Finite Group Databasebeta

← #135373

Subgroups — #135373

Order 512.

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

normal subgroups only

This group has 363 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#20809C211yesmaximal, characteristic
2#10816C211yesmaximal, characteristic
3#8730C211yesmaximal, characteristic
4#7253C212yesmaximal
5#7252C212yesmaximal
6C2×#431—21no—
7C4×#117C2211yescharacteristic
8#1843C411yescharacteristic, complemented
9#1843C411yescharacteristic, complemented
10#1832—21no—
11#1798—21no—
12#1798C2212yes—
13C4×C4⋊C8—21no—
14C4×C4⋊C8C412yescomplemented
15#1742C2211yescharacteristic
16#1740C2211yescharacteristic
17C2×#339—21no—
18C2×H(3,ℤ/4)—21no—
19#1458C2212yes—
20#1276—21no—
21#1274—21no—
22C22×M16—21no—
23C2×#117—21no—
24C2×#117D411yescharacteristic
25C2×#117C2311yescharacteristic
26C22×C42D411yescharacteristic
27#431—22no—
28#431—42no—
29#431—42no—
30C2×C4⋊C8—21no—
31C2×C4⋊C8—21no—
32C2×C4⋊C8—21no—
33C2×C4⋊C8—21no—
34C2×C4⋊C8C2×C412yes—
35C2×C4⋊C8C2×C412yes—
36C2×C22⋊C8—22no—
37C2×C4×C8—22no—
38C4×C4⋊C4—22no—
39C4×C22⋊C4—22no—
40#376C2×C411yescharacteristic
41#376D411yescharacteristic
42C43C2×C411yescharacteristic
43C43D411yescharacteristic
44#343—22no—
45#342—22no—
46#339—42no—
47H(3,ℤ/4)—24no—
48#336—22no—
49#336—22no—
50#324—22no—
51#323—22no—
52C23×C4—21no—
53C23×C4C2×D411yescharacteristic
54C2×M16—22no—
55C2×M16—41no—
56C2×M16—41no—
57C2×M16—41no—
58C2×M16—41no—
59C2×M16—41no—
60C22×C8—22no—
61C22×C8—22no—
62C22×C8—41no—
63#117—22no—
64#117—22no—
65#117—41no—
66#117—41no—
67#117—41no—
68#117—41no—
69#117—41no—
70#117—41no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—41no—
73C2×C4⋊C4#4112yes—
74C2×C22⋊C4—21no—
75C2×C22⋊C4—41no—
76C2×C22⋊C4#4112yes—
77C2×C42—21no—
78C2×C42—21no—
79C2×C42—21no—
80C2×C42—22no—
81C2×C42—22no—
82C2×C42—41no—
83C2×C42C22⋊C411yescharacteristic
84C2×C42C22⋊C411yescharacteristic
85C2×C42C22⋊C411yescharacteristic
86C2×C42C22⋊C411yescharacteristic
87C2×C42D812yes—
88C2×C42SD1612yes—
89C2×C42C22×C411yescharacteristic
90C2×C42C2×D411yescharacteristic
91C4⋊C8—42no—
92C4⋊C8—42no—
93C4⋊C8—42no—
94C4⋊C8—42no—
95C4⋊C8—42no—
96C4⋊C8—42no—
97C4⋊C8—42no—
98C22⋊C8—42no—
99C22⋊C8—42no—
100C22⋊C8—42no—
101C4×C8—42no—
102C4×C8—44no—
103#95—22no—
104#95—22no—
105C24#12311yescharacteristic
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—21no—
116C22×C4—44no—
117C22×C4—81no—
118C22×C4C2×C22⋊C411yescharacteristic
119C22×C4#11712yes—
120C22×C4#12311yescharacteristic
121C22×C4#12312yes—
122C22×C4C2×D811yescharacteristic
123C22×C4C2×SD1611yescharacteristic
124C22×C4#13711yescharacteristic
125C22×C4#13711yescharacteristic
126M16—42no—
127M16—42no—
128M16—81no—
129M16—81no—
130C2×C8—42no—
131C2×C8—42no—
132C2×C8—42no—
133C2×C8—42no—
134C2×C8—42no—
135C2×C8—42no—
136C2×C8—42no—
137C2×C8—81no—
138C2×C8—81no—
139C2×C8—82no—
140C4⋊C4—42no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C4⋊C4—81no—
144C22⋊C4—42no—
145C22⋊C4—42no—
146C22⋊C4—42no—
147C22⋊C4—81no—
148C42—21no—
149C42—21no—
150C42—21no—
151C42—21no—
152C42—21no—
153C42—21no—
154C42—21no—
155C42—22no—
156C42—22no—
157C42—21no—
158C42—21no—
159C42—44no—
160C42—44no—
161C42—42no—
162C42—81no—
163C42#10212yes—
164C42#10212yes—
165C42#10412yes—
166C23—21no—
167C23—21no—
168C23—42no—
169C23#38111yescharacteristic
170C23#48011yescharacteristic
171C23#48311yescharacteristic
172C2×C4—24no—
173C2×C4—21no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—44no—
189C2×C4—44no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—82no—
197C2×C4—162no—
198C2×C4#41011yescharacteristic
199C2×C4#41011yescharacteristic
200C2×C4C2×#10211yescharacteristic
201C2×C4#41911yescharacteristic
202C2×C4C2×#10411yescharacteristic
203C2×C4#42111yescharacteristic
204C2×C4#48011yescharacteristic
205C2×C4#48311yescharacteristic
206C8—82no—
207C8—82no—
208C8—82no—
209C8—162no—
210C22—22no—
211C22—21no—
212C22—21no—
213C22—41no—
214C22—41no—
215C22—42no—
216C22—81no—
217C22#182011yescharacteristic
218C22#182311yescharacteristic
219C22#182611yescharacteristic
220C4—24no—
221C4—21no—
222C4—21no—
223C4—42no—
224C4—42no—
225C4—42no—
226C4—42no—
227C4—81no—
228C4—162no—
229C4#150211yescharacteristic
230C4#150611yescharacteristic
231C4#151511yescharacteristic
232C4#151711yescharacteristic
233C2—21no—
234C2—21no—
235C2—41no—
236C2—41no—
237C2#871611yescharacteristic
238C2#1286412yes—
subgroupG/H
centerC2×C4#410
commutatorC2×C42C22×C4
Frattini, cosocleC2×#117C23
socleC22#1820
prime radicalC24#123
LCSG ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#117 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #1843C4semidirect
9. #1843C4semidirect
14. C4×C4⋊C8C4semidirect

Maximal subgroups

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

subgroupindex
#208092
#108162
#87302
#72532
#72522