Finite Group Databasebeta

← #142048

Subgroups — #142048

Order 512.

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

normal subgroups only

This group has 374 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#33210C211yesmaximal, characteristic
2#12607C211yesmaximal, characteristic
3#10825C211yesmaximal, characteristic
4#7255C212yesmaximal
5#7254C212yesmaximal
6C2×#498C411yescharacteristic, complemented
7C2×C4×M16C2211yescharacteristic
8#2857—21no—
9C2×#349C2212yes—
10C2×#347—21no—
11#1990—21no—
12#1943—21no—
13#1943C412yescomplemented
14#1845C411yescharacteristic, complemented
15#1809—21no—
16#1809C2212yes—
17C2×#344—21no—
18C2×#339—21no—
19C2×#338C2211yescharacteristic
20C2×H(3,ℤ/4)C2211yescharacteristic
21#1376—21no—
22C22×M16—21no—
23C22×M16D411yescharacteristic
24C22×M16C2311yescharacteristic
25C2×C4⋊Q8—21no—
26C2×C22⋊Q8—41no—
27C2×#117—21no—
28C22×C42D411yescharacteristic
29C2×M32—22no—
30#498—42no—
31C2×#107—21no—
32C2×#107C2×C412yes—
33C2×#106—21no—
34C2×#106C2×C412yes—
35C4×M16—42no—
36C2×#97C2×C411yescharacteristic
37C2×#97D411yescharacteristic
38C2×C4×C8C2×C411yescharacteristic
39C2×C4×C8D411yescharacteristic
40#391—41no—
41#389—22no—
42#389—41no—
43#384—21no—
44#380—22no—
45#380—41no—
46#349—22no—
47#349Q814yescomplemented
48#347—42no—
49#344—42no—
50#340—22no—
51#340—22no—
52#339—24no—
53#338—22no—
54H(3,ℤ/4)—22no—
55C22×Q8—41no—
56C23×C4—21no—
57C23×C4C2×D411yescharacteristic
58C2×M16—21no—
59C2×M16—21no—
60C2×M16—22no—
61C2×M16—22no—
62C2×M16—41no—
63C2×M16—41no—
64C2×M16—41no—
65C2×M16—41no—
66C2×M16—41no—
67C2×M16—41no—
68C2×M16—41no—
69C2×M16—41no—
70C2×M16C2×Q812yes—
71C2×M16#4112yes—
72C22×C8—41no—
73C22×C8C22⋊C411yescharacteristic
74C22×C8C22⋊C411yescharacteristic
75C22×C8C22×C411yescharacteristic
76C22×C8C2×D411yescharacteristic
77C4⋊Q8—42no—
78C4⋊Q8—42no—
79C22⋊Q8—84no—
80#117—22no—
81#117—41no—
82#117—41no—
83#117—41no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C22⋊C4—41no—
92C2×C22⋊C4—42no—
93C2×C42—22no—
94C2×C42—21no—
95C2×C42—41no—
96C2×C42—41no—
97C2×C42C22⋊C411yescharacteristic
98C2×C42C22⋊C411yescharacteristic
99M32—24no—
100M32—42no—
101C2×C16—42no—
102#107—42no—
103#107—42no—
104#106—42no—
105#106—42no—
106#97—22no—
107#97—41no—
108C4×C8—22no—
109C4×C8—41no—
110#95—42no—
111#95—42no—
112C24AGL(1,ℤ/8)11yescharacteristic
113C2×Q8—82no—
114C2×Q8—82no—
115C2×Q8—82no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4—81no—
128C22×C4C2×C22⋊C411yescharacteristic
129C22×C4#12411yescharacteristic
130C22×C4C4⋊Q811yescharacteristic
131C22×C4AGL(1,ℤ/8)11yescharacteristic
132C22×C4#13711yescharacteristic
133C22×C4#13711yescharacteristic
134M16—42no—
135M16—42no—
136M16—42no—
137M16—42no—
138M16—42no—
139M16—42no—
140M16—81no—
141M16—81no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—42no—
152C2×C8—81no—
153C2×C8—81no—
154C2×C8#9912yes—
155C2×C8C4×Q812yes—
156C2×C8C22⋊Q812yes—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—42no—
162C4⋊C4—82no—
163C4⋊C4—81no—
164C4⋊C4—82no—
165C4⋊C4—82no—
166C4⋊C4—84no—
167C22⋊C4—42no—
168C22⋊C4—84no—
169C22⋊C4—81no—
170C42—21no—
171C42—21no—
172C42—21no—
173C42—42no—
174C42—42no—
175C42—81no—
176C42#10412yes—
177C16—44no—
178C23—21no—
179C23—21no—
180C23—41no—
181C23—41no—
182C23#41811yescharacteristic
183C23#48911yescharacteristic
184C23#50111yescharacteristic
185Q8—164no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
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—21no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—84no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4—82no—
206C2×C4—82no—
207C2×C4—82no—
208C2×C4—82no—
209C2×C4—82no—
210C2×C4—161no—
211C2×C4—161no—
212C2×C4#39111yescharacteristic
213C2×C4C2×#9911yescharacteristic
214C2×C4#41011yescharacteristic
215C2×C4#41911yescharacteristic
216C2×C4C2×#10411yescharacteristic
217C2×C4#42111yescharacteristic
218C2×C4#48911yescharacteristic
219C2×C4#50111yescharacteristic
220C8—22no—
221C8—22no—
222C8—22no—
223C8—41no—
224C8—82no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—41no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22—81no—
234C22#150711yescharacteristic
235C22#151411yescharacteristic
236C22#199311yescharacteristic
237C4—24no—
238C4—21no—
239C4—21no—
240C4—81no—
241C4—162no—
242C4—162no—
243C4—162no—
244C4#150611yescharacteristic
245C4#151111yescharacteristic
246C4#198611yescharacteristic
247C4#199611yescharacteristic
248C2—21no—
249C2—21no—
250C2—41no—
251C2—41no—
252C2#988811yescharacteristic
253C2#1183711yescharacteristic
254C2#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
6. C2×#498C4semidirect
13. #1943C4semidirect
14. #1845C4semidirect
47. #349Q8semidirect

Maximal subgroups

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

subgroupindex
#332102
#126072
#108252
#72552
#72542