Finite Group Databasebeta

← #140579

Subgroups — #140579

Order 512.

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

normal subgroups only

This group has 430 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
1C4×#495C211yesmaximal, characteristic
2#12591C211yesmaximal, characteristic, complemented
3C2×#1334C212yesmaximal, complemented
4C2×#1333C212yesmaximal
5#8807C211yesmaximal, characteristic, complemented
6C2×#495C411yescharacteristic
7C2×C4×Q16C2211yescharacteristic
8C2×C4×SD16—21no—
9C2×C4×D8C2211yescharacteristic
10C4×#124—21no—
11C2×#346C2212yes—
12#1970C411yescharacteristic
13#1969C411yescharacteristic
14#1968—21no—
15#1958—21no—
16#1846C411yescharacteristic
17#1831—21no—
18C42×C8C2211yescharacteristic
19C2×#335—21no—
20C2×#334C2212yes—
21#1374—21no—
22#1374—21no—
23#1334—24no—
24#1333—24no—
25C22×Q16C2×C411yescharacteristic
26C22×SD16—21no—
27C22×D8C2×C411yescharacteristic
28C2×#124—21no—
29C2×C4×Q8—21no—
30C2×C4×D4—21no—
31C22×C16—22no—
32#495—22no—
33#495—22no—
34C4×Q16—22no—
35C4×Q16—22no—
36C4×SD16—42no—
37C4×SD16—42no—
38C4×D8—22no—
39C4×D8—22no—
40C2×#107C2×C411yescharacteristic
41C2×#107C2×C411yescharacteristic
42C2×#106—21no—
43C2×C4⋊C8—22no—
44C2×#103—21no—
45C2×#103—21no—
46C2×#102—21no—
47C2×#102—21no—
48C2×C4×C8C2×C411yescharacteristic
49C2×C4×C8C2×C411yescharacteristic
50C2×C4×C8D411yescharacteristic
51C2×C4×C8D411yescharacteristic
52C2×C4×C8D411yescharacteristic
53C2×C4×C8C2311yescharacteristic
54#391—21no—
55#390—21no—
56#388—42no—
57#382—21no—
58C4×C22⋊C4—42no—
59C43D411yescharacteristic
60#346—22no—
61#346D414yes—
62#336—42no—
63#335—42no—
64#334—22no—
65#334—22no—
66C22×Q8—21no—
67C22×D4—21no—
68C23×C4—41no—
69C2×Q16—22no—
70C2×Q16—21no—
71C2×Q16—22no—
72C2×Q16M1612yes—
73C2×SD16—24no—
74C2×SD16—42no—
75C2×SD16—42no—
76C2×D8—22no—
77C2×D8—21no—
78C2×D8—22no—
79C2×D8M1612yes—
80C22×C8—42no—
81C22×C8C22⋊C411yescharacteristic
82C22×C8C22⋊C411yescharacteristic
83C22×C8C22×C411yescharacteristic
84C22×C8C2×D411yescharacteristic
85#124—24no—
86#124—42no—
87C4×Q8—41no—
88C4×Q8—41no—
89C4×Q8—41no—
90C4×Q8—41no—
91C4×D4—41no—
92C4×D4—41no—
93C4×D4—41no—
94C4×D4—41no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—41no—
98C2×C22⋊C4—42no—
99C2×C22⋊C4—42no—
100C2×C22⋊C4—41no—
101C2×C42—41no—
102C2×C42C22⋊C411yescharacteristic
103C2×C42C22⋊C411yescharacteristic
104C2×C42D812yes—
105C2×C42SD1612yes—
106C2×C42C2×D411yescharacteristic
107C2×C16—24no—
108C2×C16—42no—
109C2×C16—42no—
110#107—22no—
111#107—22no—
112#106—42no—
113C4⋊C8—42no—
114C4⋊C8—42no—
115#103—42no—
116#103—42no—
117#102—42no—
118#102—42no—
119C4×C8—21no—
120C4×C8—21no—
121C4×C8—22no—
122C4×C8—41no—
123C4×C8—41no—
124C4×C8D812yes—
125C4×C8SD1612yes—
126C4×C8C2×D412yes—
127C4×C8#4112yes—
128#95—42no—
129#95—42no—
130C24—41no—
131C2×Q8—22no—
132C2×Q8—41no—
133C2×Q8—41no—
134C2×Q8—41no—
135C2×Q8—41no—
136C2×Q8—41no—
137C2×D4—22no—
138C2×D4—41no—
139C2×D4—41no—
140C2×D4—41no—
141C2×D4—41no—
142C2×D4—41no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—84no—
147C22×C4—81no—
148C22×C4—81no—
149C22×C4#10212yes—
150C22×C4#10212yes—
151C22×C4C2×C22⋊C411yescharacteristic
152C22×C4C2×D811yescharacteristic
153C22×C4C2×SD1611yescharacteristic
154Q16—24no—
155Q16—42no—
156SD16—48no—
157D8—24no—
158D8—42no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—84no—
166C2×C8—82no—
167C2×C8#10212yes—
168C2×C8#10212yes—
169C2×C8C4×D412yes—
170C2×C8#12112yes—
171C2×C8C2×M1611yescharacteristic
172C2×C8C2×M1611yescharacteristic
173C2×C8#13112yes—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—81no—
179C4⋊C4—81no—
180C22⋊C4—48no—
181C22⋊C4—84no—
182C22⋊C4—81no—
183C22⋊C4—81no—
184C42—21no—
185C42—21no—
186C42—21no—
187C42—21no—
188C42—42no—
189C42—42no—
190C42—81no—
191C42—81no—
192C42#12411yescharacteristic
193C42#12711yescharacteristic
194C42C2×D811yescharacteristic
195C42C2×SD1611yescharacteristic
196C16—44no—
197C23—42no—
198C23—82no—
199C23—82no—
200C23—81no—
201C23—81no—
202C23C2×#10211yescharacteristic
203Q8—42no—
204Q8—42no—
205Q8—81no—
206Q8—81no—
207D4—42no—
208D4—42no—
209D4—81no—
210D4—81no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—42no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—162no—
227C2×C4—162no—
228C2×C4#39011yescharacteristic
229C2×C4#40711yescharacteristic
230C2×C4#40811yescharacteristic
231C2×C4C2×#10211yescharacteristic
232C2×C4#42111yescharacteristic
233C2×C4#42111yescharacteristic
234C2×C4#48611yescharacteristic
235C2×C4#48611yescharacteristic
236C2×C4#49211yescharacteristic
237C2×C4#49311yescharacteristic
238C8—22no—
239C8—22no—
240C8—22no—
241C8—162no—
242C8#43512yes—
243C8#43512yes—
244C22—84no—
245C22—82no—
246C22—162no—
247C22—161no—
248C22—161no—
249C22#148011yescharacteristic
250C22#148611yescharacteristic
251C22#195711yescharacteristic
252C22#214512yes—
253C22#214812yes—
254C4—21no—
255C4—21no—
256C4—21no—
257C4—21no—
258C4—42no—
259C4—42no—
260C4—82no—
261C4—161no—
262C4—161no—
263C4#148011yescharacteristic
264C4#148611yescharacteristic
265C4#195511yescharacteristic
266C4#196311yescharacteristic
267C2—162no—
268C2#964411yescharacteristic
269C2#1165612yes—
270C2#1166512yes—
271C2#1321111yescharacteristic
272C2#1321411yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC22×D8C2×C4
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12591C2semidirect
3. C2×#1334C2semidirect
5. #8807C2semidirect

Maximal subgroups

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

subgroupindex
C4×#4952
#125912
C2×#13342
C2×#13332
#88072