Finite Group Databasebeta

← #6558633

Subgroups — #6558633

Order 512.

1792 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36477C211yesmaximal, characteristic, complemented
2#36127C211yesmaximal, characteristic
3#30366C212yesmaximal, complemented
4#30304C211yesmaximal, characteristic, complemented
5#29165C212yesmaximal, complemented
6#29141C211yesmaximal, characteristic, complemented
7#29121C211yesmaximal, characteristic, complemented
8#28865C214yesmaximal, complemented
9#28230C212yesmaximal, complemented
10#28150C212yesmaximal, complemented
11#26983C214yesmaximal, complemented
12#25501C212yesmaximal, complemented
13#23799C211yesmaximal, characteristic, complemented
14#23576C212yesmaximal, complemented
15#22762C212yesmaximal, complemented
16#22671C212yesmaximal, complemented
17#20568C211yesmaximal, characteristic, complemented
18C2×#547—21no—
19C2×#532C2212yes—
20C2×#468C2214yes—
21C2×#440C2214yes—
22C2×C4×SD16C2212yes—
23C2×C4×SD16C2212yes—
24C2×#436C2212yes—
25#2868C2211yescharacteristic
26#2859C2211yescharacteristic
27#2857C2212yes—
28#2855C2212yes—
29#2608—22no—
30#2587—22no—
31#2570—21no—
32#2566—21no—
33#2547—21no—
34#2538C2211yescharacteristic
35#2538C2211yescharacteristic
36#2537—21no—
37#2537C2212yes—
38#2497C2212yes—
39#2432—22no—
40#2429C2212yes—
41#2422—22no—
42#2422C2214yes—
43#2370—21no—
44#2357—21no—
45#2355C2211yescharacteristic
46Q8×C4⋊C4C2211yescharacteristic
47D4×C4⋊C4—21no—
48D4×C4⋊C4C2212yes—
49C4×C4⋊Q8C2211yescharacteristic
50C4×#127C2211yescharacteristic
51C4×C22⋊Q8C2212yes—
52C42×D4C2211yescharacteristic
53#2099C2212yes—
54#2096C2211yescharacteristic
55#2096C2211yescharacteristic
56#2094C2212yes—
57#2083C2214yes—
58#2077C2214yes—
59#2057C2212yes—
60#2057C2212yes—
61#2041C2212yes—
62#2036C2214yes—
63#2033C2214yes—
64#2024C2212yes—
65#2024C2212yes—
66#2014C2211yescharacteristic
67#2014C2211yescharacteristic
68#2013C2212yes—
69#2000C2214yes—
70#1956C2212yes—
71#1944C2214yes—
72#1935C2214yes—
73#1934C2214yes—
74#1920C2214yes—
75#1895C2214yes—
76#1894C2214yes—
77#1877C2214yes—
78#1863C2214yes—
79#1862C2214yes—
80#1845C2212yes—
81#1834C2212yes—
82#1834C2212yes—
83#1809C2214yes—
84#1803C2212yes—
85#1803C2212yes—
86#1802C2212yes—
87#1802C2212yes—
88#1770C2212yes—
89C4×C4⋊C8C2211yescharacteristic
90#1762C2214yes—
91#1753C2214yes—
92#1749C2212yes—
93#1744C2211yescharacteristic
94#1728C2212yes—
95C22×#41—22no—
96C22×SD16C2312yes—
97C22×SD16C2312yes—
98C22×M16C2314yes—
99#547—48no—
100#532—48no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#127C2311yescharacteristic
106C2×#125—21no—
107C2×#124—22no—
108C2×#123—24no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8C2312yes—
111C2×C22⋊Q8C2312yes—
112C2×#121—22no—
113C2×#121—22no—
114C2×#121C2314yes—
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2312yes—
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4—22no—
121C2×C4×D4D411yescharacteristic
122C2×C4×D4D411yescharacteristic
123C2×C4×D4C2312yes—
124C2×C4×D4C2314yes—
125C2×C4×D4C2311yescharacteristic
126C22×C4⋊C4—22no—
127C22×C42C2312yes—
128#468—48no—
129#440—48no—
130C4×SD16—28no—
131C4×SD16—28no—
132#436—28no—
133C2×#107C2314yes—
134C2×#107C2314yes—
135C2×#106C2312yes—
136C2×#106C2312yes—
137C2×C4⋊C8C2312yes—
138C2×#103C2312yes—
139C2×#103C2312yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#102C2314yes—
143C2×#102C2312yes—
144C2×#102C2312yes—
145C2×#102C2314yes—
146C2×C22⋊C8C2314yes—
147C2×#97C2314yes—
148C2×C4×C8C2312yes—
149C2×C4×C8C2312yes—
150#400—22no—
151#398—22no—
152#397—24no—
153#396—22no—
154#396—22no—
155#396—22no—
156#396—22no—
157#394—24no—
158#394—24no—
159#391—22no—
160#391C2311yescharacteristic
161#391C2311yescharacteristic
162#391C2312yes—
163#390—21no—
164#390—21no—
165#390—21no—
166#390—21no—
167#390C2312yes—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—22no—
174#389—21no—
175#389—22no—
176#389—22no—
177#389C2312yes—
178#389C2312yes—
179#389C2312yes—
180#389C2314yes—
181#388—22no—
182#388—22no—
183#387—22no—
184#387—22no—
185#387—22no—
186#387—22no—
187#387—22no—
188#385—22no—
189#385—22no—
190#385—24no—
191#385—24no—
192#384—21no—
193#384—21no—
194#384—21no—
195#384C2311yescharacteristic
196#384C2311yescharacteristic
197#384C2311yescharacteristic
198#384C2311yescharacteristic
199#384C2312yes—
200#384C2312yes—
201#382—21no—
202#382—21no—
203#381—22no—
204#380—22no—
205#380—22no—
206#380—22no—
207#380C2312yes—
208#380C2314yes—
209#380C2312yes—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C22⋊C4—22no—
220C4×C22⋊C4D412yes—
221C4×C22⋊C4D412yes—
222C4×C22⋊C4C2312yes—
223C43C2311yescharacteristic
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#336C2311yescharacteristic
231#336C2311yescharacteristic
232#336C2312yes—
233#336C2312yes—
234#336C2314yes—
235C2×#41—48no—
236C2×#41—48no—
237C2×#41—84no—
238C22×Q8—22no—
239C22×Q8C2411yescharacteristic
240C22×Q8C2411yescharacteristic
241C22×D4—24no—
242C22×D4—21no—
243C22×D4—21no—
244C22×D4C2×D411yescharacteristic
245C22×D4C2412yes—
246C23×C4—22no—
247C23×C4—24no—
248C23×C4C2×D412yes—
249C23×C4C2×D412yes—
250C23×C4C2412yes—
251C2×SD16—24no—
252C2×SD16—24no—
253C2×SD16—24no—
254C2×SD16—24no—
255C2×SD16—24no—
256C2×SD16—42no—
257C2×M16—28no—
258C2×M16—44no—
259C2×M16—44no—
260C22×C8C2414yes—
261C22×C8C2412yes—
262C22×C8C2412yes—
263C4⋊Q8—22no—
264C4⋊Q8—44no—
265C4⋊Q8—41no—
266C4⋊Q8—44no—
267C4⋊Q8—82no—
268#127—24no—
269#125—44no—
270#124—48no—
271#123—416no—
272C22⋊Q8—24no—
273C22⋊Q8—42no—
274C22⋊Q8—48no—
275C22⋊Q8—44no—
276#121—28no—
277#121—44no—
278#121—48no—
279#121—48no—
280C4×Q8—24no—
281C4×Q8—24no—
282C4×Q8—44no—
283C4×D4—28no—
284C4×D4—24no—
285C4×D4—24no—
286C4×D4—24no—
287C4×D4—24no—
288C4×D4—24no—
289C4×D4—44no—
290C4×D4—48no—
291C4×D4—44no—
292C4×D4—44no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—48no—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2414yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—44no—
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2412yes—
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355#107—44no—
356#107—44no—
357#106—24no—
358#106—24no—
359C4⋊C8—24no—
360#103—24no—
361#103—24no—
362#103—44no—
363#103—44no—
364#102—28no—
365#102—24no—
366#102—24no—
367#102—44no—
368C22⋊C8—28no—
369#97—44no—
370C4×C8—42no—
371C4×C8#4118yescomplemented
372#95—21no—
373#95—22no—
374#95—21no—
375#95—22no—
376#95—21no—
377#95—22no—
378#95—22no—
379#95—21no—
380#95—22no—
381#95—24no—
382#95—24no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95C2×D412yes—
387#95C2×D412yes—
388#95C2×D412yes—
389C24—22no—
390C24C22×D412yes—
391#41—832no—
392C2×Q8—21no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—44no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q82−1+412yes—
403C2×D4—24no—
404C2×D4—24no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—48no—
416C2×D4—48no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—84no—
420C2×D4H(5,𝔽2)14yes—
421C22×C4—21no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—84no—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C2511yescharacteristic
455SD16—216no—
456SD16—48no—
457M16—88no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8C2×#4114yes—
466C2×C8C2×#4114yes—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—48no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—81no—
495C4⋊C4—81no—
496C4⋊C4—81no—
497C4⋊C4—81no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C4⋊C4—82no—
503C22⋊C4—24no—
504C22⋊C4—24no—
505C22⋊C4—24no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—44no—
509C22⋊C4—48no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—82no—
514C22⋊C4—82no—
515C22⋊C4—82no—
516C42—24no—
517C42—22no—
518C42—21no—
519C42—21no—
520C42—42no—
521C42—42no—
522C42—41no—
523C42—41no—
524C42—42no—
525C42—42no—
526C42—82no—
527C42—82no—
528C42AGL(1,ℤ/8)14yes—
529C42#13714yes—
530C42C2×#4114yes—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—24no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23C23×D411yescharacteristic
546Q8—24no—
547Q8—42no—
548Q8—42no—
549Q8—42no—
550Q8—88no—
551D4—24no—
552D4—24no—
553D4—48no—
554D4—48no—
555D4—44no—
556D4—42no—
557D4—42no—
558D4—816no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—48no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—84no—
606C2×C4—816no—
607C2×C4—88no—
608C2×C4#53212yes—
609C2×C4#53212yes—
610C2×C4C2×AGL(1,ℤ/8)12yes—
611C2×C4C2×AGL(1,ℤ/8)12yes—
612C2×C4C2×AGL(1,ℤ/8)12yes—
613C2×C4C2×#13712yes—
614C2×C4C2×#13712yes—
615C2×C4C2×#13712yes—
616C2×C4C22×#4112yes—
617C2×C4C2×H(5,𝔽2)12yes—
618C2×C4C2×2−1+411yescharacteristic
619C2×C4#58511yescharacteristic
620C8—24no—
621C8—24no—
622C8—44no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—44no—
627C22—42no—
628C22—44no—
629C22—42no—
630C22—44no—
631C22—42no—
632C22—42no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22C2×#53211yescharacteristic
638C22#348411yescharacteristic
639C22#348911yescharacteristic
640C22C22×AGL(1,ℤ/8)11yescharacteristic
641C22C22×#13711yescharacteristic
642C22#358511yescharacteristic
643C22#358611yescharacteristic
644C4—22no—
645C4—22no—
646C4—22no—
647C4—22no—
648C4—22no—
649C4—42no—
650C4—42no—
651C4—42no—
652C4—42no—
653C4—82no—
654C4—81no—
655C4—81no—
656C4—81no—
657C4—82no—
658C4—84no—
659C4—82no—
660C4#318214yes—
661C4#349212yes—
662C4#349412yes—
663C2—82no—
664C2—84no—
665C2#3592311yescharacteristic
666C2C2×#318211yescharacteristic
667C2#6072111yescharacteristic
668C2#6158211yescharacteristic
669C2#6158611yescharacteristic
670C2#6211111yescharacteristic
671C2#6211411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36127C2
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. #36477C2semidirect
3. #30366C2semidirect
4. #30304C2semidirect
5. #29165C2semidirect
6. #29141C2semidirect
7. #29121C2semidirect
8. #28865C2semidirect
9. #28230C2semidirect
10. #28150C2semidirect
11. #26983C2semidirect
12. #25501C2semidirect
13. #23799C2semidirect
14. #23576C2semidirect
15. #22762C2semidirect
16. #22671C2semidirect
17. #20568C2semidirect
371. C4×C8#41semidirect

Maximal subgroups

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

subgroupindex
#364772
#361272
#303662
#303042
#291652
#291412
#291212
#288652
#282302
#281502
#269832
#255012
#237992
#235762
#227622
#226712
#205682