Finite Group Databasebeta

← #141064

Subgroups — #141064

Order 512.

393 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 393 conjugacy classes of subgroups — too many to draw. Only its 118 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#33240C211yesmaximal, characteristic
2C4×C4⋊C16C211yesmaximal, characteristic, complemented
3C2×#1363C212yesmaximal, complemented
4C2×#1360C212yesmaximal
5#8805C211yesmaximal, characteristic, complemented
6C2×#499C2211yescharacteristic
7C2×#495C2211yescharacteristic
8C2×#488—21no—
9#2861—21no—
10C2×C4⋊C16C2212yes—
11C2×C4×C16—21no—
12#1970C411yescharacteristic
13#1970C411yescharacteristic
14#1969C411yescharacteristic
15#1969C411yescharacteristic
16#1968—21no—
17#1968—21no—
18#1842—21no—
19C42×C8C2211yescharacteristic
20C2×#334—21no—
21C2×#334C2212yes—
22#1374—21no—
23#1363—24no—
24#1360—24no—
25C22×Q16C2×C411yescharacteristic
26C22×SD16—21no—
27C22×D8C2×C411yescharacteristic
28C2×#125—21no—
29C2×#124—21no—
30C22×C16—22no—
31#499—22no—
32#499—22no—
33#495—22no—
34#495—22no—
35#488—42no—
36#488—42no—
37C2×#107C2×C411yescharacteristic
38C2×#107C2×C411yescharacteristic
39C2×#106—21no—
40C2×C4⋊C8—22no—
41C2×C4⋊C8—42no—
42C2×#103—21no—
43C2×#103—21no—
44C2×#102—21no—
45C2×#102—21no—
46C2×C4×C8C2×C411yescharacteristic
47C2×C4×C8C2×C411yescharacteristic
48C2×C4×C8D411yescharacteristic
49C2×C4×C8D411yescharacteristic
50C2×C4×C8D411yescharacteristic
51C2×C4×C8C2311yescharacteristic
52#391—21no—
53#391—21no—
54#390—21no—
55#390—21no—
56#388—42no—
57#388—42no—
58C43D411yescharacteristic
59C4⋊C16—22no—
60C4⋊C16D414yes—
61C4×C16—42no—
62#334—22no—
63#334—22no—
64#334—42no—
65C22×Q8—21no—
66C22×D4—21no—
67C2×Q16—22no—
68C2×Q16—41no—
69C2×Q16—41no—
70C2×SD16—42no—
71C2×SD16—42no—
72C2×SD16—41no—
73C2×SD16—41no—
74C2×D8—22no—
75C2×D8—41no—
76C2×D8—41no—
77C22×C8—42no—
78C22×C8C22⋊C411yescharacteristic
79C22×C8C22⋊C411yescharacteristic
80C22×C8C22×C411yescharacteristic
81C22×C8C2×D411yescharacteristic
82#125—41no—
83#125—41no—
84#125—41no—
85#125—41no—
86#124—41no—
87#124—41no—
88#124—41no—
89#124—41no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C22⋊C4—42no—
95C2×C22⋊C4—42no—
96C2×C22⋊C4—41no—
97C2×C22⋊C4—41no—
98C2×C42C22⋊C411yescharacteristic
99C2×C42C22⋊C411yescharacteristic
100C2×C42D812yes—
101C2×C42SD1612yes—
102C2×C42C2×D411yescharacteristic
103C2×C16—24no—
104C2×C16—42no—
105C2×C16—42no—
106#107—22no—
107#107—22no—
108#106—42no—
109C4⋊C8—42no—
110C4⋊C8—42no—
111C4⋊C8—84no—
112#103—42no—
113#103—42no—
114#102—42no—
115#102—42no—
116C4×C8—22no—
117C4×C8—21no—
118C4×C8—21no—
119C4×C8—41no—
120C4×C8—41no—
121C4×C8D812yes—
122C4×C8SD1612yes—
123C4×C8C2×D412yes—
124C4×C8#4112yes—
125#95—42no—
126#95—42no—
127C24—41no—
128C2×Q8—41no—
129C2×Q8—41no—
130C2×Q8—41no—
131C2×Q8—41no—
132C2×Q8—41no—
133C2×Q8—41no—
134C2×D4—41no—
135C2×D4—41no—
136C2×D4—41no—
137C2×D4—41no—
138C2×D4—41no—
139C2×D4—41no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4#10212yes—
144C22×C4#10212yes—
145C22×C4C2×C22⋊C411yescharacteristic
146C22×C4C2×D811yescharacteristic
147C22×C4C2×SD1611yescharacteristic
148Q16—82no—
149SD16—82no—
150SD16—82no—
151D8—82no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—84no—
159C2×C8—82no—
160C2×C8#10212yes—
161C2×C8#10212yes—
162C2×C8C4×D412yes—
163C2×C8#12112yes—
164C2×C8#13512yes—
165C2×C8AGL(1,ℤ/8)11yescharacteristic
166C2×C8#13711yescharacteristic
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—81no—
172C4⋊C4—81no—
173C4⋊C4—81no—
174C4⋊C4—81no—
175C22⋊C4—84no—
176C22⋊C4—84no—
177C22⋊C4—81no—
178C22⋊C4—81no—
179C22⋊C4—81no—
180C22⋊C4—81no—
181C42—24no—
182C42—24no—
183C42—21no—
184C42—21no—
185C42—21no—
186C42—21no—
187C42#12411yescharacteristic
188C42#12711yescharacteristic
189C42C2×D811yescharacteristic
190C42C2×SD1611yescharacteristic
191C16—44no—
192C23—81no—
193C23—81no—
194C23—82no—
195C23—82no—
196C23—81no—
197C23C2×#10211yescharacteristic
198Q8—81no—
199Q8—81no—
200Q8—81no—
201Q8—81no—
202D4—81no—
203D4—81no—
204D4—81no—
205D4—81no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4#35914yes—
220C2×C4#36014yes—
221C2×C4#39011yescharacteristic
222C2×C4#41311yescharacteristic
223C2×C4#41311yescharacteristic
224C2×C4C2×#10211yescharacteristic
225C2×C4#41611yescharacteristic
226C2×C4#41711yescharacteristic
227C2×C4#48611yescharacteristic
228C2×C4#48611yescharacteristic
229C2×C4#49211yescharacteristic
230C2×C4#49311yescharacteristic
231C8—22no—
232C8—22no—
233C8—22no—
234C8—162no—
235C8#46312yes—
236C8#46412yes—
237C22—161no—
238C22—161no—
239C22—162no—
240C22—162no—
241C22—161no—
242C22#154011yescharacteristic
243C22#154711yescharacteristic
244C22#195711yescharacteristic
245C22C2×#35912yes—
246C22C2×#36012yes—
247C4—21no—
248C4—21no—
249C4—21no—
250C4—21no—
251C4—24no—
252C4—24no—
253C4—161no—
254C4—161no—
255C4—161no—
256C4#154011yescharacteristic
257C4#154711yescharacteristic
258C4#196711yescharacteristic
259C4#198311yescharacteristic
260C2—321no—
261C2#969811yescharacteristic
262C2#1194412yes—
263C2#1197712yes—
264C2#1320811yescharacteristic
265C2#1321011yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC22×D8C2×C4
LCSG ▷ C22×C8 ▷ C2×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. C4×C4⋊C16C2semidirect
3. C2×#1363C2semidirect
5. #8805C2semidirect

Maximal subgroups

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

subgroupindex
#332402
C4×C4⋊C162
C2×#13632
C2×#13602
#88052