Finite Group Databasebeta

← #323288

Subgroups — #323288

Order 512.

778 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 778 conjugacy classes of subgroups — too many to draw. Only its 226 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#33310C211yesmaximal, characteristic, complemented
2#29218C212yesmaximal, complemented
3#29139C212yesmaximal
4C4×#492C211yesmaximal, characteristic
5#10686C212yesmaximal, complemented
6#10677C212yesmaximal
7#9820C212yesmaximal
8#8815C212yesmaximal, complemented
9C8×C4⋊C8C211yesmaximal, characteristic, complemented
10C2×#498—22no—
11C2×#492C2211yescharacteristic
12C2×C4×SD16—22no—
13#2868C2212yes—
14C4×C4⋊Q8C2211yescharacteristic
15C4×#127C2211yescharacteristic
16#2100—24no—
17#2023—24no—
18#1990C2214yes—
19#1987—24no—
20#1968—22no—
21#1957C2214yes—
22#1957C2211yescharacteristic
23#1956C2211yescharacteristic
24#1844—22no—
25#1844C2212yes—
26#1844C2212yes—
27#1844C2211yescharacteristic
28#1834C2214yes—
29#1834C2214yes—
30#1802—24no—
31C4×C4⋊C8C2212yes—
32C42×C8C2211yescharacteristic
33C2×C82C2212yes—
34#1273C2212yes—
35C22×SD16—22no—
36C2×C4⋊Q8—22no—
37C2×C4⋊Q8—22no—
38C2×C4⋊Q8—22no—
39C2×C4⋊Q8C2311yescharacteristic
40C2×#127C2311yescharacteristic
41C2×C4×Q8—21no—
42C2×C4×D4—22no—
43C2×C4×D4—21no—
44#498—48no—
45#492—22no—
46#492Q814yescomplemented
47C4×SD16—48no—
48C2×#106—24no—
49C2×#106—24no—
50C2×#106—22no—
51C2×C4⋊C8D414yes—
52C2×#103—22no—
53C2×#102—22no—
54C2×#102—28no—
55C2×C4×C8D412yes—
56C2×C4×C8D412yes—
57C2×C4×C8C2314yes—
58C2×C4×C8C2311yescharacteristic
59C2×C4×C8C2311yescharacteristic
60#391—21no—
61#390—21no—
62#390—22no—
63#389—22no—
64#384—22no—
65#384—22no—
66#384—22no—
67#384C2311yescharacteristic
68#384C2311yescharacteristic
69#384C2314yes—
70C4×C4⋊C4—22no—
71C43C2311yescharacteristic
72#340—28no—
73#340—28no—
74#340—28no—
75#336D414yes—
76C82—24no—
77C22×Q8—21no—
78C22×Q8—42no—
79C22×D4—21no—
80C22×D4—22no—
81C23×C4—41no—
82C2×SD16—28no—
83C2×SD16—44no—
84C2×SD16—44no—
85C22×C8C2×D418yes—
86C22×C8C2×D412yes—
87C22×C8C2×D412yes—
88C4⋊Q8—21no—
89C4⋊Q8—22no—
90C4⋊Q8—48no—
91C4⋊Q8—48no—
92C4⋊Q8—84no—
93C4⋊Q8C2×Q812yes—
94#127—21no—
95#127—22no—
96#127C2×Q812yes—
97C4×Q8—44no—
98C4×D4—44no—
99C4×D4—48no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—42no—
112C2×C22⋊C4—41no—
113C2×C22⋊C4—42no—
114C2×C42—41no—
115C2×C42C2×D411yescharacteristic
116C2×C42C2×D412yes—
117C2×C42C2×D411yescharacteristic
118C2×C42C2×D412yes—
119C2×C42C2411yescharacteristic
120#106—48no—
121#106—44no—
122#106—48no—
123C4⋊C8—28no—
124#103—44no—
125#102—44no—
126#102—88no—
127C4×C8—24no—
128C4×C8—24no—
129C4×C8—22no—
130C4×C8—24no—
131C4×C8SD1618yes—
132C4×C8C2×Q812yes—
133C4×C8#4112yes—
134#95—42no—
135C24—41no—
136C2×Q8—22no—
137C2×Q8—41no—
138C2×Q8—42no—
139C2×Q8—42no—
140C2×Q8—84no—
141C2×Q8—84no—
142C2×Q8—84no—
143C2×D4—22no—
144C2×D4—24no—
145C2×D4—41no—
146C2×D4—42no—
147C2×D4—42no—
148C2×D4—44no—
149C2×D4—42no—
150C2×D4—44no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—44no—
155C22×C4—82no—
156C22×C4—84no—
157C22×C4#12711yescharacteristic
158C22×C4#12712yes—
159C22×C4#12711yescharacteristic
160C22×C4C22×D411yescharacteristic
161C22×C4C22×D412yes—
162SD16—416no—
163C2×C8—22no—
164C2×C8—22no—
165C2×C8—28no—
166C2×C8—28no—
167C2×C8—28no—
168C2×C8C22⋊Q814yes—
169C2×C8C2×SD1614yes—
170C2×C8C2×SD1614yes—
171C2×C8AGL(1,ℤ/8)14yes—
172C4⋊C4—24no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—44no—
177C4⋊C4—48no—
178C4⋊C4—48no—
179C4⋊C4—48no—
180C4⋊C4—44no—
181C4⋊C4—84no—
182C4⋊C4—84no—
183C4⋊C4—82no—
184C4⋊C4—84no—
185C22⋊C4—82no—
186C22⋊C4—84no—
187C42—22no—
188C42—24no—
189C42—82no—
190C42C2×SD1614yes—
191C42#13518yes—
192C42C22×Q811yescharacteristic
193C42C2×#4111yescharacteristic
194C422−1+412yes—
195C23—42no—
196C23—81no—
197C23—82no—
198C23—81no—
199C23—82no—
200C23C2×#12711yescharacteristic
201Q8—44no—
202Q8—82no—
203Q8—168no—
204D4—48no—
205D4—44no—
206D4—84no—
207D4—82no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—42no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4—81no—
215C2×C4—81no—
216C2×C4—82no—
217C2×C4—84no—
218C2×C4—84no—
219C2×C4—84no—
220C2×C4—82no—
221C2×C4—164no—
222C2×C4#49212yes—
223C2×C4#49212yes—
224C2×C4#49514yes—
225C2×C4#49514yes—
226C2×C4C2×C22⋊Q811yescharacteristic
227C2×C4#53611yescharacteristic
228C2×C4#54712yes—
229C2×C4D4×Q812yes—
230C2×C4C22×SD1612yes—
231C2×C4C2×#13514yes—
232C2×C4C2×AGL(1,ℤ/8)12yes—
233C2×C4#57714yes—
234C8—24no—
235C8—48no—
236C8#47618yes—
237C22—82no—
238C22—84no—
239C22—161no—
240C22—162no—
241C22—161no—
242C22#254211yescharacteristic
243C22C2×#49211yescharacteristic
244C22C2×#49512yes—
245C22#315111yescharacteristic
246C22#315412yes—
247C4—24no—
248C4—22no—
249C4—82no—
250C4—161no—
251C4—164no—
252C4—161no—
253C4#170814yes—
254C4C2×#47612yes—
255C4#322712yes—
256C4#334014yes—
257C4#338114yes—
258C2—162no—
259C2C2×#170811yescharacteristic
260C2#1962512yes—
261C2#1965411yescharacteristic
262C2#2850911yescharacteristic
263C2#2852912yes—
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
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
1. #33310C2semidirect
2. #29218C2semidirect
5. #10686C2semidirect
8. #8815C2semidirect
9. C8×C4⋊C8C2semidirect
46. #492Q8semidirect

Maximal subgroups

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

subgroupindex
#333102
#292182
#291392
C4×#4922
#106862
#106772
#98202
#88152
C8×C4⋊C82