Finite Group Databasebeta

← #6633252

Subgroups — #6633252

Order 512.

1884 conjugacy classes of subgroups.

The subgroup lattice has 579 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37298C211yesmaximal, characteristic, complemented
2#35254C211yesmaximal, characteristic, complemented
3#32904C212yesmaximal, complemented
4#31658C214yesmaximal, complemented
5#31648C214yesmaximal, complemented
6#30752C214yesmaximal, complemented
7#30733C214yesmaximal, complemented
8#29712C212yesmaximal, complemented
9#29340C212yesmaximal, complemented
10#25734C212yesmaximal, complemented
11#23941C212yesmaximal, complemented
12#23939C212yesmaximal, complemented
13#21921C211yesmaximal, characteristic, complemented
14C2×#540C2212yes—
15C2×#535C2211yescharacteristic
16C2×C4×#41C2211yescharacteristic
17C2×#496C2212yes—
18C2×#493C2212yes—
19C2×#460C2212yescomplemented
20C2×#459C2212yescomplemented
21C2×#449C2214yes—
22C2×#447C2214yes—
23#2856C2212yes—
24#2854—22no—
25#2682—22no—
26#2682—22no—
27#2675—24no—
28#2624—22no—
29#2619C2212yes—
30#2566C2214yescomplemented
31#2541C2214yescomplemented
32#2448C2214yes—
33#2445C2214yes—
34#2418—24no—
35#2316C2211yescharacteristic
36#2311C2211yescharacteristic
37#2309C2211yescharacteristic
38#2296C2211yescharacteristic
39C2×#390C2211yescharacteristic
40#2101C2214yescomplemented
41#2071C2214yes—
42#2054C2214yes—
43#2054C2214yes—
44#2041C2214yescomplemented
45#2026C2212yescomplemented
46#2025C2214yes—
47#2025C2212yescomplemented
48#2014C2212yescomplemented
49#2012C2214yes—
50#2011C2214yes—
51#2011C2214yes—
52#2011C2214yes—
53#2011C2212yescomplemented
54#1961C2212yes—
55#1958C2212yes—
56#1936C2214yescomplemented
57#1934C2214yescomplemented
58#1917C2214yescomplemented
59#1916C2214yescomplemented
60#1880C2214yescomplemented
61#1879C2214yes—
62#1879C2214yes—
63#1879C2214yescomplemented
64#1879C2214yescomplemented
65#1877C2214yescomplemented
66#1875C2214yes—
67#1874C2214yes—
68#1818C2212yes—
69#1803C2212yescomplemented
70#1802C2212yescomplemented
71#1797C2212yes—
72#1750C2212yes—
73C22×#41C2311yescharacteristic
74C23×D4C2311yescharacteristic
75C22×SD16C2314yes—
76C22×D8C2314yes—
77C22×D8C2314yes—
78C22×D8C2314yes—
79#540—48no—
80#535—42no—
81#535—42no—
82C2×#127C2312yes—
83C2×#127C2312yes—
84C2×#127C2312yes—
85C2×#127C2312yes—
86C2×#125C2312yes—
87C2×#124C2312yes—
88C2×#123—24no—
89C2×#123—24no—
90C2×#121—22no—
91C2×#121—22no—
92C2×#121—22no—
93C2×#121—24no—
94C2×#121—24no—
95C2×#121—24no—
96C2×#121C2314yes—
97C2×#120—22no—
98C4×#41—42no—
99C4×#41—42no—
100C2×C4×Q8C2312yes—
101C2×C4×D4—22no—
102C2×C4×D4C2312yes—
103C2×C4×D4C2314yes—
104C2×#117—21no—
105C2×#117C2311yescharacteristic
106C2×#117C2311yescharacteristic
107C2×#117C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C22⋊C4—22no—
110C22×C42—21no—
111C22×C42C2311yescharacteristic
112#496—44no—
113#493—44no—
114#460—44no—
115#459—44no—
116#449—48no—
117#447—48no—
118C2×C4⋊C8C2312yes—
119C2×C4⋊C8C2312yes—
120C2×#103C2314yes—
121C2×#102C2314yes—
122C2×#102C2314yes—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×#102C2314yes—
128C2×C22⋊C8C2314yes—
129C2×#97C2312yes—
130C2×C4×C8C2312yes—
131#399—24no—
132#397—24no—
133#396—24no—
134#396—24no—
135#394—24no—
136#394—24no—
137#394—24no—
138#392—24no—
139#392—24no—
140#392—24no—
141#391C2312yes—
142#390—22no—
143#390—21no—
144#390—22no—
145#390—22no—
146#390—21no—
147#390C2314yes—
148#390C2314yes—
149#390C2314yes—
150#390C2314yes—
151#390C2312yes—
152#389—22no—
153#389D412yes—
154#389D412yes—
155#389C2314yes—
156#388—24no—
157#388—24no—
158#386—22no—
159#386—24no—
160#385—24no—
161#385—24no—
162#385D414yes—
163#384C2312yes—
164#382C2312yes—
165#380—22no—
166#380—22no—
167#379—22no—
168C4×C4⋊C4C2314yes—
169#340C2314yes—
170#340C2314yes—
171#340C2314yes—
172#340C2314yes—
173#336C2314yes—
174#336C2314yes—
175#336C2312yes—
176#336C2312yes—
177#336C2314yes—
178C25—21no—
179C2×#41—22no—
180C2×#41—22no—
181C2×#41—24no—
182C2×#41—42no—
183C22×Q8C2411yescharacteristic
184C22×D4—21no—
185C22×D4—24no—
186C22×D4—21no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—22no—
190C22×D4—22no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—41no—
194C22×D4—41no—
195C22×D4—42no—
196C22×D4C2414yes—
197C22×D4C2412yes—
198C22×D4C2411yescharacteristic
199C23×C4—21no—
200C23×C4—21no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4C2×D412yes—
204C23×C4C2411yescharacteristic
205C2×SD16—28no—
206C2×SD16—44no—
207C2×SD16—44no—
208C2×D8—28no—
209C2×D8—44no—
210C2×D8—44no—
211C2×D8—44no—
212C2×D8—44no—
213C2×D8—44no—
214C2×D8—44no—
215C2×D8—44no—
216C2×D8—44no—
217C22×C8C2414yes—
218C22×C8C2414yes—
219#127—24no—
220#127—42no—
221#127—44no—
222#127—42no—
223#127—42no—
224#127—42no—
225#127—42no—
226#125—44no—
227#124—24no—
228#124—42no—
229#123—416no—
230#123—88no—
231#121—28no—
232#121—28no—
233#121—416no—
234#121—48no—
235#121—84no—
236#121—84no—
237#121—88no—
238#121—88no—
239#120—48no—
240C4×Q8—24no—
241C4×Q8—42no—
242C4×D4—24no—
243C4×D4—42no—
244C4×D4—44no—
245C4×D4—44no—
246C4×D4—48no—
247#117—24no—
248#117—41no—
249#117—41no—
250#117—41no—
251#117—41no—
252#117—44no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4C2×D412yes—
261C2×C4⋊C4C2×D414yes—
262C2×C4⋊C4C2×D412yes—
263C2×C4⋊C4C2×D414yes—
264C2×C4⋊C4C2412yes—
265C2×C4⋊C4C2412yes—
266C2×C4⋊C4C2414yes—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—84no—
282C2×C22⋊C4C2×D412yes—
283C2×C22⋊C4C2×D412yes—
284C2×C42—21no—
285C2×C42—22no—
286C2×C42—41no—
287C2×C42—41no—
288C2×C42—44no—
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42C2412yes—
292C2×C42C2412yes—
293C2×C42C2412yes—
294C4⋊C8—42no—
295C4⋊C8—42no—
296#103—44no—
297#102—44no—
298#102—44no—
299#102—44no—
300#102—44no—
301#102—44no—
302#102—44no—
303#102—44no—
304C22⋊C8—28no—
305#97—42no—
306C4×C8—42no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95C2×D414yes—
311#95C2×D414yes—
312C24—21no—
313C24—22no—
314C24—24no—
315C24—42no—
316C24—42no—
317C24—42no—
318C24—82no—
319C24—82no—
320C24C22×D411yescharacteristic
321#41—48no—
322#41—48no—
323C2×Q8—21no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8H(5,𝔽2)12yes—
327C2×D4—24no—
328C2×D4—22no—
329C2×D4—21no—
330C2×D4—24no—
331C2×D4—24no—
332C2×D4—24no—
333C2×D4—24no—
334C2×D4—24no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—24no—
339C2×D4—24no—
340C2×D4—24no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—48no—
344C2×D4—48no—
345C2×D4—44no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—48no—
349C2×D4—42no—
350C2×D4—44no—
351C2×D4—84no—
352C2×D4—84no—
353C2×D4—82no—
354C2×D4—84no—
355C2×D4—84no—
356C2×D4—84no—
357C2×D4—82no—
358C2×D4—84no—
359C2×D4—84no—
360C2×D4—82no—
361C2×D4—82no—
362C2×D4—84no—
363C2×D4—84no—
364C2×D4—84no—
365C2×D4—84no—
366C2×D4H(5,𝔽2)12yes—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—21no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—44no—
380C22×C4—48no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—44no—
384C22×C4—82no—
385C22×C4—84no—
386C22×C4—84no—
387C22×C4—82no—
388C22×C4C22×D414yes—
389C22×C4C22×D414yes—
390C22×C4C22×D412yes—
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D412yes—
393C22×C4H(5,𝔽2)14yes—
394C22×C4C2511yescharacteristic
395SD16—88no—
396D8—88no—
397D8—88no—
398D8—88no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C4⋊C4—24no—
406C4⋊C4—28no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—42no—
413C4⋊C4—44no—
414C4⋊C4—44no—
415C4⋊C4—42no—
416C4⋊C4—48no—
417C4⋊C4—82no—
418C4⋊C4—82no—
419C4⋊C4—84no—
420C22⋊C4—24no—
421C22⋊C4—42no—
422C22⋊C4—48no—
423C22⋊C4—44no—
424C22⋊C4—48no—
425C22⋊C4—48no—
426C22⋊C4—42no—
427C22⋊C4—48no—
428C22⋊C4—48no—
429C22⋊C4—48no—
430C22⋊C4—84no—
431C22⋊C4—84no—
432C22⋊C4—84no—
433C22⋊C4—84no—
434C22⋊C4—84no—
435C22⋊C4—84no—
436C42—22no—
437C42—22no—
438C42—22no—
439C42—22no—
440C42—24no—
441C42—41no—
442C42—41no—
443C42—44no—
444C42—42no—
445C23—22no—
446C23—21no—
447C23—21no—
448C23—22no—
449C23—22no—
450C23—42no—
451C23—41no—
452C23—41no—
453C23—42no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—42no—
459C23—42no—
460C23—44no—
461C23—42no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—84no—
467C23—88no—
468C23—164no—
469C23—162no—
470C23—162no—
471C23#53512yes—
472C23C23×D411yescharacteristic
473Q8—24no—
474Q8—42no—
475D4—24no—
476D4—48no—
477D4—42no—
478D4—44no—
479D4—44no—
480D4—44no—
481D4—44no—
482D4—44no—
483D4—44no—
484D4—84no—
485D4—164no—
486D4—168no—
487D4—168no—
488D4—164no—
489D4—168no—
490D4—168no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—48no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—84no—
519C2×C4—88no—
520C2×C4—168no—
521C2×C4—162no—
522C2×C4#53512yes—
523C2×C4#57614yes—
524C2×C4C2×H(5,𝔽2)12yes—
525C2×C4C2×H(5,𝔽2)11yescharacteristic
526C2×C4C2×H(5,𝔽2)11yescharacteristic
527C2×C4#58512yes—
528C8—44no—
529C8—44no—
530C22—22no—
531C22—24no—
532C22—42no—
533C22—41no—
534C22—41no—
535C22—42no—
536C22—44no—
537C22—82no—
538C22—81no—
539C22—81no—
540C22—82no—
541C22—84no—
542C22—84no—
543C22—84no—
544C22—84no—
545C22—84no—
546C22—84no—
547C22—84no—
548C22—84no—
549C22—82no—
550C22—84no—
551C22—82no—
552C22—168no—
553C22C2×#53511yescharacteristic
554C22#348411yescharacteristic
555C22#348711yescharacteristic
556C22C2×#57612yes—
557C22#358512yes—
558C4—22no—
559C4—24no—
560C4—42no—
561C4—44no—
562C4—42no—
563C4—44no—
564C4—84no—
565C4—82no—
566C4—82no—
567C4#353412yes—
568C4#353412yes—
569C2—42no—
570C2—82no—
571C2—164no—
572C2—162no—
573C2#3671011yescharacteristic
574C2#6083212yes—
575C2#6161712yes—
576C2#6193211yescharacteristic
577C2#6194211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37298C2semidirect
2. #35254C2semidirect
3. #32904C2semidirect
4. #31658C2semidirect
5. #31648C2semidirect
6. #30752C2semidirect
7. #30733C2semidirect
8. #29712C2semidirect
9. #29340C2semidirect
10. #25734C2semidirect
11. #23941C2semidirect
12. #23939C2semidirect
13. #21921C2semidirect
19. C2×#460C22semidirect
20. C2×#459C22semidirect
30. #2566C22semidirect
31. #2541C22semidirect
40. #2101C22semidirect
44. #2041C22semidirect
45. #2026C22semidirect
47. #2025C22semidirect
48. #2014C22semidirect
53. #2011C22semidirect
56. #1936C22semidirect
57. #1934C22semidirect
58. #1917C22semidirect
59. #1916C22semidirect
60. #1880C22semidirect
63. #1879C22semidirect
64. #1879C22semidirect
65. #1877C22semidirect
69. #1803C22semidirect
70. #1802C22semidirect

Maximal subgroups

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

subgroupindex
#372982
#352542
#329042
#316582
#316482
#307522
#307332
#297122
#293402
#257342
#239412
#239392
#219212