Finite Group Databasebeta

← #149758

Subgroups — #149758

Order 512.

351 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 351 conjugacy classes of subgroups — too many to draw. Only its 56 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#20730C211yesmaximal, characteristic
2#11522C212yesmaximal
3#7350C212yesmaximal
4#7341C212yesmaximal
5C2×#519C411yescharacteristic
6#2921C411yescharacteristic
7#2921C2211yescharacteristic
8#2165—22no—
9#1822C411yescharacteristic
10#1822C2211yescharacteristic
11#1801—22no—
12#1784C411yescharacteristic
13#1784C2211yescharacteristic
14C4×#101—22no—
15#1327—22no—
16#1320C2214yes—
17C23×Q8C2×C411yescharacteristic
18#519—42no—
19#519—42no—
20C2×C4×Q8—21no—
21C2×C4×Q8—21no—
22C2×C4×Q8—22no—
23C2×#117—22no—
24C2×#117C2×C411yescharacteristic
25C2×#117C2×C411yescharacteristic
26#433—42no—
27#433—42no—
28#423—21no—
29#423D412yes—
30C2×#101—22no—
31C2×#101—22no—
32#407C2×C411yescharacteristic
33#407C2×C411yescharacteristic
34#407C2×C411yescharacteristic
35#407C2311yescharacteristic
36C4×M16—21no—
37C4×M16D412yes—
38#364—42no—
39#349—24no—
40#347—42no—
41#343—22no—
42#343—22no—
43C22×Q8—21no—
44C22×Q8—21no—
45C22×Q8—41no—
46C22×Q8—41no—
47C22×Q8—44no—
48C23×C4M1612yes—
49C23×C4C22×C411yescharacteristic
50C2×M16—21no—
51C2×M16—21no—
52C2×M16C22⋊C412yes—
53C2×M16C2×D412yes—
54C4×Q8—42no—
55C4×Q8—42no—
56C4×Q8—48no—
57C4×Q8—42no—
58C4×Q8—42no—
59#117—22no—
60#117—22no—
61#117—48no—
62#117—41no—
63#117—41no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—21no—
68C2×C22⋊C4—21no—
69C2×C22⋊C4—21no—
70C2×C42—22no—
71C2×C42—22no—
72C2×C42C22⋊C411yescharacteristic
73C2×C42C22⋊C411yescharacteristic
74C2×C42M1612yes—
75M32—44no—
76C4⋊C8—42no—
77C4⋊C8—42no—
78#101—44no—
79#101—44no—
80C22⋊C8—41no—
81C22⋊C8—41no—
82C22⋊C8—41no—
83C22⋊C8—41no—
84#97—22no—
85#97—41no—
86C4×C8—22no—
87C4×C8—41no—
88C24C2×M1611yescharacteristic
89C2×Q8—24no—
90C2×Q8—42no—
91C2×Q8—42no—
92C2×Q8—44no—
93C2×Q8—44no—
94C2×Q8—48no—
95C2×Q8—44no—
96C2×Q8—41no—
97C2×Q8—41no—
98C22×C4—22no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—22no—
104C22×C4—44no—
105C22×C4—42no—
106C22×C4—42no—
107C22×C4C2×C22⋊C411yescharacteristic
108C22×C4C2×M1611yescharacteristic
109M16—42no—
110M16—42no—
111M16—42no—
112M16—42no—
113C2×C8—22no—
114C2×C8—22no—
115C2×C8—41no—
116C2×C8—41no—
117C4⋊C4—24no—
118C4⋊C4—44no—
119C4⋊C4—44no—
120C4⋊C4—41no—
121C4⋊C4—41no—
122C22⋊C4—24no—
123C22⋊C4—42no—
124C42—21no—
125C42—21no—
126C42—42no—
127C42—44no—
128C42—44no—
129C42#9912yes—
130C42#10012yes—
131C16—84no—
132C23—21no—
133C23—41no—
134C23—81no—
135C23#40711yescharacteristic
136Q8—44no—
137Q8—48no—
138Q8—44no—
139C2×C4—22no—
140C2×C4—24no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—21no—
144C2×C4—42no—
145C2×C4—42no—
146C2×C4—44no—
147C2×C4—42no—
148C2×C4—42no—
149C2×C4—41no—
150C2×C4—44no—
151C2×C4—42no—
152C2×C4—81no—
153C2×C4—81no—
154C2×C4—82no—
155C2×C4C2×#9911yescharacteristic
156C2×C4C2×#10011yescharacteristic
157C8—42no—
158C8—42no—
159C8—81no—
160C8—81no—
161C22—21no—
162C22—41no—
163C22—42no—
164C22—41no—
165C22—82no—
166C22#146211yescharacteristic
167C4—21no—
168C4—42no—
169C4—42no—
170C4—44no—
171C4—41no—
172C4—82no—
173C4#212012yes—
174C2—21no—
175C2—41no—
176C2—81no—
177C2#1150111yescharacteristic
subgroupG/H
center, socleC2#11501
commutatorC23×C4C22×C4
Frattini, cosocle#407C23
prime radicalC24C2×M16
LCSG ▷ C23×C4 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C42 ◁ #2921 ◁ G
LECSG ▷ #407 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#207302
#115222
#73502
#73412