Finite Group Databasebeta

← #6562266

Subgroups — #6562266

Order 512.

1523 conjugacy classes of subgroups.

The subgroup lattice has 625 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36287C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic
3#30078C212yesmaximal, complemented
4#29938C211yesmaximal, characteristic
5#29371C212yesmaximal, complemented
6#29132C212yesmaximal, complemented
7#27537C214yesmaximal
8#26528C212yesmaximal, complemented
9#25856C211yesmaximal, characteristic
10#25324C212yesmaximal, complemented
11#24918C214yesmaximal
12#24212C211yesmaximal, characteristic
13#23756C211yesmaximal, characteristic
14#23685C212yesmaximal, complemented
15#23221C212yesmaximal, complemented
16#23004C212yesmaximal, complemented
17#22643C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C2×#487C2212yes—
20C2×#462C2212yes—
21C2×#451C2212yes—
22C2×#441C2214yes—
23#2859C2212yes—
24#2857C2211yescharacteristic
25#2612—22no—
26#2567C2211yescharacteristic
27#2556—22no—
28#2552C2211yescharacteristic
29#2506—21no—
30#2497C2212yes—
31#2441—21no—
32#2423C2212yes—
33#2370—21no—
34#2369—21no—
35#2369—21no—
36#2369C2211yescharacteristic
37#2365C2212yes—
38#2365C2212yes—
39Q8×C4⋊C4C2211yescharacteristic
40#2346—21no—
41#2345—22no—
42Q8×C22⋊C4C2214yes—
43#2331—22no—
44#2316—21no—
45C4×C4⋊Q8C2212yes—
46C4×C4⋊Q8C2212yes—
47C4×C22⋊Q8C2211yescharacteristic
48#2295C2212yes—
49#2285C2211yescharacteristic
50#2274—21no—
51C4×#117C2211yescharacteristic
52#2099C2212yes—
53#2099C2211yescharacteristic
54#2096C2214yes—
55#2096C2211yescharacteristic
56#2084C2211yescharacteristic
57#2084C2212yes—
58#2077C2214yes—
59#2077C2212yes—
60#2072C2214yes—
61#2072C2211yescharacteristic
62#2066C2214yes—
63#2056C2214yes—
64#2043C2212yes—
65#2043C2212yes—
66#2043C2212yes—
67#2035C2212yes—
68#2016C2212yes—
69#1987C2212yes—
70#1981C2212yes—
71#1956C2212yes—
72#1948C2212yes—
73#1939C2212yes—
74#1935C2214yes—
75#1935C2214yes—
76#1927C2214yes—
77#1924C2214yes—
78#1878C2214yes—
79#1878C2214yes—
80#1865C2214yes—
81#1865C2214yes—
82#1865C2214yes—
83#1839C2212yes—
84#1803C2214yes—
85#1803C2214yes—
86#1797C2211yescharacteristic
87#1787C2214yes—
88#1787C2212yes—
89C4×#106C2212yes—
90#1770C2212yes—
91#1767C2212yes—
92#1763C2214yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2312yes—
96C22×Q16C2314yes—
97C2×C4⋊Q8—28no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×#125—21no—
104C2×#125—22no—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8C2311yescharacteristic
107C2×C4×Q8C2311yescharacteristic
108C2×C4×Q8C2312yes—
109C2×C4×Q8C2314yes—
110C2×C4×Q8C2314yes—
111C2×#117—21no—
112C2×#117D411yescharacteristic
113C2×#117D411yescharacteristic
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C42C2311yescharacteristic
117#487—28no—
118#462—28no—
119#451—44no—
120#441—48no—
121C2×#106C2314yes—
122C2×#106C2312yes—
123C2×#106C2312yes—
124C2×C4⋊C8C2312yes—
125C2×C4⋊C8C2314yes—
126C2×#103C2312yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2312yes—
135C2×C22⋊C8C2312yes—
136C2×C22⋊C8C2312yes—
137C2×#97C2314yes—
138C2×C4×C8C2312yes—
139#400—22no—
140#399—22no—
141#398—22no—
142#398—22no—
143#396—22no—
144#395—22no—
145#391—21no—
146#391—24no—
147#391—24no—
148#391—22no—
149#391C2314yes—
150#391C2314yes—
151#391C2312yes—
152#391C2311yescharacteristic
153#389—21no—
154#389—21no—
155#389—22no—
156#389—22no—
157#389—21no—
158#389—21no—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—22no—
163#389—21no—
164#389—22no—
165#389—22no—
166#389—22no—
167#389C2312yes—
168#389C2312yes—
169#389C2311yescharacteristic
170#389C2312yes—
171#389C2312yes—
172#388—22no—
173#388—22no—
174#387—21no—
175#387—22no—
176#387—22no—
177#387—24no—
178#387—24no—
179#387—22no—
180#387—22no—
181#385—21no—
182#384—21no—
183#384C2311yescharacteristic
184#384C2311yescharacteristic
185#384C2312yes—
186#384C2312yes—
187#384C2311yescharacteristic
188#382—21no—
189#382—22no—
190#380—21no—
191#380—21no—
192#380—24no—
193#380C2312yes—
194#380C2312yes—
195#380C2311yescharacteristic
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C22⋊C4—21no—
215C4×C22⋊C4—24no—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4C2312yes—
218#376—22no—
219#376D412yes—
220C43C2312yes—
221#340C2314yes—
222#340C2312yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#336C2312yes—
229#336C2314yes—
230#336C2311yescharacteristic
231#336C2311yescharacteristic
232#336C2314yes—
233C22×Q8—28no—
234C22×Q8—22no—
235C22×Q8—21no—
236C22×Q8—44no—
237C22×Q8—42no—
238C22×Q8—42no—
239C22×Q8C2414yes—
240C22×Q8C2411yescharacteristic
241C23×C4—22no—
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2411yescharacteristic
245C2×Q16—28no—
246C2×Q16—28no—
247C2×Q16—44no—
248C22×C8C2414yes—
249C22×C8C2412yes—
250C22×C8C2412yes—
251C4⋊Q8—28no—
252C4⋊Q8—28no—
253C4⋊Q8—28no—
254C4⋊Q8—24no—
255C4⋊Q8—24no—
256C4⋊Q8#4118yes—
257#125—44no—
258#125—84no—
259C22⋊Q8—22no—
260C22⋊Q8—41no—
261C22⋊Q8—82no—
262C4×Q8—24no—
263C4×Q8—24no—
264C4×Q8—42no—
265C4×Q8—44no—
266C4×Q8—44no—
267C4×Q8—44no—
268C4×Q8—44no—
269#117—24no—
270#117—24no—
271#117—44no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2411yescharacteristic
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2411yescharacteristic
310C2×C22⋊C4—22no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4C2×D412yes—
315C2×C22⋊C4C2×D412yes—
316C2×C42—24no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—22no—
324C2×C42C2×D412yes—
325C2×C42C2×D412yes—
326C2×C42C2×D412yes—
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D412yes—
330C2×C42C2412yes—
331C2×C42C2412yes—
332C2×C42C2411yescharacteristic
333C2×C42C2411yescharacteristic
334#106—42no—
335#106—42no—
336#106—44no—
337C4⋊C8—24no—
338C4⋊C8—44no—
339#103—28no—
340#103—28no—
341#103—28no—
342#103—24no—
343#103—44no—
344#103—44no—
345#103—42no—
346#103—44no—
347#103—44no—
348C22⋊C8—42no—
349C22⋊C8—42no—
350#97—44no—
351C4×C8—24no—
352#95—21no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—24no—
357#95—24no—
358#95—21no—
359#95—21no—
360#95—24no—
361#95—24no—
362#95—22no—
363#95—21no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—21no—
369#95—22no—
370#95—22no—
371#95C2×D412yes—
372#95C2×D412yes—
373#95C2×D412yes—
374#95C2×D412yes—
375C24C22×D411yescharacteristic
376C2×Q8—28no—
377C2×Q8—24no—
378C2×Q8—21no—
379C2×Q8—22no—
380C2×Q8—24no—
381C2×Q8—24no—
382C2×Q8—24no—
383C2×Q8—22no—
384C2×Q8—24no—
385C2×Q8—42no—
386C2×Q8—82no—
387C2×Q8—82no—
388C2×Q8—81no—
389C2×Q8—81no—
390C2×Q8—88no—
391C2×Q8—88no—
392C2×Q8C2×#4118yes—
393C2×Q8H(5,𝔽2)12yes—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—24no—
398C22×C4—28no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—41no—
413C22×C4—82no—
414C22×C4—82no—
415C22×C4#13714yes—
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D412yes—
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C2511yescharacteristic
426Q16—416no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—24no—
430C2×C8—22no—
431C2×C8—24no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—24no—
436C4⋊C4—28no—
437C4⋊C4—28no—
438C4⋊C4—28no—
439C4⋊C4—28no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—21no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—44no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—44no—
466C4⋊C4—42no—
467C4⋊C4—82no—
468C4⋊C4—84no—
469C4⋊C4—84no—
470C4⋊C4—81no—
471C4⋊C4—82no—
472C4⋊C4—82no—
473C4⋊C4—82no—
474C4⋊C4—82no—
475C4⋊C4#13714yes—
476C22⋊C4—24no—
477C22⋊C4—24no—
478C22⋊C4—42no—
479C22⋊C4—44no—
480C22⋊C4—81no—
481C22⋊C4—82no—
482C42—22no—
483C42—22no—
484C42—24no—
485C42—24no—
486C42—24no—
487C42—22no—
488C42—22no—
489C42—24no—
490C42—22no—
491C42—42no—
492C42—44no—
493C42—42no—
494C42—44no—
495C42—42no—
496C42—82no—
497C42—84no—
498C42—84no—
499C42C2×#4114yes—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23C2×#13712yes—
507C23C23×D411yescharacteristic
508Q8—24no—
509Q8—28no—
510Q8—28no—
511Q8—44no—
512Q8—42no—
513Q8—44no—
514Q8—88no—
515Q8—162no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—28no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—28no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—44no—
540C2×C4—41no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—88no—
568C2×C4#52512yes—
569C2×C4#52512yes—
570C2×C4C2×#13712yes—
571C2×C4C2×#13714yes—
572C2×C4C2×#13712yes—
573C2×C4C2×#13712yes—
574C2×C4#57712yes—
575C2×C4C22×#4112yes—
576C2×C4C2×H(5,𝔽2)11yescharacteristic
577C2×C4#58512yes—
578C2×C4#58511yescharacteristic
579C8—42no—
580C8—44no—
581C8—42no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22C2×#52511yescharacteristic
590C22#348811yescharacteristic
591C22#350011yescharacteristic
592C22C22×#13711yescharacteristic
593C22C22×#13711yescharacteristic
594C22C2×#57711yescharacteristic
595C22#358611yescharacteristic
596C4—24no—
597C4—22no—
598C4—22no—
599C4—22no—
600C4—42no—
601C4—42no—
602C4—42no—
603C4—41no—
604C4—48no—
605C4—42no—
606C4—82no—
607C4—82no—
608C4—81no—
609C4—82no—
610C4—81no—
611C4—81no—
612C4—82no—
613C4#311314yes—
614C4#351312yes—
615C4#352112yes—
616C2—42no—
617C2#3598211yescharacteristic
618C2C2×#311311yescharacteristic
619C2#6058211yescharacteristic
620C2#6196611yescharacteristic
621C2#6200211yescharacteristic
622C2#6254211yescharacteristic
623C2#6257711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
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
3. #30078C2semidirect
5. #29371C2semidirect
6. #29132C2semidirect
8. #26528C2semidirect
10. #25324C2semidirect
14. #23685C2semidirect
15. #23221C2semidirect
16. #23004C2semidirect

Maximal subgroups

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

subgroupindex
#362872
#342162
#300782
#299382
#293712
#291322
#275372
#265282
#258562
#253242
#249182
#242122
#237562
#236852
#232212
#230042
#226432