Finite Group Databasebeta

← #128981

Subgroups — #128981

Order 512.

308 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 90 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#22241C211yesmaximal, characteristic
2#11736C212yesmaximal
3#8165C212yesmaximal
4#7286C211yesmaximal, characteristic
5#7284C211yesmaximal, characteristic
6C2×#446C411yescharacteristic
7C2×C8×Q8C2211yescharacteristic
8#2399C2211yescharacteristic
9#1985C411yescharacteristic
10#1985C2211yescharacteristic
11#1917—22no—
12#1916—22no—
13#1832C411yescharacteristic
14C4×#101C2212yes—
15#1746C411yescharacteristic
16#1461C2212yes—
17#1451—22no—
18#1354C414yes—
19#1346C414yes—
20#1283—22no—
21C2×C4×Q8C2×C411yescharacteristic
22C2×C4×Q8C2311yescharacteristic
23#446—42no—
24C8×Q8—42no—
25#431—22no—
26C2×C4⋊C8—21no—
27C2×C4⋊C8C2×C411yescharacteristic
28C2×C4⋊C8C2×C411yescharacteristic
29C2×C4⋊C8D412yes—
30C2×#101—22no—
31C4×M16—22no—
32C2×#97—21no—
33C2×#97C2×C411yescharacteristic
34C2×C4×C8C2×C411yescharacteristic
35C2×C4×C8D412yes—
36#398—21no—
37#398C2×C412yes—
38#395—21no—
39#395C2×C412yes—
40C4×C4⋊C4—22no—
41#376C2×C411yescharacteristic
42#343C2×C414yes—
43#336—22no—
44#336—22no—
45#336—21no—
46#336—21no—
47#336—22no—
48#324—22no—
49#324C2×C414yes—
50#322—44no—
51C22×Q8C4211yescharacteristic
52C22×Q8#4111yescharacteristic
53C2×M16—24no—
54C22×C8—21no—
55C22×C8—21no—
56C22×C8C22⋊C412yes—
57C22×C8C22⋊C411yescharacteristic
58C22×C8C22⋊C411yescharacteristic
59C4×Q8—41no—
60C4×Q8—41no—
61C4×Q8—41no—
62C4×Q8—41no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4C4211yescharacteristic
67C2×C4⋊C4C22×C412yes—
68C2×C4⋊C4#4111yescharacteristic
69C2×C42—21no—
70C2×C42—21no—
71C2×C42C22×C411yescharacteristic
72C2×C42C2×D412yes—
73C4⋊C8—41no—
74C4⋊C8—42no—
75C4⋊C8—41no—
76C4⋊C8—42no—
77C4⋊C8—42no—
78#101—44no—
79C22⋊C8—24no—
80#97—41no—
81#97—42no—
82#97—42no—
83C4×C8—24no—
84C4×C8—22no—
85C4×C8—42no—
86C4×C8—42no—
87#95—21no—
88#95—22no—
89#95—22no—
90#95C4212yes—
91C2×Q8—24no—
92C2×Q8—41no—
93C2×Q8—41no—
94C2×Q8—81no—
95C22×C4—22no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4C2×C4211yescharacteristic
99C22×C4C2×C22⋊C411yescharacteristic
100C22×C4#11711yescharacteristic
101C22×C4C4×D412yes—
102C22×C4C4×D412yes—
103M16—44no—
104M16—44no—
105C2×C8—21no—
106C2×C8—22no—
107C2×C8—41no—
108C2×C8—42no—
109C2×C8—41no—
110C2×C8—44no—
111C2×C8—44no—
112C2×C8—41no—
113C2×C8—41no—
114C2×C8—42no—
115C2×C8—41no—
116C2×C8—41no—
117C2×C8#9912yes—
118C4⋊C4—42no—
119C4⋊C4—41no—
120C4⋊C4—41no—
121C4⋊C4—42no—
122C4⋊C4—44no—
123C4⋊C4—82no—
124C42—22no—
125C42—21no—
126C42—22no—
127C42—21no—
128C42—41no—
129C42—41no—
130C42—41no—
131C42—41no—
132C23C4×C22⋊C411yescharacteristic
133Q8—44no—
134Q8—82no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—42no—
144C2×C4—41no—
145C2×C4—42no—
146C2×C4—41no—
147C2×C4—42no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4C2×#9911yescharacteristic
154C2×C4#41011yescharacteristic
155C2×C4#44414yes—
156C8—22no—
157C8—41no—
158C8—42no—
159C8—81no—
160C8—84no—
161C8—84no—
162C8—81no—
163C22—21no—
164C22—21no—
165C22C4×#9911yescharacteristic
166C22#176512yes—
167C4—24no—
168C4—21no—
169C4—41no—
170C4—42no—
171C4—41no—
172C4—82no—
173C4—81no—
174C4—81no—
175C4#213311yescharacteristic
176C4#213511yescharacteristic
177C2—22no—
178C2#823911yescharacteristic
179C2#1262411yescharacteristic
180C2#1263111yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×Q8C23
socleC22C4×#99
prime radicalC23C4×C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#222412
#117362
#81652
#72862
#72842