Finite Group Databasebeta

← #129449

Subgroups — #129449

Order 512.

1052 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 1052 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×#2547C211yesmaximal, characteristic
2C2×#1319C214yesmaximal, complemented
3#8301C212yesmaximal
4C22×C4⋊Q8C411yescharacteristic
5C22×#124C412yes—
6C22×C4×D4C2211yescharacteristic
7#2547—28no—
8C2×#397—22no—
9C2×#391C2212yes—
10C2×#380—22no—
11C2×C4×C4⋊C4C411yescharacteristic
12#1779—22no—
13C2×#324C2214yes—
14C2×#323—22no—
15#1319—28no—
16C23×Q8C2×C412yes—
17C23×D4C2×C411yescharacteristic
18C24×C4D412yes—
19C2×C4⋊Q8—24no—
20C2×C4⋊Q8—42no—
21C2×C4⋊Q8—42no—
22C2×#124—24no—
23C2×#124—24no—
24C2×#124—48no—
25C2×C4×D4—24no—
26C2×C4×D4—42no—
27C2×C4×D4—42no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4C2311yescharacteristic
32C22×C22⋊C4—24no—
33C22×C22⋊C4D412yes—
34C22×C42C2×C411yescharacteristic
35C22×C42C2×C412yes—
36C2×C22⋊C8—24no—
37C2×C22⋊C8—44no—
38#397—216no—
39#391—24no—
40#391D418yes—
41#380—216no—
42C4×C4⋊C4—24no—
43C2×#95—21no—
44C2×#95—24no—
45#324—28no—
46#323—44no—
47C25C22⋊C412yes—
48C22×Q8—22no—
49C22×Q8—28no—
50C22×Q8—44no—
51C22×Q8—42no—
52C22×Q8—42no—
53C22×Q8C22⋊C414yes—
54C22×D4—24no—
55C22×D4—22no—
56C22×D4—24no—
57C22×D4—41no—
58C22×D4—41no—
59C23×C4—22no—
60C23×C4—22no—
61C23×C4—24no—
62C23×C4—24no—
63C23×C4—42no—
64C23×C4—48no—
65C23×C4C22⋊C411yescharacteristic
66C23×C4C22⋊C411yescharacteristic
67C23×C4C22×C411yescharacteristic
68C23×C4C2×D412yes—
69C22×C8—44no—
70C4⋊Q8—416no—
71#124—416no—
72#124—416no—
73C4×D4—416no—
74C2×C4⋊C4—24no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—44no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4C2×D414yes—
85C2×C22⋊C4—216no—
86C2×C22⋊C4—24no—
87C2×C22⋊C4—24no—
88C2×C22⋊C4—42no—
89C2×C22⋊C4—48no—
90C2×C22⋊C4—48no—
91C2×C42—22no—
92C2×C42—22no—
93C2×C42—22no—
94C2×C42—22no—
95C2×C42—44no—
96C2×C42C22⋊C414yes—
97C22⋊C8—44no—
98C22⋊C8—44no—
99C22⋊C8—88no—
100#95—24no—
101#95—216no—
102C24—22no—
103C24—22no—
104C24—22no—
105C24—22no—
106C24—22no—
107C24—22no—
108C24—44no—
109C24—44no—
110C24#10014yes—
111C24C2×C22⋊C411yescharacteristic
112C2×Q8—28no—
113C2×Q8—44no—
114C2×Q8—44no—
115C2×Q8—44no—
116C2×Q8—44no—
117C2×Q8—44no—
118C2×Q8—88no—
119C2×Q8—88no—
120C2×D4—28no—
121C2×D4—48no—
122C2×D4—48no—
123C2×D4—44no—
124C2×D4—44no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—22no—
130C22×C4—24no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—24no—
136C22×C4—24no—
137C22×C4—48no—
138C22×C4—416no—
139C22×C4—42no—
140C22×C4—44no—
141C22×C4—44no—
142C22×C4—44no—
143C22×C4—44no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—44no—
147C22×C4—44no—
148C22×C4—48no—
149C22×C4—84no—
150C22×C4—88no—
151C22×C4#9912yes—
152C22×C4#9912yes—
153C22×C4C2×C22⋊C412yes—
154C22×C4#12014yes—
155C22×C4#13114yes—
156C2×C8—84no—
157C2×C8—88no—
158C4⋊C4—44no—
159C4⋊C4—48no—
160C4⋊C4—48no—
161C4⋊C4—48no—
162C22⋊C4—48no—
163C22⋊C4—432no—
164C42—28no—
165C42—44no—
166C42—44no—
167C23—22no—
168C23—22no—
169C23—24no—
170C23—24no—
171C23—24no—
172C23—48no—
173C23—44no—
174C23—48no—
175C23—44no—
176C23—44no—
177C23—44no—
178C23—42no—
179C23—42no—
180C23—42no—
181C23—42no—
182C23—84no—
183C23—82no—
184C23—84no—
185C23—82no—
186C23#37911yescharacteristic
187C23#40812yes—
188C23C2×#9911yescharacteristic
189C23C2×#9911yescharacteristic
190C23C2×#10012yes—
191Q8—88no—
192Q8—88no—
193D4—416no—
194C2×C4—24no—
195C2×C4—24no—
196C2×C4—22no—
197C2×C4—24no—
198C2×C4—24no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—24no—
204C2×C4—42no—
205C2×C4—44no—
206C2×C4—42no—
207C2×C4—44no—
208C2×C4—42no—
209C2×C4—44no—
210C2×C4—44no—
211C2×C4—44no—
212C2×C4—44no—
213C2×C4—44no—
214C2×C4—44no—
215C2×C4—816no—
216C2×C4—82no—
217C2×C4—88no—
218C2×C4—84no—
219C2×C4—84no—
220C2×C4#35414yes—
221C2×C4#40814yes—
222C8—164no—
223C22—24no—
224C22—24no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—48no—
230C22—48no—
231C22—44no—
232C22—84no—
233C22—84no—
234C22—84no—
235C22—84no—
236C22—82no—
237C22—82no—
238C22#147112yes—
239C22#147112yes—
240C22#178011yescharacteristic
241C22#178111yescharacteristic
242C22#178311yescharacteristic
243C22C2×#35412yes—
244C22#213512yes—
245C4—24no—
246C4—24no—
247C4—44no—
248C4—42no—
249C4—84no—
250C4—84no—
251C2—21no—
252C2—22no—
253C2—21no—
254C2—84no—
255C2#830711yescharacteristic
256C2#1156414yes—
257C2#1269411yescharacteristic
258C2#1270411yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC23×D4C2×C4
LCSG ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1319C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25472
C2×#13192
#83012