Finite Group Databasebeta

← #6436857

Subgroups — #6436857

Order 512.

1790 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36640C211yesmaximal, characteristic, complemented
2C2×#2293C211yesmaximal, characteristic
3#29998C212yesmaximal, complemented
4#29994C212yesmaximal, complemented
5#29986C214yesmaximal, complemented
6#29973C214yesmaximal, complemented
7#29956C214yesmaximal
8#23784C218yesmaximal, complemented
9#21365C212yesmaximal, complemented
10#20810C211yesmaximal, characteristic, complemented
11#20738C212yesmaximal
12C22×C4×D4C2211yescharacteristic
13C22×#117—21no—
14C2×#432C2212yes—
15#2864—22no—
16#2859C2211yescharacteristic, complemented
17#2859C2211yescharacteristic, complemented
18#2859C2212yescomplemented
19#2851—21no—
20#2851—21no—
21#2851C2214yes—
22C4×#125—21no—
23C4×#125—21no—
24C4×#125C2214yescomplemented
25#2296—22no—
26#2293—21no—
27#2293—21no—
28#2293—22no—
29#2293C2212yes—
30#2293C2212yes—
31#2293C2214yes—
32C2×#390C2212yes—
33C2×#389C2212yes—
34C2×#388—22no—
35#2285C2212yes—
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C22⋊C4—21no—
38C2×C43C2211yescharacteristic
39#2101C2218yescomplemented
40#2092C2218yes—
41#2088C2218yescomplemented
42#2086C2218yescomplemented
43#2067C2218yes—
44#2065C2218yes—
45#2059C2218yescomplemented
46#2025C2218yes—
47#1948C2214yescomplemented
48#1947C2214yescomplemented
49#1853C2214yes—
50#1852C2214yes—
51#1842C2212yescomplemented
52#1831C2212yescomplemented
53#1824C2214yes—
54#1802C2214yes—
55#1802C2214yes—
56#1797C2214yes—
57#1786C2214yes—
58#1776C2218yescomplemented
59#1767C2214yescomplemented
60#1762C2218yes—
61C23×D4C2311yescharacteristic
62C24×C4—21no—
63C2×#125—21no—
64C2×#125—21no—
65C2×#125C2314yes—
66C2×C4×D4—21no—
67C2×C4×D4—21no—
68C2×C4×D4—24no—
69C2×C4×D4Q818yes—
70C2×C4×D4C2312yes—
71C2×C4×D4C2312yes—
72C2×#117—22no—
73C2×#117—24no—
74C2×#117—22no—
75C2×#117—22no—
76C2×#117—48no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2312yes—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—22no—
82C22×C22⋊C4—22no—
83C22×C42—21no—
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D412yes—
87C22×C42C2312yes—
88C22×C42C2311yescharacteristic
89#432—28no—
90C2×#107C2318yes—
91C2×#106C2318yes—
92C2×C4⋊C8C2314yes—
93C2×C4⋊C8C2318yes—
94C2×#102C2318yes—
95C2×#102C2318yes—
96C2×C22⋊C8C2314yes—
97C2×#97C2314yes—
98#390—22no—
99#390—22no—
100#390C2314yes—
101#390C2314yes—
102#389—22no—
103#389—22no—
104#389—22no—
105#389—22no—
106#389—22no—
107#389—22no—
108#389—22no—
109#389—22no—
110#389C2314yes—
111#389C2314yes—
112#389C2314yes—
113#389C2314yes—
114#388—24no—
115#388—24no—
116#388—24no—
117#388—24no—
118#387—24no—
119#387—24no—
120#387—24no—
121#387—24no—
122#385—24no—
123#385—24no—
124#384C2312yes—
125#384C2312yes—
126#382—21no—
127#382—21no—
128#382—22no—
129#382C2314yes—
130#380—22no—
131#380C2314yes—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—24no—
134C4×C4⋊C4—24no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4C2314yes—
138C4×C4⋊C4C2312yes—
139C4×C4⋊C4C2312yes—
140C4×C22⋊C4—22no—
141C4×C22⋊C4—22no—
142C4×C22⋊C4—24no—
143C2×#95—22no—
144C2×#95—21no—
145C43D411yescharacteristic
146C43D411yescharacteristic
147C43D412yes—
148C43C2312yes—
149C43C2311yescharacteristic
150C43C2311yescharacteristic
151#340C2318yes—
152#340C2318yes—
153#340C2318yes—
154#340C2318yes—
155#336C2318yes—
156C25—21no—
157C22×D4—21no—
158C22×D4—22no—
159C22×D4—44no—
160C22×D4C2×Q814yes—
161C22×D4C2412yes—
162C23×C4—21no—
163C23×C4—28no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—44no—
168C23×C4—44no—
169C23×C4C2×D412yes—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D412yes—
173C23×C4C2411yescharacteristic
174C22×C8C2418yes—
175#125—48no—
176#125—44no—
177#125—44no—
178C4×D4—28no—
179C4×D4—28no—
180C4×D4—24no—
181C4×D4—24no—
182C4×D4C2×Q818yes—
183C4×D4C2×Q818yes—
184#117—416no—
185#117—48no—
186#117—48no—
187#117—48no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—48no—
203C2×C4⋊C4C2×Q814yes—
204C2×C4⋊C4C2414yes—
205C2×C4⋊C4C2414yes—
206C2×C4⋊C4C2414yes—
207C2×C4⋊C4C2412yes—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—48no—
217C2×C22⋊C4—48no—
218C2×C42—24no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—24no—
222C2×C42—42no—
223C2×C42—44no—
224C2×C42—44no—
225C2×C42—44no—
226C2×C42C2×D412yes—
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D412yes—
231C2×C42C2×D412yes—
232C2×C42C2×D412yes—
233C2×C42C2×D412yes—
234C2×C42C2×D412yes—
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×Q814yes—
241C2×C42#4114yes—
242C2×C42C2412yes—
243C2×C42C2412yes—
244C2×C42C2411yescharacteristic
245C2×C42C2411yescharacteristic
246#107—48no—
247#106—48no—
248C4⋊C8—28no—
249C4⋊C8—48no—
250#102—48no—
251#102—48no—
252C22⋊C8—28no—
253#97—28no—
254#95—24no—
255#95—24no—
256#95—22no—
257#95—22no—
258#95—24no—
259#95—24no—
260#95—24no—
261#95—24no—
262#95—24no—
263#95—24no—
264C24—24no—
265C24—22no—
266C24—42no—
267C24—42no—
268C24—44no—
269C24—44no—
270C24C22×D411yescharacteristic
271C2×D4—24no—
272C2×D4—22no—
273C2×D4—24no—
274C2×D4—24no—
275C2×D4—48no—
276C2×D4—44no—
277C2×D4—48no—
278C2×D4C22×Q814yes—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—28no—
288C22×C4—28no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4—432no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4C22⋊Q814yes—
307C22×C4C22⋊Q814yes—
308C22×C4C22×D412yes—
309C22×C4C22×D412yes—
310C22×C4C22×D412yes—
311C22×C4C22×D411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D411yescharacteristic
315C22×C4C22×D412yes—
316C22×C4C22×D411yescharacteristic
317C22×C4C22×Q812yes—
318C22×C4C2×#4112yes—
319C22×C4C2511yescharacteristic
320C2×C8—28no—
321C2×C8—28no—
322C2×C8—28no—
323C4⋊C4—24no—
324C4⋊C4—22no—
325C4⋊C4—28no—
326C4⋊C4—28no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—44no—
330C4⋊C4—48no—
331C4⋊C4—44no—
332C4⋊C4—44no—
333C4⋊C4—48no—
334C4⋊C4—48no—
335C4⋊C4—48no—
336C4⋊C4—48no—
337C4⋊C4—48no—
338C4⋊C4—44no—
339C4⋊C4—44no—
340C4⋊C4C22×Q814yes—
341C22⋊C4—28no—
342C22⋊C4—28no—
343C22⋊C4—48no—
344C22⋊C4—48no—
345C22⋊C4—48no—
346C22⋊C4—48no—
347C22⋊C4—48no—
348C42—24no—
349C42—22no—
350C42—22no—
351C42—22no—
352C42—24no—
353C42—24no—
354C42—42no—
355C42—42no—
356C42—42no—
357C42—42no—
358C42—42no—
359C42—42no—
360C42—42no—
361C42—44no—
362C42—48no—
363C42—44no—
364C42C22⋊Q814yes—
365C42C22⋊Q814yes—
366C42AGL(1,ℤ/8)14yes—
367C42C22×Q812yes—
368C42C22×Q812yes—
369C42C2×#4112yes—
370C42C2×#4112yes—
371C23—22no—
372C23—21no—
373C23—21no—
374C23—24no—
375C23—48no—
376C23—48no—
377C23—48no—
378C23—44no—
379C23—42no—
380C23—44no—
381C23—42no—
382C23—88no—
383C23—88no—
384C23C2×C22⋊Q812yes—
385C23#57514yes—
386C23C23×D411yescharacteristic
387D4—28no—
388D4—48no—
389D4—42no—
390D4—42no—
391C2×C4—28no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—28no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—48no—
413C2×C4—416no—
414C2×C4—48no—
415C2×C4—48no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—416no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—88no—
429C2×C4C2×C22⋊Q812yes—
430C2×C4C2×C22⋊Q812yes—
431C2×C4C2×C22⋊Q812yes—
432C2×C4C2×C22⋊Q812yes—
433C2×C4C2×C22⋊Q812yes—
434C2×C4C2×AGL(1,ℤ/8)14yes—
435C2×C4C2×AGL(1,ℤ/8)12yes—
436C2×C4#57512yes—
437C2×C4C23×Q811yescharacteristic
438C2×C4C22×#4111yescharacteristic
439C2×C4#58514yes—
440C8—48no—
441C22—24no—
442C22—24no—
443C22—22no—
444C22—22no—
445C22—44no—
446C22—44no—
447C22—44no—
448C22—816no—
449C22—84no—
450C22—84no—
451C22#308014yes—
452C22C22×C22⋊Q811yescharacteristic
453C22#344912yes—
454C22C22×AGL(1,ℤ/8)11yescharacteristic
455C22C2×#57512yes—
456C22C2×#57511yescharacteristic
457C4—24no—
458C4—22no—
459C4—22no—
460C4—22no—
461C4—22no—
462C4—42no—
463C4—42no—
464C4—42no—
465C4—44no—
466C4—84no—
467C4—84no—
468C4—84no—
469C4#307714yes—
470C4#345512yes—
471C4#345812yes—
472C2—24no—
473C2—84no—
474C2#3463411yescharacteristic
475C2C2×#307711yescharacteristic
476C2C2×#308011yescharacteristic
477C2#6096212yes—
478C2#6097012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36640C2semidirect
3. #29998C2semidirect
4. #29994C2semidirect
5. #29986C2semidirect
6. #29973C2semidirect
8. #23784C2semidirect
9. #21365C2semidirect
10. #20810C2semidirect
16. #2859C22semidirect
17. #2859C22semidirect
18. #2859C22semidirect
24. C4×#125C22semidirect
39. #2101C22semidirect
41. #2088C22semidirect
42. #2086C22semidirect
45. #2059C22semidirect
47. #1948C22semidirect
48. #1947C22semidirect
51. #1842C22semidirect
52. #1831C22semidirect
58. #1776C22semidirect
59. #1767C22semidirect

Maximal subgroups

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

subgroupindex
#366402
C2×#22932
#299982
#299942
#299862
#299732
#299562
#237842
#213652
#208102
#207382