Finite Group Databasebeta

← #479719

Subgroups — #479719

Order 512.

994 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 216 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#15150C212yesmaximal
2#11324C214yesmaximal, complemented
3#11315C214yesmaximal, complemented
4#11147C211yesmaximal, characteristic
5#11096C214yesmaximal
6C22×#107C2214yes—
7C22×C22⋊C8C2212yes—
8C2×#397—22no—
9C2×#397—22no—
10C2×#384C2212yes—
11C2×#380C2214yes—
12C2×C4×C22⋊C4C2211yescharacteristic
13C2×#340C2214yes—
14C2×#340C2214yes—
15C2×#340C2214yes—
16C2×#336C2214yes—
17#1445—22no—
18#1445—22no—
19C4⋊#95C2212yes—
20C4⋊#95C2214yes—
21C24×C4C2311yescharacteristic
22C23×C8C2314yes—
23C22×C4⋊C4—22no—
24C22×C4⋊C4—22no—
25C22×C4⋊C4C2314yes—
26C22×C4⋊C4C2314yes—
27C22×C22⋊C4—24no—
28C22×C22⋊C4D411yescharacteristic
29C22×C22⋊C4D411yescharacteristic
30C22×C42C2312yes—
31C2×#107—44no—
32C2×#107—44no—
33C2×#107—44no—
34C2×C22⋊C8—28no—
35C2×C22⋊C8—42no—
36#397—48no—
37#397—48no—
38#384—42no—
39#384—42no—
40#380—44no—
41#380—44no—
42C4×C22⋊C4—42no—
43C2×#95—22no—
44C2×#95—24no—
45C2×#95—22no—
46C2×#95—24no—
47C2×#95—24no—
48C2×#95D412yes—
49#340—28no—
50#340—44no—
51#340—44no—
52#336—28no—
53C25C2×D411yescharacteristic
54C23×C4—24no—
55C23×C4—22no—
56C23×C4—24no—
57C23×C4—24no—
58C23×C4—24no—
59C23×C4—24no—
60C23×C4C2×D411yescharacteristic
61C23×C4C2×D412yes—
62C23×C4C2×D412yes—
63C23×C4C2411yescharacteristic
64C22×C8—24no—
65C22×C8—24no—
66C22×C8—24no—
67C22×C8—24no—
68C22×C8—24no—
69C22×C8—24no—
70C22×C8#4118yes—
71C2×C4⋊C4—24no—
72C2×C4⋊C4—24no—
73C2×C4⋊C4—24no—
74C2×C4⋊C4—24no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4—24no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—44no—
84C2×C4⋊C4—44no—
85C2×C4⋊C4—44no—
86C2×C4⋊C4—44no—
87C2×C4⋊C4—44no—
88C2×C4⋊C4—44no—
89C2×C22⋊C4—24no—
90C2×C22⋊C4—41no—
91C2×C22⋊C4—41no—
92C2×C22⋊C4—41no—
93C2×C22⋊C4—48no—
94C2×C22⋊C4—48no—
95C2×C22⋊C4—48no—
96C2×C22⋊C4D814yes—
97C2×C42—24no—
98C2×C42—22no—
99C2×C42—22no—
100C2×C42—24no—
101#107—416no—
102C22⋊C8—48no—
103#95—42no—
104#95—48no—
105#95—44no—
106#95—44no—
107#95—48no—
108#95—48no—
109C24—21no—
110C24—22no—
111C24—24no—
112C24—22no—
113C24—21no—
114C24—24no—
115C24C2×D812yes—
116C24C22×D411yescharacteristic
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—24no—
120C22×C4—24no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—24no—
124C22×C4—24no—
125C22×C4—21no—
126C22×C4—22no—
127C22×C4—24no—
128C22×C4—44no—
129C22×C4—44no—
130C22×C4—44no—
131C22×C4—42no—
132C22×C4—44no—
133C22×C4—44no—
134C22×C4—42no—
135C22×C4—44no—
136C22×C4—44no—
137C22×C4—44no—
138C22×C4—44no—
139C22×C4—44no—
140C22×C4—44no—
141C22×C4—44no—
142C22×C4—44no—
143C22×C4—44no—
144C22×C4—44no—
145C22×C4—44no—
146C22×C4—44no—
147C22×C4—44no—
148C22×C4—44no—
149C22×C4—44no—
150C22×C4C2×D814yes—
151C22×C4AGL(1,ℤ/8)14yes—
152C22×C4C2×#4114yes—
153C22×C4H(5,𝔽2)14yes—
154C22×C4H(5,𝔽2)12yes—
155C22×C42−1+414yes—
156C2×C8—28no—
157C2×C8—28no—
158C2×C8—28no—
159C2×C8—44no—
160C2×C8—44no—
161C2×C8—44no—
162C2×C8—44no—
163C4⋊C4—44no—
164C4⋊C4—44no—
165C4⋊C4—44no—
166C4⋊C4—44no—
167C4⋊C4—44no—
168C4⋊C4—44no—
169C4⋊C4—44no—
170C4⋊C4—44no—
171C4⋊C4—88no—
172C4⋊C4—88no—
173C22⋊C4—44no—
174C22⋊C4—44no—
175C22⋊C4—816no—
176C42—44no—
177C42—42no—
178C42—42no—
179C23—28no—
180C23—22no—
181C23—24no—
182C23—44no—
183C23—44no—
184C23—44no—
185C23—41no—
186C23—42no—
187C23—42no—
188C23—44no—
189C23—42no—
190C23—44no—
191C23—41no—
192C23#53411yescharacteristic
193C23#53512yes—
194C23#54714yes—
195C23C22×D811yescharacteristic
196C23C2×AGL(1,ℤ/8)12yes—
197C23#57511yescharacteristic
198C23#57814yes—
199C2×C4—28no—
200C2×C4—28no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—44no—
204C2×C4—44no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—44no—
211C2×C4—41no—
212C2×C4—84no—
213C2×C4—84no—
214C2×C4—84no—
215C2×C4—84no—
216C2×C4—84no—
217C2×C4—84no—
218C2×C4—84no—
219C2×C4—84no—
220C2×C4—84no—
221C2×C4—84no—
222C2×C4—84no—
223C2×C4—84no—
224C2×C4—84no—
225C2×C4—84no—
226C2×C4—84no—
227C2×C4—84no—
228C2×C4#53914yes—
229C2×C4#54214yes—
230C2×C4#54214yes—
231C2×C4#55112yes—
232C2×C4#55314yes—
233C2×C4#56014yes—
234C2×C4#56312yes—
235C2×C4#56314yes—
236C8—48no—
237C22—44no—
238C22—48no—
239C22—42no—
240C22—81no—
241C22—82no—
242C22—84no—
243C22—81no—
244C22#271712yes—
245C22#271814yes—
246C22#281511yescharacteristic
247C22#318611yescharacteristic
248C22#319211yescharacteristic
249C22#320212yes—
250C22#320512yes—
251C22#321412yes—
252C22#322314yes—
253C22#333414yes—
254C22#334314yes—
255C22#334914yes—
256C22#335314yes—
257C4—44no—
258C4—44no—
259C4—81no—
260C4—164no—
261C4—164no—
262C4#275714yes—
263C4#283814yes—
264C2—82no—
265C2#1722911yescharacteristic
266C2#3225812yes—
267C2#3227112yes—
268C2#3227814yes—
269C2#3230114yes—
270C2#3275811yescharacteristic
271C2#3276811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11324C2semidirect
3. #11315C2semidirect

Maximal subgroups

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

subgroupindex
#151502
#113242
#113152
#111472
#110962