Finite Group Databasebeta

← #6563132

Subgroups — #6563132

Order 512.

1633 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36481C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic
3#30132C212yesmaximal, complemented
4#29922C211yesmaximal, characteristic
5#29501C212yesmaximal, complemented
6#28161C212yesmaximal, complemented
7#26528C212yesmaximal, complemented
8#25722C211yesmaximal, characteristic
9#25370C212yesmaximal, complemented
10#25173C214yesmaximal
11#24918C214yesmaximal
12#24212C211yesmaximal, characteristic
13#23805C212yesmaximal, complemented
14#23756C211yesmaximal, characteristic
15#23187C212yesmaximal, complemented
16#23004C212yesmaximal, complemented
17#22636C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C2×#494C2212yes—
20C2×#462C2212yes—
21C2×#462C2214yes—
22C2×#451C2212yes—
23C2×#451C2214yes—
24C2×#441C2214yes—
25C2×#441C2214yes—
26#2863C2212yes—
27#2857C2211yescharacteristic
28#2606—22no—
29#2552C2211yescharacteristic
30#2546C2211yescharacteristic
31#2522—22no—
32#2507—21no—
33#2441—21no—
34#2429C2212yes—
35#2423C2212yes—
36#2370—21no—
37#2369—21no—
38#2369—21no—
39#2365C2212yes—
40#2365C2211yescharacteristic
41#2365C2212yes—
42Q8×C4⋊C4C2211yescharacteristic
43#2347—21no—
44#2345—22no—
45Q8×C22⋊C4C2214yes—
46#2331—22no—
47#2316—21no—
48C4×C4⋊Q8C2212yes—
49C4×C4⋊Q8C2212yes—
50C4×C4⋊Q8C2212yes—
51C4×C22⋊Q8C2211yescharacteristic
52#2285C2211yescharacteristic
53#2274—21no—
54C4×#117C2211yescharacteristic
55#2099C2211yescharacteristic
56#2096C2214yes—
57#2096C2212yes—
58#2096C2211yescharacteristic
59#2084C2211yescharacteristic
60#2077C2212yes—
61#2072C2211yescharacteristic
62#2072C2212yes—
63#2066C2214yes—
64#2056C2214yes—
65#2043C2212yes—
66#2035C2212yes—
67#2035C2214yes—
68#2016C2212yes—
69#2016C2212yes—
70#2016C2214yes—
71#2016C2212yes—
72#1981C2212yes—
73#1981C2212yes—
74#1971C2214yes—
75#1956C2212yes—
76#1948C2212yes—
77#1939C2212yes—
78#1935C2214yes—
79#1924C2214yes—
80#1878C2214yes—
81#1878C2214yes—
82#1878C2214yes—
83#1878C2214yes—
84#1865C2214yes—
85#1865C2214yes—
86#1831C2212yes—
87#1803C2214yes—
88#1797C2211yescharacteristic
89#1787C2212yes—
90C4×#106C2212yes—
91#1770C2212yes—
92C4×#103C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94#1745C2212yes—
95C23×Q8C2312yes—
96C22×Q16C2314yes—
97C22×Q16C2314yes—
98C22×Q16C2314yes—
99C2×C4⋊Q8—28no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×#125—21no—
107C2×C22⋊Q8C2311yescharacteristic
108C2×C22⋊Q8C2312yes—
109C2×C4×Q8C2311yescharacteristic
110C2×C4×Q8C2314yes—
111C2×C4×Q8C2314yes—
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2312yes—
114C2×#117—21no—
115C2×#117D411yescharacteristic
116C2×#117D411yescharacteristic
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C42C2311yescharacteristic
120#494—28no—
121#462—28no—
122#462—48no—
123#451—48no—
124#451—44no—
125#441—48no—
126#441—48no—
127C2×#106C2312yes—
128C2×#106C2314yes—
129C2×#106C2312yes—
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2314yes—
132C2×#103C2314yes—
133C2×#103C2312yes—
134C2×#103C2314yes—
135C2×#103C2312yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×C22⋊C8C2312yes—
142C2×C22⋊C8C2312yes—
143C2×#97C2314yes—
144C2×C4×C8C2312yes—
145#399—22no—
146#398—22no—
147#398—22no—
148#398—22no—
149#398—22no—
150#396—22no—
151#391—24no—
152#391—24no—
153#391C2311yescharacteristic
154#391C2314yes—
155#391C2312yes—
156#391C2314yes—
157#391C2312yes—
158#391C2312yes—
159#389—22no—
160#389—22no—
161#389—21no—
162#389—21no—
163#389—21no—
164#389—21no—
165#389—22no—
166#389—22no—
167#389—22no—
168#389—22no—
169#389—22no—
170#389—22no—
171#389—21no—
172#389—21no—
173#389—22no—
174#389—22no—
175#389C2312yes—
176#389C2311yescharacteristic
177#389C2312yes—
178#388—22no—
179#388—22no—
180#387—21no—
181#387—22no—
182#387—22no—
183#387—22no—
184#387—24no—
185#387—24no—
186#387—22no—
187#387—22no—
188#385—21no—
189#384—21no—
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#384C2312yes—
193#384C2311yescharacteristic
194#383—22no—
195#382—21no—
196#380—21no—
197#380—21no—
198#380—21no—
199#380—24no—
200#380C2311yescharacteristic
201#380C2312yes—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
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—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C22⋊C4—21no—
221C4×C22⋊C4—24no—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4C2312yes—
224#376D412yes—
225C43C2312yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#336C2312yes—
232#336C2311yescharacteristic
233#336C2311yescharacteristic
234#336C2314yes—
235#336C2314yes—
236C22×Q8—28no—
237C22×Q8—22no—
238C22×Q8—42no—
239C22×Q8—44no—
240C22×Q8—42no—
241C22×Q8C2412yes—
242C22×Q8C2414yes—
243C22×Q8C2411yescharacteristic
244C23×C4—22no—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2411yescharacteristic
248C2×Q16—28no—
249C2×Q16—28no—
250C2×Q16—28no—
251C2×Q16—28no—
252C2×Q16—28no—
253C2×Q16—44no—
254C2×Q16—44no—
255C2×Q16—44no—
256C2×Q16—44no—
257C22×C8C2412yes—
258C22×C8C2414yes—
259C22×C8C2412yes—
260C4⋊Q8—28no—
261C4⋊Q8—28no—
262C4⋊Q8—28no—
263C4⋊Q8—24no—
264C4⋊Q8—24no—
265C4⋊Q8—24no—
266C4⋊Q8—42no—
267C4⋊Q8#4118yes—
268#125—44no—
269C22⋊Q8—22no—
270C22⋊Q8—41no—
271C22⋊Q8—44no—
272C4×Q8—24no—
273C4×Q8—44no—
274C4×Q8—44no—
275C4×Q8—44no—
276C4×Q8—44no—
277C4×Q8—44no—
278C4×Q8—44no—
279#117—24no—
280#117—24no—
281#117—44no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C22⋊C4—22no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4C2×D412yes—
325C2×C22⋊C4C2×D412yes—
326C2×C42—24no—
327C2×C42—22no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D412yes—
339C2×C42C2×D412yes—
340C2×C42C2412yes—
341C2×C42C2412yes—
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344#106—24no—
345#106—42no—
346#106—44no—
347C4⋊C8—24no—
348C4⋊C8—44no—
349#103—28no—
350#103—24no—
351#103—24no—
352#103—28no—
353#103—44no—
354#103—44no—
355#103—44no—
356#103—44no—
357#103—44no—
358C22⋊C8—24no—
359C22⋊C8—42no—
360#97—28no—
361C4×C8—24no—
362#95—21no—
363#95—24no—
364#95—24no—
365#95—21no—
366#95—21no—
367#95—22no—
368#95—22no—
369#95—24no—
370#95—24no—
371#95—22no—
372#95—21no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—21no—
377#95—22no—
378#95—22no—
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—21no—
389C2×Q8—22no—
390C2×Q8—24no—
391C2×Q8—22no—
392C2×Q8—24no—
393C2×Q8—24no—
394C2×Q8—24no—
395C2×Q8—22no—
396C2×Q8—24no—
397C2×Q8—24no—
398C2×Q8—22no—
399C2×Q8—82no—
400C2×Q8—82no—
401C2×Q8—88no—
402C2×Q8—88no—
403C2×Q8C2×#4118yes—
404C2×Q8H(5,𝔽2)12yes—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—28no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—41no—
424C22×C4—82no—
425C22×C4—82no—
426C22×C4#13714yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D412yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D412yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C2511yescharacteristic
437Q16—416no—
438Q16—416no—
439Q16—88no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—24no—
444C2×C8—24no—
445C2×C8—22no—
446C2×C8—24no—
447C2×C8—22no—
448C2×C8H(5,𝔽2)14yes—
449C4⋊C4—28no—
450C4⋊C4—28no—
451C4⋊C4—28no—
452C4⋊C4—28no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—81no—
480C4⋊C4—82no—
481C4⋊C4—82no—
482C4⋊C4—84no—
483C4⋊C4—82no—
484C4⋊C4—84no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4#13714yes—
488C22⋊C4—24no—
489C22⋊C4—24no—
490C22⋊C4—42no—
491C22⋊C4—44no—
492C22⋊C4—81no—
493C22⋊C4—82no—
494C42—22no—
495C42—22no—
496C42—24no—
497C42—24no—
498C42—24no—
499C42—22no—
500C42—24no—
501C42—22no—
502C42—22no—
503C42—42no—
504C42—42no—
505C42—44no—
506C42—42no—
507C42—84no—
508C42—82no—
509C42—84no—
510C42—82no—
511C42C2×#4114yes—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23C2×#13712yes—
519C23C23×D411yescharacteristic
520Q8—28no—
521Q8—24no—
522Q8—28no—
523Q8—42no—
524Q8—44no—
525Q8—44no—
526Q8—44no—
527Q8—44no—
528Q8—88no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—28no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—28no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—44no—
553C2×C4—41no—
554C2×C4—44no—
555C2×C4—41no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—88no—
581C2×C4#52512yes—
582C2×C4#52512yes—
583C2×C4C2×#13712yes—
584C2×C4C2×#13714yes—
585C2×C4C2×#13712yes—
586C2×C4C2×#13712yes—
587C2×C4#57812yes—
588C2×C4C22×#4112yes—
589C2×C4C2×H(5,𝔽2)12yes—
590C2×C4C2×H(5,𝔽2)11yescharacteristic
591C2×C4#58511yescharacteristic
592C8—24no—
593C8—44no—
594C8—42no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22C2×#52511yescharacteristic
603C22#348811yescharacteristic
604C22#349811yescharacteristic
605C22C22×#13711yescharacteristic
606C22C22×#13711yescharacteristic
607C22C2×#57811yescharacteristic
608C22#358611yescharacteristic
609C4—24no—
610C4—22no—
611C4—22no—
612C4—22no—
613C4—42no—
614C4—41no—
615C4—48no—
616C4—42no—
617C4—42no—
618C4—42no—
619C4—82no—
620C4—81no—
621C4—82no—
622C4—81no—
623C4—82no—
624C4—81no—
625C4—82no—
626C4#311314yes—
627C4#351312yes—
628C4#351912yes—
629C2—42no—
630C2#3599511yescharacteristic
631C2C2×#311311yescharacteristic
632C2#6058311yescharacteristic
633C2#6196611yescharacteristic
634C2#6198411yescharacteristic
635C2#6242411yescharacteristic
636C2#6245411yescharacteristic
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. #30132C2semidirect
5. #29501C2semidirect
6. #28161C2semidirect
7. #26528C2semidirect
9. #25370C2semidirect
13. #23805C2semidirect
15. #23187C2semidirect
16. #23004C2semidirect

Maximal subgroups

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

subgroupindex
#364812
#342162
#301322
#299222
#295012
#281612
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