Finite Group Databasebeta

← #142061

Subgroups — #142061

Order 512.

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

normal subgroups only

This group has 345 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#33214C211yesmaximal, characteristic
2#12606C211yesmaximal, characteristic
3#10804C211yesmaximal, characteristic
4#7263C212yesmaximal
5#7262C212yesmaximal
6C2×#499C411yescharacteristic, complemented
7C2×#405C2211yescharacteristic
8#2629—21no—
9#2112C2212yes—
10C2×#347—21no—
11#1992—21no—
12#1944—21no—
13#1941C412yescomplemented
14#1844C411yescharacteristic, complemented
15#1809—21no—
16#1807C2212yes—
17C2×#344—21no—
18C2×#341C2211yescharacteristic
19#1735C2211yescharacteristic
20C2×H(3,ℤ/4)—21no—
21#1375—21no—
22C22×M16—21no—
23C22×M16D411yescharacteristic
24C23×C8C2311yescharacteristic
25C2×#125—21no—
26C2×#117—21no—
27C2×#117D411yescharacteristic
28C2×M32—22no—
29#499—42no—
30C2×#107—21no—
31C2×#106—21no—
32C2×#106C2×C412yes—
33C2×#106C2×C412yes—
34#405—42no—
35C2×#97D411yescharacteristic
36C2×#97D411yescharacteristic
37C2×C4×C8C2×C411yescharacteristic
38C2×C4×C8C2×C411yescharacteristic
39#398—42no—
40#396—42no—
41#389—22no—
42#384—21no—
43#380—22no—
44C22⋊C16—22no—
45C22⋊C16Q814yes—
46#347—42no—
47#344—24no—
48#341—22no—
49#340—22no—
50#340—22no—
51#340—22no—
52H(3,ℤ/4)—42no—
53C23×C4C2×D411yescharacteristic
54C2×M16—21no—
55C2×M16—21no—
56C2×M16—22no—
57C2×M16—22no—
58C2×M16—41no—
59C2×M16—41no—
60C2×M16—41no—
61C2×M16—41no—
62C2×M16—41no—
63C2×M16—41no—
64C22×C8—41no—
65C22×C8—41no—
66C22×C8—41no—
67C22×C8C22⋊C411yescharacteristic
68C22×C8C22⋊C411yescharacteristic
69C22×C8C22×C411yescharacteristic
70C22×C8C2×D411yescharacteristic
71C22×C8C2×Q812yes—
72C22×C8#4112yes—
73#125—42no—
74#125—42no—
75#117—22no—
76#117—22no—
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—41no—
92C2×C42C22⋊C411yescharacteristic
93C2×C42C22⋊C411yescharacteristic
94M32—42no—
95M32—42no—
96C2×C16—24no—
97#107—42no—
98#106—42no—
99#106—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—81no—
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—42no—
126M16—42no—
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—81no—
142C2×C8—81no—
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—42no—
152C4⋊C4—81no—
153C4⋊C4—84no—
154C4⋊C4—162no—
155C4⋊C4—162no—
156C22⋊C4—42no—
157C22⋊C4—42no—
158C22⋊C4—81no—
159C22⋊C4—162no—
160C42—21no—
161C42—21no—
162C42—21no—
163C42—42no—
164C42—81no—
165C42#9912yes—
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—82no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4—82no—
193C2×C4—82no—
194C2×C4—162no—
195C2×C4#39111yescharacteristic
196C2×C4C2×#9911yescharacteristic
197C2×C4#41011yescharacteristic
198C2×C4#41911yescharacteristic
199C2×C4C2×#10411yescharacteristic
200C2×C4#42111yescharacteristic
201C2×C4#48911yescharacteristic
202C2×C4#50111yescharacteristic
203C8—22no—
204C8—22no—
205C8—22no—
206C8—41no—
207C8—82no—
208C22—22no—
209C22—21no—
210C22—21no—
211C22—22no—
212C22—22no—
213C22—81no—
214C22—81no—
215C22#150511yescharacteristic
216C22#151011yescharacteristic
217C22#199311yescharacteristic
218C4—21no—
219C4—21no—
220C4—42no—
221C4—81no—
222C4—162no—
223C4—162no—
224C4—162no—
225C4#150811yescharacteristic
226C4#151511yescharacteristic
227C4#198611yescharacteristic
228C4#199611yescharacteristic
229C2—21no—
230C2—21no—
231C2—41no—
232C2—41no—
233C2#988911yescharacteristic
234C2#1329112yes—
subgroupG/H
center, socleC22#1993
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24AGL(1,ℤ/8)
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. C2×#499C4semidirect
13. #1941C4semidirect
14. #1844C4semidirect

Maximal subgroups

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

subgroupindex
#332142
#126062
#108042
#72632
#72622