Finite Group Databasebeta

← #124366

Subgroups — #124366

Order 512.

370 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 110 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#21264C211yesmaximal, characteristic
2#12894C211yesmaximal, characteristic, complemented
3#12590C211yesmaximal, characteristic, complemented
4#7118C212yesmaximal, complemented
5#7102C212yesmaximal
6C2×#431—21no—
7C42×D4C411yescharacteristic
8C2×C4×C16—21no—
9#1954C411yescharacteristic
10#1954C2211yescharacteristic
11#1939C411yescharacteristic
12#1939C2211yescharacteristic
13#1923—21no—
14#1923—21no—
15C4×C4⋊C8C411yescharacteristic
16C4×C22⋊C8—21no—
17C42×C8C2211yescharacteristic
18#1374—21no—
19#1374—21no—
20#1374—21no—
21#1374C2212yes—
22#1374C2212yes—
23C2×C4×D4C811yescharacteristic
24C2×C4×D4C811yescharacteristic
25C2×C4×D4C2×C411yescharacteristic
26C22×C42—21no—
27C22×C16—22no—
28C22×C16—22no—
29#431—44no—
30C2×C4⋊C8—21no—
31C2×C4⋊C8—21no—
32C2×C4⋊C8C2×C411yescharacteristic
33C2×C4⋊C8C2×C411yescharacteristic
34C2×C22⋊C8—21no—
35C2×C22⋊C8—21no—
36C2×C22⋊C8—21no—
37C2×C22⋊C8—21no—
38C2×C4×C8—21no—
39C2×C4×C8C2×C411yescharacteristic
40C2×C4×C8D411yescharacteristic
41C2×C4×C8D411yescharacteristic
42C2×C4×C8D411yescharacteristic
43C2×C4×C8D411yescharacteristic
44C2×C4×C8C2311yescharacteristic
45C4×C4⋊C4C811yescharacteristic
46C4×C4⋊C4C811yescharacteristic
47C4×C4⋊C4C2×C411yescharacteristic
48C4×C22⋊C4—21no—
49C4×C22⋊C4—21no—
50C4×C22⋊C4—21no—
51C43C2×C411yescharacteristic
52C4×C16—24no—
53#336—21no—
54#336—21no—
55#336—21no—
56C22×D4C2×C811yescharacteristic
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—22no—
67C4×D4—41no—
68C4×D4—41no—
69C4×D4—42no—
70C4×D4C1614yescomplemented
71C2×C4⋊C4C2×C811yescharacteristic
72C2×C4⋊C4C2×C811yescharacteristic
73C2×C4⋊C4C2×C811yescharacteristic
74C2×C22⋊C4—21no—
75C2×C22⋊C4—21no—
76C2×C22⋊C4—21no—
77C2×C42—21no—
78C2×C42—21no—
79C2×C42—21no—
80C2×C42—42no—
81C2×C42—42no—
82C2×C42C22⋊C411yescharacteristic
83C2×C42C22⋊C411yescharacteristic
84C2×C42C2×C811yescharacteristic
85C2×C42C2×C811yescharacteristic
86C2×C42M1611yescharacteristic
87C2×C42M1611yescharacteristic
88C2×C42C22×C411yescharacteristic
89C2×C16—24no—
90C2×C16—42no—
91C2×C16—42no—
92C2×C16—42no—
93C2×C16—42no—
94C2×C16—42no—
95C4⋊C8—41no—
96C4⋊C8—41no—
97C4⋊C8—42no—
98C4⋊C8—42no—
99C22⋊C8—42no—
100C22⋊C8—42no—
101C22⋊C8—42no—
102C22⋊C8—42no—
103C4×C8—22no—
104C4×C8—22no—
105C4×C8—22no—
106C4×C8—41no—
107C4×C8—41no—
108C4×C8—41no—
109C4×C8—42no—
110#95—21no—
111#95—21no—
112#95—21no—
113C24—21no—
114C2×D4—21no—
115C2×D4—41no—
116C2×D4—41no—
117C2×D4C2×C1612yes—
118C22×C4—21no—
119C22×C4—24no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4C22⋊C811yescharacteristic
134C22×C4C22⋊C811yescharacteristic
135C22×C4C22⋊C811yescharacteristic
136C22×C4C22⋊C811yescharacteristic
137C22×C4C2×C22⋊C411yescharacteristic
138C22×C4C22×C811yescharacteristic
139C22×C4C2×M1611yescharacteristic
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—41no—
150C2×C8—41no—
151C2×C8—41no—
152C2×C8—41no—
153C2×C8—41no—
154C2×C8—41no—
155C2×C8#10412yes—
156C2×C8#13512yes—
157C2×C8AGL(1,ℤ/8)12yes—
158C4⋊C4—21no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4C2×C1612yes—
162C22⋊C4—24no—
163C22⋊C4—42no—
164C22⋊C4—42no—
165C42—22no—
166C42—22no—
167C42—21no—
168C42—21no—
169C42—21no—
170C42—21no—
171C42—21no—
172C42—21no—
173C42—41no—
174C42—42no—
175C42—42no—
176C42—41no—
177C42—41no—
178C42C2×C1612yes—
179C42M3212yes—
180C16—44no—
181C16—82no—
182C23—22no—
183C23—41no—
184C23—41no—
185C23—41no—
186C23—41no—
187C23—41no—
188C23—41no—
189C23C2×C22⋊C811yescharacteristic
190D4—42no—
191D4—42no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—48no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—84no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4C22⋊C1612yes—
221C2×C4C22⋊C1612yes—
222C2×C4#34912yes—
223C2×C4#35012yes—
224C2×C4#41011yescharacteristic
225C2×C4#41311yescharacteristic
226C2×C4#41611yescharacteristic
227C2×C4#41811yescharacteristic
228C2×C4C2×#10411yescharacteristic
229C2×C4#42111yescharacteristic
230C2×C4C22×C1611yescharacteristic
231C2×C4C2×M3211yescharacteristic
232C8—22no—
233C8—22no—
234C8—22no—
235C8—41no—
236C8—81no—
237C8—81no—
238C22—42no—
239C22—42no—
240C22—42no—
241C22—81no—
242C22—81no—
243C22—81no—
244C22—81no—
245C22#145811yescharacteristic
246C22#147611yescharacteristic
247C22#147811yescharacteristic
248C22C2×C22⋊C1611yescharacteristic
249C22C2×#34911yescharacteristic
250C22C2×#35011yescharacteristic
251C22#211711yescharacteristic
252C4—22no—
253C4—22no—
254C4—22no—
255C4—22no—
256C4—21no—
257C4—21no—
258C4—42no—
259C4—41no—
260C4—81no—
261C4—81no—
262C4#150511yescharacteristic
263C4#152111yescharacteristic
264C4#153211yescharacteristic
265C4#155311yescharacteristic
266C2—82no—
267C2#758011yescharacteristic
268C2#1145911yescharacteristic
269C2#1146411yescharacteristic
270C2#1161311yescharacteristic
271C2#1161911yescharacteristic
272C2#1162911yescharacteristic
273C2#1163411yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC2×C4C22×C16
Frattini, cosocleC2×C4×C8C23
socleC23C2×C22⋊C8
prime radicalC22×D4C2×C8
LCSG ▷ C2×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
70. C4×D4C16semidirect

Maximal subgroups

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

subgroupindex
#212642
#128942
#125902
#71182
#71022