Finite Group Databasebeta

← #134147

Subgroups — #134147

Order 512.

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

normal subgroups only

This group has 960 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×#2641C211yesmaximal, characteristic
2#11505C214yesmaximal, complemented
3#8554C212yesmaximal, complemented
4C22×#127C411yescharacteristic
5C22×#125C2211yescharacteristic
6#2641—28no—
7C2×#394—22no—
8C2×#394—22no—
9C2×#390C412yes—
10C2×#390C2212yes—
11C2×#376C411yescharacteristic
12#1821—22no—
13#1463—22no—
14#1463C2214yes—
15#1318—28no—
16C23×D4—21no—
17C23×D4—21no—
18C23×D4C2×C412yes—
19C2×#127—24no—
20C2×#127—42no—
21C2×#127—42no—
22C2×#125—24no—
23C2×#125—42no—
24C2×#125—42no—
25C22×C4⋊C4C2×C411yescharacteristic
26C22×C4⋊C4D412yes—
27C22×C4⋊C4D412yes—
28C22×C4⋊C4C2311yescharacteristic
29C22×C22⋊C4—24no—
30C22×C22⋊C4—24no—
31C22×C42C2×C411yescharacteristic
32C22×C42C2×C412yes—
33#407—24no—
34#394—216no—
35#394—216no—
36#390—24no—
37#390—24no—
38#390—24no—
39#390D418yes—
40#376—24no—
41C2×#95—21no—
42C2×#95—21no—
43#343—44no—
44#324—28no—
45#324—44no—
46C25—22no—
47C22×D4—24no—
48C22×D4—24no—
49C22×D4—22no—
50C22×D4—44no—
51C22×D4—42no—
52C22×D4—42no—
53C22×D4—42no—
54C22×D4—42no—
55C22×D4—44no—
56C22×D4—42no—
57C22×D4—42no—
58C22×D4—84no—
59C22×D4—84no—
60C22×D4C22⋊C414yes—
61C23×C4—22no—
62C23×C4C22⋊C412yes—
63C23×C4C22⋊C411yescharacteristic
64C23×C4C22⋊C411yescharacteristic
65C23×C4C22×C411yescharacteristic
66C23×C4C2×D412yes—
67C2×M16—44no—
68#127—416no—
69#125—416no—
70C2×C4⋊C4—24no—
71C2×C4⋊C4—24no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—24no—
74C2×C4⋊C4—24no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4C2×D414yes—
82C2×C22⋊C4—216no—
83C2×C22⋊C4—216no—
84C2×C22⋊C4—88no—
85C2×C22⋊C4—88no—
86C2×C42—22no—
87C2×C42—22no—
88C2×C42—22no—
89C2×C42—22no—
90C2×C42—44no—
91C2×C42C22⋊C414yes—
92C22⋊C8—44no—
93C22⋊C8—44no—
94#95—24no—
95#95—24no—
96C24—24no—
97C24—48no—
98C24—48no—
99C24—44no—
100C24—44no—
101C24—42no—
102C24—42no—
103C24C2×C22⋊C411yescharacteristic
104C2×D4—28no—
105C2×D4—44no—
106C2×D4—44no—
107C2×D4—44no—
108C2×D4—48no—
109C2×D4—48no—
110C2×D4—44no—
111C2×D4—44no—
112C2×D4—88no—
113C2×D4—88no—
114C2×D4—88no—
115C2×D4—88no—
116C2×D4—88no—
117C2×D4—88no—
118C2×D4—88no—
119C2×D4—88no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—24no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—24no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—44no—
136C22×C4—44no—
137C22×C4—44no—
138C22×C4—44no—
139C22×C4—44no—
140C22×C4#9912yes—
141C22×C4#9912yes—
142C22×C4C2×C22⋊C412yes—
143C22×C4#12014yes—
144M16—88no—
145C2×C8—84no—
146C4⋊C4—44no—
147C4⋊C4—48no—
148C4⋊C4—48no—
149C4⋊C4—44no—
150C22⋊C4—816no—
151C22⋊C4—816no—
152C42—28no—
153C42—44no—
154C42—44no—
155C23—22no—
156C23—22no—
157C23—22no—
158C23—48no—
159C23—48no—
160C23—44no—
161C23—44no—
162C23—44no—
163C23—84no—
164C23—84no—
165C23—84no—
166C23—84no—
167C23—816no—
168C23—816no—
169C23—84no—
170C23—84no—
171C23#37911yescharacteristic
172C23C2×#9911yescharacteristic
173C23C2×#9911yescharacteristic
174D4—88no—
175D4—88no—
176D4—816no—
177D4—816no—
178C2×C4—24no—
179C2×C4—24no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—24no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—44no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—44no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—44no—
194C2×C4—44no—
195C2×C4—44no—
196C2×C4—44no—
197C2×C4—42no—
198C2×C4—44no—
199C2×C4—82no—
200C2×C4—88no—
201C2×C4—82no—
202C2×C4#41314yes—
203C8—164no—
204C22—28no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—88no—
212C22—88no—
213C22—84no—
214C22—84no—
215C22—84no—
216C22—168no—
217C22—168no—
218C22#178111yescharacteristic
219C22#178512yes—
220C22#212812yes—
221C22#213412yes—
222C4—24no—
223C4—44no—
224C4—42no—
225C4—42no—
226C4—84no—
227C4—82no—
228C2—22no—
229C2—22no—
230C2—22no—
231C2—164no—
232C2#855211yescharacteristic
233C2#1269711yescharacteristic
234C2#1270311yescharacteristic
subgroupG/H
center, socleC22#1781
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC22×#127C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×#125 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #11505C2semidirect
3. #8554C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26412
#115052
#85542