Finite Group Databasebeta

← #129452

Subgroups — #129452

Order 512.

1140 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 1140 conjugacy classes of subgroups — too many to draw. Only its 110 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×#2548C211yesmaximal, characteristic
2C2×#1318C214yesmaximal, complemented
3#8303C212yesmaximal, complemented
4C22×#127C411yescharacteristic
5C22×#124C412yes—
6C22×C4×Q8C2211yescharacteristic
7#2548—28no—
8C2×#394—22no—
9C2×#390—22no—
10C2×#390C2212yes—
11C2×C4×C4⋊C4C411yescharacteristic
12#1817—22no—
13C2×#324—22no—
14C2×#324C2214yes—
15#1318—28no—
16C23×Q8C2×C411yescharacteristic
17C23×D4—21no—
18C23×D4—21no—
19C23×D4C2×C412yes—
20C2×#127—24no—
21C2×#127—42no—
22C2×#127—42no—
23C2×#124—24no—
24C2×#124—24no—
25C2×#124—48no—
26C2×C4×Q8—24no—
27C2×C4×Q8—42no—
28C2×C4×Q8—42no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4D412yes—
31C22×C4⋊C4C2311yescharacteristic
32C22×C22⋊C4—24no—
33C22×C22⋊C4—24no—
34C22×C42C2×C411yescharacteristic
35C22×C42C2×C412yes—
36C22×C42D412yes—
37C2×C22⋊C8—24no—
38#394—216no—
39#390—216no—
40#390—24no—
41#390D418yes—
42C4×C4⋊C4—24no—
43C2×#95—21no—
44#336—44no—
45#324—28no—
46#324—44no—
47C25—22no—
48C22×Q8—24no—
49C22×Q8—22no—
50C22×Q8—24no—
51C22×Q8—41no—
52C22×Q8—41no—
53C22×D4—24no—
54C22×D4—24no—
55C22×D4—22no—
56C22×D4—28no—
57C22×D4—44no—
58C22×D4—42no—
59C22×D4—42no—
60C22×D4—42no—
61C22×D4—42no—
62C22×D4—48no—
63C22×D4—42no—
64C22×D4—42no—
65C22×D4—84no—
66C22×D4C22⋊C414yes—
67C23×C4—22no—
68C23×C4C22⋊C412yes—
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22⋊C411yescharacteristic
71C23×C4C22×C411yescharacteristic
72C23×C4C2×D412yes—
73C22×C8—44no—
74#127—416no—
75#124—416no—
76#124—416no—
77C4×Q8—416no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—44no—
85C2×C4⋊C4C2×D414yes—
86C2×C22⋊C4—216no—
87C2×C22⋊C4—216no—
88C2×C22⋊C4—48no—
89C2×C22⋊C4—48no—
90C2×C22⋊C4—88no—
91C2×C42—22no—
92C2×C42—22no—
93C2×C42—22no—
94C2×C42—24no—
95C2×C42—22no—
96C2×C42—24no—
97C2×C42—42no—
98C2×C42—44no—
99C2×C42C22⋊C414yes—
100C22⋊C8—44no—
101C22⋊C8—44no—
102#95—24no—
103C24—24no—
104C24—48no—
105C24—48no—
106C24—44no—
107C24—42no—
108C24—42no—
109C24—44no—
110C24C2×C22⋊C411yescharacteristic
111C2×Q8—28no—
112C2×Q8—48no—
113C2×Q8—48no—
114C2×Q8—44no—
115C2×Q8—44no—
116C2×D4—28no—
117C2×D4—44no—
118C2×D4—44no—
119C2×D4—44no—
120C2×D4—48no—
121C2×D4—44no—
122C2×D4—48no—
123C2×D4—416no—
124C2×D4—44no—
125C2×D4—88no—
126C2×D4—88no—
127C2×D4—88no—
128C2×D4—84no—
129C2×D4—88no—
130C2×D4—88no—
131C2×D4—88no—
132C2×D4—84no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—22no—
138C22×C4—24no—
139C22×C4—21no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—21no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—24no—
150C22×C4—42no—
151C22×C4—44no—
152C22×C4—44no—
153C22×C4—44no—
154C22×C4—44no—
155C22×C4#9912yes—
156C22×C4#9912yes—
157C22×C4#10014yes—
158C22×C4C2×C22⋊C412yes—
159C22×C4#12014yes—
160C22×C4#13114yes—
161C2×C8—84no—
162C2×C8—88no—
163C4⋊C4—44no—
164C4⋊C4—48no—
165C4⋊C4—48no—
166C22⋊C4—432no—
167C22⋊C4—816no—
168C42—28no—
169C42—48no—
170C42—44no—
171C42—44no—
172C23—22no—
173C23—22no—
174C23—48no—
175C23—48no—
176C23—44no—
177C23—44no—
178C23—44no—
179C23—84no—
180C23—84no—
181C23—816no—
182C23—816no—
183C23—84no—
184C23—84no—
185C23—84no—
186C23—84no—
187C23#37911yescharacteristic
188C23#40812yes—
189C23C2×#9911yescharacteristic
190C23C2×#9911yescharacteristic
191C23C2×#10012yes—
192Q8—416no—
193D4—88no—
194D4—88no—
195D4—816no—
196D4—816no—
197C2×C4—24no—
198C2×C4—24no—
199C2×C4—22no—
200C2×C4—24no—
201C2×C4—24no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—24no—
206C2×C4—24no—
207C2×C4—22no—
208C2×C4—24no—
209C2×C4—24no—
210C2×C4—42no—
211C2×C4—44no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—44no—
218C2×C4—42no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—82no—
223C2×C4—88no—
224C2×C4#35514yes—
225C2×C4#40814yes—
226C8—164no—
227C22—24no—
228C22—24no—
229C22—21no—
230C22—21no—
231C22—21no—
232C22—21no—
233C22—88no—
234C22—88no—
235C22—84no—
236C22—84no—
237C22—84no—
238C22—168no—
239C22—168no—
240C22#147112yes—
241C22#147112yes—
242C22#178011yescharacteristic
243C22#178111yescharacteristic
244C22#178311yescharacteristic
245C22#212812yes—
246C22C2×#35512yes—
247C4—24no—
248C4—24no—
249C4—44no—
250C4—44no—
251C4—42no—
252C4—84no—
253C2—21no—
254C2—21no—
255C2—22no—
256C2—164no—
257C2#830711yescharacteristic
258C2#1156714yes—
259C2#1269711yescharacteristic
260C2#1269911yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC22×#127C4
LCSG ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ 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
2. C2×#1318C2semidirect
3. #8303C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25482
C2×#13182
#83032