Finite Group Databasebeta

← #6568666

Subgroups — #6568666

Order 512.

1824 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36290C211yesmaximal, characteristic, complemented
2#36179C211yesmaximal, characteristic
3#30230C211yesmaximal, characteristic, complemented
4#30189C212yesmaximal, complemented
5#29334C212yesmaximal, complemented
6#29225C211yesmaximal, characteristic, complemented
7#29130C211yesmaximal, characteristic, complemented
8#27966C212yesmaximal, complemented
9#27822C212yesmaximal, complemented
10#27736C212yesmaximal, complemented
11#27717C214yesmaximal, complemented
12#27717C214yesmaximal, complemented
13#23819C211yesmaximal, characteristic, complemented
14C2×#2100C212yesmaximal, complemented
15#22485C212yesmaximal, complemented
16#22485C212yesmaximal, complemented
17#20558C211yesmaximal, characteristic, complemented
18C2×#547—21no—
19C2×#547—21no—
20C2×#547C2212yes—
21C2×C8×D4C2212yes—
22#2868C2211yescharacteristic
23#2859C2211yescharacteristic
24#2856C2212yes—
25#2851C2212yes—
26#2571C2211yescharacteristic
27#2552C2211yescharacteristic
28#2542—21no—
29#2542—21no—
30#2542C2212yes—
31#2509—21no—
32#2507—21no—
33#2505—22no—
34#2501—22no—
35#2501—22no—
36#2501C2214yes—
37#2473—22no—
38#2442C2212yes—
39#2442C2212yes—
40#2349—21no—
41D4×C4⋊C4—21no—
42D4×C4⋊C4C2211yescharacteristic
43D4×C4⋊C4C2211yescharacteristic
44C4×#127C2211yescharacteristic
45C4×#125C2211yescharacteristic
46C2×#389C2212yes—
47C2×#384C2212yes—
48C42×D4C2211yescharacteristic
49#2100C2211yescharacteristic
50#2100C2211yescharacteristic
51#2100C2212yes—
52#2100C2214yes—
53#2100C2212yes—
54#2100C2212yes—
55#2100C2212yes—
56#2100C2212yes—
57#2100C2214yes—
58#2100C2214yes—
59#2073C2212yes—
60#2073C2214yes—
61#2041C2211yescharacteristic
62#2041C2211yescharacteristic
63#2041C2212yes—
64#2025C2212yes—
65#2025C2214yes—
66#1992C2212yes—
67#1992C2214yes—
68#1990C2212yes—
69#1990C2212yes—
70#1988C2212yes—
71#1988C2212yes—
72#1936C2214yes—
73#1936C2214yes—
74#1936C2214yes—
75#1893C2214yes—
76#1893C2214yes—
77#1874C2214yes—
78#1874C2214yes—
79#1836C2212yes—
80#1836C2214yes—
81#1823C2214yes—
82#1823C2214yes—
83#1802C2212yes—
84#1802C2212yes—
85#1770C2212yes—
86#1767C2211yescharacteristic
87C4×#102C2212yes—
88C4×#102C2212yes—
89#1752C2214yes—
90#1749C2212yes—
91#1744C2211yescharacteristic
92C2×#340C2214yes—
93C2×#340C2214yes—
94C2×#336C2212yes—
95C22×#41—22no—
96C23×C8C2314yes—
97#547—28no—
98#547—28no—
99#547—48no—
100#547—48no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2312yes—
104C2×#127C2311yescharacteristic
105C2×#125—21no—
106C2×#125C2311yescharacteristic
107C2×#125C2312yes—
108C2×#123—24no—
109C2×#123—24no—
110C2×#123—24no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—22no—
114C2×#121—22no—
115C2×#121—22no—
116C2×#121C2314yes—
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4D411yescharacteristic
122C2×C4×D4D411yescharacteristic
123C2×C4×D4C2312yes—
124C2×C4×D4C2311yescharacteristic
125C22×C4⋊C4—22no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4—22no—
128C22×C4⋊C4C2314yes—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2312yes—
131C22×C42C2312yes—
132C8×D4—28no—
133C2×C4⋊C8C2312yes—
134C2×#102C2312yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2312yes—
138C2×#102C2312yes—
139C2×#102C2312yes—
140C2×C22⋊C8C2314yes—
141C2×#97C2314yes—
142C2×#97C2312yes—
143C2×C4×C8C2312yes—
144#400—22no—
145#400—22no—
146#400—22no—
147#399—24no—
148#399—24no—
149#399—24no—
150#398—22no—
151#398—22no—
152#398—22no—
153#390—21no—
154#390—21no—
155#390—22no—
156#390—22no—
157#390C2312yes—
158#390C2314yes—
159#389—21no—
160#389—21no—
161#389—21no—
162#389—21no—
163#389—21no—
164#389—21no—
165#389—22no—
166#389—22no—
167#389C2311yescharacteristic
168#389C2311yescharacteristic
169#389C2312yes—
170#389C2312yes—
171#389C2312yes—
172#389C2312yes—
173#387—21no—
174#387—21no—
175#387—22no—
176#387—22no—
177#385—22no—
178#385—22no—
179#385—22no—
180#385—22no—
181#385—22no—
182#385—22no—
183#384—21no—
184#384—21no—
185#384—21no—
186#384—22no—
187#384—22no—
188#384C2311yescharacteristic
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#384C2312yes—
192#384C2312yes—
193#384C2314yes—
194#382—21no—
195#382—21no—
196#382C2311yescharacteristic
197#382C2312yes—
198#380—22no—
199#380—22no—
200#380C2312yes—
201#380C2312yes—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2311yescharacteristic
209C4×C22⋊C4D412yes—
210C4×C22⋊C4D412yes—
211C4×C22⋊C4C2312yes—
212C2×#95—22no—
213C43C2311yescharacteristic
214#340C2314yes—
215#340C2314yes—
216#340C2312yes—
217#340C2314yes—
218#340C2314yes—
219#340C2312yes—
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#336C2314yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2311yescharacteristic
233C2×#41—48no—
234C2×#41—48no—
235C2×#41—48no—
236C22×Q8—22no—
237C22×D4—24no—
238C22×D4—21no—
239C22×D4—21no—
240C22×D4C2×D411yescharacteristic
241C22×D4C2412yes—
242C23×C4—24no—
243C23×C4—22no—
244C23×C4—22no—
245C23×C4—22no—
246C23×C4C2×D412yes—
247C23×C4C2×D412yes—
248C23×C4C2412yes—
249C22×C8—28no—
250C22×C8—44no—
251C22×C8—44no—
252C22×C8C2412yes—
253C22×C8C2414yes—
254C22×C8C2412yes—
255C4⋊Q8—24no—
256C4⋊Q8—24no—
257C4⋊Q8—44no—
258C4⋊Q8—44no—
259#127—42no—
260#125—22no—
261#125—41no—
262#125—44no—
263#125—82no—
264#123—416no—
265#123—416no—
266#123—416no—
267C22⋊Q8—48no—
268C22⋊Q8—48no—
269C22⋊Q8—48no—
270#121—28no—
271#121—28no—
272#121—48no—
273#121—48no—
274C4×D4—24no—
275C4×D4—24no—
276C4×D4—24no—
277C4×D4—24no—
278C4×D4—44no—
279C4×D4—44no—
280C4×D4—44no—
281C4×D4—44no—
282C4×D4#4118yes—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—28no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—42no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—48no—
315C2×C4⋊C4—48no—
316C2×C4⋊C4—84no—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2414yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2411yescharacteristic
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C42—21no—
340C2×C42—21no—
341C2×C42—44no—
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D412yes—
347C2×C42C2×D412yes—
348C2×C42C2411yescharacteristic
349C2×C42C2412yes—
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352C4⋊C8—24no—
353#102—24no—
354#102—44no—
355#102—42no—
356#102—42no—
357#102—44no—
358#102—42no—
359C22⋊C8—28no—
360#97—42no—
361#97—44no—
362C4×C8—24no—
363#95—21no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—21no—
368#95—21no—
369#95—21no—
370#95—21no—
371#95—21no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95C2×D412yes—
378#95C2×D412yes—
379#95C2×D412yes—
380C24—22no—
381C24C22×D412yes—
382#41—832no—
383C2×Q8—44no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—42no—
393C2×D4—42no—
394C2×D4—42no—
395C2×D4—42no—
396C2×D4—48no—
397C2×D4—48no—
398C2×D4—48no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4#12414yes—
402C2×D4AGL(1,ℤ/8)14yes—
403C2×D4AGL(1,ℤ/8)14yes—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—44no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—82no—
431C22×C4—82no—
432C22×C4—82no—
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D412yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C2×#4118yes—
443C22×C4C2511yescharacteristic
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—24no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—88no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—21no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—48no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C4—82no—
489C4⋊C4—82no—
490C4⋊C4—82no—
491C4⋊C4—81no—
492C4⋊C4—81no—
493C4⋊C4—82no—
494C4⋊C4#12414yes—
495C4⋊C42−1+412yes—
496C22⋊C4—24no—
497C22⋊C4—24no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—48no—
506C22⋊C4#12418yes—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—21no—
512C42—21no—
513C42—24no—
514C42—22no—
515C42—42no—
516C42—42no—
517C42—42no—
518C42—42no—
519C42C2×#4114yes—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23C2×#12414yes—
532C23C2×AGL(1,ℤ/8)14yes—
533C23C2×AGL(1,ℤ/8)14yes—
534C23C23×D411yescharacteristic
535Q8—88no—
536D4—42no—
537D4—42no—
538D4—42no—
539D4—42no—
540D4—84no—
541D4—84no—
542D4—816no—
543D4#490116yes—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—28no—
553C2×C4—28no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—48no—
568C2×C4—48no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—88no—
593C2×C4—816no—
594C2×C4—88no—
595C2×C4—164no—
596C2×C4C2×#12412yes—
597C2×C4C2×#12414yes—
598C2×C4C2×#12412yes—
599C2×C4C2×AGL(1,ℤ/8)12yes—
600C2×C4C2×AGL(1,ℤ/8)12yes—
601C2×C4#57712yes—
602C2×C4#57812yes—
603C2×C4C22×#4112yes—
604C2×C4C2×2−1+411yescharacteristic
605C2×C4#58512yes—
606C2×C4#58511yescharacteristic
607C8—42no—
608C8—44no—
609C8—42no—
610C22—24no—
611C22—24no—
612C22—24no—
613C22—24no—
614C22—24no—
615C22—24no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22C2×#49018yes—
624C22C22×#12411yescharacteristic
625C22#348511yescharacteristic
626C22#348911yescharacteristic
627C22C22×AGL(1,ℤ/8)11yescharacteristic
628C22C22×AGL(1,ℤ/8)11yescharacteristic
629C22C2×#57711yescharacteristic
630C22C2×#57811yescharacteristic
631C4—22no—
632C4—22no—
633C4—24no—
634C4—22no—
635C4—42no—
636C4—42no—
637C4—41no—
638C4—42no—
639C4—41no—
640C4—82no—
641C4—81no—
642C4—81no—
643C4—82no—
644C4—81no—
645C4—82no—
646C4—84no—
647C4C2×#49014yes—
648C4#351512yes—
649C4#352112yes—
650C2—28no—
651C2—162no—
652C2#3604211yescharacteristic
653C2C22×#49011yescharacteristic
654C2#6062311yescharacteristic
655C2#6172811yescharacteristic
656C2#6173711yescharacteristic
657C2#6212811yescharacteristic
658C2#6213211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36179C2
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. #36290C2semidirect
3. #30230C2semidirect
4. #30189C2semidirect
5. #29334C2semidirect
6. #29225C2semidirect
7. #29130C2semidirect
8. #27966C2semidirect
9. #27822C2semidirect
10. #27736C2semidirect
11. #27717C2semidirect
12. #27717C2semidirect
13. #23819C2semidirect
14. C2×#2100C2semidirect
15. #22485C2semidirect
16. #22485C2semidirect
17. #20558C2semidirect

Maximal subgroups

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

subgroupindex
#362902
#361792
#302302
#301892
#293342
#292252
#291302
#279662
#278222
#277362
#277172
#238192
C2×#21002
#224852
#205582