Finite Group Databasebeta

← #6558640

Subgroups — #6558640

Order 512.

1559 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36539C211yesmaximal, characteristic
2#36151C211yesmaximal, characteristic, complemented
3#30365C212yesmaximal, complemented
4#30306C211yesmaximal, characteristic, complemented
5#29165C212yesmaximal, complemented
6#29142C211yesmaximal, characteristic, complemented
7#29126C211yesmaximal, characteristic, complemented
8#28872C214yesmaximal, complemented
9#28229C212yesmaximal
10#28148C212yesmaximal, complemented
11#26989C214yesmaximal, complemented
12#25503C212yesmaximal, complemented
13#23797C211yesmaximal, characteristic, complemented
14#23577C212yesmaximal, complemented
15#22765C212yesmaximal, complemented
16#22675C212yesmaximal, complemented
17#20568C211yesmaximal, characteristic, complemented
18C2×#533C2212yes—
19C2×#468C2214yes—
20C2×#441C2214yes—
21C2×C4×Q16C2212yes—
22C2×C4×SD16C2212yes—
23C2×#436C2212yes—
24#2864C2211yescharacteristic
25#2862C2211yescharacteristic
26#2857C2212yes—
27#2854C2212yes—
28#2602—22no—
29#2594—22no—
30#2576—22no—
31#2536C2212yes—
32#2536C2211yescharacteristic
33#2535C2211yescharacteristic
34#2494C2212yes—
35#2489—22no—
36#2453—22no—
37#2426C2212yes—
38#2421C2214yes—
39#2415—22no—
40#2374—22no—
41#2362—22no—
42#2357C2211yescharacteristic
43#2351C2212yes—
44#2351C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×#124C2211yescharacteristic
47C4×C22⋊Q8C2212yes—
48C42×D4C2211yescharacteristic
49#2099C2212yes—
50#2097C2212yes—
51#2096C2211yescharacteristic
52#2093C2211yescharacteristic
53#2085C2214yes—
54#2076C2214yes—
55#2058C2212yes—
56#2057C2212yes—
57#2043C2212yes—
58#2035C2214yes—
59#2034C2214yes—
60#2024C2212yes—
61#2023C2212yes—
62#2016C2211yescharacteristic
63#2015C2212yes—
64#2013C2211yescharacteristic
65#2001C2214yes—
66#1958C2212yes—
67#1943C2214yes—
68#1936C2214yes—
69#1933C2214yes—
70#1920C2214yes—
71#1894C2214yes—
72#1894C2214yes—
73#1880C2214yes—
74#1863C2214yes—
75#1863C2214yes—
76#1846C2212yes—
77#1835C2212yes—
78#1835C2212yes—
79#1809C2214yes—
80#1805C2212yes—
81#1805C2212yes—
82#1805C2212yes—
83#1804C2212yes—
84#1770C2212yes—
85C4×C4⋊C8C2211yescharacteristic
86#1763C2214yes—
87#1753C2214yes—
88#1749C2212yes—
89#1744C2211yescharacteristic
90#1728C2212yes—
91C22×Q16C2312yes—
92C22×SD16C2312yes—
93C22×M16C2314yes—
94#533—48no—
95C2×C4⋊Q8C2311yescharacteristic
96C2×C4⋊Q8C2312yes—
97C2×#125—22no—
98C2×#124C2311yescharacteristic
99C2×#123—22no—
100C2×C22⋊Q8—22no—
101C2×C22⋊Q8—22no—
102C2×C22⋊Q8C2312yes—
103C2×C22⋊Q8C2314yes—
104C2×#121C2312yes—
105C2×C4×Q8—22no—
106C2×C4×Q8C2312yes—
107C2×C4×Q8C2311yescharacteristic
108C2×C4×Q8C2314yes—
109C2×C4×Q8C2312yes—
110C2×C4×D4D412yes—
111C2×C4×D4C2311yescharacteristic
112C2×C4×D4C2311yescharacteristic
113C2×#117—22no—
114C22×C42C2312yes—
115#468—48no—
116#441—48no—
117C4×Q16—28no—
118C4×SD16—28no—
119#436—28no—
120C2×#107C2312yes—
121C2×#107C2314yes—
122C2×#106C2312yes—
123C2×#106C2314yes—
124C2×C4⋊C8C2312yes—
125C2×#103C2314yes—
126C2×#103C2312yes—
127C2×#103C2312yes—
128C2×#103C2312yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#102C2312yes—
132C2×#102C2314yes—
133C2×C22⋊C8C2314yes—
134C2×#97C2314yes—
135C2×C4×C8C2312yes—
136C2×C4×C8C2312yes—
137#401—22no—
138#400—24no—
139#398—21no—
140#398—22no—
141#397—21no—
142#397—22no—
143#397—24no—
144#396—22no—
145#394—21no—
146#394—22no—
147#394—22no—
148#392—21no—
149#391—22no—
150#391—22no—
151#391—22no—
152#391C2312yes—
153#391C2311yescharacteristic
154#390—22no—
155#390C2311yescharacteristic
156#390C2312yes—
157#389—22no—
158#389—24no—
159#389—22no—
160#389—22no—
161#389C2312yes—
162#389C2312yes—
163#389C2312yes—
164#389C2314yes—
165#388—22no—
166#388—22no—
167#388—22no—
168#388—24no—
169#388—24no—
170#388—22no—
171#388—22no—
172#388—22no—
173#388—22no—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—24no—
179#386—22no—
180#386—22no—
181#385—22no—
182#384C2311yescharacteristic
183#382—21no—
184#382—22no—
185#382C2312yes—
186#382C2311yescharacteristic
187#382C2311yescharacteristic
188#382C2312yes—
189#382C2311yescharacteristic
190#381—21no—
191#380—22no—
192#380C2312yes—
193#380C2314yes—
194#380C2312yes—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4D412yes—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2311yescharacteristic
202C4×C22⋊C4—22no—
203C4×C22⋊C4—22no—
204C4×C22⋊C4—21no—
205C4×C22⋊C4D412yes—
206C4×C22⋊C4D412yes—
207C4×C22⋊C4C2312yes—
208C43C2311yescharacteristic
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#336C2311yescharacteristic
216#336C2311yescharacteristic
217#336C2312yes—
218#336C2312yes—
219#336C2314yes—
220C22×Q8—22no—
221C22×Q8C2412yes—
222C22×Q8C2411yescharacteristic
223C22×D4—22no—
224C22×D4C2×D411yescharacteristic
225C22×D4C2411yescharacteristic
226C23×C4—21no—
227C23×C4C2×D412yes—
228C23×C4C2×D412yes—
229C23×C4C2412yes—
230C2×Q16—24no—
231C2×Q16—24no—
232C2×Q16—24no—
233C2×SD16—24no—
234C2×SD16—24no—
235C2×SD16—42no—
236C2×M16—28no—
237C2×M16—44no—
238C2×M16—44no—
239C22×C8C2414yes—
240C22×C8C2412yes—
241C22×C8C2412yes—
242C4⋊Q8—24no—
243C4⋊Q8—44no—
244#125—48no—
245#124—22no—
246#124—41no—
247#123—48no—
248C22⋊Q8—28no—
249C22⋊Q8—24no—
250C22⋊Q8—42no—
251C22⋊Q8—44no—
252C22⋊Q8—48no—
253C22⋊Q8—48no—
254#121—44no—
255C4×Q8—28no—
256C4×Q8—24no—
257C4×Q8—24no—
258C4×Q8—24no—
259C4×Q8—44no—
260C4×Q8—44no—
261C4×Q8—48no—
262C4×D4—24no—
263C4×D4—24no—
264C4×D4—44no—
265#117—48no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4C2×D411yescharacteristic
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2411yescharacteristic
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2414yes—
284C2×C4⋊C4C2411yescharacteristic
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4C2×D412yes—
299C2×C22⋊C4C2×D412yes—
300C2×C22⋊C4C2×D412yes—
301C2×C42—24no—
302C2×C42—21no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—44no—
307C2×C42C2×D412yes—
308C2×C42C2×D412yes—
309C2×C42C2×D412yes—
310C2×C42C2×D412yes—
311C2×C42C2412yes—
312C2×C42C2411yescharacteristic
313C2×C42C2411yescharacteristic
314C2×C42C2411yescharacteristic
315#107—24no—
316#107—44no—
317#106—24no—
318#106—44no—
319C4⋊C8—24no—
320#103—28no—
321#103—24no—
322#103—24no—
323#103—24no—
324#103—44no—
325#103—44no—
326#102—24no—
327#102—44no—
328C22⋊C8—28no—
329#97—44no—
330C4×C8—42no—
331C4×C8#4118yes—
332#95—22no—
333#95—21no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—21no—
338#95—22no—
339#95—24no—
340#95—22no—
341#95—24no—
342#95—22no—
343#95—24no—
344#95—22no—
345#95—21no—
346#95—21no—
347#95—24no—
348#95—22no—
349#95—22no—
350#95C2×D411yescharacteristic
351#95C2×D411yescharacteristic
352#95C2×D412yes—
353#95C2×D412yes—
354C24—21no—
355C24C22×D412yes—
356C2×Q8—22no—
357C2×Q8—21no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—44no—
365C2×Q8—44no—
366C2×Q8—44no—
367C2×Q8H(5,𝔽2)14yes—
368C2×Q82−1+412yes—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—21no—
373C2×D4—21no—
374C2×D4—42no—
375C2×D4—44no—
376C2×D4—82no—
377C2×D4—82no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—82no—
398C22×C4C22×D412yes—
399C22×C4C22×D412yes—
400C22×C4C22×D412yes—
401C22×C4C22×D412yes—
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D412yes—
405C22×C4C2511yescharacteristic
406Q16—216no—
407SD16—48no—
408M16—88no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—24no—
414C2×C8—24no—
415C2×C8—24no—
416C2×C8C2×#4114yes—
417C2×C8C2×#4114yes—
418C4⋊C4—28no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—42no—
430C4⋊C4—48no—
431C4⋊C4—48no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—48no—
435C4⋊C4—44no—
436C4⋊C4—42no—
437C4⋊C4—44no—
438C4⋊C4—44no—
439C4⋊C4—44no—
440C4⋊C4—82no—
441C4⋊C4—82no—
442C22⋊C4—24no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—44no—
446C22⋊C4—48no—
447C22⋊C4—44no—
448C22⋊C4—44no—
449C22⋊C4—48no—
450C22⋊C4—44no—
451C22⋊C4—42no—
452C22⋊C4—44no—
453C22⋊C4—82no—
454C22⋊C4—82no—
455C22⋊C4—82no—
456C42—24no—
457C42—24no—
458C42—22no—
459C42—21no—
460C42—21no—
461C42—42no—
462C42—42no—
463C42—42no—
464C42—48no—
465C42—42no—
466C42—44no—
467C42—82no—
468C42—82no—
469C42C2×#4114yes—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—42no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—42no—
482C23C23×D411yescharacteristic
483Q8—24no—
484Q8—24no—
485Q8—24no—
486Q8—42no—
487Q8—42no—
488Q8—42no—
489Q8—88no—
490D4—44no—
491D4—42no—
492D4—42no—
493D4—164no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—48no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—84no—
535C2×C4—162no—
536C2×C4#53212yes—
537C2×C4#53212yes—
538C2×C4#57514yes—
539C2×C4C22×#4112yes—
540C2×C4C2×H(5,𝔽2)12yes—
541C2×C4C2×2−1+411yescharacteristic
542C2×C4#58511yescharacteristic
543C8—24no—
544C8—24no—
545C8—44no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—82no—
557C22—82no—
558C22C2×#53211yescharacteristic
559C22#348411yescharacteristic
560C22#348911yescharacteristic
561C22C2×#57512yes—
562C22#358612yes—
563C4—22no—
564C4—22no—
565C4—24no—
566C4—42no—
567C4—44no—
568C4—42no—
569C4—42no—
570C4—42no—
571C4—42no—
572C4—81no—
573C4—82no—
574C4—81no—
575C4—82no—
576C4—84no—
577C4—82no—
578C4#349212yes—
579C4#349412yes—
580C2—82no—
581C2—161no—
582C2#3592311yescharacteristic
583C2#6069311yescharacteristic
584C2#6072011yescharacteristic
585C2#6158312yes—
586C2#6211812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #36151C2semidirect
3. #30365C2semidirect
4. #30306C2semidirect
5. #29165C2semidirect
6. #29142C2semidirect
7. #29126C2semidirect
8. #28872C2semidirect
10. #28148C2semidirect
11. #26989C2semidirect
12. #25503C2semidirect
13. #23797C2semidirect
14. #23577C2semidirect
15. #22765C2semidirect
16. #22675C2semidirect
17. #20568C2semidirect

Maximal subgroups

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

subgroupindex
#365392
#361512
#303652
#303062
#291652
#291422
#291262
#288722
#282292
#281482
#269892
#255032
#237972
#235772
#227652
#226752
#205682