Finite Group Databasebeta

← #137898

Subgroups — #137898

Order 512.

310 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 88 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#33497C211yesmaximal, characteristic
2#22239C211yesmaximal, characteristic
3#11639C212yesmaximal
4#11494C211yesmaximal, characteristic
5#7144C212yesmaximal
6C2×C8×Q8C2211yescharacteristic
7C2×C8×Q8C2211yescharacteristic
8#2399C411yescharacteristic
9#2183—24no—
10#2163—22no—
11#2163C2212yes—
12#2162C2212yes—
13#1985C411yescharacteristic
14#1985C411yescharacteristic
15#1916—22no—
16#1916—22no—
17#1832C411yescharacteristic
18#1746C2211yescharacteristic
19Q8⋊C16—24no—
20#1322—22no—
21C2×C4×Q8C2×C411yescharacteristic
22C2×C4×Q8C2×C411yescharacteristic
23C8×Q8—22no—
24C8×Q8—22no—
25C8×Q8—22no—
26C8×Q8—22no—
27C2×C4⋊C8—21no—
28C2×C4⋊C8C2×C411yescharacteristic
29C2×C4⋊C8C2×C411yescharacteristic
30C2×C4⋊C8D412yes—
31C2×#97C2×C411yescharacteristic
32C2×C4×C8—21no—
33C2×C4×C8D412yes—
34C2×C4×C8C2311yescharacteristic
35#398—21no—
36#398C812yes—
37#395—21no—
38#395C812yes—
39C4×C4⋊C4—22no—
40#376C2×C411yescharacteristic
41C4⋊C16—24no—
42C4⋊C16—22no—
43C4⋊C16—22no—
44C22⋊C16—22no—
45C22⋊C16—22no—
46#346—24no—
47#346—22no—
48C4×C16D414yes—
49#336—22no—
50#336—22no—
51#336—21no—
52#336—21no—
53#336—22no—
54C22×Q8C2×C811yescharacteristic
55C22×Q8M1611yescharacteristic
56C22×C8—21no—
57C22×C8—21no—
58C22×C8C22⋊C412yes—
59C22×C8C2×D411yescharacteristic
60C22×C8C2×D411yescharacteristic
61C4×Q8—22no—
62C4×Q8—22no—
63C4×Q8—22no—
64C4×Q8—22no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4C2×C811yescharacteristic
69C2×C4⋊C4C2×C812yes—
70C2×C4⋊C4M1611yescharacteristic
71C2×C42—21no—
72C2×C42—21no—
73C2×C42C22⋊C412yes—
74C2×C42C22×C411yescharacteristic
75C2×C16—24no—
76C2×C16—42no—
77C4⋊C8—24no—
78C4⋊C8—22no—
79C4⋊C8—22no—
80C4⋊C8—41no—
81C4⋊C8—41no—
82#97—22no—
83C4×C8—22no—
84C4×C8—22no—
85C4×C8—42no—
86C4×C8C2×D412yes—
87C4×C8#4112yes—
88#95—22no—
89#95—22no—
90#95—21no—
91#95C2×C812yes—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—42no—
95C2×Q8—42no—
96C22×C4—21no—
97C22×C4—22no—
98C22×C4—21no—
99C22×C4C22⋊C812yes—
100C22×C4C22⋊C812yes—
101C22×C4C2×C22⋊C411yescharacteristic
102C22×C4C22×C811yescharacteristic
103C22×C4C2×M1611yescharacteristic
104C2×C8—22no—
105C2×C8—41no—
106C2×C8—41no—
107C2×C8—42no—
108C2×C8—41no—
109C2×C8—41no—
110C2×C8—42no—
111C2×C8—41no—
112C2×C8—41no—
113C2×C8#12011yescharacteristic
114C2×C8#12112yes—
115C2×C8#12311yescharacteristic
116C4⋊C4—24no—
117C4⋊C4—22no—
118C4⋊C4—22no—
119C4⋊C4—44no—
120C4⋊C4—44no—
121C4⋊C4—42no—
122C42—22no—
123C42—22no—
124C42—41no—
125C42—41no—
126C42—41no—
127C42—41no—
128C42C4×D412yes—
129C42#13112yes—
130C16—44no—
131C16—82no—
132C23C2×C22⋊C811yescharacteristic
133Q8—24no—
134Q8—42no—
135Q8—44no—
136C2×C4—22no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—22no—
140C2×C4—22no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—42no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—41no—
153C2×C4#38611yescharacteristic
154C2×C4#40811yescharacteristic
155C2×C4C8×D412yes—
156C2×C4#43512yes—
157C8—22no—
158C8—21no—
159C8—21no—
160C8—42no—
161C8—81no—
162C8—81no—
163C22—21no—
164C22—21no—
165C22#186911yescharacteristic
166C22#211712yes—
167C4—21no—
168C4—22no—
169C4—22no—
170C4—42no—
171C4—42no—
172C4—42no—
173C4—82no—
174C4—81no—
175C4#190911yescharacteristic
176C4#191211yescharacteristic
177C2—22no—
178C2#910011yescharacteristic
179C2#1298111yescharacteristic
180C2#1298411yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC22#1869
prime radicalC23C2×C22⋊C8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#334972
#222392
#116392
#114942
#71442