Finite Group Databasebeta

← #6563967

Subgroups — #6563967

Order 512.

1639 conjugacy classes of subgroups.

The subgroup lattice has 639 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36288C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic
3#30042C212yesmaximal, complemented
4#29923C211yesmaximal, characteristic
5#29370C212yesmaximal, complemented
6#28373C212yesmaximal, complemented
7#27228C214yesmaximal
8#26466C212yesmaximal, complemented
9#25722C211yesmaximal, characteristic
10#25370C212yesmaximal, complemented
11#25161C214yesmaximal
12#24212C211yesmaximal, characteristic
13#23757C211yesmaximal, characteristic
14#23679C212yesmaximal, complemented
15#23184C212yesmaximal, complemented
16#23096C212yesmaximal, complemented
17#22639C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C2×#494C2212yes—
20C2×#477C2214yes—
21C2×#462C2212yes—
22C2×#462C2214yes—
23C2×#451C2212yes—
24C2×#451C2214yes—
25C2×C4×Q16C2214yes—
26#2862C2212yes—
27#2857C2211yescharacteristic
28#2612—22no—
29#2574—22no—
30#2551C2211yescharacteristic
31#2546C2211yescharacteristic
32#2508—21no—
33#2492C2212yes—
34#2439—21no—
35#2429C2212yes—
36#2369—21no—
37#2369—21no—
38#2369—21no—
39#2365C2212yes—
40#2365C2212yes—
41#2365C2211yescharacteristic
42#2355C2211yescharacteristic
43#2346—21no—
44#2345—22no—
45Q8×C22⋊C4C2214yes—
46#2331—22no—
47#2316—21no—
48C4×C4⋊Q8C2212yes—
49C4×C4⋊Q8C2212yes—
50C4×C22⋊Q8C2211yescharacteristic
51#2295C2212yes—
52#2285C2211yescharacteristic
53#2274—21no—
54C4×#117C2211yescharacteristic
55#2100C2211yescharacteristic
56#2095C2212yes—
57#2095C2211yescharacteristic
58#2083C2211yescharacteristic
59#2076C2212yes—
60#2072C2211yescharacteristic
61#2072C2212yes—
62#2068C2214yes—
63#2056C2214yes—
64#2043C2212yes—
65#2036C2212yes—
66#2036C2214yes—
67#2024C2214yes—
68#2016C2214yes—
69#2016C2212yes—
70#2016C2212yes—
71#2016C2212yes—
72#1981C2212yes—
73#1981C2212yes—
74#1969C2214yes—
75#1969C2214yes—
76#1956C2212yes—
77#1947C2212yes—
78#1939C2212yes—
79#1928C2214yes—
80#1923C2214yes—
81#1878C2214yes—
82#1878C2214yes—
83#1878C2214yes—
84#1878C2214yes—
85#1863C2214yes—
86#1839C2212yes—
87#1797C2211yescharacteristic
88#1787C2212yes—
89C4×#107C2212yes—
90C8×C4⋊C4C2212yes—
91C4×C4⋊C8C2212yes—
92C4×#103C2212yes—
93C4×#103C2214yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2312yes—
96C22×Q16C2314yes—
97C22×Q16C2314yes—
98C22×Q16C2314yes—
99C2×C4⋊Q8—28no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108C2×C22⋊Q8C2311yescharacteristic
109C2×C22⋊Q8C2312yes—
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2312yes—
112C2×C4×Q8C2314yes—
113C2×C4×Q8C2314yes—
114C2×#117—21no—
115C2×#117D411yescharacteristic
116C2×#117D411yescharacteristic
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C42C2311yescharacteristic
120#494—28no—
121#477—48no—
122#462—28no—
123#462—48no—
124#451—48no—
125#451—44no—
126C4×Q16—48no—
127C2×#107C2312yes—
128C2×#107C2312yes—
129C2×#107C2314yes—
130C2×C4⋊C8C2314yes—
131C2×C4⋊C8C2312yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2312yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2312yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×C22⋊C8C2312yes—
142C2×C22⋊C8C2312yes—
143C2×C4×C8C2312yes—
144C2×C4×C8C2314yes—
145#400—22no—
146#398—22no—
147#398—22no—
148#396—22no—
149#395—22no—
150#393—22no—
151#391—22no—
152#391—22no—
153#391—24no—
154#391—24no—
155#391—22no—
156#391C2314yes—
157#391C2314yes—
158#391C2312yes—
159#391C2312yes—
160#391C2311yescharacteristic
161#391C2312yes—
162#391C2312yes—
163#389—22no—
164#389—22no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—22no—
170#389—22no—
171#389—22no—
172#389—22no—
173#389—21no—
174#389—21no—
175#389—22no—
176#389C2312yes—
177#389C2311yescharacteristic
178#388—22no—
179#388—22no—
180#387—22no—
181#387—21no—
182#387—22no—
183#387—22no—
184#387—22no—
185#387—24no—
186#387—24no—
187#385—21no—
188#384C2311yescharacteristic
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#382—21no—
192#382—22no—
193#382—21no—
194#382—22no—
195#380—21no—
196#380—21no—
197#380—21no—
198#380—24no—
199#380C2311yescharacteristic
200#380C2312yes—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C22⋊C4—21no—
220C4×C22⋊C4—24no—
221C4×C22⋊C4D412yes—
222C4×C22⋊C4C2312yes—
223#376—22no—
224#376D412yes—
225C43C2312yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#340C2312yes—
231#336C2312yes—
232#336C2311yescharacteristic
233#336C2311yescharacteristic
234#336C2314yes—
235#336C2314yes—
236C22×Q8—28no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—42no—
240C22×Q8—44no—
241C22×Q8—42no—
242C22×Q8C2411yescharacteristic
243C22×Q8C2412yes—
244C22×Q8C2414yes—
245C23×C4—22no—
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2411yescharacteristic
249C2×Q16—28no—
250C2×Q16—28no—
251C2×Q16—28no—
252C2×Q16—28no—
253C2×Q16—44no—
254C2×Q16—44no—
255C2×Q16—44no—
256C2×Q16—44no—
257C2×Q16—44no—
258C22×C8C2412yes—
259C22×C8C2414yes—
260C22×C8C2412yes—
261C4⋊Q8—28no—
262C4⋊Q8—28no—
263C4⋊Q8—28no—
264C4⋊Q8—24no—
265C4⋊Q8—24no—
266C4⋊Q8—44no—
267C4⋊Q8—44no—
268C4⋊Q8—44no—
269C4⋊Q8#4118yes—
270C22⋊Q8—22no—
271C22⋊Q8—41no—
272C22⋊Q8—44no—
273C4×Q8—24no—
274C4×Q8—24no—
275C4×Q8—42no—
276C4×Q8—44no—
277C4×Q8—44no—
278C4×Q8—44no—
279C4×Q8—44no—
280#117—42no—
281#117—44no—
282#117Q1618yescomplemented
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—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2411yescharacteristic
318C2×C22⋊C4—22no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4C2×D412yes—
323C2×C22⋊C4C2×D412yes—
324C2×C42—24no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—21no—
328C2×C42—21no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—22no—
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2×D412yes—
338C2×C42C2412yes—
339C2×C42C2412yes—
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342#107—24no—
343#107—24no—
344#107—44no—
345C4⋊C8—28no—
346C4⋊C8—24no—
347#103—28no—
348#103—28no—
349#103—24no—
350#103—28no—
351#103—42no—
352#103—44no—
353#103—44no—
354#103—44no—
355#103—44no—
356C22⋊C8—24no—
357C22⋊C8—42no—
358C4×C8—24no—
359C4×C8—44no—
360#95—21no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—24no—
365#95—24no—
366#95—21no—
367#95—22no—
368#95—24no—
369#95—24no—
370#95—22no—
371#95—21no—
372#95—21no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—21no—
379#95—22no—
380#95—22no—
381#95C2×D412yes—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385C24C22×D411yescharacteristic
386C2×Q8—28no—
387C2×Q8—24no—
388C2×Q8—22no—
389C2×Q8—24no—
390C2×Q8—24no—
391C2×Q8—24no—
392C2×Q8—24no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—24no—
396C2×Q8—21no—
397C2×Q8—24no—
398C2×Q8—21no—
399C2×Q8—22no—
400C2×Q8—44no—
401C2×Q8—82no—
402C2×Q8—82no—
403C2×Q8—82no—
404C2×Q8—82no—
405C2×Q8—88no—
406C2×Q8—88no—
407C2×Q8C2×#4118yes—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—24no—
412C22×C4—28no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—41no—
427C22×C4—82no—
428C22×C4—82no—
429C22×C4C2×Q1614yes—
430C22×C4C22×D412yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D412yes—
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C2511yescharacteristic
440Q16—416no—
441Q16—88no—
442Q16—88no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—22no—
449C2×C8—24no—
450C2×C8—22no—
451C2×C8H(5,𝔽2)14yes—
452C4⋊C4—28no—
453C4⋊C4—28no—
454C4⋊C4—28no—
455C4⋊C4—28no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—44no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—81no—
479C4⋊C4—84no—
480C4⋊C4—84no—
481C4⋊C4—82no—
482C4⋊C4—82no—
483C4⋊C4—82no—
484C4⋊C4—82no—
485C4⋊C4C2×Q1614yes—
486C4⋊C4C2×Q1614yes—
487C4⋊C4H(5,𝔽2)12yes—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—44no—
491C22⋊C4—81no—
492C22⋊C4—82no—
493C22⋊C4C2×Q1618yes—
494C42—22no—
495C42—22no—
496C42—24no—
497C42—22no—
498C42—22no—
499C42—24no—
500C42—22no—
501C42—42no—
502C42—42no—
503C42—44no—
504C42—42no—
505C42—44no—
506C42—42no—
507C42—82no—
508C42—84no—
509C42—84no—
510C42C2×Q1618yes—
511C42C2×#4114yes—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23C22×Q1612yes—
519C23C23×D411yescharacteristic
520Q8—28no—
521Q8—28no—
522Q8—44no—
523Q8—44no—
524Q8—44no—
525Q8—42no—
526Q8—44no—
527Q8—42no—
528Q8—88no—
529Q8—164no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—28no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—28no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—88no—
581C2×C4#52512yes—
582C2×C4#52512yes—
583C2×C4C22×Q1612yes—
584C2×C4C22×Q1614yes—
585C2×C4C22×Q1614yes—
586C2×C4C22×Q1612yes—
587C2×C4#57512yes—
588C2×C4#57812yes—
589C2×C4C22×#4112yes—
590C2×C4C2×H(5,𝔽2)12yes—
591C2×C4C2×H(5,𝔽2)11yescharacteristic
592C2×C4#58511yescharacteristic
593C8—24no—
594C8—42no—
595C8—44no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22C2×#52511yescharacteristic
604C22#348811yescharacteristic
605C22#349811yescharacteristic
606C22C23×Q1611yescharacteristic
607C22C2×#57511yescharacteristic
608C22C2×#57811yescharacteristic
609C22#358611yescharacteristic
610C4—24no—
611C4—22no—
612C4—22no—
613C4—22no—
614C4—24no—
615C4—42no—
616C4—42no—
617C4—41no—
618C4—48no—
619C4—42no—
620C4—82no—
621C4—81no—
622C4—82no—
623C4—81no—
624C4—82no—
625C4—82no—
626C4—81no—
627C4#310414yes—
628C4#351412yes—
629C4#351912yes—
630C2—42no—
631C2#3600411yescharacteristic
632C2C2×#310411yescharacteristic
633C2#6058311yescharacteristic
634C2#6195211yescharacteristic
635C2#6198511yescharacteristic
636C2#6241611yescharacteristic
637C2#6245611yescharacteristic
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. #30042C2semidirect
5. #29370C2semidirect
6. #28373C2semidirect
8. #26466C2semidirect
10. #25370C2semidirect
14. #23679C2semidirect
15. #23184C2semidirect
16. #23096C2semidirect
282. #117Q16semidirect

Maximal subgroups

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

subgroupindex
#362882
#342162
#300422
#299232
#293702
#283732
#272282
#264662
#257222
#253702
#251612
#242122
#237572
#236792
#231842
#230962
#226392