Finite Group Databasebeta

← #135385

Subgroups — #135385

Order 512.

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

normal subgroups only

This group has 342 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#20821C211yesmaximal, characteristic
2#12850C211yesmaximal, characteristic
3#10800C211yesmaximal, characteristic
4#7255C212yesmaximal
5#7254C212yesmaximal
6C2×#405C2211yescharacteristic
7#2314—21no—
8C2×#349C2212yes—
9#1846C411yescharacteristic, complemented
10#1845C411yescharacteristic, complemented
11#1836—21no—
12#1809C2212yes—
13#1808—21no—
14#1776C412yescomplemented
15C4×#107—21no—
16C2×#338C2211yescharacteristic
17C2×H(3,ℤ/4)—21no—
18C2×H(3,ℤ/4)C2211yescharacteristic
19#1729—21no—
20#1376—21no—
21#1375—21no—
22C22×M16—21no—
23C22×M16C2311yescharacteristic
24C23×C8D411yescharacteristic
25C2×#117—21no—
26C2×#117D411yescharacteristic
27C2×M32—22no—
28C2×#107—21no—
29C2×#107—21no—
30C2×#107C2×C412yes—
31C2×#106C2×C412yes—
32#405—22no—
33#405—22no—
34C2×#97C2×C411yescharacteristic
35C2×#97D411yescharacteristic
36C2×C4×C8C2×C411yescharacteristic
37C2×C4×C8D411yescharacteristic
38#388—42no—
39#387—42no—
40#384—21no—
41#382—21no—
42#380—22no—
43C4×C4⋊C4—22no—
44#349—22no—
45#349—22no—
46#340—22no—
47#340—22no—
48#338—22no—
49H(3,ℤ/4)—22no—
50H(3,ℤ/4)—42no—
51#336—24no—
52C23×C4C2×D411yescharacteristic
53C2×M16—41no—
54C2×M16—41no—
55C2×M16—41no—
56C2×M16—41no—
57C2×M16—41no—
58C2×M16—41no—
59C2×M16—41no—
60C2×M16—41no—
61C2×M16#4112yes—
62C2×M16#4112yes—
63C22×C8—21no—
64C22×C8—21no—
65C22×C8—22no—
66C22×C8—22no—
67C22×C8—41no—
68C22×C8C22⋊C411yescharacteristic
69C22×C8C22⋊C411yescharacteristic
70C22×C8C22×C411yescharacteristic
71C22×C8C2×D411yescharacteristic
72#117—21no—
73#117—21no—
74#117—41no—
75#117—41no—
76#117—41no—
77#117—41no—
78#117SD1612yes—
79#117Q1612yes—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—42no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—41no—
87C2×C22⋊C4—42no—
88C2×C42—22no—
89C2×C42—42no—
90C2×C42C22⋊C411yescharacteristic
91C2×C42C22⋊C411yescharacteristic
92M32—42no—
93M32—42no—
94C2×C16—42no—
95#107—42no—
96#107—42no—
97#107—42no—
98#106—42no—
99#97—22no—
100#97—22no—
101C4×C8—22no—
102C4×C8—22no—
103#95—42no—
104#95—42no—
105#95—42no—
106C24AGL(1,ℤ/8)11yescharacteristic
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—42no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4C2×C22⋊C411yescharacteristic
117C22×C4#12311yescharacteristic
118C22×C4#12311yescharacteristic
119C22×C4C2×SD1611yescharacteristic
120C22×C4C2×Q1611yescharacteristic
121C22×C4AGL(1,ℤ/8)11yescharacteristic
122M16—42no—
123M16—42no—
124M16—81no—
125M16—81no—
126M16—81no—
127M16—81no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—42no—
138C2×C8—42no—
139C2×C8—42no—
140C2×C8—42no—
141C2×C8—42no—
142C2×C8#10412yes—
143C2×C8#11712yes—
144C2×C8#12312yes—
145C4⋊C4—22no—
146C4⋊C4—22no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4—42no—
150C4⋊C4—42no—
151C4⋊C4—81no—
152C4⋊C4—81no—
153C4⋊C4—162no—
154C22⋊C4—22no—
155C22⋊C4—22no—
156C22⋊C4—81no—
157C22⋊C4—81no—
158C22⋊C4—162no—
159C42—21no—
160C42—21no—
161C42—42no—
162C42—42no—
163C42—162no—
164C42#10312yes—
165C42#10312yes—
166C16—82no—
167C23—21no—
168C23—21no—
169C23—41no—
170C23—41no—
171C23#41811yescharacteristic
172C23#48111yescharacteristic
173C23#48411yescharacteristic
174C2×C4—21no—
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—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4—82no—
193C2×C4—82no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4—82no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4#38111yescharacteristic
200C2×C4#41011yescharacteristic
201C2×C4#41011yescharacteristic
202C2×C4C2×#10311yescharacteristic
203C2×C4C2×#10411yescharacteristic
204C2×C4#42111yescharacteristic
205C2×C4#48111yescharacteristic
206C2×C4#48411yescharacteristic
207C8—22no—
208C8—22no—
209C8—22no—
210C8—41no—
211C8—82no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—41no—
217C22—41no—
218C22—41no—
219C22—41no—
220C22—81no—
221C22#150511yescharacteristic
222C22#151411yescharacteristic
223C22#182411yescharacteristic
224C4—21no—
225C4—21no—
226C4—42no—
227C4—42no—
228C4—162no—
229C4—162no—
230C4—162no—
231C4#150311yescharacteristic
232C4#151811yescharacteristic
233C4#182011yescharacteristic
234C4#182611yescharacteristic
235C2—21no—
236C2—21no—
237C2—41no—
238C2—41no—
239C2#871711yescharacteristic
240C2#1177011yescharacteristic
241C2#1177111yescharacteristic
subgroupG/H
center, socleC22#1505
commutatorC22×C8C22×C4
Frattini, cosocleC22×M16C23
prime radicalC24AGL(1,ℤ/8)
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G
LECSG ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
9. #1846C4semidirect
10. #1845C4semidirect
14. #1776C4semidirect

Maximal subgroups

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

subgroupindex
#208212
#128502
#108002
#72552
#72542