Finite Group Databasebeta

← #134288

Subgroups — #134288

Order 512.

560 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 110 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#8602C212yesmaximal, complemented
4#7190C212yesmaximal, complemented
5C24×C8C411yescharacteristic
6C2×#407C411yescharacteristic
7C2×#405C411yescharacteristic
8C2×#405C2211yescharacteristic
9#2280C2211yescharacteristic
10#1752—22no—
11C8×C22⋊C4—22no—
12#1729C411yescharacteristic
13#1729C2211yescharacteristic
14#1564—22no—
15#1458C2214yes—
16#1317—22no—
17#1316—22no—
18#1275—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#407C814yescomplemented
35C2×C22⋊C8—22no—
36#405—42no—
37#405—42no—
38C2×#97—21no—
39C2×#97D412yes—
40C2×C4×C8—21no—
41C2×C4×C8D412yes—
42#381D411yescharacteristic
43#381Q811yescharacteristic
44#380—22no—
45#379D411yescharacteristic
46#379Q811yescharacteristic
47C4×C22⋊C4—22no—
48#336—22no—
49#336—22no—
50#335—42no—
51#334—42no—
52#324—24no—
53#323—24no—
54C25C4⋊C411yescharacteristic
55C23×C4—21no—
56C23×C4—21no—
57C23×C4—21no—
58C23×C4—21no—
59C23×C4—41no—
60C23×C4—41no—
61C23×C4—41no—
62C23×C4—41no—
63C23×C4C4⋊C411yescharacteristic
64C23×C4C2×C812yes—
65C23×C4M1612yes—
66C23×C4C22×C411yescharacteristic
67C2×M16—21no—
68C2×M16—41no—
69C2×M16—41no—
70C2×M16—41no—
71C2×M16—41no—
72C2×M16C2×C812yes—
73C2×M16C2×C812yes—
74C2×M16M1612yes—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8—22no—
78C22×C8—22no—
79C22×C8—24no—
80C22×C8—24no—
81C22×C8—21no—
82C22×C8—21no—
83C22×C8—21no—
84C22×C8—48no—
85C22×C8—48no—
86C22×C8—42no—
87C22×C8—42no—
88C22×C8—42no—
89C22×C8—42no—
90C22×C8—41no—
91C22×C8—41no—
92C22×C8C22⋊C412yes—
93C22×C8C22⋊C411yescharacteristic
94C22×C8C22⋊C411yescharacteristic
95C22×C8M1612yes—
96#117—41no—
97#117—41no—
98#117—41no—
99#117—41no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4#4112yes—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4C4⋊C411yescharacteristic
105C2×C22⋊C4C4⋊C411yescharacteristic
106C2×C22⋊C4C2×D411yescharacteristic
107C2×C22⋊C4C2×Q811yescharacteristic
108C2×C42—21no—
109C2×C42C2×D412yes—
110C4⋊C8—44no—
111C22⋊C8—28no—
112C22⋊C8—44no—
113#97—42no—
114#97—42no—
115C4×C8—22no—
116C4×C8—22no—
117C4×C8—44no—
118C4×C8—42no—
119#95—22no—
120#95—22no—
121C24—21no—
122C24—21no—
123C24—41no—
124C24—41no—
125C24—44no—
126C24C4⋊C812yes—
127C24C2×C4⋊C411yescharacteristic
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4C4⋊C812yes—
157C22×C4C2×C22⋊C411yescharacteristic
158C22×C4#11711yescharacteristic
159C22×C4#12112yes—
160C22×C4C22⋊Q812yes—
161C22×C4C22×C811yescharacteristic
162C22×C4C2×M1611yescharacteristic
163C22×C4C2×M1611yescharacteristic
164C22×C4C2×M1611yescharacteristic
165M16—42no—
166M16—42no—
167M16—42no—
168M16—42no—
169C2×C8—24no—
170C2×C8—24no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—22no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—22no—
177C2×C8—22no—
178C2×C8—22no—
179C2×C8—48no—
180C2×C8—48no—
181C2×C8—42no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8—42no—
187C2×C8—84no—
188C2×C8C22⋊C814yes—
189C2×C8#9912yes—
190C2×C8#10012yes—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C22⋊C4—41no—
194C22⋊C4—41no—
195C22⋊C4—44no—
196C22⋊C4—44no—
197C22⋊C4—41no—
198C22⋊C4—41no—
199C42—22no—
200C42—22no—
201C42—41no—
202C42—41no—
203C23—24no—
204C23—22no—
205C23—21no—
206C23—21no—
207C23—42no—
208C23—44no—
209C23—42no—
210C23—42no—
211C23—42no—
212C23—44no—
213C23—44no—
214C23—48no—
215C23—44no—
216C23—41no—
217C23—41no—
218C23#38011yescharacteristic
219C23C2×C4⋊C811yescharacteristic
220C23#42311yescharacteristic
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—44no—
235C2×C4—44no—
236C2×C4—42no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4C2×C22⋊C811yescharacteristic
251C2×C4#40711yescharacteristic
252C2×C4C2×#9911yescharacteristic
253C2×C4#41011yescharacteristic
254C2×C4C2×#10011yescharacteristic
255C2×C4#41311yescharacteristic
256C2×C4#43111yescharacteristic
257C2×C4#43211yescharacteristic
258C8—24no—
259C8—22no—
260C8—22no—
261C8—42no—
262C8—42no—
263C8—164no—
264C22—22no—
265C22—24no—
266C22—21no—
267C22—21no—
268C22—41no—
269C22—41no—
270C22—44no—
271C22—42no—
272C22—42no—
273C22—44no—
274C22—44no—
275C22—48no—
276C22—44no—
277C22—42no—
278C22—42no—
279C22#179711yescharacteristic
280C22#179911yescharacteristic
281C22#180011yescharacteristic
282C4—21no—
283C4—21no—
284C4—42no—
285C4—42no—
286C4—42no—
287C4—81no—
288C4—81no—
289C4#145911yescharacteristic
290C4#146011yescharacteristic
291C4#146211yescharacteristic
292C4#146411yescharacteristic
293C2—21no—
294C2—21no—
295C2—44no—
296C2—41no—
297C2—41no—
298C2#855911yescharacteristic
299C2#1280512yes—
subgroupG/H
centerC2×C4#410
commutatorC22×C4C22×C8
Frattini, cosocleC2×#117C23
socleC22#1799
prime radicalC25C4⋊C4
LCSG ▷ C22×C4 ▷ 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. #8602C2semidirect
4. #7190C2semidirect
34. #407C8semidirect

Maximal subgroups

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

subgroupindex
#203852
#114252
#86022
#71902