Finite Group Databasebeta

← #6566080

Subgroups — #6566080

Order 512.

2345 conjugacy classes of subgroups.

The subgroup lattice has 730 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3334C212yesmaximal, complemented
2#36179C211yesmaximal, characteristic, complemented
3#34308C211yesmaximal, characteristic, complemented
4#29864C211yesmaximal, characteristic, complemented
5#29754C212yesmaximal, complemented
6#28689C211yesmaximal, characteristic, complemented
7#28572C212yesmaximal, complemented
8#28201C211yesmaximal, characteristic, complemented
9#27644C212yesmaximal, complemented
10#25449C212yesmaximal, complemented
11#25200C214yesmaximal, complemented
12#24811C212yesmaximal, complemented
13#24576C214yesmaximal, complemented
14#23676C211yesmaximal, characteristic, complemented
15C2×#2083C212yesmaximal, complemented
16#22696C211yesmaximal, characteristic, complemented
17#22485C212yesmaximal, complemented
18C2×#547—21no—
19C2×#547—21no—
20C2×#547C2212yes—
21C2×#547C2211yescharacteristic
22C2×#520—21no—
23#3334—216no—
24C2×#500C2212yes—
25C2×#480C2214yes—
26C2×#480C2214yes—
27C2×#474C2212yes—
28C2×#474C2212yes—
29C2×#466C2214yes—
30C2×C4×D8C2212yes—
31C2×C8×D4C2212yes—
32C22×#107C2214yes—
33#2868C2211yescharacteristic
34#2856C2212yes—
35#2542—21no—
36#2542—21no—
37#2542C2212yes—
38#2501—22no—
39#2501—22no—
40#2501C2214yes—
41#2365C2211yescharacteristic
42#2361C2212yes—
43#2360—21no—
44#2359—22no—
45D4×C4⋊C4C2211yescharacteristic
46D4×C4⋊C4C2211yescharacteristic
47#2344C2212yes—
48#2339—21no—
49#2323—21no—
50#2322—22no—
51C4×C4⋊Q8C2211yescharacteristic
52C4×#127C2211yescharacteristic
53C4×#121C2212yes—
54#2293C2211yescharacteristic
55C2×#389C2212yes—
56C2×#384C2212yes—
57C42×D4C2211yescharacteristic
58#2083C2212yes—
59#2083C2214yes—
60#2083C2212yes—
61#2083C2212yes—
62#2083C2212yes—
63#2083C2212yes—
64#2083C2211yescharacteristic
65#2083C2214yes—
66#2071C2212yes—
67#2071C2211yescharacteristic
68#2071C2212yes—
69#2067C2212yes—
70#2067C2214yes—
71#2067C2214yes—
72#2054C2212yes—
73#2041C2212yes—
74#2041C2211yescharacteristic
75#2011C2211yescharacteristic
76#1990C2212yes—
77#1988C2212yes—
78#1947C2214yes—
79#1947C2214yes—
80#1939C2212yes—
81#1925C2214yes—
82#1925C2214yes—
83#1893C2214yes—
84#1879C2214yes—
85#1874C2214yes—
86#1850C2214yes—
87#1831C2211yescharacteristic
88#1823C2214yes—
89#1802C2212yes—
90#1802C2214yes—
91#1797C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×#102C2212yes—
94C2×#340C2214yes—
95C2×#336C2212yes—
96C22×#41—21no—
97C22×#41—22no—
98C22×D8C2312yes—
99C23×C8C2314yes—
100#547—28no—
101#547—24no—
102#547—28no—
103#547—48no—
104#547—48no—
105#547D418yescomplemented
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8—21no—
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2311yescharacteristic
110C2×#127C2311yescharacteristic
111C2×#123—22no—
112C2×#123—24no—
113C2×#123—24no—
114C2×#123—24no—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8—22no—
117C2×C22⋊Q8—22no—
118C2×C22⋊Q8—22no—
119C2×#121—22no—
120C2×#121—22no—
121C2×#121C2314yes—
122C2×#121C2312yes—
123#520—48no—
124C2×C4×Q8—21no—
125C2×C4×Q8—21no—
126C2×C4×D4—21no—
127C2×C4×D4—22no—
128C2×C4×D4—22no—
129C2×C4×D4—21no—
130C2×C4×D4—21no—
131C2×C4×D4D411yescharacteristic
132C2×C4×D4D411yescharacteristic
133C2×C4×D4C2312yes—
134C2×C4×D4C2311yescharacteristic
135C2×C4×D4C2312yes—
136C2×C4×D4C2311yescharacteristic
137C2×#117—22no—
138C2×#117—21no—
139C22×C4⋊C4—22no—
140C22×C4⋊C4—22no—
141C22×C4⋊C4—22no—
142C22×C4⋊C4—21no—
143C22×C4⋊C4C2314yes—
144C22×C4⋊C4C2312yes—
145C22×C4⋊C4C2312yes—
146C22×C42C2312yes—
147#500—28no—
148#480—216no—
149#480—216no—
150#474—28no—
151#474—28no—
152#466—216no—
153C4×D8—28no—
154C8×D4—28no—
155C2×#107—216no—
156C2×#107C2314yes—
157C2×#107C2314yes—
158C2×#107C2314yes—
159C2×#107C2312yes—
160C2×#107C2314yes—
161C2×C4⋊C8C2312yes—
162C2×C4⋊C8C2312yes—
163C2×C4⋊C8C2314yes—
164C2×#102C2312yes—
165C2×#102C2314yes—
166C2×#102C2312yes—
167C2×#102C2314yes—
168C2×#102C2312yes—
169C2×#102C2312yes—
170C2×#102C2314yes—
171C2×C22⋊C8C2314yes—
172C2×C4×C8C2312yes—
173#399—24no—
174#399—24no—
175#399—24no—
176#391—21no—
177#391—21no—
178#390—21no—
179#390—21no—
180#390—22no—
181#390—22no—
182#390C2312yes—
183#390C2311yescharacteristic
184#390C2314yes—
185#390C2311yescharacteristic
186#389—21no—
187#389—21no—
188#389—22no—
189#389—22no—
190#389C2312yes—
191#389C2311yescharacteristic
192#389C2312yes—
193#389C2312yes—
194#389C2311yescharacteristic
195#388—21no—
196#388—22no—
197#388—21no—
198#388—22no—
199#387—22no—
200#387—22no—
201#387—22no—
202#386—22no—
203#386—22no—
204#385—22no—
205#385—22no—
206#385—22no—
207#384—21no—
208#384—21no—
209#384—22no—
210#384—22no—
211#384C2311yescharacteristic
212#384C2311yescharacteristic
213#384C2314yes—
214#384C2312yes—
215#384C2312yes—
216#381—22no—
217#380—21no—
218#380C2312yes—
219C4×C4⋊C4—21no—
220C4×C4⋊C4D411yescharacteristic
221C4×C4⋊C4D411yescharacteristic
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2311yescharacteristic
225C4×C4⋊C4C2311yescharacteristic
226C4×C22⋊C4—21no—
227C4×C22⋊C4—22no—
228C4×C22⋊C4D412yes—
229C4×C22⋊C4D412yes—
230C4×C22⋊C4C2312yes—
231C2×#95—22no—
232C43C2311yescharacteristic
233#340C2312yes—
234#340C2314yes—
235#340C2314yes—
236#340C2312yes—
237#340C2314yes—
238#340C2314yes—
239#340C2314yes—
240#336C2311yescharacteristic
241#336C2312yes—
242#336C2312yes—
243#336C2311yescharacteristic
244#336C2314yes—
245C2×#41—28no—
246C2×#41—48no—
247C2×#41—44no—
248C2×#41—48no—
249C2×#41—44no—
250C2×#41—48no—
251C22×Q8—21no—
252C22×Q8—22no—
253C22×D4—22no—
254C22×D4—24no—
255C22×D4—21no—
256C22×D4—21no—
257C22×D4C2×D411yescharacteristic
258C22×D4C2412yes—
259C22×D4C2411yescharacteristic
260C23×C4—22no—
261C23×C4—21no—
262C23×C4—24no—
263C23×C4—22no—
264C23×C4—22no—
265C23×C4—22no—
266C23×C4C2×D412yes—
267C23×C4C2×D412yes—
268C23×C4C2412yes—
269C2×D8—24no—
270C2×D8—24no—
271C2×D8—42no—
272C22×C8—28no—
273C22×C8—28no—
274C22×C8—44no—
275C22×C8C2414yes—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C4⋊Q8—24no—
279C4⋊Q8—22no—
280C4⋊Q8—24no—
281C4⋊Q8—44no—
282C4⋊Q8—44no—
283C4⋊Q8C2×D414yes—
284#127—42no—
285#123—216no—
286#123—416no—
287#123—416no—
288#123—416no—
289C22⋊Q8—28no—
290C22⋊Q8—48no—
291C22⋊Q8—48no—
292C22⋊Q8—48no—
293#121—24no—
294#121—28no—
295#121—28no—
296#121—48no—
297#121—48no—
298#121C2×D418yes—
299C4×Q8—44no—
300C4×Q8—44no—
301C4×D4—24no—
302C4×D4—24no—
303C4×D4—22no—
304C4×D4—24no—
305C4×D4—44no—
306C4×D4—44no—
307C4×D4—44no—
308C4×D4—48no—
309C4×D4—48no—
310C4×D4—44no—
311C4×D4D818yescomplemented
312C4×D4C2×D414yes—
313C4×D4C2×D414yes—
314C4×D4#4114yes—
315#117—48no—
316#117—44no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—28no—
321C2×C4⋊C4—28no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—44no—
341C2×C4⋊C4—48no—
342C2×C4⋊C4—48no—
343C2×C4⋊C4—48no—
344C2×C4⋊C4—44no—
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D418yes—
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2411yescharacteristic
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2414yes—
353C2×C4⋊C4C2411yescharacteristic
354C2×C4⋊C4C2411yescharacteristic
355C2×C4⋊C4C2412yes—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—24no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4C2×D412yes—
371C2×C22⋊C4C2×D412yes—
372C2×C22⋊C4C2×D412yes—
373C2×C42—22no—
374C2×C42—21no—
375C2×C42—22no—
376C2×C42—44no—
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D412yes—
381C2×C42C2×D412yes—
382C2×C42C2×D411yescharacteristic
383C2×C42C2412yes—
384C2×C42C2411yescharacteristic
385C2×C42C2411yescharacteristic
386C2×C42C2411yescharacteristic
387#107—24no—
388#107—28no—
389#107—28no—
390#107—28no—
391#107—28no—
392C4⋊C8—24no—
393C4⋊C8—24no—
394C4⋊C8—44no—
395#102—24no—
396#102—28no—
397#102—24no—
398#102—24no—
399#102—28no—
400#102—42no—
401#102—44no—
402C22⋊C8—28no—
403C4×C8—24no—
404#95—21no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—21no—
410#95—22no—
411#95—22no—
412#95—21no—
413#95C2×D412yes—
414#95C2×D412yes—
415#95C2×D412yes—
416C24—21no—
417C24—22no—
418C24C22×D412yes—
419#41—432no—
420#41—832no—
421C2×Q8—24no—
422C2×Q8—44no—
423C2×Q8—42no—
424C2×Q8—44no—
425C2×Q8—42no—
426C2×Q8—44no—
427C2×D4—28no—
428C2×D4—22no—
429C2×D4—21no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4—42no—
441C2×D4—44no—
442C2×D4—44no—
443C2×D4—48no—
444C2×D4—48no—
445C2×D4—48no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4#12114yes—
449C2×D4C2×D814yes—
450C2×D4AGL(1,ℤ/8)14yes—
451C2×D4C22×D412yes—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—28no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—82no—
481C22×C4—82no—
482C22×C4—82no—
483C22×C4—82no—
484C22×C4C2×D818yes—
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D412yes—
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D412yes—
489C22×C4C22×D414yes—
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D412yes—
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C2×#4114yes—
496C22×C4C2511yescharacteristic
497D8—48no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—24no—
502C2×C8—22no—
503C2×C8—24no—
504C2×C8—22no—
505C2×C8—24no—
506C2×C8—416no—
507C2×C82−1+414yes—
508C4⋊C4—28no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—21no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—24no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—24no—
520C4⋊C4—42no—
521C4⋊C4—44no—
522C4⋊C4—48no—
523C4⋊C4—44no—
524C4⋊C4—48no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—42no—
532C4⋊C4—44no—
533C4⋊C4—48no—
534C4⋊C4—42no—
535C4⋊C4—44no—
536C4⋊C4—81no—
537C4⋊C4—82no—
538C4⋊C4#12114yes—
539C4⋊C4C2×D814yes—
540C4⋊C4C22×D412yes—
541C4⋊C4C22×D414yes—
542C22⋊C4—24no—
543C22⋊C4—24no—
544C22⋊C4—28no—
545C22⋊C4—44no—
546C22⋊C4—48no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—48no—
550C22⋊C4—42no—
551C22⋊C4—42no—
552C22⋊C4—44no—
553C22⋊C4—48no—
554C22⋊C4—42no—
555C22⋊C4—81no—
556C22⋊C4—82no—
557C22⋊C4#12118yes—
558C22⋊C4C2×D818yes—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—24no—
563C42—22no—
564C42—42no—
565C42—42no—
566C42—41no—
567C42—44no—
568C42—44no—
569C42—42no—
570C42C2×D814yes—
571C42C22×D412yes—
572C42C2×#4112yes—
573C23—22no—
574C23—22no—
575C23—22no—
576C23—22no—
577C23—22no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23C2×#12114yes—
592C23C22×D814yes—
593C23C2×AGL(1,ℤ/8)14yes—
594C23C23×D411yescharacteristic
595Q8—48no—
596Q8—88no—
597D4—24no—
598D4—416no—
599D4—42no—
600D4—42no—
601D4—42no—
602D4—42no—
603D4—42no—
604D4—84no—
605D4—84no—
606D4—816no—
607D4#459116yes—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—22no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—21no—
616C2×C4—22no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—28no—
621C2×C4—28no—
622C2×C4—21no—
623C2×C4—22no—
624C2×C4—24no—
625C2×C4—22no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—48no—
631C2×C4—48no—
632C2×C4—416no—
633C2×C4—42no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—48no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—42no—
647C2×C4—41no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—41no—
653C2×C4—816no—
654C2×C4—88no—
655C2×C4—164no—
656C2×C4—164no—
657C2×C4C2×#12112yes—
658C2×C4C2×#12114yes—
659C2×C4C2×#12112yes—
660C2×C4C22×D812yes—
661C2×C4C22×D814yes—
662C2×C4C22×D812yes—
663C2×C4C22×D812yes—
664C2×C4C2×AGL(1,ℤ/8)12yes—
665C2×C4#57812yes—
666C2×C4C23×D411yescharacteristic
667C2×C4C22×#4111yescharacteristic
668C2×C4C2×2−1+412yes—
669C2×C4#58512yes—
670C8—24no—
671C8—42no—
672C8—44no—
673C22—24no—
674C22—24no—
675C22—24no—
676C22—24no—
677C22—24no—
678C22—24no—
679C22—42no—
680C22—42no—
681C22—42no—
682C22—81no—
683C22—81no—
684C22—81no—
685C22—81no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22C2×#45918yes—
694C22C22×#12111yescharacteristic
695C22#348511yescharacteristic
696C22#349911yescharacteristic
697C22C23×D811yescharacteristic
698C22C22×AGL(1,ℤ/8)11yescharacteristic
699C22C2×#57811yescharacteristic
700C22#358611yescharacteristic
701C4—22no—
702C4—22no—
703C4—22no—
704C4—24no—
705C4—22no—
706C4—24no—
707C4—41no—
708C4—42no—
709C4—42no—
710C4—44no—
711C4—81no—
712C4—82no—
713C4—84no—
714C4—81no—
715C4—82no—
716C4C2×#45914yes—
717C4#348912yes—
718C4#351512yes—
719C2—28no—
720C2—82no—
721C2—162no—
722C2#3602011yescharacteristic
723C2C22×#45911yescharacteristic
724C2#6043211yescharacteristic
725C2#6169411yescharacteristic
726C2#6173711yescharacteristic
727C2#6247111yescharacteristic
728C2#6250311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
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
1. C2×#3334C2semidirect
2. #36179C2semidirect
3. #34308C2semidirect
4. #29864C2semidirect
5. #29754C2semidirect
6. #28689C2semidirect
7. #28572C2semidirect
8. #28201C2semidirect
9. #27644C2semidirect
10. #25449C2semidirect
11. #25200C2semidirect
12. #24811C2semidirect
13. #24576C2semidirect
14. #23676C2semidirect
15. C2×#2083C2semidirect
16. #22696C2semidirect
17. #22485C2semidirect
105. #547D4semidirect
311. C4×D4D8semidirect

Maximal subgroups

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

subgroupindex
C2×#33342
#361792
#343082
#298642
#297542
#286892
#285722
#282012
#276442
#254492
#252002
#248112
#245762
#236762
C2×#20832
#226962
#224852