Finite Group Databasebeta

← #122425

Subgroups — #122425

Order 512.

273 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 273 conjugacy classes of subgroups — too many to draw. Only its 72 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#21715C211yesmaximal, characteristic
2#11494C212yesmaximal
3#7066C212yesmaximal
4#7061C212yesmaximal
5C2×C8×Q8C411yescharacteristic
6C2×C8×Q8C2211yescharacteristic
7#2399C411yescharacteristic
8#2163—22no—
9#1985C411yescharacteristic
10#1985C2211yescharacteristic
11#1940—22no—
12#1843C2211yescharacteristic
13C8×C4⋊C4—22no—
14#1726C411yescharacteristic
15#1322—22no—
16#1322C2214yes—
17C2×C4×Q8C2×C411yescharacteristic
18C2×C4×Q8C2×C411yescharacteristic
19C8×Q8—41no—
20C8×Q8—41no—
21C8×Q8—41no—
22C8×Q8—41no—
23C2×C4⋊C8—21no—
24C2×C4⋊C8—22no—
25C2×C4⋊C8C2×C411yescharacteristic
26C2×C4⋊C8D412yes—
27C2×C4⋊C8C2311yescharacteristic
28C2×#97—21no—
29C2×#97C2×C411yescharacteristic
30C2×C4×C8—21no—
31C2×C4×C8—21no—
32C2×C4×C8C2×C411yescharacteristic
33C2×C4×C8D412yes—
34#398—21no—
35#398C812yes—
36#395—21no—
37#395C812yes—
38C4×C4⋊C4—22no—
39#376C2×C411yescharacteristic
40C4⋊C16—42no—
41C22⋊C16—24no—
42#346—42no—
43#336—22no—
44#336—22no—
45C22×Q8C2×C811yescharacteristic
46C22×Q8M1611yescharacteristic
47C22×C8—21no—
48C22×C8—21no—
49C22×C8C22⋊C411yescharacteristic
50C22×C8C22⋊C411yescharacteristic
51C22×C8C2×D412yes—
52C4×Q8—41no—
53C4×Q8—41no—
54C4×Q8—41no—
55C4×Q8—41no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4C2×C811yescharacteristic
60C2×C4⋊C4C2×C812yes—
61C2×C4⋊C4M1611yescharacteristic
62C2×C42—21no—
63C2×C42—21no—
64C2×C42C22⋊C412yes—
65C2×C42C22×C411yescharacteristic
66C2×C16—44no—
67C4⋊C8—42no—
68C4⋊C8—41no—
69C4⋊C8—41no—
70C4⋊C8—44no—
71C4⋊C8—42no—
72#97—41no—
73#97—42no—
74C4×C8—24no—
75C4×C8—22no—
76C4×C8—22no—
77C4×C8—21no—
78C4×C8—21no—
79C4×C8—41no—
80C4×C8—41no—
81#95—22no—
82#95—22no—
83#95—21no—
84#95C2×C812yes—
85C2×Q8—41no—
86C2×Q8—42no—
87C2×Q8—42no—
88C2×Q8—41no—
89C22×C4—22no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4C22⋊C812yes—
93C22×C4C22⋊C812yes—
94C22×C4C2×C22⋊C411yescharacteristic
95C22×C4C22×C811yescharacteristic
96C22×C4C2×M1611yescharacteristic
97C2×C8—22no—
98C2×C8—22no—
99C2×C8—21no—
100C2×C8—22no—
101C2×C8—41no—
102C2×C8—41no—
103C2×C8—41no—
104C2×C8—42no—
105C2×C8—42no—
106C2×C8—41no—
107C2×C8#9911yescharacteristic
108C2×C8#9911yescharacteristic
109C4⋊C4—42no—
110C4⋊C4—41no—
111C4⋊C4—41no—
112C4⋊C4—44no—
113C4⋊C4—44no—
114C4⋊C4—42no—
115C42—22no—
116C42—22no—
117C42—21no—
118C42—22no—
119C42—21no—
120C42—22no—
121C42—41no—
122C42—41no—
123C16—84no—
124C23C2×C22⋊C811yescharacteristic
125Q8—42no—
126Q8—44no—
127Q8—42no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—24no—
131C2×C4—22no—
132C2×C4—22no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—42no—
140C2×C4—41no—
141C2×C4—42no—
142C2×C4—41no—
143C2×C4—42no—
144C2×C4—41no—
145C2×C4—42no—
146C2×C4—41no—
147C2×C4C2×#9911yescharacteristic
148C2×C4#41311yescharacteristic
149C8—21no—
150C8—42no—
151C8—42no—
152C8—41no—
153C8—81no—
154C8#35612yes—
155C22—21no—
156C22—21no—
157C22#145911yescharacteristic
158C22#211712yes—
159C4—21no—
160C4—44no—
161C4—42no—
162C4—41no—
163C4—42no—
164C4—42no—
165C4—41no—
166C4—81no—
167C4#212611yescharacteristic
168C4C2×#35611yescharacteristic
169C2—22no—
170C2#1146911yescharacteristic
171C2#1147811yescharacteristic
172C2#1148311yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4⋊C8C23
socleC22#1459
prime radicalC23C2×C22⋊C8
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#217152
#114942
#70662
#70612