Finite Group Databasebeta

← #149812

Subgroups — #149812

Order 512.

380 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 62 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#20800C211yesmaximal, characteristic
2#11512C212yesmaximal
3#7371C212yesmaximal, complemented
4#7370C212yesmaximal, complemented
5C2×#428—22no—
6C2×C4×M16C411yescharacteristic
7#2285C411yescharacteristic
8C2×#384C2211yescharacteristic
9#1847—22no—
10#1845C411yescharacteristic
11#1845C2211yescharacteristic
12#1844C411yescharacteristic
13#1844C2211yescharacteristic
14#1463—22no—
15#1325—22no—
16#1325C2214yes—
17C22×M16—21no—
18C2×#117—21no—
19C22×C4⋊C4—41no—
20C22×C4⋊C4D412yes—
21C22×C42C2×C411yescharacteristic
22#428—44no—
23#428—44no—
24C2×#107—22no—
25C2×#107—22no—
26C2×#106—22no—
27C2×#106—22no—
28#407—42no—
29C4×M16—22no—
30C4×M16—22no—
31C2×#97—21no—
32C2×C4×C8C2×C411yescharacteristic
33C2×C4×C8C2×C411yescharacteristic
34#384—41no—
35#384C2×C411yescharacteristic
36#384C2×C411yescharacteristic
37#384C2×C411yescharacteristic
38#384D412yes—
39#384C2311yescharacteristic
40#382—21no—
41#380—41no—
42C4×C4⋊C4—41no—
43#324—24no—
44#324—42no—
45C23×C4—21no—
46C23×C4—22no—
47C23×C4C22⋊C411yescharacteristic
48C2×M16—24no—
49C2×M16—24no—
50C2×M16—24no—
51C2×M16—82no—
52C22×C8—21no—
53C22×C8—21no—
54#117—42no—
55#117—42no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—42no—
60C2×C4⋊C4—42no—
61C2×C4⋊C4—41no—
62C2×C4⋊C4—42no—
63C2×C4⋊C4—41no—
64C2×C4⋊C4—41no—
65C2×C4⋊C4—41no—
66C2×C4⋊C4—84no—
67C2×C4⋊C4C22⋊C412yes—
68C2×C4⋊C4C2×D412yes—
69C2×C22⋊C4—41no—
70C2×C42—21no—
71C2×C42—22no—
72C2×C42—21no—
73C2×C42—21no—
74C2×C42—41no—
75C2×C42C22⋊C411yescharacteristic
76C2×C42C22×C411yescharacteristic
77#107—44no—
78#107—44no—
79#106—44no—
80#106—44no—
81C22⋊C8—44no—
82C22⋊C8—82no—
83#97—24no—
84C4×C8—21no—
85C4×C8—21no—
86C4×C8M1612yescomplemented
87C4×C8M1612yescomplemented
88#95—42no—
89C24#9911yescharacteristic
90C22×C4—22no—
91C22×C4—21no—
92C22×C4—22no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—21no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—42no—
101C22×C4—42no—
102C22×C4—42no—
103C22×C4—42no—
104C22×C4—41no—
105C22×C4—41no—
106C22×C4—84no—
107C22×C4#9911yescharacteristic
108C22×C4C2×C22⋊C411yescharacteristic
109M16—216no—
110M16—164no—
111C2×C8—22no—
112C2×C8—22no—
113C2×C8—22no—
114C2×C8—22no—
115C2×C8—22no—
116C2×C8—22no—
117C2×C8—84no—
118C4⋊C4—42no—
119C4⋊C4—42no—
120C4⋊C4—42no—
121C4⋊C4—42no—
122C4⋊C4—82no—
123C4⋊C4—82no—
124C4⋊C4—82no—
125C4⋊C4—82no—
126C4⋊C4—82no—
127C4⋊C4—84no—
128C22⋊C4—82no—
129C42—22no—
130C42—22no—
131C42—21no—
132C42—21no—
133C42—81no—
134C42—81no—
135C42#10112yes—
136C42C2×M1611yescharacteristic
137C42C2×M1611yescharacteristic
138C23—21no—
139C23—21no—
140C23—41no—
141C23—41no—
142C23#35312yes—
143C23C2×#9911yescharacteristic
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—24no—
147C2×C4—24no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—22no—
155C2×C4—44no—
156C2×C4—44no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—84no—
162C2×C4—82no—
163C2×C4—82no—
164C2×C4—82no—
165C2×C4—82no—
166C2×C4—81no—
167C2×C4—81no—
168C2×C4—162no—
169C2×C4#35612yes—
170C2×C4#40711yescharacteristic
171C2×C4C2×#10111yescharacteristic
172C8—24no—
173C8—24no—
174C8—164no—
175C22—21no—
176C22—21no—
177C22—22no—
178C22—22no—
179C22—22no—
180C22—42no—
181C22—42no—
182C22#146311yescharacteristic
183C22C2×#35311yescharacteristic
184C22C2×#35611yescharacteristic
185C4—22no—
186C4—22no—
187C4—21no—
188C4—22no—
189C4—21no—
190C4—42no—
191C4—82no—
192C4—82no—
193C4—161no—
194C4—161no—
195C2—21no—
196C2—21no—
197C2—42no—
198C2#1151211yescharacteristic
199C2#1358411yescharacteristic
200C2#1359011yescharacteristic
subgroupG/H
center, socleC22C2×#353
commutatorC2×C42C22×C4
Frattini, cosocle#384C23
prime radicalC24#99
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#384 ◁ G
LECSG ▷ #384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7371C2semidirect
4. #7370C2semidirect
86. C4×C8M16semidirect
87. C4×C8M16semidirect

Maximal subgroups

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

subgroupindex
#208002
#115122
#73712
#73702