Finite Group Databasebeta

← #6621064

Subgroups — #6621064

Order 512.

2211 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×#3373C212yesmaximal, complemented
2Q8×#127C211yesmaximal, characteristic, complemented
3C4×#559C211yesmaximal, characteristic, complemented
4#33197C212yesmaximal, complemented
5#33196C211yesmaximal, characteristic, complemented
6#28726C212yesmaximal, complemented
7#28716C211yesmaximal, characteristic, complemented
8#28253C212yesmaximal, complemented
9#28251C211yesmaximal, characteristic
10#25334C214yesmaximal, complemented
11#25318C212yesmaximal, complemented
12#24692C214yesmaximal, complemented
13#24678C212yesmaximal, complemented
14#22722C211yesmaximal, characteristic, complemented
15#22523C212yesmaximal, complemented
16#21503C212yesmaximal, complemented
17C4×#460C211yesmaximal, characteristic, complemented
18C2×#559C2211yescharacteristic
19C2×D4×Q8—21no—
20C2×D4×Q8—21no—
21C2×D4×Q8C2212yes—
22#3373—216no—
23C2×#489C2214yes—
24C2×#486C2212yes—
25C2×#475C2212yes—
26C2×#476C2214yes—
27C2×#460C2211yescharacteristic
28C2×#463C2214yes—
29C2×#459C2212yes—
30C2×#459C2212yes—
31C2×#446C2212yes—
32C2×#442C2214yes—
33C2×C4×SD16C2212yes—
34#2868C2211yescharacteristic
35#2868C2212yes—
36#2542—21no—
37#2542—22no—
38#2542—21no—
39#2542—22no—
40#2542C2212yes—
41#2542C2214yes—
42#2374—22no—
43#2367—21no—
44#2367C2212yes—
45#2366C2211yescharacteristic
46#2357—21no—
47#2357C2212yes—
48#2355C2211yescharacteristic
49#2355C2211yescharacteristic
50C4×#127C2211yescharacteristic
51C4×#127C2211yescharacteristic
52C4×#127C2212yes—
53C4×#124—21no—
54C4×#124C2212yes—
55C4×#123—22no—
56C42×Q8C2211yescharacteristic
57C42×Q8C2211yescharacteristic
58C42×D4—21no—
59C42×D4C2212yes—
60#2072C2214yes—
61#2071C2214yes—
62#2071C2214yes—
63#2055C2214yes—
64#2054C2214yes—
65#2054C2214yes—
66#1993C2214yes—
67#1990C2212yes—
68#1988C2212yes—
69#1987C2214yes—
70#1982C2211yescharacteristic
71#1948C2214yes—
72#1948C2212yes—
73#1948C2212yes—
74#1948C2211yescharacteristic
75#1940C2214yes—
76#1939C2212yes—
77#1926C2214yes—
78#1926C2212yes—
79#1880C2212yes—
80#1879C2214yes—
81#1842C2211yescharacteristic
82#1842C2212yes—
83#1804C2214yes—
84#1803C2211yescharacteristic
85#1803C2212yes—
86#1803C2212yes—
87#1802C2212yes—
88#1802C2212yes—
89#1802C2212yes—
90#1802C2212yes—
91#1802C2214yes—
92#1767C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94#1762C2214yes—
95C4×#102C2212yes—
96C23×Q8—22no—
97C22×SD16C2312yes—
98C22×D8C2314yes—
99#559—24no—
100#559D418yescomplemented
101D4×Q8—28no—
102D4×Q8—28no—
103D4×Q8—48no—
104D4×Q8—48no—
105C2×C4⋊Q8—22no—
106C2×C4⋊Q8—22no—
107C2×C4⋊Q8—21no—
108C2×C4⋊Q8—21no—
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2314yes—
111C2×#127C2311yescharacteristic
112C2×#127C2311yescharacteristic
113C2×#124—21no—
114C2×#124C2312yes—
115C2×#123—22no—
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×Q8C2311yescharacteristic
125C2×C4×Q8C2311yescharacteristic
126C2×C4×Q8C2311yescharacteristic
127C2×C4×Q8C2311yescharacteristic
128C2×C4×D4—21no—
129C2×C4×D4—21no—
130C2×C4×D4—22no—
131C2×C4×D4—22no—
132C2×C4×D4—21no—
133C2×C4×D4—21no—
134C2×C4×D4—21no—
135C2×C4×D4—21no—
136C2×C4×D4C2312yes—
137C2×C4×D4C2312yes—
138C2×C4×D4C2314yes—
139C2×C4×D4C2312yes—
140C2×C4×D4C2312yes—
141C22×C42—22no—
142#489—216no—
143#486—28no—
144#476—216no—
145#475—28no—
146#463—216no—
147#460—24no—
148#459—28no—
149#459—28no—
150#446—28no—
151#442—216no—
152C4×SD16—28no—
153C2×#106C2314yes—
154C2×#106C2312yes—
155C2×C4⋊C8C2312yes—
156C2×C4⋊C8C2311yescharacteristic
157C2×C4⋊C8C2311yescharacteristic
158C2×C4⋊C8C2312yes—
159C2×C4⋊C8C2314yes—
160C2×C4⋊C8C2312yes—
161C2×C4⋊C8C2314yes—
162C2×#103C2314yes—
163C2×#103C2312yes—
164C2×#102C2312yes—
165C2×#102C2314yes—
166C2×#102C2314yes—
167C2×#102C2314yes—
168C2×#102C2314yes—
169C2×#102C2312yes—
170C2×#102C2312yes—
171C2×#97C2314yes—
172C2×C4×C8C2312yes—
173#399—24no—
174#399—24no—
175#399—24no—
176#391—21no—
177#391C2312yes—
178#390—21no—
179#390—21no—
180#390—21no—
181#390—21no—
182#390—22no—
183#390—22no—
184#390—21no—
185#390C2312yes—
186#390C2312yes—
187#390C2314yes—
188#390C2312yes—
189#388—22no—
190#388—22no—
191#388—22no—
192#388—22no—
193#388—22no—
194#388—22no—
195#387—22no—
196#387—22no—
197#386—22no—
198#385—22no—
199#384—21no—
200#384—22no—
201#384—22no—
202#384—21no—
203#384C2311yescharacteristic
204#384C2312yes—
205#384C2311yescharacteristic
206#384C2314yes—
207#384C2312yes—
208#382—21no—
209#382C2312yes—
210#381—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4D412yes—
218C4×C4⋊C4D411yescharacteristic
219C4×C4⋊C4D412yes—
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2311yescharacteristic
225C4×C4⋊C4C2311yescharacteristic
226C4×C4⋊C4C2312yes—
227C4×C22⋊C4—22no—
228C4×C22⋊C4—22no—
229C4×C22⋊C4—22no—
230C4×C22⋊C4—22no—
231C4×C22⋊C4—22no—
232C4×C22⋊C4—22no—
233C4×C22⋊C4—22no—
234C43—21no—
235C43C2311yescharacteristic
236C43C2312yes—
237#340C2314yes—
238#340C2314yes—
239#340C2314yes—
240#340C2312yes—
241#340C2314yes—
242#340C2314yes—
243#340C2312yes—
244#336C2312yes—
245#336C2314yes—
246C22×Q8—24no—
247C22×Q8—22no—
248C22×Q8—48no—
249C22×Q8—48no—
250C22×Q8—48no—
251C22×Q8C2×D411yescharacteristic
252C22×Q8C2411yescharacteristic
253C22×D4—21no—
254C22×D4—21no—
255C22×D4—21no—
256C22×D4—21no—
257C22×D4C2412yes—
258C22×D4C2412yes—
259C23×C4—22no—
260C23×C4—24no—
261C23×C4—22no—
262C23×C4—22no—
263C23×C4—22no—
264C2×SD16—24no—
265C2×SD16—24no—
266C2×SD16—42no—
267C2×D8—28no—
268C2×D8—28no—
269C2×D8—44no—
270C22×C8C2414yes—
271C22×C8C2412yes—
272C22×C8C2412yes—
273C4⋊Q8—24no—
274C4⋊Q8—28no—
275C4⋊Q8—28no—
276C4⋊Q8—24no—
277C4⋊Q8—44no—
278C4⋊Q8—48no—
279C4⋊Q8—48no—
280C4⋊Q8—44no—
281#127—22no—
282#127—22no—
283#127C2×D414yes—
284#127#4114yes—
285#124—28no—
286#124—24no—
287#124C2×D418yes—
288#123—216no—
289C22⋊Q8—48no—
290C22⋊Q8—416no—
291C22⋊Q8—416no—
292C22⋊Q8—48no—
293C22⋊Q8—416no—
294C22⋊Q8—48no—
295C4×Q8—24no—
296C4×Q8—24no—
297C4×Q8—22no—
298C4×Q8—22no—
299C4×Q8—22no—
300C4×Q8—22no—
301C4×Q8—41no—
302C4×Q8C2×D414yes—
303C4×Q8C2×D414yes—
304C4×Q8#4114yes—
305C4×D4—28no—
306C4×D4—24no—
307C4×D4—24no—
308C4×D4—28no—
309C4×D4—24no—
310C4×D4—28no—
311C4×D4—24no—
312C4×D4—24no—
313C4×D4—24no—
314C4×D4—42no—
315C4×D4—44no—
316C4×D4—48no—
317C4×D4—48no—
318C4×D4—44no—
319C4×D4—44no—
320C4×D4—44no—
321C4×D4—82no—
322C4×D4C2×D418yes—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2×D412yes—
340C2×C4⋊C4C2×D412yes—
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2×D412yes—
343C2×C4⋊C4C2×D411yescharacteristic
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2414yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2411yescharacteristic
348C2×C4⋊C4C2411yescharacteristic
349C2×C4⋊C4C2411yescharacteristic
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—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C42—21no—
366C2×C42—21no—
367C2×C42—21no—
368C2×C42—21no—
369C2×C42—21no—
370C2×C42—21no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—84no—
374C2×C42C2×D412yes—
375C2×C42C2×D412yes—
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D412yes—
379C2×C42C2×D412yes—
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2412yes—
383C2×C42C2411yescharacteristic
384C2×C42C2412yes—
385C2×C42C2411yescharacteristic
386C2×C42C2411yescharacteristic
387#106—24no—
388#106—28no—
389C4⋊C8—24no—
390C4⋊C8—28no—
391C4⋊C8—24no—
392C4⋊C8—22no—
393C4⋊C8—22no—
394C4⋊C8—44no—
395C4⋊C8—42no—
396#103—24no—
397#103—28no—
398#102—28no—
399#102—28no—
400#102—28no—
401#102—28no—
402#102—24no—
403#102—24no—
404#102—24no—
405#97—28no—
406C4×C8—24no—
407#95—21no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—21no—
415#95—21no—
416#95C2×D412yes—
417#95C2×D412yes—
418#95C2×D412yes—
419C24—22no—
420C24—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—48no—
429C2×Q8—48no—
430C2×Q8—48no—
431C2×Q8—44no—
432C2×Q8—44no—
433C2×Q8—44no—
434C2×Q8—832no—
435C2×Q8#12114yes—
436C2×Q8C22×D412yes—
437C2×D4—24no—
438C2×D4—24no—
439C2×D4—24no—
440C2×D4—24no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—81no—
458C2×D4—81no—
459C2×D4C22×D414yes—
460C2×D4C2×#4114yes—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—28no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—44no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—44no—
474C22×C4—44no—
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×C4C22×D412yes—
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D412yes—
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C2511yescharacteristic
494SD16—48no—
495D8—416no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—24no—
499C2×C8—24no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—24no—
504C2×C8—22no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—21no—
515C4⋊C4—22no—
516C4⋊C4—24no—
517C4⋊C4—22no—
518C4⋊C4—21no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—44no—
522C4⋊C4—48no—
523C4⋊C4—44no—
524C4⋊C4—48no—
525C4⋊C4—44no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—44no—
529C4⋊C4—42no—
530C4⋊C4—48no—
531C4⋊C4—44no—
532C4⋊C4—81no—
533C4⋊C4#12114yes—
534C4⋊C4#12118yes—
535C4⋊C4AGL(1,ℤ/8)14yes—
536C4⋊C4C22×D412yes—
537C4⋊C42−1+412yes—
538C22⋊C4—28no—
539C22⋊C4—28no—
540C22⋊C4—28no—
541C22⋊C4—44no—
542C22⋊C4—48no—
543C22⋊C4—44no—
544C22⋊C4—48no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—48no—
550C22⋊C4—44no—
551C22⋊C4—82no—
552C22⋊C4—82no—
553C42—24no—
554C42—24no—
555C42—22no—
556C42—24no—
557C42—24no—
558C42—22no—
559C42—22no—
560C42—21no—
561C42—22no—
562C42—21no—
563C42—41no—
564C42—42no—
565C42—42no—
566C42—42no—
567C42—42no—
568C42—42no—
569C42—81no—
570C42—82no—
571C42—82no—
572C42AGL(1,ℤ/8)14yes—
573C42C22×D412yes—
574C42C22×D414yes—
575C42C2×#4114yes—
576C42C2×#4112yes—
577C42C2×#4112yes—
578C422−1+412yes—
579C23—24no—
580C23—24no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23C23×D411yescharacteristic
594Q8—28no—
595Q8—22no—
596Q8—22no—
597Q8—41no—
598Q8—41no—
599Q8—816no—
600Q8—88no—
601D4—24no—
602D4—24no—
603D4—24no—
604D4—24no—
605D4—48no—
606D4—48no—
607D4—42no—
608D4—42no—
609D4—42no—
610D4—42no—
611D4—48no—
612D4—84no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—22no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—22no—
619C2×C4—21no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—21no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—21no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—41no—
644C2×C4—42no—
645C2×C4—41no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—88no—
653C2×C4—88no—
654C2×C4—816no—
655C2×C4—88no—
656C2×C4—88no—
657C2×C4C2×#12114yes—
658C2×C4C2×#12114yes—
659C2×C4C2×#12112yes—
660C2×C4C2×#12112yes—
661C2×C4#54812yes—
662C2×C4#54812yes—
663C2×C4C2×AGL(1,ℤ/8)14yes—
664C2×C4C2×AGL(1,ℤ/8)12yes—
665C2×C4C2×AGL(1,ℤ/8)12yes—
666C2×C4C2×AGL(1,ℤ/8)12yes—
667C2×C4C2×AGL(1,ℤ/8)12yes—
668C2×C4#57712yes—
669C2×C4C23×D411yescharacteristic
670C2×C4C22×#4111yescharacteristic
671C2×C4C22×#4111yescharacteristic
672C2×C4C2×2−1+411yescharacteristic
673C2×C4C2×2−1+411yescharacteristic
674C2×C4#58511yescharacteristic
675C8—44no—
676C8—42no—
677C8—42no—
678C22—44no—
679C22—44no—
680C22—44no—
681C22—44no—
682C22—44no—
683C22—44no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22C22×#12111yescharacteristic
693C22C2×#54811yescharacteristic
694C22#348911yescharacteristic
695C22C22×AGL(1,ℤ/8)11yescharacteristic
696C22C22×AGL(1,ℤ/8)11yescharacteristic
697C22C2×#57711yescharacteristic
698C22#358611yescharacteristic
699C4—24no—
700C4—22no—
701C4—22no—
702C4—22no—
703C4—24no—
704C4—24no—
705C4—42no—
706C4—42no—
707C4—42no—
708C4—41no—
709C4—42no—
710C4—82no—
711C4—81no—
712C4—84no—
713C4—82no—
714C4#303814yes—
715C4#337314yescomplemented
716C4D4×#4111yescharacteristic
717C4#348211yescharacteristic
718C4#349911yescharacteristic
719C4#351811yescharacteristic
720C2—84no—
721C2—84no—
722C2#3644411yescharacteristic
723C2C2×#303811yescharacteristic
724C2#6042911yescharacteristic
725C2C2×#337311yescharacteristic
726C2#6131711yescharacteristic
727C2#6211111yescharacteristic
728C2#6212811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#28251C2
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×#3373C2semidirect
2. Q8×#127C2semidirect
3. C4×#559C2semidirect
4. #33197C2semidirect
5. #33196C2semidirect
6. #28726C2semidirect
7. #28716C2semidirect
8. #28253C2semidirect
10. #25334C2semidirect
11. #25318C2semidirect
12. #24692C2semidirect
13. #24678C2semidirect
14. #22722C2semidirect
15. #22523C2semidirect
16. #21503C2semidirect
17. C4×#460C2semidirect
100. #559D4semidirect
715. C4#3373semidirect

Maximal subgroups

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

subgroupindex
C2×#33732
Q8×#1272
C4×#5592
#331972
#331962
#287262
#287162
#282532
#282512
#253342
#253182
#246922
#246782
#227222
#225232
#215032
C4×#4602