Finite Group Databasebeta

← #124367

Subgroups — #124367

Order 512.

321 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 321 conjugacy classes of subgroups — too many to draw. Only its 94 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#21266C211yesmaximal, characteristic
2#12894C211yesmaximal, characteristic, complemented
3#12590C211yesmaximal, characteristic, complemented
4#7118C212yesmaximal, complemented
5#7102C212yesmaximal
6C2×#433—21no—
7#2293C411yescharacteristic
8C2×C4×C16—21no—
9#1954C411yescharacteristic
10#1954C2211yescharacteristic
11#1939C411yescharacteristic
12#1939C2211yescharacteristic
13#1923—21no—
14#1923—21no—
15#1767C411yescharacteristic
16#1750—21no—
17C42×C8C2211yescharacteristic
18#1374—21no—
19#1374—21no—
20#1374—21no—
21#1374C2212yes—
22#1374C2212yes—
23C2×C4×D4C2×C411yescharacteristic
24C2×#117—21no—
25C22×C16—22no—
26C22×C16—22no—
27#433—44no—
28C2×C4⋊C8—21no—
29C2×C4⋊C8—21no—
30C2×C4⋊C8C2×C411yescharacteristic
31C2×C4⋊C8C2×C411yescharacteristic
32C2×C22⋊C8—21no—
33C2×C22⋊C8—21no—
34C2×C22⋊C8—21no—
35C2×C22⋊C8—21no—
36C2×#97—21no—
37C2×C4×C8C2×C411yescharacteristic
38C2×C4×C8D411yescharacteristic
39C2×C4×C8D411yescharacteristic
40C2×C4×C8D411yescharacteristic
41C2×C4×C8D411yescharacteristic
42C2×C4×C8C2311yescharacteristic
43#390C811yescharacteristic
44#390C811yescharacteristic
45#389C811yescharacteristic
46#389C811yescharacteristic
47#388—21no—
48#388—21no—
49C4×C4⋊C4C2×C411yescharacteristic
50C4×C22⋊C4—21no—
51C43C2×C411yescharacteristic
52C4×C16—24no—
53#336—21no—
54#336—21no—
55#336—21no—
56C22×D4C2×C811yescharacteristic
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—
66#117—42no—
67#117—42no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4C2×C811yescharacteristic
71C2×C4⋊C4C2×C811yescharacteristic
72C2×C4⋊C4C2×C811yescharacteristic
73C2×C22⋊C4—21no—
74C2×C22⋊C4—21no—
75C2×C22⋊C4—21no—
76C2×C22⋊C4—21no—
77C2×C22⋊C4—21no—
78C2×C42—21no—
79C2×C42C22⋊C411yescharacteristic
80C2×C42C22⋊C411yescharacteristic
81C2×C42C2×C811yescharacteristic
82C2×C42C2×C811yescharacteristic
83C2×C42M1611yescharacteristic
84C2×C42M1611yescharacteristic
85C2×C42C22×C411yescharacteristic
86C2×C16—24no—
87C2×C16—42no—
88C2×C16—42no—
89C2×C16—42no—
90C2×C16—42no—
91C2×C16—42no—
92C4⋊C8—41no—
93C4⋊C8—41no—
94C4⋊C8—42no—
95C4⋊C8—42no—
96C22⋊C8—42no—
97C22⋊C8—42no—
98C22⋊C8—42no—
99C22⋊C8—42no—
100#97—42no—
101C4×C8—22no—
102C4×C8—22no—
103C4×C8—22no—
104C4×C8—41no—
105C4×C8—41no—
106C4×C8—41no—
107#95—21no—
108#95—21no—
109#95—21no—
110C24—21no—
111C2×D4—22no—
112C2×D4—41no—
113C2×D4—41no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—41no—
118C22×C4—41no—
119C22×C4—41no—
120C22×C4—41no—
121C22×C4—41no—
122C22×C4—41no—
123C22×C4C22⋊C811yescharacteristic
124C22×C4C22⋊C811yescharacteristic
125C22×C4C22⋊C811yescharacteristic
126C22×C4C22⋊C811yescharacteristic
127C22×C4C2×C22⋊C411yescharacteristic
128C22×C4C22×C811yescharacteristic
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—41no—
140C2×C8—41no—
141C2×C8—41no—
142C2×C8—41no—
143C2×C8—41no—
144C2×C8—41no—
145C2×C8#10412yes—
146C2×C8#13512yes—
147C2×C8AGL(1,ℤ/8)12yes—
148C4⋊C4—22no—
149C4⋊C4—41no—
150C4⋊C4—41no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C22⋊C4—42no—
154C22⋊C4—42no—
155C22⋊C4—42no—
156C22⋊C4—42no—
157C22⋊C4—42no—
158C42—22no—
159C42—22no—
160C42—21no—
161C42—21no—
162C42—21no—
163C42—21no—
164C42—21no—
165C42—21no—
166C42—22no—
167C42—41no—
168C42—41no—
169C42—41no—
170C16—44no—
171C16—82no—
172C23—41no—
173C23—41no—
174C23—41no—
175C23—41no—
176C23—41no—
177C23—41no—
178C23—41no—
179C23C2×C22⋊C811yescharacteristic
180D4—82no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—81no—
206C2×C4—81no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4#35012yes—
210C2×C4#35012yes—
211C2×C4#41011yescharacteristic
212C2×C4#41311yescharacteristic
213C2×C4#41611yescharacteristic
214C2×C4#41811yescharacteristic
215C2×C4C2×#10411yescharacteristic
216C2×C4#42111yescharacteristic
217C2×C4#50412yes—
218C8—22no—
219C8—22no—
220C8—22no—
221C8—41no—
222C8—81no—
223C8—81no—
224C22—81no—
225C22—81no—
226C22—81no—
227C22—81no—
228C22—81no—
229C22—81no—
230C22—81no—
231C22#145811yescharacteristic
232C22#147611yescharacteristic
233C22#147811yescharacteristic
234C22#211311yescharacteristic
235C22#211511yescharacteristic
236C22C2×#35011yescharacteristic
237C22C2×#35011yescharacteristic
238C4—22no—
239C4—22no—
240C4—22no—
241C4—41no—
242C4—41no—
243C4—41no—
244C4—81no—
245C4—81no—
246C4—81no—
247C4#150511yescharacteristic
248C4#152111yescharacteristic
249C4#153211yescharacteristic
250C4#155311yescharacteristic
251C2—161no—
252C2#758011yescharacteristic
253C2#1146111yescharacteristic
254C2#1146211yescharacteristic
255C2#1161611yescharacteristic
256C2#1161811yescharacteristic
257C2#1163011yescharacteristic
258C2#1163311yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC23C2×C22⋊C8
prime radicalC22×D4C2×C8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12894C2semidirect
3. #12590C2semidirect
4. #7118C2semidirect

Maximal subgroups

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

subgroupindex
#212662
#128942
#125902
#71182
#71022