Finite Group Databasebeta

← #6513527

Subgroups — #6513527

Order 512.

2672 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3100C211yesmaximal, characteristic, complemented
2C2×#2975C211yesmaximal, characteristic, complemented
3#35165C211yesmaximal, characteristic, complemented
4C2×#2537C211yesmaximal, characteristic, complemented
5#28231C214yesmaximal, complemented
6#28149C214yesmaximal, complemented
7#25392C214yesmaximal, complemented
8#25347C214yesmaximal, complemented
9#24558C214yesmaximal, complemented
10#24384C214yesmaximal, complemented
11#21981C211yesmaximal, characteristic, complemented
12#21859C211yesmaximal, characteristic, complemented
13#20732C211yesmaximal, characteristic, complemented
14C2×#535—21no—
15C2×#532C2211yescharacteristic
16C22×#121—21no—
17C22×#121C2212yes—
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20#3100—28no—
21C2×#480C2212yes—
22C2×#474C2212yescomplemented
23C2×#469C2218yescomplemented
24C2×#459C2212yescomplemented
25C2×#447C2212yes—
26#2975—28no—
27C2×C4×D8C2214yescomplemented
28C2×#436C2214yescomplemented
29C2×#436C2214yescomplemented
30C2×#431C2211yescharacteristic
31C2×#431C2211yescharacteristic
32C2×#407C2214yes—
33#2847C2214yes—
34#2617—22no—
35#2547C2212yescomplemented
36#2544—22no—
37#2541C2212yescomplemented
38#2537—22no—
39#2537—22no—
40#2537C2214yes—
41#2537C2214yes—
42#2537C2212yescomplemented
43#2537C2212yescomplemented
44#2535—22no—
45C2×#394—22no—
46#2311—21no—
47#2285C2211yescharacteristic
48C2×#384C2211yescharacteristic
49C2×#380—21no—
50C2×#380C2212yes—
51#2076C2218yescomplemented
52#2034C2218yescomplemented
53#2034C2218yescomplemented
54#1999C2218yescomplemented
55#1990C2212yescomplemented
56#1957C2212yescomplemented
57#1948C2214yescomplemented
58#1895C2218yescomplemented
59#1895C2218yescomplemented
60#1874C2214yes—
61#1853C2214yes—
62#1844C2212yescomplemented
63#1834C2212yescomplemented
64#1823C2212yes—
65#1809C2218yescomplemented
66#1802C2212yescomplemented
67#1802C2214yescomplemented
68#1802C2212yescomplemented
69#1786C2212yes—
70C4×#102C2214yescomplemented
71C22×#41—21no—
72C23×D4—22no—
73C23×D4D411yescharacteristic
74C23×D4C2311yescharacteristic
75C24×C4—21no—
76C24×C4C2312yes—
77C22×D8C2314yes—
78C22×M16C2318yes—
79#535—48no—
80#532—24no—
81#532—24no—
82C2×C4⋊Q8C2311yescharacteristic
83C2×#127—22no—
84C2×#127C2311yescharacteristic
85C2×#124—22no—
86C2×#124—21no—
87C2×#121—24no—
88C2×#121—24no—
89C2×#121—22no—
90C2×#121—22no—
91C2×#121—21no—
92C2×#121—24no—
93C2×#121—48no—
94C2×#121—48no—
95C2×#121C2312yes—
96C2×#121C2318yes—
97C2×#121C2314yes—
98C2×#121C2314yes—
99C2×#120—24no—
100C2×C4×D4—24no—
101C2×C4×D4—21no—
102C2×C4×D4—21no—
103C2×C4×D4—21no—
104C2×C4×D4—24no—
105C2×C4×D4—24no—
106C2×C4×D4—24no—
107C2×C4×D4D412yes—
108C2×C4×D4D412yes—
109C2×C4×D4C2312yes—
110C2×C4×D4C2312yes—
111C2×C4×D4C2312yes—
112C2×C4×D4C2312yes—
113C2×C4×D4C2312yes—
114C2×#117—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4D411yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2312yes—
120C22×C22⋊C4—22no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—22no—
123C22×C22⋊C4D412yes—
124C22×C42C2311yescharacteristic
125#480—28no—
126#474—28no—
127#469—416no—
128#459—28no—
129#447—28no—
130C4×D8—216no—
131#436—216no—
132#436—216no—
133#431—24no—
134#431D418yescomplemented
135C2×#107C2314yes—
136C2×#106C2318yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2312yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2318yes—
143C2×#102C2314yes—
144#407—216no—
145C2×C22⋊C8C2318yes—
146C2×C22⋊C8C2312yes—
147C2×C22⋊C8C2312yes—
148C2×C4×C8C2312yes—
149C2×C4×C8C2312yes—
150#397—24no—
151#394—24no—
152#394—24no—
153#394—28no—
154#394—28no—
155#392—24no—
156#391—22no—
157#390—22no—
158#386—24no—
159#386—24no—
160#386—24no—
161#384—21no—
162#384—21no—
163#384C2311yescharacteristic
164#384C2312yes—
165#384C2311yescharacteristic
166#384C2312yes—
167#382—21no—
168#380—21no—
169#380—24no—
170#380—22no—
171#380—22no—
172#380—24no—
173#380—24no—
174#380C2314yes—
175#380C2318yes—
176#380C2314yes—
177#380C2312yes—
178C4×C4⋊C4—21no—
179C4×C4⋊C4C2312yes—
180C2×#95—22no—
181#340C2314yes—
182#340C2318yes—
183#340C2314yes—
184#340C2318yes—
185C25—21no—
186C25C2×D412yes—
187C2×#41—44no—
188C2×#41—44no—
189C2×#41—44no—
190C22×Q8—21no—
191C22×D4—24no—
192C22×D4—22no—
193C22×D4—28no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—21no—
200C22×D4—22no—
201C22×D4—21no—
202C22×D4—21no—
203C22×D4—22no—
204C22×D4—42no—
205C22×D4—48no—
206C22×D4—48no—
207C22×D4—84no—
208C22×D4C2×D412yes—
209C22×D4C2×D412yes—
210C22×D4C2412yes—
211C22×D4C2412yes—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—21no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4C2×D412yes—
225C23×C4C2×D414yes—
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2414yes—
229C23×C4C2411yescharacteristic
230C2×D8—28no—
231C2×D8—28no—
232C2×D8—44no—
233C2×M16—216no—
234C2×M16—48no—
235C2×M16—48no—
236C22×C8C2414yes—
237C22×C8C2414yes—
238C4⋊Q8—22no—
239C4⋊Q8—22no—
240#127—22no—
241#127—22no—
242#127—48no—
243#124—48no—
244#124—44no—
245#121—216no—
246#121—24no—
247#121—24no—
248#121—48no—
249#121—416no—
250#121—416no—
251#121—416no—
252#121—48no—
253#121—48no—
254#121—44no—
255#121—48no—
256#121—48no—
257#120—416no—
258C4×D4—28no—
259C4×D4—28no—
260C4×D4—24no—
261C4×D4—24no—
262C4×D4—24no—
263C4×D4—24no—
264C4×D4—24no—
265C4×D4—24no—
266C4×D4—24no—
267C4×D4—24no—
268C4×D4—24no—
269C4×D4—24no—
270C4×D4—44no—
271C4×D4—44no—
272C4×D4—44no—
273#117—44no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—82no—
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2414yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—48no—
311C2×C22⋊C4—48no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4C2×D414yes—
314C2×C22⋊C4C2×D414yes—
315C2×C42—22no—
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D414yes—
319C2×C42#4114yes—
320C2×C42C2411yescharacteristic
321C2×C42C2411yescharacteristic
322#107—28no—
323#106—48no—
324C4⋊C8—24no—
325C4⋊C8C2×D418yescomplemented
326#102—28no—
327#102—28no—
328#102—28no—
329#102—44no—
330#102—48no—
331C22⋊C8—24no—
332C22⋊C8—216no—
333C22⋊C8C2×D418yes—
334C4×C8—24no—
335C4×C8C2×D418yescomplemented
336#95—28no—
337#95—22no—
338#95—22no—
339#95—24no—
340#95—24no—
341C24—24no—
342C24—24no—
343C24—22no—
344C24—24no—
345C24—24no—
346C24—22no—
347C24—42no—
348C24—42no—
349C24—42no—
350C24—42no—
351C24—42no—
352C24—42no—
353C24—42no—
354C24—42no—
355C24—42no—
356C24—42no—
357C24C22×D414yes—
358C24C22×D411yescharacteristic
359#41—816no—
360C2×Q8—42no—
361C2×Q8—42no—
362C2×Q8—42no—
363C2×D4—24no—
364C2×D4—24no—
365C2×D4—24no—
366C2×D4—24no—
367C2×D4—24no—
368C2×D4—24no—
369C2×D4—24no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—24no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—24no—
379C2×D4—48no—
380C2×D4—48no—
381C2×D4—42no—
382C2×D4—416no—
383C2×D4—416no—
384C2×D4—44no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—48no—
388C2×D4—48no—
389C2×D4—42no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—42no—
393C2×D4—42no—
394C2×D4—416no—
395C2×D4—81no—
396C2×D4—81no—
397C2×D4—88no—
398C2×D4—84no—
399C2×D4—84no—
400C2×D4—84no—
401C2×D4—84no—
402C2×D4—832no—
403C2×D4H(5,𝔽2)14yes—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—42no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—42no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—44no—
428C22×C4—42no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—88no—
439C22×C4—88no—
440C22×C4—816no—
441C22×C4#12114yes—
442C22×C4#12114yes—
443C22×C4AGL(1,ℤ/8)14yes—
444C22×C4AGL(1,ℤ/8)14yes—
445C22×C4C22×D414yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C2×#4112yes—
453C22×C4C2511yescharacteristic
454D8—416no—
455M16—816no—
456C2×C8—24no—
457C2×C8—24no—
458C2×C8—24no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8C22×D418yes—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—28no—
470C4⋊C4—24no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—81no—
479C4⋊C4—81no—
480C4⋊C4—82no—
481C4⋊C4—82no—
482C22⋊C4—28no—
483C22⋊C4—28no—
484C22⋊C4—416no—
485C22⋊C4—416no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—84no—
494C22⋊C4—84no—
495C42—44no—
496C42#12114yes—
497C42#12114yes—
498C42C22×D412yes—
499C42C22×D412yes—
500C42C2×#4112yes—
501C42C2×#4112yes—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—48no—
515C23—48no—
516C23—48no—
517C23—48no—
518C23—48no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—48no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—84no—
534C23—84no—
535C23—84no—
536C23—84no—
537C23—84no—
538C23—84no—
539C23—84no—
540C23—84no—
541C23C2×#12112yes—
542C23C2×AGL(1,ℤ/8)12yes—
543C23C2×AGL(1,ℤ/8)12yes—
544C23C23×D411yescharacteristic
545Q8—84no—
546D4—24no—
547D4—24no—
548D4—48no—
549D4—42no—
550D4—44no—
551D4—42no—
552D4—44no—
553D4—48no—
554D4—48no—
555D4—84no—
556D4—832no—
557D4—88no—
558D4—84no—
559D4—168no—
560D4—168no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—44no—
573C2×C4—48no—
574C2×C4—48no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—88no—
591C2×C4—84no—
592C2×C4—88no—
593C2×C4—84no—
594C2×C4—88no—
595C2×C4—88no—
596C2×C4—88no—
597C2×C4#469116yescomplemented
598C2×C4C2×#12112yes—
599C2×C4C2×#12112yes—
600C2×C4C2×#12112yes—
601C2×C4C2×#12112yes—
602C2×C4C2×#12112yes—
603C2×C4C2×AGL(1,ℤ/8)12yes—
604C2×C4C2×AGL(1,ℤ/8)12yes—
605C2×C4C2×AGL(1,ℤ/8)12yes—
606C2×C4C2×AGL(1,ℤ/8)12yes—
607C2×C4C23×D411yescharacteristic
608C2×C4C22×#4111yescharacteristic
609C2×C4C2×H(5,𝔽2)12yes—
610C2×C4#58512yes—
611C8—28no—
612C8—44no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—44no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22—44no—
629C22—816no—
630C22—816no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22C2×#46914yes—
643C22C22×#12111yescharacteristic
644C22#348411yescharacteristic
645C22#348511yescharacteristic
646C22C22×AGL(1,ℤ/8)11yescharacteristic
647C22C22×AGL(1,ℤ/8)11yescharacteristic
648C22#358511yescharacteristic
649C22#358611yescharacteristic
650C4—22no—
651C4—22no—
652C4—22no—
653C4—44no—
654C4—82no—
655C4—82no—
656C4—82no—
657C4—84no—
658C4—82no—
659C4C2×#46914yes—
660C4C2×#46914yes—
661C4#348812yes—
662C4#349412yes—
663C2—24no—
664C2—84no—
665C2—84no—
666C2—162no—
667C2#3551411yescharacteristic
668C2C22×#46911yescharacteristic
669C2#6045611yescharacteristic
670C2#6158211yescharacteristic
671C2#6158511yescharacteristic
672C2#6171711yescharacteristic
673C2#6172011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3100C2semidirect
2. C2×#2975C2semidirect
3. #35165C2semidirect
4. C2×#2537C2semidirect
5. #28231C2semidirect
6. #28149C2semidirect
7. #25392C2semidirect
8. #25347C2semidirect
9. #24558C2semidirect
10. #24384C2semidirect
11. #21981C2semidirect
12. #21859C2semidirect
13. #20732C2semidirect
22. C2×#474C22semidirect
23. C2×#469C22semidirect
24. C2×#459C22semidirect
27. C2×C4×D8C22semidirect
28. C2×#436C22semidirect
29. C2×#436C22semidirect
35. #2547C22semidirect
37. #2541C22semidirect
42. #2537C22semidirect
43. #2537C22semidirect
51. #2076C22semidirect
52. #2034C22semidirect
53. #2034C22semidirect
54. #1999C22semidirect
55. #1990C22semidirect
56. #1957C22semidirect
57. #1948C22semidirect
58. #1895C22semidirect
59. #1895C22semidirect
62. #1844C22semidirect
63. #1834C22semidirect
65. #1809C22semidirect
66. #1802C22semidirect
67. #1802C22semidirect
68. #1802C22semidirect
70. C4×#102C22semidirect
134. #431D4semidirect
325. C4⋊C8C2×D4semidirect
335. C4×C8C2×D4semidirect
597. C2×C4#469semidirect

Maximal subgroups

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

subgroupindex
C2×#31002
C2×#29752
#351652
C2×#25372
#282312
#281492
#253922
#253472
#245582
#243842
#219812
#218592
#207322