Finite Group Databasebeta

← #6481934

Subgroups — #6481934

Order 512.

1746 conjugacy classes of subgroups.

The subgroup lattice has 525 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35181C211yesmaximal, characteristic, complemented
2C2×#2354C211yesmaximal, characteristic
3#33151C214yesmaximal, complemented
4#29956C214yesmaximal
5#26091C214yesmaximal, complemented
6#26087C214yesmaximal, complemented
7#26025C214yesmaximal, complemented
8#26021C214yesmaximal, complemented
9#21982C211yesmaximal, characteristic, complemented
10#21981C211yesmaximal, characteristic, complemented
11#21951C211yesmaximal, characteristic, complemented
12#20732C211yesmaximal, characteristic
13#20732C211yesmaximal, characteristic
14C22×C4×D4C2211yescharacteristic
15C22×#117—21no—
16C2×#431C2211yescharacteristic
17C2×#431C2211yescharacteristic
18#2851C2214yes—
19#2646—22no—
20#2571C2212yescomplemented
21#2571C2212yescomplemented
22#2570—22no—
23#2570—22no—
24#2552C2212yescomplemented
25#2552C2212yescomplemented
26#2354—22no—
27#2354—22no—
28#2354C2214yes—
29#2354C2214yes—
30#2317—21no—
31#2317—21no—
32C2×#390C2212yes—
33C2×#388—22no—
34#2285C2211yescharacteristic
35#2285C2211yescharacteristic
36C2×#384C2211yescharacteristic
37C2×#380—21no—
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C4⋊C4C2211yescharacteristic
40#2092C2218yes—
41#2067C2218yes—
42#2065C2218yes—
43#2025C2218yes—
44#1989C2218yescomplemented
45#1949C2218yescomplemented
46#1948C2214yescomplemented
47#1947C2214yescomplemented
48#1939C2214yescomplemented
49#1939C2214yescomplemented
50#1921C2218yescomplemented
51#1919C2218yescomplemented
52#1853C2214yes—
53#1852C2214yes—
54#1845C2212yescomplemented
55#1844C2212yescomplemented
56#1834C2212yescomplemented
57#1834C2212yescomplemented
58#1823C2212yes—
59#1823C2212yes—
60#1802C2212yes—
61#1802C2212yes—
62#1802C2212yes—
63#1802C2212yes—
64#1797C2214yes—
65#1786C2212yes—
66#1786C2212yes—
67C4×#102C2214yes—
68C4×#102C2214yes—
69C23×D4C2311yescharacteristic
70C24×C4—21no—
71C2×#125C2314yes—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—24no—
75C2×C4×D4Q818yes—
76C2×C4×D4C2312yes—
77C2×C4×D4C2312yes—
78C2×#117—21no—
79C2×#117—21no—
80C2×#117—24no—
81C2×#117—24no—
82C2×#117—48no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4D411yescharacteristic
86C22×C4⋊C4D411yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2312yes—
89C22×C22⋊C4—21no—
90C22×C22⋊C4—22no—
91C22×C22⋊C4—22no—
92C22×C42—22no—
93C22×C42C2312yes—
94C22×C42C2311yescharacteristic
95#431—24no—
96#431—24no—
97C2×#107C2318yes—
98C2×#106C2318yes—
99C2×C4⋊C8C2312yes—
100C2×C4⋊C8C2312yes—
101C2×C4⋊C8C2318yes—
102C2×#102C2314yes—
103C2×#102C2314yes—
104C2×#102C2314yes—
105C2×#102C2314yes—
106C2×C22⋊C8C2312yes—
107C2×C22⋊C8C2312yes—
108C2×C4×C8C2312yes—
109C2×C4×C8C2312yes—
110#400—24no—
111#400—24no—
112#398—24no—
113#398—24no—
114#390—24no—
115#390C2318yes—
116#389—22no—
117#389—22no—
118#389—22no—
119#389—22no—
120#389C2314yes—
121#389C2314yes—
122#388—28no—
123#388—28no—
124#387—24no—
125#387—24no—
126#385—24no—
127#385—24no—
128#384—22no—
129#384—22no—
130#384—21no—
131#384—21no—
132#384C2311yescharacteristic
133#384C2311yescharacteristic
134#384C2311yescharacteristic
135#384C2311yescharacteristic
136#384C2312yes—
137#384C2312yes—
138#382—22no—
139#382—22no—
140#382—21no—
141#382—21no—
142#382C2314yes—
143#380—21no—
144#380—21no—
145#380—24no—
146#380—24no—
147#380C2312yes—
148#380C2312yes—
149C4×C4⋊C4—21no—
150C4×C4⋊C4—21no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4D412yes—
153C4×C4⋊C4D412yes—
154C4×C4⋊C4C2312yes—
155C4×C4⋊C4C2312yes—
156C4×C4⋊C4C2318yes—
157C4×C4⋊C4C2312yes—
158C4×C4⋊C4C2312yes—
159C2×#95—22no—
160C2×#95D412yes—
161#340C2314yes—
162#340C2314yes—
163#340C2314yes—
164#340C2314yes—
165#340C2318yes—
166#340C2318yes—
167#336C2318yes—
168C25—21no—
169C22×D4—21no—
170C22×D4—22no—
171C22×D4—42no—
172C22×D4—42no—
173C22×D4C2×Q814yes—
174C22×D4C2412yes—
175C23×C4—21no—
176C23×C4—28no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—44no—
181C23×C4—44no—
182C23×C4C2×D412yes—
183C23×C4C2×D412yes—
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2411yescharacteristic
187C22×C8C2414yes—
188C22×C8C2414yes—
189#125—48no—
190C4×D4—28no—
191C4×D4—28no—
192C4×D4—24no—
193C4×D4—24no—
194C4×D4C2×Q818yes—
195C4×D4C2×Q818yes—
196#117—416no—
197#117—416no—
198#117—44no—
199#117—44no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—82no—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4C2×Q814yes—
229C2×C4⋊C4C2414yes—
230C2×C4⋊C4C2412yes—
231C2×C4⋊C4C2412yes—
232C2×C4⋊C4C2412yes—
233C2×C4⋊C4C2412yes—
234C2×C4⋊C4C2412yes—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—48no—
243C2×C22⋊C4—84no—
244C2×C42—28no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—44no—
248C2×C42—48no—
249C2×C42C2×D414yes—
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42C2×Q814yes—
253C2×C42#4114yes—
254C2×C42C2414yes—
255C2×C42C2411yescharacteristic
256C2×C42C2411yescharacteristic
257#107—48no—
258#106—48no—
259C4⋊C8—24no—
260C4⋊C8—24no—
261C4⋊C8—48no—
262#102—44no—
263#102—44no—
264#102—44no—
265#102—44no—
266C22⋊C8—24no—
267C22⋊C8—24no—
268C4×C8—24no—
269C4×C8—24no—
270#95—28no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—24no—
276#95—24no—
277#95—24no—
278#95—24no—
279#95C2×D414yes—
280#95C2×D414yes—
281C24—24no—
282C24—22no—
283C24—42no—
284C24—42no—
285C24—42no—
286C24—42no—
287C24—42no—
288C24—42no—
289C24C22×D411yescharacteristic
290C2×D4—24no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—48no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—48no—
300C2×D4C22×Q814yes—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—28no—
308C22×C4—28no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—432no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4C22⋊Q814yes—
331C22×C4C22⋊Q814yes—
332C22×C4#13514yes—
333C22×C4AGL(1,ℤ/8)14yes—
334C22×C4C22×D414yes—
335C22×C4C22×D414yes—
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C22×Q812yes—
342C22×C4C2×#4112yes—
343C22×C4C2511yescharacteristic
344C2×C8—24no—
345C2×C8—24no—
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—24no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—22no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—28no—
361C4⋊C4—44no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—44no—
365C4⋊C4—44no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—81no—
369C4⋊C4—81no—
370C4⋊C4—81no—
371C4⋊C4—81no—
372C4⋊C4—84no—
373C4⋊C4—84no—
374C4⋊C4—82no—
375C4⋊C4—82no—
376C4⋊C4—84no—
377C4⋊C4—84no—
378C4⋊C4C22×Q814yes—
379C22⋊C4—28no—
380C22⋊C4—28no—
381C22⋊C4—416no—
382C22⋊C4—44no—
383C22⋊C4—44no—
384C22⋊C4—88no—
385C42—28no—
386C42—44no—
387C42—416no—
388C42—44no—
389C42—44no—
390C42C22⋊Q814yes—
391C42C22⋊Q814yes—
392C42C22×Q812yes—
393C42C22×Q812yes—
394C42C2×#4112yes—
395C42C2×#4112yes—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—24no—
401C23—48no—
402C23—48no—
403C23—48no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—84no—
411C23—84no—
412C23—84no—
413C23—84no—
414C23C2×C22⋊Q812yes—
415C23C2×#13512yes—
416C23C2×AGL(1,ℤ/8)12yes—
417C23C23×D411yescharacteristic
418D4—24no—
419D4—24no—
420D4—48no—
421D4—42no—
422D4—42no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—44no—
439C2×C4—48no—
440C2×C4—48no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—416no—
444C2×C4—416no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—84no—
461C2×C4—88no—
462C2×C4#478116yes—
463C2×C4C2×C22⋊Q812yes—
464C2×C4C2×C22⋊Q812yes—
465C2×C4C2×C22⋊Q812yes—
466C2×C4C2×C22⋊Q812yes—
467C2×C4C2×C22⋊Q812yes—
468C2×C4C2×#13512yes—
469C2×C4C2×#13512yes—
470C2×C4C2×AGL(1,ℤ/8)12yes—
471C2×C4C2×AGL(1,ℤ/8)12yes—
472C2×C4C23×Q811yescharacteristic
473C2×C4C22×#4111yescharacteristic
474C2×C4#58512yes—
475C2×C4#58512yes—
476C8—44no—
477C8—44no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—44no—
485C22—44no—
486C22—44no—
487C22—816no—
488C22—82no—
489C22—82no—
490C22—82no—
491C22—82no—
492C22C2×#47814yes—
493C22C22×C22⋊Q811yescharacteristic
494C22#348511yescharacteristic
495C22#348711yescharacteristic
496C22C22×#13511yescharacteristic
497C22C22×AGL(1,ℤ/8)11yescharacteristic
498C22#358511yescharacteristic
499C22#358611yescharacteristic
500C4—22no—
501C4—22no—
502C4—22no—
503C4—44no—
504C4—44no—
505C4—44no—
506C4—82no—
507C4—82no—
508C4—82no—
509C4—82no—
510C4—84no—
511C4C2×#47814yes—
512C4C2×#47814yes—
513C4#345812yes—
514C4#349312yes—
515C2—24no—
516C2—84no—
517C2#3515111yescharacteristic
518C2C22×#47811yescharacteristic
519C2#6049211yescharacteristic
520C2#6170011yescharacteristic
521C2#6171711yescharacteristic
522C2#6188211yescharacteristic
523C2#6189011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35181C2semidirect
3. #33151C2semidirect
5. #26091C2semidirect
6. #26087C2semidirect
7. #26025C2semidirect
8. #26021C2semidirect
9. #21982C2semidirect
10. #21981C2semidirect
11. #21951C2semidirect
20. #2571C22semidirect
21. #2571C22semidirect
24. #2552C22semidirect
25. #2552C22semidirect
44. #1989C22semidirect
45. #1949C22semidirect
46. #1948C22semidirect
47. #1947C22semidirect
48. #1939C22semidirect
49. #1939C22semidirect
50. #1921C22semidirect
51. #1919C22semidirect
54. #1845C22semidirect
55. #1844C22semidirect
56. #1834C22semidirect
57. #1834C22semidirect

Maximal subgroups

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

subgroupindex
#351812
C2×#23542
#331512
#299562
#260912
#260872
#260252
#260212
#219822
#219812
#219512
#207322