Finite Group Databasebeta

← #125631

Subgroups — #125631

Order 512.

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

normal subgroups only

This group has 270 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#33267C211yesmaximal, characteristic
2#12879C211yesmaximal, characteristic
3#8785C211yesmaximal, characteristic
4#7121C212yesmaximal
5#7113C212yesmaximal
6#2863—21no—
7C2×C4⋊C16—21no—
8#1969C411yescharacteristic
9#1969C411yescharacteristic
10#1969C2211yescharacteristic
11#1942C411yescharacteristic
12#1942C411yescharacteristic
13#1942C2211yescharacteristic
14#1933—21no—
15#1933—21no—
16#1933—21no—
17#1831—21no—
18C42×C8C2211yescharacteristic
19#1374—21no—
20#1374C2212yes—
21#1277—21no—
22#1277C2212yes—
23C22×Q16C2×C411yescharacteristic
24C22×C16—22no—
25C2×#107C2×C411yescharacteristic
26C2×#107C2×C411yescharacteristic
27C2×#107C2×C411yescharacteristic
28C2×C4⋊C8—22no—
29C2×#103—21no—
30C2×#103—21no—
31C2×#103—21no—
32C2×C4×C8C2×C411yescharacteristic
33C2×C4×C8C2×C411yescharacteristic
34C2×C4×C8D411yescharacteristic
35C2×C4×C8D411yescharacteristic
36C2×C4×C8D411yescharacteristic
37C2×C4×C8C2311yescharacteristic
38#391—21no—
39#391—21no—
40#391—21no—
41#389—21no—
42#389—21no—
43#389—21no—
44#387—41no—
45#387—41no—
46#387—41no—
47#383—41no—
48C43D411yescharacteristic
49C4⋊C16—42no—
50#340—21no—
51#340—21no—
52#340—21no—
53#336—41no—
54#336—41no—
55C22×Q8—21no—
56C2×Q16—22no—
57C2×Q16—41no—
58C2×Q16—41no—
59C22×C8—42no—
60C22×C8C22⋊C411yescharacteristic
61C22×C8C22⋊C411yescharacteristic
62C22×C8C22×C411yescharacteristic
63C22×C8C2×D411yescharacteristic
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—41no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4—41no—
70C2×C42C22⋊C411yescharacteristic
71C2×C42C22⋊C411yescharacteristic
72C2×C42D811yescharacteristic
73C2×C42D811yescharacteristic
74C2×C42SD1611yescharacteristic
75C2×C42SD1611yescharacteristic
76C2×C42C2×D411yescharacteristic
77C2×C16—42no—
78C2×C16—42no—
79C2×C16—42no—
80#107—22no—
81#107—41no—
82#107—41no—
83C4⋊C8—42no—
84C4⋊C8—42no—
85#103—42no—
86#103—42no—
87#103—41no—
88#103—41no—
89C4×C8—22no—
90C4×C8—21no—
91C4×C8—21no—
92C4×C8—41no—
93C4×C8—41no—
94C4×C8—41no—
95C4×C8—41no—
96#95—41no—
97#95—41no—
98#95—41no—
99#95—41no—
100#95—41no—
101#95—41no—
102#95—41no—
103#95—41no—
104#95—41no—
105C2×Q8—41no—
106C2×Q8—41no—
107C2×Q8—41no—
108C2×Q8—41no—
109C2×Q8—41no—
110C2×Q8—41no—
111C22×C4—41no—
112C22×C4—41no—
113C22×C4—41no—
114C22×C4—41no—
115C22×C4#10211yescharacteristic
116C22×C4#10211yescharacteristic
117C22×C4#10211yescharacteristic
118C22×C4#10211yescharacteristic
119C22×C4C2×C22⋊C411yescharacteristic
120C22×C4C2×D811yescharacteristic
121C22×C4C2×SD1611yescharacteristic
122Q16—82no—
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—82no—
134C2×C8—82no—
135C2×C8—82no—
136C2×C8#13112yes—
137C2×C8AGL(1,ℤ/8)11yescharacteristic
138C2×C8#13711yescharacteristic
139C4⋊C4—41no—
140C4⋊C4—41no—
141C4⋊C4—41no—
142C4⋊C4—41no—
143C4⋊C4—41no—
144C4⋊C4—41no—
145C4⋊C4—82no—
146C4⋊C4—82no—
147C4⋊C4—82no—
148C42—22no—
149C42—21no—
150C42—21no—
151C42—21no—
152C42—21no—
153C42—21no—
154C42—21no—
155C42—22no—
156C42—41no—
157C42—41no—
158C16—82no—
159C23C2×#10211yescharacteristic
160Q8—81no—
161Q8—81no—
162Q8—81no—
163Q8—81no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—81no—
181C2×C4—81no—
182C2×C4—81no—
183C2×C4—81no—
184C2×C4—81no—
185C2×C4—81no—
186C2×C4—81no—
187C2×C4—81no—
188C2×C4—81no—
189C2×C4—81no—
190C2×C4—81no—
191C2×C4—81no—
192C2×C4#35912yes—
193C2×C4#40811yescharacteristic
194C2×C4#41011yescharacteristic
195C2×C4#41311yescharacteristic
196C2×C4#41711yescharacteristic
197C2×C4#42111yescharacteristic
198C2×C4#42111yescharacteristic
199C2×C4#51012yes—
200C8—22no—
201C8—21no—
202C8—21no—
203C8—41no—
204C8—41no—
205C8—162no—
206C22#148011yescharacteristic
207C22#151711yescharacteristic
208C22#154711yescharacteristic
209C22#214111yescharacteristic
210C22C2×#35911yescharacteristic
211C22#214511yescharacteristic
212C22#214811yescharacteristic
213C4—22no—
214C4—22no—
215C4—41no—
216C4—41no—
217C4—41no—
218C4—41no—
219C4—161no—
220C4—161no—
221C4—161no—
222C4—161no—
223C4#147311yescharacteristic
224C4#149511yescharacteristic
225C4#151211yescharacteristic
226C4#153511yescharacteristic
227C2#773211yescharacteristic
228C2#1165011yescharacteristic
229C2#1165611yescharacteristic
230C2#1185911yescharacteristic
231C2#1186111yescharacteristic
232C2#1197811yescharacteristic
233C2#1198711yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#332672
#128792
#87852
#71212
#71132