Finite Group Databasebeta

← #124707

Subgroups — #124707

Order 512.

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

normal subgroups only

This group has 247 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#21344C211yesmaximal, characteristic
2#12896C211yesmaximal, characteristic
3#12595C211yesmaximal, characteristic
4#7133C212yesmaximal
5#7124C212yesmaximal
6#2295C411yescharacteristic
7C2×#347—21no—
8#1985C411yescharacteristic
9#1985C2211yescharacteristic
10#1940C411yescharacteristic
11#1940C2211yescharacteristic
12#1916—21no—
13#1916—21no—
14#1768—21no—
15C4×C4⋊C8C411yescharacteristic
16#1746—21no—
17C4×#97C2211yescharacteristic
18#1377C2212yes—
19#1377C2212yes—
20#1376—21no—
21#1376—21no—
22#1375—21no—
23C2×C4×Q8C2×C411yescharacteristic
24C2×M32—22no—
25C2×M32—22no—
26C2×C4⋊C8—21no—
27C2×C4⋊C8—21no—
28C2×C4⋊C8C2×C411yescharacteristic
29C2×C4⋊C8C2×C411yescharacteristic
30C2×#97C2×C411yescharacteristic
31C2×#97D411yescharacteristic
32C2×#97D411yescharacteristic
33C2×#97C2311yescharacteristic
34C2×C4×C8—21no—
35C2×C4×C8D411yescharacteristic
36C2×C4×C8D411yescharacteristic
37#391—21no—
38#389—21no—
39#387—22no—
40C4×C4⋊C4—21no—
41C4×C4⋊C4C2×C411yescharacteristic
42#376—21no—
43C43C2×C411yescharacteristic
44#347—24no—
45#336—21no—
46#336—21no—
47#336—21no—
48#336—21no—
49#336—21no—
50#336—21no—
51#336—21no—
52C22×Q8M1611yescharacteristic
53C22×C8—21no—
54C22×C8—21no—
55C22×C8C22⋊C411yescharacteristic
56C22×C8C22⋊C411yescharacteristic
57C22×C8C2×D411yescharacteristic
58C22×C8C2×D411yescharacteristic
59C4×Q8—41no—
60C4×Q8—41no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4M1611yescharacteristic
65C2×C42—21no—
66C2×C42—21no—
67C2×C42—21no—
68C2×C42—21no—
69C2×C42C22⋊C411yescharacteristic
70C2×C42C22⋊C411yescharacteristic
71C2×C42C22×C411yescharacteristic
72M32—24no—
73M32—42no—
74M32—42no—
75M32—42no—
76C2×C16—42no—
77C2×C16—42no—
78C4⋊C8—41no—
79C4⋊C8—41no—
80C4⋊C8—42no—
81C4⋊C8—42no—
82#97—22no—
83#97—22no—
84#97—41no—
85#97—41no—
86C4×C8—22no—
87C4×C8—41no—
88C4×C8—42no—
89#95—22no—
90#95—22no—
91#95—22no—
92#95M1612yes—
93C2×Q8—41no—
94C2×Q8—81no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—22no—
99C22×C4—21no—
100C22×C4C2×C22⋊C411yescharacteristic
101C22×C4C2×M1611yescharacteristic
102C22×C4C2×M1611yescharacteristic
103C2×C8—21no—
104C2×C8—21no—
105C2×C8—21no—
106C2×C8—21no—
107C2×C8—21no—
108C2×C8—21no—
109C2×C8—21no—
110C2×C8—21no—
111C2×C8—21no—
112C2×C8—21no—
113C2×C8—41no—
114C2×C8—41no—
115C2×C8—41no—
116C2×C8—41no—
117C2×C8—41no—
118C2×C8—41no—
119C2×C8#13512yes—
120C2×C8#13712yes—
121C4⋊C4—41no—
122C4⋊C4—44no—
123C4⋊C4—42no—
124C4⋊C4—81no—
125C42—21no—
126C42—21no—
127C42—21no—
128C42—21no—
129C42—21no—
130C42—42no—
131C42—41no—
132C42—41no—
133C42—41no—
134C42—41no—
135C42—81no—
136C42#10412yes—
137C16—44no—
138C16—82no—
139C23#40711yescharacteristic
140Q8—82no—
141C2×C4—22no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—22no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—42no—
158C2×C4#41011yescharacteristic
159C2×C4#41311yescharacteristic
160C2×C4#41611yescharacteristic
161C2×C4#41911yescharacteristic
162C2×C4C2×#10411yescharacteristic
163C2×C4#42111yescharacteristic
164C8—22no—
165C8—22no—
166C8—41no—
167C8—41no—
168C8—81no—
169C8—81no—
170C22—21no—
171C22—21no—
172C22#146411yescharacteristic
173C22#148811yescharacteristic
174C22#149111yescharacteristic
175C4—42no—
176C4—42no—
177C4—41no—
178C4—41no—
179C4—82no—
180C4—81no—
181C4—81no—
182C4#150011yescharacteristic
183C4#151511yescharacteristic
184C4#152911yescharacteristic
185C4#155411yescharacteristic
186C2—21no—
187C2—21no—
188C2#761111yescharacteristic
189C2#1152211yescharacteristic
190C2#1152411yescharacteristic
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
#213442
#128962
#125952
#71332
#71242