Finite Group Databasebeta

← #134166

Subgroups — #134166

Order 512.

668 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 82 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#35674C211yesmaximal, characteristic
2#11510C212yesmaximal
3#11504C212yesmaximal
4#8743C212yesmaximal
5C22×C4⋊Q8C411yescharacteristic
6C2×#519C411yescharacteristic
7#2484—24no—
8#2446—24no—
9#2446—24no—
10#2310C411yescharacteristic
11#2310C2211yescharacteristic
12#2310C2211yescharacteristic
13#2296C411yescharacteristic
14#2296C2211yescharacteristic
15#1464C2214yes—
16#1319—24no—
17#1317—24no—
18#1285—24no—
19C23×Q8C2×C411yescharacteristic
20C23×Q8C2×C411yescharacteristic
21C2×C4⋊Q8—22no—
22C2×C4⋊Q8—22no—
23C2×C4⋊Q8—44no—
24#519—44no—
25C2×C4×Q8—22no—
26C2×C4×Q8—22no—
27C2×#117—22no—
28C2×#117C2×C411yescharacteristic
29C2×#117C2×C411yescharacteristic
30C2×#117C2×C411yescharacteristic
31C2×#117C2311yescharacteristic
32C22×C4⋊C4—22no—
33C22×C42C2×C411yescharacteristic
34#432—24no—
35#400—28no—
36#398—28no—
37#391—22no—
38#391—22no—
39#391—22no—
40#391—22no—
41#391—22no—
42#391D414yes—
43#389—22no—
44#389—22no—
45#387—28no—
46#387—28no—
47#385—22no—
48#385—22no—
49#381—22no—
50#381—22no—
51#381D414yes—
52C4×C4⋊C4—22no—
53C4×C4⋊C4D414yes—
54#324—24no—
55#323—24no—
56C22×Q8—24no—
57C22×Q8—21no—
58C22×Q8—21no—
59C22×Q8—21no—
60C22×Q8—42no—
61C22×Q8—42no—
62C22×Q8—42no—
63C22×Q8—42no—
64C22×Q8—42no—
65C22×Q8C22⋊C412yes—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4C22⋊C412yes—
69C23×C4C22×C411yescharacteristic
70C4⋊Q8—48no—
71C4⋊Q8—44no—
72C4⋊Q8—44no—
73C4×Q8—48no—
74C4×Q8—44no—
75C4×Q8—44no—
76#117—42no—
77#117—48no—
78#117—42no—
79#117—42no—
80#117—42no—
81C2×C4⋊C4—28no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—24no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—24no—
88C2×C4⋊C4—48no—
89C2×C4⋊C4C2×D412yes—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4C22⋊C412yes—
93C2×C22⋊C4C2×D412yes—
94C2×C42—24no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42—42no—
98C2×C42C22⋊C412yes—
99C2×C42C22⋊C412yes—
100C2×C42C22⋊C412yes—
101C2×C42C2×D412yes—
102C4⋊C8—44no—
103C22⋊C8—44no—
104#97—44no—
105#95—24no—
106#95—28no—
107#95—24no—
108#95—28no—
109#95—28no—
110C24C2×C22⋊C411yescharacteristic
111C2×Q8—24no—
112C2×Q8—22no—
113C2×Q8—416no—
114C2×Q8—42no—
115C2×Q8—42no—
116C2×Q8—42no—
117C2×Q8—42no—
118C2×Q8—42no—
119C2×Q8—42no—
120C2×Q8—42no—
121C2×Q8—42no—
122C2×Q8—42no—
123C2×Q8—42no—
124C2×Q8—88no—
125C2×Q8#10014yes—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—24no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—24no—
137C22×C4—44no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4C2×C22⋊C411yescharacteristic
144C22×C4C2×C22⋊C411yescharacteristic
145C22×C4#12014yes—
146C2×C8—84no—
147C4⋊C4—42no—
148C4⋊C4—48no—
149C4⋊C4—48no—
150C4⋊C4—416no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—48no—
154C4⋊C4—42no—
155C22⋊C4—42no—
156C22⋊C4—42no—
157C22⋊C4—42no—
158C22⋊C4—42no—
159C42—24no—
160C42—42no—
161C42—42no—
162C42—42no—
163C42—48no—
164C42—42no—
165C42—42no—
166C23—21no—
167C23—21no—
168C23—42no—
169C23#37911yescharacteristic
170C23#41312yes—
171Q8—44no—
172Q8—48no—
173Q8—44no—
174Q8—84no—
175Q8—84no—
176C2×C4—28no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—24no—
182C2×C4—24no—
183C2×C4—24no—
184C2×C4—42no—
185C2×C4—48no—
186C2×C4—48no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—44no—
191C2×C4—44no—
192C2×C4—44no—
193C2×C4—44no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—84no—
201C2×C4#41012yes—
202C2×C4C2×#10012yes—
203C8—164no—
204C22—22no—
205C22—22no—
206C22—42no—
207C22—42no—
208C22—42no—
209C22#178211yescharacteristic
210C22#178311yescharacteristic
211C22#178511yescharacteristic
212C4—22no—
213C4—22no—
214C4—44no—
215C4—44no—
216C4—44no—
217C4—82no—
218C4—82no—
219C4—82no—
220C4—82no—
221C2—21no—
222C2—21no—
223C2—42no—
224C2#855311yescharacteristic
225C2#1272412yes—
subgroupG/H
center, socleC22#1782
commutatorC23×C4C22×C4
Frattini, cosocleC2×#117C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2296 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#356742
#115102
#115042
#87432