Finite Group Databasebeta

← #134146

Subgroups — #134146

Order 512.

700 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 86 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
1C2×#2642C211yesmaximal, characteristic
2#11506C214yesmaximal
3#8554C212yesmaximal
4C22×C4⋊Q8C411yescharacteristic
5C22×#125C2211yescharacteristic
6#2642—28no—
7C2×#400—22no—
8C2×#400—22no—
9C2×#391C412yes—
10C2×#391C2212yes—
11C2×#376C411yescharacteristic
12#1821—22no—
13#1463—22no—
14#1463C2214yes—
15#1319—28no—
16C23×Q8C2×C412yes—
17C2×C4⋊Q8—24no—
18C2×C4⋊Q8—42no—
19C2×C4⋊Q8—42no—
20C2×#125—24no—
21C2×#125—42no—
22C2×#125—42no—
23C22×C4⋊C4—21no—
24C22×C4⋊C4—21no—
25C22×C4⋊C4C2×C411yescharacteristic
26C22×C4⋊C4D412yes—
27C22×C4⋊C4D412yes—
28C22×C4⋊C4C2311yescharacteristic
29C22×C42C2×C411yescharacteristic
30C22×C42C2×C412yes—
31#407—24no—
32#400—216no—
33#400—216no—
34#391—24no—
35#391—24no—
36#391—24no—
37#391D418yes—
38#376—24no—
39C2×#95—21no—
40C2×#95—21no—
41C2×#95—24no—
42C2×#95—24no—
43#343—44no—
44#324—28no—
45#324—44no—
46C22×Q8—22no—
47C22×Q8—44no—
48C22×Q8—44no—
49C22×Q8—42no—
50C22×Q8—42no—
51C22×Q8C22⋊C414yes—
52C23×C4—22no—
53C23×C4—22no—
54C23×C4C22⋊C412yes—
55C23×C4C22⋊C411yescharacteristic
56C23×C4C22⋊C411yescharacteristic
57C23×C4C22×C411yescharacteristic
58C23×C4C2×D412yes—
59C2×M16—44no—
60C4⋊Q8—416no—
61#125—416no—
62C2×C4⋊C4—24no—
63C2×C4⋊C4—24no—
64C2×C4⋊C4—24no—
65C2×C4⋊C4—24no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—24no—
68C2×C4⋊C4—24no—
69C2×C4⋊C4—41no—
70C2×C4⋊C4—41no—
71C2×C4⋊C4—42no—
72C2×C4⋊C4—42no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4C2×D414yes—
80C2×C42—22no—
81C2×C42—22no—
82C2×C42—22no—
83C2×C42—22no—
84C2×C42—44no—
85C2×C42C22⋊C414yes—
86C22⋊C8—44no—
87C22⋊C8—44no—
88#95—24no—
89#95—24no—
90#95—216no—
91#95—216no—
92C24C2×C22⋊C411yescharacteristic
93C2×Q8—28no—
94C2×Q8—44no—
95C2×Q8—44no—
96C2×Q8—44no—
97C2×Q8—44no—
98C2×Q8—44no—
99C2×Q8—88no—
100C2×Q8—88no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—24no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—24no—
111C22×C4—22no—
112C22×C4—24no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4—44no—
118C22×C4—44no—
119C22×C4—44no—
120C22×C4—44no—
121C22×C4—44no—
122C22×C4—44no—
123C22×C4—44no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4#9912yes—
127C22×C4#9912yes—
128C22×C4C2×C22⋊C412yes—
129C22×C4#12014yes—
130M16—88no—
131C2×C8—84no—
132C4⋊C4—44no—
133C4⋊C4—48no—
134C4⋊C4—48no—
135C4⋊C4—44no—
136C4⋊C4—48no—
137C4⋊C4—48no—
138C42—28no—
139C42—44no—
140C42—44no—
141C23—22no—
142C23—22no—
143C23—22no—
144C23#37911yescharacteristic
145C23C2×#9911yescharacteristic
146C23C2×#9911yescharacteristic
147Q8—88no—
148Q8—88no—
149C2×C4—24no—
150C2×C4—24no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—24no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—44no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—44no—
164C2×C4—44no—
165C2×C4—44no—
166C2×C4—44no—
167C2×C4—44no—
168C2×C4—42no—
169C2×C4—44no—
170C2×C4—44no—
171C2×C4—44no—
172C2×C4—44no—
173C2×C4—82no—
174C2×C4—88no—
175C2×C4—84no—
176C2×C4—84no—
177C2×C4—82no—
178C2×C4#41314yes—
179C8—164no—
180C22—28no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22—21no—
186C22—21no—
187C22#178111yescharacteristic
188C22#178512yes—
189C22#212912yes—
190C22#213512yes—
191C4—24no—
192C4—44no—
193C4—42no—
194C4—42no—
195C4—84no—
196C4—84no—
197C4—82no—
198C2—22no—
199C2—22no—
200C2—22no—
201C2#855211yescharacteristic
202C2#1269811yescharacteristic
203C2#1270411yescharacteristic
subgroupG/H
center, socleC22#1781
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×#125 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#26422
#115062
#85542