Finite Group Databasebeta

← #6562257

Subgroups — #6562257

Order 512.

1653 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36272C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic, complemented
3#30079C212yesmaximal, complemented
4#29936C211yesmaximal, characteristic, complemented
5#29369C212yesmaximal, complemented
6#29132C212yesmaximal, complemented
7#27535C214yesmaximal, complemented
8#26532C212yesmaximal, complemented
9#25856C211yesmaximal, characteristic, complemented
10#25320C212yesmaximal, complemented
11#24910C214yesmaximal
12#24209C211yesmaximal, characteristic
13#23754C211yesmaximal, characteristic
14#23683C212yesmaximal, complemented
15#23222C212yesmaximal, complemented
16#23002C212yesmaximal, complemented
17#22643C211yesmaximal, characteristic, complemented
18C2×#552—21no—
19C22×C4⋊Q8C2211yescharacteristic
20C2×#487C2212yes—
21C2×#461C2212yes—
22C2×#448C2212yes—
23C2×#440C2214yes—
24#2859C2212yes—
25#2857C2211yescharacteristic
26#2611—22no—
27#2553—22no—
28#2552C2211yescharacteristic
29#2542C2211yescharacteristic
30#2496C2212yes—
31#2489—21no—
32#2422C2212yes—
33#2415—21no—
34#2369—21no—
35#2369—21no—
36#2365C2211yescharacteristic
37#2365C2212yes—
38#2365C2212yes—
39#2360—21no—
40D4×C4⋊C4C2211yescharacteristic
41#2345—22no—
42Q8×C22⋊C4C2214yes—
43#2331—22no—
44#2316—21no—
45C4×C4⋊Q8C2212yes—
46C4×C4⋊Q8C2212yes—
47C4×#121C2211yescharacteristic
48#2293C2212yes—
49#2285C2211yescharacteristic
50#2274—21no—
51C4×#117C2211yescharacteristic
52#2099C2212yes—
53#2099C2211yescharacteristic
54#2095C2214yes—
55#2093C2211yescharacteristic
56#2083C2211yescharacteristic
57#2083C2212yes—
58#2075C2214yes—
59#2075C2212yes—
60#2072C2214yes—
61#2071C2211yescharacteristic
62#2068C2214yes—
63#2055C2214yes—
64#2043C2212yes—
65#2043C2212yes—
66#2043C2212yes—
67#2033C2212yes—
68#2013C2212yes—
69#1988C2212yes—
70#1981C2212yes—
71#1956C2212yes—
72#1947C2212yes—
73#1939C2212yes—
74#1935C2214yes—
75#1935C2214yes—
76#1928C2214yes—
77#1924C2214yes—
78#1880C2214yes—
79#1877C2214yes—
80#1863C2214yes—
81#1863C2214yes—
82#1863C2214yes—
83#1839C2212yes—
84#1803C2214yes—
85#1803C2214yes—
86#1797C2211yescharacteristic
87#1787C2214yes—
88#1787C2212yes—
89C4×#107C2212yes—
90#1770C2212yes—
91#1767C2212yes—
92#1763C2214yes—
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2312yes—
96C22×SD16C2314yes—
97#552—48no—
98C2×C4⋊Q8—28no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×#126—22no—
105C2×#125—21no—
106C2×#125—22no—
107C2×#123—22no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—21no—
111C2×#121C2311yescharacteristic
112C2×C4×Q8—21no—
113C2×C4×Q8C2314yes—
114C2×C4×Q8C2314yes—
115C2×C4×D4—21no—
116C2×C4×D4C2311yescharacteristic
117C2×C4×D4C2312yes—
118C2×#117—21no—
119C2×#117—22no—
120C2×#117D411yescharacteristic
121C2×#117D411yescharacteristic
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C42C2311yescharacteristic
126#487—28no—
127#461—28no—
128#448—44no—
129#440—48no—
130C2×#107C2314yes—
131C2×#107C2312yes—
132C2×#107C2312yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#102C2312yes—
143C2×#102C2312yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2312yes—
146C2×#97C2314yes—
147C2×C4×C8C2312yes—
148#400—22no—
149#399—22no—
150#398—22no—
151#397—22no—
152#397—22no—
153#394—22no—
154#391—21no—
155#391—21no—
156#391—24no—
157#391—24no—
158#391C2314yes—
159#391C2314yes—
160#390C2312yes—
161#390C2311yescharacteristic
162#389—22no—
163#389—22no—
164#389—21no—
165#389—22no—
166#389—22no—
167#389—22no—
168#389—22no—
169#389—21no—
170#389—22no—
171#389—22no—
172#389C2312yes—
173#389C2312yes—
174#389C2311yescharacteristic
175#389C2312yes—
176#389C2312yes—
177#388—21no—
178#388—22no—
179#388—22no—
180#387—24no—
181#387—24no—
182#387—22no—
183#387—22no—
184#386—21no—
185#385—22no—
186#385—22no—
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#384C2312yes—
190#384C2312yes—
191#384C2311yescharacteristic
192#382—21no—
193#382—22no—
194#380—21no—
195#380—21no—
196#380—21no—
197#380—24no—
198#380C2312yes—
199#380C2312yes—
200#380C2311yescharacteristic
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C22⋊C4—24no—
216C4×C22⋊C4—21no—
217C4×C22⋊C4—22no—
218C4×C22⋊C4D412yes—
219C4×C22⋊C4C2312yes—
220#376D412yes—
221C43C2312yes—
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2314yes—
232#336C2311yescharacteristic
233#336C2314yes—
234C22×Q8—28no—
235C22×Q8—22no—
236C22×Q8—21no—
237C22×Q8—44no—
238C22×Q8—42no—
239C22×Q8—42no—
240C22×Q8C2414yes—
241C22×D4—21no—
242C22×D4C2411yescharacteristic
243C23×C4—22no—
244C23×C4—21no—
245C23×C4—22no—
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2411yescharacteristic
249C2×SD16—28no—
250C2×SD16—28no—
251C2×SD16—44no—
252C22×C8C2414yes—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C4⋊Q8—28no—
256C4⋊Q8—28no—
257C4⋊Q8—28no—
258C4⋊Q8—24no—
259C4⋊Q8—24no—
260C4⋊Q8#4118yes—
261#126—48no—
262#125—48no—
263#125—44no—
264#123—48no—
265C22⋊Q8—44no—
266C22⋊Q8—44no—
267C22⋊Q8—44no—
268#121—22no—
269#121—41no—
270C4×Q8—44no—
271C4×Q8—44no—
272C4×Q8—44no—
273C4×Q8—44no—
274C4×Q8—44no—
275C4×D4—24no—
276C4×D4—24no—
277C4×D4—42no—
278C4×D4—44no—
279#117—24no—
280#117—24no—
281#117—48no—
282#117—44no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—28no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C22⋊C4—22no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4C2×D412yes—
330C2×C22⋊C4C2×D412yes—
331C2×C42—24no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—21no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42C2×D412yes—
338C2×C42C2×D412yes—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42C2412yes—
344C2×C42C2412yes—
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347#107—42no—
348#107—42no—
349#107—44no—
350C4⋊C8—24no—
351C4⋊C8—44no—
352#103—28no—
353#103—28no—
354#103—28no—
355#103—44no—
356#103—44no—
357#103—44no—
358#103—44no—
359#102—24no—
360#102—42no—
361C22⋊C8—42no—
362C22⋊C8—42no—
363#97—44no—
364C4×C8—24no—
365#95—24no—
366#95—24no—
367#95—22no—
368#95—21no—
369#95—21no—
370#95—22no—
371#95—24no—
372#95—24no—
373#95—22no—
374#95—22no—
375#95—21no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95C2×D412yes—
381#95C2×D412yes—
382#95C2×D412yes—
383#95C2×D412yes—
384C24—21no—
385C24C22×D411yescharacteristic
386C2×Q8—28no—
387C2×Q8—24no—
388C2×Q8—24no—
389C2×Q8—24no—
390C2×Q8—24no—
391C2×Q8—24no—
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8—82no—
396C2×Q8—82no—
397C2×Q8—88no—
398C2×Q8—88no—
399C2×Q8C2×#4118yes—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—21no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—81no—
406C2×D4H(5,𝔽2)12yes—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—28no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—41no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—82no—
431C22×C4—82no—
432C22×C4—82no—
433C22×C4AGL(1,ℤ/8)14yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C2511yescharacteristic
444SD16—416no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—22no—
449C2×C8—24no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—24no—
454C4⋊C4—28no—
455C4⋊C4—28no—
456C4⋊C4—28no—
457C4⋊C4—28no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—82no—
487C4⋊C4—81no—
488C4⋊C4—84no—
489C4⋊C4—84no—
490C4⋊C4#13714yes—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—44no—
494C22⋊C4—42no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—42no—
498C22⋊C4—44no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C22⋊C4—44no—
502C22⋊C4—42no—
503C42—22no—
504C42—22no—
505C42—24no—
506C42—24no—
507C42—24no—
508C42—22no—
509C42—22no—
510C42—24no—
511C42—22no—
512C42—42no—
513C42—44no—
514C42—44no—
515C42—42no—
516C42—84no—
517C42—84no—
518C42C2×#4114yes—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—22no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23C2×AGL(1,ℤ/8)12yes—
533C23C23×D411yescharacteristic
534Q8—28no—
535Q8—28no—
536Q8—44no—
537Q8—44no—
538Q8—88no—
539Q8—84no—
540D4—24no—
541D4—42no—
542D4—84no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—28no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—28no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—88no—
592C2×C4—88no—
593C2×C4—84no—
594C2×C4#52512yes—
595C2×C4#52512yes—
596C2×C4C2×AGL(1,ℤ/8)12yes—
597C2×C4C2×AGL(1,ℤ/8)12yes—
598C2×C4C2×#13714yes—
599C2×C4C2×#13712yes—
600C2×C4#57812yes—
601C2×C4C22×#4112yes—
602C2×C4C2×H(5,𝔽2)11yescharacteristic
603C2×C4#58512yes—
604C2×C4#58511yescharacteristic
605C8—42no—
606C8—44no—
607C8—42no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22C2×#52511yescharacteristic
623C22#348811yescharacteristic
624C22#350011yescharacteristic
625C22C22×AGL(1,ℤ/8)11yescharacteristic
626C22C22×#13711yescharacteristic
627C22C2×#57811yescharacteristic
628C22#358611yescharacteristic
629C4—24no—
630C4—22no—
631C4—22no—
632C4—22no—
633C4—42no—
634C4—42no—
635C4—42no—
636C4—41no—
637C4—48no—
638C4—82no—
639C4—82no—
640C4—81no—
641C4—82no—
642C4—81no—
643C4—81no—
644C4—82no—
645C4#311214yes—
646C4#351312yes—
647C4#352112yes—
648C2—42no—
649C2—82no—
650C2#3598211yescharacteristic
651C2C2×#311211yescharacteristic
652C2#6058311yescharacteristic
653C2#6196211yescharacteristic
654C2#6200011yescharacteristic
655C2#6253811yescharacteristic
656C2#6257711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
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. #34216C2semidirect
3. #30079C2semidirect
4. #29936C2semidirect
5. #29369C2semidirect
6. #29132C2semidirect
7. #27535C2semidirect
8. #26532C2semidirect
9. #25856C2semidirect
10. #25320C2semidirect
14. #23683C2semidirect
15. #23222C2semidirect
16. #23002C2semidirect
17. #22643C2semidirect

Maximal subgroups

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

subgroupindex
#362722
#342162
#300792
#299362
#293692
#291322
#275352
#265322
#258562
#253202
#249102
#242092
#237542
#236832
#232222
#230022
#226432