Finite Group Databasebeta

← #133232

Subgroups — #133232

Order 512.

717 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 717 conjugacy classes of subgroups — too many to draw. Only its 82 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×#2821C211yesmaximal, characteristic
2#11815C214yesmaximal, complemented
3#8556C211yesmaximal, characteristic
4#8556C211yesmaximal, characteristic
5C22×C4⋊Q8C411yescharacteristic
6C22×#126C2211yescharacteristic
7#2821—28no—
8C2×#400—21no—
9C2×#399—21no—
10C2×#398—21no—
11C2×#397—21no—
12C2×#395C2212yes—
13C2×#393C412yes—
14C2×#383C411yescharacteristic
15#1821—21no—
16#1821C2212yes—
17#1463—21no—
18#1463C2212yes—
19#1462—21no—
20#1462—21no—
21#1354—28no—
22C23×Q8C2×C412yes—
23C2×C4⋊Q8—24no—
24C2×C4⋊Q8—42no—
25C2×C4⋊Q8—42no—
26C2×#126—24no—
27C2×#126—42no—
28C2×#126—42no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4—22no—
32C22×C4⋊C4D411yescharacteristic
33C22×C4⋊C4D411yescharacteristic
34C22×C4⋊C4C2311yescharacteristic
35C22×C22⋊C4—22no—
36C22×C22⋊C4C2×C411yescharacteristic
37C22×C22⋊C4D411yescharacteristic
38C22×C22⋊C4D411yescharacteristic
39C22×C42C2×C411yescharacteristic
40#407—22no—
41#407—22no—
42#400—28no—
43#399—28no—
44#398—28no—
45#397—28no—
46#395—24no—
47#395D418yes—
48#393—24no—
49#393—24no—
50#383—24no—
51C2×#95—22no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—22no—
55C2×#95C2×C412yes—
56#343—24no—
57#343—42no—
58#324—24no—
59#324—42no—
60#323—42no—
61#323—42no—
62C25C22⋊C411yescharacteristic
63C22×Q8—22no—
64C22×Q8—42no—
65C22×Q8—42no—
66C22×Q8—84no—
67C22×Q8C22⋊C414yes—
68C23×C4—22no—
69C23×C4—22no—
70C23×C4C22⋊C411yescharacteristic
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22⋊C411yescharacteristic
73C23×C4C22×C411yescharacteristic
74C23×C4C2×D411yescharacteristic
75C23×C4C2×D411yescharacteristic
76C2×M16—42no—
77C2×M16—42no—
78C4⋊Q8—416no—
79#126—416no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—28no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—84no—
96C2×C4⋊C4C2×D414yes—
97C2×C22⋊C4—22no—
98C2×C22⋊C4—22no—
99C2×C22⋊C4—22no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—28no—
103C2×C22⋊C4—41no—
104C2×C22⋊C4—41no—
105C2×C22⋊C4—41no—
106C2×C22⋊C4—41no—
107C2×C22⋊C4—84no—
108C2×C42—22no—
109C2×C42—22no—
110C2×C42—22no—
111C22⋊C8—42no—
112C22⋊C8—42no—
113C22⋊C8—42no—
114C22⋊C8—42no—
115#95—24no—
116#95—24no—
117#95—28no—
118#95—22no—
119#95—28no—
120#95C22⋊C414yes—
121C24—21no—
122C24—21no—
123C24—21no—
124C24—42no—
125C24—42no—
126C24—42no—
127C24C2×C22⋊C411yescharacteristic
128C2×Q8—44no—
129C2×Q8—44no—
130C2×Q8—84no—
131C2×Q8—84no—
132C2×Q8—88no—
133C2×Q8—88no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—24no—
145C22×C4—21no—
146C22×C4—24no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—44no—
156C22×C4—44no—
157C22×C4—44no—
158C22×C4—44no—
159C22×C4—44no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4#9912yes—
163C22×C4#9912yes—
164C22×C4C2×C22⋊C412yes—
165C22×C4#12012yes—
166C22×C4#12012yes—
167M16—84no—
168M16—84no—
169C2×C8—82no—
170C2×C8—82no—
171C4⋊C4—44no—
172C4⋊C4—44no—
173C4⋊C4—44no—
174C4⋊C4—48no—
175C4⋊C4—48no—
176C4⋊C4—88no—
177C22⋊C4—44no—
178C22⋊C4—44no—
179C22⋊C4—44no—
180C22⋊C4—88no—
181C42—28no—
182C23—22no—
183C23—22no—
184C23—22no—
185C23—42no—
186C23—42no—
187C23—42no—
188C23—41no—
189C23—42no—
190C23—42no—
191C23—88no—
192C23—81no—
193C23—81no—
194C23—81no—
195C23—81no—
196C23#37911yescharacteristic
197C23C2×#9911yescharacteristic
198C23C2×#9911yescharacteristic
199Q8—88no—
200Q8—88no—
201C2×C4—24no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—44no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—42no—
223C2×C4—44no—
224C2×C4—42no—
225C2×C4—41no—
226C2×C4—44no—
227C2×C4—81no—
228C2×C4—81no—
229C2×C4—81no—
230C2×C4—88no—
231C2×C4—84no—
232C2×C4—84no—
233C2×C4—81no—
234C8—162no—
235C8—162no—
236C22—28no—
237C22—21no—
238C22—21no—
239C22—21no—
240C22—21no—
241C22—21no—
242C22—21no—
243C22—84no—
244C22—81no—
245C22—81no—
246C22—81no—
247C22—162no—
248C22—162no—
249C22#178111yescharacteristic
250C22#179612yes—
251C22#212212yes—
252C22#213512yes—
253C4—42no—
254C4—42no—
255C4—42no—
256C4—42no—
257C4—42no—
258C4—84no—
259C4—84no—
260C4—81no—
261C2—22no—
262C2—22no—
263C2—22no—
264C2—161no—
265C2#852011yescharacteristic
266C2#1269111yescharacteristic
267C2#1270411yescharacteristic
subgroupG/H
center, socleC22#1781
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×#126 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11815C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28212
#118152
#85562