Finite Group Databasebeta

← #323281

Subgroups — #323281

Order 512.

756 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 258 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#29206C212yesmaximal
3#28512C212yesmaximal
4C4×#498C211yesmaximal, characteristic
5#10685C214yesmaximal
6#9819C212yesmaximal
7#8813C212yesmaximal
8C8×C4⋊C8C211yesmaximal, characteristic
9C2×#498—22no—
10C2×#498C2211yescharacteristic
11C2×#475—24no—
12#2868C2211yescharacteristic
13#2868C2211yescharacteristic
14C4×C4⋊Q8C2212yes—
15#2096—24no—
16#2058—24no—
17#1990C2214yes—
18#1990C2212yes—
19#1956C2214yes—
20#1948—24no—
21#1941—22no—
22#1844—22no—
23#1844C2214yes—
24#1844C2212yes—
25#1844C2212yes—
26#1844C2214yes—
27#1844C2211yescharacteristic
28C4×#106—22no—
29C4×C4⋊C8C2212yes—
30C42×C8C2211yescharacteristic
31C2×C82C2212yes—
32#1273C2212yes—
33C2×C4⋊Q8—22no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—22no—
37C2×C4⋊Q8C2312yes—
38C2×C4×Q8—22no—
39#498—22no—
40#498—48no—
41#498Q814yescomplemented
42#475—416no—
43C2×#106—22no—
44C2×#106—28no—
45C2×#106—24no—
46C2×#106—24no—
47C2×#106—22no—
48C2×#106—28no—
49C2×C4⋊C8D414yes—
50C2×#103—28no—
51C2×C4×C8D412yes—
52C2×C4×C8D412yes—
53C2×C4×C8C2314yes—
54C2×C4×C8C2311yescharacteristic
55C2×C4×C8C2311yescharacteristic
56#391—22no—
57#389—22no—
58#389—22no—
59#384—22no—
60#384—21no—
61#384—21no—
62#384—22no—
63#384C2312yes—
64#384C2312yes—
65#384C2312yes—
66C4×C4⋊C4—22no—
67C4×C4⋊C4—22no—
68C43C2311yescharacteristic
69#340—28no—
70#340—24no—
71#336D414yes—
72C82—24no—
73C22×Q8—22no—
74C22×Q8—41no—
75C22×Q8—41no—
76C22×C8C2×D418yes—
77C22×C8C2×D412yes—
78C22×C8C2×D412yes—
79C4⋊Q8—22no—
80C4⋊Q8—22no—
81C4⋊Q8—22no—
82C4⋊Q8—48no—
83C4⋊Q8—44no—
84C4⋊Q8—44no—
85C4⋊Q8—84no—
86C4⋊Q8C2×Q814yes—
87C4×Q8—48no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—24no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—41no—
102C2×C42—42no—
103C2×C42C2×D412yes—
104C2×C42C2×D411yescharacteristic
105C2×C42C2×D412yes—
106C2×C42C2×D411yescharacteristic
107C2×C42C2411yescharacteristic
108#106—28no—
109#106—416no—
110#106—48no—
111#106—48no—
112#106—44no—
113#106—88no—
114C4⋊C8SD16116yes—
115#103—416no—
116C4×C8—24no—
117C4×C8—24no—
118C4×C8—22no—
119C4×C8—24no—
120C4×C8SD1618yes—
121C4×C8C2×Q812yes—
122C4×C8#4112yes—
123#95—42no—
124#95—42no—
125C2×Q8—24no—
126C2×Q8—42no—
127C2×Q8—44no—
128C2×Q8—44no—
129C2×Q8—84no—
130C2×Q8—82no—
131C2×Q8—84no—
132C2×Q8—82no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4#12711yescharacteristic
138C22×C4#12712yes—
139C22×C4#12711yescharacteristic
140C22×C4C22×D411yescharacteristic
141C22×C4C22×D412yes—
142C2×C8—22no—
143C2×C8—28no—
144C2×C8—22no—
145C2×C8—28no—
146C2×C8C22⋊Q814yes—
147C2×C8C2×SD1614yes—
148C2×C8C2×SD16116yes—
149C2×C8C2×SD1614yes—
150C2×C8#13714yes—
151C4⋊C4—24no—
152C4⋊C4—24no—
153C4⋊C4—42no—
154C4⋊C4—42no—
155C4⋊C4—48no—
156C4⋊C4—44no—
157C4⋊C4—44no—
158C4⋊C4—44no—
159C4⋊C4—44no—
160C4⋊C4—44no—
161C4⋊C4—48no—
162C4⋊C4—84no—
163C4⋊C4—84no—
164C4⋊C4—84no—
165C4⋊C4—82no—
166C4⋊C4—82no—
167C42—22no—
168C42—24no—
169C42—84no—
170C42C2×SD1614yes—
171C42C2×SD1618yes—
172C42C22×Q811yescharacteristic
173C42C2×#4111yescharacteristic
174C422−1+412yes—
175C23C2×#12711yescharacteristic
176Q8—48no—
177Q8—84no—
178Q8—164no—
179Q8—164no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—44no—
184C2×C4—82no—
185C2×C4—82no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—84no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4—84no—
192C2×C4—82no—
193C2×C4#49214yes—
194C2×C4#49212yes—
195C2×C4#49214yes—
196C2×C4#49212yes—
197C2×C4C2×C22⋊Q811yescharacteristic
198C2×C4#53611yescharacteristic
199C2×C4#54712yes—
200C2×C4D4×Q812yes—
201C2×C4C22×SD1612yes—
202C2×C4C22×SD1614yes—
203C2×C4C2×#13712yes—
204C2×C4#57714yes—
205C8—216no—
206C8—24no—
207C8#47518yescomplemented
208C22#254211yescharacteristic
209C22C2×#49211yescharacteristic
210C22C2×#49212yes—
211C22#315211yescharacteristic
212C22#315412yes—
213C4—24no—
214C4—22no—
215C4—84no—
216C4—162no—
217C4—162no—
218C4—162no—
219C4#170514yes—
220C4C2×#47512yes—
221C4#322912yes—
222C4#333514yes—
223C4Q8×SD1614yes—
224C2C2×#170511yescharacteristic
225C2#1962212yes—
226C2#1965111yescharacteristic
227C2#2851211yescharacteristic
228C2#2853012yes—
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
207. C8#475semidirect

Maximal subgroups

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

subgroupindex
#333102
#292062
#285122
C4×#4982
#106852
#98192
#88132
C8×C4⋊C82