Finite Group Databasebeta

← #135387

Subgroups — #135387

Order 512.

370 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 84 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#20824C211yesmaximal, characteristic
2#12849C211yesmaximal, characteristic
3#10805C211yesmaximal, characteristic
4#7255C212yesmaximal
5#7254C212yesmaximal
6C2×C4×M16C2211yescharacteristic
7#2285—21no—
8C2×#349C2212yes—
9#1845C411yescharacteristic, complemented
10#1844C411yescharacteristic, complemented
11#1834—21no—
12#1809—21no—
13#1809C2212yes—
14#1776—21no—
15#1776C412yescomplemented
16C2×#344—21no—
17C2×#339—21no—
18C2×#338C2211yescharacteristic
19C2×H(3,ℤ/4)C2211yescharacteristic
20#1376—21no—
21#1375—21no—
22C22×M16—21no—
23C22×M16D411yescharacteristic
24C22×M16C2311yescharacteristic
25C2×#117—21no—
26C2×#117—41no—
27C22×C42D411yescharacteristic
28C2×M32—22no—
29C2×#107—21no—
30C2×#107C2×C412yes—
31C2×#106—21no—
32C2×#106C2×C412yes—
33C4×M16—22no—
34C4×M16—22no—
35C2×#97C2×C411yescharacteristic
36C2×#97D411yescharacteristic
37C2×C4×C8C2×C411yescharacteristic
38C2×C4×C8D411yescharacteristic
39#384—21no—
40#384—21no—
41#382—41no—
42#380—22no—
43#380—41no—
44C4×C4⋊C4—22no—
45C4×C4⋊C4—41no—
46#349—22no—
47#349—22no—
48#344—24no—
49#340—22no—
50#340—22no—
51#339—42no—
52#338—22no—
53H(3,ℤ/4)—22no—
54C23×C4—21no—
55C23×C4C2×D411yescharacteristic
56C2×M16—21no—
57C2×M16—21no—
58C2×M16—22no—
59C2×M16—22no—
60C2×M16—41no—
61C2×M16—41no—
62C2×M16—41no—
63C2×M16—41no—
64C2×M16—41no—
65C2×M16—41no—
66C2×M16—41no—
67C2×M16—41no—
68C2×M16#4112yes—
69C2×M16#4112yes—
70C22×C8—41no—
71C22×C8C22⋊C411yescharacteristic
72C22×C8C22⋊C411yescharacteristic
73C22×C8C22×C411yescharacteristic
74C22×C8C2×D411yescharacteristic
75#117—22no—
76#117—41no—
77#117—41no—
78#117—41no—
79#117—84no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C22⋊C4—41no—
88C2×C22⋊C4—42no—
89C2×C42—21no—
90C2×C42—21no—
91C2×C42—21no—
92C2×C42—41no—
93C2×C42—42no—
94C2×C42C22⋊C411yescharacteristic
95C2×C42C22⋊C411yescharacteristic
96C2×C42SD1612yes—
97C2×C42Q1612yes—
98M32—42no—
99M32—42no—
100C2×C16—42no—
101#107—42no—
102#107—42no—
103#106—42no—
104#106—42no—
105#97—22no—
106#97—22no—
107C4×C8—22no—
108C4×C8—22no—
109#95—42no—
110#95—42no—
111C24AGL(1,ℤ/8)11yescharacteristic
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4—81no—
125C22×C4C2×C22⋊C411yescharacteristic
126C22×C4#12311yescharacteristic
127C22×C4#12311yescharacteristic
128C22×C4C2×SD1611yescharacteristic
129C22×C4C2×Q1611yescharacteristic
130C22×C4AGL(1,ℤ/8)11yescharacteristic
131M16—42no—
132M16—42no—
133M16—42no—
134M16—42no—
135M16—42no—
136M16—42no—
137M16—81no—
138M16—81no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—42no—
149C2×C8—81no—
150C2×C8—81no—
151C2×C8#10412yes—
152C2×C8#11712yes—
153C2×C8#12312yes—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—42no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C4⋊C4—82no—
160C4⋊C4—82no—
161C4⋊C4—81no—
162C4⋊C4—82no—
163C4⋊C4—82no—
164C22⋊C4—42no—
165C22⋊C4—84no—
166C22⋊C4—81no—
167C42—21no—
168C42—21no—
169C42—22no—
170C42—22no—
171C42—42no—
172C42—81no—
173C42—84no—
174C42#10312yes—
175C42#10312yes—
176C16—82no—
177C23—21no—
178C23—21no—
179C23—41no—
180C23—41no—
181C23#41811yescharacteristic
182C23#48111yescharacteristic
183C23#48411yescharacteristic
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—44no—
199C2×C4—44no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—82no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4—82no—
206C2×C4—82no—
207C2×C4—82no—
208C2×C4—82no—
209C2×C4—161no—
210C2×C4—161no—
211C2×C4#38111yescharacteristic
212C2×C4#41011yescharacteristic
213C2×C4#41011yescharacteristic
214C2×C4C2×#10311yescharacteristic
215C2×C4C2×#10411yescharacteristic
216C2×C4#42111yescharacteristic
217C2×C4#48111yescharacteristic
218C2×C4#48411yescharacteristic
219C8—22no—
220C8—22no—
221C8—22no—
222C8—41no—
223C8—82no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—41no—
229C22—41no—
230C22—41no—
231C22—41no—
232C22—81no—
233C22#150511yescharacteristic
234C22#151411yescharacteristic
235C22#182411yescharacteristic
236C4—24no—
237C4—21no—
238C4—21no—
239C4—42no—
240C4—162no—
241C4—162no—
242C4—162no—
243C4#150311yescharacteristic
244C4#151811yescharacteristic
245C4#182011yescharacteristic
246C4#182611yescharacteristic
247C2—21no—
248C2—21no—
249C2—41no—
250C2—41no—
251C2#871711yescharacteristic
252C2#1183711yescharacteristic
253C2#1183811yescharacteristic
subgroupG/H
center, socleC22#1514
commutatorC22×C8C22×C4
Frattini, cosocleC22×M16C23
prime radicalC24AGL(1,ℤ/8)
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×M16 ◁ G
LECSG ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
9. #1845C4semidirect
10. #1844C4semidirect
15. #1776C4semidirect

Maximal subgroups

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

subgroupindex
#208242
#128492
#108052
#72552
#72542