Finite Group Databasebeta

← #128366

Subgroups — #128366

Order 512.

572 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 114 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
1#20385C211yesmaximal, characteristic
2#11425C212yesmaximal
3#8157C212yesmaximal, complemented
4#7188C212yesmaximal, complemented
5C24×C8C411yescharacteristic
6C2×#407C411yescharacteristic
7C2×#405C411yescharacteristic
8C2×#405C2211yescharacteristic
9#2280C2211yescharacteristic
10#1752—22no—
11C8×C22⋊C4—22no—
12#1729C411yescharacteristic
13#1729C2211yescharacteristic
14#1458C2214yes—
15#1452—22no—
16#1317C414yes—
17#1316C414yes—
18#1281—22no—
19C24×C4C2×C411yescharacteristic
20C22×M16C2×C411yescharacteristic
21C22×M16C2×C411yescharacteristic
22C23×C8—22no—
23C23×C8—21no—
24C23×C8—21no—
25C23×C8—22no—
26C23×C8—42no—
27C23×C8—42no—
28C23×C8C2×C411yescharacteristic
29C23×C8C2×C411yescharacteristic
30C2×#117C2×C411yescharacteristic
31C2×#117C2311yescharacteristic
32#431—24no—
33#407—22no—
34#407—22no—
35C2×C22⋊C8—22no—
36#405—42no—
37#405—42no—
38C2×#97—21no—
39C2×#97D412yes—
40C2×C4×C8—21no—
41C2×C4×C8D412yes—
42#381C2×C412yes—
43#380—22no—
44#379C2×C412yes—
45C4×C22⋊C4—22no—
46#336—22no—
47#336—22no—
48#324—24no—
49#323C2×C418yes—
50#322—44no—
51C25C4211yescharacteristic
52C23×C4—21no—
53C23×C4—21no—
54C23×C4—21no—
55C23×C4—21no—
56C23×C4—41no—
57C23×C4—41no—
58C23×C4—41no—
59C23×C4—41no—
60C23×C4C4211yescharacteristic
61C23×C4M1612yes—
62C23×C4M1612yes—
63C23×C4C22×C411yescharacteristic
64C2×M16—21no—
65C2×M16—21no—
66C2×M16—21no—
67C2×M16—41no—
68C2×M16—41no—
69C2×M16—41no—
70C2×M16—41no—
71C2×M16M1612yes—
72C22×C8—21no—
73C22×C8—21no—
74C22×C8—22no—
75C22×C8—22no—
76C22×C8—24no—
77C22×C8—24no—
78C22×C8—21no—
79C22×C8—21no—
80C22×C8—21no—
81C22×C8—48no—
82C22×C8—48no—
83C22×C8—42no—
84C22×C8—42no—
85C22×C8—42no—
86C22×C8—42no—
87C22×C8—41no—
88C22×C8—41no—
89C22×C8C22⋊C412yes—
90C22×C8C22⋊C411yescharacteristic
91C22×C8C22⋊C411yescharacteristic
92C22×C8M1612yes—
93#117—41no—
94#117—41no—
95#117—41no—
96#117—41no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4#4112yes—
99C2×C22⋊C4—24no—
100C2×C22⋊C4C4212yes—
101C2×C22⋊C4C22×C412yes—
102C2×C42—21no—
103C2×C42C2×D412yes—
104C4⋊C8—44no—
105C22⋊C8—28no—
106C22⋊C8—28no—
107#97—42no—
108#97—42no—
109C4×C8—24no—
110C4×C8—44no—
111C4×C8—42no—
112#95—24no—
113C24—21no—
114C24—21no—
115C24—41no—
116C24—41no—
117C24—44no—
118C24#9712yes—
119C24C2×C4211yescharacteristic
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4#9712yes—
148C22×C4C2×C22⋊C411yescharacteristic
149C22×C4#11711yescharacteristic
150C22×C4C4×D414yes—
151C22×C4C2×M1611yescharacteristic
152C22×C4C2×M1611yescharacteristic
153C22×C4C2×M1611yescharacteristic
154C22×C4C2×M1611yescharacteristic
155M16—42no—
156M16—42no—
157M16—42no—
158M16—42no—
159C2×C8—24no—
160C2×C8—24no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—22no—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—48no—
171C2×C8—48no—
172C2×C8—42no—
173C2×C8—41no—
174C2×C8—41no—
175C2×C8—41no—
176C2×C8—41no—
177C2×C8—42no—
178C2×C8—48no—
179C2×C8#9912yes—
180C2×C8#10012yes—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C22⋊C4—42no—
184C22⋊C4—48no—
185C22⋊C4—42no—
186C42—22no—
187C42—22no—
188C42—41no—
189C42—41no—
190C23—24no—
191C23—22no—
192C23—21no—
193C23—21no—
194C23—42no—
195C23—44no—
196C23—42no—
197C23—42no—
198C23—42no—
199C23—44no—
200C23—44no—
201C23—48no—
202C23—44no—
203C23—41no—
204C23—41no—
205C23C4×C22⋊C411yescharacteristic
206C23C2×#9711yescharacteristic
207C23C4×M1611yescharacteristic
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—24no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—44no—
230C2×C4—42no—
231C2×C4—44no—
232C2×C4—42no—
233C2×C4#40711yescharacteristic
234C2×C4#40711yescharacteristic
235C2×C4C2×#9911yescharacteristic
236C2×C4#41011yescharacteristic
237C2×C4C2×#10011yescharacteristic
238C2×C4#41311yescharacteristic
239C2×C4#43211yescharacteristic
240C2×C4#43211yescharacteristic
241C8—22no—
242C8—22no—
243C8—42no—
244C8—42no—
245C8—42no—
246C8—88no—
247C22—22no—
248C22—24no—
249C22—21no—
250C22—21no—
251C22—41no—
252C22—41no—
253C22—44no—
254C22—42no—
255C22—42no—
256C22—44no—
257C22—44no—
258C22—48no—
259C22—44no—
260C22—42no—
261C22—42no—
262C22#174911yescharacteristic
263C22C4×#9911yescharacteristic
264C22C4×#10011yescharacteristic
265C4—21no—
266C4—21no—
267C4—44no—
268C4—42no—
269C4—82no—
270C4#146211yescharacteristic
271C4#146411yescharacteristic
272C4#146511yescharacteristic
273C4#146611yescharacteristic
274C2—21no—
275C2—21no—
276C2—44no—
277C2—41no—
278C2—41no—
279C2#815211yescharacteristic
280C2#1262312yes—
subgroupG/H
centerC2×C4C2×#99
commutatorC24C2×C42
Frattini, cosocleC2×#117C23
socleC22C4×#99
prime radicalC25C42
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#405 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8157C2semidirect
4. #7188C2semidirect

Maximal subgroups

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

subgroupindex
#203852
#114252
#81572
#71882