Finite Group Databasebeta

← #149769

Subgroups — #149769

Order 512.

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

normal subgroups only

This group has 301 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#21831C211yesmaximal, characteristic
2#11521C212yesmaximal
3#7351C212yesmaximal
4#7350C212yesmaximal
5C2×#533C411yescharacteristic
6#2920C411yescharacteristic
7#2920C2211yescharacteristic
8#2165—22no—
9#1998—22no—
10#1997—22no—
11#1822C411yescharacteristic
12#1822C411yescharacteristic
13#1822C2211yescharacteristic
14#1822C2211yescharacteristic
15#1327—22no—
16#1320C2214yes—
17#533—41no—
18#533—41no—
19#533—41no—
20#533—41no—
21C2×C4⋊Q8—21no—
22C2×#125—21no—
23C2×C22⋊Q8—22no—
24C2×C4×Q8—22no—
25C2×#117C2×C411yescharacteristic
26C2×#117C2×C411yescharacteristic
27C22×C42C2×C411yescharacteristic
28#432—41no—
29#432—41no—
30#431—41no—
31#431—41no—
32#423—21no—
33#423D412yes—
34#407C2×C411yescharacteristic
35#407C2×C411yescharacteristic
36#407C2×C411yescharacteristic
37#407C2311yescharacteristic
38C4×M16—21no—
39C4×M16D412yes—
40#364—42no—
41#349—24no—
42#347—42no—
43#343—22no—
44#343—22no—
45#343—22no—
46#343—22no—
47C22×Q8—22no—
48C23×C4—21no—
49C23×C4C22×C411yescharacteristic
50C2×M16—21no—
51C2×M16—21no—
52C2×M16C22⋊C412yes—
53C2×M16C2×D412yes—
54C4⋊Q8—41no—
55C4⋊Q8—41no—
56C4⋊Q8—41no—
57C4⋊Q8—41no—
58#125—42no—
59#125—41no—
60#125—41no—
61C22⋊Q8—42no—
62C22⋊Q8—42no—
63C22⋊Q8—42no—
64C22⋊Q8—42no—
65C4×Q8—42no—
66C4×Q8—42no—
67C4×Q8—42no—
68C4×Q8—42no—
69#117—41no—
70#117—41no—
71#117—41no—
72#117—41no—
73C2×C4⋊C4—24no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C22⋊C4—21no—
78C2×C22⋊C4M1612yes—
79C2×C42—21no—
80C2×C42—21no—
81C2×C42—21no—
82C2×C42—21no—
83C2×C42—21no—
84C2×C42—21no—
85C2×C42C22⋊C411yescharacteristic
86C2×C42C22⋊C411yescharacteristic
87C2×C42M1612yes—
88M32—44no—
89C4⋊C8—42no—
90C4⋊C8—42no—
91C22⋊C8—41no—
92C22⋊C8—41no—
93C22⋊C8—41no—
94C22⋊C8—41no—
95#97—22no—
96#97—41no—
97C4×C8—22no—
98C4×C8—41no—
99C24C2×M1611yescharacteristic
100C2×Q8—42no—
101C2×Q8—42no—
102C2×Q8—84no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—42no—
114C22×C4—82no—
115C22×C4C2×C22⋊C411yescharacteristic
116C22×C4C2×M1611yescharacteristic
117M16—42no—
118M16—42no—
119M16—42no—
120M16—42no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8—41no—
124C2×C8—41no—
125C4⋊C4—44no—
126C4⋊C4—44no—
127C4⋊C4—42no—
128C4⋊C4—42no—
129C4⋊C4—41no—
130C4⋊C4—41no—
131C4⋊C4—41no—
132C4⋊C4—41no—
133C22⋊C4—42no—
134C22⋊C4—42no—
135C42—21no—
136C42—21no—
137C42—21no—
138C42—21no—
139C42—21no—
140C42—21no—
141C42—42no—
142C42—42no—
143C42#9911yescharacteristic
144C42#9911yescharacteristic
145C42#10011yescharacteristic
146C42#10011yescharacteristic
147C16—84no—
148C23—21no—
149C23—41no—
150C23—81no—
151C23#40711yescharacteristic
152Q8—84no—
153Q8—84no—
154C2×C4—21no—
155C2×C4—22no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—41no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—81no—
169C2×C4—81no—
170C2×C4—81no—
171C2×C4—82no—
172C2×C4—82no—
173C2×C4C2×#9911yescharacteristic
174C2×C4C2×#10011yescharacteristic
175C8—42no—
176C8—42no—
177C8—81no—
178C8—81no—
179C22—21no—
180C22—41no—
181C22—41no—
182C22—41no—
183C22—41no—
184C22—82no—
185C22#146211yescharacteristic
186C4—22no—
187C4—21no—
188C4—41no—
189C4—41no—
190C4—41no—
191C4—41no—
192C4—82no—
193C4—82no—
194C4#212312yes—
195C2—21no—
196C2—41no—
197C2—81no—
198C2#1150411yescharacteristic
subgroupG/H
center, socleC2#11504
commutatorC23×C4C22×C4
Frattini, cosocle#407C23
prime radicalC24C2×M16
LCSG ▷ C23×C4 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C42 ◁ #2920 ◁ 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
#218312
#115212
#73512
#73502