Finite Group Databasebeta

← #133229

Subgroups — #133229

Order 512.

862 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 862 conjugacy classes of subgroups — too many to draw. Only its 90 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×#2503C211yesmaximal, characteristic
2#11814C214yesmaximal, complemented
3#8555C211yesmaximal, characteristic
4#8550C211yesmaximal, characteristic
5C22×C22⋊Q8C2211yescharacteristic
6C23×C4⋊C4C411yescharacteristic
7#2503—28no—
8C2×#400C412yes—
9C2×#400C2212yes—
10C2×#391—21no—
11C2×#382—21no—
12C2×#381—21no—
13C2×#381—21no—
14C2×C4×C22⋊C4C411yescharacteristic
15#1821C2212yes—
16#1784—21no—
17#1463—21no—
18#1463C2212yes—
19#1462—21no—
20#1462—21no—
21#1353—28no—
22C23×Q8D411yescharacteristic
23C24×C4—21no—
24C24×C4C2×C411yescharacteristic
25C2×C22⋊Q8—24no—
26C2×C22⋊Q8—42no—
27C2×C22⋊Q8—42no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4—24no—
30C22×C4⋊C4—42no—
31C22×C4⋊C4—42no—
32C22×C4⋊C4C2×C411yescharacteristic
33C22×C4⋊C4C2×C412yes—
34C22×C4⋊C4D411yescharacteristic
35C22×C4⋊C4C2311yescharacteristic
36C22×C22⋊C4—21no—
37C22×C22⋊C4D411yescharacteristic
38C22×C22⋊C4D411yescharacteristic
39C22×C42—21no—
40C2×#101—42no—
41#407—22no—
42#407—22no—
43#400—24no—
44#400—24no—
45#400—24no—
46#400D418yes—
47#391—28no—
48#382—28no—
49#381—28no—
50#381—28no—
51C4×C22⋊C4—24no—
52C2×#95—22no—
53C2×#95—22no—
54C2×#95—22no—
55C2×#95—22no—
56C2×#95C2×C412yes—
57#343—24no—
58#324—24no—
59#324—42no—
60#323—42no—
61#323—42no—
62C25C22⋊C411yescharacteristic
63C22×Q8—22no—
64C22×Q8—22no—
65C22×Q8—41no—
66C22×Q8—44no—
67C23×C4—22no—
68C23×C4—22no—
69C23×C4—22no—
70C23×C4—24no—
71C23×C4—22no—
72C23×C4—22no—
73C23×C4—44no—
74C23×C4—42no—
75C23×C4—48no—
76C23×C4—42no—
77C23×C4C22⋊C411yescharacteristic
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22×C411yescharacteristic
81C23×C4C2×D411yescharacteristic
82C23×C4C2×D411yescharacteristic
83C2×M16—42no—
84C2×M16—42no—
85C22⋊Q8—416no—
86C2×C4⋊C4—24no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—416no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4C22⋊C414yes—
102C2×C4⋊C4C2×D414yes—
103C2×C22⋊C4—24no—
104C2×C22⋊C4—22no—
105C2×C22⋊C4—22no—
106C2×C22⋊C4—22no—
107C2×C22⋊C4—22no—
108C2×C22⋊C4—41no—
109C2×C22⋊C4—41no—
110C2×C22⋊C4—44no—
111C2×C42—24no—
112C2×C42—44no—
113#101—84no—
114C22⋊C8—42no—
115C22⋊C8—42no—
116C22⋊C8—42no—
117C22⋊C8—42no—
118#95—28no—
119#95—28no—
120#95—22no—
121#95—28no—
122#95—28no—
123#95C22⋊C414yes—
124C24—24no—
125C24—21no—
126C24—21no—
127C24—42no—
128C24—42no—
129C24#9912yes—
130C24C2×C22⋊C411yescharacteristic
131C2×Q8—44no—
132C2×Q8—48no—
133C2×Q8—44no—
134C2×Q8—82no—
135C2×Q8—84no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—24no—
139C22×C4—28no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—24no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—24no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—42no—
155C22×C4—48no—
156C22×C4—48no—
157C22×C4—44no—
158C22×C4—44no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—44no—
163C22×C4—44no—
164C22×C4—48no—
165C22×C4—44no—
166C22×C4—416no—
167C22×C4—48no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—88no—
172C22×C4—84no—
173C22×C4—84no—
174C22×C4#9912yes—
175C22×C4C2×C22⋊C412yes—
176C22×C4#12012yes—
177C22×C4#12012yes—
178M16—84no—
179M16—84no—
180C2×C8—82no—
181C2×C8—82no—
182C4⋊C4—44no—
183C4⋊C4—44no—
184C4⋊C4—44no—
185C4⋊C4—48no—
186C4⋊C4—44no—
187C4⋊C4—44no—
188C22⋊C4—44no—
189C22⋊C4—44no—
190C22⋊C4—48no—
191C42—48no—
192C23—22no—
193C23—22no—
194C23—22no—
195C23—24no—
196C23—24no—
197C23—22no—
198C23—22no—
199C23—42no—
200C23—42no—
201C23—42no—
202C23—416no—
203C23—42no—
204C23—42no—
205C23—42no—
206C23#35514yes—
207C23#37911yescharacteristic
208C23C2×#9911yescharacteristic
209C23C2×#9911yescharacteristic
210Q8—88no—
211C2×C4—24no—
212C2×C4—24no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—24no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—416no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—44no—
226C2×C4—44no—
227C2×C4—48no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—44no—
231C2×C4—42no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—44no—
235C2×C4—81no—
236C2×C4—81no—
237C2×C4—81no—
238C2×C4—88no—
239C2×C4—84no—
240C2×C4—816no—
241C2×C4—84no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4#35214yes—
246C8—162no—
247C8—162no—
248C22—28no—
249C22—21no—
250C22—21no—
251C22—21no—
252C22—21no—
253C22—21no—
254C22—21no—
255C22—24no—
256C22—44no—
257C22—48no—
258C22—48no—
259C22—44no—
260C22—42no—
261C22—42no—
262C22#178111yescharacteristic
263C22#179612yes—
264C22C2×#35212yes—
265C22C2×#35512yes—
266C4—24no—
267C4—42no—
268C4—42no—
269C4—42no—
270C4—84no—
271C4—81no—
272C4—81no—
273C4—84no—
274C2—22no—
275C2—22no—
276C2—22no—
277C2—44no—
278C2#852011yescharacteristic
279C2#1268811yescharacteristic
280C2#1269911yescharacteristic
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×C22⋊Q8 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11814C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25032
#118142
#85552
#85502