Finite Group Databasebeta

← #124539

Subgroups — #124539

Order 512.

309 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 309 conjugacy classes of subgroups — too many to draw. Only its 90 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#12879C211yesmaximal, 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—42no—
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—42no—
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—41no—
99C22⋊C8—42no—
100C22⋊C8—41no—
101#97—42no—
102C4×C8—21no—
103C4×C8—22no—
104C4×C8—21no—
105C4×C8—41no—
106C4×C8—41no—
107C4×C8—41no—
108C4×C8—41no—
109#95—21no—
110#95—21no—
111#95—21no—
112C24—21no—
113C2×D4—22no—
114C2×D4—41no—
115C2×D4—41no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—41no—
120C22×C4—41no—
121C22×C4—41no—
122C22×C4—41no—
123C22×C4—41no—
124C22×C4—41no—
125C22×C4C22⋊C811yescharacteristic
126C22×C4C22⋊C811yescharacteristic
127C22×C4C22⋊C811yescharacteristic
128C22×C4C22⋊C811yescharacteristic
129C22×C4C2×C22⋊C411yescharacteristic
130C22×C4C22×C811yescharacteristic
131C22×C4C2×M1611yescharacteristic
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—21no—
141C2×C8—21no—
142C2×C8—41no—
143C2×C8—41no—
144C2×C8—41no—
145C2×C8—41no—
146C2×C8—41no—
147C2×C8—41no—
148C2×C8AGL(1,ℤ/8)11yescharacteristic
149C2×C8AGL(1,ℤ/8)12yes—
150C2×C8#13711yescharacteristic
151C4⋊C4—22no—
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C22⋊C4—42no—
157C22⋊C4—42no—
158C22⋊C4—42no—
159C22⋊C4—42no—
160C22⋊C4—42no—
161C42—22no—
162C42—21no—
163C42—21no—
164C42—21no—
165C42—21no—
166C42—21no—
167C42—21no—
168C42—22no—
169C42—41no—
170C42—41no—
171C42—41no—
172C42—41no—
173C16—82no—
174C16—82no—
175C23—41no—
176C23—41no—
177C23—41no—
178C23—41no—
179C23—41no—
180C23—41no—
181C23—41no—
182C23C2×C22⋊C811yescharacteristic
183D4—82no—
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—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4—81no—
212C2×C4—81no—
213C2×C4#35012yes—
214C2×C4#41011yescharacteristic
215C2×C4#41311yescharacteristic
216C2×C4#41711yescharacteristic
217C2×C4#41811yescharacteristic
218C2×C4#42111yescharacteristic
219C2×C4#42111yescharacteristic
220C2×C4#50412yes—
221C8—21no—
222C8—21no—
223C8—22no—
224C8—41no—
225C8—41no—
226C8—81no—
227C8—81no—
228C22—81no—
229C22—81no—
230C22—81no—
231C22—81no—
232C22—81no—
233C22—81no—
234C22—81no—
235C22#145811yescharacteristic
236C22#147711yescharacteristic
237C22#147811yescharacteristic
238C22#211311yescharacteristic
239C22#211511yescharacteristic
240C22C2×#35011yescharacteristic
241C22#211711yescharacteristic
242C4—22no—
243C4—22no—
244C4—41no—
245C4—41no—
246C4—41no—
247C4—41no—
248C4—81no—
249C4—81no—
250C4—81no—
251C4#151211yescharacteristic
252C4#151411yescharacteristic
253C4#153511yescharacteristic
254C4#155711yescharacteristic
255C2—161no—
256C2#759411yescharacteristic
257C2#1146111yescharacteristic
258C2#1146411yescharacteristic
259C2#1162411yescharacteristic
260C2#1162511yescharacteristic
261C2#1163011yescharacteristic
262C2#1163511yescharacteristic
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. #12879C2semidirect
4. #7118C2semidirect

Maximal subgroups

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

subgroupindex
#212662
#128942
#128792
#71182
#71022