Finite Group Databasebeta

← #496391

Subgroups — #496391

Order 512.

659 conjugacy classes of subgroups.

The subgroup lattice has 279 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60193C211yesmaximal, characteristic
2#33427C212yesmaximal
3#13270C214yesmaximal, complemented
4#13269C214yesmaximal, complemented
5#12851C212yesmaximal
6#12850C212yesmaximal
7#3458C411yescharacteristic
8#2962—24no—
9#2962—24no—
10#2961C412yes—
11#2961C2212yes—
12#2960C412yes—
13#2960C2212yes—
14#2905C411yescharacteristic
15C2×#424C411yescharacteristic
16C2×#424C411yescharacteristic
17#2873C2211yescharacteristic
18#2873C2211yescharacteristic
19C2×#405C2211yescharacteristic
20#2166—24no—
21#2165—24no—
22#2153C2214yes—
23C2×#349C2214yes—
24#2110C2214yes—
25#1378C2218yes—
26#1376C2214yes—
27#1375C2214yes—
28C22×M16C2×C411yescharacteristic
29C22×M16C2×C411yescharacteristic
30C22×M16C2311yescharacteristic
31C23×C8C2×C411yescharacteristic
32#544—22no—
33#544—22no—
34#544—22no—
35#543—22no—
36#533—22no—
37#533—22no—
38#533C2×C412yes—
39#533C2×C412yes—
40C2×#117C2×C411yescharacteristic
41C2×#117C2×C411yescharacteristic
42C2×#117C2×C411yescharacteristic
43C2×M32C2318yes—
44#446—24no—
45#446—24no—
46#446—24no—
47#446—24no—
48#446—24no—
49#446—24no—
50C8×Q8—24no—
51C8×Q8—24no—
52#424—22no—
53#424—22no—
54#424—22no—
55#424—22no—
56#424C2×C412yes—
57#424C2×C412yes—
58#424C2×C412yes—
59#424C2×C412yes—
60#423C2×C412yes—
61#423C2×C412yes—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—22no—
64#405—22no—
65#405—22no—
66#405—22no—
67#405C2×C412yes—
68#405C2×C412yes—
69#405D414yes—
70#405C2312yes—
71#405C2312yes—
72C4×M16—22no—
73C4×M16D414yes—
74C2×#97C2×C411yescharacteristic
75C2×#97C2311yescharacteristic
76C2×C4×C8C2×C411yescharacteristic
77C2×C4×C8C2311yescharacteristic
78#364—28no—
79#349—24no—
80#349—24no—
81#347—28no—
82C23×C4C22×C411yescharacteristic
83C2×M16—21no—
84C2×M16—24no—
85C2×M16—42no—
86C2×M16—42no—
87C2×M16C22⋊C412yes—
88C2×M16C22×C412yes—
89C2×M16C22×C412yes—
90C2×M16C2×D412yes—
91C2×M16C2×D412yes—
92C22×C8—21no—
93C22×C8—21no—
94C22×C8—21no—
95C22×C8—42no—
96C22×C8C22⋊C412yes—
97C22×C8C22×C411yescharacteristic
98C22×C8C22×C411yescharacteristic
99C22×C8C22×C411yescharacteristic
100C22×C8C2411yescharacteristic
101C4⋊Q8—24no—
102C4⋊Q8—24no—
103C4⋊Q8—22no—
104C4⋊Q8—22no—
105#125—24no—
106#125—24no—
107#125—22no—
108#125—22no—
109C22⋊Q8—24no—
110C22⋊Q8—24no—
111C22⋊Q8—24no—
112C22⋊Q8—24no—
113C22⋊Q8—24no—
114C22⋊Q8—24no—
115C4×Q8—24no—
116C4×Q8—24no—
117#117—21no—
118#117—21no—
119#117—21no—
120#117—21no—
121#117—21no—
122#117—21no—
123#117C22⋊C412yes—
124#117M1612yes—
125#117M1612yes—
126#117M1612yes—
127#117C22×C412yes—
128#117C22×C412yes—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C22⋊C4—21no—
133C2×C22⋊C4—21no—
134C2×C22⋊C4M1612yes—
135C2×C42—21no—
136C2×C42C22⋊C412yes—
137C2×C42C22×C411yescharacteristic
138C2×C42C22×C411yescharacteristic
139M32—28no—
140M32—28no—
141C2×C16—28no—
142C4⋊C8—24no—
143C4⋊C8—24no—
144C4⋊C8—24no—
145C4⋊C8—24no—
146C4⋊C8—24no—
147C4⋊C8—24no—
148#97—22no—
149#97—22no—
150#97—22no—
151#97—22no—
152#97C22⋊C414yes—
153#97C2×D414yes—
154C4×C8—22no—
155C4×C8—22no—
156C4×C8—22no—
157C4×C8—22no—
158C4×C8C22⋊C414yes—
159C4×C8C2×D414yes—
160C24C2×M1611yescharacteristic
161C2×Q8—24no—
162C2×Q8—24no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4C2×C22⋊C411yescharacteristic
170C22×C4C2×C22⋊C411yescharacteristic
171C22×C4C2×M1611yescharacteristic
172C22×C4C2×M1611yescharacteristic
173C22×C4C2×M1611yescharacteristic
174C22×C4C23×C411yescharacteristic
175M16—24no—
176M16—24no—
177M16—24no—
178M16—24no—
179M16—42no—
180M16—42no—
181C2×C8—22no—
182C2×C8—21no—
183C2×C8—22no—
184C2×C8—21no—
185C2×C8—22no—
186C2×C8—22no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8C2×C22⋊C412yes—
192C2×C8C2×C22⋊C414yes—
193C2×C8C22×D412yes—
194C2×C82−1+414yes—
195C4⋊C4—24no—
196C4⋊C4—24no—
197C4⋊C4—24no—
198C4⋊C4—24no—
199C4⋊C4—24no—
200C4⋊C4—24no—
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—22no—
206C4⋊C4—22no—
207C22⋊C4—22no—
208C22⋊C4—22no—
209C22⋊C4—22no—
210C22⋊C4—22no—
211C22⋊C4—22no—
212C22⋊C4C2×M1614yes—
213C42—22no—
214C42—21no—
215C42—21no—
216C42—22no—
217C42—22no—
218C42C2×C22⋊C414yes—
219C42C2×M1612yes—
220C42C2×M1612yes—
221C16—48no—
222C23—22no—
223C23—81no—
224C23#40711yescharacteristic
225C23#40711yescharacteristic
226C23C22×M1611yescharacteristic
227Q8—84no—
228Q8—84no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—24no—
237C2×C4—24no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—81no—
241C2×C4—81no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4#40711yescharacteristic
245C2×C4#40711yescharacteristic
246C2×C4C22×C22⋊C411yescharacteristic
247C2×C4#51912yes—
248C2×C4C22×M1611yescharacteristic
249C2×C4#56812yes—
250C8—22no—
251C8—24no—
252C8—41no—
253C8—41no—
254C8—42no—
255C8#53614yes—
256C22—21no—
257C22—21no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—82no—
262C22C2×#40711yescharacteristic
263C22#298312yes—
264C4—21no—
265C4—21no—
266C4—44no—
267C4—44no—
268C4—42no—
269C4—42no—
270C4#231011yescharacteristic
271C4#287911yescharacteristic
272C4#298312yes—
273C2—21no—
274C2—22no—
275C2—41no—
276C2—41no—
277C2#2182211yescharacteristic
subgroupG/H
centerC4#2310
commutatorC22×C4C23×C4
Frattini, cosocleC22×C8C24
socleC2#21822
prime radicalC24C2×M16
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C22×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13270C2semidirect
4. #13269C2semidirect

Maximal subgroups

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

subgroupindex
#601932
#334272
#132702
#132692
#128512
#128502