Finite Group Databasebeta

← #353084

Subgroups — #353084

Order 512.

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

normal subgroups only

This group has 642 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#33310C211yesmaximal, characteristic
2#33291C211yesmaximal, characteristic
3#30305C212yesmaximal
4#30304C212yesmaximal
5#8821C214yesmaximal
6#8819C214yesmaximal
7#7843C211yesmaximal, characteristic
8C2×#498—22no—
9#2868C2212yes—
10#2859C2211yescharacteristic
11#2859C2211yescharacteristic
12#2084—24no—
13#2083—24no—
14#2058—24no—
15#2057—24no—
16#1991—22no—
17#1991—22no—
18#1845C2214yes—
19#1844—22no—
20#1844C2214yes—
21#1844C2214yes—
22#1834C2212yes—
23#1834C2212yes—
24#1834C2214yes—
25#1834C2214yes—
26#1744C2212yes—
27C42×C8C2211yescharacteristic
28C2×#322C2214yes—
29C2×C4⋊Q8—22no—
30C2×C4⋊Q8—22no—
31C2×C4⋊Q8—22no—
32C2×#125—21no—
33C2×#125—21no—
34#498—48no—
35C2×#107—28no—
36C2×#106—28no—
37C2×#106—28no—
38C2×C4×C8D412yes—
39C2×C4×C8D412yes—
40C2×C4×C8C2312yes—
41C2×C4×C8C2314yes—
42#389—22no—
43#389—22no—
44#389—22no—
45#389—22no—
46#384—22no—
47#384—22no—
48#384—22no—
49#384C2311yescharacteristic
50#384C2312yes—
51#384C2311yescharacteristic
52#384C2314yes—
53#382—21no—
54#382—21no—
55C43C2311yescharacteristic
56#340—28no—
57#340—28no—
58#340—24no—
59#340—24no—
60#336D414yes—
61#336D414yes—
62#322Q8116yescomplemented
63C22×Q8—42no—
64C22×C8C2×D418yes—
65C22×C8C2×D414yes—
66C4⋊Q8—48no—
67C4⋊Q8—84no—
68C4⋊Q8—84no—
69#125—44no—
70#125—44no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—24no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C42C2×D412yes—
88C2×C42C2×D412yes—
89C2×C42C2×D411yescharacteristic
90C2×C42C2×D411yescharacteristic
91C2×C42C2411yescharacteristic
92#107—88no—
93#106—416no—
94#106—88no—
95C4×C8—24no—
96C4×C8—24no—
97C4×C8C2×Q818yes—
98C4×C8C2×Q8116yes—
99#95—42no—
100#95—41no—
101#95—41no—
102C2×Q8—84no—
103C2×Q8—84no—
104C2×Q8—84no—
105C22×C4—42no—
106C22×C4—41no—
107C22×C4—41no—
108C22×C4—44no—
109C22×C4#12712yes—
110C22×C4#12711yescharacteristic
111C22×C4#12711yescharacteristic
112C22×C4C22×D411yescharacteristic
113C22×C4C22×D412yes—
114C2×C8—24no—
115C2×C8—24no—
116C2×C8—28no—
117C2×C8—28no—
118C2×C8C4⋊Q8116yes—
119C2×C8C4⋊Q818yes—
120C4⋊C4—48no—
121C4⋊C4—44no—
122C4⋊C4—42no—
123C4⋊C4—42no—
124C4⋊C4—44no—
125C4⋊C4—48no—
126C4⋊C4—44no—
127C4⋊C4—48no—
128C4⋊C4—44no—
129C4⋊C4—44no—
130C4⋊C4—84no—
131C4⋊C4—84no—
132C4⋊C4—84no—
133C4⋊C4—84no—
134C4⋊C4—82no—
135C4⋊C4—82no—
136C42—24no—
137C42—24no—
138C42AGL(1,ℤ/8)14yes—
139C42#13714yes—
140C42C22×Q814yes—
141C23C2×#12711yescharacteristic
142Q8—168no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—82no—
148C2×C4—84no—
149C2×C4—81no—
150C2×C4—81no—
151C2×C4—84no—
152C2×C4—84no—
153C2×C4—82no—
154C2×C4—82no—
155C2×C4—82no—
156C2×C4—82no—
157C2×C4#49614yes—
158C2×C4#49712yes—
159C2×C4#49712yes—
160C2×C4C2×C4⋊Q814yes—
161C2×C4C2×C4⋊Q812yes—
162C2×C4C2×AGL(1,ℤ/8)12yes—
163C2×C4C2×#13712yes—
164C2×C4#57714yes—
165C2×C4#57714yes—
166C8—216no—
167C8—28no—
168C22#286811yescharacteristic
169C22C2×#49611yescharacteristic
170C22C2×#49711yescharacteristic
171C22#315412yes—
172C22#315412yes—
173C4—24no—
174C4—22no—
175C4—22no—
176C4—162no—
177C4—164no—
178C4—161no—
179C4—161no—
180C4#316318yes—
181C4#316514yes—
182C2#1969712yes—
183C2#1970211yescharacteristic
184C2#1970311yescharacteristic
185C2#3331411yescharacteristic
186C2#3331812yes—
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
62. #322Q8semidirect

Maximal subgroups

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

subgroupindex
#333102
#332912
#303052
#303042
#88212
#88192
#78432