Finite Group Databasebeta

← #124849

Subgroups — #124849

Order 512.

343 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 343 conjugacy classes of subgroups — too many to draw. Only its 70 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#21265C211yesmaximal, characteristic
2#12659C211yesmaximal, characteristic, complemented
3#12595C211yesmaximal, characteristic, complemented
4#7124C212yesmaximal, complemented
5#7105C212yesmaximal
6C2×#431—21no—
7C42×D4C411yescharacteristic
8C2×#364—21no—
9C2×#347—21no—
10C2×#347—21no—
11#1955C411yescharacteristic
12#1955C2211yescharacteristic
13#1940C411yescharacteristic
14#1940C2211yescharacteristic
15#1924—21no—
16#1924—21no—
17C4×C4⋊C8C411yescharacteristic
18C4×C22⋊C8—21no—
19C4×#97C2211yescharacteristic
20#1377C2212yes—
21#1377C2212yes—
22#1376—21no—
23C2×C4×D4—21no—
24C2×C4×D4C2×C411yescharacteristic
25C22×C42—21no—
26C2×M32—22no—
27C2×M32—22no—
28#431—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×#97C2×C411yescharacteristic
38C2×#97D411yescharacteristic
39C2×#97D411yescharacteristic
40C2×#97C2311yescharacteristic
41C2×C4×C8—21no—
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#364—24no—
51#347—24no—
52#347—42no—
53#336—21no—
54#336—21no—
55#336—21no—
56C22×D4M1611yescharacteristic
57C23×C4—21no—
58C23×C4—21no—
59C23×C4—21no—
60C22×C8—21no—
61C22×C8—21no—
62C22×C8C22⋊C411yescharacteristic
63C22×C8C22⋊C411yescharacteristic
64C22×C8C2×D411yescharacteristic
65C22×C8C2×D411yescharacteristic
66C4×D4—41no—
67C4×D4—41no—
68C4×D4—44no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4M1611yescharacteristic
71C2×C22⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—21no—
74C2×C42—21no—
75C2×C42—21no—
76C2×C42—21no—
77C2×C42—21no—
78C2×C42—21no—
79C2×C42—42no—
80C2×C42—42no—
81C2×C42C22⋊C411yescharacteristic
82C2×C42C22⋊C411yescharacteristic
83C2×C42C22×C411yescharacteristic
84M32—24no—
85M32—24no—
86M32—42no—
87M32—42no—
88C2×C16—42no—
89C2×C16—42no—
90C4⋊C8—41no—
91C4⋊C8—41no—
92C4⋊C8—42no—
93C4⋊C8—42no—
94C22⋊C8—42no—
95C22⋊C8—42no—
96C22⋊C8—41no—
97C22⋊C8—42no—
98C22⋊C8—41no—
99#97—22no—
100#97—21no—
101#97—21no—
102#97—41no—
103#97—41no—
104C4×C8—22no—
105C4×C8—22no—
106C4×C8—42no—
107#95—21no—
108#95—21no—
109#95M1611yescharacteristic
110#95M1611yescharacteristic
111C24—21no—
112C2×D4—41no—
113C2×D4—42no—
114C22×C4—21no—
115C22×C4—24no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4—41no—
126C22×C4—41no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4C2×C22⋊C411yescharacteristic
132C22×C4C2×M1611yescharacteristic
133C22×C4C2×M1611yescharacteristic
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—41no—
144C2×C8—41no—
145C2×C8—41no—
146C2×C8—41no—
147C2×C8—41no—
148C2×C8—41no—
149C2×C8#10412yes—
150C2×C8AGL(1,ℤ/8)11yescharacteristic
151C2×C8AGL(1,ℤ/8)12yes—
152C2×C8#13711yescharacteristic
153C4⋊C4—41no—
154C4⋊C4—42no—
155C22⋊C4—42no—
156C22⋊C4—42no—
157C22⋊C4—42no—
158C42—21no—
159C42—21no—
160C42—21no—
161C42—21no—
162C42—21no—
163C42—42no—
164C42—42no—
165C42—42no—
166C42—42no—
167C42—41no—
168C42—41no—
169C42#10412yes—
170C16—44no—
171C16—44no—
172C23—22no—
173C23—41no—
174C23—41no—
175C23—41no—
176C23—41no—
177C23—41no—
178C23—41no—
179C23#40711yescharacteristic
180D4—42no—
181D4—42no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—48no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—84no—
203C2×C4—81no—
204C2×C4—81no—
205C2×C4—81no—
206C2×C4—81no—
207C2×C4#41011yescharacteristic
208C2×C4#41311yescharacteristic
209C2×C4#41711yescharacteristic
210C2×C4#41811yescharacteristic
211C2×C4C2×#10411yescharacteristic
212C2×C4C2×#10411yescharacteristic
213C8—22no—
214C8—21no—
215C8—21no—
216C8—22no—
217C8—41no—
218C8—81no—
219C8—81no—
220C22—21no—
221C22—21no—
222C22—42no—
223C22—42no—
224C22—42no—
225C22—81no—
226C22—81no—
227C22—81no—
228C22—81no—
229C22#146411yescharacteristic
230C22#148911yescharacteristic
231C22#149011yescharacteristic
232C4—24no—
233C4—42no—
234C4—41no—
235C4—41no—
236C4—81no—
237C4—81no—
238C4—81no—
239C4#150511yescharacteristic
240C4#150911yescharacteristic
241C4#152411yescharacteristic
242C4#155711yescharacteristic
243C2—21no—
244C2—21no—
245C2—82no—
246C2#762811yescharacteristic
247C2#1152111yescharacteristic
248C2#1152411yescharacteristic
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. #12659C2semidirect
3. #12595C2semidirect
4. #7124C2semidirect

Maximal subgroups

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

subgroupindex
#212652
#126592
#125952
#71242
#71052