Finite Group Databasebeta

← #6526699

Subgroups — #6526699

Order 512.

2180 conjugacy classes of subgroups.

The subgroup lattice has 735 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3234C211yesmaximal, characteristic, complemented
2C2×#3109C211yesmaximal, characteristic
3C2×#2566C211yesmaximal, characteristic, complemented
4#34206C211yesmaximal, characteristic, complemented
5#28117C212yesmaximal, complemented
6#28114C212yesmaximal, complemented
7#27651C212yesmaximal, complemented
8#27645C212yesmaximal, complemented
9#25461C214yesmaximal, complemented
10#25330C214yesmaximal, complemented
11#24823C214yesmaximal, complemented
12#24688C214yesmaximal, complemented
13#21955C211yesmaximal, characteristic, complemented
14#21838C211yesmaximal, characteristic, complemented
15#20738C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C2×#520—21no—
19C22×C4×D4C2211yescharacteristic
20#3234—28no—
21C2×#501C2212yescomplemented
22C2×#490C2212yescomplemented
23#3109—28no—
24C2×#481C2212yes—
25C2×#481C2212yes—
26C2×#480C2212yes—
27C2×#478C2214yescomplemented
28C2×#476C2212yes—
29C2×#459C2212yes—
30C2×#447C2212yes—
31C2×#442C2214yes—
32C2×#432C2211yescharacteristic
33C2×#432C2211yescharacteristic
34#2566—22no—
35#2566—22no—
36#2566C2214yes—
37#2566C2214yes—
38C2×#396—21no—
39C2×#396—21no—
40#2367—22no—
41#2366C2212yescomplemented
42#2365C2212yescomplemented
43#2354C2212yescomplemented
44#2352—22no—
45D4×C4⋊C4C2212yescomplemented
46#2325—21no—
47#2325—21no—
48#2311—21no—
49C4×#121C2212yes—
50C4×#121C2212yes—
51C2×#390C2211yescharacteristic
52C2×#390C2211yescharacteristic
53C2×#385—21no—
54C2×#385—21no—
55#2285C2211yescharacteristic
56C2×#384C2211yescharacteristic
57C2×#382C2211yescharacteristic
58C2×C4×C4⋊C4C2211yescharacteristic
59#2085C2214yescomplemented
60#2084C2214yescomplemented
61#2073C2214yescomplemented
62#2071C2214yescomplemented
63#2067C2214yes—
64#2065C2214yes—
65#2054C2214yes—
66#2054C2214yes—
67#1992C2212yescomplemented
68#1992C2212yescomplemented
69#1989C2214yescomplemented
70#1949C2214yescomplemented
71#1948C2214yescomplemented
72#1940C2214yescomplemented
73#1940C2214yescomplemented
74#1926C2214yes—
75#1925C2214yes—
76#1879C2214yes—
77#1879C2214yes—
78#1824C2212yes—
79#1824C2212yes—
80#1823C2212yes—
81#1804C2214yescomplemented
82#1802C2212yes—
83#1802C2212yes—
84#1802C2214yescomplemented
85#1797C2212yes—
86#1797C2212yes—
87#1786C2212yes—
88#1762C2214yes—
89C22×#41—21no—
90C23×D4C2311yescharacteristic
91C24×C4—21no—
92C22×D8C2314yes—
93#532—24no—
94#532D418yes—
95C2×C4⋊Q8C2311yescharacteristic
96C2×#127C2311yescharacteristic
97C2×#125—21no—
98C2×#125—21no—
99C2×#125—24no—
100C2×#125—24no—
101C2×#125C2312yes—
102C2×#125C2312yes—
103C2×#124—21no—
104C2×#121—21no—
105C2×#121C2312yes—
106#520—48no—
107C2×C4×Q8—22no—
108C2×C4×D4—22no—
109C2×C4×D4—22no—
110C2×C4×D4—22no—
111C2×C4×D4—21no—
112C2×C4×D4—21no—
113C2×C4×D4—21no—
114C2×C4×D4—24no—
115C2×C4×D4—24no—
116C2×C4×D4C2312yes—
117C2×C4×D4C2312yes—
118C2×C4×D4C2312yes—
119C2×C4×D4C2312yes—
120C2×C4×D4C2312yes—
121C2×#117—21no—
122C2×#117—22no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—22no—
128C22×C4⋊C4D411yescharacteristic
129C22×C4⋊C4D411yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2311yescharacteristic
133C22×C22⋊C4—21no—
134C22×C22⋊C4—21no—
135C22×C22⋊C4—21no—
136C22×C22⋊C4—21no—
137C22×C22⋊C4—21no—
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140C22×C42C2311yescharacteristic
141#501—28no—
142#490—28no—
143#481—28no—
144#481—28no—
145#480—28no—
146#478—216no—
147#476—28no—
148#459—28no—
149#447—28no—
150#442—216no—
151#432—24no—
152#432—24no—
153C2×#107C2314yes—
154C2×#106C2314yes—
155C2×#106C2314yes—
156C2×C4⋊C8C2312yes—
157C2×C4⋊C8C2312yes—
158C2×C4⋊C8C2314yes—
159C2×C4⋊C8C2314yes—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C2×#102C2314yes—
163C2×#102C2314yes—
164C2×#102C2314yes—
165C2×#102C2314yes—
166C2×C22⋊C8C2312yes—
167C2×C22⋊C8C2312yes—
168C2×#97C2312yes—
169C2×#97C2312yes—
170#396—24no—
171#396—24no—
172#396—24no—
173#396—24no—
174#391—22no—
175#390—22no—
176#390—22no—
177#390—22no—
178#390C2312yes—
179#390C2312yes—
180#390C2314yes—
181#390C2314yes—
182#389C2312yes—
183#389C2312yes—
184#388—22no—
185#388—22no—
186#388—22no—
187#388—22no—
188#387—22no—
189#387—22no—
190#386—22no—
191#386—22no—
192#386—22no—
193#386—22no—
194#385—24no—
195#385—24no—
196#385—24no—
197#385—24no—
198#385—22no—
199#385—22no—
200#384—22no—
201#384C2314yes—
202#384C2311yescharacteristic
203#384C2311yescharacteristic
204#382—22no—
205#382—21no—
206#382C2314yes—
207#380—21no—
208#380—22no—
209#380—22no—
210#380C2312yes—
211C4×C4⋊C4—21no—
212C4×C4⋊C4D412yes—
213C4×C4⋊C4D412yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2312yes—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—22no—
221C2×#95—21no—
222C2×#95—21no—
223C2×#95D411yescharacteristic
224C2×#95D411yescharacteristic
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#336C2314yes—
232#336C2314yes—
233C25—21no—
234C2×#41—28no—
235C2×#41—44no—
236C2×#41—44no—
237C22×Q8—21no—
238C22×D4—22no—
239C22×D4—24no—
240C22×D4—21no—
241C22×D4—21no—
242C22×D4—21no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—42no—
246C22×D4C2412yes—
247C22×D4C2412yes—
248C23×C4—22no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4—22no—
255C23×C4—22no—
256C23×C4—44no—
257C23×C4—82no—
258C23×C4—82no—
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2411yescharacteristic
266C2×D8—28no—
267C2×D8—28no—
268C2×D8—44no—
269C22×C8C2414yes—
270C22×C8C2414yes—
271C4⋊Q8—22no—
272C4⋊Q8C2×D414yes—
273#127—22no—
274#127C2×D414yes—
275#125—24no—
276#125—24no—
277#125—24no—
278#125—24no—
279#125—44no—
280#125—44no—
281#124—28no—
282#121—28no—
283#121—24no—
284#121C2×D418yes—
285C4×Q8—48no—
286C4×D4—28no—
287C4×D4—24no—
288C4×D4—24no—
289C4×D4—24no—
290C4×D4—24no—
291C4×D4—24no—
292C4×D4—44no—
293C4×D4—44no—
294C4×D4—48no—
295C4×D4—48no—
296C4×D4—48no—
297C4×D4—44no—
298C4×D4—44no—
299C4×D4C2×D418yes—
300#117—48no—
301#117—44no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—48no—
326C2×C4⋊C4—42no—
327C2×C4⋊C4—44no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4—82no—
330C2×C4⋊C4—82no—
331C2×C4⋊C4C2×D412yes—
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2×D412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2412yes—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—44no—
357C2×C22⋊C4—44no—
358C2×C22⋊C4—44no—
359C2×C22⋊C4—82no—
360C2×C22⋊C4—82no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—42no—
365C2×C42—42no—
366C2×C42C2×D412yes—
367C2×C42C2×D412yes—
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D414yes—
371C2×C42#4114yes—
372C2×C42C2412yes—
373C2×C42C2412yes—
374C2×C42C2411yescharacteristic
375C2×C42C2411yescharacteristic
376#107—28no—
377#106—28no—
378#106—28no—
379C4⋊C8—24no—
380C4⋊C8—24no—
381C4⋊C8—44no—
382C4⋊C8—44no—
383#102—28no—
384#102—28no—
385#102—28no—
386#102—28no—
387#102—28no—
388#102—44no—
389C22⋊C8—24no—
390C22⋊C8—24no—
391#97—24no—
392#97—24no—
393#95—24no—
394#95—24no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95C2×D412yes—
402#95C2×D412yes—
403#95C2×D412yes—
404#95C2×D412yes—
405C24—24no—
406C24—22no—
407C24—22no—
408C24—42no—
409C24—42no—
410C24—42no—
411C24—42no—
412C24—42no—
413C24—42no—
414C24C22×D411yescharacteristic
415#41—432no—
416C2×Q8—24no—
417C2×Q8—42no—
418C2×Q8—42no—
419C2×D4—24no—
420C2×D4—28no—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—24no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—24no—
430C2×D4—22no—
431C2×D4—48no—
432C2×D4—48no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4—44no—
437C2×D4—44no—
438C2×D4—42no—
439C2×D4—81no—
440C2×D4—81no—
441C2×D4C22×D414yes—
442C2×D42−1+414yes—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—28no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—82no—
483C22×C4—82no—
484C22×C4—82no—
485C22×C4—82no—
486C22×C4—82no—
487C22×C4—82no—
488C22×C4—82no—
489C22×C4—82no—
490C22×C4—816no—
491C22×C4#12114yes—
492C22×C4#12114yes—
493C22×C4AGL(1,ℤ/8)14yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C2×#4112yes—
505C22×C4C2511yescharacteristic
506D8—416no—
507C2×C8—24no—
508C2×C8—24no—
509C2×C8—24no—
510C2×C8—24no—
511C2×C8—24no—
512C2×C8—24no—
513C4⋊C4—24no—
514C4⋊C4—24no—
515C4⋊C4—24no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—44no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—81no—
539C4⋊C4—81no—
540C4⋊C4—84no—
541C4⋊C4—84no—
542C4⋊C4C22×D414yes—
543C22⋊C4—28no—
544C22⋊C4—28no—
545C22⋊C4—48no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—48no—
551C22⋊C4—44no—
552C22⋊C4—82no—
553C22⋊C4—82no—
554C22⋊C4—84no—
555C22⋊C4—84no—
556C22⋊C4—82no—
557C22⋊C4—82no—
558C42—24no—
559C42—24no—
560C42—42no—
561C42—42no—
562C42—44no—
563C42—44no—
564C42—44no—
565C42#12114yes—
566C42#12114yes—
567C42C22×D412yes—
568C42C22×D412yes—
569C42C2×#4112yes—
570C42C2×#4112yes—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—24no—
576C23—24no—
577C23—48no—
578C23—48no—
579C23—48no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—84no—
593C23—84no—
594C23—84no—
595C23—84no—
596C23C2×#12112yes—
597C23C2×AGL(1,ℤ/8)12yes—
598C23#57512yes—
599C23C23×D411yescharacteristic
600Q8—48no—
601D4—24no—
602D4—28no—
603D4—24no—
604D4—48no—
605D4—48no—
606D4—416no—
607D4—42no—
608D4—44no—
609D4—42no—
610D4—84no—
611C2×C4—22no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—22no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—28no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—24no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—416no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—88no—
658C2×C4—88no—
659C2×C4—82no—
660C2×C4—82no—
661C2×C4—84no—
662C2×C4—84no—
663C2×C4—84no—
664C2×C4—164no—
665C2×C4—164no—
666C2×C4C2×#12112yes—
667C2×C4C2×#12112yes—
668C2×C4C2×#12112yes—
669C2×C4C2×#12112yes—
670C2×C4C2×#12112yes—
671C2×C4C2×AGL(1,ℤ/8)14yes—
672C2×C4C23×D411yescharacteristic
673C2×C4C22×#4111yescharacteristic
674C2×C4C2×2−1+412yes—
675C2×C4#58512yes—
676C8—44no—
677C8—44no—
678C22—22no—
679C22—22no—
680C22—22no—
681C22—22no—
682C22—22no—
683C22—22no—
684C22—44no—
685C22—44no—
686C22—44no—
687C22—44no—
688C22—44no—
689C22—44no—
690C22—816no—
691C22—82no—
692C22—82no—
693C22—82no—
694C22—82no—
695C22—82no—
696C22—82no—
697C22—82no—
698C22—82no—
699C22#303614yes—
700C22C22×#12111yescharacteristic
701C22#348511yescharacteristic
702C22#348611yescharacteristic
703C22C22×AGL(1,ℤ/8)11yescharacteristic
704C22C2×#57511yescharacteristic
705C22#358511yescharacteristic
706C22#358511yescharacteristic
707C4—22no—
708C4—22no—
709C4—24no—
710C4—22no—
711C4—22no—
712C4—22no—
713C4—42no—
714C4—42no—
715C4—42no—
716C4—44no—
717C4—82no—
718C4—82no—
719C4—82no—
720C4—82no—
721C4—82no—
722C4#348912yes—
723C4#349512yes—
724C2—24no—
725C2—84no—
726C2—84no—
727C2#3563311yescharacteristic
728C2C2×#303611yescharacteristic
729C2#6042611yescharacteristic
730C2#6170711yescharacteristic
731C2#6171311yescharacteristic
732C2#6182511yescharacteristic
733C2#6183111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3109C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3234C2semidirect
3. C2×#2566C2semidirect
4. #34206C2semidirect
5. #28117C2semidirect
6. #28114C2semidirect
7. #27651C2semidirect
8. #27645C2semidirect
9. #25461C2semidirect
10. #25330C2semidirect
11. #24823C2semidirect
12. #24688C2semidirect
13. #21955C2semidirect
14. #21838C2semidirect
15. #20738C2semidirect
21. C2×#501C22semidirect
22. C2×#490C22semidirect
27. C2×#478C22semidirect
41. #2366C22semidirect
42. #2365C22semidirect
43. #2354C22semidirect
45. D4×C4⋊C4C22semidirect
59. #2085C22semidirect
60. #2084C22semidirect
61. #2073C22semidirect
62. #2071C22semidirect
67. #1992C22semidirect
68. #1992C22semidirect
69. #1989C22semidirect
70. #1949C22semidirect
71. #1948C22semidirect
72. #1940C22semidirect
73. #1940C22semidirect
81. #1804C22semidirect
84. #1802C22semidirect

Maximal subgroups

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

subgroupindex
C2×#32342
C2×#31092
C2×#25662
#342062
#281172
#281142
#276512
#276452
#254612
#253302
#248232
#246882
#219552
#218382
#207382