Finite Group Databasebeta

← #97067

Subgroups — #97067

Order 512.

1256 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1256 conjugacy classes of subgroups — too many to draw. Only its 158 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
1C22×#382C211yesmaximal, characteristic
2#8698C214yesmaximal
3#8299C212yesmaximal
4C23×C4⋊C4C411yescharacteristic
5C23×C4⋊C4C2211yescharacteristic
6C23×C42C411yescharacteristic
7C2×#382—28no—
8C2×#382—24no—
9C22×#95C412yes—
10C22×#95C2212yes—
11#1779C2214yes—
12C2×#324—24no—
13#1280—28no—
14C24×C4C2×C411yescharacteristic
15C24×C4C2×C412yes—
16C24×C4C2×C412yes—
17C24×C4C2×C411yescharacteristic
18C24×C4C2311yescharacteristic
19C22×C4⋊C4—24no—
20C22×C4⋊C4—22no—
21C22×C4⋊C4—24no—
22C22×C4⋊C4—44no—
23C22×C4⋊C4—44no—
24C22×C4⋊C4D414yes—
25C22×C42—24no—
26C22×C42—22no—
27C22×C42—24no—
28C22×C42—22no—
29C22×C42C814yescomplemented
30C2×C22⋊C8—28no—
31C2×C22⋊C8—24no—
32#382—232no—
33C2×#95—24no—
34C2×#95—24no—
35C2×#95—24no—
36C2×#95—24no—
37C2×#95—24no—
38C2×#95D418yes—
39#324—48no—
40C25C22×C411yescharacteristic
41C23×C4—22no—
42C23×C4—22no—
43C23×C4—24no—
44C23×C4—24no—
45C23×C4—22no—
46C23×C4—22no—
47C23×C4—22no—
48C23×C4—22no—
49C23×C4—22no—
50C23×C4—22no—
51C23×C4—22no—
52C23×C4—42no—
53C23×C4—42no—
54C23×C4—44no—
55C23×C4—44no—
56C23×C4—44no—
57C23×C4—44no—
58C23×C4—44no—
59C23×C4—44no—
60C23×C4—44no—
61C23×C4C22⋊C412yes—
62C23×C4C22⋊C414yes—
63C23×C4C22⋊C412yes—
64C23×C4C22⋊C414yes—
65C23×C4C2×C812yes—
66C23×C4C2×C814yes—
67C23×C4M1612yes—
68C23×C4M1614yes—
69C23×C4C2×D414yes—
70C23×C4C2×D412yes—
71C22×C8—44no—
72C2×C4⋊C4—28no—
73C2×C4⋊C4—28no—
74C2×C4⋊C4—28no—
75C2×C4⋊C4—44no—
76C2×C4⋊C4—416no—
77C2×C4⋊C4—416no—
78C2×C4⋊C4—44no—
79C2×C4⋊C4—44no—
80C2×C42—28no—
81C2×C42—28no—
82C2×C42—216no—
83C2×C42—216no—
84C2×C42—28no—
85C22⋊C8—48no—
86C22⋊C8—416no—
87#95—216no—
88#95—216no—
89C24—22no—
90C24—24no—
91C24—22no—
92C24—24no—
93C24C2×C22⋊C411yescharacteristic
94C24C2×C22⋊C412yes—
95C24C22×C811yescharacteristic
96C24C2×M1611yescharacteristic
97C24C2×M1612yes—
98C22×C4—28no—
99C22×C4—216no—
100C22×C4—28no—
101C22×C4—22no—
102C22×C4—24no—
103C22×C4—22no—
104C22×C4—24no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—28no—
108C22×C4—28no—
109C22×C4—24no—
110C22×C4—24no—
111C22×C4—24no—
112C22×C4—24no—
113C22×C4—24no—
114C22×C4—24no—
115C22×C4—24no—
116C22×C4—28no—
117C22×C4—22no—
118C22×C4—24no—
119C22×C4—24no—
120C22×C4—28no—
121C22×C4—44no—
122C22×C4—44no—
123C22×C4—44no—
124C22×C4—48no—
125C22×C4—48no—
126C22×C4—44no—
127C22×C4—44no—
128C22×C4—44no—
129C22×C4—44no—
130C22×C4—44no—
131C22×C4—44no—
132C22×C4—48no—
133C22×C4—48no—
134C22×C4—44no—
135C22×C4—44no—
136C22×C4—44no—
137C22×C4—44no—
138C22×C4—44no—
139C22×C4—44no—
140C22×C4—44no—
141C22×C4—44no—
142C22×C4—44no—
143C22×C4—44no—
144C22×C4—44no—
145C22×C4—44no—
146C22×C4—88no—
147C22×C4—88no—
148C22×C4—816no—
149C22×C4C22⋊C814yes—
150C22×C4C22⋊C818yes—
151C22×C4#9914yes—
152C22×C4#10114yes—
153C22×C4#12014yes—
154C2×C8—84no—
155C2×C8—88no—
156C4⋊C4—416no—
157C4⋊C4—416no—
158C42—232no—
159C23—28no—
160C23—28no—
161C23—216no—
162C23—22no—
163C23—22no—
164C23—24no—
165C23—24no—
166C23—24no—
167C23—24no—
168C23—24no—
169C23—22no—
170C23—22no—
171C23—22no—
172C23—22no—
173C23—22no—
174C23—22no—
175C23#37911yescharacteristic
176C23C2×C22⋊C811yescharacteristic
177C23C2×C22⋊C812yes—
178C23#40711yescharacteristic
179C23#40712yes—
180C23C2×#9912yes—
181C23#41014yes—
182C23C2×#10112yes—
183C23#41314yes—
184C2×C4—216no—
185C2×C4—216no—
186C2×C4—24no—
187C2×C4—24no—
188C2×C4—24no—
189C2×C4—24no—
190C2×C4—24no—
191C2×C4—24no—
192C2×C4—28no—
193C2×C4—24no—
194C2×C4—28no—
195C2×C4—48no—
196C2×C4—48no—
197C2×C4—48no—
198C2×C4—48no—
199C2×C4—48no—
200C2×C4—44no—
201C2×C4—44no—
202C2×C4—48no—
203C2×C4—42no—
204C2×C4—44no—
205C2×C4—44no—
206C2×C4—44no—
207C2×C4—44no—
208C2×C4—84no—
209C2×C4—88no—
210C2×C4—88no—
211C2×C4—88no—
212C2×C4—88no—
213C2×C4—84no—
214C2×C4—88no—
215C2×C4#32418yes—
216C8—164no—
217C22—28no—
218C22—28no—
219C22—216no—
220C22—22no—
221C22—24no—
222C22—22no—
223C22—24no—
224C22—24no—
225C22—24no—
226C22—24no—
227C22—22no—
228C22—22no—
229C22—22no—
230C22—22no—
231C22—22no—
232C22—22no—
233C22C2×#32412yes—
234C22#145814yes—
235C22#146312yes—
236C22#146414yes—
237C22#177911yescharacteristic
238C22#178111yescharacteristic
239C22#178212yes—
240C22#178411yescharacteristic
241C22#178512yes—
242C4—28no—
243C4—44no—
244C4—48no—
245C4—82no—
246C4—84no—
247C2—22no—
248C2—24no—
249C2—22no—
250C2—24no—
251C2#829911yescharacteristic
252C2#830012yes—
253C2#854411yescharacteristic
254C2#855211yescharacteristic
255C2#855412yes—
subgroupG/H
center, socleC23#379
commutatorC24C22×C8
Frattini, cosocleC24×C4C23
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
29. C22×C42C8semidirect

Maximal subgroups

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

subgroupindex
C22×#3822
#86982
#82992