Finite Group Databasebeta

← #6500756

Subgroups — #6500756

Order 512.

3194 conjugacy classes of subgroups.

The subgroup lattice has 555 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2408C211yesmaximal, characteristic
2#34186C211yesmaximal, characteristic, complemented
3#27585C212yesmaximal, complemented
4#27582C212yesmaximal, complemented
5#24890C218yesmaximal, complemented
6#24886C218yesmaximal, complemented
7#24009C214yesmaximal, complemented
8#21944C212yesmaximal, complemented
9#21742C211yesmaximal, characteristic, complemented
10#20598C212yesmaximal, complemented
11C24×D4C2211yescharacteristic
12C22×#121C2212yes—
13C22×#120—21no—
14C22×C4×D4C2211yescharacteristic
15C2×#482C2218yescomplemented
16C2×#418C2218yescomplemented
17C2×#407C2212yes—
18#2408—22no—
19#2408—22no—
20#2408C2214yes—
21#2408C2214yes—
22C2×#394—21no—
23C2×#392—21no—
24#2357—22no—
25#2357—22no—
26#2354C2214yescomplemented
27#2354C2214yescomplemented
28#2312—21no—
29#2312—21no—
30#2309—21no—
31C2×#390C2212yes—
32#2285C2211yescharacteristic, complemented
33#2285C2211yescharacteristic, complemented
34#2280C2212yes—
35#2280C2212yescomplemented
36C2×#380C2211yescharacteristic
37C2×#379—21no—
38#2274—21no—
39#2094C2214yescomplemented
40#2093C2214yescomplemented
41#2071C2214yescomplemented
42#2071C2214yescomplemented
43#2034C2218yescomplemented
44#2033C2218yescomplemented
45#1926C2218yescomplemented
46#1925C2218yescomplemented
47#1875C2218yescomplemented
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—24no—
72C2×#121—48no—
73C2×#121C2318yes—
74C2×#120—24no—
75C2×#120—24no—
76C2×#120—48no—
77C2×C4×D4—28no—
78C2×C4×D4D412yes—
79C2×C4×D4D412yes—
80C2×C4×D4C2312yes—
81C2×C4×D4C2312yes—
82C2×#117—21no—
83C2×#117—21no—
84C2×#117C2314yes—
85C2×#117C2314yescomplemented
86C22×C4⋊C4D411yescharacteristic
87C22×C4⋊C4C2312yes—
88C22×C22⋊C4—21no—
89C22×C22⋊C4—22no—
90C22×C22⋊C4—22no—
91C22×C22⋊C4—22no—
92C22×C22⋊C4D411yescharacteristic
93C22×C22⋊C4D411yescharacteristic
94C22×C42C2311yescharacteristic
95#482—416no—
96C2×#107C2318yescomplemented
97C2×#106C2318yescomplemented
98#418—416no—
99C2×#102C2318yes—
100C2×#102C2318yes—
101C2×#102C2318yes—
102C2×#102C2318yes—
103#407—28no—
104C2×C22⋊C8C2314yes—
105C2×C22⋊C8C2314yes—
106C2×C22⋊C8C2318yescomplemented
107#394—24no—
108#394—24no—
109#392—24no—
110#392—24no—
111#390—24no—
112#390—24no—
113#390—22no—
114#390—22no—
115#390C2314yes—
116#390C2314yes—
117#388—24no—
118#388—24no—
119#388—24no—
120#388—24no—
121#384C2312yescomplemented
122#384C2312yescomplemented
123#382—21no—
124#382—21no—
125#381—22no—
126#381—22no—
127#380—21no—
128#380—21no—
129#380—22no—
130#380C2314yes—
131#380C2314yes—
132#380C2312yes—
133#380C2312yes—
134#380C2314yescomplemented
135#379—22no—
136#379—22no—
137#379—24no—
138#379—24no—
139C4×C4⋊C4—24no—
140C4×C4⋊C4—24no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4—21no—
143C4×C4⋊C4C2312yescomplemented
144C4×C4⋊C4C2312yescomplemented
145C4×C22⋊C4—22no—
146C4×C22⋊C4—22no—
147C4×C22⋊C4—22no—
148C4×C22⋊C4—22no—
149C2×#95—21no—
150#340C2318yescomplemented
151#340C2318yescomplemented
152#336C2314yescomplemented
153#336C2314yescomplemented
154C25—22no—
155C25—24no—
156C25—48no—
157C25—44no—
158C25—44no—
159C25—44no—
160C25—84no—
161C25C2×D411yescharacteristic
162C25C2×D411yescharacteristic
163C22×D4—216no—
164C22×D4—24no—
165C22×D4—28no—
166C22×D4—28no—
167C22×D4—24no—
168C22×D4—22no—
169C22×D4—24no—
170C22×D4—28no—
171C22×D4—28no—
172C22×D4—28no—
173C22×D4—44no—
174C22×D4—48no—
175C22×D4—42no—
176C22×D4—464no—
177C22×D4—416no—
178C22×D4—48no—
179C22×D4—48no—
180C22×D4—48no—
181C22×D4—44no—
182C22×D4—48no—
183C22×D4—84no—
184C22×D4C2×D412yes—
185C22×D4C2×D412yes—
186C22×D4C2414yes—
187C23×C4—24no—
188C23×C4—22no—
189C23×C4—44no—
190C23×C4—44no—
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D412yes—
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D412yes—
197C23×C4#4118yes—
198C23×C4C2412yes—
199C23×C4C2411yescharacteristic
200C23×C4C2412yes—
201C2×M16—28no—
202C2×M16—48no—
203C22×C8C2418yes—
204#121—216no—
205#121—48no—
206#121—416no—
207#120—416no—
208#120—416no—
209C4×D4—28no—
210C4×D4—28no—
211C4×D4—24no—
212C4×D4—24no—
213C4×D4—24no—
214C4×D4—24no—
215#117—44no—
216#117—44no—
217#117—48no—
218#117—48no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4C2414yes—
229C2×C4⋊C4C2414yes—
230C2×C4⋊C4C2414yes—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—48no—
247C2×C22⋊C4—48no—
248C2×C22⋊C4—84no—
249C2×C22⋊C4C2×D412yes—
250C2×C22⋊C4C2×D412yes—
251C2×C22⋊C4C2×D412yes—
252C2×C22⋊C4C2×D412yes—
253C2×C42—24no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2411yescharacteristic
261C2×C42C2411yescharacteristic
262#107—48no—
263#106—48no—
264#102—216no—
265#102—216no—
266#102—48no—
267#102—48no—
268C22⋊C8—28no—
269C22⋊C8—28no—
270C22⋊C8—48no—
271#95—24no—
272#95—22no—
273#95—22no—
274#95—24no—
275#95—24no—
276#95—22no—
277#95—22no—
278C24—22no—
279C24—22no—
280C24—28no—
281C24—22no—
282C24—24no—
283C24—22no—
284C24—22no—
285C24—42no—
286C24—42no—
287C24—42no—
288C24—42no—
289C24—416no—
290C24—44no—
291C24—44no—
292C24—44no—
293C24—48no—
294C24—48no—
295C24—432no—
296C24—48no—
297C24—44no—
298C24—44no—
299C24—44no—
300C24—88no—
301C24—816no—
302C24—832no—
303C24—832no—
304C24—816no—
305C24#12314yes—
306C24C22×D412yes—
307C24C22×D412yes—
308C24C22×D411yescharacteristic
309C2×D4—232no—
310C2×D4—28no—
311C2×D4—24no—
312C2×D4—24no—
313C2×D4—24no—
314C2×D4—24no—
315C2×D4—28no—
316C2×D4—28no—
317C2×D4—24no—
318C2×D4—432no—
319C2×D4—432no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—416no—
323C2×D4—416no—
324C2×D4—44no—
325C2×D4—416no—
326C2×D4—464no—
327C2×D4—432no—
328C2×D4—416no—
329C2×D4—416no—
330C2×D4—44no—
331C2×D4—48no—
332C2×D4—88no—
333C2×D4—832no—
334C2×D4H(5,𝔽2)18yes—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—28no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—28no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—416no—
349C22×C4—42no—
350C22×C4—44no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—44no—
354C22×C4—44no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—88no—
358C22×C4#12314yes—
359C22×C4#12318yes—
360C22×C4AGL(1,ℤ/8)14yes—
361C22×C4C22×D412yes—
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D412yes—
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C2×#4114yes—
369C22×C4C2×#4114yes—
370C22×C4C2×#4114yes—
371C22×C4C2511yescharacteristic
372M16—88no—
373C2×C8—28no—
374C2×C8—28no—
375C2×C8—28no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—28no—
379C4⋊C4—28no—
380C4⋊C4—28no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4—82no—
385C4⋊C4—82no—
386C22⋊C4—24no—
387C22⋊C4—24no—
388C22⋊C4—24no—
389C22⋊C4—24no—
390C22⋊C4—416no—
391C22⋊C4—48no—
392C22⋊C4—44no—
393C22⋊C4—44no—
394C22⋊C4—48no—
395C22⋊C4—48no—
396C22⋊C4—84no—
397C22⋊C4—84no—
398C22⋊C4—84no—
399C22⋊C4—84no—
400C22⋊C4—84no—
401C22⋊C4—84no—
402C42—22no—
403C42—22no—
404C42—22no—
405C42—22no—
406C42—44no—
407C42—44no—
408C42—84no—
409C42—84no—
410C23—22no—
411C23—22no—
412C23—24no—
413C23—28no—
414C23—24no—
415C23—22no—
416C23—22no—
417C23—24no—
418C23—22no—
419C23—28no—
420C23—24no—
421C23—22no—
422C23—22no—
423C23—44no—
424C23—44no—
425C23—48no—
426C23—48no—
427C23—432no—
428C23—416no—
429C23—48no—
430C23—416no—
431C23—416no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23—44no—
439C23—82no—
440C23—82no—
441C23—84no—
442C23—88no—
443C23—88no—
444C23—88no—
445C23—816no—
446C23—816no—
447C23—832no—
448C23—832no—
449C23—8128no—
450C23—88no—
451C23C2×#12314yes—
452C23C2×#12312yes—
453C23C2×AGL(1,ℤ/8)12yes—
454C23#57512yes—
455C23C23×D411yescharacteristic
456D4—216no—
457D4—48no—
458D4—48no—
459D4—416no—
460D4—432no—
461D4—48no—
462D4—832no—
463C2×C4—28no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—28no—
471C2×C4—28no—
472C2×C4—28no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—48no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—84no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—88no—
496C2×C4C2×#12312yes—
497C2×C4C2×#12314yes—
498C2×C4C2×AGL(1,ℤ/8)12yes—
499C2×C4C2×AGL(1,ℤ/8)12yes—
500C2×C4C22×#4112yes—
501C2×C4C2×H(5,𝔽2)14yes—
502C8—48no—
503C22—22no—
504C22—22no—
505C22—24no—
506C22—24no—
507C22—24no—
508C22—24no—
509C22—24no—
510C22—416no—
511C22—44no—
512C22—44no—
513C22—42no—
514C22—48no—
515C22—42no—
516C22—42no—
517C22—44no—
518C22—48no—
519C22—44no—
520C22—48no—
521C22—88no—
522C22—832no—
523C22—864no—
524C22—84no—
525C22—84no—
526C22—84no—
527C22—84no—
528C22#309414yes—
529C22C22×#12311yescharacteristic
530C22#348412yes—
531C22C22×AGL(1,ℤ/8)11yescharacteristic
532C22C2×#57511yescharacteristic
533C22#358512yes—
534C4—22no—
535C4—24no—
536C4—22no—
537C4—22no—
538C4—42no—
539C4—41no—
540C4—41no—
541C4—84no—
542C4—84no—
543C4—84no—
544C4#349014yes—
545C2—24no—
546C2—44no—
547C2—82no—
548C2—88no—
549C2#3535011yescharacteristic
550C2C2×#309411yescharacteristic
551C2#6051811yescharacteristic
552C2#6157612yes—
553C2#6158212yes—
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. #34186C2semidirect
3. #27585C2semidirect
4. #27582C2semidirect
5. #24890C2semidirect
6. #24886C2semidirect
7. #24009C2semidirect
8. #21944C2semidirect
9. #21742C2semidirect
10. #20598C2semidirect
15. C2×#482C22semidirect
16. C2×#418C22semidirect
26. #2354C22semidirect
27. #2354C22semidirect
32. #2285C22semidirect
33. #2285C22semidirect
35. #2280C22semidirect
39. #2094C22semidirect
40. #2093C22semidirect
41. #2071C22semidirect
42. #2071C22semidirect
43. #2034C22semidirect
44. #2033C22semidirect
45. #1926C22semidirect
46. #1925C22semidirect
47. #1875C22semidirect
48. #1870C22semidirect
49. #1870C22semidirect
50. #1854C22semidirect
51. #1853C22semidirect
52. #1852C22semidirect
53. #1817C22semidirect
54. #1809C22semidirect
57. #1786C22semidirect
58. #1779C22semidirect
85. C2×#117C23semidirect
96. C2×#107C23semidirect
97. C2×#106C23semidirect
106. C2×C22⋊C8C23semidirect
121. #384C23semidirect
122. #384C23semidirect
134. #380C23semidirect
143. C4×C4⋊C4C23semidirect
144. C4×C4⋊C4C23semidirect
150. #340C23semidirect
151. #340C23semidirect
152. #336C23semidirect
153. #336C23semidirect

Maximal subgroups

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

subgroupindex
C2×#24082
#341862
#275852
#275822
#248902
#248862
#240092
#219442
#217422
#205982