Finite Group Databasebeta

← #141114

Subgroups — #141114

Order 512.

498 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 114 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#33172C211yesmaximal, characteristic
2#12783C211yesmaximal, characteristic
3C2×#1350C214yesmaximal
4C4×#341C211yesmaximal, characteristic
5C2×#494C411yescharacteristic
6C2×#467—21no—
7C2×#467C2212yes—
8C2×C4×M16—21no—
9C22×C4×C8C2211yescharacteristic
10#2857—21no—
11C2×C4⋊C16—21no—
12C2×C22⋊C16C2212yes—
13#1969—21no—
14#1969C412yes—
15#1956—21no—
16#1845C411yescharacteristic
17C2×#341—21no—
18C2×#341C2212yes—
19#1374—21no—
20#1350—28no—
21C22×Q16—21no—
22C22×Q16C2×C412yes—
23C22×M16—22no—
24C23×C8D411yescharacteristic
25C23×C8C2311yescharacteristic
26C2×C4⋊Q8—21no—
27C2×C22⋊Q8—22no—
28C22×C42D411yescharacteristic
29C22×C16—22no—
30#494—42no—
31#467—24no—
32#467—24no—
33#467—44no—
34C2×#107—21no—
35C2×#107C2×C412yes—
36C2×#103—22no—
37C2×#103—22no—
38C4×M16—24no—
39C4×M16—42no—
40C2×#97—41no—
41C2×C4×C8—42no—
42C2×C4×C8—41no—
43C2×C4×C8C2×C411yescharacteristic
44C2×C4×C8C2×C411yescharacteristic
45C2×C4×C8D411yescharacteristic
46C2×C4×C8D411yescharacteristic
47#391—22no—
48#389—41no—
49#389—41no—
50#384—21no—
51#380—41no—
52#380—41no—
53C4⋊C16—42no—
54C22⋊C16—24no—
55#341—24no—
56#341—22no—
57#341D414yes—
58C22×Q8—22no—
59C23×C4SD1611yescharacteristic
60C23×C4SD1611yescharacteristic
61C23×C4C2×D411yescharacteristic
62C2×Q16—24no—
63C2×Q16—42no—
64C2×Q16—42no—
65C2×Q16—42no—
66C2×Q16—42no—
67C2×Q16—42no—
68C2×M16—24no—
69C2×M16—24no—
70C2×M16—24no—
71C2×M16—42no—
72C2×M16—42no—
73C2×M16—42no—
74C22×C8—22no—
75C22×C8—22no—
76C22×C8—41no—
77C22×C8—41no—
78C22×C8—41no—
79C22×C8—42no—
80C22×C8C22⋊C411yescharacteristic
81C22×C8C22⋊C411yescharacteristic
82C22×C8C22×C411yescharacteristic
83C22×C8C2×D411yescharacteristic
84C22×C8C2×D412yes—
85C22×C8#4112yes—
86C4⋊Q8—42no—
87C4⋊Q8—42no—
88C22⋊Q8—44no—
89C22⋊Q8—44no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—42no—
96C2×C22⋊C4—42no—
97C2×C42—22no—
98C2×C42—22no—
99C2×C42C22⋊C411yescharacteristic
100C2×C42C22⋊C411yescharacteristic
101C2×C42D811yescharacteristic
102C2×C42D811yescharacteristic
103C2×C16—42no—
104C2×C16—42no—
105C2×C16—42no—
106#107—24no—
107#107—42no—
108#103—44no—
109#103—42no—
110#103—42no—
111#97—44no—
112C4×C8—22no—
113C4×C8—22no—
114C4×C8—41no—
115C4×C8—41no—
116C4×C8—44no—
117#95—42no—
118#95—42no—
119C24C2×SD1611yescharacteristic
120C2×Q8—42no—
121C2×Q8—42no—
122C2×Q8—42no—
123C2×Q8—42no—
124C2×Q8—42no—
125C2×Q8—42no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—41no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4#10211yescharacteristic
139C22×C4#10211yescharacteristic
140C22×C4#10211yescharacteristic
141C22×C4#10211yescharacteristic
142C22×C4C2×C22⋊C411yescharacteristic
143C22×C4#12112yes—
144C22×C4C2×D811yescharacteristic
145C22×C4AGL(1,ℤ/8)12yes—
146Q16—84no—
147Q16—84no—
148M16—28no—
149M16—44no—
150M16—44no—
151M16—44no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—42no—
162C2×C8—42no—
163C2×C8—44no—
164C2×C8—44no—
165C2×C8—44no—
166C2×C8—44no—
167C2×C8#10412yes—
168C2×C8C4×D412yes—
169C2×C8#12411yescharacteristic
170C2×C8#12711yescharacteristic
171C4⋊C4—42no—
172C4⋊C4—42no—
173C4⋊C4—82no—
174C4⋊C4—82no—
175C4⋊C4—82no—
176C4⋊C4—82no—
177C4⋊C4—84no—
178C22⋊C4—84no—
179C42—22no—
180C42—22no—
181C42—21no—
182C42—21no—
183C42#9912yes—
184C42#10412yes—
185C16—82no—
186C23—21no—
187C23—21no—
188C23—21no—
189C23—21no—
190C23—21no—
191C23—21no—
192C23C2×#10211yescharacteristic
193C23#46012yes—
194Q8—82no—
195Q8—82no—
196Q8—82no—
197Q8—82no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—44no—
212C2×C4—44no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—82no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4#32712yes—
220C2×C4#32712yes—
221C2×C4#35912yes—
222C2×C4#36212yes—
223C2×C4#36212yes—
224C2×C4#39011yescharacteristic
225C2×C4C2×#9911yescharacteristic
226C2×C4#41011yescharacteristic
227C2×C4#41811yescharacteristic
228C2×C4C2×#10411yescharacteristic
229C2×C4C2×#10411yescharacteristic
230C2×C4#45912yes—
231C8—21no—
232C8—22no—
233C8—21no—
234C8—22no—
235C8—48no—
236C8—41no—
237C22—22no—
238C22—22no—
239C22—22no—
240C22—41no—
241C22—41no—
242C22—41no—
243C22—41no—
244C22C2×#32711yescharacteristic
245C22#150211yescharacteristic
246C22#198211yescharacteristic
247C22C2×#35911yescharacteristic
248C22#214311yescharacteristic
249C22C2×#36211yescharacteristic
250C22C2×#36211yescharacteristic
251C4—22no—
252C4—22no—
253C4—22no—
254C4—41no—
255C4—162no—
256C4—162no—
257C4#135014yescomplemented
258C4#150511yescharacteristic
259C4#150711yescharacteristic
260C4#196211yescharacteristic
261C4#196411yescharacteristic
262C2—42no—
263C2#971211yescharacteristic
264C2C2×#135011yescharacteristic
265C2#1171211yescharacteristic
266C2#1174811yescharacteristic
267C2#1175211yescharacteristic
268C2#1325111yescharacteristic
269C2#1325611yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24C2×SD16
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
257. C4#1350semidirect

Maximal subgroups

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

subgroupindex
#331722
#127832
C2×#13502
C4×#3412