Finite Group Databasebeta

← #294898

Subgroups — #294898

Order 512.

780 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 170 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#39642C211yesmaximal, characteristic
2#33225C211yesmaximal, characteristic
3#29799C211yesmaximal, characteristic
4#29225C211yesmaximal, characteristic
5#10065C214yesmaximal
6#9871C211yesmaximal, characteristic
7#9702C211yesmaximal, characteristic
8#9060C214yesmaximal
9#8176C211yesmaximal, characteristic
10C2×#487C2214yes—
11C2×#462C2212yes—
12#2868C2211yescharacteristic
13#2868C2211yescharacteristic
14#2713—24no—
15#2647—21no—
16#2567C2212yes—
17#2552—22no—
18#2511—22no—
19#2506—22no—
20#2369C2212yes—
21#2330—21no—
22C4×#125C2211yescharacteristic
23#2295C2211yescharacteristic
24#2100—22no—
25#2043—22no—
26#2016—22no—
27#1992—21no—
28#1988C2214yes—
29#1981C2212yes—
30#1956—24no—
31#1836—21no—
32#1831C2211yescharacteristic
33#1767C2211yescharacteristic
34C42×C8C2211yescharacteristic
35C2×#333C2214yes—
36C2×Q8⋊C8C2212yes—
37#1279C2214yes—
38#1279C2212yes—
39C22×Q16—22no—
40C2×C4⋊Q8—22no—
41C2×C4⋊Q8D414yes—
42C2×C4⋊Q8C2312yes—
43C2×C4⋊Q8C2312yes—
44C2×#125—24no—
45C2×#125D411yescharacteristic
46C2×C4×Q8C2311yescharacteristic
47#487—28no—
48#487—44no—
49#462—44no—
50C2×C4⋊C8C2312yes—
51C2×C4⋊C8C2312yes—
52C2×#103—28no—
53C2×#103—22no—
54C2×#103—24no—
55C2×#97—21no—
56C2×C4×C8—22no—
57C2×C4×C8C2312yes—
58#400—24no—
59#400—24no—
60#398—24no—
61#398—24no—
62#395—24no—
63#391—22no—
64#391—24no—
65#391D412yes—
66#389—24no—
67#389—22no—
68#389—22no—
69#389—22no—
70#389—22no—
71#389D412yes—
72#389D412yes—
73#387—24no—
74#387—24no—
75#387—22no—
76#387—22no—
77#384—22no—
78#384—21no—
79#384—21no—
80#384—22no—
81#384—21no—
82#384—21no—
83#384—22no—
84#382D411yescharacteristic
85C4×C4⋊C4—22no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4C2312yes—
90C4×C4⋊C4C2311yescharacteristic
91#376—21no—
92C43C2311yescharacteristic
93#340—24no—
94#340—24no—
95#336—22no—
96#336—21no—
97#333—44no—
98Q8⋊C8—24no—
99C22×Q8—22no—
100C22×Q8C2×D411yescharacteristic
101C22×Q8C2×D414yes—
102C2×Q16—44no—
103C2×Q16—84no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—22no—
107C4⋊Q8—24no—
108C4⋊Q8—24no—
109C4⋊Q8—24no—
110C4⋊Q8—24no—
111C4⋊Q8—48no—
112C4⋊Q8—84no—
113#125—22no—
114#125—88no—
115#125Q1614yescomplemented
116C4×Q8—22no—
117C4×Q8—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4C2×D411yescharacteristic
134C2×C4⋊C4C2×D412yes—
135C2×C4⋊C4C2×D412yes—
136C2×C4⋊C4C2×D412yes—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42—22no—
140C2×C42—24no—
141C2×C42C2×D412yes—
142C2×C42C2×D412yes—
143C2×C42C2×D411yescharacteristic
144C2×C42C2×D411yescharacteristic
145C2×C42C2411yescharacteristic
146C4⋊C8—22no—
147C4⋊C8—22no—
148C4⋊C8—24no—
149#103—44no—
150#103—416no—
151#103—84no—
152#97—42no—
153C4×C8—22no—
154C4×C8—44no—
155C4×C8#4114yes—
156#95—24no—
157#95—24no—
158#95—24no—
159#95—24no—
160#95—22no—
161#95—22no—
162#95—22no—
163#95—22no—
164#95—22no—
165#95—21no—
166#95—22no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—28no—
170C2×Q8—24no—
171C2×Q8—44no—
172C2×Q8—44no—
173C2×Q8—42no—
174C2×Q8—42no—
175C2×Q8—44no—
176C2×Q8—44no—
177C22×C4—22no—
178C22×C4—21no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—24no—
182C22×C4—21no—
183C22×C4—42no—
184C22×C4#12012yes—
185C22×C4#12012yes—
186C22×C4C22×D411yescharacteristic
187C22×C4C22×D411yescharacteristic
188C22×C4C22×D411yescharacteristic
189Q16—88no—
190C2×C8—24no—
191C2×C8—44no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—42no—
195C2×C8—44no—
196C2×C8—42no—
197C4⋊C4—22no—
198C4⋊C4—28no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—24no—
202C4⋊C4—24no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—44no—
207C4⋊C4—44no—
208C4⋊C4—44no—
209C4⋊C4—48no—
210C4⋊C4—44no—
211C4⋊C4—44no—
212C4⋊C4—84no—
213C4⋊C4—84no—
214C4⋊C4—84no—
215C4⋊C4—84no—
216C4⋊C4—84no—
217C4⋊C4—84no—
218C4⋊C4C2×Q1614yes—
219C42—24no—
220C42—22no—
221C42—24no—
222C42—42no—
223C42—44no—
224C42—42no—
225C42—84no—
226C42C2×Q1612yes—
227C42#13712yes—
228C42C2×#4112yes—
229C42H(5,𝔽2)12yes—
230C23C2×#12011yescharacteristic
231Q8—42no—
232Q8—42no—
233Q8—88no—
234Q8—84no—
235Q8—82no—
236Q8—82no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—24no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—44no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—44no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—44no—
258C2×C4—82no—
259C2×C4—84no—
260C2×C4#45114yes—
261C2×C4#45514yes—
262C2×C4#53211yescharacteristic
263C2×C4#53511yescharacteristic
264C2×C4#54612yes—
265C2×C4#55012yes—
266C2×C4C22×Q1611yescharacteristic
267C2×C4C2×#13711yescharacteristic
268C2×C4#57714yes—
269C2×C4#57812yes—
270C8—44no—
271C8—82no—
272C8—82no—
273C22#243111yescharacteristic
274C22C2×#45111yescharacteristic
275C22#301012yes—
276C22#301111yescharacteristic
277C22C2×#45512yes—
278C4—24no—
279C4—24no—
280C4—22no—
281C4—44no—
282C4—82no—
283C4—82no—
284C4—81no—
285C4—81no—
286C4—84no—
287C4—162no—
288C4#317412yes—
289C4#322212yes—
290C4#339512yes—
291C4#339812yes—
292C2#1863312yes—
293C2#1869012yes—
294C2#2559011yescharacteristic
295C2#2560711yescharacteristic
296C2#2563411yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#120
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
115. #125Q16semidirect

Maximal subgroups

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

subgroupindex
#396422
#332252
#297992
#292252
#100652
#98712
#97022
#90602
#81762