Finite Group Databasebeta

← #137889

Subgroups — #137889

Order 512.

364 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 92 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#33486C211yesmaximal, characteristic
2#22209C211yesmaximal, characteristic
3#11639C212yesmaximal, complemented
4#11488C211yesmaximal, characteristic
5#7144C212yesmaximal, complemented
6C2×C8×Q8C2211yescharacteristic
7C2×C8×D4C2211yescharacteristic
8#2398C411yescharacteristic
9D4×C16—24no—
10#2163C2212yes—
11#2162C2212yes—
12C2×C22⋊C16—22no—
13#1985C411yescharacteristic
14#1955C411yescharacteristic
15#1923—22no—
16#1850—22no—
17#1843C411yescharacteristic
18#1746C2211yescharacteristic
19Q8⋊C16—24no—
20#1314—22no—
21C23×C8D412yes—
22C2×C4×Q8C2×C411yescharacteristic
23C2×C4×D4C2×C411yescharacteristic
24C22×C16—24no—
25C8×Q8—22no—
26C8×Q8—22no—
27C8×D4—22no—
28C8×D4—22no—
29C2×C4⋊C8—21no—
30C2×C4⋊C8—22no—
31C2×C4⋊C8C2×C411yescharacteristic
32C2×C4⋊C8C2×C411yescharacteristic
33C2×C22⋊C8—22no—
34C2×C22⋊C8D412yes—
35C2×#97C2×C411yescharacteristic
36C2×C4×C8—21no—
37C2×C4×C8C2311yescharacteristic
38#400—21no—
39#397C812yes—
40#394—21no—
41#393C812yes—
42C4×C22⋊C4—22no—
43#376C2×C411yescharacteristic
44C4⋊C16—22no—
45C4⋊C16—22no—
46C22⋊C16—24no—
47C22⋊C16—22no—
48C22⋊C16—22no—
49#346—22no—
50C4×C16D414yes—
51#336—22no—
52#336—21no—
53#336—21no—
54C22×Q8C2×C811yescharacteristic
55C22×D4M1611yescharacteristic
56C23×C4C22⋊C412yes—
57C22×C8—22no—
58C22×C8—22no—
59C22×C8—21no—
60C22×C8—21no—
61C22×C8—82no—
62C22×C8C22⋊C412yes—
63C22×C8C2×D411yescharacteristic
64C22×C8C2×D411yescharacteristic
65C4×Q8—22no—
66C4×Q8—22no—
67C4×D4—22no—
68C4×D4—22no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4C2×C811yescharacteristic
71C2×C4⋊C4M1611yescharacteristic
72C2×C22⋊C4—22no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4C2×C812yes—
75C2×C42—21no—
76C2×C42—21no—
77C2×C42C22×C411yescharacteristic
78C2×C16—24no—
79C2×C16—42no—
80C2×C16—84no—
81C4⋊C8—22no—
82C4⋊C8—22no—
83C4⋊C8—41no—
84C4⋊C8—41no—
85C4⋊C8—82no—
86C22⋊C8—24no—
87C22⋊C8—82no—
88#97—22no—
89C4×C8—42no—
90C4×C8C2×D412yes—
91C4×C8#4112yes—
92#95—22no—
93#95—22no—
94#95—21no—
95#95C2×C812yes—
96C24C22⋊C812yes—
97C2×Q8—22no—
98C2×Q8—42no—
99C2×D4—24no—
100C2×D4—22no—
101C22×C4—21no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—21no—
106C22×C4—42no—
107C22×C4—42no—
108C22×C4C22⋊C812yes—
109C22×C4C2×C22⋊C411yescharacteristic
110C22×C4C22×C811yescharacteristic
111C22×C4C2×M1611yescharacteristic
112C2×C8—22no—
113C2×C8—41no—
114C2×C8—41no—
115C2×C8—44no—
116C2×C8—42no—
117C2×C8—41no—
118C2×C8—42no—
119C2×C8—41no—
120C2×C8—41no—
121C2×C8—41no—
122C2×C8—82no—
123C2×C8#12011yescharacteristic
124C2×C8#12112yes—
125C2×C8#12311yescharacteristic
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—42no—
129C22⋊C4—24no—
130C22⋊C4—44no—
131C22⋊C4—44no—
132C42—41no—
133C42—41no—
134C42—41no—
135C42—41no—
136C42C4×D412yes—
137C42#13112yes—
138C16—44no—
139C16—82no—
140C23—22no—
141C23—22no—
142C23—42no—
143C23—42no—
144C23#35014yes—
145C23C2×C22⋊C811yescharacteristic
146Q8—24no—
147Q8—42no—
148D4—28no—
149C2×C4—22no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—41no—
165C2×C4—82no—
166C2×C4#38611yescharacteristic
167C2×C4#40811yescharacteristic
168C2×C4C8×D412yes—
169C2×C4#43512yes—
170C8—22no—
171C8—21no—
172C8—21no—
173C8—42no—
174C8—81no—
175C8—81no—
176C22—21no—
177C22—21no—
178C22—24no—
179C22—44no—
180C22—42no—
181C22—42no—
182C22—82no—
183C22#186911yescharacteristic
184C22C2×#35012yes—
185C4—21no—
186C4—22no—
187C4—22no—
188C4—42no—
189C4—42no—
190C4—82no—
191C4—81no—
192C4#190911yescharacteristic
193C4#191111yescharacteristic
194C2—22no—
195C2—44no—
196C2#909911yescharacteristic
197C2#1298111yescharacteristic
198C2#1298311yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC22#1869
prime radicalC22×D4M16
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #11639C2semidirect
5. #7144C2semidirect

Maximal subgroups

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

subgroupindex
#334862
#222092
#116392
#114882
#71442