Finite Group Databasebeta

← #124679

Subgroups — #124679

Order 512.

321 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 321 conjugacy classes of subgroups — too many to draw. Only its 68 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#21271C211yesmaximal, characteristic
2#12896C211yesmaximal, characteristic, complemented
3#12595C211yesmaximal, characteristic, complemented
4#7124C212yesmaximal, complemented
5#7105C212yesmaximal
6C2×#432—21no—
7C42×D4C411yescharacteristic
8C2×#347—21no—
9#1955C411yescharacteristic
10#1955C2211yescharacteristic
11#1940C411yescharacteristic
12#1940C2211yescharacteristic
13#1923—21no—
14#1923—21no—
15#1767C411yescharacteristic
16#1749—21no—
17C4×#97C2211yescharacteristic
18#1377C2212yes—
19#1377C2212yes—
20#1376—21no—
21#1376—21no—
22#1375—21no—
23C2×C4×D4—21no—
24C2×C4×D4C2×C411yescharacteristic
25C22×C42—21no—
26C2×M32—22no—
27C2×M32—22no—
28#432—44no—
29C2×C4⋊C8—21no—
30C2×C4⋊C8—21no—
31C2×C4⋊C8C2×C411yescharacteristic
32C2×C4⋊C8C2×C411yescharacteristic
33C2×C22⋊C8—21no—
34C2×C22⋊C8—21no—
35C2×C22⋊C8—21no—
36C2×C22⋊C8—21no—
37C2×#97—21no—
38C2×#97C2×C411yescharacteristic
39C2×#97D411yescharacteristic
40C2×#97D411yescharacteristic
41C2×#97C2311yescharacteristic
42C2×C4×C8D411yescharacteristic
43C2×C4×C8D411yescharacteristic
44C4×C4⋊C4—21no—
45C4×C4⋊C4C2×C411yescharacteristic
46C4×C22⋊C4—21no—
47C4×C22⋊C4—21no—
48C4×C22⋊C4—21no—
49C43C2×C411yescharacteristic
50#347—24no—
51#336—21no—
52#336—21no—
53#336—21no—
54C22×D4M1611yescharacteristic
55C23×C4—21no—
56C23×C4—21no—
57C23×C4—21no—
58C22×C8—21no—
59C22×C8—21no—
60C22×C8C22⋊C411yescharacteristic
61C22×C8C22⋊C411yescharacteristic
62C22×C8C2×D411yescharacteristic
63C22×C8C2×D411yescharacteristic
64C4×D4—41no—
65C4×D4—41no—
66C4×D4—44no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4M1611yescharacteristic
69C2×C22⋊C4—21no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—21no—
72C2×C42—21no—
73C2×C42—21no—
74C2×C42—21no—
75C2×C42—21no—
76C2×C42—21no—
77C2×C42—42no—
78C2×C42—42no—
79C2×C42C22⋊C411yescharacteristic
80C2×C42C22⋊C411yescharacteristic
81C2×C42C22×C411yescharacteristic
82M32—24no—
83M32—42no—
84M32—42no—
85M32—42no—
86C2×C16—42no—
87C2×C16—42no—
88C4⋊C8—41no—
89C4⋊C8—41no—
90C4⋊C8—42no—
91C4⋊C8—42no—
92C22⋊C8—42no—
93C22⋊C8—42no—
94C22⋊C8—42no—
95C22⋊C8—42no—
96#97—22no—
97#97—22no—
98#97—41no—
99#97—41no—
100#97—42no—
101C4×C8—22no—
102C4×C8—41no—
103#95—21no—
104#95—21no—
105#95M1611yescharacteristic
106#95M1611yescharacteristic
107C24—21no—
108C2×D4—41no—
109C2×D4—42no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4—41no—
120C22×C4—41no—
121C22×C4—41no—
122C22×C4—41no—
123C22×C4—41no—
124C22×C4—41no—
125C22×C4—81no—
126C22×C4—81no—
127C22×C4C2×C22⋊C411yescharacteristic
128C22×C4C2×M1611yescharacteristic
129C22×C4C2×M1611yescharacteristic
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—41no—
141C2×C8—41no—
142C2×C8—41no—
143C2×C8—41no—
144C2×C8—41no—
145C2×C8—41no—
146C2×C8#13512yes—
147C2×C8AGL(1,ℤ/8)12yes—
148C4⋊C4—41no—
149C4⋊C4—42no—
150C22⋊C4—24no—
151C22⋊C4—42no—
152C22⋊C4—42no—
153C42—21no—
154C42—21no—
155C42—21no—
156C42—21no—
157C42—21no—
158C42—42no—
159C42—42no—
160C42—42no—
161C42—41no—
162C42—41no—
163C42—81no—
164C42#10412yes—
165C16—44no—
166C16—82no—
167C23—22no—
168C23—41no—
169C23—41no—
170C23—41no—
171C23—41no—
172C23—41no—
173C23—41no—
174C23#40711yescharacteristic
175D4—42no—
176D4—42no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—84no—
197C2×C4—84no—
198C2×C4—81no—
199C2×C4—81no—
200C2×C4—81no—
201C2×C4—81no—
202C2×C4#41011yescharacteristic
203C2×C4#41311yescharacteristic
204C2×C4#41611yescharacteristic
205C2×C4#41811yescharacteristic
206C2×C4C2×#10411yescharacteristic
207C2×C4#42111yescharacteristic
208C8—22no—
209C8—22no—
210C8—41no—
211C8—41no—
212C8—81no—
213C8—81no—
214C22—21no—
215C22—21no—
216C22—42no—
217C22—42no—
218C22—42no—
219C22—81no—
220C22—81no—
221C22—81no—
222C22—81no—
223C22#146411yescharacteristic
224C22#148811yescharacteristic
225C22#149011yescharacteristic
226C4—42no—
227C4—42no—
228C4—42no—
229C4—41no—
230C4—41no—
231C4—81no—
232C4—81no—
233C4#150011yescharacteristic
234C4#151411yescharacteristic
235C4#152911yescharacteristic
236C4#155311yescharacteristic
237C2—21no—
238C2—21no—
239C2—82no—
240C2#760811yescharacteristic
241C2#1152111yescharacteristic
242C2#1152311yescharacteristic
subgroupG/H
centerC2×C4#410
commutatorC2×C42C22×C4
Frattini, cosocleC2×#97C23
socleC22#1464
prime radicalC22×D4M16
LCSG ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#97 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12896C2semidirect
3. #12595C2semidirect
4. #7124C2semidirect

Maximal subgroups

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

subgroupindex
#212712
#128962
#125952
#71242
#71052