Finite Group Databasebeta

← #6472895

Subgroups — #6472895

Order 512.

1858 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37307C211yesmaximal, characteristic, complemented
2#37297C211yesmaximal, characteristic
3#33055C212yesmaximal, complemented
4#33042C212yesmaximal, complemented
5#30197C212yesmaximal, complemented
6#30194C212yesmaximal, complemented
7#27720C214yesmaximal, complemented
8#27702C214yesmaximal, complemented
9#27279C214yesmaximal, complemented
10#27254C214yesmaximal, complemented
11#21770C212yesmaximal, complemented
12#20975C212yesmaximal, complemented
13#20383C211yesmaximal, characteristic, complemented
14C22×#131C2211yescharacteristic
15C2×#535C2212yes—
16C2×#490C2212yes—
17C2×#486C2212yes—
18C2×#418C2212yescomplemented
19C22×#102C2212yescomplemented
20C2×#408C2211yescharacteristic
21C2×#405C2211yescharacteristic, complemented
22C22×C4×C8C2211yescharacteristic, complemented
23#2857C2212yescomplemented
24#2856C2212yes—
25#2806—22no—
26#2748—22no—
27#2733—22no—
28#2732—22no—
29#2683—22no—
30#2679—22no—
31#2669—22no—
32#2669—22no—
33#2629C2212yescomplemented
34#2619C2212yes—
35#2500C2214yes—
36#2441C2214yescomplemented
37#2440C2214yescomplemented
38#2433C2214yes—
39#2311C2212yes—
40#2297C2211yescharacteristic
41#2101C2214yescomplemented
42#2100C2214yescomplemented
43#2071C2214yescomplemented
44#2071C2214yescomplemented
45#2041C2214yes—
46#2041C2214yes—
47#2025C2214yescomplemented
48#2023C2214yescomplemented
49#1991C2212yescomplemented
50#1990C2212yescomplemented
51#1957C2212yes—
52#1957C2212yes—
53#1936C2214yescomplemented
54#1934C2214yescomplemented
55#1934C2214yescomplemented
56#1934C2214yescomplemented
57#1895C2214yes—
58#1895C2214yescomplemented
59#1895C2214yescomplemented
60#1893C2214yes—
61#1893C2214yescomplemented
62#1893C2214yescomplemented
63#1875C2214yes—
64#1874C2214yes—
65#1836C2212yescomplemented
66#1834C2212yescomplemented
67#1825C2214yescomplemented
68#1786C2214yescomplemented
69#1753C2214yescomplemented
70C8×C22⋊C4C2214yescomplemented
71#1736C2212yescomplemented
72#1735C2212yescomplemented
73#1729C2211yescharacteristic, complemented
74#1728C2211yescharacteristic, complemented
75C22×#41C2311yescharacteristic
76C23×D4C2312yes—
77C2×#131—24no—
78C2×#131—42no—
79C2×#131—42no—
80C22×M16C2312yes—
81C22×M16C2311yescharacteristic
82C22×M16C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2312yes—
85C23×C8C2311yescharacteristic
86#535—28no—
87#535—28no—
88C2×C4⋊Q8C2312yes—
89C2×#127—22no—
90C2×#127C2312yes—
91C2×#127C2312yes—
92C2×#125C2312yes—
93C2×#124—22no—
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8—24no—
96C2×#121—22no—
97C2×#121—22no—
98C2×#121C2314yes—
99C2×#121C2314yes—
100C2×#120—22no—
101C2×#120—22no—
102C2×#120—24no—
103C2×#120—24no—
104C2×#117C2312yes—
105C2×#117C2312yes—
106C2×#117C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C42C2311yescharacteristic
109#490—44no—
110#486—44no—
111#418—44no—
112C2×#102—44no—
113C2×#102C2314yes—
114C2×#102C2314yes—
115C2×#102C2314yes—
116C2×#102C2314yes—
117C2×#102C2314yes—
118C2×#102C2314yes—
119C2×#102C2314yes—
120C2×#102C2314yes—
121#408—24no—
122C2×C22⋊C8C2314yes—
123#405—42no—
124C2×#97C2312yes—
125C2×C4×C8—42no—
126C2×C4×C8C2312yes—
127C2×C4×C8C2312yes—
128C2×C4×C8C2312yes—
129#399—24no—
130#398—24no—
131#398—24no—
132#396—24no—
133#396—24no—
134#395—24no—
135#394—24no—
136#394—24no—
137#394—24no—
138#392—24no—
139#392—24no—
140#392—24no—
141#391—22no—
142#391—22no—
143#391D412yes—
144#390—22no—
145#390—22no—
146#390—22no—
147#390D412yes—
148#390C2314yes—
149#390C2314yes—
150#389—22no—
151#389—24no—
152#389C2314yes—
153#389C2314yes—
154#388—24no—
155#388—24no—
156#387—24no—
157#386—22no—
158#386—22no—
159#386D412yes—
160#386D412yes—
161#385—24no—
162#384C2312yes—
163#384C2312yes—
164#384C2312yes—
165#382C2312yes—
166#380—22no—
167#380C2314yes—
168#380C2314yes—
169C4×C22⋊C4C2314yes—
170#340C2314yes—
171#340C2314yes—
172#340C2314yes—
173#340C2314yes—
174#340C2314yes—
175#340C2314yes—
176#340C2314yes—
177#340C2314yes—
178#336C2314yes—
179#336C2314yes—
180C25—22no—
181C2×#41—41no—
182C2×#41—41no—
183C2×#41—41no—
184C2×#41—41no—
185C2×#41#4114yes—
186C2×#41#4114yes—
187C22×Q8—22no—
188C22×Q8C2×D411yescharacteristic
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—42no—
198C22×D4—42no—
199C22×D4—42no—
200C22×D4—42no—
201C22×D4C2×D411yescharacteristic
202C22×D4C2×D412yes—
203C22×D4C2414yes—
204C23×C4—22no—
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209#131—416no—
210C2×M16—24no—
211C2×M16—22no—
212C2×M16—22no—
213C2×M16—24no—
214C2×M16—41no—
215C2×M16—42no—
216C2×M16—41no—
217C2×M16—41no—
218C2×M16—41no—
219C22×C8—22no—
220C22×C8—22no—
221C22×C8—22no—
222C22×C8—24no—
223C22×C8—22no—
224C22×C8—24no—
225C22×C8—41no—
226C22×C8—42no—
227C22×C8—41no—
228C22×C8C2412yes—
229C22×C8C2412yes—
230C22×C8C2412yes—
231C22×C8C2411yescharacteristic
232C22×C8C2411yescharacteristic
233C4⋊Q8—42no—
234C4⋊Q8—42no—
235#127—42no—
236#127—42no—
237#127—42no—
238#127—42no—
239#127—48no—
240#125—42no—
241#125—42no—
242#124—48no—
243C22⋊Q8—88no—
244C22⋊Q8—84no—
245#121—28no—
246#121—28no—
247#121—28no—
248#121—28no—
249#121—84no—
250#121—84no—
251#120—48no—
252#120—48no—
253#120—88no—
254#120—88no—
255#117—24no—
256#117—24no—
257#117—41no—
258#117—41no—
259#117—42no—
260#117—42no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4C2×D412yes—
274C2×C4⋊C4C2414yes—
275C2×C4⋊C4C2414yes—
276C2×C4⋊C4C2414yes—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4C2×D414yes—
288C2×C22⋊C4C2×D414yes—
289C2×C22⋊C4C2×D412yes—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—41no—
293C2×C42—41no—
294C2×C42C2×D412yes—
295C2×C42C2412yes—
296C2×C42C2412yes—
297#102—28no—
298#102—28no—
299#102—44no—
300#102—44no—
301#102—44no—
302#102—44no—
303#102—44no—
304#102—44no—
305C22⋊C8—28no—
306#97—24no—
307C4×C8—24no—
308C4×C8—42no—
309C4×C8—42no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95C2×D414yes—
316#95C2×D414yes—
317C24—24no—
318C24—44no—
319C24—82no—
320C24—82no—
321C24—84no—
322C24—84no—
323C24C22×D412yes—
324C24C22×D411yescharacteristic
325#41—44no—
326#41—44no—
327#41—44no—
328#41—44no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—44no—
332C2×Q8—82no—
333C2×Q8—82no—
334C2×Q8#12414yes—
335C2×D4—24no—
336C2×D4—24no—
337C2×D4—24no—
338C2×D4—24no—
339C2×D4—24no—
340C2×D4—24no—
341C2×D4—41no—
342C2×D4—42no—
343C2×D4—41no—
344C2×D4—42no—
345C2×D4—48no—
346C2×D4—48no—
347C2×D4—48no—
348C2×D4—48no—
349C2×D4—44no—
350C2×D4—44no—
351C2×D4—44no—
352C2×D4—84no—
353C2×D4—84no—
354C2×D4—82no—
355C2×D4—82no—
356C2×D4—84no—
357C2×D4—84no—
358C2×D4—82no—
359C2×D4—82no—
360C2×D4—82no—
361C2×D4—82no—
362C2×D4—84no—
363C2×D4—84no—
364C2×D4—84no—
365C2×D4—84no—
366C2×D4—84no—
367C2×D4#12414yes—
368C2×D4#12418yes—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—82no—
385C22×C4—84no—
386C22×C4C22×D414yes—
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D412yes—
389C22×C4C22×D412yes—
390C22×C4C22×D412yes—
391C22×C4C2×#4114yes—
392C22×C4C2×#4114yes—
393C22×C4C2×#4112yes—
394C22×C4C2×#4112yes—
395C22×C4C2511yescharacteristic
396M16—48no—
397M16—44no—
398M16—44no—
399C2×C8—28no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—48no—
414C2×C8—44no—
415C2×C8H(5,𝔽2)12yes—
416C2×C8H(5,𝔽2)12yes—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—42no—
424C4⋊C4—48no—
425C4⋊C4—44no—
426C4⋊C4—82no—
427C4⋊C4—82no—
428C4⋊C4—82no—
429C4⋊C4—82no—
430C4⋊C4—84no—
431C4⋊C4—84no—
432C4⋊C4—84no—
433C4⋊C4—82no—
434C22⋊C4—28no—
435C22⋊C4—28no—
436C22⋊C4—42no—
437C22⋊C4—44no—
438C22⋊C4—48no—
439C22⋊C4—48no—
440C22⋊C4—48no—
441C22⋊C4—48no—
442C22⋊C4—84no—
443C22⋊C4—84no—
444C22⋊C4—82no—
445C22⋊C4—84no—
446C22⋊C4—84no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—22no—
451C42—42no—
452C42—44no—
453C42—82no—
454C23—21no—
455C23—22no—
456C23—21no—
457C23—22no—
458C23—24no—
459C23—22no—
460C23—24no—
461C23—22no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—88no—
470C23—168no—
471C23—164no—
472C23—164no—
473C23—168no—
474C23C2×#12414yes—
475C23C2×#12412yes—
476C23C23×D411yescharacteristic
477Q8—44no—
478Q8—164no—
479D4—48no—
480D4—44no—
481D4—44no—
482D4—44no—
483D4—44no—
484D4—44no—
485D4—88no—
486D4—164no—
487D4—164no—
488D4—168no—
489D4—168no—
490D4—168no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—164no—
522C2×C4C2×#12414yes—
523C2×C4C2×#12412yes—
524C2×C4#57614yes—
525C2×C4#57714yes—
526C2×C4C22×#4111yescharacteristic
527C2×C4C22×#4111yescharacteristic
528C2×C4C2×H(5,𝔽2)11yescharacteristic
529C2×C4C2×H(5,𝔽2)11yescharacteristic
530C2×C4#58511yescharacteristic
531C2×C4#58511yescharacteristic
532C8—22no—
533C8—22no—
534C8—42no—
535C8—42no—
536C8—42no—
537C22—24no—
538C22—22no—
539C22—24no—
540C22—28no—
541C22—41no—
542C22—42no—
543C22—41no—
544C22—42no—
545C22—42no—
546C22—44no—
547C22—84no—
548C22—84no—
549C22—84no—
550C22—84no—
551C22—84no—
552C22—84no—
553C22—84no—
554C22—1616no—
555C22C22×#12411yescharacteristic
556C22#349611yescharacteristic
557C22#349611yescharacteristic
558C22C2×#57612yes—
559C22C2×#57712yes—
560C4—24no—
561C4—24no—
562C4—24no—
563C4—22no—
564C4—44no—
565C4—84no—
566C4—84no—
567C4—84no—
568C4#345911yescharacteristic
569C4#345911yescharacteristic
570C4#349011yescharacteristic
571C4#349111yescharacteristic
572C2—42no—
573C2—44no—
574C2—164no—
575C2#3499311yescharacteristic
576C2#6062212yes—
577C2#6226811yescharacteristic
578C2#6227011yescharacteristic
579C2#6234212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#37297C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37307C2semidirect
3. #33055C2semidirect
4. #33042C2semidirect
5. #30197C2semidirect
6. #30194C2semidirect
7. #27720C2semidirect
8. #27702C2semidirect
9. #27279C2semidirect
10. #27254C2semidirect
11. #21770C2semidirect
12. #20975C2semidirect
13. #20383C2semidirect
18. C2×#418C22semidirect
19. C22×#102C22semidirect
21. C2×#405C22semidirect
22. C22×C4×C8C22semidirect
23. #2857C22semidirect
33. #2629C22semidirect
36. #2441C22semidirect
37. #2440C22semidirect
41. #2101C22semidirect
42. #2100C22semidirect
43. #2071C22semidirect
44. #2071C22semidirect
47. #2025C22semidirect
48. #2023C22semidirect
49. #1991C22semidirect
50. #1990C22semidirect
53. #1936C22semidirect
54. #1934C22semidirect
55. #1934C22semidirect
56. #1934C22semidirect
58. #1895C22semidirect
59. #1895C22semidirect
61. #1893C22semidirect
62. #1893C22semidirect
65. #1836C22semidirect
66. #1834C22semidirect
67. #1825C22semidirect
68. #1786C22semidirect
69. #1753C22semidirect
70. C8×C22⋊C4C22semidirect
71. #1736C22semidirect
72. #1735C22semidirect
73. #1729C22semidirect
74. #1728C22semidirect

Maximal subgroups

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

subgroupindex
#373072
#372972
#330552
#330422
#301972
#301942
#277202
#277022
#272792
#272542
#217702
#209752
#203832