Finite Group Databasebeta

← #142049

Subgroups — #142049

Order 512.

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

normal subgroups only

This group has 352 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#33211C211yesmaximal, characteristic
2#12607C211yesmaximal, characteristic
3#10825C211yesmaximal, characteristic
4#7255C212yesmaximal
5#7254C212yesmaximal
6C2×#499C411yescharacteristic, complemented
7C2×C4×M16C2211yescharacteristic
8#2851—21no—
9C2×#349C2212yes—
10C2×#347—21no—
11#1991—21no—
12#1944—21no—
13#1943C412yescomplemented
14#1846C411yescharacteristic, 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×#125—21no—
26C2×#117—21no—
27C22×C42D411yescharacteristic
28C2×M32—22no—
29#499—42no—
30C2×#107—21no—
31C2×#107C2×C412yes—
32C2×#106—21no—
33C2×#106C2×C412yes—
34C4×M16—42no—
35C2×#97C2×C411yescharacteristic
36C2×#97D411yescharacteristic
37C2×C4×C8C2×C411yescharacteristic
38C2×C4×C8D411yescharacteristic
39#389—22no—
40#387—42no—
41#385—42no—
42#382—21no—
43#380—22no—
44#349—22no—
45#349Q814yes—
46#347—42no—
47#344—42no—
48#340—22no—
49#340—22no—
50#339—24no—
51#338—22no—
52H(3,ℤ/4)—22no—
53C23×C4—21no—
54C23×C4C2×D411yescharacteristic
55C2×M16—21no—
56C2×M16—21no—
57C2×M16—22no—
58C2×M16—22no—
59C2×M16—41no—
60C2×M16—41no—
61C2×M16—41no—
62C2×M16—41no—
63C2×M16—41no—
64C2×M16—41no—
65C2×M16—41no—
66C2×M16—41no—
67C2×M16C2×Q812yes—
68C2×M16#4112yes—
69C22×C8—41no—
70C22×C8C22⋊C411yescharacteristic
71C22×C8C22⋊C411yescharacteristic
72C22×C8C22×C411yescharacteristic
73C22×C8C2×D411yescharacteristic
74#125—42no—
75#125—42no—
76#117—22no—
77#117—41no—
78#117—41no—
79#117—41no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—42no—
85C2×C22⋊C4—41no—
86C2×C22⋊C4—42no—
87C2×C42—22no—
88C2×C42—21no—
89C2×C42—41no—
90C2×C42—41no—
91C2×C42C22⋊C411yescharacteristic
92C2×C42C22⋊C411yescharacteristic
93M32—24no—
94M32—42no—
95C2×C16—42no—
96#107—42no—
97#107—42no—
98#106—42no—
99#106—42no—
100#97—22no—
101#97—41no—
102C4×C8—22no—
103C4×C8—41no—
104#95—42no—
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×C4—82no—
118C22×C4—81no—
119C22×C4C2×C22⋊C411yescharacteristic
120C22×C4#12411yescharacteristic
121C22×C4C4⋊Q811yescharacteristic
122C22×C4AGL(1,ℤ/8)11yescharacteristic
123C22×C4#13711yescharacteristic
124C22×C4#13711yescharacteristic
125M16—42no—
126M16—42no—
127M16—42no—
128M16—42no—
129M16—42no—
130M16—42no—
131M16—81no—
132M16—81no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—42no—
143C2×C8—81no—
144C2×C8—81no—
145C2×C8#9912yes—
146C2×C8C4×Q812yes—
147C2×C8C22⋊Q812yes—
148C4⋊C4—42no—
149C4⋊C4—42no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—81no—
154C4⋊C4—84no—
155C4⋊C4—84no—
156C22⋊C4—42no—
157C22⋊C4—84no—
158C22⋊C4—81no—
159C42—21no—
160C42—21no—
161C42—21no—
162C42—42no—
163C42—42no—
164C42—81no—
165C42#10412yes—
166C16—44no—
167C23—21no—
168C23—21no—
169C23—42no—
170C23#41811yescharacteristic
171C23#48911yescharacteristic
172C23#50111yescharacteristic
173C2×C4—21no—
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—22no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—41no—
187C2×C4—84no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4—82no—
193C2×C4—82no—
194C2×C4—82no—
195C2×C4—162no—
196C2×C4#39111yescharacteristic
197C2×C4C2×#9911yescharacteristic
198C2×C4#41011yescharacteristic
199C2×C4#41911yescharacteristic
200C2×C4C2×#10411yescharacteristic
201C2×C4#42111yescharacteristic
202C2×C4#48911yescharacteristic
203C2×C4#50111yescharacteristic
204C8—22no—
205C8—22no—
206C8—22no—
207C8—41no—
208C8—82no—
209C22—22no—
210C22—21no—
211C22—21no—
212C22—41no—
213C22—41no—
214C22—42no—
215C22—81no—
216C22#150711yescharacteristic
217C22#151411yescharacteristic
218C22#199311yescharacteristic
219C4—21no—
220C4—21no—
221C4—42no—
222C4—81no—
223C4—162no—
224C4—162no—
225C4—162no—
226C4#150611yescharacteristic
227C4#151111yescharacteristic
228C4#198611yescharacteristic
229C4#199611yescharacteristic
230C2—21no—
231C2—21no—
232C2—41no—
233C2—41no—
234C2#988811yescharacteristic
235C2#1183612yes—
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×#499C4semidirect
13. #1943C4semidirect
14. #1846C4semidirect

Maximal subgroups

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

subgroupindex
#332112
#126072
#108252
#72552
#72542