Finite Group Databasebeta

← #149783

Subgroups — #149783

Order 512.

406 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 66 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
1C2×#1992C211yesmaximal, characteristic
2#11512C212yesmaximal
3#7358C214yesmaximal, complemented
4C22×#125C411yescharacteristic
5C22×#97C411yescharacteristic
6C2×#384C2211yescharacteristic
7#1992—22no—
8#1992C412yes—
9#1992C2212yes—
10C2×#340—22no—
11#1463—22no—
12#1325—22no—
13#1325C2214yes—
14C23×C8—21no—
15C2×#125—24no—
16C2×#125—21no—
17C2×#125—42no—
18C2×#125—41no—
19C2×#125C2×C412yes—
20C22×C4⋊C4—21no—
21C22×C4⋊C4—41no—
22C22×C4⋊C4—41no—
23C22×C4⋊C4D412yes—
24C22×C42C2×C411yescharacteristic
25#407—42no—
26C2×#97—21no—
27C2×#97—42no—
28C2×#97C2×C412yes—
29#384—41no—
30#384C2×C411yescharacteristic
31#384D412yes—
32#384C2311yescharacteristic
33#340—24no—
34#340—24no—
35#324—24no—
36#324—42no—
37C23×C4—21no—
38C23×C4—22no—
39C23×C4—41no—
40C23×C4C22⋊C411yescharacteristic
41C2×M16—82no—
42C22×C8—24no—
43C22×C8—22no—
44C22×C8—41no—
45C22×C8—41no—
46C22×C8—42no—
47#125—24no—
48#125—22no—
49#125—42no—
50#125—48no—
51#125—42no—
52#125M1614yescomplemented
53C2×C4⋊C4—24no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—42no—
57C2×C4⋊C4—42no—
58C2×C4⋊C4—48no—
59C2×C4⋊C4—42no—
60C2×C4⋊C4—41no—
61C2×C4⋊C4—42no—
62C2×C4⋊C4—42no—
63C2×C4⋊C4—41no—
64C2×C4⋊C4—42no—
65C2×C4⋊C4—41no—
66C2×C4⋊C4—84no—
67C2×C4⋊C4C22⋊C412yes—
68C2×C4⋊C4C2×D412yes—
69C2×C42—21no—
70C2×C42—22no—
71C2×C42—22no—
72C2×C42C22⋊C411yescharacteristic
73C2×C42C22×C411yescharacteristic
74C22⋊C8—44no—
75C22⋊C8—82no—
76#97—42no—
77#97—42no—
78C24#9911yescharacteristic
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—22no—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—24no—
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—22no—
88C22×C4—22no—
89C22×C4—42no—
90C22×C4—42no—
91C22×C4—44no—
92C22×C4—42no—
93C22×C4—42no—
94C22×C4—42no—
95C22×C4—82no—
96C22×C4—82no—
97C22×C4—84no—
98C22×C4#9911yescharacteristic
99C22×C4C2×C22⋊C411yescharacteristic
100M16—164no—
101C2×C8—24no—
102C2×C8—44no—
103C2×C8—44no—
104C2×C8—42no—
105C2×C8—42no—
106C2×C8—84no—
107C4⋊C4—24no—
108C4⋊C4—44no—
109C4⋊C4—42no—
110C4⋊C4—42no—
111C4⋊C4—44no—
112C4⋊C4—48no—
113C4⋊C4—44no—
114C4⋊C4—42no—
115C4⋊C4—82no—
116C4⋊C4—82no—
117C4⋊C4—84no—
118C4⋊C4—82no—
119C42—24no—
120C42—21no—
121C42—21no—
122C42#10012yes—
123C42C2×M1612yes—
124C23—21no—
125C23—22no—
126C23—22no—
127C23—21no—
128C23#35411yescharacteristic
129C23#35411yescharacteristic
130C23C2×#9911yescharacteristic
131C2×C4—22no—
132C2×C4—22no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—22no—
140C2×C4—22no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—48no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4—44no—
149C2×C4—88no—
150C2×C4—82no—
151C2×C4—82no—
152C2×C4—84no—
153C2×C4—82no—
154C2×C4—82no—
155C2×C4—162no—
156C2×C4#35612yes—
157C2×C4#40711yescharacteristic
158C2×C4C2×#10011yescharacteristic
159C8—44no—
160C8—164no—
161C22—24no—
162C22—24no—
163C22—21no—
164C22—21no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—21no—
169C22#141212yes—
170C22#141312yes—
171C22#146211yescharacteristic
172C22C2×#35411yescharacteristic
173C22C2×#35611yescharacteristic
174C4—21no—
175C4—22no—
176C4—21no—
177C4—42no—
178C4—42no—
179C4—84no—
180C4—82no—
181C4—82no—
182C2—21no—
183C2—21no—
184C2—22no—
185C2—22no—
186C2#1150611yescharacteristic
187C2C2×#141211yescharacteristic
188C2C2×#141311yescharacteristic
subgroupG/H
center, socleC22C2×#354
commutatorC2×C42C22×C4
Frattini, cosocle#384C23
prime radicalC24#99
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#384 ◁ G
LECSG ▷ #384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #7358C2semidirect
52. #125M16semidirect

Maximal subgroups

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

subgroupindex
C2×#19922
#115122
#73582