Finite Group Databasebeta

← #496490

Subgroups — #496490

Order 512.

822 conjugacy classes of subgroups.

The subgroup lattice has 353 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3177C211yesmaximal, characteristic
2C2×#2164C211yesmaximal, characteristic
3#20803C211yesmaximal, characteristic
4#13284C211yesmaximal, characteristic, complemented
5#13281C211yesmaximal, characteristic, complemented
6#13207C211yesmaximal, characteristic, complemented
7#13203C211yesmaximal, characteristic, complemented
8#12994C212yesmaximal
9#12993C212yesmaximal
10#12814C212yesmaximal, complemented
11#12812C212yesmaximal, complemented
12C2×#541—21no—
13#3177—24no—
14#3177C418yescomplemented
15C2×#499C2211yescharacteristic
16C2×#494C2211yescharacteristic
17C2×#491—21no—
18C2×#470—21no—
19C2×#467C2212yes—
20C2×#441—21no—
21C2×C4×Q16C2212yes—
22C2×#428—21no—
23C2×#427C2211yescharacteristic
24C22×#107C2211yescharacteristic
25C2×#405C2211yescharacteristic
26#2317—21no—
27#2164—24no—
28C2×C4⋊C16C2212yes—
29C2×#358C2212yes—
30C2×#358C2212yes—
31C2×#358C2212yes—
32C2×#358C2212yes—
33C2×C22⋊C16C2212yes—
34C2×#346C2211yescharacteristic
35C2×C4×C16C2211yescharacteristic
36#1847—21no—
37#1847—21no—
38#1846C2211yescharacteristic
39#1846C2211yescharacteristic
40#1845C2211yescharacteristic
41#1845C2211yescharacteristic
42#1380C2212yes—
43#1380C2212yes—
44#1380C2212yes—
45#1380C2212yes—
46C22×Q16—21no—
47C22×Q16C2312yes—
48C22×M16D411yescharacteristic
49C23×C8C2311yescharacteristic
50#541—28no—
51#541—44no—
52C2×C4⋊Q8—21no—
53C2×#126—42no—
54C2×#125—21no—
55C2×#125—42no—
56C2×C22⋊Q8—22no—
57C2×C4×Q8—22no—
58C2×#117—21no—
59C2×#117D411yescharacteristic
60C22×C4⋊C4—21no—
61C22×C16C2312yes—
62C22×C16C2312yes—
63#499—22no—
64#499C2×C414yescomplemented
65#494—22no—
66#494C2×C414yescomplemented
67#491—28no—
68#470—28no—
69#467—24no—
70#467C2×C418yes—
71#441—28no—
72C4×Q16—24no—
73C4×Q16C2×C418yescomplemented
74#428—44no—
75#427—42no—
76C2×#107—21no—
77C2×#107—21no—
78C2×#107—21no—
79C2×#107—42no—
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2312yes—
85C2×#106—21no—
86C2×#106—21no—
87C2×#106—21no—
88C2×#106—21no—
89C2×#103—22no—
90C2×#103—22no—
91#405C2×C414yes—
92#405D414yes—
93C2×#97D411yescharacteristic
94C2×#97D411yescharacteristic
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97#389—42no—
98#385—42no—
99#384—21no—
100#384—21no—
101#382—21no—
102#382—21no—
103C4⋊C16—24no—
104#358—42no—
105#358—42no—
106#358—42no—
107#358—42no—
108C22⋊C16—24no—
109#346—22no—
110C4×C16—22no—
111C22×Q8—22no—
112C23×C4—41no—
113C23×C4C2×D411yescharacteristic
114C2×Q16—24no—
115C2×Q16—22no—
116C2×Q16—22no—
117C2×Q16—22no—
118C2×Q16—22no—
119C2×Q16—22no—
120C2×Q16—42no—
121C2×Q16—42no—
122C2×Q16C22×C414yes—
123C2×M16—22no—
124C2×M16—22no—
125C2×M16C22⋊C414yes—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8C22×C412yes—
129C22×C8C2×D411yescharacteristic
130C22×C8C2×D411yescharacteristic
131C22×C8C2×D412yes—
132C22×C8C2×D411yescharacteristic
133C22×C8C2411yescharacteristic
134C4⋊Q8—24no—
135C4⋊Q8—42no—
136#126—416no—
137#125—24no—
138#125—416no—
139#125—42no—
140C22⋊Q8—28no—
141C22⋊Q8—44no—
142C4×Q8—28no—
143C4×Q8—44no—
144#117—42no—
145#117—42no—
146#117C22⋊C414yes—
147#117SD1614yes—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4D812yes—
163C2×C22⋊C4—42no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4D812yes—
166C2×C42—42no—
167C2×C42—41no—
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C16—22no—
171C2×C16—22no—
172C2×C16—22no—
173C2×C16—22no—
174C2×C16—22no—
175C2×C16—22no—
176#107—22no—
177#107—22no—
178#107—22no—
179#107—22no—
180#107—22no—
181#107—42no—
182#107—42no—
183#107—42no—
184#107C22×C414yes—
185#106—24no—
186#106—42no—
187#106—42no—
188#106—42no—
189#103—28no—
190#103—28no—
191#97C22⋊C414yes—
192#97C22⋊C414yes—
193C4×C8C22×C412yes—
194C4×C8C22×C412yes—
195C4×C8C2×D412yes—
196C4×C8C2×D412yes—
197#95—42no—
198#95—42no—
199C24C2×D811yescharacteristic
200C2×Q8—24no—
201C2×Q8—42no—
202C2×Q8—42no—
203C2×Q8—42no—
204C2×Q8—42no—
205C2×Q8—42no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—82no—
217C22×C4—162no—
218C22×C4C2×C22⋊C412yes—
219C22×C4C2×D812yes—
220C22×C4C2×D812yes—
221C22×C4C2×D811yescharacteristic
222C22×C4C2×SD1612yes—
223C22×C4C22×D411yescharacteristic
224Q16—24no—
225Q16—24no—
226Q16—42no—
227Q16—42no—
228Q16—48no—
229M16—28no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—24no—
235C2×C8—24no—
236C2×C8C2×C22⋊C412yes—
237C2×C8C2×C22⋊C412yes—
238C2×C8C2×C22⋊C412yes—
239C2×C8AGL(1,ℤ/8)12yes—
240C2×C8#13712yes—
241C2×C8C23×C411yescharacteristic
242C2×C8C22×D411yescharacteristic
243C2×C8H(5,𝔽2)12yes—
244C2×C82−1+412yes—
245C4⋊C4—24no—
246C4⋊C4—42no—
247C4⋊C4—48no—
248C4⋊C4—48no—
249C4⋊C4—48no—
250C4⋊C4—48no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—82no—
260C4⋊C4—162no—
261C4⋊C4—162no—
262C4⋊C4#10218yes—
263C22⋊C4—48no—
264C22⋊C4—82no—
265C22⋊C4#10218yes—
266C42—48no—
267C42—82no—
268C42C2×C22⋊C412yes—
269C42C2×C22⋊C412yes—
270C42C2×SD1612yes—
271C42C2×SD1612yes—
272C16—42no—
273C16—42no—
274C23—22no—
275C23—22no—
276C23—21no—
277C23—21no—
278C23C2×#10212yes—
279C23C22×D811yescharacteristic
280Q8—44no—
281Q8—44no—
282Q8—82no—
283Q8—82no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—164no—
309C2×C4C2×#10214yes—
310C2×C4C2×#10214yes—
311C2×C4C2×#10212yes—
312C2×C4C22×C22⋊C411yescharacteristic
313C2×C4#53511yescharacteristic
314C2×C4#53611yescharacteristic
315C2×C4C22×SD1611yescharacteristic
316C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
317C2×C4C2×#13711yescharacteristic
318C8—22no—
319C8—22no—
320C8#41714yes—
321C8#52012yes—
322C8#53712yes—
323C22—22no—
324C22—24no—
325C22—42no—
326C22—41no—
327C22—41no—
328C22C22×#10211yescharacteristic
329C22#320511yescharacteristic
330C22#322611yescharacteristic
331C22#341612yes—
332C22#341712yes—
333C4—22no—
334C4—24no—
335C4—24no—
336C4—84no—
337C4—84no—
338C4—161no—
339C4—161no—
340C4—161no—
341C4—161no—
342C4#231111yescharacteristic
343C4C2×#41711yescharacteristic
344C4#320611yescharacteristic
345C4#322711yescharacteristic
346C2—42no—
347C2#2185911yescharacteristic
348C2#3338212yes—
349C2#3378811yescharacteristic
350C2#3378911yescharacteristic
351C2#3380812yes—
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13284C2semidirect
5. #13281C2semidirect
6. #13207C2semidirect
7. #13203C2semidirect
10. #12814C2semidirect
11. #12812C2semidirect
14. #3177C4semidirect
64. #499C2×C4semidirect
66. #494C2×C4semidirect
73. C4×Q16C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31772
C2×#21642
#208032
#132842
#132812
#132072
#132032
#129942
#129932
#128142
#128122