990C2212yes—
68#1988C2212yes—
69#1987C2214yes—
70#1982C2211yescharacteristic
71#1947C2212yes—
72#1947C2214yes—
73#1947C2212yes—
74#1947C2211yescharacteristic
75#1940C2214yes—
76#1939C2212yes—
77#1925C2214yes—
78#1925C2212yes—
79#1880C2214yes—
80#1879C2212yes—
81#1842C2211yescharacteristic
82#1842C2212yes—
83#1804C2214yes—
84#1803C2212yes—
85#1803C2212yes—
86#1803C2212yes—
87#1803C2214yes—
88#1802C2211yescharacteristic
89#1802C2212yes—
90#1802C2212yes—
91#1802C2212yes—
92#1767C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94#1762C2214yes—
95C4×#102C2212yes—
96C23×Q8—22no—
97C22×SD16C2314yes—
98C22×D8C2312yes—
99#556—24no—
100#556D418yescomplemented
101D4×Q8—28no—
102D4×Q8—28no—
103D4×Q8—48no—
104D4×Q8—48no—
105C2×C4⋊Q8—22no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8—22no—
108C2×C4⋊Q8—21no—
109C2×C4⋊Q8C2311yescharacteristic
110C2×C4⋊Q8C2314yes—
111C2×C4⋊Q8C2312yes—
112C2×#127C2311yescharacteristic
113C2×#126—22no—
114C2×#124C2312yes—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8—22no—
117C2×C22⋊Q8—24no—
118C2×C22⋊Q8—24no—
119C2×C22⋊Q8—22no—
120C2×C22⋊Q8—22no—
121C2×C22⋊Q8—24no—
122C2×C4×Q8D411yescharacteristic
123C2×C4×Q8D411yescharacteristic
124C2×C4×Q8C2312yes—
125C2×C4×Q8C2312yes—
126C2×C4×Q8C2312yes—
127C2×C4×Q8C2311yescharacteristic
128C2×C4×D4—22no—
129C2×C4×D4—21no—
130C2×C4×D4—22no—
131C2×C4×D4—21no—
132C2×C4×D4C2312yes—
133C2×C4×D4C2314yes—
134C2×C4×D4C2311yescharacteristic
135C2×C4×D4C2311yescharacteristic
136C2×C4×D4C2311yescharacteristic
137C2×#117—21no—
138C2×#117—21no—
139C22×C42—21no—
140#489—216no—
141#487—28no—
142#477—216no—
143#474—28no—
144#463—216no—
145#461—28no—
146#460—28no—
147#459—24no—
148#446—28no—
149#440—216no—
150C4×D8—28no—
151C2×#107C2312yes—
152C2×#107C2314yes—
153C2×C4⋊C8C2314yes—
154C2×C4⋊C8C2311yescharacteristic
155C2×C4⋊C8C2312yes—
156C2×C4⋊C8C2312yes—
157C2×C4⋊C8C2311yescharacteristic
158C2×C4⋊C8C2314yes—
159C2×C4⋊C8C2312yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#102C2314yes—
164C2×#102C2314yes—
165C2×#102C2312yes—
166C2×#102C2312yes—
167C2×#102C2312yes—
168C2×#102C2312yes—
169C2×#97C2314yes—
170C2×C4×C8C2312yes—
171#399—24no—
172#399—24no—
173#399—24no—
174#391—21no—
175#391C2312yes—
176#391C2312yes—
177#390—22no—
178#390—21no—
179#390—21no—
180#390—22no—
181#390C2314yes—
182#390C2312yes—
183#390C2312yes—
184#389—21no—
185#389—21no—
186#389—21no—
187#388—22no—
188#388—22no—
189#388—22no—
190#388—22no—
191#388—22no—
192#387—22no—
193#387—22no—
194#387—22no—
195#387—22no—
196#385—22no—
197#384—22no—
198#384—21no—
199#384—21no—
200#384—22no—
201#384C2312yes—
202#384C2311yescharacteristic
203#384C2311yescharacteristic
204#384C2312yes—
205#384C2314yes—
206#383—22no—
207#382C2312yes—
208#380—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4D412yes—
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4D412yes—
219C4×C4⋊C4D411yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2311yescharacteristic
225C4×C4⋊C4C2311yescharacteristic
226C4×C4⋊C4C2312yes—
227C4×C22⋊C4—21no—
228C4×C22⋊C4—22no—
229C4×C22⋊C4—22no—
230C4×C22⋊C4—21no—
231C4×C22⋊C4—21no—
232C4×C22⋊C4—22no—
233#376—22no—
234C43—22no—
235C43C2311yescharacteristic
236C43C2312yes—
237#340C2314yes—
238#340C2314yes—
239#340C2314yes—
240#340C2314yes—
241#340C2312yes—
242#340C2312yes—
243#340C2314yes—
244#336C2314yes—
245#336C2312yes—
246C22×Q8—24no—
247C22×Q8—22no—
248C22×Q8—48no—
249C22×Q8—48no—
250C22×Q8—48no—
251C22×Q8C2×D411yescharacteristic
252C22×Q8C2412yes—
253C22×D4—21no—
254C22×D4—21no—
255C22×D4C2411yescharacteristic
256C22×D4C2412yes—
257C23×C4—21no—
258C23×C4—24no—
259C23×C4—22no—
260C23×C4—21no—
261C23×C4—21no—
262C2×SD16—28no—
263C2×SD16—28no—
264C2×SD16—44no—
265C2×D8—24no—
266C2×D8—24no—
267C2×D8—42no—
268C22×C8C2412yes—
269C22×C8C2414yes—
270C22×C8C2412yes—
271C4⋊Q8—24no—
272C4⋊Q8—28no—
273C4⋊Q8—22no—
274C4⋊Q8—24no—
275C4⋊Q8—28no—
276C4⋊Q8—44no—
277C4⋊Q8—48no—
278C4⋊Q8—48no—
279C4⋊Q8—44no—
280C4⋊Q8C2×D414yes—
281#127—22no—
282#127#4114yes—
283#126—216no—
284#124—24no—
285#124C2×D418yes—
286C22⋊Q8—28no—
287C22⋊Q8—48no—
288C22⋊Q8—416no—
289C22⋊Q8—416no—
290C22⋊Q8—48no—
291C22⋊Q8—416no—
292C22⋊Q8—48no—
293C4×Q8—24no—
294C4×Q8—24no—
295C4×Q8—24no—
296C4×Q8—24no—
297C4×Q8—24no—
298C4×Q8—24no—
299C4×Q8—42no—
300C4×Q8C2×D418yes—
301C4×Q8C2×D414yes—
302C4×Q8#4114yes—
303C4×D4—22no—
304C4×D4—24no—
305C4×D4—28no—
306C4×D4—22no—
307C4×D4—24no—
308C4×D4—28no—
309C4×D4—22no—
310C4×D4—22no—
311C4×D4—41no—
312C4×D4—44no—
313C4×D4—48no—
314C4×D4—48no—
315C4×D4—44no—
316C4×D4C2×D414yes—
317#117—28no—
318#117—44no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2×D412yes—
338C2×C4⋊C4C2×D412yes—
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2×D412yes—
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2414yes—
344C2×C4⋊C4C2411yescharacteristic
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2411yescharacteristic
347C2×C4⋊C4C2411yescharacteristic
348C2×C22⋊C4—22no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—22no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—21no—
366C2×C42—21no—
367C2×C42—21no—
368C2×C42—22no—
369C2×C42—44no—
370C2×C42C2×D412yes—
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D412yes—
373C2×C42C2×D412yes—
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D412yes—
377C2×C42C2×D411yescharacteristic
378C2×C42C2411yescharacteristic
379C2×C42C2412yes—
380C2×C42C2411yescharacteristic
381C2×C42C2412yes—
382C2×C42C2411yescharacteristic
383#107—24no—
384#107—28no—
385C4⋊C8—24no—
386C4⋊C8—28no—
387C4⋊C8—22no—
388C4⋊C8—24no—
389C4⋊C8—22no—
390C4⋊C8—42no—
391C4⋊C8—44no—
392#103—28no—
393#103—28no—
394#103—28no—
395#102—24no—
396#102—28no—
397#102—24no—
398#102—28no—
399#102—24no—
400#102—24no—
401#97—28no—
402C4×C8—24no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—21no—
407#95—21no—
408#95—21no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95C2×D412yes—
415#95C2×D412yes—
416#95C2×D412yes—
417C24—21no—
418C24—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—48no—
427C2×Q8—48no—
428C2×Q8—48no—
429C2×Q8—44no—
430C2×Q8—44no—
431C2×Q8—44no—
432C2×Q8—832no—
433C2×Q8#12114yes—
434C2×Q8C22×D414yes—
435C2×D4—24no—
436C2×D4—24no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—22no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—22no—
445C2×D4—21no—
446C2×D4—22no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4C22×D412yes—
452C2×D4C2×#4114yes—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—42no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—48no—
467C22×C4—44no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—42no—
474C22×C4—81no—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D412yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C2511yescharacteristic
486SD16—416no—
487D8—48no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—24no—
491C2×C8—22no—
492C2×C8—24no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—24no—
497C4⋊C4—24no—
498C4⋊C4—28no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—22no—
506C4⋊C4—24no—
507C4⋊C4—21no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—24no—
511C4⋊C4—21no—
512C4⋊C4—22no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—44no—
516C4⋊C4—48no—
517C4⋊C4—44no—
518C4⋊C4—48no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—42no—
522C4⋊C4—44no—
523C4⋊C4—42no—
524C4⋊C4—48no—
525C4⋊C4—44no—
526C4⋊C4—81no—
527C4⋊C4#12114yes—
528C4⋊C4#12118yes—
529C4⋊C4AGL(1,ℤ/8)14yes—
530C4⋊C4C22×D412yes—
531C4⋊C42−1+412yes—
532C22⋊C4—28no—
533C22⋊C4—24no—
534C22⋊C4—28no—
535C22⋊C4—42no—
536C22⋊C4—48no—
537C22⋊C4—44no—
538C22⋊C4—48no—
539C22⋊C4—44no—
540C22⋊C4—42no—
541C22⋊C4—44no—
542C22⋊C4—48no—
543C22⋊C4—44no—
544C22⋊C4—82no—
545C42—28no—
546C42—24no—
547C42—22no—
548C42—24no—
549C42—24no—
550C42—22no—
551C42—21no—
552C42—22no—
553C42—21no—
554C42—22no—
555C42—42no—
556C42—41no—
557C42—44no—
558C42—44no—
559C42—42no—
560C42—42no—
561C42—42no—
562C42—44no—
563C42—82no—
564C42AGL(1,ℤ/8)14yes—
565C42C22×D412yes—
566C42C22×D414yes—
567C42C2×#4112yes—
568C42C2×#4114yes—
569C42C2×#4112yes—
570C422−1+412yes—
571C23—22no—
572C23—24no—
573C23—41no—
574C23—42no—
575C23—42no—
576C23—41no—
577C23—41no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—42no—
585C23C23×D411yescharacteristic
586Q8—28no—
587Q8—24no—
588Q8—24no—
589Q8—42no—
590Q8—42no—
591Q8—816no—
592Q8—88no—
593D4—22no—
594D4—24no—
595D4—24no—
596D4—22no—
597D4—48no—
598D4—48no—
599D4—41no—
600D4—42no—
601D4—42no—
602D4—41no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—21no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—24no—
618C2×C4—21no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—48no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—41no—
643C2×C4—84no—
644C2×C4—816no—
645C2×C4—84no—
646C2×C4—88no—
647C2×C4C2×#12114yes—
648C2×C4C2×#12112yes—
649C2×C4C2×#12114yes—
650C2×C4C2×#12112yes—
651C2×C4#54812yes—
652C2×C4#54812yes—
653C2×C4C2×AGL(1,ℤ/8)14yes—
654C2×C4C2×AGL(1,ℤ/8)12yes—
655C2×C4C2×AGL(1,ℤ/8)12yes—
656C2×C4C2×AGL(1,ℤ/8)12yes—
657C2×C4C2×AGL(1,ℤ/8)12yes—
658C2×C4#57812yes—
659C2×C4C23×D411yescharacteristic
660C2×C4C22×#4111yescharacteristic
661C2×C4C22×#4111yescharacteristic
662C2×C4C2×2−1+411yescharacteristic
663C2×C4C2×2−1+411yescharacteristic
664C2×C4#58511yescharacteristic
665C8—42no—
666C8—44no—
667C8—42no—
668C22—42no—
669C22—42no—
670C22—44no—
671C22—44no—
672C22—44no—
673C22—42no—
674C22—82no—
675C22—81no—
676C22—81no—
677C22—81no—
678C22—82no—
679C22—81no—
680C22—82no—
681C22—82no—
682C22C22×#12111yescharacteristic
683C22C2×#54811yescharacteristic
684C22#348911yescharacteristic
685C22C22×AGL(1,ℤ/8)11yescharacteristic
686C22C22×AGL(1,ℤ/8)11yescharacteristic
687C22C2×#57811yescharacteristic
688C22#358611yescharacteristic
689C4—24no—
690C4—22no—
691C4—22no—
692C4—22no—
693C4—24no—
694C4—24no—
695C4—44no—
696C4—42no—
697C4—42no—
698C4—41no—
699C4—42no—
700C4—81no—
701C4—82no—
702C4—84no—
703C4—82no—
704C4#303814yes—
705C4#336714yescomplemented
706C4D4×#4111yescharacteristic
707C4#348211yescharacteristic
708C4#349911yescharacteristic
709C4#351811yescharacteristic
710C2—82no—
711C2—84no—
712C2#3644411yescharacteristic
713C2C2×#303811yescharacteristic
714C2#6043211yescharacteristic
715C2C2×#336711yescharacteristic
716C2#6131711yescharacteristic
717C2#6210711yescharacteristic
718C2#6213211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#459C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3367C2semidirect
2. Q8×#127C2semidirect
3. C4×#556C2semidirect
4. #33198C2semidirect
5. #33195C2semidirect
6. #28726C2semidirect
7. #28717C2semidirect
9. #28252C2semidirect
10. #25334C2semidirect
11. #25320C2semidirect
12. #24694C2semidirect
13. #24676C2semidirect
14. #22722C2semidirect
15. #22523C2semidirect
16. #21505C2semidirect
100. #556D4semidirect
705. C4#3367semidirect

Maximal subgroups

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

subgroupindex
C2×#33672
Q8×#1272
C4×#5562
#331982
#331952
#287262
#287172
#282532
#282522
#253342
#253202
#246942
#246762
#227222
#225232
#215052
C4×#4592