Finite Group Databasebeta

← #124706

Subgroups — #124706

Order 512.

267 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 267 conjugacy classes of subgroups — too many to draw. Only its 68 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#21346C211yesmaximal, characteristic
2#12896C211yesmaximal, characteristic
3#12595C211yesmaximal, characteristic
4#7133C212yesmaximal
5#7124C212yesmaximal
6C42×Q8C411yescharacteristic
7C2×#347—21no—
8#1985C411yescharacteristic
9#1985C2211yescharacteristic
10#1940C411yescharacteristic
11#1940C2211yescharacteristic
12#1916—21no—
13#1916—21no—
14#1767—21no—
15#1767C411yescharacteristic
16#1745—21no—
17C4×#97C2211yescharacteristic
18#1377C2212yes—
19#1377C2212yes—
20#1376—21no—
21#1376—21no—
22#1375—21no—
23C2×C4×Q8—21no—
24C2×C4×Q8C2×C411yescharacteristic
25C2×M32—22no—
26C2×M32—22no—
27C2×C4⋊C8—21no—
28C2×C4⋊C8—21no—
29C2×C4⋊C8C2×C411yescharacteristic
30C2×C4⋊C8C2×C411yescharacteristic
31C2×#97—21no—
32C2×#97C2×C411yescharacteristic
33C2×#97D411yescharacteristic
34C2×#97D411yescharacteristic
35C2×#97C2311yescharacteristic
36C2×C4×C8D411yescharacteristic
37C2×C4×C8D411yescharacteristic
38C4×C4⋊C4—21no—
39C4×C4⋊C4—21no—
40C4×C4⋊C4—21no—
41C4×C4⋊C4—21no—
42C4×C4⋊C4C2×C411yescharacteristic
43C43—21no—
44C43C2×C411yescharacteristic
45#347—24no—
46#336—21no—
47#336—21no—
48#336—21no—
49#336—21no—
50#336—21no—
51#336—21no—
52#336—21no—
53C22×Q8M1611yescharacteristic
54C22×C8—21no—
55C22×C8—21no—
56C22×C8C22⋊C411yescharacteristic
57C22×C8C22⋊C411yescharacteristic
58C22×C8C2×D411yescharacteristic
59C22×C8C2×D411yescharacteristic
60C4×Q8—41no—
61C4×Q8—41no—
62C4×Q8—44no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4M1611yescharacteristic
68C2×C42—21no—
69C2×C42—21no—
70C2×C42—21no—
71C2×C42—21no—
72C2×C42—21no—
73C2×C42—21no—
74C2×C42—21no—
75C2×C42—21no—
76C2×C42C22⋊C411yescharacteristic
77C2×C42C22⋊C411yescharacteristic
78C2×C42C22×C411yescharacteristic
79M32—24no—
80M32—42no—
81M32—42no—
82M32—42no—
83C2×C16—42no—
84C2×C16—42no—
85C4⋊C8—41no—
86C4⋊C8—41no—
87C4⋊C8—42no—
88C4⋊C8—42no—
89#97—22no—
90#97—22no—
91#97—41no—
92#97—41no—
93#97—42no—
94C4×C8—22no—
95C4×C8—41no—
96#95—21no—
97#95—21no—
98#95M1611yescharacteristic
99#95M1611yescharacteristic
100C2×Q8—42no—
101C2×Q8—41no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4C2×C22⋊C411yescharacteristic
109C22×C4C2×M1611yescharacteristic
110C22×C4C2×M1611yescharacteristic
111C2×C8—21no—
112C2×C8—21no—
113C2×C8—21no—
114C2×C8—21no—
115C2×C8—21no—
116C2×C8—21no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—41no—
122C2×C8—41no—
123C2×C8—41no—
124C2×C8—41no—
125C2×C8—41no—
126C2×C8—41no—
127C2×C8#13512yes—
128C2×C8#13712yes—
129C4⋊C4—24no—
130C4⋊C4—41no—
131C4⋊C4—42no—
132C4⋊C4—42no—
133C4⋊C4—42no—
134C42—21no—
135C42—21no—
136C42—21no—
137C42—21no—
138C42—21no—
139C42—42no—
140C42—42no—
141C42—42no—
142C42—41no—
143C42—41no—
144C42—41no—
145C42—41no—
146C42—81no—
147C42—81no—
148C42—81no—
149C42#10412yes—
150C16—44no—
151C16—82no—
152C23#40711yescharacteristic
153Q8—42no—
154Q8—42no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4#41011yescharacteristic
178C2×C4#41311yescharacteristic
179C2×C4#41611yescharacteristic
180C2×C4#41911yescharacteristic
181C2×C4C2×#10411yescharacteristic
182C2×C4#42111yescharacteristic
183C8—22no—
184C8—22no—
185C8—41no—
186C8—41no—
187C8—81no—
188C8—81no—
189C22—21no—
190C22—21no—
191C22#146411yescharacteristic
192C22#148811yescharacteristic
193C22#149111yescharacteristic
194C4—42no—
195C4—42no—
196C4—42no—
197C4—42no—
198C4—41no—
199C4—41no—
200C4—81no—
201C4—81no—
202C4#150011yescharacteristic
203C4#151511yescharacteristic
204C4#152911yescharacteristic
205C4#155411yescharacteristic
206C2—21no—
207C2—21no—
208C2#761111yescharacteristic
209C2#1152111yescharacteristic
210C2#1152311yescharacteristic
subgroupG/H
centerC2×C4#410
commutatorC2×C42C22×C4
Frattini, cosocleC2×#97C23
socleC22#1464
prime radicalC23#407
LCSG ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#97 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#213462
#128962
#125952
#71332
#71242