Finite Group Databasebeta

← #97040

Subgroups — #97040

Order 512.

912 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 912 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2621C211yesmaximal, characteristic
2#8300C212yesmaximal
3#7595C214yesmaximal, complemented
4C22×C4⋊Q8C411yescharacteristic
5C22×C22⋊Q8C412yes—
6C22×#117C2211yescharacteristic
7#2621—28no—
8C2×#398—22no—
9C2×#397—22no—
10C2×#391C2212yes—
11C2×#382C411yescharacteristic
12#1817C2214yes—
13C2×#324—22no—
14C2×#323—22no—
15#1272—28no—
16C23×Q8C2×C412yes—
17C24×C4C2×C411yescharacteristic
18C2×C4⋊Q8—24no—
19C2×C4⋊Q8—44no—
20C2×C22⋊Q8—24no—
21C2×C22⋊Q8—24no—
22C2×C22⋊Q8—44no—
23C2×C22⋊Q8—44no—
24C2×#117—24no—
25C2×#117—44no—
26C22×C4⋊C4—22no—
27C22×C4⋊C4—22no—
28C22×C4⋊C4C2×C412yes—
29C22×C4⋊C4D412yes—
30C22×C22⋊C4—22no—
31C22×C22⋊C4D412yes—
32C22×C42C2×C411yescharacteristic
33C22×C42C2311yescharacteristic
34C2×C22⋊C8—24no—
35#398—216no—
36#397—216no—
37#391—24no—
38#391D418yes—
39#382—24no—
40C2×#95—22no—
41C2×#95—24no—
42#336—28no—
43#324—44no—
44#323—44no—
45C25C22⋊C411yescharacteristic
46C22×Q8—22no—
47C22×Q8—28no—
48C22×Q8—42no—
49C22×Q8—42no—
50C22×Q8—44no—
51C22×Q8C22⋊C414yes—
52C23×C4—22no—
53C23×C4—22no—
54C23×C4—22no—
55C23×C4—22no—
56C23×C4—22no—
57C23×C4—24no—
58C23×C4—42no—
59C23×C4C22⋊C411yescharacteristic
60C23×C4C22⋊C412yes—
61C23×C4C22×C411yescharacteristic
62C23×C4C2×D412yes—
63C22×C8—44no—
64C4⋊Q8—416no—
65C22⋊Q8—416no—
66C22⋊Q8—416no—
67#117—416no—
68C2×C4⋊C4—28no—
69C2×C4⋊C4—28no—
70C2×C4⋊C4—24no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—24no—
73C2×C4⋊C4—42no—
74C2×C4⋊C4—42no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4—44no—
77C2×C4⋊C4—44no—
78C2×C4⋊C4—44no—
79C2×C4⋊C4—44no—
80C2×C4⋊C4C22⋊C414yes—
81C2×C22⋊C4—28no—
82C2×C22⋊C4—24no—
83C2×C22⋊C4—24no—
84C2×C22⋊C4—42no—
85C2×C22⋊C4—44no—
86C2×C22⋊C4—44no—
87C2×C42—24no—
88C2×C42—22no—
89C2×C42—42no—
90C2×C42C2×D414yes—
91C22⋊C8—44no—
92C22⋊C8—44no—
93#95—28no—
94#95—216no—
95C24—21no—
96C24—21no—
97C24—22no—
98C24—22no—
99C24—44no—
100C24#10012yes—
101C24C2×C22⋊C411yescharacteristic
102C2×Q8—28no—
103C2×Q8—44no—
104C2×Q8—44no—
105C2×Q8—44no—
106C2×Q8—44no—
107C2×Q8—44no—
108C2×Q8—88no—
109C2×Q8—88no—
110C22×C4—22no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—28no—
114C22×C4—22no—
115C22×C4—21no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—24no—
123C22×C4—24no—
124C22×C4—48no—
125C22×C4—44no—
126C22×C4—48no—
127C22×C4—44no—
128C22×C4—44no—
129C22×C4—44no—
130C22×C4—48no—
131C22×C4—44no—
132C22×C4—44no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4#9914yes—
138C22×C4#10012yes—
139C22×C4C2×C22⋊C412yes—
140C22×C4#12014yes—
141C22×C4#13114yes—
142C2×C8—84no—
143C2×C8—88no—
144C4⋊C4—48no—
145C4⋊C4—44no—
146C4⋊C4—44no—
147C4⋊C4—416no—
148C4⋊C4—416no—
149C22⋊C4—48no—
150C22⋊C4—416no—
151C42—28no—
152C42—44no—
153C23—24no—
154C23—22no—
155C23—24no—
156C23—44no—
157C23—44no—
158C23—44no—
159C23—42no—
160C23—41no—
161C23—41no—
162C23—82no—
163C23—82no—
164C23—84no—
165C23#37911yescharacteristic
166C23#40812yes—
167C23C2×#9912yes—
168C23C2×#10011yescharacteristic
169C23C2×#10011yescharacteristic
170Q8—88no—
171Q8—88no—
172C2×C4—24no—
173C2×C4—24no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—28no—
177C2×C4—24no—
178C2×C4—24no—
179C2×C4—24no—
180C2×C4—22no—
181C2×C4—416no—
182C2×C4—44no—
183C2×C4—44no—
184C2×C4—42no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—44no—
188C2×C4—48no—
189C2×C4—44no—
190C2×C4—44no—
191C2×C4—44no—
192C2×C4—44no—
193C2×C4—82no—
194C2×C4—82no—
195C2×C4—84no—
196C2×C4—84no—
197C2×C4—84no—
198C2×C4—84no—
199C2×C4#40814yes—
200C8—164no—
201C22—22no—
202C22—22no—
203C22—22no—
204C22—22no—
205C22—24no—
206C22—44no—
207C22—42no—
208C22—84no—
209C22—84no—
210C22—82no—
211C22—82no—
212C22#147112yes—
213C22#147112yes—
214C22#178011yescharacteristic
215C22#178111yescharacteristic
216C22#178311yescharacteristic
217C22#179614yes—
218C4—24no—
219C4—44no—
220C4—44no—
221C4—42no—
222C4—84no—
223C4—84no—
224C2—22no—
225C2—22no—
226C2—82no—
227C2#830711yescharacteristic
228C2#832712yes—
229C2#852012yes—
230C2#852412yes—
subgroupG/H
center, socleC23#379
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7595C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26212
#83002
#75952