Finite Group Databasebeta

← #6519480

Subgroups — #6519480

Order 512.

2138 conjugacy classes of subgroups.

The subgroup lattice has 677 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35274C211yesmaximal, characteristic
2#34966C211yesmaximal, characteristic, complemented
3#29354C212yesmaximal, complemented
4#29348C212yesmaximal, complemented
5#28756C212yesmaximal, complemented
6#28749C212yesmaximal, complemented
7#27780C214yesmaximal, complemented
8#27760C214yesmaximal, complemented
9#27264C214yesmaximal, complemented
10#27254C214yesmaximal, complemented
11#21888C211yesmaximal, characteristic, complemented
12#21810C211yesmaximal, characteristic, complemented
13#21395C211yesmaximal, characteristic, complemented
14C2×#1982C211yesmaximal, characteristic, complemented
15#20546C211yesmaximal, characteristic, complemented
16C2×#535—22no—
17C2×#532C2211yescharacteristic
18C22×#127C2211yescharacteristic
19C2×#490C2214yescomplemented
20C2×#486C2214yescomplemented
21C2×#432C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C22×#102C2214yes—
24#2867C2212yes—
25#2858C2212yes—
26#2657—21no—
27#2657—21no—
28#2606—21no—
29#2605—21no—
30#2570C2212yescomplemented
31#2550C2212yescomplemented
32#2541—21no—
33#2541—21no—
34#2541C2214yes—
35#2442C2212yes—
36#2440C2212yes—
37#2438—22no—
38#2433—22no—
39#2433C2212yes—
40#2433C2212yes—
41#2402—21no—
42#2399—21no—
43C4×C4⋊Q8C2212yescomplemented
44C4×#125C2212yescomplemented
45C4×C22⋊Q8—22no—
46#2283C2211yescharacteristic
47C4×#117C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49#2073C2214yescomplemented
50#2071C2214yescomplemented
51#2041C2214yescomplemented
52#2041C2214yescomplemented
53#2041C2214yescomplemented
54#2041C2214yescomplemented
55#2025C2214yescomplemented
56#2023C2214yescomplemented
57#1987C2211yescharacteristic, complemented
58#1987C2211yescharacteristic, complemented
59#1987C2212yescomplemented
60#1982C2212yes—
61#1982C2212yes—
62#1982C2212yes—
63#1982C2212yes—
64#1982C2212yes—
65#1982C2212yes—
66#1982C2211yescharacteristic, complemented
67#1982C2212yescomplemented
68#1982C2211yescharacteristic, complemented
69#1961C2214yes—
70#1961C2214yescomplemented
71#1957C2214yescomplemented
72#1936C2214yescomplemented
73#1936C2214yescomplemented
74#1934C2214yescomplemented
75#1934C2214yescomplemented
76#1893C2214yes—
77#1874C2214yes—
78#1804C2214yescomplemented
79#1786C2214yes—
80#1771C2214yescomplemented
81#1767C2212yescomplemented
82C4×C4⋊C8C2212yescomplemented
83#1762C2212yescomplemented
84#1762C2212yescomplemented
85#1752C2212yes—
86C8×C22⋊C4C2212yes—
87C22×#41—21no—
88C23×D4—21no—
89C23×D4—21no—
90C23×D4C2312yes—
91C23×C8C2312yes—
92#535—416no—
93#532—24no—
94#532—24no—
95C2×C4⋊Q8C2311yescharacteristic
96C2×C4⋊Q8C2312yes—
97C2×#127—22no—
98C2×#127—22no—
99C2×#127—21no—
100C2×#127—21no—
101C2×#127—22no—
102C2×#127—22no—
103C2×#127—24no—
104C2×#127—24no—
105C2×#127C2312yes—
106C2×#127C2311yescharacteristic
107C2×#127C2312yes—
108C2×#127C2314yes—
109C2×#127C2314yes—
110C2×#125C2312yes—
111C2×#124—22no—
112C2×#124—22no—
113C2×#124—21no—
114C2×C22⋊Q8—22no—
115C2×#121—21no—
116C2×#121—24no—
117C2×#121—24no—
118C2×#121C2312yes—
119C2×#120—24no—
120C2×#120—24no—
121C2×C4×Q8—22no—
122C2×C4×D4—21no—
123C2×C4×D4C2312yes—
124C2×#117D412yes—
125C22×C4⋊C4C2312yes—
126C22×C42—21no—
127C22×C42C2311yescharacteristic
128#490—48no—
129#486—48no—
130#432—24no—
131C2×C4⋊C8—24no—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2312yes—
136C2×#102—48no—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×C22⋊C8C2312yes—
150C2×#97C2312yes—
151C2×#97C2314yes—
152C2×C4×C8C2314yes—
153#400—22no—
154#400—22no—
155#398—22no—
156#398—22no—
157#398—22no—
158#398—22no—
159#395—22no—
160#395—22no—
161#392—24no—
162#392—24no—
163#392—22no—
164#392—22no—
165#392—22no—
166#392—22no—
167#391—22no—
168#391—22no—
169#390—22no—
170#390—22no—
171#390C2314yes—
172#390C2314yes—
173#389—22no—
174#389—22no—
175#389C2312yes—
176#389C2312yes—
177#389C2312yes—
178#389C2312yes—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#385—22no—
184#385—22no—
185#384C2312yes—
186#383—21no—
187#382C2312yes—
188#382C2312yes—
189#381—21no—
190#380—22no—
191#380C2312yes—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4D411yescharacteristic
198C4×C4⋊C4D411yescharacteristic
199C4×C4⋊C4C2311yescharacteristic
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2311yescharacteristic
202C4×C4⋊C4C2311yescharacteristic
203C4×C4⋊C4C2311yescharacteristic
204C4×C4⋊C4C2312yes—
205C4×C22⋊C4—21no—
206C4×C22⋊C4D412yes—
207C4×C22⋊C4C2311yescharacteristic
208C4×C22⋊C4C2311yescharacteristic
209#376—21no—
210#376D411yescharacteristic
211#376D411yescharacteristic
212C2×#95—21no—
213C43C2312yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#336C2314yes—
219#336C2314yes—
220C25—22no—
221C2×#41—48no—
222C2×#41—44no—
223C22×Q8—21no—
224C22×Q8—22no—
225C22×D4—22no—
226C22×D4—21no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4—21no—
230C22×D4—22no—
231C22×D4—24no—
232C22×D4—24no—
233C22×D4—24no—
234C22×D4—24no—
235C22×D4—22no—
236C22×D4—22no—
237C22×D4—24no—
238C22×D4—24no—
239C22×D4—44no—
240C22×D4—44no—
241C22×D4—48no—
242C22×D4—44no—
243C22×D4—44no—
244C22×D4—82no—
245C22×D4—82no—
246C22×D4C2412yes—
247C22×D4C2414yes—
248C23×C4—22no—
249C23×C4—22no—
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2411yescharacteristic
253C22×C8—24no—
254C22×C8—44no—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C22×C8C2414yes—
258C4⋊Q8—24no—
259C4⋊Q8—22no—
260C4⋊Q8—22no—
261C4⋊Q8—42no—
262#127—24no—
263#127—24no—
264#127—24no—
265#127—24no—
266#127—22no—
267#127—22no—
268#127—44no—
269#127—44no—
270#127—44no—
271#127—44no—
272#127—48no—
273#127—48no—
274#127—44no—
275#127—44no—
276#127—84no—
277#127—84no—
278#125—24no—
279#125—42no—
280#124—48no—
281#124—48no—
282#124—44no—
283C22⋊Q8—48no—
284#121—24no—
285#121—24no—
286#121—416no—
287#121—416no—
288#121—44no—
289#120—416no—
290#120—416no—
291C4×Q8—84no—
292C4×D4—24no—
293C4×D4—24no—
294C4×D4—44no—
295#117—28no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4C2×D412yes—
324C2×C22⋊C4C2×D412yes—
325C2×C42—21no—
326C2×C42—22no—
327C2×C42—21no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—82no—
331C2×C42C2×D412yes—
332C2×C42C2×D412yes—
333C2×C42C2×D412yes—
334C2×C42C2×D412yes—
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42#4114yes—
338C2×C42#4114yes—
339C2×C42C2412yes—
340C2×C42C2412yes—
341C2×C42C2411yescharacteristic
342C2×C42C2411yescharacteristic
343C4⋊C8—24no—
344C4⋊C8—22no—
345C4⋊C8—22no—
346C4⋊C8—24no—
347#102—28no—
348#102—28no—
349#102—28no—
350#102—44no—
351#102—44no—
352#102—44no—
353#102—44no—
354#102—44no—
355#102—44no—
356#102—44no—
357#102—44no—
358#102—44no—
359C22⋊C8—24no—
360#97—24no—
361#97—44no—
362C4×C8—44no—
363#95—22no—
364#95—21no—
365#95—21no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—21no—
376#95—21no—
377#95C2×D412yes—
378#95C2×D412yes—
379#95C2×D411yescharacteristic
380#95C2×D411yescharacteristic
381#95C2×D411yescharacteristic
382#95C2×D411yescharacteristic
383C24—24no—
384C24—22no—
385C24—44no—
386C24—48no—
387C24—84no—
388C24—82no—
389C24—82no—
390C24C22×D411yescharacteristic
391#41—816no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—42no—
395C2×Q8—44no—
396C2×Q8—82no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—24no—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—24no—
404C2×D4—24no—
405C2×D4—24no—
406C2×D4—24no—
407C2×D4—416no—
408C2×D4—48no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—48no—
413C2×D4—48no—
414C2×D4—44no—
415C2×D4—48no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—48no—
424C2×D4—48no—
425C2×D4—42no—
426C2×D4—44no—
427C2×D4—42no—
428C2×D4—44no—
429C2×D4—84no—
430C2×D4—84no—
431C2×D4—84no—
432C2×D4—82no—
433C2×D4—82no—
434C2×D4—816no—
435C2×D4—816no—
436C2×D4—84no—
437C2×D4—84no—
438C2×D4—84no—
439C2×D4—84no—
440C2×D4—84no—
441C2×D4—84no—
442C2×D4—84no—
443C2×D4—84no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—44no—
455C22×C4—48no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—82no—
459C22×C4#12414yes—
460C22×C4#12414yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C2×#4112yes—
471C22×C4C2×#4112yes—
472C22×C4C2511yescharacteristic
473C2×C8—24no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—24no—
477C2×C8—24no—
478C2×C8—24no—
479C2×C8—24no—
480C2×C8—84no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C4⋊C4—82no—
503C4⋊C4H(5,𝔽2)14yes—
504C22⋊C4—24no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4AGL(1,ℤ/8)18yes—
513C42—24no—
514C42—24no—
515C42—24no—
516C42—24no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—44no—
522C42—42no—
523C42—42no—
524C42—81no—
525C42—81no—
526C42#12414yes—
527C42#12414yes—
528C42C2×#4112yes—
529C42C2×#4112yes—
530C42C2×#4112yes—
531C42C2×#4112yes—
532C23—22no—
533C23—21no—
534C23—21no—
535C23—42no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—42no—
544C23—44no—
545C23—42no—
546C23—44no—
547C23—88no—
548C23—88no—
549C23—88no—
550C23—168no—
551C23—168no—
552C23C2×#12412yes—
553C23C2×AGL(1,ℤ/8)14yes—
554C23C23×D411yescharacteristic
555Q8—88no—
556Q8—84no—
557D4—44no—
558D4—44no—
559D4—44no—
560D4—42no—
561D4—44no—
562D4—42no—
563D4—44no—
564D4—88no—
565D4—88no—
566D4—88no—
567D4—84no—
568D4—88no—
569D4—88no—
570D4—84no—
571D4—88no—
572D4—168no—
573D4—168no—
574D4—168no—
575D4—168no—
576C2×C4—28no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—28no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—88no—
611C2×C4—88no—
612C2×C4C2×#12412yes—
613C2×C4C2×#12412yes—
614C2×C4C2×#12412yes—
615C2×C4C2×#12412yes—
616C2×C4C2×#12412yes—
617C2×C4C2×AGL(1,ℤ/8)14yes—
618C2×C4C2×AGL(1,ℤ/8)14yes—
619C2×C4#57612yes—
620C2×C4#57712yes—
621C2×C4C22×#4111yescharacteristic
622C2×C4C22×#4111yescharacteristic
623C2×C4C2×H(5,𝔽2)12yes—
624C2×C4#58512yes—
625C8—44no—
626C8—42no—
627C8—42no—
628C22—24no—
629C22—24no—
630C22—22no—
631C22—22no—
632C22—84no—
633C22—84no—
634C22—84no—
635C22—84no—
636C22—84no—
637C22—84no—
638C22—84no—
639C22—82no—
640C22—82no—
641C22—84no—
642C22—84no—
643C22—82no—
644C22—1616no—
645C22#313814yes—
646C22C22×#12411yescharacteristic
647C22#348812yes—
648C22C22×AGL(1,ℤ/8)12yes—
649C22C2×#57611yescharacteristic
650C22C2×#57711yescharacteristic
651C4—24no—
652C4—24no—
653C4—22no—
654C4—22no—
655C4—22no—
656C4—22no—
657C4—42no—
658C4—44no—
659C4—42no—
660C4—82no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4#313614yes—
665C4#350612yes—
666C4#350811yescharacteristic
667C4#351011yescharacteristic
668C2—24no—
669C2—162no—
670C2—164no—
671C2#3555611yescharacteristic
672C2C2×#313611yescharacteristic
673C2C2×#313811yescharacteristic
674C2#6197912yes—
675C2#6198312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35274C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34966C2semidirect
3. #29354C2semidirect
4. #29348C2semidirect
5. #28756C2semidirect
6. #28749C2semidirect
7. #27780C2semidirect
8. #27760C2semidirect
9. #27264C2semidirect
10. #27254C2semidirect
11. #21888C2semidirect
12. #21810C2semidirect
13. #21395C2semidirect
14. C2×#1982C2semidirect
15. #20546C2semidirect
19. C2×#490C22semidirect
20. C2×#486C22semidirect
30. #2570C22semidirect
31. #2550C22semidirect
43. C4×C4⋊Q8C22semidirect
44. C4×#125C22semidirect
49. #2073C22semidirect
50. #2071C22semidirect
51. #2041C22semidirect
52. #2041C22semidirect
53. #2041C22semidirect
54. #2041C22semidirect
55. #2025C22semidirect
56. #2023C22semidirect
57. #1987C22semidirect
58. #1987C22semidirect
59. #1987C22semidirect
66. #1982C22semidirect
67. #1982C22semidirect
68. #1982C22semidirect
70. #1961C22semidirect
71. #1957C22semidirect
72. #1936C22semidirect
73. #1936C22semidirect
74. #1934C22semidirect
75. #1934C22semidirect
78. #1804C22semidirect
80. #1771C22semidirect
81. #1767C22semidirect
82. C4×C4⋊C8C22semidirect
83. #1762C22semidirect
84. #1762C22semidirect

Maximal subgroups

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

subgroupindex
#352742
#349662
#293542
#293482
#287562
#287492
#277802
#277602
#272642
#272542
#218882
#218102
#213952
C2×#19822
#205462