Finite Group Databasebeta

← #137913

Subgroups — #137913

Order 512.

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

normal subgroups only

This group has 352 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#33496C211yesmaximal, characteristic, complemented
2#22209C211yesmaximal, characteristic
3#11631C212yesmaximal
4#11494C211yesmaximal, characteristic, complemented
5#7144C212yesmaximal, complemented
6C2×C8×Q8C2211yescharacteristic
7C2×C8×D4C2211yescharacteristic
8#2398C411yescharacteristic
9Q8×C16—24no—
10#2163C2212yes—
11#2162—22no—
12#2162C2212yes—
13#1985C411yescharacteristic
14#1955C411yescharacteristic
15#1923—22no—
16#1850—22no—
17#1843C411yescharacteristic
18#1746C2211yescharacteristic
19D4⋊C16—24no—
20#1322—22no—
21C23×C8—21no—
22C2×C4×Q8C2×C411yescharacteristic
23C2×C4×D4C2×C411yescharacteristic
24C8×Q8—22no—
25C8×Q8—22no—
26C8×D4—22no—
27C8×D4—22no—
28C2×C4⋊C8—22no—
29C2×C4⋊C8C2×C411yescharacteristic
30C2×C4⋊C8C2×C411yescharacteristic
31C2×C4⋊C8D412yes—
32C2×C22⋊C8—21no—
33C2×C22⋊C8—22no—
34C2×#97C2×C411yescharacteristic
35C2×C4×C8D412yes—
36C2×C4×C8C2311yescharacteristic
37#400C812yes—
38#397—21no—
39#394C812yes—
40#393—21no—
41C4×C22⋊C4—22no—
42#376C2×C411yescharacteristic
43C4⋊C16—24no—
44C4⋊C16—22no—
45C4⋊C16—22no—
46C22⋊C16—22no—
47C22⋊C16—22no—
48#346—22no—
49C4×C16—24no—
50C4×C16D414yes—
51#336—21no—
52#336—21no—
53#336—22no—
54C22×Q8M1611yescharacteristic
55C22×D4C2×C811yescharacteristic
56C23×C4—21no—
57C22×C8—21no—
58C22×C8—21no—
59C22×C8—42no—
60C22×C8—82no—
61C22×C8C22⋊C412yes—
62C22×C8C2×D411yescharacteristic
63C22×C8C2×D411yescharacteristic
64C4×Q8—22no—
65C4×Q8—22no—
66C4×D4—22no—
67C4×D4—22no—
68C2×C4⋊C4C2×C811yescharacteristic
69C2×C4⋊C4C2×C812yes—
70C2×C4⋊C4M1611yescharacteristic
71C2×C22⋊C4—22no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—21no—
74C2×C42—21no—
75C2×C42C22⋊C412yes—
76C2×C42C22×C411yescharacteristic
77C2×C16—24no—
78C2×C16—42no—
79C4⋊C8—24no—
80C4⋊C8—22no—
81C4⋊C8—22no—
82C4⋊C8—82no—
83C22⋊C8—41no—
84C22⋊C8—41no—
85C22⋊C8—82no—
86#97—22no—
87C4×C8—22no—
88C4×C8—22no—
89C4×C8C2×D412yes—
90C4×C8#4112yes—
91#95—22no—
92#95—22no—
93#95—21no—
94#95C2×C812yes—
95C24—21no—
96C2×Q8—24no—
97C2×Q8—22no—
98C2×D4—22no—
99C2×D4—42no—
100C22×C4—21no—
101C22×C4—22no—
102C22×C4—42no—
103C22×C4—42no—
104C22×C4—41no—
105C22×C4—41no—
106C22×C4C22⋊C812yes—
107C22×C4C22⋊C812yes—
108C22×C4C2×C22⋊C411yescharacteristic
109C22×C4C22×C811yescharacteristic
110C22×C4C2×M1611yescharacteristic
111C2×C8—22no—
112C2×C8—41no—
113C2×C8—41no—
114C2×C8—42no—
115C2×C8—41no—
116C2×C8—42no—
117C2×C8—41no—
118C2×C8—41no—
119C2×C8—41no—
120C2×C8—84no—
121C2×C8#12011yescharacteristic
122C2×C8#12112yes—
123C2×C8#12311yescharacteristic
124C4⋊C4—24no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C22⋊C4—44no—
128C22⋊C4—44no—
129C22⋊C4—42no—
130C42—22no—
131C42—22no—
132C42—41no—
133C42—41no—
134C42C4×D412yes—
135C42#13112yes—
136C16—44no—
137C16—82no—
138C23—22no—
139C23—42no—
140C23—41no—
141C23—42no—
142C23—41no—
143C23C2×C22⋊C811yescharacteristic
144Q8—28no—
145D4—24no—
146D4—42no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—22no—
153C2×C4—41no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—41no—
160C2×C4—82no—
161C2×C4—81no—
162C2×C4—81no—
163C2×C4#35014yes—
164C2×C4#38611yescharacteristic
165C2×C4#40811yescharacteristic
166C2×C4C8×D412yes—
167C2×C4#43512yes—
168C8—22no—
169C8—21no—
170C8—21no—
171C8—42no—
172C8—81no—
173C8—81no—
174C22—21no—
175C22—21no—
176C22—44no—
177C22—42no—
178C22—82no—
179C22—81no—
180C22—81no—
181C22#186911yescharacteristic
182C22C2×#35012yes—
183C4—21no—
184C4—22no—
185C4—24no—
186C4—22no—
187C4—42no—
188C4—82no—
189C4—81no—
190C4#191011yescharacteristic
191C4#191211yescharacteristic
192C2—22no—
193C2—82no—
194C2#910211yescharacteristic
195C2#1298111yescharacteristic
196C2#1298311yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC22#1869
prime radicalC22×D4C2×C8
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
1. #33496C2semidirect
4. #11494C2semidirect
5. #7144C2semidirect

Maximal subgroups

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

subgroupindex
#334962
#222092
#116312
#114942
#71442