Finite Group Databasebeta

← #150658

Subgroups — #150658

Order 512.

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

normal subgroups only

This group has 379 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×#2061C211yesmaximal, characteristic
2#11511C211yesmaximal, characteristic
3#8718C211yesmaximal, characteristic
4#7442C214yesmaximal, complemented
5C2×#544C411yescharacteristic
6C2×#428—21no—
7C2×#382C2211yescharacteristic
8#2061—21no—
9#2061—21no—
10#2061C412yes—
11#2061C2212yes—
12#1749—21no—
13C2×#343C411yescharacteristic
14C2×#339—21no—
15#1463—21no—
16#1324—21no—
17#1324C2212yes—
18#1295—21no—
19#1295C2212yes—
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⋊C4C2×C411yescharacteristic
30C22×C4⋊C4D411yescharacteristic
31C22×C42D411yescharacteristic
32#428—22no—
33#428—22no—
34#428—42no—
35C2×#107—41no—
36C2×#106—41no—
37#407—41no—
38C2×C22⋊C8—41no—
39C2×#97—22no—
40#382—41no—
41#382C2×C411yescharacteristic
42#382D411yescharacteristic
43#382D411yescharacteristic
44#382C2311yescharacteristic
45C2×#95—41no—
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⋊C4C22⋊C411yescharacteristic
90C2×C4⋊C4C22⋊C411yescharacteristic
91C2×C4⋊C4C22×C411yescharacteristic
92C2×C4⋊C4C2×D411yescharacteristic
93C2×C22⋊C4—41no—
94C2×C42—22no—
95C2×C42—21no—
96C2×C42—41no—
97C2×C42—41no—
98C2×C42C22⋊C411yescharacteristic
99C2×C42C2×D411yescharacteristic
100#107—42no—
101#107—42no—
102#106—42no—
103#106—42no—
104C22⋊C8—44no—
105C22⋊C8—42no—
106C22⋊C8—81no—
107#97—42no—
108#97—42no—
109#95—41no—
110#95—41no—
111#95—41no—
112#95—41no—
113C24#9911yescharacteristic
114C2×Q8—42no—
115C2×Q8—42no—
116C2×Q8—162no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—22no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—81no—
137C22×C4—82no—
138C22×C4—82no—
139C22×C4#9911yescharacteristic
140C22×C4C2×C22⋊C411yescharacteristic
141M16—24no—
142M16—44no—
143M16—42no—
144M16—162no—
145C2×C8—42no—
146C2×C8—44no—
147C2×C8—44no—
148C2×C8—82no—
149C2×C8—82no—
150C2×C8—82no—
151C4⋊C4—22no—
152C4⋊C4—22no—
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4—42no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4—42no—
161C4⋊C4—42no—
162C4⋊C4—44no—
163C4⋊C4—44no—
164C4⋊C4—81no—
165C22⋊C4—42no—
166C22⋊C4—42no—
167C42—21no—
168C42—21no—
169C42—21no—
170C42—42no—
171C42—42no—
172C42—42no—
173C42#13712yes—
174C23—21no—
175C23—22no—
176C23—21no—
177C23—41no—
178C23AΣL(1,𝔽16)11yescharacteristic
179C23AΣL(1,𝔽16)11yescharacteristic
180C23C2×#9911yescharacteristic
181Q8—162no—
182Q8—162no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—84no—
204C2×C4—82no—
205C2×C4—82no—
206C2×C4—82no—
207C2×C4—82no—
208C2×C4—81no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4—81no—
212C2×C4—161no—
213C2×C4—161no—
214C2×C4#35612yes—
215C2×C4#41911yescharacteristic
216C2×C4#42111yescharacteristic
217C8—44no—
218C8—84no—
219C8—162no—
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—42no—
228C22—42no—
229C22#140412yes—
230C22#151111yescharacteristic
231C22C2×AΣL(1,𝔽16)11yescharacteristic
232C22C2×#35611yescharacteristic
233C4—21no—
234C4—22no—
235C4—21no—
236C4—42no—
237C4—81no—
238C4—81no—
239C4—81no—
240C4—82no—
241C4—81no—
242C4—82no—
243C4—81no—
244C2—21no—
245C2—21no—
246C2—22no—
247C2—41no—
248C2#1181511yescharacteristic
249C2#1356511yescharacteristic
250C2C2×#140411yescharacteristic
subgroupG/H
center, socleC22C2×AΣL(1,𝔽16)
commutatorC2×C4⋊C4C22×C4
Frattini, cosocle#382C23
prime radicalC24#99
LCSG ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#382 ◁ G
LECSG ▷ #382 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7442C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20612
#115112
#87182
#74422