Finite Group Databasebeta

← #6555800

Subgroups — #6555800

Order 512.

1599 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic, complemented
2#34331C211yesmaximal, characteristic
3#30046C212yesmaximal, complemented
4#29799C211yesmaximal, characteristic, complemented
5#29560C212yesmaximal, complemented
6#29368C212yesmaximal, complemented
7#29258C211yesmaximal, characteristic
8#28438C211yesmaximal, characteristic
9#26614C214yesmaximal, complemented
10#26317C212yesmaximal, complemented
11#26186C214yesmaximal, complemented
12#25629C212yesmaximal
13#23807C212yesmaximal, complemented
14#23777C211yesmaximal, characteristic, complemented
15#23190C212yesmaximal
16#23004C212yesmaximal, complemented
17#22832C211yesmaximal, characteristic, complemented
18C2×#559C2211yescharacteristic
19C2×#492C2212yes—
20C2×#462C2212yes—
21C2×#461C2214yes—
22C2×#441C2214yes—
23C2×#440C2214yes—
24#2868C2211yescharacteristic
25#2862C2212yes—
26#2582—21no—
27#2571—21no—
28#2570—22no—
29#2568C2212yes—
30#2511—22no—
31#2446C2214yes—
32#2367—21no—
33#2367C2212yes—
34#2366C2211yescharacteristic
35#2365C2212yes—
36#2359—22no—
37#2355—21no—
38#2355—21no—
39#2355C2211yescharacteristic
40#2354—21no—
41Q8×C4⋊C4C2211yescharacteristic
42#2349C2212yes—
43#2335—22no—
44#2326—21no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2212yes—
47C4×#127C2211yescharacteristic
48C4×#124C2212yes—
49#2295C2211yescharacteristic
50#2293C2212yes—
51#2275—21no—
52C42×Q8C2211yescharacteristic
53#2100C2211yescharacteristic
54#2097C2214yes—
55#2095C2211yescharacteristic
56#2094C2212yes—
57#2087C2212yes—
58#2074C2214yes—
59#2068C2212yes—
60#2057C2211yescharacteristic
61#2043C2212yes—
62#2043C2212yes—
63#2024C2211yescharacteristic
64#2023C2212yes—
65#2016C2212yes—
66#2016C2212yes—
67#2015C2214yes—
68#2015C2214yes—
69#2013C2212yes—
70#2013C2212yes—
71#1984C2212yes—
72#1982C2212yes—
73#1981C2212yes—
74#1973C2214yes—
75#1956C2212yes—
76#1948C2212yes—
77#1935C2214yes—
78#1928C2214yes—
79#1927C2214yes—
80#1917C2214yes—
81#1880C2214yes—
82#1878C2214yes—
83#1877C2214yes—
84#1877C2214yes—
85#1877C2214yes—
86#1865C2214yes—
87#1864C2214yes—
88#1839C2212yes—
89#1831C2211yescharacteristic
90C4×#107C2212yes—
91#1770C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×#103C2212yes—
94#1745C2212yes—
95C22×Q16C2314yes—
96C22×SD16C2314yes—
97C22×SD16C2314yes—
98#559—24no—
99#559—24no—
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×#127C2311yescharacteristic
105C2×#124—21no—
106C2×#124C2312yes—
107C2×#123—22no—
108C2×C4×Q8—21no—
109C2×C4×Q8D411yescharacteristic
110C2×C4×Q8D411yescharacteristic
111C2×C4×Q8C2311yescharacteristic
112C2×C4×Q8C2314yes—
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2311yescharacteristic
115C2×C4×D4—21no—
116C2×C4×D4—21no—
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4C2312yes—
120C2×C4×D4C2312yes—
121C2×C4×D4C2312yes—
122C2×#117—22no—
123#492—28no—
124#462—28no—
125#461—48no—
126#441—48no—
127#440—48no—
128C2×#107C2312yes—
129C2×#107C2312yes—
130C2×#106C2314yes—
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2312yes—
133C2×#103C2312yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2312yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#97C2314yes—
143C2×C4×C8C2312yes—
144#400—24no—
145#400—24no—
146#398—24no—
147#391—22no—
148#391—21no—
149#391—21no—
150#391—22no—
151#391—22no—
152#391C2314yes—
153#391C2314yes—
154#391C2311yescharacteristic
155#391C2314yes—
156#391C2311yescharacteristic
157#390—21no—
158#390—21no—
159#390—21no—
160#390C2312yes—
161#390C2312yes—
162#389—22no—
163#389—21no—
164#389—22no—
165#389—22no—
166#389—22no—
167#389C2312yes—
168#389C2311yescharacteristic
169#389C2311yescharacteristic
170#389C2312yes—
171#388—22no—
172#388—22no—
173#388—22no—
174#388—22no—
175#388—22no—
176#388—22no—
177#387—22no—
178#387—24no—
179#387—21no—
180#387—21no—
181#387—24no—
182#387—24no—
183#386—22no—
184#385—22no—
185#385—22no—
186#385—22no—
187#384—21no—
188#384—21no—
189#384—21no—
190#384—21no—
191#384—21no—
192#384C2311yescharacteristic
193#384C2311yescharacteristic
194#382—22no—
195#382—22no—
196#382—22no—
197#382C2312yes—
198#380—22no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4D412yes—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D412yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C22⋊C4—22no—
216C4×C22⋊C4—22no—
217C4×C22⋊C4—22no—
218C4×C22⋊C4—22no—
219#376—21no—
220C2×#95—22no—
221C43C2311yescharacteristic
222C43C2312yes—
223#340C2312yes—
224#340C2312yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#336C2311yescharacteristic
229#336C2314yes—
230#336C2312yes—
231#336C2312yes—
232#336C2311yescharacteristic
233#336C2314yes—
234#336C2314yes—
235C22×Q8—21no—
236C22×Q8C2×D411yescharacteristic
237C22×Q8C2414yes—
238C22×Q8C2411yescharacteristic
239C22×D4—21no—
240C22×D4—21no—
241C22×D4C2412yes—
242C23×C4—22no—
243C23×C4—22no—
244C23×C4—22no—
245C2×Q16—28no—
246C2×Q16—44no—
247C2×Q16—44no—
248C2×SD16—28no—
249C2×SD16—28no—
250C2×SD16—28no—
251C2×SD16—44no—
252C2×SD16—44no—
253C2×SD16—44no—
254C22×C8C2412yes—
255C22×C8C2414yes—
256C22×C8C2412yes—
257C4⋊Q8—24no—
258C4⋊Q8—24no—
259C4⋊Q8—42no—
260C4⋊Q8—44no—
261C4⋊Q8—44no—
262#127—22no—
263#127#4114yes—
264#124—28no—
265#124—24no—
266#124—24no—
267#123—216no—
268C4×Q8—24no—
269C4×Q8—22no—
270C4×Q8—22no—
271C4×Q8—22no—
272C4×Q8—42no—
273C4×Q8—44no—
274C4×Q8—44no—
275C4×Q8—44no—
276C4×Q8—44no—
277C4×Q8#4114yes—
278C4×D4—28no—
279C4×D4—24no—
280C4×D4—24no—
281C4×D4—44no—
282C4×D4—44no—
283C4×D4—44no—
284C4×D4—44no—
285C4×D4—82no—
286#117—84no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2411yescharacteristic
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C42—21no—
327C2×C42—21no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—24no—
332C2×C42C2×D412yes—
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2×D412yes—
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2411yescharacteristic
341C2×C42C2412yes—
342C2×C42C2412yes—
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345#107—24no—
346#107—42no—
347#106—44no—
348C4⋊C8—24no—
349C4⋊C8—24no—
350#103—24no—
351#103—28no—
352#103—28no—
353#103—44no—
354#103—42no—
355#103—44no—
356#103—44no—
357#102—44no—
358#102—44no—
359#97—28no—
360C4×C8—24no—
361#95—21no—
362#95—22no—
363#95—22no—
364#95—21no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95—22no—
369#95—24no—
370#95—24no—
371#95—22no—
372#95—21no—
373#95—24no—
374#95—22no—
375#95—21no—
376#95—22no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95C2×D412yes—
381#95C2×D412yes—
382#95C2×D412yes—
383C24—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—21no—
388C2×Q8—24no—
389C2×Q8—24no—
390C2×Q8—22no—
391C2×Q8—24no—
392C2×Q8—24no—
393C2×Q8—22no—
394C2×Q8—24no—
395C2×Q8—24no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8C2×#4112yes—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—22no—
403C2×D4—24no—
404C2×D4—24no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—81no—
408C2×D4—81no—
409C2×D4C2×#4114yes—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—28no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—82no—
422C22×C4—82no—
423C22×C4—82no—
424C22×C4—82no—
425C22×C4—82no—
426C22×C4—82no—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C2511yescharacteristic
438Q16—416no—
439SD16—416no—
440SD16—88no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—22no—
449C2×C8H(5,𝔽2)14yes—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—24no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—21no—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—44no—
468C4⋊C4—42no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—81no—
476C4⋊C4—82no—
477C4⋊C4—81no—
478C4⋊C4—82no—
479C4⋊C4—84no—
480C4⋊C4—82no—
481C4⋊C4—82no—
482C4⋊C4C2×#4112yes—
483C22⋊C4—28no—
484C22⋊C4—28no—
485C22⋊C4—28no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—82no—
490C22⋊C4—82no—
491C22⋊C4—82no—
492C42—24no—
493C42—24no—
494C42—22no—
495C42—24no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—24no—
500C42—22no—
501C42—22no—
502C42—41no—
503C42—42no—
504C42—42no—
505C42—42no—
506C42—44no—
507C42—82no—
508C42—84no—
509C42—82no—
510C42#13714yes—
511C42C2×#4112yes—
512C42C2×#4112yes—
513C23—24no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23C23×D411yescharacteristic
521Q8—24no—
522Q8—44no—
523Q8—42no—
524Q8—44no—
525Q8—44no—
526Q8—44no—
527Q8—44no—
528Q8—84no—
529D4—28no—
530D4—48no—
531D4—44no—
532D4—84no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—88no—
581C2×C4—164no—
582C2×C4—164no—
583C2×C4#52512yes—
584C2×C4#52512yes—
585C2×C4C2×#13712yes—
586C2×C4C2×#13712yes—
587C2×C4C2×#13712yes—
588C2×C4#57512yes—
589C2×C4#57712yes—
590C2×C4C22×#4111yescharacteristic
591C2×C4C22×#4111yescharacteristic
592C2×C4C2×H(5,𝔽2)12yes—
593C2×C4#58512yes—
594C8—24no—
595C8—44no—
596C8—42no—
597C22—44no—
598C22—44no—
599C22—44no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22C2×#52511yescharacteristic
605C22#348711yescharacteristic
606C22#349811yescharacteristic
607C22C22×#13711yescharacteristic
608C22C2×#57511yescharacteristic
609C22C2×#57711yescharacteristic
610C22#358611yescharacteristic
611C4—22no—
612C4—22no—
613C4—22no—
614C4—22no—
615C4—42no—
616C4—42no—
617C4—42no—
618C4—42no—
619C4—42no—
620C4—41no—
621C4—42no—
622C4—84no—
623C4—81no—
624C4—82no—
625C4—82no—
626C4—82no—
627C4—81no—
628C4#311614yes—
629C4#350612yes—
630C4#351312yes—
631C2—84no—
632C2#3584411yescharacteristic
633C2C2×#311611yescharacteristic
634C2#6059111yescharacteristic
635C2#6189311yescharacteristic
636C2#6189911yescharacteristic
637C2#6242211yescharacteristic
638C2#6245611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #36598C2semidirect
3. #30046C2semidirect
4. #29799C2semidirect
5. #29560C2semidirect
6. #29368C2semidirect
9. #26614C2semidirect
10. #26317C2semidirect
11. #26186C2semidirect
13. #23807C2semidirect
14. #23777C2semidirect
16. #23004C2semidirect
17. #22832C2semidirect

Maximal subgroups

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

subgroupindex
#365982
#343312
#300462
#297992
#295602
#293682
#292582
#284382
#266142
#263172
#261862
#256292
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