Finite Group Databasebeta

← #6495668

Subgroups — #6495668

Order 512.

1912 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35293C211yesmaximal, characteristic, complemented
2#34255C211yesmaximal, characteristic, complemented
3#33116C212yesmaximal, complemented
4#32911C212yesmaximal, complemented
5#29982C212yesmaximal, complemented
6#29794C212yesmaximal, complemented
7#29558C212yesmaximal, complemented
8#29552C212yesmaximal, complemented
9#29376C214yesmaximal, complemented
10#29343C212yesmaximal, complemented
11#28319C212yesmaximal, complemented
12#24062C212yesmaximal, complemented
13#23930C212yesmaximal, complemented
14#22158C214yesmaximal, complemented
15#20780C211yesmaximal, characteristic, complemented
16C2×#532C2212yes—
17C2×#518C2211yescharacteristic
18C2×#482C2214yes—
19C2×#459C2214yes—
20C2×#447C2214yes—
21#2867C2211yescharacteristic, complemented
22#2862C2211yescharacteristic, complemented
23#2858C2212yescomplemented
24#2619C2212yes—
25#2619C2212yes—
26#2544—22no—
27#2544—22no—
28#2431—22no—
29#2431—22no—
30#2357—22no—
31#2354—22no—
32#2342—22no—
33#2342—22no—
34#2328C2212yescomplemented
35#2328C2212yescomplemented
36#2319C2212yes—
37#2319C2212yes—
38#2317C2211yescharacteristic
39#2309C2211yescharacteristic
40#2296C2211yescharacteristic
41#2293C2212yes—
42#2293C2212yes—
43C2×#390C2211yescharacteristic
44#2274C2211yescharacteristic
45C4×#117C2211yescharacteristic
46#2101C2214yescomplemented
47#2092C2214yes—
48#2058C2212yescomplemented
49#2057C2212yescomplemented
50#2041C2212yescomplemented
51#2041C2212yescomplemented
52#2041C2214yescomplemented
53#2023C2212yes—
54#2023C2212yes—
55#2011C2212yes—
56#2011C2212yes—
57#2011C2214yes—
58#1982C2214yescomplemented
59#1972C2214yes—
60#1970C2214yes—
61#1949C2214yescomplemented
62#1936C2214yescomplemented
63#1934C2214yescomplemented
64#1926C2214yescomplemented
65#1925C2214yescomplemented
66#1879C2214yes—
67#1879C2214yes—
68#1879C2214yes—
69#1879C2214yes—
70#1875C2214yes—
71#1874C2214yes—
72#1864C2214yescomplemented
73#1862C2214yescomplemented
74#1840C2212yescomplemented
75#1831C2212yescomplemented
76#1817C2212yes—
77#1804C2214yes—
78#1798C2212yes—
79#1768C2212yescomplemented
80#1750C2212yes—
81#1745C2211yescharacteristic, complemented
82#1744C2211yescharacteristic, complemented
83C22×#41—22no—
84C23×D4C2311yescharacteristic
85C22×D8C2314yes—
86C22×D8C2314yes—
87#532—48no—
88C2×C4⋊Q8C2312yes—
89C2×#127—22no—
90C2×#127—22no—
91C2×#127C2312yes—
92C2×#127C2312yes—
93C2×#127C2312yes—
94C2×#124—22no—
95C2×#124—22no—
96C2×#124—22no—
97C2×#121—22no—
98C2×#121C2314yes—
99#518—28no—
100C2×C4×D4—22no—
101C2×C4×D4—22no—
102C2×C4×D4C2312yes—
103C2×C4×D4C2312yes—
104C2×C4×D4C2314yes—
105C2×#117—22no—
106C2×#117—22no—
107C2×#117C2311yescharacteristic
108C2×#117C2311yescharacteristic
109C2×#117C2311yescharacteristic
110C2×#117C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C22⋊C4—22no—
113C22×C22⋊C4—22no—
114C22×C42C2311yescharacteristic
115#482—48no—
116#459—216no—
117#447—48no—
118C2×#107C2314yes—
119C2×#106C2314yes—
120C2×C4⋊C8C2314yes—
121C2×#102C2314yes—
122C2×#102C2314yes—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×C22⋊C8C2314yes—
130C2×#97C2312yes—
131C2×C4×C8C2312yes—
132#394—24no—
133#394—22no—
134#394—22no—
135#392—24no—
136#392—22no—
137#392—22no—
138#391—22no—
139#391—22no—
140#390—21no—
141#390—21no—
142#390—22no—
143#390—22no—
144#390—22no—
145#390C2312yes—
146#390C2312yes—
147#390C2312yes—
148#390C2314yes—
149#390C2314yes—
150#390C2312yes—
151#389C2312yes—
152#389C2312yes—
153#389C2312yes—
154#389C2312yes—
155#388—22no—
156#388—22no—
157#388—22no—
158#388—22no—
159#388—22no—
160#388—22no—
161#388—22no—
162#388—22no—
163#386—24no—
164#386—24no—
165#386—24no—
166#386—24no—
167#385—22no—
168#385—21no—
169#385—21no—
170#384—21no—
171#382—21no—
172#382C2312yes—
173#382C2312yes—
174#381—22no—
175#381—22no—
176#380—22no—
177#380—22no—
178#379—21no—
179#379—21no—
180C4×C4⋊C4—22no—
181C4×C4⋊C4—22no—
182C4×C4⋊C4D412yes—
183C4×C4⋊C4D412yes—
184C4×C4⋊C4C2312yes—
185C4×C4⋊C4C2312yes—
186C4×C22⋊C4—22no—
187C4×C22⋊C4—22no—
188C4×C22⋊C4—22no—
189C4×C22⋊C4D412yes—
190C4×C22⋊C4D412yes—
191#376C2311yescharacteristic
192#376C2311yescharacteristic
193C43C2311yescharacteristic
194C43C2311yescharacteristic
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2312yes—
200#336C2312yes—
201#336C2312yes—
202#336C2312yes—
203#336C2314yes—
204#336C2314yes—
205C25—21no—
206C2×#41—48no—
207C2×#41—84no—
208C2×#41—84no—
209C22×Q8—22no—
210C22×D4—24no—
211C22×D4—24no—
212C22×D4—21no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—24no—
218C22×D4—24no—
219C22×D4—42no—
220C22×D4C2414yes—
221C22×D4C2412yes—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—21no—
225C23×C4—24no—
226C23×C4C2×D412yes—
227C23×C4C2411yescharacteristic
228C2×D8—28no—
229C2×D8—28no—
230C2×D8—44no—
231C2×D8—44no—
232C2×D8—44no—
233C2×D8—44no—
234C22×C8C2414yes—
235C22×C8C2414yes—
236C4⋊Q8—44no—
237#127—28no—
238#127—42no—
239#127—42no—
240#127—44no—
241#127—84no—
242#127—84no—
243#124—48no—
244#124—84no—
245#124—84no—
246#121—28no—
247#121—28no—
248#121—48no—
249C4×D4—216no—
250C4×D4—24no—
251C4×D4—24no—
252C4×D4—24no—
253C4×D4—44no—
254C4×D4—44no—
255C4×D4—48no—
256C4×D4#4118yescomplemented
257#117—24no—
258#117—24no—
259#117—24no—
260#117—42no—
261#117—84no—
262#117—84no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4C2×D411yescharacteristic
272C2×C4⋊C4C2×D411yescharacteristic
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2414yes—
276C2×C4⋊C4C2412yes—
277C2×C22⋊C4—216no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—84no—
297C2×C22⋊C4C2×D412yes—
298C2×C22⋊C4C2×D411yescharacteristic
299C2×C22⋊C4C2×D411yescharacteristic
300C2×C42—21no—
301C2×C42—24no—
302C2×C42—21no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42C2×D412yes—
306C2×C42C2×D412yes—
307C2×C42C2×D412yes—
308C2×C42C2×D412yes—
309C2×C42C2×D412yes—
310C2×C42C2411yescharacteristic
311C2×C42C2411yescharacteristic
312C2×C42C2411yescharacteristic
313C2×C42C2411yescharacteristic
314C2×C42C2411yescharacteristic
315C2×C42C2411yescharacteristic
316#107—44no—
317#106—44no—
318C4⋊C8—28no—
319#102—28no—
320#102—28no—
321#102—44no—
322#102—44no—
323#102—44no—
324#102—44no—
325#102—44no—
326#102—44no—
327C22⋊C8—28no—
328#97—42no—
329C4×C8—42no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—24no—
339#95C2×D412yes—
340#95C2×D412yes—
341#95C2×D412yes—
342#95C2×D412yes—
343C24—24no—
344C24—22no—
345C24—24no—
346C24—42no—
347C24—42no—
348C24—82no—
349C24—82no—
350C24C22×D411yescharacteristic
351#41—832no—
352C2×Q8—44no—
353C2×Q8—44no—
354C2×Q8—44no—
355C2×D4—24no—
356C2×D4—22no—
357C2×D4—24no—
358C2×D4—24no—
359C2×D4—24no—
360C2×D4—24no—
361C2×D4—24no—
362C2×D4—24no—
363C2×D4—24no—
364C2×D4—24no—
365C2×D4—416no—
366C2×D4—42no—
367C2×D4—44no—
368C2×D4—44no—
369C2×D4—44no—
370C2×D4—44no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4—48no—
374C2×D4—48no—
375C2×D4—42no—
376C2×D4—44no—
377C2×D4—44no—
378C2×D4—84no—
379C2×D4—84no—
380C2×D4—84no—
381C2×D4—84no—
382C2×D4—84no—
383C2×D4—84no—
384C2×D4C2×#4114yes—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—28no—
390C22×C4—24no—
391C22×C4—28no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—42no—
396C22×C4—48no—
397C22×C4—42no—
398C22×C4—82no—
399C22×C4—82no—
400C22×C4—82no—
401C22×C4—82no—
402C22×C4—84no—
403C22×C4—84no—
404C22×C4—82no—
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D412yes—
409C22×C4C22×D412yes—
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D412yes—
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D412yes—
414C22×C4H(5,𝔽2)14yes—
415C22×C4C2511yescharacteristic
416D8—88no—
417D8—88no—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C4⋊C4—24no—
425C4⋊C4—22no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—42no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—82no—
436C4⋊C4—82no—
437C4⋊C4—82no—
438C4⋊C4C2×#4114yes—
439C22⋊C4—24no—
440C22⋊C4—28no—
441C22⋊C4—24no—
442C22⋊C4—28no—
443C22⋊C4—41no—
444C22⋊C4—41no—
445C22⋊C4—48no—
446C22⋊C4—42no—
447C22⋊C4—48no—
448C22⋊C4—44no—
449C22⋊C4—44no—
450C22⋊C4—82no—
451C22⋊C4—82no—
452C22⋊C4—82no—
453C22⋊C4—82no—
454C22⋊C4—82no—
455C22⋊C4—82no—
456C22⋊C4—84no—
457C22⋊C4—84no—
458C22⋊C4—84no—
459C22⋊C4—84no—
460C42—24no—
461C42—24no—
462C42—24no—
463C42—24no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—42no—
470C42—42no—
471C42—42no—
472C42—82no—
473C42—82no—
474C42C2×#4114yes—
475C23—22no—
476C23—21no—
477C23—21no—
478C23—22no—
479C23—24no—
480C23—48no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—88no—
495C23—164no—
496C23—162no—
497C23—162no—
498C23#53512yes—
499C23C23×D411yescharacteristic
500Q8—88no—
501D4—28no—
502D4—48no—
503D4—44no—
504D4—44no—
505D4—44no—
506D4—44no—
507D4—44no—
508D4—88no—
509D4—88no—
510D4—816no—
511D4—168no—
512D4—164no—
513D4—164no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—28no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—48no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—88no—
549C2×C4—88no—
550C2×C4—816no—
551C2×C4—162no—
552C2×C4#52514yes—
553C2×C4#53512yes—
554C2×C4#57614yes—
555C2×C4C22×#4112yes—
556C2×C4C2×H(5,𝔽2)12yes—
557C2×C4#58512yes—
558C8—44no—
559C8—44no—
560C22—24no—
561C22—22no—
562C22—41no—
563C22—41no—
564C22—42no—
565C22—44no—
566C22—44no—
567C22—44no—
568C22—82no—
569C22—82no—
570C22—82no—
571C22—82no—
572C22—84no—
573C22—84no—
574C22—84no—
575C22—84no—
576C22—84no—
577C22—84no—
578C22—84no—
579C22—168no—
580C22C2×#52511yescharacteristic
581C22C2×#53511yescharacteristic
582C22#344911yescharacteristic
583C22C2×#57612yes—
584C22#358512yes—
585C4—24no—
586C4—24no—
587C4—22no—
588C4—41no—
589C4—41no—
590C4—41no—
591C4—41no—
592C4—42no—
593C4—42no—
594C4—44no—
595C4—82no—
596C4—82no—
597C4—84no—
598C4#349414yes—
599C2—42no—
600C2—84no—
601C2—164no—
602C2#3530111yescharacteristic
603C2#6059012yes—
604C2#6081112yes—
605C2#6101111yescharacteristic
606C2#6101711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
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
1. #35293C2semidirect
2. #34255C2semidirect
3. #33116C2semidirect
4. #32911C2semidirect
5. #29982C2semidirect
6. #29794C2semidirect
7. #29558C2semidirect
8. #29552C2semidirect
9. #29376C2semidirect
10. #29343C2semidirect
11. #28319C2semidirect
12. #24062C2semidirect
13. #23930C2semidirect
14. #22158C2semidirect
15. #20780C2semidirect
21. #2867C22semidirect
22. #2862C22semidirect
23. #2858C22semidirect
34. #2328C22semidirect
35. #2328C22semidirect
46. #2101C22semidirect
48. #2058C22semidirect
49. #2057C22semidirect
50. #2041C22semidirect
51. #2041C22semidirect
52. #2041C22semidirect
58. #1982C22semidirect
61. #1949C22semidirect
62. #1936C22semidirect
63. #1934C22semidirect
64. #1926C22semidirect
65. #1925C22semidirect
72. #1864C22semidirect
73. #1862C22semidirect
74. #1840C22semidirect
75. #1831C22semidirect
79. #1768C22semidirect
81. #1745C22semidirect
82. #1744C22semidirect
256. C4×D4#41semidirect

Maximal subgroups

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

subgroupindex
#352932
#342552
#331162
#329112
#299822
#297942
#295582
#295522
#293762
#293432
#283192
#240622
#239302
#221582
#207802