Finite Group Databasebeta

← #6396452

Subgroups — #6396452

Order 512.

2606 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3205C211yesmaximal, characteristic
2#34210C211yesmaximal, characteristic, complemented
3#34207C211yesmaximal, characteristic, complemented
4#23099C214yesmaximal, complemented
5#23001C214yesmaximal, complemented
6#22858C218yesmaximal, complemented
7#22269C212yesmaximal, complemented
8#22219C212yesmaximal, complemented
9#21981C211yesmaximal, characteristic, complemented
10#21690C212yesmaximal, complemented
11#21646C212yesmaximal, complemented
12#21401C211yesmaximal, characteristic, complemented
13C2×#1957C211yesmaximal, characteristic, complemented
14#20473C211yesmaximal, characteristic, complemented
15C2×#532C2211yescharacteristic
16C22×#127C2211yescharacteristic
17C2×#520—21no—
18C2×#518—21no—
19#3205C4116yescomplemented
20C2×#493C2212yes—
21C2×#486C2212yes—
22C2×#459C2214yes—
23C2×#447C2214yes—
24C2×#442C2218yescomplemented
25C2×#431C2211yescharacteristic
26C22×#102C2214yes—
27C22×C4×C8C2211yescharacteristic
28#2366—21no—
29#2366—21no—
30#2366C2212yes—
31#2366C2212yes—
32#2361—22no—
33#2361C2214yes—
34#2354—21no—
35#2354C2212yescomplemented
36D4×C4⋊C4—21no—
37D4×C4⋊C4C2212yescomplemented
38D4×C22⋊C4—22no—
39D4×C22⋊C4C2214yes—
40#2328—22no—
41#2326—22no—
42#2311—21no—
43#2309—21no—
44C4×C4⋊Q8C2212yescomplemented
45C4×#127C2212yes—
46C4×#127C2212yes—
47C4×#127C2212yescomplemented
48#2285C2211yescharacteristic
49C2×#384C2211yescharacteristic
50C4×#117C2211yescharacteristic
51#1990C2211yescharacteristic, complemented
52#1990C2211yescharacteristic, complemented
53#1972C2218yescomplemented
54#1957C2214yes—
55#1957C2212yes—
56#1957C2212yes—
57#1957C2211yescharacteristic, complemented
58#1957C2211yescharacteristic, complemented
59#1948C2214yescomplemented
60#1936C2218yescomplemented
61#1926C2218yescomplemented
62#1893C2214yes—
63#1879C2218yescomplemented
64#1879C2218yescomplemented
65#1864C2218yescomplemented
66#1853C2214yes—
67#1844C2212yescomplemented
68#1834C2212yescomplemented
69#1770C2212yescomplemented
70#1770C2212yescomplemented
71#1762C2218yescomplemented
72C4×#102C2214yescomplemented
73#1752C2214yes—
74#1744C2212yescomplemented
75C4×#97C2212yescomplemented
76C22×#41—21no—
77C23×D4—21no—
78C23×D4—21no—
79C23×D4C2312yes—
80C22×D8C2318yes—
81C23×C8C2312yes—
82#532—24no—
83#532C2×C418yescomplemented
84C2×C4⋊Q8C2311yescharacteristic
85C2×#127—21no—
86C2×#127—21no—
87C2×#127—24no—
88C2×#127C2×C418yescomplemented
89C2×#127C2312yes—
90C2×#127C2311yescharacteristic
91C2×#127C2312yes—
92C2×#124—21no—
93C2×#121—21no—
94C2×#121C2312yes—
95#520—48no—
96#518—48no—
97C2×C4×Q8—22no—
98C2×C4×D4—22no—
99C2×C4×D4—22no—
100C2×C4×D4—24no—
101C2×C4×D4—21no—
102C2×C4×D4—22no—
103C2×C4×D4—22no—
104C2×C4×D4—22no—
105C2×C4×D4—22no—
106C2×C4×D4—22no—
107C2×C4×D4—22no—
108C2×C4×D4C2312yes—
109C2×C4×D4C2314yes—
110C2×C4×D4C2314yes—
111C2×C4×D4C2314yes—
112C2×#117—21no—
113C2×#117—22no—
114C2×#117D411yescharacteristic
115C2×#117D411yescharacteristic
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—22no—
119C22×C4⋊C4C2312yes—
120C22×C22⋊C4—24no—
121C22×C42C2311yescharacteristic
122#493C2×C4116yescomplemented
123#486C2×C4116yescomplemented
124#459C2×C4132yescomplemented
125#447C2×C4132yescomplemented
126#442—416no—
127#431C2×C418yescomplemented
128C2×#106C2318yes—
129C2×C4⋊C8C2312yes—
130C2×#102—48no—
131C2×#102C2318yes—
132C2×#102C2314yes—
133C2×#102C2318yes—
134C2×#102C2314yes—
135C2×#102C2318yes—
136C2×C22⋊C8C2312yes—
137C2×#97C2314yes—
138C2×#97C2314yes—
139C2×C4×C8—24no—
140C2×C4×C8C2311yescharacteristic
141C2×C4×C8C2311yescharacteristic
142C2×C4×C8C2312yes—
143C2×C4×C8C2312yes—
144#391—22no—
145#390—22no—
146#390—22no—
147#390—22no—
148#390—22no—
149#390—22no—
150#390—22no—
151#390—22no—
152#390—22no—
153#390C2314yes—
154#390C2314yes—
155#390C2314yes—
156#389—22no—
157#389C2314yes—
158#388—24no—
159#388—24no—
160#386—24no—
161#386—24no—
162#386—24no—
163#386—24no—
164#386—24no—
165#385—24no—
166#385—24no—
167#384—21no—
168#384—21no—
169#384C2311yescharacteristic
170#384C2311yescharacteristic
171#384C2312yes—
172#384C2312yes—
173#382—21no—
174#380—21no—
175#380—22no—
176#380—22no—
177#380C2312yes—
178#379—24no—
179C4×C4⋊C4—22no—
180C4×C4⋊C4—21no—
181C4×C4⋊C4D411yescharacteristic
182C4×C4⋊C4D411yescharacteristic
183C4×C4⋊C4C2314yes—
184C4×C4⋊C4C2312yes—
185C4×C4⋊C4C2311yescharacteristic
186C4×C4⋊C4C2311yescharacteristic
187C4×C22⋊C4D412yes—
188C4×C22⋊C4C2312yes—
189#376D412yes—
190C43C2312yes—
191#340C2318yes—
192#340C2314yes—
193#340C2314yes—
194#336C2314yes—
195#336C2314yes—
196C25—22no—
197C2×#41—28no—
198C2×#41—44no—
199C2×#41—44no—
200C22×Q8—21no—
201C22×D4—22no—
202C22×D4—28no—
203C22×D4—28no—
204C22×D4—21no—
205C22×D4—21no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—24no—
212C22×D4—44no—
213C22×D4—44no—
214C22×D4—44no—
215C22×D4—44no—
216C22×D4—82no—
217C22×D4—82no—
218C22×D4C22×C418yes—
219C22×D4C2412yes—
220C22×D4C2414yes—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2411yescharacteristic
229C2×D8—48no—
230C2×D8—48no—
231C2×D8C22×C4132yes—
232C22×C8—24no—
233C22×C8—24no—
234C22×C8—42no—
235C22×C8C2412yes—
236C22×C8C2414yes—
237C22×C8C2412yes—
238C4⋊Q8—22no—
239C4⋊Q8C22×C414yes—
240#127—28no—
241#127—28no—
242#127—24no—
243#127—24no—
244#127—22no—
245#127C22×C418yes—
246#127C22×C418yes—
247#127C22×C414yes—
248#124—28no—
249#121—28no—
250#121—24no—
251#121C22×C418yes—
252C4×Q8—48no—
253C4×D4—28no—
254C4×D4—24no—
255C4×D4—48no—
256C4×D4—48no—
257C4×D4—48no—
258C4×D4—48no—
259C4×D4—48no—
260C4×D4—48no—
261C4×D4—416no—
262C4×D4—48no—
263C4×D4—48no—
264C4×D4—84no—
265C4×D4—84no—
266C4×D4—84no—
267C4×D4C22×C418yes—
268#117—24no—
269#117—24no—
270#117—48no—
271#117—44no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—48no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—82no—
287C2×C4⋊C4C2×D411yescharacteristic
288C2×C4⋊C4C2×D411yescharacteristic
289C2×C4⋊C4C2×D411yescharacteristic
290C2×C4⋊C4C2×D411yescharacteristic
291C2×C4⋊C4C2414yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—416no—
308C2×C22⋊C4C2×D412yes—
309C2×C22⋊C4C2×D412yes—
310C2×C42—22no—
311C2×C42—24no—
312C2×C42C22×C414yes—
313C2×C42C2×D412yes—
314C2×C42C2×D412yes—
315C2×C42C2×D412yes—
316C2×C42C2×D412yes—
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42#4114yes—
320C2×C42C2412yes—
321C2×C42C2412yes—
322C2×C42C2411yescharacteristic
323C2×C42C2411yescharacteristic
324#106—48no—
325C4⋊C8C22×C418yes—
326#102—216no—
327#102—28no—
328#102—48no—
329#102—44no—
330#102C22×C4132yes—
331C22⋊C8C22×C418yes—
332#97—28no—
333#97—44no—
334C4×C8—24no—
335C4×C8—22no—
336C4×C8—22no—
337C4×C8C22×C418yes—
338#95—24no—
339#95—22no—
340#95—22no—
341#95—24no—
342#95C2×D412yes—
343#95C2×D412yes—
344#95C2×D412yes—
345#95C2×D412yes—
346C24—22no—
347C24—24no—
348C24—28no—
349C24—44no—
350C24—44no—
351C24—82no—
352C24—82no—
353C24—82no—
354C24—82no—
355C24C22×D411yescharacteristic
356#41—432no—
357C2×Q8—24no—
358C2×Q8—42no—
359C2×Q8—42no—
360C2×D4—28no—
361C2×D4—24no—
362C2×D4—24no—
363C2×D4—28no—
364C2×D4—28no—
365C2×D4—28no—
366C2×D4—28no—
367C2×D4—28no—
368C2×D4—22no—
369C2×D4—24no—
370C2×D4—28no—
371C2×D4—24no—
372C2×D4—24no—
373C2×D4—28no—
374C2×D4—416no—
375C2×D4—44no—
376C2×D4—44no—
377C2×D4—44no—
378C2×D4—44no—
379C2×D4—42no—
380C2×D4—44no—
381C2×D4—44no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—44no—
385C2×D4—44no—
386C2×D4—42no—
387C2×D4—432no—
388C2×D4—88no—
389C2×D4—816no—
390C2×D4—82no—
391C2×D4—82no—
392C2×D4—82no—
393C2×D4—82no—
394C2×D4C23×C418yes—
395C2×D4C23×C414yes—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—28no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—48no—
406C22×C4—48no—
407C22×C4—48no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—84no—
412C22×C4—84no—
413C22×C4—84no—
414C22×C4—84no—
415C22×C4—84no—
416C22×C4—84no—
417C22×C4—82no—
418C22×C4—82no—
419C22×C4C4×D414yes—
420C22×C4C4×D414yes—
421C22×C4AGL(1,ℤ/8)14yes—
422C22×C4AGL(1,ℤ/8)14yes—
423C22×C4C23×C412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C2×#4112yes—
434C22×C4C2511yescharacteristic
435D8—432no—
436C2×C8—22no—
437C2×C8—24no—
438C2×C8—22no—
439C2×C8—24no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—48no—
444C2×C8C23×C418yes—
445C2×C8H(5,𝔽2)14yes—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—28no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—48no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—44no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—82no—
464C4⋊C4—82no—
465C4⋊C4C23×C414yes—
466C22⋊C4—24no—
467C22⋊C4—24no—
468C22⋊C4—28no—
469C22⋊C4—28no—
470C22⋊C4—48no—
471C22⋊C4—48no—
472C22⋊C4—48no—
473C22⋊C4—44no—
474C22⋊C4—84no—
475C22⋊C4—84no—
476C22⋊C4—84no—
477C22⋊C4—84no—
478C22⋊C4—84no—
479C22⋊C4—84no—
480C42—24no—
481C42—24no—
482C42—24no—
483C42—24no—
484C42—24no—
485C42—24no—
486C42—48no—
487C42—44no—
488C42C4×D414yes—
489C42C4×D414yes—
490C42C23×C412yes—
491C42C23×C412yes—
492C42C2×#4112yes—
493C42C2×#4112yes—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—24no—
499C23—28no—
500C23—42no—
501C23—44no—
502C23—44no—
503C23—42no—
504C23—42no—
505C23—44no—
506C23—42no—
507C23—44no—
508C23—42no—
509C23—44no—
510C23—44no—
511C23—42no—
512C23—416no—
513C23—88no—
514C23—88no—
515C23—164no—
516C23—164no—
517C23—164no—
518C23—164no—
519C23C2×C4×D412yes—
520C23C2×AGL(1,ℤ/8)12yes—
521C23C2×AGL(1,ℤ/8)12yes—
522C23C23×D411yescharacteristic
523Q8—48no—
524D4—28no—
525D4—216no—
526D4—416no—
527D4—416no—
528D4—416no—
529D4—48no—
530D4—48no—
531D4—416no—
532D4—44no—
533D4—48no—
534D4—88no—
535D4—88no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—416no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—168no—
569C2×C4—168no—
570C2×C4—168no—
571C2×C4—164no—
572C2×C4#442116yescomplemented
573C2×C4C2×C4×D412yes—
574C2×C4C2×C4×D412yes—
575C2×C4C2×C4×D412yes—
576C2×C4C2×C4×D412yes—
577C2×C4C2×C4×D412yes—
578C2×C4C2×AGL(1,ℤ/8)12yes—
579C2×C4C2×AGL(1,ℤ/8)12yes—
580C2×C4C2×AGL(1,ℤ/8)12yes—
581C2×C4C2×AGL(1,ℤ/8)12yes—
582C2×C4#57612yes—
583C2×C4#57712yes—
584C2×C4C24×C411yescharacteristic
585C2×C4C22×#4111yescharacteristic
586C2×C4C2×H(5,𝔽2)12yes—
587C2×C4#58512yes—
588C8—28no—
589C8—24no—
590C8—42no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—48no—
598C22—44no—
599C22—44no—
600C22—48no—
601C22—48no—
602C22—44no—
603C22—84no—
604C22—82no—
605C22—82no—
606C22—84no—
607C22—84no—
608C22—82no—
609C22—82no—
610C22—84no—
611C22—1616no—
612C22C2×#44214yes—
613C22C22×C4×D411yescharacteristic
614C22#349811yescharacteristic
615C22#350011yescharacteristic
616C22C22×AGL(1,ℤ/8)11yescharacteristic
617C22C22×AGL(1,ℤ/8)11yescharacteristic
618C22C2×#57611yescharacteristic
619C22C2×#57711yescharacteristic
620C4—22no—
621C4—22no—
622C4—42no—
623C4—42no—
624C4—42no—
625C4—42no—
626C4—44no—
627C4—82no—
628C4—82no—
629C4—84no—
630C4C2×#44214yes—
631C4C2×#44214yes—
632C4#294614yes—
633C4C4×H(5,𝔽2)12yes—
634C4#349412yes—
635C2—24no—
636C2—84no—
637C2—88no—
638C2#3427511yescharacteristic
639C2C22×#44211yescharacteristic
640C2C2×#294611yescharacteristic
641C2#6242111yescharacteristic
642C2#6242311yescharacteristic
643C2#6253511yescharacteristic
644C2#6254111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC2×#3205C2
LCSG ▷ C2×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. #34210C2semidirect
3. #34207C2semidirect
4. #23099C2semidirect
5. #23001C2semidirect
6. #22858C2semidirect
7. #22269C2semidirect
8. #22219C2semidirect
9. #21981C2semidirect
10. #21690C2semidirect
11. #21646C2semidirect
12. #21401C2semidirect
13. C2×#1957C2semidirect
14. #20473C2semidirect
19. #3205C4semidirect
24. C2×#442C22semidirect
35. #2354C22semidirect
37. D4×C4⋊C4C22semidirect
44. C4×C4⋊Q8C22semidirect
47. C4×#127C22semidirect
51. #1990C22semidirect
52. #1990C22semidirect
53. #1972C22semidirect
57. #1957C22semidirect
58. #1957C22semidirect
59. #1948C22semidirect
60. #1936C22semidirect
61. #1926C22semidirect
63. #1879C22semidirect
64. #1879C22semidirect
65. #1864C22semidirect
67. #1844C22semidirect
68. #1834C22semidirect
69. #1770C22semidirect
70. #1770C22semidirect
71. #1762C22semidirect
72. C4×#102C22semidirect
74. #1744C22semidirect
75. C4×#97C22semidirect
83. #532C2×C4semidirect
88. C2×#127C2×C4semidirect
122. #493C2×C4semidirect
123. #486C2×C4semidirect
124. #459C2×C4semidirect
125. #447C2×C4semidirect
127. #431C2×C4semidirect
572. C2×C4#442semidirect

Maximal subgroups

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

subgroupindex
C2×#32052
#342102
#342072
#230992
#230012
#228582
#222692
#222192
#219812
#216902
#216462
#214012
C2×#19572
#204732