Finite Group Databasebeta

← #323289

Subgroups — #323289

Order 512.

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

normal subgroups only

This group has 684 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
2#29219C212yesmaximal
3#29141C212yesmaximal
4C4×#498C211yesmaximal, characteristic
5#10681C214yesmaximal
6#9819C212yesmaximal
7#8815C212yesmaximal
8C8×C4⋊C8C211yesmaximal, characteristic
9C2×#498—22no—
10C2×#498C2211yescharacteristic
11#2868C2211yescharacteristic
12#2868C2211yescharacteristic
13C4×C4⋊Q8C2212yes—
14#2099—24no—
15#2024—24no—
16#1990C2214yes—
17#1990C2212yes—
18#1988—24no—
19#1956C2214yes—
20#1941—22no—
21#1844—22no—
22#1844C2212yes—
23#1844C2212yes—
24#1844C2211yescharacteristic
25#1834C2214yes—
26#1834C2214yes—
27#1803—24no—
28C4×#106—22no—
29C4×C4⋊C8C2212yes—
30C42×C8C2211yescharacteristic
31C2×C82C2212yes—
32#1273C2212yes—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8—22no—
36C2×C4⋊Q8—22no—
37C2×C4⋊Q8C2312yes—
38C2×C4×Q8—22no—
39#498—22no—
40#498—48no—
41#498Q814yescomplemented
42C2×#106—22no—
43C2×#106—24no—
44C2×#106—24no—
45C2×#106—22no—
46C2×C4⋊C8D414yes—
47C2×#103—28no—
48C2×C4×C8D412yes—
49C2×C4×C8D412yes—
50C2×C4×C8C2314yes—
51C2×C4×C8C2311yescharacteristic
52C2×C4×C8C2311yescharacteristic
53#391—22no—
54#389—22no—
55#389—22no—
56#384—22no—
57#384—22no—
58#384—21no—
59#384—21no—
60#384C2312yes—
61#384C2312yes—
62#384C2312yes—
63C4×C4⋊C4—22no—
64C4×C4⋊C4—22no—
65C43C2311yescharacteristic
66#340—28no—
67#340—28no—
68#340—28no—
69#340—24no—
70#336D414yes—
71C82—24no—
72C22×Q8—22no—
73C22×Q8—41no—
74C22×Q8—41no—
75C22×C8C2×D418yes—
76C22×C8C2×D412yes—
77C22×C8C2×D412yes—
78C4⋊Q8—22no—
79C4⋊Q8—22no—
80C4⋊Q8—22no—
81C4⋊Q8—44no—
82C4⋊Q8—44no—
83C4⋊Q8—48no—
84C4⋊Q8—84no—
85C4⋊Q8C2×Q814yes—
86C4×Q8—48no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—24no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—42no—
101C2×C42—42no—
102C2×C42C2×D412yes—
103C2×C42C2×D411yescharacteristic
104C2×C42C2×D411yescharacteristic
105C2×C42C2×D412yes—
106C2×C42C2411yescharacteristic
107#106—28no—
108#106—48no—
109#106—48no—
110#106—44no—
111C4⋊C8—28no—
112#103—88no—
113C4×C8—24no—
114C4×C8—24no—
115C4×C8—22no—
116C4×C8—24no—
117C4×C8SD1618yes—
118C4×C8C2×Q812yes—
119C4×C8#4112yes—
120#95—42no—
121#95—42no—
122C2×Q8—24no—
123C2×Q8—42no—
124C2×Q8—44no—
125C2×Q8—44no—
126C2×Q8—84no—
127C2×Q8—82no—
128C2×Q8—82no—
129C2×Q8—84no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4#12711yescharacteristic
135C22×C4#12712yes—
136C22×C4#12711yescharacteristic
137C22×C4C22×D411yescharacteristic
138C22×C4C22×D412yes—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—28no—
142C2×C8—28no—
143C2×C8—28no—
144C2×C8C22⋊Q814yes—
145C2×C8C2×SD1614yes—
146C2×C8C2×SD1614yes—
147C2×C8#13714yes—
148C4⋊C4—24no—
149C4⋊C4—24no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4—44no—
153C4⋊C4—44no—
154C4⋊C4—44no—
155C4⋊C4—48no—
156C4⋊C4—44no—
157C4⋊C4—48no—
158C4⋊C4—44no—
159C4⋊C4—84no—
160C4⋊C4—82no—
161C4⋊C4—82no—
162C4⋊C4—84no—
163C4⋊C4—84no—
164C42—22no—
165C42—24no—
166C42—84no—
167C42C2×SD1614yes—
168C42#13518yes—
169C42C22×Q811yescharacteristic
170C42C2×#4111yescharacteristic
171C422−1+412yes—
172C23C2×#12711yescharacteristic
173Q8—48no—
174Q8—84no—
175Q8—164no—
176Q8—164no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—44no—
181C2×C4—82no—
182C2×C4—82no—
183C2×C4—82no—
184C2×C4—82no—
185C2×C4—82no—
186C2×C4—84no—
187C2×C4—82no—
188C2×C4—84no—
189C2×C4—82no—
190C2×C4#49212yes—
191C2×C4#49212yes—
192C2×C4#49514yes—
193C2×C4#49514yes—
194C2×C4C2×C22⋊Q811yescharacteristic
195C2×C4#53611yescharacteristic
196C2×C4#54712yes—
197C2×C4D4×Q812yes—
198C2×C4C22×SD1612yes—
199C2×C4C2×#13514yes—
200C2×C4C2×#13712yes—
201C2×C4#57714yes—
202C8—24no—
203C8—48no—
204C8#47518yes—
205C22#254211yescharacteristic
206C22C2×#49211yescharacteristic
207C22C2×#49512yes—
208C22#315211yescharacteristic
209C22#315412yes—
210C4—24no—
211C4—22no—
212C4—84no—
213C4—162no—
214C4—162no—
215C4—162no—
216C4#170814yes—
217C4C2×#47512yes—
218C4#322912yes—
219C4#334014yes—
220C4#338114yes—
221C2C2×#170811yescharacteristic
222C2#1962612yes—
223C2#1965111yescharacteristic
224C2#2851211yescharacteristic
225C2#2852912yes—
subgroupG/H
center, socle, prime radicalC23C2×#127
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
41. #498Q8semidirect

Maximal subgroups

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

subgroupindex
#333102
#292192
#291412
C4×#4982
#106812
#98192
#88152
C8×C4⋊C82