Finite Group Databasebeta

← #353080

Subgroups — #353080

Order 512.

618 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 234 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#33291C211yesmaximal, characteristic
2#33289C211yesmaximal, characteristic
3#30305C212yesmaximal
4#30304C212yesmaximal
5#8821C214yesmaximal
6#8817C212yesmaximal
7#8814C212yesmaximal
8#7841C211yesmaximal, characteristic
9#2868C2212yes—
10#2859C2212yes—
11#2084—24no—
12#2083—24no—
13#2058—24no—
14#2057—24no—
15#1991—24no—
16#1990—24no—
17#1845C2214yes—
18#1844C2214yes—
19#1834C2212yes—
20#1834C2212yes—
21#1834C2211yescharacteristic
22#1834C2211yescharacteristic
23#1834C2212yes—
24#1834C2214yes—
25#1834C2214yes—
26#1744C2211yescharacteristic
27#1744C2211yescharacteristic
28C42×C8C2211yescharacteristic
29C2×#322C2212yes—
30C2×C82C2212yes—
31C2×C4⋊Q8—22no—
32C2×C4⋊Q8—22no—
33C2×C4⋊Q8—22no—
34C2×#125—22no—
35C2×#107—28no—
36C2×#106—28no—
37C2×C4×C8D414yes—
38C2×C4×C8C2311yescharacteristic
39C2×C4×C8C2311yescharacteristic
40C2×C4×C8C2312yes—
41C2×C4×C8C2312yes—
42#389—22no—
43#389—22no—
44#389—22no—
45#389—22no—
46#384—22no—
47#384—22no—
48#384—22no—
49#384C2312yes—
50#384C2314yes—
51#384C2312yes—
52#382—22no—
53C43C2311yescharacteristic
54#340—28no—
55#340—28no—
56#340—24no—
57#340—24no—
58#340—22no—
59#340—22no—
60#340—24no—
61#336D414yes—
62#336D412yes—
63#336D412yes—
64#322Q818yescomplemented
65C82Q818yescomplemented
66C22×Q8—42no—
67C22×C8C2×D414yes—
68C22×C8C2×D414yes—
69C22×C8C2×D412yes—
70C22×C8C2×D412yes—
71C4⋊Q8—48no—
72C4⋊Q8—84no—
73C4⋊Q8—84no—
74#125—48no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—24no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C42C2×D411yescharacteristic
90C2×C42C2×D412yes—
91C2×C42C2×D411yescharacteristic
92C2×C42C2×D412yes—
93C2×C42C2411yescharacteristic
94#107—88no—
95#106—88no—
96C4×C8—28no—
97C4×C8C2×Q814yes—
98C4×C8C2×Q814yes—
99C4×C8C2×Q814yes—
100C4×C8C2×Q814yes—
101C4×C8C2×Q814yes—
102C4×C8C2×Q814yes—
103#95—42no—
104#95—42no—
105C2×Q8—84no—
106C2×Q8—84no—
107C2×Q8—84no—
108C22×C4—42no—
109C22×C4—42no—
110C22×C4—44no—
111C22×C4#12712yes—
112C22×C4#12712yes—
113C22×C4C22×D411yescharacteristic
114C22×C4C22×D411yescharacteristic
115C22×C4C22×D411yescharacteristic
116C2×C8—24no—
117C2×C8—24no—
118C2×C8—24no—
119C2×C8—28no—
120C2×C8—24no—
121C2×C8C4⋊Q814yes—
122C2×C8C4⋊Q814yes—
123C2×C8C4⋊Q818yes—
124C2×C8C4⋊Q818yes—
125C4⋊C4—44no—
126C4⋊C4—44no—
127C4⋊C4—48no—
128C4⋊C4—44no—
129C4⋊C4—48no—
130C4⋊C4—44no—
131C4⋊C4—48no—
132C4⋊C4—44no—
133C4⋊C4—44no—
134C4⋊C4—84no—
135C4⋊C4—84no—
136C4⋊C4—84no—
137C4⋊C4—84no—
138C4⋊C4—84no—
139C42—22no—
140C42—24no—
141C42—22no—
142C42#13518yes—
143C42C22×Q812yes—
144C42C22×Q812yes—
145C23C2×#12711yescharacteristic
146Q8—168no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—82no—
152C2×C4—82no—
153C2×C4—84no—
154C2×C4—84no—
155C2×C4—82no—
156C2×C4—82no—
157C2×C4—84no—
158C2×C4—82no—
159C2×C4—82no—
160C2×C4#49514yes—
161C2×C4#49514yes—
162C2×C4C2×C4⋊Q811yescharacteristic
163C2×C4C2×C4⋊Q811yescharacteristic
164C2×C4C2×C4⋊Q812yes—
165C2×C4C2×C4⋊Q812yes—
166C2×C4C2×#13514yes—
167C2×C4#57612yes—
168C2×C4#57714yes—
169C2×C4#57812yes—
170C8—24no—
171C8—24no—
172C8—28no—
173C8—28no—
174C22#286811yescharacteristic
175C22C2×#49512yes—
176C22#315412yes—
177C22#315511yescharacteristic
178C22#315611yescharacteristic
179C4—24no—
180C4—24no—
181C4—162no—
182C4—164no—
183C4—162no—
184C4#316012yes—
185C4#316012yes—
186C4#316314yes—
187C4#316414yes—
188C2#1965812yes—
189C2#1965912yes—
190C2#3331311yescharacteristic
191C2#3331811yescharacteristic
192C2#3331911yescharacteristic
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
64. #322Q8semidirect
65. C82Q8semidirect

Maximal subgroups

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

subgroupindex
#332912
#332892
#303052
#303042
#88212
#88172
#88142
#78412