Finite Group Databasebeta

← #449606

Subgroups — #449606

Order 512.

1924 conjugacy classes of subgroups; only the 272 normal subgroups are shown.

normal subgroups only

This group has 1924 conjugacy classes of subgroups — too many to draw. Only its 272 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
1C2×#2486C212yesmaximal
2C2×#1852C214yesmaximal, complemented
3#11328C214yesmaximal, complemented
4#11199C214yesmaximal, complemented
5#10964C211yesmaximal, characteristic
6C22×#123—22no—
7C22×C4×D4C2211yescharacteristic
8C23×C4⋊C4C2212yes—
9C22×#107C2214yes—
10C22×C22⋊C8C2214yes—
11#2486—216no—
12C2×#398—22no—
13C2×#397—22no—
14C2×#389C2214yes—
15C2×#381—22no—
16C2×#380C2212yes—
17C2×#380C2214yes—
18#1852—216no—
19C2×#340C2218yes—
20C2×#340C2214yes—
21C2×#336C2212yes—
22C23×D4D411yescharacteristic
23C24×C4—22no—
24C24×C4C2312yes—
25C23×C8C2314yes—
26C2×#123—216no—
27C2×#123—48no—
28C2×#123—48no—
29C2×C4×D4—28no—
30C2×C4×D4—42no—
31C22×C4⋊C4—22no—
32C22×C4⋊C4—24no—
33C22×C4⋊C4—28no—
34C22×C4⋊C4—28no—
35C22×C4⋊C4—44no—
36C22×C4⋊C4D411yescharacteristic
37C22×C4⋊C4C2314yes—
38C22×C4⋊C4C2314yes—
39C22×C22⋊C4—24no—
40C22×C22⋊C4—22no—
41C22×C22⋊C4D412yes—
42C22×C42C2311yescharacteristic
43C2×#107—216no—
44C2×#107—44no—
45C2×C22⋊C8—28no—
46C2×C22⋊C8—216no—
47#398—216no—
48#397—216no—
49#389—28no—
50#389—28no—
51#381—216no—
52#380—28no—
53#380—28no—
54#380D4116yes—
55C2×#95—22no—
56C2×#95—24no—
57C2×#95—24no—
58#340—28no—
59#340—216no—
60#336—42no—
61C25C2×D412yes—
62C22×D4—24no—
63C22×D4—24no—
64C22×D4—41no—
65C22×D4—41no—
66C22×D4D814yes—
67C23×C4—24no—
68C23×C4—28no—
69C23×C4—24no—
70C23×C4—28no—
71C23×C4—28no—
72C23×C4—24no—
73C23×C4—24no—
74C23×C4—44no—
75C23×C4—44no—
76C23×C4—44no—
77C23×C4—44no—
78C23×C4—44no—
79C23×C4—44no—
80C23×C4C2×D411yescharacteristic
81C23×C4C2×D412yes—
82C23×C4C2×D418yes—
83C23×C4C2×D411yescharacteristic
84C23×C4C2411yescharacteristic
85C22×C8—28no—
86C22×C8—28no—
87C22×C8—24no—
88C22×C8—24no—
89C22×C8#4118yes—
90#123—464no—
91C4×D4—416no—
92C2×C4⋊C4—28no—
93C2×C4⋊C4—28no—
94C2×C4⋊C4—28no—
95C2×C4⋊C4—28no—
96C2×C4⋊C4—28no—
97C2×C4⋊C4—216no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—432no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—44no—
104C2×C4⋊C4—48no—
105C2×C4⋊C4—44no—
106C2×C4⋊C4—48no—
107C2×C4⋊C4C2×D418yes—
108C2×C4⋊C4C2×D418yes—
109C2×C22⋊C4—28no—
110C2×C22⋊C4—216no—
111C2×C22⋊C4—24no—
112C2×C22⋊C4—24no—
113C2×C22⋊C4—42no—
114C2×C22⋊C4—44no—
115C2×C22⋊C4—44no—
116C2×C22⋊C4—48no—
117C2×C22⋊C4—48no—
118C2×C42—24no—
119C2×C42—22no—
120#107—416no—
121C22⋊C8—232no—
122#95—28no—
123#95—216no—
124#95—216no—
125C24—22no—
126C24—22no—
127C24—28no—
128C24—22no—
129C24—22no—
130C24—28no—
131C24#12014yes—
132C24C2×D814yes—
133C24AGL(1,ℤ/8)14yes—
134C24C22×D411yescharacteristic
135C2×D4—216no—
136C2×D4—44no—
137C2×D4—44no—
138C2×D4—44no—
139C2×D4—48no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—21no—
143C22×C4—22no—
144C22×C4—28no—
145C22×C4—22no—
146C22×C4—21no—
147C22×C4—28no—
148C22×C4—28no—
149C22×C4—216no—
150C22×C4—216no—
151C22×C4—22no—
152C22×C4—28no—
153C22×C4—28no—
154C22×C4—21no—
155C22×C4—24no—
156C22×C4—44no—
157C22×C4—44no—
158C22×C4—48no—
159C22×C4—48no—
160C22×C4—44no—
161C22×C4—44no—
162C22×C4—416no—
163C22×C4—416no—
164C22×C4—416no—
165C22×C4—44no—
166C22×C4—44no—
167C22×C4—48no—
168C22×C4—44no—
169C22×C4—44no—
170C22×C4—44no—
171C22×C4—44no—
172C22×C4—44no—
173C22×C4—44no—
174C22×C4—48no—
175C22×C4—88no—
176C22×C4—88no—
177C22×C4—88no—
178C22×C4—88no—
179C22×C4#12014yes—
180C22×C4C2×D812yes—
181C22×C4C22×D414yes—
182C22×C4C2×#4114yes—
183C22×C4H(5,𝔽2)12yes—
184C22×C42−1+414yes—
185C2×C8—216no—
186C2×C8—28no—
187C2×C8—44no—
188C2×C8—48no—
189C2×C8—44no—
190C4⋊C4—216no—
191C4⋊C4—216no—
192C4⋊C4—44no—
193C4⋊C4—416no—
194C4⋊C4—432no—
195C4⋊C4—44no—
196C4⋊C4—44no—
197C4⋊C4—48no—
198C22⋊C4—48no—
199C22⋊C4—416no—
200C22⋊C4—432no—
201C42—44no—
202C23—216no—
203C23—216no—
204C23—24no—
205C23—28no—
206C23—24no—
207C23—24no—
208C23—24no—
209C23—24no—
210C23—24no—
211C23—28no—
212C23—48no—
213C23—48no—
214C23—44no—
215C23—44no—
216C23—44no—
217C23—44no—
218C23#44718yes—
219C23C2×#12012yes—
220C23#53411yescharacteristic
221C23#54714yes—
222C23C22×D811yescharacteristic
223C23C2×AGL(1,ℤ/8)12yes—
224C23#57511yescharacteristic
225C23#57814yes—
226D4—416no—
227C2×C4—24no—
228C2×C4—28no—
229C2×C4—22no—
230C2×C4—24no—
231C2×C4—41no—
232C2×C4—44no—
233C2×C4—41no—
234C2×C4—44no—
235C2×C4—42no—
236C2×C4—416no—
237C2×C4—416no—
238C2×C4—41no—
239C2×C4—48no—
240C2×C4—48no—
241C2×C4—42no—
242C2×C4—48no—
243C2×C4—48no—
244C2×C4—48no—
245C2×C4—44no—
246C2×C4—48no—
247C2×C4—832no—
248C2×C4—88no—
249C2×C4—88no—
250C2×C4—84no—
251C2×C4—84no—
252C2×C4—84no—
253C2×C4—84no—
254C2×C4#52514yes—
255C2×C4#53614yes—
256C2×C4#53914yes—
257C2×C4D4214yes—
258C2×C4#54718yes—
259C2×C4#55912yes—
260C2×C4#56312yes—
261C8—48no—
262C22—216no—
263C22—28no—
264C22—48no—
265C22—48no—
266C22—48no—
267C22—44no—
268C22—44no—
269C22—48no—
270C22—48no—
271C22#240311yescharacteristic
272C22#240614yes—
273C22#272412yes—
274C22C2×#44712yes—
275C22#300612yes—
276C22#301114yes—
277C22#318611yescharacteristic
278C22#319211yescharacteristic
279C22#320212yes—
280C22#333414yes—
281C22#334318yes—
282C22#335214yes—
283C4—44no—
284C4—42no—
285C4—88no—
286C4—88no—
287C4—81no—
288C4#245514yes—
289C4#268014yes—
290C2—48no—
291C2#1594911yescharacteristic
292C2#2391811yescharacteristic
293C2#2392811yescharacteristic
294C2#2408814yes—
295C2#2411014yes—
296C2#3245812yes—
297C2#3246912yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1852C2semidirect
3. #11328C2semidirect
4. #11199C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24862
C2×#18522
#113282
#111992
#109642