Finite Group Databasebeta

← #150660

Subgroups — #150660

Order 512.

398 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 58 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×#2060C211yesmaximal, characteristic
2#11512C211yesmaximal, characteristic
3#8723C211yesmaximal, characteristic
4#7443C214yesmaximal, complemented
5C2×#544C411yescharacteristic
6C2×#428—21no—
7C2×#384C2211yescharacteristic
8#2060—21no—
9#2060—21no—
10#2060C412yes—
11#2060C2212yes—
12#1797—21no—
13C2×#343C411yescharacteristic
14C2×#339—21no—
15#1463—21no—
16#1325—21no—
17#1325C2212yes—
18#1296—21no—
19#1296C2212yes—
20C22×M16—21no—
21#544—21no—
22#544—41no—
23#544—42no—
24#544C2×C412yes—
25C2×C4⋊Q8—41no—
26C2×#125—41no—
27C2×C22⋊Q8—41no—
28C2×#117—21no—
29C22×C4⋊C4—41no—
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C4⋊C4D411yescharacteristic
32C22×C42D411yescharacteristic
33#428—22no—
34#428—22no—
35#428—42no—
36C2×#107—41no—
37C2×#106—41no—
38C2×C4⋊C8—22no—
39#407—41no—
40C2×C22⋊C8—41no—
41#384—41no—
42#384C2×C411yescharacteristic
43#384D411yescharacteristic
44#384D411yescharacteristic
45#384C2311yescharacteristic
46#343—21no—
47#343C2×C412yes—
48#339—22no—
49#339—22no—
50#336—41no—
51#324—22no—
52#324—41no—
53C22×Q8—41no—
54C23×C4—21no—
55C23×C4—21no—
56C23×C4—21no—
57C23×C4C22⋊C411yescharacteristic
58C2×M16—24no—
59C2×M16—22no—
60C2×M16—42no—
61C2×M16—41no—
62C2×M16—81no—
63C22×C8—41no—
64C22×C8—42no—
65C4⋊Q8—42no—
66C4⋊Q8—42no—
67C4⋊Q8—44no—
68#125—42no—
69#125—42no—
70#125—44no—
71C22⋊Q8—42no—
72C22⋊Q8—42no—
73C22⋊Q8—42no—
74C22⋊Q8—42no—
75#117—22no—
76#117—22no—
77#117—41no—
78#117—41no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—82no—
94C2×C4⋊C4—82no—
95C2×C4⋊C4C22⋊C411yescharacteristic
96C2×C4⋊C4C22⋊C411yescharacteristic
97C2×C4⋊C4C22×C411yescharacteristic
98C2×C4⋊C4C2×D411yescharacteristic
99C2×C22⋊C4—41no—
100C2×C42—22no—
101C2×C42—21no—
102C2×C42—41no—
103C2×C42—41no—
104C2×C42C22⋊C411yescharacteristic
105C2×C42C2×D411yescharacteristic
106#107—42no—
107#107—42no—
108#106—42no—
109#106—42no—
110C4⋊C8—42no—
111C4⋊C8—42no—
112C22⋊C8—42no—
113C22⋊C8—82no—
114C22⋊C8—81no—
115C24#9911yescharacteristic
116C2×Q8—42no—
117C2×Q8—42no—
118C2×Q8—162no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—22no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—82no—
141C22×C4—82no—
142C22×C4#9911yescharacteristic
143C22×C4C2×C22⋊C411yescharacteristic
144M16—24no—
145M16—44no—
146M16—42no—
147M16—162no—
148C2×C8—42no—
149C2×C8—44no—
150C2×C8—82no—
151C2×C8—82no—
152C2×C8—82no—
153C2×C8—82no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4—42no—
161C4⋊C4—41no—
162C4⋊C4—41no—
163C4⋊C4—42no—
164C4⋊C4—42no—
165C4⋊C4—44no—
166C4⋊C4—44no—
167C4⋊C4—81no—
168C4⋊C4—81no—
169C4⋊C4—81no—
170C4⋊C4—81no—
171C4⋊C4—81no—
172C4⋊C4—82no—
173C4⋊C4—82no—
174C22⋊C4—42no—
175C22⋊C4—42no—
176C42—21no—
177C42—21no—
178C42—21no—
179C42—42no—
180C42—42no—
181C42—42no—
182C42#13712yes—
183C23—21no—
184C23—22no—
185C23—21no—
186C23—41no—
187C23AΣL(1,𝔽16)11yescharacteristic
188C23AΣL(1,𝔽16)11yescharacteristic
189C23C2×#9911yescharacteristic
190Q8—162no—
191Q8—162no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—44no—
206C2×C4—44no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—82no—
216C2×C4—82no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4—161no—
224C2×C4—161no—
225C2×C4#35612yes—
226C2×C4#41911yescharacteristic
227C2×C4#42111yescharacteristic
228C8—44no—
229C8—162no—
230C8—162no—
231C22—21no—
232C22—21no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—42no—
239C22—42no—
240C22#140212yes—
241C22#151111yescharacteristic
242C22C2×AΣL(1,𝔽16)11yescharacteristic
243C22C2×#35611yescharacteristic
244C4—22no—
245C4—22no—
246C4—21no—
247C4—22no—
248C4—21no—
249C4—42no—
250C4—82no—
251C4—81no—
252C4—81no—
253C4—81no—
254C4—81no—
255C4—82no—
256C2—21no—
257C2—21no—
258C2—22no—
259C2—41no—
260C2#1181511yescharacteristic
261C2C2×#140211yescharacteristic
262C2#1357311yescharacteristic
subgroupG/H
center, socleC22C2×AΣL(1,𝔽16)
commutatorC2×C4⋊C4C22×C4
Frattini, cosocle#384C23
prime radicalC24#99
LCSG ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#384 ◁ G
LECSG ▷ #384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7443C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20602
#115122
#87232
#74432