Finite Group Databasebeta

← #134131

Subgroups — #134131

Order 512.

744 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 744 conjugacy classes of subgroups — too many to draw. Only its 94 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×#2660C211yesmaximal, characteristic
2#11512C214yesmaximal
3#8554C212yesmaximal
4C22×C4⋊Q8C411yescharacteristic
5C22×#125C2211yescharacteristic
6#2660—28no—
7C2×#398—22no—
8C2×#395—22no—
9C2×#384C2212yes—
10C2×#382C412yes—
11C2×#376C411yescharacteristic
12#1821—22no—
13#1463—22no—
14#1463C2214yes—
15#1325—28no—
16C23×Q8—21no—
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—24no—
25C22×C4⋊C4C2×C411yescharacteristic
26C22×C4⋊C4C2×C412yes—
27C22×C4⋊C4D412yes—
28C22×C4⋊C4D412yes—
29C22×C4⋊C4C2311yescharacteristic
30C22×C42C2×C411yescharacteristic
31C22×C42C2×C412yes—
32#407—24no—
33#398—216no—
34#395—216no—
35#384—24no—
36#384D418yes—
37#382—24no—
38#382—24no—
39#376—24no—
40C2×#95—21no—
41C2×#95—24no—
42C2×#95—21no—
43#343—44no—
44#324—28no—
45#324—44no—
46C22×Q8—24no—
47C22×Q8—42no—
48C22×Q8—42no—
49C22×Q8—84no—
50C23×C4—22no—
51C23×C4—22no—
52C23×C4C22⋊C412yes—
53C23×C4C22⋊C411yescharacteristic
54C23×C4C22⋊C411yescharacteristic
55C23×C4C22×C411yescharacteristic
56C23×C4C2×D412yes—
57C2×M16—44no—
58C4⋊Q8—416no—
59#125—416no—
60C2×C4⋊C4—24no—
61C2×C4⋊C4—24no—
62C2×C4⋊C4—24no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—24no—
66C2×C4⋊C4—24no—
67C2×C4⋊C4—216no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4—41no—
70C2×C4⋊C4—42no—
71C2×C4⋊C4—42no—
72C2×C4⋊C4—41no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4—42no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—88no—
79C2×C4⋊C4C22⋊C414yes—
80C2×C4⋊C4C2×D414yes—
81C2×C42—22no—
82C2×C42—22no—
83C2×C42—22no—
84C2×C42—22no—
85C2×C42—28no—
86C2×C42C22⋊C414yes—
87C22⋊C8—44no—
88C22⋊C8—44no—
89#95—24no—
90#95—24no—
91#95—216no—
92C24C2×C22⋊C411yescharacteristic
93C2×Q8—48no—
94C2×Q8—88no—
95C2×Q8—88no—
96C2×Q8—88no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—24no—
100C22×C4—28no—
101C22×C4—24no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—24no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—44no—
113C22×C4—44no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4—44no—
117C22×C4—44no—
118C22×C4—48no—
119C22×C4—42no—
120C22×C4—42no—
121C22×C4#9912yes—
122C22×C4#9912yes—
123C22×C4C2×C22⋊C412yes—
124C22×C4#12014yes—
125M16—88no—
126C2×C8—84no—
127C4⋊C4—44no—
128C4⋊C4—48no—
129C4⋊C4—48no—
130C4⋊C4—44no—
131C4⋊C4—44no—
132C4⋊C4—44no—
133C4⋊C4—48no—
134C4⋊C4—816no—
135C42—28no—
136C42—28no—
137C42—24no—
138C42#10118yes—
139C23—22no—
140C23—22no—
141C23—22no—
142C23#37911yescharacteristic
143C23C2×#9911yescharacteristic
144C23C2×#9911yescharacteristic
145Q8—816no—
146C2×C4—24no—
147C2×C4—24no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—24no—
153C2×C4—28no—
154C2×C4—44no—
155C2×C4—416no—
156C2×C4—44no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—44no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—44no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—48no—
166C2×C4—44no—
167C2×C4—44no—
168C2×C4—42no—
169C2×C4—44no—
170C2×C4—82no—
171C2×C4—84no—
172C2×C4—84no—
173C2×C4—82no—
174C2×C4C2×#10114yes—
175C8—164no—
176C22—28no—
177C22—21no—
178C22—21no—
179C22—21no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22#178111yescharacteristic
184C22#178412yes—
185C22#212812yes—
186C22#213512yes—
187C4—24no—
188C4—48no—
189C4—44no—
190C4—42no—
191C4—42no—
192C4—84no—
193C4—82no—
194C2—22no—
195C2—22no—
196C2—22no—
197C2#855111yescharacteristic
198C2#1269711yescharacteristic
199C2#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×#26602
#115122
#85542