Finite Group Databasebeta

← #142133

Subgroups — #142133

Order 512.

290 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 106 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#33257C211yesmaximal, characteristic
2#12898C211yesmaximal, characteristic
3#8273C211yesmaximal, characteristic
4#7275C212yesmaximal
5#7275C212yesmaximal
6#2864—21no—
7C2×C4⋊C16—21no—
8C2×C4⋊C16—21no—
9C2×C4⋊C16C2212yes—
10#1942C411yescharacteristic
11#1942C411yescharacteristic
12#1942C411yescharacteristic
13#1942C411yescharacteristic
14#1941—21no—
15#1941—21no—
16#1846C2211yescharacteristic
17#1846C2211yescharacteristic
18#1835—21no—
19#1767—21no—
20C42×C8C2211yescharacteristic
21C2×#334C2212yes—
22#1277—21no—
23C22×C16—22no—
24C2×#107C2×C411yescharacteristic
25C2×#107C2×C411yescharacteristic
26C2×#107C2×C411yescharacteristic
27C2×#107C2×C411yescharacteristic
28C2×#106—21no—
29C2×#106—21no—
30C2×C4⋊C8—22no—
31C2×#97—41no—
32C2×C4×C8C2×C411yescharacteristic
33C2×C4×C8C2×C411yescharacteristic
34C2×C4×C8D411yescharacteristic
35C2×C4×C8D411yescharacteristic
36C2×C4×C8D411yescharacteristic
37C2×C4×C8C2311yescharacteristic
38#389—21no—
39#389—21no—
40#389—21no—
41#389—21no—
42#387—41no—
43#387—41no—
44#387—41no—
45#387—41no—
46#382—21no—
47#382—21no—
48C43D411yescharacteristic
49C4⋊C16—24no—
50C4⋊C16—42no—
51C4⋊C16—42no—
52#340—21no—
53#340—21no—
54#340—21no—
55#340—21no—
56#336—41no—
57#334—22no—
58#334Q814yes—
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—21no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4—41no—
70C2×C4⋊C4—41no—
71C2×C4⋊C4—41no—
72C2×C42C22⋊C411yescharacteristic
73C2×C42C22⋊C411yescharacteristic
74C2×C42D811yescharacteristic
75C2×C42D811yescharacteristic
76C2×C42SD1611yescharacteristic
77C2×C42SD1611yescharacteristic
78C2×C42C2×D411yescharacteristic
79C2×C16—42no—
80C2×C16—42no—
81C2×C16—42no—
82#107—41no—
83#107—41no—
84#107—41no—
85#107—41no—
86#106—41no—
87#106—41no—
88#106—41no—
89#106—41no—
90C4⋊C8—24no—
91C4⋊C8—42no—
92#97—82no—
93C4×C8—21no—
94C4×C8—21no—
95C4×C8—21no—
96C4×C8—21no—
97C4×C8—22no—
98C4×C8—41no—
99C4×C8—41no—
100C4×C8C2×Q812yes—
101C4×C8#4112yes—
102#95—41no—
103#95—41no—
104#95—41no—
105#95—41no—
106#95—41no—
107#95—41no—
108#95—41no—
109#95—41no—
110C22×C4—41no—
111C22×C4—41no—
112C22×C4—41no—
113C22×C4—41no—
114C22×C4#10211yescharacteristic
115C22×C4#10211yescharacteristic
116C22×C4#10211yescharacteristic
117C22×C4#10211yescharacteristic
118C22×C4C2×C22⋊C411yescharacteristic
119C22×C4C2×D811yescharacteristic
120C22×C4C2×SD1611yescharacteristic
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—44no—
130C2×C8—82no—
131C2×C8—82no—
132C2×C8C4×Q812yes—
133C2×C8#12411yescharacteristic
134C2×C8C4⋊Q811yescharacteristic
135C2×C8AGL(1,ℤ/8)11yescharacteristic
136C2×C8AGL(1,ℤ/8)11yescharacteristic
137C2×C8#13711yescharacteristic
138C2×C8#13711yescharacteristic
139C4⋊C4—41no—
140C4⋊C4—41no—
141C4⋊C4—41no—
142C4⋊C4—41no—
143C4⋊C4—41no—
144C4⋊C4—41no—
145C4⋊C4—41no—
146C4⋊C4—41no—
147C4⋊C4—82no—
148C4⋊C4—82no—
149C4⋊C4—82no—
150C4⋊C4—82no—
151C42—22no—
152C42—22no—
153C42—21no—
154C42—21no—
155C42—21no—
156C42—21no—
157C42—41no—
158C42—41no—
159C42C22⋊Q812yes—
160C42#13712yes—
161C16—82no—
162C23C2×#10211yescharacteristic
163C2×C4—21no—
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—81no—
178C2×C4—81no—
179C2×C4—81no—
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#35912yes—
190C2×C4#36012yes—
191C2×C4#39111yescharacteristic
192C2×C4#41311yescharacteristic
193C2×C4#41311yescharacteristic
194C2×C4#41711yescharacteristic
195C2×C4#41711yescharacteristic
196C2×C4#41911yescharacteristic
197C2×C4#47412yes—
198C2×C4#47512yes—
199C8—22no—
200C8—41no—
201C8—41no—
202C8—84no—
203C8#49011yescharacteristic
204C8#49111yescharacteristic
205C8#50111yescharacteristic
206C8#50111yescharacteristic
207C22#154711yescharacteristic
208C22#154711yescharacteristic
209C22#198811yescharacteristic
210C22C2×#35911yescharacteristic
211C22#214311yescharacteristic
212C22C2×#36011yescharacteristic
213C22#214511yescharacteristic
214C4—22no—
215C4—22no—
216C4—22no—
217C4—22no—
218C4—41no—
219C4—41no—
220C4—161no—
221C4—161no—
222C4—161no—
223C4—161no—
224C4#155911yescharacteristic
225C4#155911yescharacteristic
226C4#199211yescharacteristic
227C4#199811yescharacteristic
228C2#990411yescharacteristic
229C2#1197711yescharacteristic
230C2#1197811yescharacteristic
231C2#1197911yescharacteristic
232C2#1198011yescharacteristic
233C2#1327411yescharacteristic
234C2#1327811yescharacteristic
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 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#332572
#128982
#82732
#72752