Finite Group Databasebeta

← #134128

Subgroups — #134128

Order 512.

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

normal subgroups only

This group has 768 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×#2859C211yesmaximal, characteristic
2#11512C212yesmaximal
3#11511C212yesmaximal
4#8554C212yesmaximal
5C22×#125C411yescharacteristic
6C22×#125C2211yescharacteristic
7#2859—24no—
8#2859—24no—
9C2×#389—22no—
10C2×#389—22no—
11C2×#384C411yescharacteristic
12C2×#384C2211yescharacteristic
13C2×#382C411yescharacteristic
14C2×#382C2211yescharacteristic
15C2×C43C411yescharacteristic
16#1821—22no—
17#1463—22no—
18#1463C2214yes—
19#1325—24no—
20#1324—24no—
21C2×#125—22no—
22C2×#125—22no—
23C2×#125—22no—
24C2×#125—22no—
25C2×#125—42no—
26C2×#125—42no—
27C2×#125—42no—
28C2×#125—42no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4—22no—
32C22×C4⋊C4—22no—
33C22×C4⋊C4C2×C411yescharacteristic
34C22×C4⋊C4C2×C411yescharacteristic
35C22×C4⋊C4C2×C411yescharacteristic
36C22×C4⋊C4D412yes—
37C22×C4⋊C4D412yes—
38C22×C4⋊C4C2311yescharacteristic
39C22×C42—21no—
40C22×C42—21no—
41C22×C42C2×C411yescharacteristic
42C22×C42C2×C411yescharacteristic
43C22×C42C2×C411yescharacteristic
44#407—24no—
45#389—216no—
46#389—216no—
47#384—22no—
48#384—22no—
49#384—22no—
50#384D414yes—
51#382—22no—
52#382—22no—
53#382—22no—
54#382D414yes—
55C2×#95—22no—
56C2×#95—22no—
57C43—22no—
58C43—22no—
59#343—44no—
60#324—24no—
61#324—24no—
62#324—44no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4C22⋊C412yes—
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22×C411yescharacteristic
71C23×C4C2×D412yes—
72C2×M16—44no—
73#125—48no—
74#125—48no—
75#125—48no—
76#125—48no—
77C2×C4⋊C4—24no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—28no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—28no—
87C2×C4⋊C4—24no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—48no—
100C2×C4⋊C4—48no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4C22⋊C412yes—
106C2×C4⋊C4C22⋊C412yes—
107C2×C4⋊C4C2×D412yes—
108C2×C4⋊C4C2×D412yes—
109C2×C42—22no—
110C2×C42—22no—
111C2×C42—21no—
112C2×C42—21no—
113C2×C42—21no—
114C2×C42—21no—
115C2×C42—24no—
116C2×C42—24no—
117C2×C42—24no—
118C2×C42—24no—
119C2×C42—44no—
120C2×C42—44no—
121C2×C42C22⋊C412yes—
122C2×C42C22⋊C412yes—
123C22⋊C8—44no—
124C22⋊C8—44no—
125#95—28no—
126#95—28no—
127C24C2×C22⋊C411yescharacteristic
128C22×C4—24no—
129C22×C4—24no—
130C22×C4—24no—
131C22×C4—22no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—24no—
137C22×C4—22no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—44no—
154C22×C4—44no—
155C22×C4—44no—
156C22×C4—44no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4#9912yes—
162C22×C4#9912yes—
163C22×C4C2×C22⋊C411yescharacteristic
164C22×C4C2×C22⋊C411yescharacteristic
165C22×C4#12014yes—
166M16—88no—
167C2×C8—84no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—48no—
171C4⋊C4—48no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—416no—
175C4⋊C4—416no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—48no—
181C4⋊C4—48no—
182C42—24no—
183C42—24no—
184C42—24no—
185C42—24no—
186C42—22no—
187C42—22no—
188C42—48no—
189C42—48no—
190C42#10114yes—
191C42#10114yes—
192C23—22no—
193C23—22no—
194C23—21no—
195C23—21no—
196C23#37911yescharacteristic
197C23C2×#9911yescharacteristic
198C23C2×#9911yescharacteristic
199C2×C4—24no—
200C2×C4—24no—
201C2×C4—216no—
202C2×C4—24no—
203C2×C4—24no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—24no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—24no—
212C2×C4—24no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—48no—
218C2×C4—44no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—44no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—44no—
225C2×C4—44no—
226C2×C4—44no—
227C2×C4—44no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—82no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4—82no—
236C2×C4—82no—
237C2×C4C2×#10112yes—
238C2×C4C2×#10112yes—
239C8—164no—
240C22—24no—
241C22—24no—
242C22—21no—
243C22—21no—
244C22—21no—
245C22—21no—
246C22—21no—
247C22—21no—
248C22#178111yescharacteristic
249C22#178411yescharacteristic
250C22#178411yescharacteristic
251C22#212612yes—
252C22#213212yes—
253C4—28no—
254C4—22no—
255C4—22no—
256C4—44no—
257C4—44no—
258C4—42no—
259C4—42no—
260C4—82no—
261C4—82no—
262C4—82no—
263C2—21no—
264C2—21no—
265C2—22no—
266C2—22no—
267C2#855111yescharacteristic
268C2#1269511yescharacteristic
269C2#1270111yescharacteristic
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×#28592
#115122
#115112
#85542