Finite Group Databasebeta

← #149431

Subgroups — #149431

Order 512.

916 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 916 conjugacy classes of subgroups — too many to draw. Only its 154 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×#2568C211yesmaximal, characteristic
2C2×#1394C214yesmaximal
3#10763C212yesmaximal
4C22×C4⋊Q8C411yescharacteristic
5C22×C4×Q8C2211yescharacteristic
6C22×#101C2214yes—
7#2568—28no—
8C2×#400—22no—
9C2×#391—22no—
10C2×#391C2212yes—
11C2×#382C412yes—
12C2×C4×C4⋊C4C411yescharacteristic
13C2×#343—22no—
14#1728—22no—
15#1394—28no—
16C23×Q8—21no—
17C23×Q8C2311yescharacteristic
18C22×M16—24no—
19C2×C4⋊Q8—24no—
20C2×C4⋊Q8—42no—
21C2×C4⋊Q8—42no—
22C2×C4×Q8—24no—
23C2×C4×Q8—42no—
24C2×C4×Q8—42no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4C2×C412yes—
29C22×C4⋊C4D412yes—
30C22×C42C2×C411yescharacteristic
31C22×C42C2×C412yes—
32C22×C42D412yes—
33C2×#101—28no—
34C2×#101—28no—
35C2×#101—24no—
36C2×#101Q818yescomplemented
37#400—216no—
38#391—216no—
39#391—24no—
40#391D418yes—
41#382—24no—
42#382—24no—
43C4×C4⋊C4—24no—
44C2×#95—21no—
45C2×#95—24no—
46C2×#95—24no—
47#343—44no—
48#336—44no—
49C22×Q8—24no—
50C22×Q8—41no—
51C22×Q8—41no—
52C22×Q8—42no—
53C22×Q8—42no—
54C22×Q8—48no—
55C22×Q8—42no—
56C22×Q8C2×D414yes—
57C22×Q8C2×Q814yes—
58C22×Q8#4114yes—
59C23×C4—22no—
60C23×C4—22no—
61C23×C4C22⋊C412yes—
62C23×C4C22⋊C411yescharacteristic
63C23×C4C22⋊C411yescharacteristic
64C23×C4C22×C411yescharacteristic
65C23×C4C2×D412yes—
66C2×M16—28no—
67C2×M16—44no—
68C2×M16—44no—
69C2×M16—44no—
70C2×M16—44no—
71C2×M16—44no—
72C22×C8—44no—
73C4⋊Q8—416no—
74C4×Q8—416no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4—24no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—44no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4C22⋊C414yes—
90C2×C42—22no—
91C2×C42—22no—
92C2×C42—22no—
93C2×C42—24no—
94C2×C42—22no—
95C2×C42—24no—
96C2×C42—42no—
97C2×C42—44no—
98C2×C42C22⋊C414yes—
99#101—216no—
100#101—48no—
101#95—24no—
102#95—216no—
103#95—216no—
104C24C2×C22⋊C411yescharacteristic
105C2×Q8—44no—
106C2×Q8—44no—
107C2×Q8—44no—
108C2×Q8—44no—
109C2×Q8—44no—
110C2×Q8—44no—
111C2×Q8—48no—
112C2×Q8—416no—
113C2×Q8—88no—
114C2×Q8—84no—
115C2×Q8—84no—
116C2×Q8C22⋊Q818yes—
117C2×Q8#12414yes—
118C2×Q8C4⋊Q814yes—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—22no—
123C22×C4—21no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—24no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—24no—
132C22×C4—22no—
133C22×C4—24no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—42no—
137C22×C4—44no—
138C22×C4—44no—
139C22×C4—44no—
140C22×C4—48no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—44no—
144C22×C4#9912yes—
145C22×C4#9912yes—
146C22×C4C2×C22⋊C412yes—
147C22×C4C4×Q814yes—
148C22×C4#12014yes—
149C22×C4C22⋊Q814yes—
150C22×C4#12412yes—
151C22×C4C4⋊Q812yes—
152M16—48no—
153M16—48no—
154M16—84no—
155M16—84no—
156C2×C8—48no—
157C2×C8—84no—
158C2×C8—84no—
159C4⋊C4—48no—
160C4⋊C4—44no—
161C4⋊C4—48no—
162C4⋊C4—44no—
163C4⋊C4—44no—
164C4⋊C4—48no—
165C42—28no—
166C42—48no—
167C42—44no—
168C42—44no—
169C23—22no—
170C23—22no—
171C23#37911yescharacteristic
172C23#39112yes—
173C23C2×#9911yescharacteristic
174C23C2×#9911yescharacteristic
175C23#41012yes—
176Q8—48no—
177Q8—44no—
178Q8—44no—
179Q8—816no—
180C2×C4—24no—
181C2×C4—24no—
182C2×C4—22no—
183C2×C4—24no—
184C2×C4—22no—
185C2×C4—24no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—24no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—24no—
194C2×C4—22no—
195C2×C4—24no—
196C2×C4—22no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—44no—
201C2×C4—44no—
202C2×C4—44no—
203C2×C4—48no—
204C2×C4—44no—
205C2×C4—42no—
206C2×C4—44no—
207C2×C4—44no—
208C2×C4—88no—
209C2×C4—84no—
210C2×C4—84no—
211C2×C4—82no—
212C2×C4#35514yes—
213C2×C4#39114yes—
214C2×C4#39314yes—
215C2×C4#39514yes—
216C8—88no—
217C22—21no—
218C22—21no—
219C22—21no—
220C22—21no—
221C22—24no—
222C22—24no—
223C22#144611yescharacteristic
224C22#178111yescharacteristic
225C22#178211yescharacteristic
226C22#199612yes—
227C22#199612yes—
228C22C2×#35512yes—
229C22#213512yes—
230C4—24no—
231C4—24no—
232C4—24no—
233C4—24no—
234C4—42no—
235C4—84no—
236C4—84no—
237C4—82no—
238C2—21no—
239C2—21no—
240C2—22no—
241C2#1138111yescharacteristic
242C2#1269911yescharacteristic
243C2#1270411yescharacteristic
244C2#1330114yes—
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC23×Q8C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
36. C2×#101Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25682
C2×#13942
#107632