Finite Group Databasebeta

← #6801064

Subgroups — #6801064

Order 512.

1567 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36559C211yesmaximal, characteristic
2#34951C211yesmaximal, characteristic, complemented
3#33198C211yesmaximal, characteristic, complemented
4#31822C214yesmaximal, complemented
5#31165C214yesmaximal, complemented
6#29997C212yesmaximal, complemented
7#29141C211yesmaximal, characteristic, complemented
8#29132C212yesmaximal, complemented
9#28252C212yesmaximal
10#28252C211yesmaximal, characteristic
11#28195C212yesmaximal, complemented
12#26106C212yesmaximal, complemented
13#25301C212yesmaximal, complemented
14#24730C212yesmaximal, complemented
15#22722C211yesmaximal, characteristic, complemented
16#22153C212yesmaximal
17C4×#460C211yesmaximal, characteristic
18C2×#554—21no—
19C2×#477C2212yes—
20C2×#460C2212yes—
21C2×#460C2211yescharacteristic
22C2×#461C2214yes—
23C2×#461C2212yes—
24C2×#446C2212yes—
25C2×C4×SD16C2212yes—
26#2868C2211yescharacteristic
27#2859C2212yes—
28#2585—22no—
29#2582C2211yescharacteristic
30#2568C2211yescharacteristic
31#2566—22no—
32#2563—21no—
33#2552C2211yescharacteristic
34#2546C2211yescharacteristic
35#2538—21no—
36#2538C2212yes—
37#2511C2212yes—
38#2509—21no—
39#2453—21no—
40#2452—22no—
41#2452C2214yes—
42#2446C2212yes—
43#2402—22no—
44D4×C4⋊C4—21no—
45D4×C4⋊C4C2212yes—
46C4×C4⋊Q8C2212yes—
47C4×C4⋊Q8C2211yescharacteristic
48C4×#127C2212yes—
49C4×#127C2211yescharacteristic
50C4×#125—21no—
51#2295—21no—
52#2295C2212yes—
53C42×Q8C2211yescharacteristic
54C42×Q8C2211yescharacteristic
55#2099C2214yes—
56#2088C2214yes—
57#2086C2214yes—
58#2068C2214yes—
59#2066C2214yes—
60#2055C2214yes—
61#2055C2214yes—
62#2055C2214yes—
63#2043C2214yes—
64#2024C2214yes—
65#1987C2212yes—
66#1982C2211yescharacteristic
67#1973C2214yes—
68#1956C2212yes—
69#1948C2211yescharacteristic
70#1947C2212yes—
71#1947C2212yes—
72#1940C2214yes—
73#1939C2212yes—
74#1936C2214yes—
75#1935C2214yes—
76#1935C2214yes—
77#1926C2212yes—
78#1880C2212yes—
79#1877C2214yes—
80#1862C2214yes—
81#1842C2212yes—
82#1842C2211yescharacteristic
83#1834C2212yes—
84#1803C2212yes—
85#1803C2212yes—
86#1803C2212yes—
87#1803C2212yes—
88#1803C2212yes—
89#1803C2211yescharacteristic
90#1802C2212yes—
91#1802C2214yes—
92#1767C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95C22×SD16C2312yes—
96C22×SD16C2314yes—
97#554—48no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2312yes—
103C2×#127C2311yescharacteristic
104C2×#125—21no—
105C2×#125—22no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8—24no—
108C2×C4×Q8D411yescharacteristic
109C2×C4×Q8D411yescharacteristic
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2312yes—
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2311yescharacteristic
115C2×C4×D4—21no—
116C2×C4×D4—22no—
117C2×C4×D4—21no—
118C2×C4×D4—22no—
119C2×C4×D4C2312yes—
120C2×C4×D4C2314yes—
121C22×C4⋊C4—22no—
122#477—44no—
123#461—48no—
124#461—44no—
125#460—24no—
126#460—28no—
127#446—44no—
128C4×SD16—44no—
129C2×#107C2314yes—
130C2×#106C2312yes—
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2311yescharacteristic
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2314yes—
137C2×C4⋊C8C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2312yes—
143C2×#102C2314yes—
144C2×#102C2312yes—
145C2×#102C2312yes—
146C2×#102C2312yes—
147C2×#97C2314yes—
148C2×C4×C8C2312yes—
149#400—22no—
150#400—22no—
151#400—22no—
152#400—22no—
153#398—22no—
154#398—22no—
155#398—22no—
156#398—22no—
157#396—24no—
158#396—24no—
159#393—24no—
160#391—21no—
161#391—21no—
162#391C2312yes—
163#391C2312yes—
164#391C2312yes—
165#391C2312yes—
166#391C2312yes—
167#390—22no—
168#390—21no—
169#390—21no—
170#390—22no—
171#390C2312yes—
172#390C2314yes—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—21no—
178#389—22no—
179#389—22no—
180#389—21no—
181#389—22no—
182#389—22no—
183#389C2314yes—
184#389C2314yes—
185#389C2312yes—
186#389C2312yes—
187#387—22no—
188#387—22no—
189#387—22no—
190#387—22no—
191#387—22no—
192#387—22no—
193#385—24no—
194#385—24no—
195#384—21no—
196#384—21no—
197#384C2311yescharacteristic
198#384C2312yes—
199#384C2311yescharacteristic
200#384C2312yes—
201#384C2312yes—
202#382—21no—
203#382—22no—
204#380—22no—
205#380—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4D412yes—
217C4×C4⋊C4D412yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2312yes—
223#376—22no—
224C43—21no—
225C43C2312yes—
226C43C2311yescharacteristic
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2312yes—
233#340C2312yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—22no—
237C22×Q8C2×D411yescharacteristic
238C22×Q8C2412yes—
239C22×Q8C2411yescharacteristic
240C22×D4—21no—
241C22×D4—21no—
242C22×D4C2412yes—
243C23×C4—22no—
244C23×C4—24no—
245C2×SD16—24no—
246C2×SD16—24no—
247C2×SD16—28no—
248C2×SD16—42no—
249C2×SD16—44no—
250C2×SD16—44no—
251C22×C8C2412yes—
252C22×C8C2414yes—
253C22×C8C2412yes—
254C4⋊Q8—24no—
255C4⋊Q8—22no—
256C4⋊Q8—22no—
257C4⋊Q8—42no—
258C4⋊Q8—42no—
259C4⋊Q8—42no—
260C4⋊Q8—44no—
261C4⋊Q8—82no—
262#127—22no—
263#127#4114yes—
264#125—48no—
265#125—44no—
266C22⋊Q8—416no—
267C22⋊Q8—48no—
268C4×Q8—24no—
269C4×Q8—24no—
270C4×Q8—22no—
271C4×Q8—22no—
272C4×Q8—22no—
273C4×Q8—22no—
274C4×Q8—24no—
275C4×Q8—22no—
276C4×Q8—22no—
277C4×Q8—22no—
278C4×Q8—24no—
279C4×Q8—41no—
280C4×D4—24no—
281C4×D4—24no—
282C4×D4—48no—
283C4×D4—44no—
284C4×D4—44no—
285C4×D4—44no—
286C4×D4—44no—
287C4×D4—84no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—48no—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2414yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C22⋊C4—22no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C42—21no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—22no—
334C2×C42—21no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42C2×D412yes—
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D412yes—
344C2×C42C2×D411yescharacteristic
345C2×C42C2412yes—
346C2×C42C2411yescharacteristic
347C2×C42C2412yes—
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350#107—44no—
351#106—42no—
352C4⋊C8—24no—
353C4⋊C8—22no—
354C4⋊C8—24no—
355C4⋊C8—22no—
356C4⋊C8—44no—
357C4⋊C8—44no—
358C4⋊C8—42no—
359#103—28no—
360#103—44no—
361#103—42no—
362#103—44no—
363#103—44no—
364#102—28no—
365#102—24no—
366#102—42no—
367#102—42no—
368#97—44no—
369C4×C8—24no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—21no—
375#95—21no—
376#95—21no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—24no—
386#95—24no—
387#95C2×D412yes—
388#95C2×D412yes—
389#95C2×D412yes—
390C24—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—22no—
397C2×Q8—21no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—44no—
406C2×Q8—82no—
407C2×Q8—82no—
408C2×Q8H(5,𝔽2)12yes—
409C2×D4—24no—
410C2×D4—24no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4C2×#4114yes—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—48no—
433C22×C4—84no—
434C22×C4—84no—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D412yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C2511yescharacteristic
446SD16—48no—
447SD16—88no—
448C2×C8—22no—
449C2×C8—24no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—24no—
453C2×C8—24no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—22no—
472C4⋊C4—21no—
473C4⋊C4—24no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—48no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—48no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—81no—
493C4⋊C4—82no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4AGL(1,ℤ/8)14yes—
498C22⋊C4—44no—
499C22⋊C4—48no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—84no—
503C22⋊C4—84no—
504C42—24no—
505C42—22no—
506C42—24no—
507C42—24no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—21no—
512C42—22no—
513C42—22no—
514C42—21no—
515C42—41no—
516C42—42no—
517C42—42no—
518C42—42no—
519C42—42no—
520C42—44no—
521C42—42no—
522C42—42no—
523C42—81no—
524C42#13714yes—
525C42C2×#4112yes—
526C42H(5,𝔽2)12yes—
527C422−1+412yes—
528C422−1+412yes—
529C23—24no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23C23×D411yescharacteristic
537Q8—22no—
538Q8—22no—
539Q8—44no—
540Q8—41no—
541Q8—41no—
542Q8—42no—
543Q8—42no—
544Q8—42no—
545Q8—42no—
546Q8—164no—
547D4—24no—
548D4—24no—
549D4—48no—
550D4—48no—
551D4—42no—
552D4—42no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—44no—
597C2×C4—88no—
598C2×C4—168no—
599C2×C4#53712yes—
600C2×C4#53712yes—
601C2×C4#54812yes—
602C2×C4#54812yes—
603C2×C4C2×AGL(1,ℤ/8)14yes—
604C2×C4C2×AGL(1,ℤ/8)12yes—
605C2×C4C2×#13712yes—
606C2×C4C2×#13712yes—
607C2×C4C2×#13712yes—
608C2×C4#57712yes—
609C2×C4C22×#4111yescharacteristic
610C2×C4C2×H(5,𝔽2)11yescharacteristic
611C2×C4C2×H(5,𝔽2)11yescharacteristic
612C2×C4C2×2−1+411yescharacteristic
613C2×C4C2×2−1+411yescharacteristic
614C2×C4#58511yescharacteristic
615C8—42no—
616C8—44no—
617C8—42no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22C2×#53711yescharacteristic
626C22C2×#54811yescharacteristic
627C22#348811yescharacteristic
628C22C22×AGL(1,ℤ/8)11yescharacteristic
629C22C22×#13711yescharacteristic
630C22C2×#57711yescharacteristic
631C22#358611yescharacteristic
632C4—22no—
633C4—22no—
634C4—24no—
635C4—22no—
636C4—42no—
637C4—42no—
638C4—41no—
639C4—42no—
640C4—42no—
641C4—81no—
642C4—81no—
643C4—84no—
644C4—81no—
645C4—82no—
646C4—82no—
647C4—82no—
648C4#325414yes—
649C4#337214yes—
650C4#351511yescharacteristic
651C4#351911yescharacteristic
652C4#354411yescharacteristic
653C4#354711yescharacteristic
654C2—84no—
655C2#3912611yescharacteristic
656C2C2×#325411yescharacteristic
657C2#6093111yescharacteristic
658C2C2×#337211yescharacteristic
659C2#6131711yescharacteristic
660C2#6196611yescharacteristic
661C2#6198311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34951C2semidirect
3. #33198C2semidirect
4. #31822C2semidirect
5. #31165C2semidirect
6. #29997C2semidirect
7. #29141C2semidirect
8. #29132C2semidirect
11. #28195C2semidirect
12. #26106C2semidirect
13. #25301C2semidirect
14. #24730C2semidirect
15. #22722C2semidirect

Maximal subgroups

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

subgroupindex
#365592
#349512
#331982
#318222
#311652
#299972
#291412
#291322
#282522
#281952
#261062
#253012
#247302
#227222
#221532
C4×#4602