Finite Group Databasebeta

← #134126

Subgroups — #134126

Order 512.

848 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 848 conjugacy classes of subgroups — too many to draw. Only its 102 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×#2862C211yesmaximal, characteristic
2#11511C214yesmaximal
3#8551C212yesmaximal
4C22×C4⋊Q8C411yescharacteristic
5C22×C4⋊Q8C2211yescharacteristic
6#2862—28no—
7C2×#391—22no—
8C2×#391—22no—
9C2×#382C412yes—
10C2×#382C2212yes—
11C2×C43C411yescharacteristic
12#1784—22no—
13#1463—22no—
14#1463C2214yes—
15#1324—28no—
16C23×Q8—21no—
17C23×Q8D412yes—
18C2×C4⋊Q8—24no—
19C2×C4⋊Q8—24no—
20C2×C4⋊Q8—42no—
21C2×C4⋊Q8—42no—
22C2×C4⋊Q8—42no—
23C2×C4⋊Q8—42no—
24C22×C4⋊C4—21no—
25C22×C4⋊C4C2×C411yescharacteristic
26C22×C4⋊C4C2×C412yes—
27C22×C4⋊C4D412yes—
28C22×C4⋊C4C2311yescharacteristic
29C22×C42—21no—
30C22×C42—21no—
31C22×C42C2×C411yescharacteristic
32C22×C42C2×C412yes—
33C2×#101—44no—
34#407—24no—
35#391—216no—
36#391—216no—
37#382—24no—
38#382—24no—
39#382—24no—
40#382D418yes—
41C2×#95—24no—
42C2×#95—24no—
43C43—24no—
44#324—28no—
45#324—44no—
46C22×Q8—24no—
47C22×Q8—24no—
48C22×Q8—24no—
49C22×Q8—42no—
50C22×Q8—48no—
51C22×Q8—42no—
52C22×Q8—48no—
53C22×Q8—42no—
54C23×C4—22no—
55C23×C4—22no—
56C23×C4C22⋊C411yescharacteristic
57C23×C4C22⋊C411yescharacteristic
58C23×C4C22⋊C412yes—
59C23×C4C22×C411yescharacteristic
60C23×C4C2×D412yes—
61C2×M16—44no—
62C4⋊Q8—416no—
63C4⋊Q8—416no—
64C2×C4⋊C4—24no—
65C2×C4⋊C4—24no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—24no—
69C2×C4⋊C4—41no—
70C2×C4⋊C4—41no—
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⋊C4C22⋊C414yes—
79C2×C4⋊C4C2×D414yes—
80C2×C42—22no—
81C2×C42—22no—
82C2×C42—22no—
83C2×C42—22no—
84C2×C42—28no—
85C2×C42—24no—
86C2×C42—24no—
87C2×C42—44no—
88C2×C42—44no—
89C2×C42C22⋊C414yes—
90#101—88no—
91C22⋊C8—44no—
92C22⋊C8—44no—
93#95—216no—
94#95—216no—
95C24C2×C22⋊C411yescharacteristic
96C2×Q8—48no—
97C2×Q8—48no—
98C2×Q8—416no—
99C2×Q8—416no—
100C2×Q8—48no—
101C2×Q8—84no—
102C2×Q8—88no—
103C2×Q8—88no—
104C2×Q8—84no—
105C2×Q8—84no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—24no—
113C22×C4—24no—
114C22×C4—24no—
115C22×C4—28no—
116C22×C4—24no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—24no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—44no—
124C22×C4—44no—
125C22×C4—44no—
126C22×C4—44no—
127C22×C4—48no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4#9912yes—
131C22×C4#9912yes—
132C22×C4C2×C22⋊C412yes—
133C22×C4#12014yes—
134M16—88no—
135C2×C8—84no—
136C4⋊C4—44no—
137C4⋊C4—48no—
138C4⋊C4—44no—
139C4⋊C4—44no—
140C4⋊C4—44no—
141C4⋊C4—48no—
142C42—28no—
143C42—28no—
144C42—24no—
145C42—48no—
146C42—48no—
147C42#10118yescomplemented
148C23—22no—
149C23—22no—
150C23—22no—
151C23#37911yescharacteristic
152C23C2×#9911yescharacteristic
153C23C2×#9911yescharacteristic
154Q8—816no—
155Q8—816no—
156C2×C4—24no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—24no—
160C2×C4—24no—
161C2×C4—24no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—24no—
169C2×C4—28no—
170C2×C4—44no—
171C2×C4—44no—
172C2×C4—416no—
173C2×C4—44no—
174C2×C4—44no—
175C2×C4—44no—
176C2×C4—48no—
177C2×C4—44no—
178C2×C4—42no—
179C2×C4—44no—
180C2×C4—82no—
181C2×C4—84no—
182C2×C4—84no—
183C2×C4—82no—
184C2×C4#35414yes—
185C2×C4#35514yes—
186C2×C4C2×#10114yes—
187C8—164no—
188C22—28no—
189C22—21no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—21no—
195C22#178111yescharacteristic
196C22#178412yes—
197C22C2×#35412yes—
198C22C2×#35512yes—
199C4—24no—
200C4—24no—
201C4—24no—
202C4—48no—
203C4—44no—
204C4—82no—
205C4—84no—
206C2—22no—
207C2—22no—
208C2—22no—
209C2#855111yescharacteristic
210C2#1269411yescharacteristic
211C2#1269911yescharacteristic
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×C4⋊Q8 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
147. C42#101semidirect

Maximal subgroups

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

subgroupindex
C2×#28622
#115112
#85512