Finite Group Databasebeta

← #6573185

Subgroups — #6573185

Order 512.

1497 conjugacy classes of subgroups.

The subgroup lattice has 627 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36480C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic
3#30151C212yesmaximal, complemented
4#29691C211yesmaximal, characteristic
5#29495C212yesmaximal, complemented
6#28206C212yesmaximal, complemented
7#27950C211yesmaximal, characteristic
8#27648C211yesmaximal, characteristic
9#27100C214yesmaximal
10#26790C214yesmaximal
11#26046C212yesmaximal, complemented
12#24817C212yesmaximal, complemented
13#23835C212yesmaximal, complemented
14#23571C211yesmaximal, characteristic
15#23063C212yesmaximal, complemented
16#22808C212yesmaximal, complemented
17#22632C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C2×#484C2212yes—
20C2×#477C2212yes—
21C2×#477C2214yes—
22C2×C4×Q16C2212yes—
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#2369C2211yescharacteristic
34#2365—21no—
35#2360—22no—
36#2360C2214yes—
37#2355—21no—
38#2355C2212yes—
39#2355C2211yescharacteristic
40Q8×C4⋊C4—21no—
41Q8×C4⋊C4C2212yes—
42#2347—21no—
43#2330—22no—
44#2317—21no—
45C4×C4⋊Q8C2212yes—
46C4×C4⋊Q8C2212yes—
47C4×C4⋊Q8C2212yes—
48C4×C22⋊Q8C2211yescharacteristic
49C2×#384C2211yescharacteristic
50#2274—21no—
51C4×#117C2211yescharacteristic
52#2087C2214yes—
53#2083C2212yes—
54#2083C2212yes—
55#2083C2214yes—
56#2083C2212yes—
57#2072C2212yes—
58#2068C2214yes—
59#2057C2211yescharacteristic
60#2057C2214yes—
61#2057C2212yes—
62#2043C2211yescharacteristic
63#2043C2212yes—
64#2036C2212yes—
65#2036C2214yes—
66#2024C2211yescharacteristic
67#2016C2211yescharacteristic
68#2001C2212yes—
69#1990C2212yes—
70#1969C2214yes—
71#1947C2212yes—
72#1939C2212yes—
73#1933C2214yes—
74#1928C2214yes—
75#1928C2214yes—
76#1928C2214yes—
77#1923C2214yes—
78#1921C2214yes—
79#1878C2214yes—
80#1863C2214yes—
81#1852C2214yes—
82#1845C2212yes—
83#1831C2212yes—
84#1824C2212yes—
85#1803C2212yes—
86#1803C2212yes—
87#1803C2214yes—
88#1797C2211yescharacteristic
89C4×#107C2214yes—
90C4×#107C2214yes—
91C8×C4⋊C4C2212yes—
92C4×#103C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94#1744C2212yes—
95C22×Q16C2312yes—
96#533—28no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×#125—21no—
103C2×C22⋊Q8—21no—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8C2312yes—
106C2×C22⋊Q8C2311yescharacteristic
107C2×C4×Q8—21no—
108C2×C4×Q8—22no—
109C2×C4×Q8—22no—
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2312yes—
112C2×C4×Q8C2314yes—
113C2×#117—21no—
114C2×#117D411yescharacteristic
115C2×#117D411yescharacteristic
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4C2312yes—
119C22×C42C2311yescharacteristic
120#484—44no—
121#477—28no—
122#477—48no—
123C4×Q16—28no—
124C2×#107C2314yes—
125C2×#107C2314yes—
126C2×#107C2312yes—
127C2×#107C2314yes—
128C2×#107C2314yes—
129C2×C4⋊C8C2312yes—
130C2×C4⋊C8C2314yes—
131C2×#103C2312yes—
132C2×#103C2314yes—
133C2×#103C2312yes—
134C2×#103C2312yes—
135C2×#103C2312yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×C22⋊C8C2312yes—
139C2×C22⋊C8C2312yes—
140C2×C4×C8C2312yes—
141C2×C4×C8C2314yes—
142#400—22no—
143#400—22no—
144#396—22no—
145#395—22no—
146#395—22no—
147#393—22no—
148#391—22no—
149#391—22no—
150#391—21no—
151#391—22no—
152#391—22no—
153#391C2314yes—
154#391C2312yes—
155#391C2311yescharacteristic
156#391C2312yes—
157#389—21no—
158#389—21no—
159#389—21no—
160#389—21no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—21no—
166#389—24no—
167#389—24no—
168#389C2311yescharacteristic
169#389C2314yes—
170#389C2312yes—
171#389C2312yes—
172#389C2312yes—
173#388—22no—
174#388—22no—
175#387—21no—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—24no—
180#387—24no—
181#387—22no—
182#387—22no—
183#385—21no—
184#385—24no—
185#384—22no—
186#384—21no—
187#384—21no—
188#384C2312yes—
189#384C2312yes—
190#384C2311yescharacteristic
191#384C2312yes—
192#383—22no—
193#382—22no—
194#382—21no—
195#380—21no—
196#380C2311yescharacteristic
197#380C2312yes—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4C2314yes—
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#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#336C2312yes—
229#336C2311yescharacteristic
230#336C2314yes—
231#336C2311yescharacteristic
232#336C2314yes—
233C22×Q8—21no—
234C22×Q8—22no—
235C22×Q8—21no—
236C22×Q8C2411yescharacteristic
237C22×Q8C2412yes—
238C23×C4—22no—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2411yescharacteristic
242C2×Q16—24no—
243C2×Q16—24no—
244C2×Q16—42no—
245C22×C8C2412yes—
246C22×C8C2414yes—
247C22×C8C2412yes—
248C4⋊Q8—24no—
249C4⋊Q8—24no—
250C4⋊Q8—24no—
251C4⋊Q8—24no—
252C4⋊Q8—42no—
253C4⋊Q8—44no—
254#125—28no—
255C22⋊Q8—28no—
256C22⋊Q8—24no—
257C22⋊Q8—24no—
258C22⋊Q8—22no—
259C22⋊Q8—41no—
260C22⋊Q8—82no—
261C4×Q8—28no—
262C4×Q8—24no—
263C4×Q8—24no—
264C4×Q8—44no—
265C4×Q8—44no—
266C4×Q8—84no—
267C4×Q8—84no—
268C4×Q8#4118yes—
269#117—28no—
270#117—42no—
271#117Q1618yescomplemented
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—82no—
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2414yes—
305C2×C4⋊C4C2411yescharacteristic
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2411yescharacteristic
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2411yescharacteristic
311C2×C22⋊C4—22no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4C2×D412yes—
316C2×C22⋊C4C2×D412yes—
317C2×C42—22no—
318C2×C42—24no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—24no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—24no—
325C2×C42C2×D412yes—
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D412yes—
329C2×C42C2×D412yes—
330C2×C42C2×D412yes—
331C2×C42C2412yes—
332C2×C42C2411yescharacteristic
333C2×C42C2412yes—
334C2×C42C2411yescharacteristic
335#107—24no—
336#107—28no—
337#107—28no—
338#107—44no—
339#107—44no—
340C4⋊C8—24no—
341C4⋊C8—44no—
342#103—24no—
343#103—28no—
344#103—24no—
345#103—24no—
346#103—24no—
347#103—44no—
348#103—44no—
349C22⋊C8—24no—
350C22⋊C8—42no—
351C4×C8—28no—
352C4×C8—24no—
353#95—24no—
354#95—24no—
355#95—21no—
356#95—21no—
357#95—22no—
358#95—21no—
359#95—24no—
360#95—22no—
361#95—22no—
362#95—21no—
363#95—24no—
364#95—22no—
365#95—21no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95C2×D412yes—
373#95C2×D412yes—
374#95C2×D412yes—
375#95C2×D412yes—
376C24C22×D411yescharacteristic
377C2×Q8—24no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—21no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—21no—
387C2×Q8—44no—
388C2×Q8—42no—
389C2×Q8—44no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—82no—
393C2×Q8—81no—
394C2×Q8—81no—
395C2×Q8C2×#4114yes—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—24no—
413C22×C4—41no—
414C22×C4—44no—
415C22×C4—82no—
416C22×C4—82no—
417C22×C4C2×Q1614yes—
418C22×C4C22×D412yes—
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D412yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C2511yescharacteristic
428Q16—48no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—22no—
435C2×C8—24no—
436C2×C8—22no—
437C2×C82−1+414yes—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—28no—
441C4⋊C4—24no—
442C4⋊C4—28no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—24no—
454C4⋊C4—42no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—44no—
461C4⋊C4—42no—
462C4⋊C4—44no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—82no—
466C4⋊C4—82no—
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⋊C4C2×#4114yes—
476C4⋊C4H(5,𝔽2)12yes—
477C22⋊C4—28no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—81no—
481C22⋊C4—82no—
482C22⋊C4C2×Q1618yes—
483C42—22no—
484C42—22no—
485C42—28no—
486C42—24no—
487C42—22no—
488C42—22no—
489C42—24no—
490C42—22no—
491C42—42no—
492C42—42no—
493C42—42no—
494C42—42no—
495C42—84no—
496C42—82no—
497C42—84no—
498C42—82no—
499C42C2×Q1618yes—
500C42C2×#4114yes—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23C22×Q1612yes—
508C23C23×D411yescharacteristic
509Q8—24no—
510Q8—24no—
511Q8—48no—
512Q8—42no—
513Q8—42no—
514Q8—42no—
515Q8—42no—
516Q8—88no—
517Q8—162no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—28no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—164no—
569C2×C4#52512yes—
570C2×C4#52512yes—
571C2×C4C22×Q1612yes—
572C2×C4C22×Q1614yes—
573C2×C4C22×Q1614yes—
574C2×C4C22×Q1612yes—
575C2×C4#57512yes—
576C2×C4#57612yes—
577C2×C4C22×#4112yes—
578C2×C4C2×H(5,𝔽2)11yescharacteristic
579C2×C4C2×2−1+412yes—
580C2×C4#58511yescharacteristic
581C8—24no—
582C8—42no—
583C8—44no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22C2×#52511yescharacteristic
592C22#348811yescharacteristic
593C22#349911yescharacteristic
594C22C23×Q1611yescharacteristic
595C22C2×#57511yescharacteristic
596C22C2×#57611yescharacteristic
597C22#358611yescharacteristic
598C4—24no—
599C4—24no—
600C4—22no—
601C4—22no—
602C4—22no—
603C4—42no—
604C4—42no—
605C4—42no—
606C4—44no—
607C4—41no—
608C4—44no—
609C4—84no—
610C4—82no—
611C4—81no—
612C4—81no—
613C4—82no—
614C4—81no—
615C4#310414yes—
616C4#349512yes—
617C4#351112yes—
618C2—42no—
619C2#3608911yescharacteristic
620C2C2×#310411yescharacteristic
621C2#6058411yescharacteristic
622C2#6195211yescharacteristic
623C2#6199511yescharacteristic
624C2#6247211yescharacteristic
625C2#6250711yescharacteristic
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. #30151C2semidirect
5. #29495C2semidirect
6. #28206C2semidirect
11. #26046C2semidirect
12. #24817C2semidirect
13. #23835C2semidirect
15. #23063C2semidirect
16. #22808C2semidirect
271. #117Q16semidirect

Maximal subgroups

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

subgroupindex
#364802
#342142
#301512
#296912
#294952
#282062
#279502
#276482
#271002
#267902
#260462
#248172
#238352
#235712
#230632
#228082
#226322