Finite Group Databasebeta

← #6500754

Subgroups — #6500754

Order 512.

3208 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
1C2×#2408C211yesmaximal, characteristic
2#34185C211yesmaximal, characteristic, complemented
3#27588C214yesmaximal, complemented
4#24882C218yesmaximal, complemented
5#24878C218yesmaximal, complemented
6#24005C212yesmaximal, complemented
7#24002C212yesmaximal, complemented
8#21944C212yesmaximal, complemented
9#21742C211yesmaximal, characteristic, complemented
10#20598C212yesmaximal, complemented
11C24×D4C2211yescharacteristic
12C22×#121C2212yes—
13C22×#120—21no—
14C22×C4×D4C2211yescharacteristic
15C2×#481C2214yescomplemented
16C2×#480C2214yescomplemented
17C2×#418C2218yescomplemented
18C2×#407C2212yes—
19#2408—22no—
20#2408—22no—
21#2408C2214yes—
22#2408C2214yes—
23C2×#394—21no—
24C2×#392—21no—
25#2353—22no—
26#2353—22no—
27#2352C2214yescomplemented
28#2352C2214yescomplemented
29#2313—21no—
30#2312—21no—
31#2308—21no—
32C2×#390C2212yes—
33#2283C2212yescomplemented
34#2280C2212yes—
35C2×#380C2211yescharacteristic
36C2×#380C2211yescharacteristic, complemented
37C2×#380C2211yescharacteristic, complemented
38C2×#379—21no—
39#2273—21no—
40#2092C2218yescomplemented
41#2073C2218yescomplemented
42#2034C2218yescomplemented
43#2033C2218yescomplemented
44#1926C2218yescomplemented
45#1925C2218yescomplemented
46#1874C2214yescomplemented
47#1874C2214yescomplemented
48#1870C2214yescomplemented
49#1870C2214yescomplemented
50#1854C2218yescomplemented
51#1853C2214yescomplemented
52#1852C2214yescomplemented
53#1817C2212yescomplemented
54#1809C2218yescomplemented
55#1786C2214yes—
56#1786C2214yes—
57#1786C2218yescomplemented
58#1779C2212yescomplemented
59C26—21no—
60C23×D4—22no—
61C23×D4—21no—
62C23×D4—22no—
63C23×D4—28no—
64C23×D4—48no—
65C23×D4D411yescharacteristic
66C23×D4C2314yes—
67C23×D4C2312yes—
68C24×C4C2311yescharacteristic
69C24×C4C2311yescharacteristic
70C22×M16C2314yes—
71C2×#121—22no—
72C2×#121—22no—
73C2×#121—48no—
74C2×#121C2314yes—
75C2×#121C2314yes—
76C2×#120—24no—
77C2×#120—24no—
78C2×#120—48no—
79C2×C4×D4—28no—
80C2×C4×D4D412yes—
81C2×C4×D4D412yes—
82C2×C4×D4C2312yes—
83C2×C4×D4C2312yes—
84C2×#117C2314yes—
85C22×C4⋊C4D411yescharacteristic
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yescomplemented
88C22×C4⋊C4C2312yescomplemented
89C22×C22⋊C4—21no—
90C22×C22⋊C4—21no—
91C22×C22⋊C4—21no—
92C22×C22⋊C4—22no—
93C22×C22⋊C4—22no—
94C22×C22⋊C4—22no—
95C22×C22⋊C4D411yescharacteristic
96C22×C22⋊C4D411yescharacteristic
97C22×C42C2311yescharacteristic
98#481—48no—
99#480—48no—
100C2×#107C2318yescomplemented
101C2×#106C2318yescomplemented
102#418—416no—
103C2×#102C2318yes—
104C2×#102C2318yes—
105C2×#102C2318yes—
106C2×#102C2318yes—
107#407—28no—
108C2×C22⋊C8C2314yes—
109C2×C22⋊C8C2314yes—
110C2×C22⋊C8C2314yescomplemented
111C2×C22⋊C8C2314yescomplemented
112#394—24no—
113#394—24no—
114#392—24no—
115#392—24no—
116#390—24no—
117#390C2318yes—
118#388—24no—
119#388—24no—
120#387—24no—
121#387—24no—
122#386—24no—
123#386—24no—
124#383—22no—
125#382C2314yescomplemented
126#381—22no—
127#381—22no—
128#380—21no—
129#380—21no—
130#380—21no—
131#380—21no—
132#380—22no—
133#380C2314yes—
134#380C2314yes—
135#380C2312yes—
136#380C2312yes—
137#380C2312yescomplemented
138#380C2312yescomplemented
139#379—22no—
140#379—24no—
141#379—24no—
142C4×C4⋊C4C2314yescomplemented
143C4×C22⋊C4—22no—
144C4×C22⋊C4—22no—
145C4×C22⋊C4—22no—
146C4×C22⋊C4—24no—
147C4×C22⋊C4—24no—
148#376—22no—
149C2×#95—21no—
150C2×#95—21no—
151C2×#95—21no—
152#340C2318yescomplemented
153#340C2318yescomplemented
154#336C2318yescomplemented
155C25—22no—
156C25—24no—
157C25—48no—
158C25—44no—
159C25—44no—
160C25—44no—
161C25—84no—
162C25C2×D411yescharacteristic
163C25C2×D411yescharacteristic
164C22×D4—216no—
165C22×D4—24no—
166C22×D4—24no—
167C22×D4—28no—
168C22×D4—24no—
169C22×D4—24no—
170C22×D4—22no—
171C22×D4—24no—
172C22×D4—28no—
173C22×D4—28no—
174C22×D4—28no—
175C22×D4—44no—
176C22×D4—48no—
177C22×D4—42no—
178C22×D4—464no—
179C22×D4—416no—
180C22×D4—48no—
181C22×D4—48no—
182C22×D4—48no—
183C22×D4—44no—
184C22×D4—48no—
185C22×D4—84no—
186C22×D4C2×D412yes—
187C22×D4C2×D412yes—
188C22×D4C2414yes—
189C23×C4—24no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—44no—
194C23×C4—44no—
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D412yes—
202C23×C4#4118yes—
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C23×C4C2412yes—
207C2×M16—28no—
208C2×M16—48no—
209C22×C8C2418yes—
210#121—28no—
211#121—28no—
212#121—44no—
213#121—44no—
214#121—48no—
215#121—48no—
216#120—416no—
217#120—416no—
218C4×D4—28no—
219C4×D4—28no—
220C4×D4—24no—
221C4×D4—24no—
222C4×D4—24no—
223C4×D4—24no—
224#117—48no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4C2414yes—
235C2×C4⋊C4C2414yes—
236C2×C4⋊C4C2414yes—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—48no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—48no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—84no—
256C2×C22⋊C4C2×D412yes—
257C2×C22⋊C4C2×D412yes—
258C2×C22⋊C4C2×D412yes—
259C2×C22⋊C4C2×D412yes—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42C2×D412yes—
264C2×C42C2412yes—
265#107—48no—
266#106—48no—
267#102—216no—
268#102—216no—
269#102—48no—
270#102—48no—
271C22⋊C8—28no—
272C22⋊C8—28no—
273C22⋊C8—44no—
274C22⋊C8—44no—
275#95—22no—
276#95—22no—
277#95—24no—
278#95—22no—
279#95—22no—
280#95—24no—
281#95—24no—
282#95—24no—
283#95—24no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—28no—
289C24—24no—
290C24—42no—
291C24—42no—
292C24—42no—
293C24—42no—
294C24—416no—
295C24—44no—
296C24—44no—
297C24—44no—
298C24—48no—
299C24—48no—
300C24—432no—
301C24—48no—
302C24—44no—
303C24—44no—
304C24—44no—
305C24—88no—
306C24—816no—
307C24—832no—
308C24—832no—
309C24—816no—
310C24#12314yes—
311C24AGL(1,ℤ/8)14yes—
312C24C22×D412yes—
313C24C22×D411yescharacteristic
314C24C22×D411yescharacteristic
315C24C22×D411yescharacteristic
316C2×D4—232no—
317C2×D4—28no—
318C2×D4—24no—
319C2×D4—24no—
320C2×D4—24no—
321C2×D4—24no—
322C2×D4—28no—
323C2×D4—28no—
324C2×D4—24no—
325C2×D4—432no—
326C2×D4—432no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—416no—
330C2×D4—44no—
331C2×D4—416no—
332C2×D4—416no—
333C2×D4—464no—
334C2×D4—432no—
335C2×D4—48no—
336C2×D4—48no—
337C2×D4—48no—
338C2×D4—48no—
339C2×D4—44no—
340C2×D4—48no—
341C2×D4—84no—
342C2×D4—84no—
343C2×D4—832no—
344C2×D4H(5,𝔽2)18yes—
345C22×C4—24no—
346C22×C4—28no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—416no—
362C22×C4—42no—
363C22×C4—44no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—88no—
374C22×C4—82no—
375C22×C4—82no—
376C22×C4—82no—
377C22×C4—82no—
378C22×C4#12314yes—
379C22×C4#12318yes—
380C22×C4C22×D412yes—
381C22×C4C22×D412yes—
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D412yes—
385C22×C4C22×D411yescharacteristic
386C22×C4C2×#4114yes—
387C22×C4C2×#4114yes—
388C22×C4C2×#4114yes—
389C22×C4C2511yescharacteristic
390M16—88no—
391C2×C8—28no—
392C2×C8—28no—
393C2×C8—28no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—28no—
397C4⋊C4—28no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—44no—
401C4⋊C4—82no—
402C4⋊C4—82no—
403C22⋊C4—24no—
404C22⋊C4—24no—
405C22⋊C4—24no—
406C22⋊C4—24no—
407C22⋊C4—48no—
408C22⋊C4—48no—
409C22⋊C4—416no—
410C22⋊C4—44no—
411C22⋊C4—48no—
412C22⋊C4—44no—
413C22⋊C4—44no—
414C22⋊C4—44no—
415C22⋊C4—88no—
416C22⋊C4—84no—
417C22⋊C4—84no—
418C42—24no—
419C42—22no—
420C42—22no—
421C42—84no—
422C42—84no—
423C23—22no—
424C23—22no—
425C23—24no—
426C23—22no—
427C23—22no—
428C23—28no—
429C23—21no—
430C23—21no—
431C23—24no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—21no—
436C23—21no—
437C23—22no—
438C23—28no—
439C23—28no—
440C23—44no—
441C23—44no—
442C23—48no—
443C23—48no—
444C23—48no—
445C23—432no—
446C23—416no—
447C23—416no—
448C23—416no—
449C23—44no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—82no—
456C23—82no—
457C23—84no—
458C23—88no—
459C23—88no—
460C23—88no—
461C23—816no—
462C23—816no—
463C23—832no—
464C23—832no—
465C23—8128no—
466C23—88no—
467C23C2×#12314yes—
468C23C2×#12312yes—
469C23C2×AGL(1,ℤ/8)12yes—
470C23C2×AGL(1,ℤ/8)12yes—
471C23C2×AGL(1,ℤ/8)12yes—
472C23#57512yes—
473C23C23×D411yescharacteristic
474D4—216no—
475D4—416no—
476D4—432no—
477D4—48no—
478D4—48no—
479D4—48no—
480D4—816no—
481D4—816no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—28no—
485C2×C4—28no—
486C2×C4—28no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—48no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—88no—
515C2×C4—164no—
516C2×C4—164no—
517C2×C4C2×#12312yes—
518C2×C4C2×#12314yes—
519C2×C4C22×#4112yes—
520C2×C4C2×H(5,𝔽2)14yes—
521C8—48no—
522C22—22no—
523C22—22no—
524C22—24no—
525C22—24no—
526C22—24no—
527C22—24no—
528C22—22no—
529C22—22no—
530C22—24no—
531C22—24no—
532C22—24no—
533C22—416no—
534C22—41no—
535C22—41no—
536C22—41no—
537C22—41no—
538C22—44no—
539C22—42no—
540C22—44no—
541C22—42no—
542C22—48no—
543C22—48no—
544C22—48no—
545C22—88no—
546C22—832no—
547C22—864no—
548C22—84no—
549C22—84no—
550C22—84no—
551C22—84no—
552C22#309414yes—
553C22C22×#12311yescharacteristic
554C22#348412yes—
555C22C22×AGL(1,ℤ/8)11yescharacteristic
556C22C2×#57511yescharacteristic
557C22#358512yes—
558C4—22no—
559C4—24no—
560C4—41no—
561C4—41no—
562C4—42no—
563C4—42no—
564C4—84no—
565C4—84no—
566C4—84no—
567C4#349014yes—
568C2—24no—
569C2—44no—
570C2—42no—
571C2—42no—
572C2—88no—
573C2#3535011yescharacteristic
574C2C2×#309411yescharacteristic
575C2#6051811yescharacteristic
576C2#6157112yes—
577C2#6158112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2408C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34185C2semidirect
3. #27588C2semidirect
4. #24882C2semidirect
5. #24878C2semidirect
6. #24005C2semidirect
7. #24002C2semidirect
8. #21944C2semidirect
9. #21742C2semidirect
10. #20598C2semidirect
15. C2×#481C22semidirect
16. C2×#480C22semidirect
17. C2×#418C22semidirect
27. #2352C22semidirect
28. #2352C22semidirect
33. #2283C22semidirect
36. C2×#380C22semidirect
37. C2×#380C22semidirect
40. #2092C22semidirect
41. #2073C22semidirect
42. #2034C22semidirect
43. #2033C22semidirect
44. #1926C22semidirect
45. #1925C22semidirect
46. #1874C22semidirect
47. #1874C22semidirect
48. #1870C22semidirect
49. #1870C22semidirect
50. #1854C22semidirect
51. #1853C22semidirect
52. #1852C22semidirect
53. #1817C22semidirect
54. #1809C22semidirect
57. #1786C22semidirect
58. #1779C22semidirect
87. C22×C4⋊C4C23semidirect
88. C22×C4⋊C4C23semidirect
100. C2×#107C23semidirect
101. C2×#106C23semidirect
110. C2×C22⋊C8C23semidirect
111. C2×C22⋊C8C23semidirect
125. #382C23semidirect
137. #380C23semidirect
138. #380C23semidirect
142. C4×C4⋊C4C23semidirect
152. #340C23semidirect
153. #340C23semidirect
154. #336C23semidirect

Maximal subgroups

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

subgroupindex
C2×#24082
#341852
#275882
#248822
#248782
#240052
#240022
#219442
#217422
#205982