Finite Group Databasebeta

← #122243

Subgroups — #122243

Order 512.

453 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 453 conjugacy classes of subgroups — too many to draw. Only its 100 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#21847C211yesmaximal, characteristic
2C2×#1322C212yesmaximal, complemented
3#7057C212yesmaximal, complemented
4#7056C212yesmaximal, complemented
5C2×C4×#41C411yescharacteristic
6C2×#431C411yescharacteristic
7C22×C4⋊C8C2211yescharacteristic
8C2×C22⋊C16—22no—
9#1984C411yescharacteristic
10#1984C2211yescharacteristic
11#1954C411yescharacteristic
12#1954C2211yescharacteristic
13#1869—22no—
14#1797—22no—
15#1322—22no—
16#1322C2214yes—
17C22×#41C811yescharacteristic
18C23×C8D412yes—
19C4×#41—42no—
20C4×#41—42no—
21C2×C4×Q8C811yescharacteristic
22C2×C4×Q8C2×C411yescharacteristic
23C2×C4×D4—21no—
24C2×C4×D4—21no—
25C2×C4×D4C811yescharacteristic
26C2×C4×D4C2×C411yescharacteristic
27C2×#117—21no—
28C2×#117C811yescharacteristic
29C22×C42—21no—
30C22×C42C2×C411yescharacteristic
31C22×C16—42no—
32#431—42no—
33C2×C4⋊C8—21no—
34C2×C4⋊C8—42no—
35C2×C4⋊C8C2×C411yescharacteristic
36C2×C4⋊C8D412yes—
37C2×C4⋊C8C2311yescharacteristic
38C2×C22⋊C8—21no—
39C2×C22⋊C8—22no—
40C2×C22⋊C8—22no—
41C2×C4×C8C2×C411yescharacteristic
42C2×C4×C8C2×C411yescharacteristic
43C22⋊C16—24no—
44C22⋊C16—42no—
45#336—22no—
46#336—22no—
47C2×#41—22no—
48C2×#41—42no—
49C2×#41—42no—
50C2×#41C1614yescomplemented
51C22×Q8C2×C811yescharacteristic
52C22×D4—21no—
53C22×D4C2×C811yescharacteristic
54C23×C4—21no—
55C23×C4—21no—
56C23×C4—21no—
57C23×C4C22⋊C411yescharacteristic
58C23×C4C2×C811yescharacteristic
59C23×C4M1611yescharacteristic
60C22×C8—22no—
61C22×C8—22no—
62C22×C8—21no—
63C22×C8—21no—
64C22×C8—82no—
65C22×C8C22⋊C412yes—
66C22×C8C2×D412yes—
67C4×Q8—42no—
68C4×Q8—41no—
69C4×Q8—41no—
70C4×D4—41no—
71C4×D4—41no—
72C4×D4—42no—
73C4×D4—44no—
74C4×D4—42no—
75C4×D4—42no—
76#117—44no—
77#117—42no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4C2×C811yescharacteristic
81C2×C4⋊C4C2×C811yescharacteristic
82C2×C22⋊C4—21no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—21no—
85C2×C42—22no—
86C2×C42—22no—
87C2×C42—21no—
88C2×C42—21no—
89C2×C42—21no—
90C2×C42—42no—
91C2×C42—42no—
92C2×C42C22⋊C411yescharacteristic
93C2×C42C2×C811yescharacteristic
94C2×C42M1611yescharacteristic
95C2×C42C22×C411yescharacteristic
96C2×C16—44no—
97C2×C16—84no—
98C4⋊C8—41no—
99C4⋊C8—42no—
100C4⋊C8—41no—
101C4⋊C8—42no—
102C22⋊C8—42no—
103C22⋊C8—82no—
104C22⋊C8—82no—
105C4×C8—41no—
106C4×C8—41no—
107C24—21no—
108C24C22⋊C811yescharacteristic
109#41—48no—
110#41—48no—
111C2×Q8—21no—
112C2×Q8—41no—
113C2×Q8—41no—
114C2×Q8C2×C1612yescomplemented
115C2×D4—24no—
116C2×D4—21no—
117C2×D4—41no—
118C2×D4—41no—
119C2×D4—42no—
120C2×D4—42no—
121C2×D4C2×C1612yescomplemented
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—21no—
125C22×C4—22no—
126C22×C4—24no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4—82no—
139C22×C4—81no—
140C22×C4C22⋊C811yescharacteristic
141C22×C4C22⋊C811yescharacteristic
142C22×C4C22⋊C811yescharacteristic
143C22×C4#9912yes—
144C22×C4#10012yes—
145C22×C4C2×C1612yes—
146C22×C4M3212yes—
147C22×C4C2×C22⋊C411yescharacteristic
148C22×C4C22×C811yescharacteristic
149C22×C4C2×M1611yescharacteristic
150C2×C8—22no—
151C2×C8—22no—
152C2×C8—44no—
153C2×C8—42no—
154C2×C8—42no—
155C2×C8—41no—
156C2×C8—41no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—41no—
160C2×C8—41no—
161C2×C8—82no—
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C22⋊C4—42no—
167C22⋊C4—42no—
168C22⋊C4—42no—
169C42—22no—
170C42—22no—
171C42—21no—
172C42—21no—
173C42—21no—
174C42—21no—
175C42—42no—
176C42—42no—
177C42—41no—
178C42—41no—
179C16—84no—
180C23—22no—
181C23—41no—
182C23—41no—
183C23—42no—
184C23—42no—
185C23—41no—
186C23—41no—
187C23#32312yes—
188C23C22⋊C1612yes—
189C23#34912yes—
190C23C2×C22⋊C811yescharacteristic
191Q8—42no—
192Q8—42no—
193D4—42no—
194D4—42no—
195D4—48no—
196C2×C4—24no—
197C2×C4—24no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4—48no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—82no—
221C2×C4—82no—
222C2×C4—84no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4#32312yes—
226C2×C4C22⋊C1612yes—
227C2×C4C2×#9911yescharacteristic
228C2×C4C2×#10011yescharacteristic
229C2×C4C22×C1611yescharacteristic
230C2×C4C2×M3211yescharacteristic
231C8—42no—
232C8—42no—
233C8—81no—
234C8—81no—
235C22—21no—
236C22—22no—
237C22—21no—
238C22—22no—
239C22—42no—
240C22—42no—
241C22—44no—
242C22—42no—
243C22—82no—
244C22—81no—
245C22—81no—
246C22#131414yes—
247C22C2×#32311yescharacteristic
248C22C2×C22⋊C1611yescharacteristic
249C22C2×#34911yescharacteristic
250C4—22no—
251C4—21no—
252C4—21no—
253C4—21no—
254C4—22no—
255C4—42no—
256C4—41no—
257C4—41no—
258C4—81no—
259C4—81no—
260C4#212711yescharacteristic
261C4#213311yescharacteristic
262C2—21no—
263C2—21no—
264C2—24no—
265C2—82no—
266C2C2×#131411yescharacteristic
267C2#1143211yescharacteristic
268C2#1143811yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC2×C4C22×C16
Frattini, cosocleC2×C4⋊C8C23
socleC22C2×#323
prime radicalC22×#41C8
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4⋊C8 ◁ G
LECSG ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1322C2semidirect
3. #7057C2semidirect
4. #7056C2semidirect
50. C2×#41C16semidirect
114. C2×Q8C2×C16semidirect
121. C2×D4C2×C16semidirect

Maximal subgroups

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

subgroupindex
#218472
C2×#13222
#70572
#70562