Finite Group Databasebeta

← #6478012

Subgroups — #6478012

Order 512.

1660 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35006C211yesmaximal, characteristic
2C2×#2369C211yesmaximal, characteristic, complemented
3#33147C211yesmaximal, characteristic, complemented
4#33107C211yesmaximal, characteristic
5#33095C212yesmaximal, complemented
6#29967C211yesmaximal, characteristic, complemented
7#29755C211yesmaximal, characteristic
8#29639C212yesmaximal, complemented
9#27130C214yesmaximal, complemented
10#27002C214yesmaximal, complemented
11#26860C214yesmaximal, complemented
12#26692C214yesmaximal, complemented
13#22076C212yesmaximal, complemented
14#21912C211yesmaximal, characteristic, complemented
15C4×#481C212yesmaximal, complemented
16C2×#539—21no—
17C22×C4⋊Q8C2211yescharacteristic
18C2×#481C2212yes—
19C2×#481C2212yes—
20#2857C2211yescharacteristic
21#2856C2211yescharacteristic
22#2854C2212yes—
23#2639—21no—
24#2618—21no—
25#2560—22no—
26#2558—22no—
27#2512C2212yes—
28#2506C2212yes—
29#2452C2212yes—
30#2445C2212yes—
31#2369—22no—
32#2369—22no—
33#2369C2214yes—
34#2369C2214yes—
35#2325—21no—
36C4×#121C2212yes—
37C2×#391—21no—
38C2×#391—21no—
39C2×#389—21no—
40C2×#389C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×C4×C4⋊C4—21no—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47#2099C2212yes—
48#2088C2214yes—
49#2084C2212yes—
50#2065C2214yes—
51#2058C2212yes—
52#2058C2214yes—
53#2058C2214yes—
54#2041C2212yes—
55#2023C2214yes—
56#2023C2214yes—
57#1988C2212yes—
58#1982C2212yes—
59#1948C2212yes—
60#1948C2212yes—
61#1941C2214yes—
62#1934C2214yes—
63#1926C2214yes—
64#1926C2214yes—
65#1926C2214yes—
66#1926C2214yes—
67#1920C2214yes—
68#1919C2214yes—
69#1919C2214yes—
70#1919C2214yes—
71#1919C2214yes—
72#1916C2214yes—
73#1916C2214yes—
74#1874C2212yes—
75#1874C2212yes—
76#1864C2214yes—
77#1853C2212yes—
78#1853C2212yes—
79#1824C2212yes—
80#1824C2212yes—
81#1817C2212yes—
82#1797C2211yescharacteristic
83#1797C2211yescharacteristic
84C4×#106C2214yes—
85C4×#102C2214yes—
86C4×C22⋊C8C2212yes—
87C23×Q8—21no—
88#539—48no—
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8—22no—
92C2×C4⋊Q8C2314yes—
93C2×C4⋊Q8C2311yescharacteristic
94C2×#127C2311yescharacteristic
95C2×#126—22no—
96C2×#125—22no—
97C2×#123—22no—
98C2×#123—22no—
99C2×C22⋊Q8—21no—
100C2×C22⋊Q8—22no—
101C2×#121—21no—
102C2×#121—21no—
103C2×#121C2312yes—
104C2×C4×D4—22no—
105C2×C4×D4C2312yes—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4D411yescharacteristic
110C22×C4⋊C4D411yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C42—21no—
116C22×C42—21no—
117C22×C42—21no—
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121#481—28no—
122#481—28no—
123C2×#106C2314yes—
124C2×#106C2314yes—
125C2×#106C2314yes—
126C2×#106C2314yes—
127C2×C4⋊C8C2312yes—
128C2×C4⋊C8C2312yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×C22⋊C8C2312yes—
134C2×C22⋊C8C2312yes—
135C2×C4×C8C2314yes—
136#400—22no—
137#398—22no—
138#397—22no—
139#396—22no—
140#396—22no—
141#394—22no—
142#394—22no—
143#393—22no—
144#391—21no—
145#391—24no—
146#391—24no—
147#391—24no—
148#391—24no—
149#390—21no—
150#390—21no—
151#390C2312yes—
152#390C2312yes—
153#390C2312yes—
154#390C2312yes—
155#389—24no—
156#389—24no—
157#389—21no—
158#389—22no—
159#389—21no—
160#389—21no—
161#389C2312yes—
162#389C2312yes—
163#389C2312yes—
164#389C2312yes—
165#389C2312yes—
166#389C2314yes—
167#389C2312yes—
168#388—22no—
169#388—22no—
170#388—22no—
171#388—22no—
172#387—22no—
173#386—22no—
174#386—22no—
175#386—22no—
176#385—22no—
177#384C2311yescharacteristic
178#384C2311yescharacteristic
179#380—21no—
180#380C2312yes—
181C4×C4⋊C4—24no—
182C4×C4⋊C4—24no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4—21no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4D412yes—
188C4×C4⋊C4D412yes—
189C4×C4⋊C4C2314yes—
190C4×C4⋊C4C2312yes—
191C4×C4⋊C4C2314yes—
192C4×C4⋊C4C2312yes—
193C4×C4⋊C4C2312yes—
194C4×C4⋊C4C2312yes—
195C4×C4⋊C4C2312yes—
196C4×C22⋊C4—22no—
197C2×#95—21no—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95D411yescharacteristic
203C2×#95D411yescharacteristic
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#336C2312yes—
213#336C2312yes—
214#336C2314yes—
215#336C2314yes—
216#336C2314yes—
217C22×Q8—28no—
218C22×Q8—24no—
219C22×Q8—21no—
220C22×Q8—82no—
221C22×Q8—82no—
222C22×D4—22no—
223C22×D4—21no—
224C22×D4—21no—
225C22×D4C2412yes—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—22no—
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2411yescharacteristic
238C22×C8C2414yes—
239C22×C8C2414yes—
240C4⋊Q8—216no—
241C4⋊Q8—28no—
242C4⋊Q8—28no—
243C4⋊Q8—42no—
244#127—42no—
245#126—48no—
246#125—48no—
247#123—48no—
248#123—48no—
249C22⋊Q8—44no—
250C22⋊Q8—48no—
251#121—24no—
252#121—24no—
253#121—44no—
254#121—44no—
255C4×D4—44no—
256C4×D4—48no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—82no—
281C2×C4⋊C4—82no—
282C2×C4⋊C4SD1618yes—
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2×D412yes—
285C2×C4⋊C4C2×D412yes—
286C2×C4⋊C4C2×D412yes—
287C2×C4⋊C4#4114yes—
288C2×C4⋊C4#4114yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—42no—
315C2×C42—42no—
316C2×C42—44no—
317C2×C42—82no—
318C2×C42—82no—
319C2×C42C2×D412yes—
320C2×C42C2×D412yes—
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2412yes—
324C2×C42C2411yescharacteristic
325C2×C42C2412yes—
326C2×C42C2411yescharacteristic
327#106—28no—
328#106—28no—
329#106—28no—
330#106—44no—
331C4⋊C8—42no—
332C4⋊C8—42no—
333#102—28no—
334#102—28no—
335#102—28no—
336#102—44no—
337C22⋊C8—24no—
338C22⋊C8—24no—
339C4×C8—44no—
340#95—24no—
341#95—22no—
342#95—22no—
343#95—24no—
344#95—22no—
345#95—22no—
346#95—24no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95C2×D412yes—
354#95C2×D412yes—
355#95C2×D412yes—
356#95C2×D412yes—
357C24—22no—
358C24C22×D411yescharacteristic
359C2×Q8—216no—
360C2×Q8—42no—
361C2×Q8—84no—
362C2×Q8—84no—
363C2×Q8—81no—
364C2×Q8—81no—
365C2×Q8—816no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—24no—
369C2×D4—24no—
370C2×D4—42no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4—44no—
374C2×D4—42no—
375C2×D4—81no—
376C2×D4—81no—
377C2×D4—81no—
378C2×D4—81no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—44no—
416C22×C4—82no—
417C22×C4—82no—
418C22×C4C2×SD1614yes—
419C22×C4C2×SD1614yes—
420C22×C4C2×SD1614yes—
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D412yes—
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D412yes—
430C22×C4C2×#4112yes—
431C22×C4C2×#4112yes—
432C22×C4C2511yescharacteristic
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—24no—
437C2×C8—24no—
438C2×C8—24no—
439C4⋊C4—24no—
440C4⋊C4—216no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—44no—
456C4⋊C4—42no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—42no—
460C4⋊C4—44no—
461C4⋊C4—81no—
462C4⋊C4—84no—
463C4⋊C4—82no—
464C4⋊C4—82no—
465C4⋊C4—81no—
466C4⋊C4C2×SD1618yes—
467C4⋊C4C2×SD1618yes—
468C4⋊C4C2×#4114yes—
469C4⋊C4C2×#4114yes—
470C4⋊C42−1+414yes—
471C22⋊C4—44no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—82no—
479C22⋊C4—82no—
480C22⋊C4—82no—
481C22⋊C4—82no—
482C42—24no—
483C42—22no—
484C42—22no—
485C42—24no—
486C42—22no—
487C42—22no—
488C42—42no—
489C42—41no—
490C42—41no—
491C42—48no—
492C42—44no—
493C42—84no—
494C42—82no—
495C42—82no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23#52512yes—
508C23C22×SD1612yes—
509C23#57512yes—
510C23C23×D411yescharacteristic
511Q8—816no—
512Q8—162no—
513D4—44no—
514D4—44no—
515D4—88no—
516D4—162no—
517D4—162no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—28no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—24no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—48no—
549C2×C4—48no—
550C2×C4—48no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—84no—
573C2×C4—164no—
574C2×C4#52512yes—
575C2×C4C22×SD1614yes—
576C2×C4C22×SD1614yes—
577C2×C4C22×SD1612yes—
578C2×C4C22×SD1612yes—
579C2×C4C22×#4111yescharacteristic
580C2×C4C22×#4111yescharacteristic
581C2×C4C2×2−1+412yes—
582C2×C4#58512yes—
583C8—44no—
584C8—44no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22—82no—
598C22#310314yes—
599C22C2×#52511yescharacteristic
600C22#348511yescharacteristic
601C22#348611yescharacteristic
602C22C23×SD1611yescharacteristic
603C22C2×#57511yescharacteristic
604C22#358511yescharacteristic
605C22#358511yescharacteristic
606C4—24no—
607C4—22no—
608C4—22no—
609C4—22no—
610C4—24no—
611C4—42no—
612C4—41no—
613C4—41no—
614C4—44no—
615C4—42no—
616C4—44no—
617C4—44no—
618C4—82no—
619C4—82no—
620C4—82no—
621C4—82no—
622C4#346112yes—
623C4#350712yes—
624C2—24no—
625C2—162no—
626C2#3509211yescharacteristic
627C2C2×#310311yescharacteristic
628C2#6056411yescharacteristic
629C2#6169711yescharacteristic
630C2#6171011yescharacteristic
631C2#6182011yescharacteristic
632C2#6183111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2369C2semidirect
3. #33147C2semidirect
5. #33095C2semidirect
6. #29967C2semidirect
8. #29639C2semidirect
9. #27130C2semidirect
10. #27002C2semidirect
11. #26860C2semidirect
12. #26692C2semidirect
13. #22076C2semidirect
14. #21912C2semidirect
15. C4×#481C2semidirect

Maximal subgroups

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

subgroupindex
#350062
C2×#23692
#331472
#331072
#330952
#299672
#297552
#296392
#271302
#270022
#268602
#266922
#220762
#219122
C4×#4812