Finite Group Databasebeta

← #353059

Subgroups — #353059

Order 512.

662 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 662 conjugacy classes of subgroups — too many to draw. Only its 250 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#33297C212yesmaximal
3#33256C211yesmaximal, characteristic
4#30304C212yesmaximal
5#8821C214yesmaximal
6#8815C212yesmaximal
7#8814C212yesmaximal
8C8×#97C211yesmaximal, characteristic
9C2×#498—22no—
10#2868C2212yes—
11#2859C2212yes—
12#2083—24no—
13#2057—24no—
14#1993—24no—
15#1992—24no—
16#1941—22no—
17#1941—22no—
18#1845C2214yes—
19#1844—22no—
20#1844C2212yes—
21#1844C2212yes—
22#1844C2211yescharacteristic
23#1844C2211yescharacteristic
24#1834C2214yes—
25#1834C2214yes—
26#1834C2214yes—
27#1834C2212yes—
28#1744C2211yescharacteristic
29C4×#97C2211yescharacteristic
30C42×C8C2211yescharacteristic
31C2×#322C2212yes—
32C2×C82C2212yes—
33C2×C4⋊Q8—22no—
34C2×C4⋊Q8—22no—
35C2×C4⋊Q8—22no—
36C2×#125—22no—
37#498—48no—
38C2×#107—28no—
39C2×#106—24no—
40C2×#106—24no—
41C2×#106—22no—
42C2×#106—22no—
43C2×#97D414yes—
44C2×C4×C8D412yes—
45C2×C4×C8D412yes—
46C2×C4×C8C2312yes—
47C2×C4×C8C2311yescharacteristic
48C2×C4×C8C2312yes—
49C2×C4×C8C2311yescharacteristic
50#389—22no—
51#389—22no—
52#389—22no—
53#389—22no—
54#384—22no—
55#384—22no—
56#384—22no—
57#384C2312yes—
58#384C2312yes—
59#384C2314yes—
60#382—22no—
61C43C2311yescharacteristic
62#340—28no—
63#340—24no—
64#340—28no—
65#340—28no—
66#336D414yes—
67#322Q818yescomplemented
68C82Q818yescomplemented
69C22×Q8—42no—
70C22×C8C2×D414yes—
71C22×C8C2×D414yes—
72C22×C8C2×D412yes—
73C22×C8C2×D412yes—
74C4⋊Q8—48no—
75C4⋊Q8—48no—
76C4⋊Q8—84no—
77#125—48no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—42no—
92C2×C42C2×D411yescharacteristic
93C2×C42C2×D412yes—
94C2×C42C2×D411yescharacteristic
95C2×C42C2×D412yes—
96C2×C42C2411yescharacteristic
97#107—88no—
98#106—48no—
99#106—44no—
100#106—48no—
101#106—44no—
102#97—28no—
103C4×C8—24no—
104C4×C8—24no—
105C4×C8C2×Q814yes—
106C4×C8C2×Q814yes—
107C4×C8C2×Q814yes—
108C4×C8C2×Q814yes—
109C4×C8C2×Q814yes—
110C4×C8C2×Q814yes—
111#95—42no—
112#95—42no—
113C2×Q8—84no—
114C2×Q8—84no—
115C2×Q8—84no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4—44no—
119C22×C4#12712yes—
120C22×C4#12712yes—
121C22×C4C22×D411yescharacteristic
122C22×C4C22×D411yescharacteristic
123C22×C4C22×D411yescharacteristic
124C2×C8—28no—
125C2×C8—22no—
126C2×C8—22no—
127C2×C8—28no—
128C2×C8C4⋊Q818yes—
129C2×C8C4⋊Q818yes—
130C2×C8C4⋊Q814yes—
131C2×C8C4⋊Q814yes—
132C2×C8AGL(1,ℤ/8)14yes—
133C2×C8#13714yes—
134C4⋊C4—44no—
135C4⋊C4—44no—
136C4⋊C4—48no—
137C4⋊C4—48no—
138C4⋊C4—44no—
139C4⋊C4—44no—
140C4⋊C4—44no—
141C4⋊C4—48no—
142C4⋊C4—44no—
143C4⋊C4—84no—
144C4⋊C4—84no—
145C4⋊C4—84no—
146C4⋊C4—84no—
147C4⋊C4—84no—
148C42—22no—
149C42—22no—
150C42—24no—
151C42#13518yes—
152C42C22×Q812yes—
153C42C22×Q812yes—
154C23C2×#12711yescharacteristic
155Q8—168no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—82no—
161C2×C4—82no—
162C2×C4—84no—
163C2×C4—84no—
164C2×C4—82no—
165C2×C4—82no—
166C2×C4—84no—
167C2×C4—82no—
168C2×C4—82no—
169C2×C4#49514yes—
170C2×C4#49514yes—
171C2×C4C2×C4⋊Q812yes—
172C2×C4C2×C4⋊Q811yescharacteristic
173C2×C4C2×C4⋊Q812yes—
174C2×C4C2×C4⋊Q811yescharacteristic
175C2×C4C2×#13514yes—
176C2×C4C2×AGL(1,ℤ/8)12yes—
177C2×C4C2×#13712yes—
178C2×C4#57714yes—
179C8—28no—
180C8—28no—
181C8—24no—
182C8#50118yes—
183C22#286811yescharacteristic
184C22C2×#49512yes—
185C22#315111yescharacteristic
186C22#315211yescharacteristic
187C22#315412yes—
188C4—24no—
189C4—24no—
190C4—162no—
191C4—164no—
192C4—162no—
193C4#316014yes—
194C4C2×#50112yes—
195C4#316314yes—
196C4#316512yes—
197C2#1962512yes—
198C2#1962612yes—
199C2#3331311yescharacteristic
200C2#3331611yescharacteristic
201C2#3331811yescharacteristic
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
67. #322Q8semidirect
68. C82Q8semidirect

Maximal subgroups

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

subgroupindex
#333102
#332972
#332562
#303042
#88212
#88152
#88142
C8×#972