Finite Group Databasebeta

← #6508993

Subgroups — #6508993

Order 512.

1858 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2485C211yesmaximal, characteristic
2#34181C211yesmaximal, characteristic
3#27614C212yesmaximal, complemented
4#27613C212yesmaximal, complemented
5#27601C212yesmaximal, complemented
6#27600C212yesmaximal, complemented
7#24951C218yesmaximal, complemented
8#24949C218yesmaximal, complemented
9#21942C212yesmaximal, complemented
10#21742C211yesmaximal, characteristic
11#20702C212yesmaximal, complemented
12C2×#518—21no—
13C22×C4×D4C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C2×#407C2212yes—
16#2485—24no—
17#2485C2218yes—
18C2×#397—21no—
19C2×#394—21no—
20#2349—24no—
21D4×C4⋊C4C2214yes—
22D4×C4⋊C4C2214yes—
23#2316—21no—
24#2315—21no—
25#2315—21no—
26C2×#389C2211yescharacteristic
27C2×#389C2211yescharacteristic
28C2×#388—21no—
29C2×#387—21no—
30#2285C2212yes—
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×#380C2211yescharacteristic
34C2×#380C2211yescharacteristic
35C2×#380C2211yescharacteristic
36#2100C2214yescomplemented
37#2100C2214yescomplemented
38#2084C2214yescomplemented
39#2083C2214yescomplemented
40#2077C2218yescomplemented
41#2075C2218yescomplemented
42#1921C2218yescomplemented
43#1919C2218yescomplemented
44#1870C2218yes—
45#1853C2214yes—
46#1853C2214yescomplemented
47#1853C2214yescomplemented
48#1852C2214yes—
49#1852C2214yescomplemented
50#1852C2214yescomplemented
51#1825C2218yescomplemented
52#1824C2214yes—
53#1823C2214yes—
54#1823C2214yescomplemented
55#1823C2214yescomplemented
56#1817C2212yes—
57#1809C2218yescomplemented
58#1779C2212yes—
59#1736C2214yescomplemented
60#1736C2214yescomplemented
61C23×D4D411yescharacteristic
62C24×C4C2311yescharacteristic
63C24×C4C2311yescharacteristic
64C22×M16C2314yes—
65#518—48no—
66C2×C4×D4—28no—
67C2×C4×D4—24no—
68C2×C4×D4—24no—
69C2×C4×D4D414yes—
70C2×C4×D4C2314yes—
71C2×#117—22no—
72C2×#117—22no—
73C2×#117—28no—
74C2×#117C2314yes—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C22⋊C4—22no—
87C22×C22⋊C4—22no—
88C22×C22⋊C4—21no—
89C22×C22⋊C4—21no—
90C22×C22⋊C4—21no—
91C22×C22⋊C4—21no—
92C22×C22⋊C4D411yescharacteristic
93C22×C22⋊C4D411yescharacteristic
94C22×C42—21no—
95C22×C42C2311yescharacteristic
96C2×#107C2318yes—
97C2×#106C2318yes—
98#407—28no—
99C2×C22⋊C8C2314yes—
100C2×C22⋊C8C2314yes—
101C2×C22⋊C8C2314yes—
102C2×C22⋊C8C2314yes—
103#397—28no—
104#394—28no—
105#389—24no—
106#389—24no—
107#389—22no—
108#389—22no—
109#389C2314yes—
110#389C2314yes—
111#388—28no—
112#387—28no—
113#385—24no—
114#385—24no—
115#385—24no—
116#385—24no—
117#384C2312yes—
118#384C2312yes—
119#382—22no—
120#380—22no—
121#380—22no—
122#380—22no—
123#380—22no—
124#380—22no—
125#380—22no—
126#380—22no—
127#380—22no—
128#380C2314yes—
129#380C2314yes—
130#380C2314yes—
131#380C2314yes—
132#380C2314yes—
133#380C2314yes—
134C4×C4⋊C4—22no—
135C4×C4⋊C4C2314yes—
136C2×#95—22no—
137C2×#95—22no—
138C2×#95—21no—
139C2×#95—21no—
140C2×#95—21no—
141C2×#95—21no—
142C2×#95—21no—
143#340C2318yes—
144#340C2318yes—
145#340C2318yes—
146#340C2318yes—
147#340C2318yes—
148#340C2318yes—
149#336C2314yes—
150#336C2314yes—
151C25C2×D411yescharacteristic
152C25C2×D411yescharacteristic
153C22×D4—24no—
154C22×D4—24no—
155C22×D4—24no—
156C22×D4C2×D414yes—
157C23×C4—24no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D412yes—
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D412yes—
176C23×C4#4118yes—
177C23×C4C2412yes—
178C23×C4C2411yescharacteristic
179C23×C4C2412yes—
180C2×M16—28no—
181C2×M16—44no—
182C2×M16—44no—
183C22×C8C2418yes—
184C4×D4—216no—
185C4×D4—28no—
186C4×D4—28no—
187C4×D4—416no—
188C4×D4—416no—
189#117—28no—
190#117—28no—
191#117—48no—
192#117—48no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—82no—
218C2×C4⋊C4C2×D414yes—
219C2×C4⋊C4C2414yes—
220C2×C4⋊C4C2414yes—
221C2×C4⋊C4C2412yes—
222C2×C4⋊C4C2412yes—
223C2×C4⋊C4C2412yes—
224C2×C4⋊C4C2412yes—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—48no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—48no—
238C2×C22⋊C4—82no—
239C2×C22⋊C4—82no—
240C2×C22⋊C4C2×D414yes—
241C2×C22⋊C4C2×D414yes—
242C2×C42—24no—
243C2×C42—24no—
244C2×C42—24no—
245C2×C42—44no—
246C2×C42C2×D412yes—
247C2×C42C2412yes—
248#107—48no—
249#106—48no—
250C22⋊C8—28no—
251C22⋊C8—28no—
252C22⋊C8—44no—
253C22⋊C8—44no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—24no—
259#95—24no—
260#95—24no—
261#95—24no—
262#95—24no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24—41no—
268C24—41no—
269C24—41no—
270C24—41no—
271C24—41no—
272C24—41no—
273C24—41no—
274C24—41no—
275C24#12314yes—
276C24AGL(1,ℤ/8)14yes—
277C24C22×D412yes—
278C24C22×D412yes—
279C24C22×D411yescharacteristic
280C2×D4—232no—
281C2×D4—28no—
282C2×D4—28no—
283C2×D4—28no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—416no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—88no—
330C22×C4—82no—
331C22×C4—82no—
332C22×C4—82no—
333C22×C4—82no—
334C22×C4—82no—
335C22×C4—82no—
336C22×C4—82no—
337C22×C4—82no—
338C22×C4#12314yes—
339C22×C4#12318yes—
340C22×C4#13714yes—
341C22×C4C22×D412yes—
342C22×C4C22×D412yes—
343C22×C4C22×D412yes—
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D412yes—
346C22×C4C2×#4114yes—
347C22×C4C2×#4114yes—
348C22×C4C2×#4114yes—
349C22×C4C2511yescharacteristic
350M16—88no—
351C2×C8—28no—
352C2×C8—28no—
353C2×C8—28no—
354C4⋊C4—28no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—28no—
359C4⋊C4—24no—
360C4⋊C4—28no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—44no—
364C4⋊C4—44no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—48no—
368C4⋊C4—82no—
369C4⋊C4—82no—
370C4⋊C4—82no—
371C4⋊C4—82no—
372C4⋊C4—82no—
373C4⋊C4—82no—
374C4⋊C4—84no—
375C22⋊C4—28no—
376C22⋊C4—28no—
377C22⋊C4—48no—
378C22⋊C4—48no—
379C22⋊C4—48no—
380C22⋊C4—48no—
381C22⋊C4—48no—
382C22⋊C4—84no—
383C22⋊C4—84no—
384C42—24no—
385C42—24no—
386C42—48no—
387C42—48no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—22no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—28no—
405C23—28no—
406C23—44no—
407C23—44no—
408C23—44no—
409C23—44no—
410C23—82no—
411C23—82no—
412C23—82no—
413C23—82no—
414C23—82no—
415C23—82no—
416C23—82no—
417C23—82no—
418C23C2×#12314yes—
419C23C2×#12312yes—
420C23C2×AGL(1,ℤ/8)14yes—
421C23C2×AGL(1,ℤ/8)12yes—
422C23C2×#13712yes—
423C23C23×D411yescharacteristic
424D4—232no—
425C2×C4—28no—
426C2×C4—22no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—28no—
433C2×C4—28no—
434C2×C4—28no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—48no—
442C2×C4—44no—
443C2×C4—48no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—84no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—84no—
466C2×C4—84no—
467C2×C4—84no—
468C2×C4—84no—
469C2×C4—164no—
470C2×C4—164no—
471C2×C4—164no—
472C2×C4—164no—
473C2×C4C2×#12312yes—
474C2×C4C2×#12314yes—
475C2×C4C2×#13714yes—
476C2×C4C22×#4112yes—
477C2×C4#58512yes—
478C2×C4#58512yes—
479C8—48no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—24no—
487C22—24no—
488C22—24no—
489C22—24no—
490C22—24no—
491C22—24no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22—88no—
501C22—88no—
502C22#309214yes—
503C22C22×#12311yescharacteristic
504C22#348511yescharacteristic
505C22#348511yescharacteristic
506C22C22×AGL(1,ℤ/8)11yescharacteristic
507C22C22×#13711yescharacteristic
508C22#358511yescharacteristic
509C22#358611yescharacteristic
510C4—24no—
511C4—22no—
512C4—24no—
513C4—42no—
514C4—42no—
515C4—84no—
516C4—82no—
517C4—84no—
518C4—82no—
519C4—82no—
520C4—82no—
521C4#349114yes—
522C2—24no—
523C2—44no—
524C2—44no—
525C2#3544811yescharacteristic
526C2C2×#309211yescharacteristic
527C2#6052011yescharacteristic
528C2#6170411yescharacteristic
529C2#6170711yescharacteristic
530C2#6172111yescharacteristic
531C2#6172311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
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
3. #27614C2semidirect
4. #27613C2semidirect
5. #27601C2semidirect
6. #27600C2semidirect
7. #24951C2semidirect
8. #24949C2semidirect
9. #21942C2semidirect
11. #20702C2semidirect
36. #2100C22semidirect
37. #2100C22semidirect
38. #2084C22semidirect
39. #2083C22semidirect
40. #2077C22semidirect
41. #2075C22semidirect
42. #1921C22semidirect
43. #1919C22semidirect
46. #1853C22semidirect
47. #1853C22semidirect
49. #1852C22semidirect
50. #1852C22semidirect
51. #1825C22semidirect
54. #1823C22semidirect
55. #1823C22semidirect
57. #1809C22semidirect
59. #1736C22semidirect
60. #1736C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24852
#341812
#276142
#276132
#276012
#276002
#249512
#249492
#219422
#217422
#207022