Finite Group Databasebeta

← #149817

Subgroups — #149817

Order 512.

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

normal subgroups only

This group has 358 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#20801C211yesmaximal, characteristic
2#11511C212yesmaximal
3#7372C214yesmaximal, complemented
4C2×#428—22no—
5C2×C4×M16C411yescharacteristic
6#2285C411yescharacteristic
7C2×#382C2211yescharacteristic
8#1847C412yes—
9#1847C2212yes—
10#1846—22no—
11#1463—22no—
12#1324—22no—
13#1324C2214yes—
14C22×M16—21no—
15C2×#117—21no—
16C22×C4⋊C4D412yes—
17C22×C42C2×C411yescharacteristic
18#428—44no—
19#428—44no—
20C2×#107—22no—
21C2×#107—22no—
22C2×#106—22no—
23C2×#106—22no—
24#407—42no—
25C4×M16—24no—
26C2×#97C2×C412yes—
27C2×C4×C8—21no—
28#384C2×C412yes—
29#382—21no—
30#382—41no—
31#382C2×C411yescharacteristic
32#382D412yes—
33#382C2311yescharacteristic
34#380—41no—
35C4×C4⋊C4—41no—
36C2×#95—41no—
37#324—24no—
38#324—42no—
39C23×C4—21no—
40C23×C4—22no—
41C23×C4C22⋊C411yescharacteristic
42C2×M16—24no—
43C2×M16—24no—
44C2×M16—22no—
45C2×M16—22no—
46C2×M16—82no—
47C22×C8—22no—
48#117—42no—
49#117—42no—
50C2×C4⋊C4—22no—
51C2×C4⋊C4—22no—
52C2×C4⋊C4—42no—
53C2×C4⋊C4—42no—
54C2×C4⋊C4—42no—
55C2×C4⋊C4C22⋊C412yes—
56C2×C4⋊C4C2×D412yes—
57C2×C22⋊C4—41no—
58C2×C42—21no—
59C2×C42—22no—
60C2×C42—22no—
61C2×C42—41no—
62C2×C42C22⋊C411yescharacteristic
63C2×C42C22×C411yescharacteristic
64#107—44no—
65#107—44no—
66#106—44no—
67#106—44no—
68C22⋊C8—44no—
69C22⋊C8—82no—
70#97—22no—
71#97M1614yescomplemented
72C4×C8—24no—
73#95—41no—
74#95—42no—
75#95—41no—
76#95—42no—
77C24#9911yescharacteristic
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—22no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—21no—
85C22×C4—24no—
86C22×C4—22no—
87C22×C4—22no—
88C22×C4—42no—
89C22×C4—42no—
90C22×C4—42no—
91C22×C4—82no—
92C22×C4—84no—
93C22×C4#9911yescharacteristic
94C22×C4C2×C22⋊C411yescharacteristic
95M16—28no—
96M16—28no—
97M16—164no—
98C2×C8—24no—
99C2×C8—24no—
100C2×C8—24no—
101C2×C8—84no—
102C4⋊C4—42no—
103C4⋊C4—42no—
104C4⋊C4—42no—
105C4⋊C4—42no—
106C4⋊C4—82no—
107C4⋊C4—84no—
108C22⋊C4—81no—
109C22⋊C4—81no—
110C42—24no—
111C42—21no—
112C42—21no—
113C42—82no—
114C42#10112yes—
115C42C2×M1612yes—
116C23—21no—
117C23—21no—
118C23—42no—
119C23#35311yescharacteristic
120C23#35311yescharacteristic
121C23C2×#9911yescharacteristic
122C2×C4—22no—
123C2×C4—22no—
124C2×C4—22no—
125C2×C4—22no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—42no—
133C2×C4—42no—
134C2×C4—42no—
135C2×C4—42no—
136C2×C4—84no—
137C2×C4—84no—
138C2×C4—82no—
139C2×C4—82no—
140C2×C4—84no—
141C2×C4—82no—
142C2×C4—162no—
143C2×C4#35612yes—
144C2×C4#40711yescharacteristic
145C2×C4C2×#10111yescharacteristic
146C8—28no—
147C8—164no—
148C22—22no—
149C22—22no—
150C22—22no—
151C22—21no—
152C22—21no—
153C22—42no—
154C22—42no—
155C22#146311yescharacteristic
156C22C2×#35311yescharacteristic
157C22C2×#35611yescharacteristic
158C4—21no—
159C4—22no—
160C4—21no—
161C4—42no—
162C4—42no—
163C4—82no—
164C4—82no—
165C4—162no—
166C2—21no—
167C2—21no—
168C2—41no—
169C2—41no—
170C2#1151211yescharacteristic
171C2#1357812yes—
subgroupG/H
center, socleC22C2×#353
commutatorC2×C42C22×C4
Frattini, cosocle#382C23
prime radicalC24#99
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#382 ◁ G
LECSG ▷ #382 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7372C2semidirect
71. #97M16semidirect

Maximal subgroups

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

subgroupindex
#208012
#115112
#73722