Finite Group Databasebeta

← #190183

Subgroups — #190183

Order 512.

1194 conjugacy classes of subgroups; only the 254 normal subgroups are shown.

normal subgroups only

This group has 1194 conjugacy classes of subgroups — too many to draw. Only its 254 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×#2316C211yesmaximal, characteristic
2C2×#1463C212yesmaximal
3#8757C218yesmaximal, complemented
4#8107C214yesmaximal
5C22×#117C412yes—
6C23×C4⋊C4C2211yescharacteristic
7C2×#407C2214yes—
8#2316—22no—
9#2316—22no—
10#2316C414yes—
11#2316C2214yes—
12C2×#389—24no—
13C2×#380C412yes—
14C2×#380C2212yes—
15#1463—24no—
16#1463C2218yes—
17C2×#324—22no—
18C2×#324—22no—
19#1305C22116yes—
20C24×C4C2×C411yescharacteristic
21C24×C4C2×C411yescharacteristic
22C24×C4C2311yescharacteristic
23C22×M16—24no—
24C2×#117—22no—
25C2×#117—22no—
26C2×#117—48no—
27C2×#117C2×C414yes—
28C2×#117C2×C414yes—
29C22×C4⋊C4—24no—
30C22×C4⋊C4—24no—
31C22×C4⋊C4—22no—
32C22×C4⋊C4—22no—
33C22×C4⋊C4—22no—
34C22×C4⋊C4C2×C412yes—
35C22×C4⋊C4D412yes—
36C22×C4⋊C4D412yes—
37C22×C4⋊C4D414yes—
38C22×C4⋊C4C2312yes—
39C22×C22⋊C4C2×C414yes—
40C22×C42—22no—
41#407—24no—
42#407—24no—
43#407—24no—
44#407C2318yes—
45C2×C22⋊C8—24no—
46C2×C22⋊C8—24no—
47#389—28no—
48#389—28no—
49#389—28no—
50#389—28no—
51#380—24no—
52#380—22no—
53#380—22no—
54#380C2×C414yes—
55#380C2×C418yes—
56#380C2314yes—
57C2×#95—22no—
58C2×#95—22no—
59#324—28no—
60#324—28no—
61C25C22×C411yescharacteristic
62C23×C4—22no—
63C23×C4—22no—
64C23×C4—24no—
65C23×C4—24no—
66C23×C4—22no—
67C23×C4—22no—
68C23×C4—24no—
69C23×C4—82no—
70C23×C4—82no—
71C23×C4C22⋊C412yes—
72C23×C4C22⋊C411yescharacteristic
73C23×C4C22⋊C411yescharacteristic
74C23×C4C22⋊C412yes—
75C23×C4C22⋊C412yes—
76C23×C4C22×C412yes—
77C23×C4C22×C412yes—
78C23×C4C22×C411yescharacteristic
79C23×C4C2×D412yes—
80C23×C4C2×D412yes—
81C23×C4#4114yes—
82C23×C4#4114yes—
83C23×C4C2411yescharacteristic
84C2×M16—28no—
85C2×M16—44no—
86C2×M16—88no—
87C22×C8—44no—
88#117—48no—
89#117—48no—
90#117—48no—
91#117—48no—
92C2×C4⋊C4—28no—
93C2×C4⋊C4—28no—
94C2×C4⋊C4—28no—
95C2×C4⋊C4—24no—
96C2×C4⋊C4—24no—
97C2×C4⋊C4—24no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—28no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—28no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—48no—
106C2×C4⋊C4—84no—
107C2×C4⋊C4—84no—
108C2×C4⋊C4C22⋊C414yes—
109C2×C4⋊C4C22⋊C414yes—
110C2×C4⋊C4C2×D414yes—
111C2×C4⋊C4C2×D414yes—
112C2×C22⋊C4—24no—
113C2×C22⋊C4—48no—
114C2×C22⋊C4C22×C418yes—
115C2×C42—24no—
116C2×C42—24no—
117C2×C42—48no—
118C22⋊C8—28no—
119C22⋊C8—28no—
120C22⋊C8—44no—
121C22⋊C8—44no—
122#95—28no—
123#95—24no—
124#95—24no—
125C24—22no—
126C24—22no—
127C24—42no—
128C24—42no—
129C24C2×C22⋊C411yescharacteristic
130C24C2×C22⋊C411yescharacteristic
131C24#11712yes—
132C24#11712yes—
133C24C23×C411yescharacteristic
134C22×C4—216no—
135C22×C4—216no—
136C22×C4—24no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—24no—
144C22×C4—24no—
145C22×C4—24no—
146C22×C4—24no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—24no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—28no—
153C22×C4—44no—
154C22×C4—48no—
155C22×C4—44no—
156C22×C4—48no—
157C22×C4—48no—
158C22×C4—48no—
159C22×C4—81no—
160C22×C4—82no—
161C22×C4—84no—
162C22×C4—81no—
163C22×C4—84no—
164C22×C4—84no—
165C22×C4—82no—
166C22×C4—82no—
167C22×C4#10014yes—
168C22×C4#10114yes—
169C22×C4C2×C22⋊C412yes—
170C22×C4C2×C22⋊C412yes—
171C22×C4C2×C22⋊C414yes—
172C22×C4C2×C22⋊C412yes—
173C22×C4#11714yes—
174C22×C4#12314yes—
175C22×C4#12314yes—
176C22×C4#12314yes—
177C22×C4#12314yes—
178C22×C4C22×D412yes—
179C22×C4C2×#4114yes—
180M16—88no—
181M16—88no—
182C2×C8—48no—
183C2×C8—88no—
184C4⋊C4—28no—
185C4⋊C4—28no—
186C4⋊C4—28no—
187C4⋊C4—28no—
188C4⋊C4—48no—
189C4⋊C4—48no—
190C4⋊C4—84no—
191C4⋊C4—84no—
192C4⋊C4—88no—
193C22⋊C4—48no—
194C22⋊C4—48no—
195C42—48no—
196C42—48no—
197C23—22no—
198C23—22no—
199C23—21no—
200C23—21no—
201C23—24no—
202C23—24no—
203C23—24no—
204C23—24no—
205C23—21no—
206C23—21no—
207C23—41no—
208C23—42no—
209C23—41no—
210C23—41no—
211C23—42no—
212C23—41no—
213C23—48no—
214C23—48no—
215C23#38112yes—
216C23#38111yescharacteristic
217C23#38111yescharacteristic
218C23C2×#10012yes—
219C23C2×#10112yes—
220C23C22×C22⋊C411yescharacteristic
221C23C2×#11712yes—
222C2×C4—28no—
223C2×C4—28no—
224C2×C4—216no—
225C2×C4—28no—
226C2×C4—28no—
227C2×C4—28no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—24no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—44no—
236C2×C4—44no—
237C2×C4—44no—
238C2×C4—44no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—48no—
244C2×C4—48no—
245C2×C4—48no—
246C2×C4—82no—
247C2×C4—82no—
248C2×C4—84no—
249C2×C4—816no—
250C2×C4#38114yes—
251C2×C4C2×#10014yes—
252C2×C4C2×#10114yes—
253C2×C4C2×#12314yes—
254C8—88no—
255C22—24no—
256C22—24no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—22no—
262C22—21no—
263C22—21no—
264C22—22no—
265C22—22no—
266C22—22no—
267C22—22no—
268C22—42no—
269C22—42no—
270C22—48no—
271C22—48no—
272C22—42no—
273C22—42no—
274C22#182114yes—
275C22#182812yes—
276C22#183012yes—
277C22C2×#38111yescharacteristic
278C22C22×#10011yescharacteristic
279C22C22×#10111yescharacteristic
280C4—24no—
281C4—24no—
282C4—24no—
283C4—24no—
284C4—44no—
285C4—88no—
286C2—21no—
287C2—22no—
288C2—22no—
289C2—21no—
290C2—24no—
291C2—42no—
292C2C2×#182111yescharacteristic
293C2C2×#182811yescharacteristic
294C2C2×#183011yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8757C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23162
C2×#14632
#87572
#81072