Finite Group Databasebeta

← #6510746

Subgroups — #6510746

Order 512.

1802 conjugacy classes of subgroups.

The subgroup lattice has 535 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35002C211yesmaximal, characteristic, complemented
2C2×#2500C211yesmaximal, characteristic
3#29109C212yesmaximal, complemented
4#29103C212yesmaximal, complemented
5#27969C212yesmaximal, complemented
6#27961C212yesmaximal, complemented
7#27720C218yesmaximal, complemented
8#27704C218yesmaximal, complemented
9#21764C212yesmaximal
10#20702C212yesmaximal, complemented
11#20524C211yesmaximal, characteristic, complemented
12C2×#543—21no—
13C22×#121C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C2×#407C2212yes—
16#2857C2212yes—
17#2857C2212yes—
18#2659—21no—
19#2659—21no—
20#2549—24no—
21#2500—24no—
22#2500C2218yes—
23#2440C2214yescomplemented
24#2440C2214yescomplemented
25C2×#394—21no—
26C2×#394—21no—
27#2315—21no—
28C2×#390C2211yescharacteristic
29C2×#390C2211yescharacteristic
30C2×#388—21no—
31C2×#388—21no—
32C2×#384C2211yescharacteristic
33C2×#382C2211yescharacteristic
34C2×#380C2212yes—
35C2×C4×C22⋊C4C2211yescharacteristic
36#2101C2218yescomplemented
37#2100C2218yescomplemented
38#2071C2214yescomplemented
39#2071C2214yescomplemented
40#2023C2214yescomplemented
41#2023C2214yescomplemented
42#1993C2214yescomplemented
43#1990C2214yescomplemented
44#1936C2218yescomplemented
45#1934C2218yescomplemented
46#1895C2218yes—
47#1874C2214yes—
48#1874C2214yes—
49#1836C2214yescomplemented
50#1834C2214yescomplemented
51#1825C2218yescomplemented
52#1824C2214yes—
53#1824C2214yes—
54#1823C2214yes—
55#1786C2214yes—
56#1753C2218yescomplemented
57#1749C2212yes—
58C4×C22⋊C8C2212yes—
59#1736C2214yescomplemented
60#1736C2214yescomplemented
61C23×D4—21no—
62C23×D4C2311yescharacteristic
63C24×C4C2311yescharacteristic
64C22×M16C2314yes—
65#543—48no—
66C2×C4⋊Q8C2312yes—
67C2×C4⋊Q8C2312yes—
68C2×C22⋊Q8—22no—
69C2×C22⋊Q8—22no—
70C2×C22⋊Q8—24no—
71C2×C22⋊Q8—24no—
72C2×#121—22no—
73C2×#121—28no—
74C2×#121C2314yes—
75C2×#117—22no—
76C2×#117—28no—
77C2×#117C2314yes—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C22⋊C4—22no—
86C22×C22⋊C4—21no—
87C22×C22⋊C4—21no—
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
96C22×C42C2311yescharacteristic
97C2×#102C2318yes—
98C2×#102C2318yes—
99#407—28no—
100C2×C22⋊C8C2314yes—
101C2×C22⋊C8C2314yes—
102C2×#97C2314yes—
103C2×C4×C8C2314yes—
104#398—24no—
105#398—24no—
106#395—24no—
107#395—24no—
108#394—28no—
109#394—28no—
110#391—22no—
111#391—22no—
112#390—22no—
113#390—22no—
114#390C2314yes—
115#390C2314yes—
116#389—22no—
117#389—22no—
118#389C2314yes—
119#389C2314yes—
120#388—28no—
121#388—28no—
122#385—24no—
123#385—24no—
124#384—22no—
125#384C2312yes—
126#384C2312yes—
127#384C2314yes—
128#382—22no—
129#382C2314yes—
130#380—22no—
131#380—22no—
132#380—22no—
133#380—22no—
134#380C2314yes—
135#380C2314yes—
136C4×C22⋊C4D414yes—
137C4×C22⋊C4C2314yes—
138C2×#95—21no—
139C2×#95—21no—
140C2×#95—22no—
141C2×#95D411yescharacteristic
142C2×#95D411yescharacteristic
143#340C2318yes—
144#340C2318yes—
145#340C2318yes—
146#340C2318yes—
147#340C2318yes—
148#340C2318yes—
149#336C2314yes—
150#336C2314yes—
151C25—21no—
152C25C2×D411yescharacteristic
153C22×Q8—22no—
154C22×Q8—22no—
155C22×D4—24no—
156C22×D4—21no—
157C22×D4—24no—
158C22×D4—42no—
159C22×D4—42no—
160C22×D4—44no—
161C22×D4—82no—
162C22×D4—82no—
163C22×D4C2412yes—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D412yes—
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4#4118yes—
178C23×C4C2411yescharacteristic
179C23×C4C2412yes—
180C2×M16—28no—
181C2×M16—44no—
182C2×M16—44no—
183C22×C8C2418yes—
184C4⋊Q8—44no—
185C4⋊Q8—44no—
186C22⋊Q8—416no—
187C22⋊Q8—416no—
188C22⋊Q8—48no—
189C22⋊Q8—48no—
190#121—28no—
191#121—28no—
192#121—48no—
193#117—28no—
194#117—28no—
195#117—48no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—82no—
217C2×C4⋊C4—82no—
218C2×C4⋊C4C2412yes—
219C2×C4⋊C4C2412yes—
220C2×C4⋊C4C2412yes—
221C2×C4⋊C4C2414yes—
222C2×C4⋊C4C2414yes—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
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—48no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—82no—
239C2×C22⋊C4—82no—
240C2×C22⋊C4C2×D414yes—
241C2×C22⋊C4C2×D414yes—
242C2×C42—24no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—44no—
246C2×C42C2×D412yes—
247C2×C42C2×D412yes—
248C2×C42C2412yes—
249C2×C42C2412yes—
250#102—48no—
251#102—48no—
252C22⋊C8—28no—
253C22⋊C8—28no—
254#97—44no—
255C4×C8—44no—
256#95—24no—
257#95—24no—
258#95—24no—
259#95—24no—
260#95—24no—
261#95—24no—
262#95C2×D414yes—
263#95C2×D414yes—
264C24—22no—
265C24—22no—
266C24—22no—
267C24—41no—
268C24—41no—
269C24—41no—
270C24—41no—
271C24—42no—
272C24—42no—
273C24—42no—
274C24—42no—
275C24—42no—
276C24—42no—
277C24—42no—
278C24#12414yes—
279C24C22×D412yes—
280C24C22×D411yescharacteristic
281C2×Q8—44no—
282C2×Q8—44no—
283C2×Q8—82no—
284C2×Q8—82no—
285C2×Q8—82no—
286C2×Q8—82no—
287C2×D4—24no—
288C2×D4—24no—
289C2×D4—24no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—48no—
293C2×D4—42no—
294C2×D4—88no—
295C2×D4—84no—
296C2×D4—84no—
297C2×D4—816no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—416no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—82no—
334C22×C4—82no—
335C22×C4—82no—
336C22×C4—82no—
337C22×C4#12414yes—
338C22×C4#12418yes—
339C22×C4AGL(1,ℤ/8)14yes—
340C22×C4#13714yes—
341C22×C4C22×D412yes—
342C22×C4C22×D412yes—
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D412yes—
345C22×C4C22×D412yes—
346C22×C4C22×D412yes—
347C22×C4C2×#4114yes—
348C22×C4C2×#4114yes—
349C22×C4C2×#4114yes—
350C22×C4C2511yescharacteristic
351M16—88no—
352C2×C8—28no—
353C2×C8—28no—
354C2×C8—28no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—28no—
359C4⋊C4—28no—
360C4⋊C4—44no—
361C4⋊C4—48no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—44no—
366C4⋊C4—48no—
367C4⋊C4—82no—
368C4⋊C4—82no—
369C4⋊C4—82no—
370C4⋊C4—82no—
371C4⋊C4—84no—
372C4⋊C4—84no—
373C4⋊C4—82no—
374C4⋊C4—82no—
375C22⋊C4—28no—
376C22⋊C4—28no—
377C22⋊C4—48no—
378C22⋊C4—48no—
379C22⋊C4—48no—
380C22⋊C4—48no—
381C22⋊C4—48no—
382C22⋊C4—48no—
383C22⋊C4—84no—
384C22⋊C4—84no—
385C42—24no—
386C42—24no—
387C42—42no—
388C42—42no—
389C42—48no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—28no—
401C23—44no—
402C23—44no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—82no—
411C23—82no—
412C23—82no—
413C23—82no—
414C23—84no—
415C23—84no—
416C23—84no—
417C23—84no—
418C23—84no—
419C23—84no—
420C23—84no—
421C23C2×#12414yes—
422C23C2×#12412yes—
423C23C2×AGL(1,ℤ/8)12yes—
424C23C2×#13712yes—
425C23C23×D411yescharacteristic
426Q8—164no—
427Q8—164no—
428D4—48no—
429D4—84no—
430D4—816no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—28no—
440C2×C4—28no—
441C2×C4—28no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—44no—
449C2×C4—48no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—84no—
470C2×C4—84no—
471C2×C4—84no—
472C2×C4—164no—
473C2×C4—164no—
474C2×C4C2×#12414yes—
475C2×C4C2×#12412yes—
476C2×C4C2×AGL(1,ℤ/8)14yes—
477C2×C4C2×#13714yes—
478C2×C4C22×#4112yes—
479C2×C4#58512yes—
480C2×C4#58512yes—
481C8—48no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—24no—
489C22—24no—
490C22—24no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—88no—
496C22—82no—
497C22—82no—
498C22—82no—
499C22—82no—
500C22—82no—
501C22—82no—
502C22—82no—
503C22—168no—
504C22#314114yes—
505C22C22×#12411yescharacteristic
506C22#348511yescharacteristic
507C22#348511yescharacteristic
508C22C22×AGL(1,ℤ/8)11yescharacteristic
509C22C22×#13711yescharacteristic
510C22#358511yescharacteristic
511C22#358511yescharacteristic
512C4—24no—
513C4—24no—
514C4—24no—
515C4—22no—
516C4—42no—
517C4—42no—
518C4—84no—
519C4—84no—
520C4—82no—
521C4—82no—
522C4—82no—
523C4#349114yes—
524C2—24no—
525C2—44no—
526C2—162no—
527C2#3545511yescharacteristic
528C2C2×#314111yescharacteristic
529C2#6061911yescharacteristic
530C2#6170711yescharacteristic
531C2#6170811yescharacteristic
532C2#6172011yescharacteristic
533C2#6172111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35002C2semidirect
3. #29109C2semidirect
4. #29103C2semidirect
5. #27969C2semidirect
6. #27961C2semidirect
7. #27720C2semidirect
8. #27704C2semidirect
10. #20702C2semidirect
11. #20524C2semidirect
23. #2440C22semidirect
24. #2440C22semidirect
36. #2101C22semidirect
37. #2100C22semidirect
38. #2071C22semidirect
39. #2071C22semidirect
40. #2023C22semidirect
41. #2023C22semidirect
42. #1993C22semidirect
43. #1990C22semidirect
44. #1936C22semidirect
45. #1934C22semidirect
49. #1836C22semidirect
50. #1834C22semidirect
51. #1825C22semidirect
56. #1753C22semidirect
59. #1736C22semidirect
60. #1736C22semidirect

Maximal subgroups

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

subgroupindex
#350022
C2×#25002
#291092
#291032
#279692
#279612
#277202
#277042
#217642
#207022
#205242