Finite Group Databasebeta

← #6528502

Subgroups — #6528502

Order 512.

1712 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35167C211yesmaximal, characteristic, complemented
2C2×#2566C211yesmaximal, characteristic
3#29109C212yesmaximal
4#29103C212yesmaximal
5#27969C212yesmaximal, complemented
6#27961C212yesmaximal, complemented
7#27825C218yesmaximal, complemented
8#27739C218yesmaximal, complemented
9#21838C212yesmaximal, complemented
10#20738C212yesmaximal
11#20534C211yesmaximal, characteristic, complemented
12C2×#544—21no—
13C22×#125C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C2×#432C2212yes—
16#2857C2212yes—
17#2857C2212yes—
18#2659—21no—
19#2659—21no—
20#2566—24no—
21#2566C2218yes—
22#2552C2214yescomplemented
23#2551C2214yescomplemented
24#2550—24no—
25C2×#396—21no—
26C2×#396—21no—
27#2316—21no—
28C2×#390C2211yescharacteristic
29C2×#390C2211yescharacteristic
30C2×#385—21no—
31C2×#385—21no—
32#2285C2212yes—
33C2×#384C2211yescharacteristic
34C2×#382C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36#2101C2218yescomplemented
37#2100C2218yescomplemented
38#2071C2214yes—
39#2071C2214yes—
40#2023C2214yes—
41#2023C2214yes—
42#1993C2214yescomplemented
43#1992C2214yescomplemented
44#1992C2214yescomplemented
45#1990C2214yescomplemented
46#1989C2218yescomplemented
47#1936C2218yes—
48#1934C2218yes—
49#1836C2214yescomplemented
50#1834C2214yescomplemented
51#1824C2214yes—
52#1824C2214yes—
53#1823C2214yes—
54#1802C2214yes—
55#1802C2214yes—
56#1786C2214yes—
57#1771C2218yescomplemented
58#1762C2218yes—
59#1749C2212yes—
60C4×C22⋊C8C2212yes—
61C23×D4C2311yescharacteristic
62C24×C4—21no—
63#544—48no—
64C2×C4⋊Q8—24no—
65C2×C4⋊Q8C2312yes—
66C2×C4⋊Q8C2312yes—
67C2×#125—24no—
68C2×#125—22no—
69C2×#125—28no—
70C2×#125C2314yes—
71C2×C22⋊Q8—22no—
72C2×C22⋊Q8—22no—
73C2×C4×D4—22no—
74C2×C4×D4—28no—
75C2×C4×D4C2314yes—
76C2×#117—22no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4D411yescharacteristic
82C22×C4⋊C4D411yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C22⋊C4—21no—
87C22×C22⋊C4—21no—
88C22×C22⋊C4—21no—
89C22×C22⋊C4—21no—
90C22×C22⋊C4—21no—
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93C22×C42C2311yescharacteristic
94#432—28no—
95C2×C4⋊C8C2314yes—
96C2×#102C2318yes—
97C2×#102C2318yes—
98C2×C22⋊C8C2314yes—
99C2×#97C2314yes—
100C2×#97C2314yes—
101C2×C4×C8C2314yes—
102#398—24no—
103#398—24no—
104#396—28no—
105#396—28no—
106#395—24no—
107#395—24no—
108#391—22no—
109#391—22no—
110#390—22no—
111#390—22no—
112#390C2314yes—
113#390C2314yes—
114#389—24no—
115#389—24no—
116#389—22no—
117#389—22no—
118#389C2314yes—
119#389C2314yes—
120#385—28no—
121#385—28no—
122#384—22no—
123#384C2314yes—
124#384C2312yes—
125#384C2312yes—
126#384C2312yes—
127#384C2312yes—
128#382—22no—
129#382—22no—
130#382C2314yes—
131#380—22no—
132#380—22no—
133#380—22no—
134#380C2314yes—
135C4×C4⋊C4—22no—
136C4×C4⋊C4D414yes—
137C4×C4⋊C4C2314yes—
138C4×C4⋊C4C2314yes—
139C2×#95—21no—
140C2×#95—21no—
141C2×#95D411yescharacteristic
142C2×#95D411yescharacteristic
143#340C2318yes—
144#340C2318yes—
145#340C2318yes—
146#340C2318yes—
147#340C2318yes—
148#340C2318yes—
149#336C2314yes—
150#336C2314yes—
151C25—21no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×D4—24no—
155C22×D4—21no—
156C22×D4—24no—
157C22×D4—42no—
158C22×D4C2412yes—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—24no—
163C23×C4—44no—
164C23×C4—82no—
165C23×C4—82no—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2411yescharacteristic
173C22×C8C2418yes—
174C4⋊Q8—44no—
175C4⋊Q8—44no—
176C4⋊Q8—416no—
177#125—28no—
178#125—28no—
179#125—416no—
180#125—48no—
181C22⋊Q8—48no—
182C22⋊Q8—48no—
183C4×D4—28no—
184C4×D4—28no—
185C4×D4—48no—
186#117—48no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—82no—
214C2×C4⋊C4—82no—
215C2×C4⋊C4C2×D414yes—
216C2×C4⋊C4C2×D414yes—
217C2×C4⋊C4C2412yes—
218C2×C4⋊C4C2412yes—
219C2×C4⋊C4C2412yes—
220C2×C4⋊C4C2414yes—
221C2×C4⋊C4C2414yes—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—82no—
232C2×C22⋊C4—82no—
233C2×C42—24no—
234C2×C42—42no—
235C2×C42—42no—
236C2×C42C2×D412yes—
237C2×C42C2×D412yes—
238C2×C42C2×D412yes—
239C2×C42#4118yes—
240C2×C42C2412yes—
241C2×C42C2412yes—
242C2×C42C2412yes—
243C4⋊C8—28no—
244#102—216no—
245#102—48no—
246C22⋊C8—28no—
247#97—28no—
248#97—44no—
249C4×C8—44no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95C2×D414yes—
257#95C2×D414yes—
258C24—24no—
259C24—22no—
260C24—42no—
261C24—42no—
262C24—42no—
263C24—42no—
264C24—42no—
265C24—42no—
266C24C22×D411yescharacteristic
267C2×Q8—44no—
268C2×Q8—44no—
269C2×Q8—82no—
270C2×Q8—82no—
271C2×Q8—82no—
272C2×Q8—82no—
273C2×D4—24no—
274C2×D4—24no—
275C2×D4—22no—
276C2×D4—24no—
277C2×D4—416no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D42−1+414yes—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—48no—
312C22×C4—84no—
313C22×C4—84no—
314C22×C4—816no—
315C22×C4#12414yes—
316C22×C4#12414yes—
317C22×C4AGL(1,ℤ/8)14yes—
318C22×C4#13714yes—
319C22×C4C22×D412yes—
320C22×C4C22×D412yes—
321C22×C4C22×D411yescharacteristic
322C22×C4C22×D412yes—
323C22×C4C22×D412yes—
324C22×C4C22×D412yes—
325C22×C4C22×D412yes—
326C22×C4C2×#4114yes—
327C22×C4C2511yescharacteristic
328C2×C8—28no—
329C2×C8—28no—
330C2×C8—28no—
331C4⋊C4—28no—
332C4⋊C4—28no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—28no—
337C4⋊C4—28no—
338C4⋊C4—44no—
339C4⋊C4—44no—
340C4⋊C4—44no—
341C4⋊C4—48no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—48no—
346C4⋊C4—48no—
347C4⋊C4—44no—
348C4⋊C4—48no—
349C4⋊C4—82no—
350C4⋊C4—82no—
351C4⋊C4—82no—
352C4⋊C4—82no—
353C4⋊C4—84no—
354C4⋊C4—84no—
355C4⋊C4—82no—
356C4⋊C4—82no—
357C22⋊C4—48no—
358C22⋊C4—48no—
359C22⋊C4—48no—
360C22⋊C4—48no—
361C22⋊C4—84no—
362C22⋊C4—84no—
363C42—24no—
364C42—24no—
365C42—42no—
366C42—42no—
367C42—48no—
368C42#12418yes—
369C42C2×#4114yes—
370C42C2×#4114yes—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—24no—
376C23—48no—
377C23—48no—
378C23—48no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—84no—
386C23—84no—
387C23—84no—
388C23—84no—
389C23C2×#12412yes—
390C23C2×AGL(1,ℤ/8)12yes—
391C23C2×#13712yes—
392C23C23×D411yescharacteristic
393Q8—164no—
394Q8—164no—
395D4—28no—
396D4—48no—
397D4—44no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—28no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—28no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—816no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—84no—
438C2×C4—84no—
439C2×C4—84no—
440C2×C4C2×#12412yes—
441C2×C4C2×#12414yes—
442C2×C4C2×#12414yes—
443C2×C4C2×AGL(1,ℤ/8)14yes—
444C2×C4C2×#13714yes—
445C2×C4C22×#4112yes—
446C2×C4C2×2−1+412yes—
447C2×C4#58512yes—
448C8—48no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—44no—
456C22—44no—
457C22—44no—
458C22—816no—
459C22—82no—
460C22—82no—
461C22—82no—
462C22—82no—
463C22#314114yes—
464C22C22×#12411yescharacteristic
465C22#348511yescharacteristic
466C22#348611yescharacteristic
467C22C22×AGL(1,ℤ/8)11yescharacteristic
468C22C22×#13711yescharacteristic
469C22#358511yescharacteristic
470C22#358511yescharacteristic
471C4—24no—
472C4—24no—
473C4—22no—
474C4—24no—
475C4—24no—
476C4—42no—
477C4—42no—
478C4—84no—
479C4—84no—
480C4—82no—
481C4—82no—
482C4—82no—
483C4#349514yes—
484C2—24no—
485C2—84no—
486C2#3564411yescharacteristic
487C2C2×#314111yescharacteristic
488C2#6061911yescharacteristic
489C2#6170711yescharacteristic
490C2#6170811yescharacteristic
491C2#6182511yescharacteristic
492C2#6182611yescharacteristic
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. #35167C2semidirect
5. #27969C2semidirect
6. #27961C2semidirect
7. #27825C2semidirect
8. #27739C2semidirect
9. #21838C2semidirect
11. #20534C2semidirect
22. #2552C22semidirect
23. #2551C22semidirect
36. #2101C22semidirect
37. #2100C22semidirect
42. #1993C22semidirect
43. #1992C22semidirect
44. #1992C22semidirect
45. #1990C22semidirect
46. #1989C22semidirect
49. #1836C22semidirect
50. #1834C22semidirect
57. #1771C22semidirect

Maximal subgroups

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

subgroupindex
#351672
C2×#25662
#291092
#291032
#279692
#279612
#278252
#277392
#218382
#207382
#205342