Finite Group Databasebeta

← #141123

Subgroups — #141123

Order 512.

478 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 100 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#33176C211yesmaximal, characteristic
2#12785C211yesmaximal, characteristic
3C2×#1342C214yesmaximal
4C4×#338C211yesmaximal, characteristic
5C2×#494C411yescharacteristic
6C2×#470—21no—
7C2×#470C2212yes—
8C2×C4×M16—21no—
9C2×C4×M16C2211yescharacteristic
10#2857—21no—
11#2150—21no—
12C2×#349C2212yes—
13#1971—21no—
14#1971C412yes—
15#1956—21no—
16#1844C411yescharacteristic
17C2×#338—21no—
18C2×#338C2212yes—
19#1377—21no—
20#1342—28no—
21C22×Q16—21no—
22C22×Q16C2×C412yes—
23C22×M16—22no—
24C22×M16D411yescharacteristic
25C22×M16C2311yescharacteristic
26C2×C4⋊Q8—21no—
27C2×C22⋊Q8—22no—
28C22×C42D411yescharacteristic
29C2×M32—22no—
30#494—42no—
31#470—24no—
32#470—24no—
33#470—44no—
34C2×#106—21no—
35C2×#106C2×C412yes—
36C2×#103—22no—
37C2×#103—22no—
38C4×M16—42no—
39C4×M16—42no—
40C4×M16—42no—
41C2×#97—22no—
42C2×#97C2×C411yescharacteristic
43C2×#97D411yescharacteristic
44C2×C4×C8—22no—
45C2×C4×C8C2×C411yescharacteristic
46C2×C4×C8D411yescharacteristic
47#391—22no—
48#389—42no—
49#384—21no—
50#380—42no—
51C4⋊C16—42no—
52#349—24no—
53#338—24no—
54#338—22no—
55#338D414yes—
56C22×Q8—22no—
57C23×C4SD1612yes—
58C23×C4C2×D411yescharacteristic
59C2×Q16—24no—
60C2×Q16—42no—
61C2×Q16—42no—
62C2×Q16—42no—
63C2×Q16—42no—
64C2×Q16—42no—
65C2×M16—24no—
66C2×M16—24no—
67C2×M16—22no—
68C2×M16—22no—
69C2×M16—41no—
70C2×M16—41no—
71C2×M16—41no—
72C2×M16—42no—
73C2×M16—42no—
74C2×M16—42no—
75C2×M16—42no—
76C2×M16C2×D412yes—
77C2×M16#4112yes—
78C22×C8—24no—
79C22×C8C22⋊C411yescharacteristic
80C22×C8C22⋊C411yescharacteristic
81C22×C8C22×C411yescharacteristic
82C22×C8C2×D411yescharacteristic
83C4⋊Q8—42no—
84C4⋊Q8—41no—
85C4⋊Q8—41no—
86C22⋊Q8—44no—
87C22⋊Q8—44no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—42no—
92C2×C22⋊C4—42no—
93C2×C42—22no—
94C2×C42—21no—
95C2×C42—21no—
96C2×C42C22⋊C411yescharacteristic
97C2×C42C22⋊C411yescharacteristic
98C2×C42D812yes—
99M32—42no—
100M32—42no—
101C2×C16—42no—
102#106—24no—
103#106—42no—
104#103—44no—
105#103—42no—
106#103—42no—
107#97—41no—
108#97—41no—
109#97—44no—
110C4×C8—22no—
111C4×C8—22no—
112C4×C8—44no—
113#95—44no—
114C24C2×SD1611yescharacteristic
115C2×Q8—42no—
116C2×Q8—42no—
117C2×Q8—42no—
118C2×Q8—42no—
119C2×Q8—42no—
120C2×Q8—42no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—44no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4#10212yes—
131C22×C4#10212yes—
132C22×C4C2×C22⋊C411yescharacteristic
133C22×C4#12112yes—
134C22×C4C2×D811yescharacteristic
135C22×C4AGL(1,ℤ/8)12yes—
136Q16—84no—
137Q16—84no—
138M16—42no—
139M16—42no—
140M16—44no—
141M16—44no—
142M16—44no—
143M16—44no—
144M16—44no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—28no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—44no—
156C2×C8—44no—
157C2×C8#10412yes—
158C2×C8C4×D412yes—
159C2×C8#12411yescharacteristic
160C2×C8#12711yescharacteristic
161C4⋊C4—42no—
162C4⋊C4—42no—
163C4⋊C4—84no—
164C4⋊C4—82no—
165C4⋊C4—82no—
166C4⋊C4—84no—
167C22⋊C4—84no—
168C42—22no—
169C42—22no—
170C42—21no—
171C42—21no—
172C42#9912yes—
173C42#10411yescharacteristic
174C42#10411yescharacteristic
175C16—82no—
176C23—21no—
177C23—21no—
178C23—42no—
179C23C2×#10211yescharacteristic
180C23#46012yes—
181Q8—82no—
182Q8—82no—
183Q8—82no—
184Q8—82no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—24no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—44no—
194C2×C4—42no—
195C2×C4—82no—
196C2×C4—84no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4#32712yes—
202C2×C4#32712yes—
203C2×C4#39011yescharacteristic
204C2×C4C2×#9911yescharacteristic
205C2×C4#41011yescharacteristic
206C2×C4#41811yescharacteristic
207C2×C4C2×#10411yescharacteristic
208C2×C4C2×#10411yescharacteristic
209C2×C4#45912yes—
210C8—21no—
211C8—22no—
212C8—21no—
213C8—22no—
214C8—48no—
215C8—41no—
216C22—22no—
217C22—22no—
218C22—42no—
219C22—42no—
220C22—81no—
221C22C2×#32711yescharacteristic
222C22#150211yescharacteristic
223C22#198211yescharacteristic
224C4—22no—
225C4—22no—
226C4—42no—
227C4—162no—
228C4—162no—
229C4#134214yescomplemented
230C4#150511yescharacteristic
231C4#150711yescharacteristic
232C4#196211yescharacteristic
233C4#196411yescharacteristic
234C2—21no—
235C2—21no—
236C2—42no—
237C2#971211yescharacteristic
238C2C2×#134212yes—
subgroupG/H
center, socleC22C2×#327
commutatorC22×C8C22×C4
Frattini, cosocleC22×M16C23
prime radicalC24C2×SD16
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ 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
229. C4#1342semidirect

Maximal subgroups

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

subgroupindex
#331762
#127852
C2×#13422
C4×#3382