Finite Group Databasebeta

← #479295

Subgroups — #479295

Order 512.

1114 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 1114 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#15125C212yesmaximal
2#11384C214yesmaximal, complemented
3#11315C214yesmaximal, complemented
4#11199C214yesmaximal
5#11147C211yesmaximal, characteristic
6C22×C4⋊Q8C2212yes—
7C22×C22⋊C8C2212yes—
8C2×#393—22no—
9C2×#380C2214yes—
10C2×C4×C22⋊C4C2211yescharacteristic
11C2×#340C2214yes—
12C2×#340C2214yes—
13C2×#340C2214yes—
14C2×#340C2214yes—
15C2×#336C2214yes—
16#1446—22no—
17#1442—22no—
18#1439—22no—
19C4⋊#95C2212yes—
20C4⋊#95C2214yes—
21C23×Q8—22no—
22C24×C4C2311yescharacteristic
23C23×C8C2314yes—
24C2×C4⋊Q8—28no—
25C2×C4⋊Q8—44no—
26C2×C4⋊Q8—42no—
27C2×C4⋊Q8—42no—
28C22×C4⋊C4—22no—
29C22×C4⋊C4C2314yes—
30C22×C4⋊C4C2314yes—
31C22×C22⋊C4—24no—
32C22×C22⋊C4D411yescharacteristic
33C22×C22⋊C4D411yescharacteristic
34C22×C42C2312yes—
35C2×C22⋊C8—44no—
36C2×C22⋊C8—42no—
37#393—48no—
38#380—28no—
39#380—44no—
40C4×C22⋊C4—42no—
41C2×#95—24no—
42C2×#95—22no—
43C2×#95—22no—
44C2×#95—24no—
45C2×#95—24no—
46C2×#95D412yes—
47#340—28no—
48#340—44no—
49#340—44no—
50#340—44no—
51#336—44no—
52C25C2×D411yescharacteristic
53C22×Q8—48no—
54C22×Q8—44no—
55C22×Q8—44no—
56C22×Q8—44no—
57C22×Q8—84no—
58C23×C4—24no—
59C23×C4—24no—
60C23×C4—22no—
61C23×C4—24no—
62C23×C4—24no—
63C23×C4—24no—
64C23×C4C2×D411yescharacteristic
65C23×C4C2×D412yes—
66C23×C4C2×D412yes—
67C23×C4C2411yescharacteristic
68C22×C8—24no—
69C22×C8—24no—
70C22×C8—24no—
71C22×C8—24no—
72C22×C8—24no—
73C22×C8—24no—
74C22×C8—24no—
75C4⋊Q8—48no—
76C4⋊Q8—416no—
77C4⋊Q8—48no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—24no—
84C2×C4⋊C4—24no—
85C2×C4⋊C4—24no—
86C2×C4⋊C4—24no—
87C2×C4⋊C4—24no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—48no—
90C2×C4⋊C4—44no—
91C2×C4⋊C4#4118yes—
92C2×C22⋊C4—24no—
93C2×C22⋊C4—41no—
94C2×C22⋊C4—41no—
95C2×C22⋊C4—41no—
96C2×C22⋊C4—48no—
97C2×C22⋊C4—48no—
98C2×C22⋊C4—84no—
99C2×C22⋊C4D814yes—
100C2×C42—24no—
101C2×C42—24no—
102C2×C42—22no—
103C2×C42—22no—
104C22⋊C8—48no—
105#95—42no—
106#95—48no—
107#95—48no—
108#95—44no—
109#95—44no—
110#95—84no—
111C24—21no—
112C24—22no—
113C24—24no—
114C24—22no—
115C24—21no—
116C24—24no—
117C24C2×D812yes—
118C24C22×D411yescharacteristic
119C2×Q8—416no—
120C2×Q8—816no—
121C2×Q8—88no—
122C2×Q8—816no—
123C2×Q8—88no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—24no—
127C22×C4—24no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—24no—
131C22×C4—24no—
132C22×C4—21no—
133C22×C4—22no—
134C22×C4—24no—
135C22×C4—28no—
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—42no—
146C22×C4—44no—
147C22×C4—42no—
148C22×C4—44no—
149C22×C4—44no—
150C22×C4—44no—
151C22×C4—44no—
152C22×C4—44no—
153C22×C4—44no—
154C22×C4—44no—
155C22×C4—44no—
156C22×C4—44no—
157C22×C4C2×D814yes—
158C22×C4#13714yes—
159C22×C4C2×#4114yes—
160C22×C4H(5,𝔽2)12yes—
161C22×C4H(5,𝔽2)14yes—
162C22×C42−1+414yes—
163C2×C8—44no—
164C2×C8—44no—
165C2×C8—44no—
166C2×C8—44no—
167C2×C8—44no—
168C2×C8—44no—
169C2×C8—44no—
170C4⋊C4—28no—
171C4⋊C4—28no—
172C4⋊C4—416no—
173C4⋊C4—44no—
174C4⋊C4—44no—
175C4⋊C4—44no—
176C4⋊C4—44no—
177C4⋊C4—44no—
178C4⋊C4—44no—
179C22⋊C4—44no—
180C22⋊C4—44no—
181C22⋊C4—816no—
182C42—44no—
183C42—42no—
184C42—42no—
185C23—22no—
186C23—24no—
187C23—28no—
188C23—41no—
189C23—44no—
190C23—41no—
191C23—42no—
192C23—42no—
193C23—44no—
194C23—42no—
195C23—44no—
196C23—44no—
197C23—44no—
198C23#53411yescharacteristic
199C23#53612yes—
200C23#54614yes—
201C23C22×D811yescharacteristic
202C23C2×#13712yes—
203C23#57511yescharacteristic
204C23#57714yes—
205Q8—832no—
206C2×C4—24no—
207C2×C4—24no—
208C2×C4—28no—
209C2×C4—28no—
210C2×C4—44no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—44no—
216C2×C4—41no—
217C2×C4—48no—
218C2×C4—42no—
219C2×C4—44no—
220C2×C4—48no—
221C2×C4—48no—
222C2×C4—84no—
223C2×C4—84no—
224C2×C4—84no—
225C2×C4—84no—
226C2×C4—84no—
227C2×C4—84no—
228C2×C4—84no—
229C2×C4—84no—
230C2×C4—84no—
231C2×C4—84no—
232C2×C4—84no—
233C2×C4—84no—
234C2×C4—84no—
235C2×C4#53814yes—
236C2×C4#54214yes—
237C2×C4#54214yes—
238C2×C4#55112yes—
239C2×C4#55614yes—
240C2×C4H(3,𝔽4)12yes—
241C2×C4#56214yes—
242C2×C4#56314yes—
243C8—84no—
244C22—42no—
245C22—44no—
246C22—48no—
247C22—84no—
248C22—81no—
249C22—82no—
250C22—81no—
251C22#268412yes—
252C22#270014yes—
253C22#281711yescharacteristic
254C22#318611yescharacteristic
255C22#319511yescharacteristic
256C22#320012yes—
257C22#322612yes—
258C22#323712yes—
259C22#324414yes—
260C22#329414yes—
261C22#331114yes—
262C22#331614yes—
263C22#332914yes—
264C4—44no—
265C4—44no—
266C4—88no—
267C4—81no—
268C4—164no—
269C4#275914yes—
270C4#283214yes—
271C2—82no—
272C2#1719811yescharacteristic
273C2#3126012yes—
274C2#3127212yes—
275C2#3173814yes—
276C2#3176514yes—
277C2#3278611yescharacteristic
278C2#3279411yescharacteristic
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. #11384C2semidirect
3. #11315C2semidirect

Maximal subgroups

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

subgroupindex
#151252
#113842
#113152
#111992
#111472