Finite Group Databasebeta

← #125056

Subgroups — #125056

Order 512.

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

normal subgroups only

This group has 304 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
1C4×#499C211yesmaximal, characteristic
2#12879C211yesmaximal, characteristic
3#8784C211yesmaximal, characteristic
4#7121C212yesmaximal
5#7120C212yesmaximal
6C2×#499C411yescharacteristic
7C4×#125—21no—
8C2×C4⋊C16—21no—
9#1991—21no—
10#1942C2211yescharacteristic
11#1942C2211yescharacteristic
12#1941—21no—
13#1846C411yescharacteristic
14#1842—21no—
15C4×#107C411yescharacteristic
16C4×#107C411yescharacteristic
17C4×#106—21no—
18C42×C8C2211yescharacteristic
19#1374—21no—
20#1374C2212yes—
21#1277—21no—
22#1277C2212yes—
23C2×#125—21no—
24C22×C16—22no—
25#499—22no—
26#499—22no—
27C2×#107C2×C411yescharacteristic
28C2×#107C2×C411yescharacteristic
29C2×#107C2×C411yescharacteristic
30C2×#107C2×C411yescharacteristic
31C2×#106—21no—
32C2×#106—21no—
33C2×C4⋊C8—22no—
34C2×C4⋊C8—41no—
35C2×C4⋊C8—41no—
36C2×C4×C8C2×C411yescharacteristic
37C2×C4×C8C2×C411yescharacteristic
38C2×C4×C8D411yescharacteristic
39C2×C4×C8D411yescharacteristic
40C2×C4×C8D411yescharacteristic
41C2×C4×C8C2311yescharacteristic
42#389—21no—
43#389—21no—
44#387—41no—
45#387—41no—
46#382—21no—
47C4×C4⋊C4—21no—
48C4×C4⋊C4—21no—
49C4×C4⋊C4—41no—
50C4×C4⋊C4—41no—
51C43D411yescharacteristic
52C4⋊C16—42no—
53#340—21no—
54#340—21no—
55#340—21no—
56#340—21no—
57C22×C8—42no—
58C22×C8C22⋊C411yescharacteristic
59C22×C8C22⋊C411yescharacteristic
60C22×C8C22×C411yescharacteristic
61C22×C8C2×D411yescharacteristic
62#125—24no—
63#125—42no—
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×C42—41no—
73C2×C42—41no—
74C2×C42C22⋊C411yescharacteristic
75C2×C42C22⋊C411yescharacteristic
76C2×C42D811yescharacteristic
77C2×C42D811yescharacteristic
78C2×C42SD1611yescharacteristic
79C2×C42SD1611yescharacteristic
80C2×C42C2×D411yescharacteristic
81C2×C16—42no—
82C2×C16—42no—
83C2×C16—42no—
84#107—21no—
85#107—21no—
86#107—41no—
87#107—41no—
88#107M1612yes—
89#107M1612yes—
90#106—24no—
91#106—42no—
92C4⋊C8—42no—
93C4⋊C8—42no—
94C4⋊C8—82no—
95C4⋊C8—82no—
96C4×C8—21no—
97C4×C8—21no—
98C4×C8—22no—
99C4×C8—41no—
100C4×C8—41no—
101C4×C8—41no—
102C4×C8SD1612yes—
103C4×C8Q1612yes—
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—21no—
130C2×C8—82no—
131C2×C8—82no—
132C2×C8—82no—
133C2×C8#10312yes—
134C2×C8C2×M1611yescharacteristic
135C2×C8C2×M1611yescharacteristic
136C2×C8C2×SD1611yescharacteristic
137C2×C8C2×Q1611yescharacteristic
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—44no—
141C4⋊C4—44no—
142C4⋊C4—41no—
143C4⋊C4—41no—
144C4⋊C4—41no—
145C4⋊C4—41no—
146C4⋊C4—41no—
147C4⋊C4—41no—
148C4⋊C4—82no—
149C4⋊C4—81no—
150C4⋊C4—81no—
151C42—22no—
152C42—22no—
153C42—21no—
154C42—21no—
155C42—21no—
156C42—21no—
157C42—21no—
158C42—21no—
159C42—41no—
160C42—82no—
161C42—82no—
162C42#10112yes—
163C42#10312yes—
164C16—82no—
165C23C2×#10211yescharacteristic
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—42no—
180C2×C4—42no—
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#33212yes—
192C2×C4#33212yes—
193C2×C4#35912yes—
194C2×C4#36012yes—
195C2×C4#40711yescharacteristic
196C2×C4#41011yescharacteristic
197C2×C4C2×#10111yescharacteristic
198C2×C4C2×#10311yescharacteristic
199C2×C4#42111yescharacteristic
200C2×C4#42111yescharacteristic
201C2×C4#50911yescharacteristic
202C2×C4#51011yescharacteristic
203C8—21no—
204C8—21no—
205C8—22no—
206C8—21no—
207C8—21no—
208C8—41no—
209C8—162no—
210C22#148611yescharacteristic
211C22#151711yescharacteristic
212C22C2×#33211yescharacteristic
213C22#213911yescharacteristic
214C22C2×#35911yescharacteristic
215C22C2×#36011yescharacteristic
216C22#214811yescharacteristic
217C4—21no—
218C4—21no—
219C4—22no—
220C4—41no—
221C4—41no—
222C4—41no—
223C4—82no—
224C4—82no—
225C4—161no—
226C4—161no—
227C4#136314yes—
228C4#146711yescharacteristic
229C4#148711yescharacteristic
230C4#151811yescharacteristic
231C4#153711yescharacteristic
232C2#765211yescharacteristic
233C2#1166011yescharacteristic
234C2#1166611yescharacteristic
235C2#1186011yescharacteristic
236C2#1186111yescharacteristic
237C2C2×#136311yescharacteristic
238C2#1194211yescharacteristic
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
C4×#4992
#128792
#87842
#71212
#71202