Finite Group Databasebeta

← #142059

Subgroups — #142059

Order 512.

346 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 82 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#33216C211yesmaximal, characteristic
2#12608C211yesmaximal, characteristic
3#10799C211yesmaximal, characteristic
4#7255C212yesmaximal
5#7254C212yesmaximal
6C2×#499C411yescharacteristic, complemented
7C2×#405C2211yescharacteristic
8#2629—21no—
9C2×#349C2212yes—
10C2×#347—21no—
11#1992—21no—
12#1943C412yescomplemented
13#1942—21no—
14#1845C411yescharacteristic, complemented
15#1809C2212yes—
16#1808—21no—
17C2×#338C2211yescharacteristic
18C2×H(3,ℤ/4)—21no—
19C2×H(3,ℤ/4)C2211yescharacteristic
20#1729—21no—
21#1376—21no—
22C22×M16—21no—
23C22×M16C2311yescharacteristic
24C23×C8D411yescharacteristic
25C2×#125—21no—
26C2×#117—21no—
27C2×#117D411yescharacteristic
28C2×M32—22no—
29#499—42no—
30C2×#107—21no—
31C2×#107—21no—
32C2×#107C2×C412yes—
33C2×#106C2×C412yes—
34#405—42no—
35C2×#97C2×C411yescharacteristic
36C2×#97D411yescharacteristic
37C2×C4×C8C2×C411yescharacteristic
38C2×C4×C8D411yescharacteristic
39#398—42no—
40#396—42no—
41#389—22no—
42#384—21no—
43#380—22no—
44#349—22no—
45#349Q814yes—
46#347—42no—
47#340—22no—
48#340—22no—
49#338—22no—
50H(3,ℤ/4)—22no—
51H(3,ℤ/4)—42no—
52#336—24no—
53C23×C4C2×D411yescharacteristic
54C2×M16—41no—
55C2×M16—41no—
56C2×M16—41no—
57C2×M16—41no—
58C2×M16—41no—
59C2×M16—41no—
60C2×M16—41no—
61C2×M16—41no—
62C2×M16C2×Q812yes—
63C2×M16#4112yes—
64C22×C8—21no—
65C22×C8—21no—
66C22×C8—22no—
67C22×C8—22no—
68C22×C8—41no—
69C22×C8C22⋊C411yescharacteristic
70C22×C8C22⋊C411yescharacteristic
71C22×C8C22×C411yescharacteristic
72C22×C8C2×D411yescharacteristic
73#125—42no—
74#125—42no—
75#117—22no—
76#117—41no—
77#117—41no—
78#117—41no—
79#117—41no—
80#117—41no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—41no—
90C2×C22⋊C4—42no—
91C2×C42—22no—
92C2×C42C22⋊C411yescharacteristic
93C2×C42C22⋊C411yescharacteristic
94M32—24no—
95M32—42no—
96C2×C16—42no—
97#107—42no—
98#107—42no—
99#107—42no—
100#106—42no—
101#97—22no—
102#97—41no—
103C4×C8—22no—
104C4×C8—41no—
105#95—42no—
106#95—42no—
107C24AGL(1,ℤ/8)11yescharacteristic
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4C2×C22⋊C411yescharacteristic
118C22×C4#12411yescharacteristic
119C22×C4C4⋊Q811yescharacteristic
120C22×C4AGL(1,ℤ/8)11yescharacteristic
121C22×C4#13711yescharacteristic
122C22×C4#13711yescharacteristic
123M16—42no—
124M16—42no—
125M16—81no—
126M16—81no—
127M16—81no—
128M16—81no—
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—21no—
138C2×C8—42no—
139C2×C8—42no—
140C2×C8—42no—
141C2×C8—42no—
142C2×C8—42no—
143C2×C8#10412yes—
144C2×C8C4×Q812yes—
145C2×C8C22⋊Q812yes—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4—42no—
150C4⋊C4—42no—
151C4⋊C4—81no—
152C4⋊C4—81no—
153C4⋊C4—84no—
154C4⋊C4—162no—
155C4⋊C4—162no—
156C22⋊C4—42no—
157C22⋊C4—81no—
158C22⋊C4—81no—
159C22⋊C4—162no—
160C42—21no—
161C42—21no—
162C42—21no—
163C42—42no—
164C42—42no—
165C42#9912yes—
166C16—44no—
167C23—21no—
168C23—21no—
169C23—41no—
170C23—41no—
171C23#41811yescharacteristic
172C23#48911yescharacteristic
173C23#50111yescharacteristic
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—41no—
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#39111yescharacteristic
200C2×C4C2×#9911yescharacteristic
201C2×C4#41011yescharacteristic
202C2×C4#41911yescharacteristic
203C2×C4C2×#10411yescharacteristic
204C2×C4#42111yescharacteristic
205C2×C4#48911yescharacteristic
206C2×C4#50111yescharacteristic
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#151011yescharacteristic
223C22#199311yescharacteristic
224C4—21no—
225C4—21no—
226C4—42no—
227C4—81no—
228C4—162no—
229C4—162no—
230C4—162no—
231C4#150811yescharacteristic
232C4#151511yescharacteristic
233C4#198611yescharacteristic
234C4#199611yescharacteristic
235C2—21no—
236C2—21no—
237C2—41no—
238C2—41no—
239C2#988911yescharacteristic
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
6. C2×#499C4semidirect
12. #1943C4semidirect
14. #1845C4semidirect

Maximal subgroups

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

subgroupindex
#332162
#126082
#107992
#72552
#72542