Finite Group Databasebeta

← #93021

Subgroups — #93021

Order 512.

249 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 249 conjugacy classes of subgroups — too many to draw. Only its 56 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#12977C211yesmaximal, characteristic
2#8614C211yesmaximal, characteristic
3#8608C211yesmaximal, characteristic
4C22×M32C411yescharacteristic, complemented
5C2×C8×D4C2211yescharacteristic
6C2×C22⋊C16—21no—
7C2×C22⋊C16C411yescharacteristic, complemented
8#1807—21no—
9#1800—21no—
10#1799—21no—
11#1566—21no—
12#1374—21no—
13#1292C412yescomplemented
14#1289C412yes—
15C23×C8C2×C411yescharacteristic
16C23×C8D411yescharacteristic
17C2×C4×D4D411yescharacteristic
18C2×M32—42no—
19C2×M32—41no—
20C22×C16—21no—
21C22×C16—21no—
22C8×D4—41no—
23C8×D4—41no—
24C2×#106—41no—
25#423—22no—
26C2×C4⋊C8C2×C411yescharacteristic
27C2×#100—41no—
28C2×#99—41no—
29C2×C22⋊C8C2×C411yescharacteristic
30C2×C22⋊C8Q811yescharacteristic
31C2×C4×C8D411yescharacteristic
32#380—22no—
33C22⋊C16—42no—
34C22⋊C16—41no—
35#343—41no—
36#342—41no—
37#340—41no—
38#334—41no—
39#334—41no—
40C22×D4D811yescharacteristic
41C23×C4C22⋊C411yescharacteristic
42C23×C4C4⋊C411yescharacteristic
43C2×M16—42no—
44C22×C8—21no—
45C22×C8—21no—
46C22×C8—21no—
47C22×C8—21no—
48C22×C8—41no—
49C22×C8—41no—
50C22×C8—41no—
51C22×C8—41no—
52C22×C8C4211yescharacteristic
53C22×C8C22⋊C411yescharacteristic
54C22×C8C4⋊C411yescharacteristic
55C22×C8C4⋊C411yescharacteristic
56C4×D4—41no—
57C4×D4—41no—
58C2×C4⋊C4—42no—
59C2×C4⋊C4SD1611yescharacteristic
60C2×C22⋊C4—42no—
61C2×C22⋊C4SD1611yescharacteristic
62C2×C22⋊C4Q1611yescharacteristic
63C2×C42C22⋊C411yescharacteristic
64M32—42no—
65M32—42no—
66C2×C16—22no—
67C2×C16—41no—
68C2×C16—41no—
69C2×C16—41no—
70C2×C16—41no—
71C2×C16—41no—
72#106—81no—
73#106—81no—
74C4⋊C8—41no—
75C4⋊C8—42no—
76C4⋊C8—42no—
77#100—82no—
78#99—82no—
79C22⋊C8—41no—
80C22⋊C8—41no—
81C4×C8—21no—
82C4×C8—21no—
83#95—42no—
84C24#10211yescharacteristic
85C24#10711yescharacteristic
86C2×D4—22no—
87C2×D4—22no—
88C2×D4—41no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—41no—
94C22×C4—41no—
95C22×C4—41no—
96C22×C4—41no—
97C22×C4—82no—
98C22×C4#9511yescharacteristic
99C22×C4#10211yescharacteristic
100C22×C4#10311yescharacteristic
101C22×C4#10311yescharacteristic
102C22×C4#10611yescharacteristic
103M16—82no—
104M16—82no—
105C2×C8—21no—
106C2×C8—21no—
107C2×C8—21no—
108C2×C8—21no—
109C2×C8—21no—
110C2×C8—42no—
111C2×C8—41no—
112C2×C8—41no—
113C2×C8—41no—
114C2×C8—41no—
115C2×C8—41no—
116C2×C8—41no—
117C2×C8—41no—
118C2×C8—81no—
119C2×C8—82no—
120C2×C8#9911yescharacteristic
121C2×C8#9911yescharacteristic
122C4⋊C4—41no—
123C4⋊C4—82no—
124C4⋊C4—82no—
125C22⋊C4—41no—
126C22⋊C4—41no—
127C22⋊C4—82no—
128C22⋊C4—82no—
129C42—21no—
130C42—21no—
131C16—42no—
132C16—81no—
133C23—21no—
134C23—21no—
135C23—21no—
136C23—21no—
137C23—21no—
138C23—41no—
139C23—41no—
140C23—41no—
141C23—41no—
142C23#34011yescharacteristic
143C23#36012yes—
144D4—44no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—42no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—81no—
161C2×C4—162no—
162C2×C4—162no—
163C2×C4—162no—
164C2×C4H(3,ℤ/4)11yescharacteristic
165C2×C4#34211yescharacteristic
166C2×C4#36012yes—
167C8—21no—
168C8—41no—
169C8—41no—
170C8—41no—
171C8—162no—
172C8AΣL(1,𝔽16)11yescharacteristic
173C8#35211yescharacteristic
174C22—21no—
175C22—21no—
176C22—22no—
177C22—42no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22—41no—
183C22—81no—
184C22—81no—
185C22—81no—
186C22#128911yescharacteristic
187C22#138611yescharacteristic
188C22#138611yescharacteristic
189C4—22no—
190C4—21no—
191C4—41no—
192C4—41no—
193C4—41no—
194C4—322no—
195C4#139011yescharacteristic
196C4#139511yescharacteristic
197C2—21no—
198C2—21no—
199C2—42no—
200C2—81no—
201C2#722711yescharacteristic
202C2#723311yescharacteristic
203C2#723611yescharacteristic
subgroupG/H
center, socleC22#1289
commutatorC22×C8C42
Frattini, cosocleC2×C8×D4C22
prime radicalC22×D4D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C8×D4 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C22×M32C4semidirect
7. C2×C22⋊C16C4semidirect
13. #1292C4semidirect

Maximal subgroups

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

subgroupindex
#129772
#86142
#86082