Finite Group Databasebeta

← #6571131

Subgroups — #6571131

Order 512.

1533 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36480C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic
3#30131C212yesmaximal, complemented
4#29939C211yesmaximal, characteristic
5#29495C212yesmaximal, complemented
6#28206C212yesmaximal, complemented
7#27524C214yesmaximal
8#27228C214yesmaximal
9#26466C212yesmaximal, complemented
10#25856C211yesmaximal, characteristic
11#25324C212yesmaximal, complemented
12#24212C211yesmaximal, characteristic
13#23816C212yesmaximal, complemented
14#23757C211yesmaximal, characteristic
15#23217C212yesmaximal, complemented
16#23096C212yesmaximal, complemented
17#22632C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C2×#487C2212yes—
20C2×#477C2214yes—
21C2×#462C2212yes—
22C2×#451C2212yes—
23#2863C2212yes—
24#2857C2211yescharacteristic
25#2605—22no—
26#2567C2211yescharacteristic
27#2551C2211yescharacteristic
28#2509—21no—
29#2497C2212yes—
30#2492C2212yes—
31#2465—22no—
32#2439—21no—
33#2369—21no—
34#2369—21no—
35#2369C2211yescharacteristic
36#2365—21no—
37#2365C2212yes—
38#2365C2212yes—
39#2355C2211yescharacteristic
40#2347—21no—
41#2345—22no—
42Q8×C22⋊C4C2214yes—
43#2331—22no—
44#2316—21no—
45C4×C4⋊Q8C2212yes—
46C4×C4⋊Q8C2212yes—
47C4×C4⋊Q8C2212yes—
48C4×C22⋊Q8C2211yescharacteristic
49#2285C2211yescharacteristic
50#2274—21no—
51C4×#117C2211yescharacteristic
52#2100C2211yescharacteristic
53#2100C2212yes—
54#2095C2211yescharacteristic
55#2083C2211yescharacteristic
56#2083C2212yes—
57#2076C2214yes—
58#2076C2212yes—
59#2072C2214yes—
60#2072C2211yescharacteristic
61#2068C2214yes—
62#2056C2214yes—
63#2043C2212yes—
64#2043C2212yes—
65#2043C2212yes—
66#2036C2212yes—
67#2024C2214yes—
68#2016C2212yes—
69#1987C2212yes—
70#1981C2212yes—
71#1969C2214yes—
72#1956C2212yes—
73#1947C2212yes—
74#1939C2212yes—
75#1933C2214yes—
76#1928C2214yes—
77#1928C2214yes—
78#1923C2214yes—
79#1878C2214yes—
80#1878C2214yes—
81#1863C2214yes—
82#1863C2214yes—
83#1831C2212yes—
84#1803C2214yes—
85#1797C2211yescharacteristic
86#1787C2214yes—
87#1787C2212yes—
88C4×#107C2212yes—
89C8×C4⋊C4C2212yes—
90C4×#103C2214yes—
91C4×#103C2214yes—
92C4×#103C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94#1744C2212yes—
95C23×Q8C2312yes—
96C22×Q16C2314yes—
97C2×C4⋊Q8—28no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8C2311yescharacteristic
107C2×C4×Q8—22no—
108C2×C4×Q8C2311yescharacteristic
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#477—48no—
119#462—28no—
120#451—44no—
121C2×#107C2312yes—
122C2×#107C2314yes—
123C2×#107C2312yes—
124C2×C4⋊C8C2312yes—
125C2×C4⋊C8C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2312yes—
132C2×#103C2314yes—
133C2×#103C2312yes—
134C2×#103C2314yes—
135C2×C22⋊C8C2312yes—
136C2×C22⋊C8C2312yes—
137C2×C4×C8C2312yes—
138C2×C4×C8C2314yes—
139#400—22no—
140#400—22no—
141#396—22no—
142#395—22no—
143#395—22no—
144#393—22no—
145#391—22no—
146#391—21no—
147#391—24no—
148#391—24no—
149#391—22no—
150#391—22no—
151#391C2314yes—
152#391C2314yes—
153#391C2312yes—
154#391C2311yescharacteristic
155#391C2312yes—
156#389—21no—
157#389—21no—
158#389—22no—
159#389—22no—
160#389—21no—
161#389—21no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—22no—
166#389—22no—
167#389—21no—
168#389C2311yescharacteristic
169#389C2312yes—
170#389C2312yes—
171#389C2312yes—
172#388—22no—
173#388—22no—
174#387—21no—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—24no—
179#387—24no—
180#387—22no—
181#387—22no—
182#385—21no—
183#384C2311yescharacteristic
184#384C2311yescharacteristic
185#384C2312yes—
186#384C2311yescharacteristic
187#383—22no—
188#382—21no—
189#382—21no—
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—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C22⋊C4—21no—
216C4×C22⋊C4—24no—
217C4×C22⋊C4D412yes—
218C4×C22⋊C4C2312yes—
219#376D412yes—
220C43C2312yes—
221#340C2314yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#340C2312yes—
228#336C2312yes—
229#336C2311yescharacteristic
230#336C2314yes—
231#336C2311yescharacteristic
232#336C2314yes—
233C22×Q8—28no—
234C22×Q8—22no—
235C22×Q8—22no—
236C22×Q8—21no—
237C22×Q8—44no—
238C22×Q8—42no—
239C22×Q8—42no—
240C22×Q8C2411yescharacteristic
241C22×Q8C2414yes—
242C23×C4—22no—
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2411yescharacteristic
246C2×Q16—28no—
247C2×Q16—44no—
248C2×Q16—44no—
249C22×C8C2414yes—
250C22×C8C2412yes—
251C22×C8C2412yes—
252C4⋊Q8—28no—
253C4⋊Q8—28no—
254C4⋊Q8—28no—
255C4⋊Q8—24no—
256C4⋊Q8—24no—
257C4⋊Q8—24no—
258C4⋊Q8—42no—
259C4⋊Q8—44no—
260C4⋊Q8#4118yes—
261C22⋊Q8—22no—
262C22⋊Q8—41no—
263C22⋊Q8—82no—
264C4×Q8—24no—
265C4×Q8—44no—
266C4×Q8—44no—
267C4×Q8—44no—
268C4×Q8—44no—
269C4×Q8—84no—
270#117—42no—
271#117—44no—
272#117Q1618yescomplemented
273C2×C4⋊C4—28no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
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—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2411yescharacteristic
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2411yescharacteristic
307C2×C4⋊C4C2411yescharacteristic
308C2×C22⋊C4—22no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4C2×D412yes—
313C2×C22⋊C4C2×D412yes—
314C2×C42—24no—
315C2×C42—22no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42C2×D412yes—
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D412yes—
326C2×C42C2×D412yes—
327C2×C42C2×D412yes—
328C2×C42C2412yes—
329C2×C42C2412yes—
330C2×C42C2411yescharacteristic
331C2×C42C2411yescharacteristic
332#107—24no—
333#107—42no—
334#107—44no—
335C4⋊C8—24no—
336C4⋊C8—44no—
337#103—28no—
338#103—28no—
339#103—28no—
340#103—28no—
341#103—44no—
342#103—44no—
343#103—42no—
344#103—42no—
345#103—44no—
346C22⋊C8—42no—
347C22⋊C8—42no—
348C4×C8—24no—
349C4×C8—44no—
350#95—24no—
351#95—24no—
352#95—21no—
353#95—21no—
354#95—22no—
355#95—24no—
356#95—24no—
357#95—22no—
358#95—21no—
359#95—22no—
360#95—22no—
361#95—21no—
362#95—22no—
363#95—22no—
364#95—21no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
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—22no—
379C2×Q8—24no—
380C2×Q8—21no—
381C2×Q8—24no—
382C2×Q8—24no—
383C2×Q8—24no—
384C2×Q8—21no—
385C2×Q8—22no—
386C2×Q8—44no—
387C2×Q8—44no—
388C2×Q8—42no—
389C2×Q8—82no—
390C2×Q8—82no—
391C2×Q8—82no—
392C2×Q8—81no—
393C2×Q8—81no—
394C2×Q8—88no—
395C2×Q8—88no—
396C2×Q8C2×#4118yes—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—24no—
401C22×C4—28no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—41no—
416C22×C4—82no—
417C22×C4—82no—
418C22×C4C2×Q1614yes—
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C2511yescharacteristic
429Q16—88no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—24no—
437C2×C8—22no—
438C2×C8—22no—
439C4⋊C4—28no—
440C4⋊C4—28no—
441C4⋊C4—28no—
442C4⋊C4—28no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—42no—
456C4⋊C4—44no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—44no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4—82no—
468C4⋊C4—82no—
469C4⋊C4—84no—
470C4⋊C4—84no—
471C4⋊C4—82no—
472C4⋊C4—81no—
473C4⋊C4C2×Q1614yes—
474C4⋊C4C2×Q1614yes—
475C4⋊C4H(5,𝔽2)12yes—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—44no—
479C22⋊C4—81no—
480C22⋊C4—82no—
481C22⋊C4C2×Q1618yes—
482C42—22no—
483C42—22no—
484C42—24no—
485C42—22no—
486C42—22no—
487C42—24no—
488C42—22no—
489C42—42no—
490C42—42no—
491C42—42no—
492C42—44no—
493C42—42no—
494C42—84no—
495C42—84no—
496C42—82no—
497C42—82no—
498C42C2×Q1618yes—
499C42C2×#4114yes—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23C22×Q1612yes—
507C23C23×D411yescharacteristic
508Q8—28no—
509Q8—28no—
510Q8—44no—
511Q8—44no—
512Q8—42no—
513Q8—42no—
514Q8—88no—
515Q8—88no—
516Q8—162no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—28no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—28no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—88no—
568C2×C4#52512yes—
569C2×C4#52512yes—
570C2×C4C22×Q1612yes—
571C2×C4C22×Q1614yes—
572C2×C4C22×Q1614yes—
573C2×C4C22×Q1612yes—
574C2×C4#57512yes—
575C2×C4#57712yes—
576C2×C4C22×#4112yes—
577C2×C4C2×H(5,𝔽2)11yescharacteristic
578C2×C4#58512yes—
579C2×C4#58511yescharacteristic
580C8—42no—
581C8—44no—
582C8—42no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22C2×#52511yescharacteristic
591C22#348811yescharacteristic
592C22#350011yescharacteristic
593C22C23×Q1611yescharacteristic
594C22C2×#57511yescharacteristic
595C22C2×#57711yescharacteristic
596C22#358611yescharacteristic
597C4—24no—
598C4—24no—
599C4—22no—
600C4—22no—
601C4—22no—
602C4—42no—
603C4—42no—
604C4—42no—
605C4—41no—
606C4—48no—
607C4—82no—
608C4—82no—
609C4—81no—
610C4—82no—
611C4—81no—
612C4—82no—
613C4—81no—
614C4#310414yes—
615C4#351412yes—
616C4#352112yes—
617C2—42no—
618C2#3607511yescharacteristic
619C2C2×#310411yescharacteristic
620C2#6058211yescharacteristic
621C2#6195211yescharacteristic
622C2#6199611yescharacteristic
623C2#6253011yescharacteristic
624C2#6258111yescharacteristic
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. #30131C2semidirect
5. #29495C2semidirect
6. #28206C2semidirect
9. #26466C2semidirect
11. #25324C2semidirect
13. #23816C2semidirect
15. #23217C2semidirect
16. #23096C2semidirect
272. #117Q16semidirect

Maximal subgroups

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

subgroupindex
#364802
#342162
#301312
#299392
#294952
#282062
#275242
#272282
#264662
#258562
#253242
#242122
#238162
#237572
#232172
#230962
#226322