Finite Group Databasebeta

← #215151

Subgroups — #215151

Order 512.

887 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 887 conjugacy classes of subgroups — too many to draw. Only its 202 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×#2664C211yesmaximal, characteristic
2#21916C212yesmaximal, complemented
3#9654C214yesmaximal, complemented
4#9652C214yesmaximal
5#8309C212yesmaximal
6#8306C212yesmaximal, complemented
7C22×#126C2211yescharacteristic
8C22×#125C2211yescharacteristic
9#2664—28no—
10C2×#400C411yescharacteristic
11C2×#399C411yescharacteristic
12C2×#398C412yes—
13C2×#398C411yescharacteristic
14C2×#397C411yescharacteristic
15C2×#396C412yes—
16C2×#389C2212yes—
17C2×#388C2212yes—
18C2×C4×C4⋊C4C2211yescharacteristic
19#1917—28no—
20#1817C2214yes—
21#1797C2214yes—
22#1749C2214yes—
23C2×#324C2218yes—
24C2×#324C2214yes—
25C2×#323C2214yes—
26C2×#126—24no—
27C2×#126—44no—
28C2×#125—24no—
29C2×#125—44no—
30C22×C4⋊C4C2×C412yes—
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C4⋊C4C2×C411yescharacteristic
33C22×C4⋊C4C2×C412yes—
34C22×C4⋊C4C2×C411yescharacteristic
35C22×C4⋊C4C2×C411yescharacteristic
36C22×C4⋊C4C2×C412yes—
37C22×C4⋊C4C2×C412yes—
38C22×C4⋊C4C2312yes—
39C22×C4⋊C4C2311yescharacteristic
40C22×C22⋊C4C2×C411yescharacteristic
41C22×C22⋊C4C2×C412yes—
42C22×C22⋊C4C2×C411yescharacteristic
43C22×C22⋊C4C2×C412yes—
44C22×C22⋊C4C2311yescharacteristic
45C22×C42C2311yescharacteristic
46C22×C42C2312yes—
47C2×C4⋊C8—24no—
48C2×C22⋊C8C2318yes—
49C2×#97—24no—
50#400—24no—
51#399—24no—
52#398—28no—
53#398—24no—
54#397—24no—
55#396—28no—
56#389—24no—
57#389—24no—
58#388—24no—
59#388—24no—
60C4×C4⋊C4D418yes—
61C2×#95C2×C412yes—
62C2×#95C2×C412yes—
63C2×#95C2×C412yes—
64C2×#95C2×C412yes—
65C2×#95C2×C411yescharacteristic
66C2×#95C2×C411yescharacteristic
67#336—24no—
68#336—28no—
69#336—24no—
70#324—48no—
71#324—44no—
72#323—44no—
73C25C22×C411yescharacteristic
74C23×C4C22×C411yescharacteristic
75C23×C4C22×C412yes—
76C23×C4C22×C411yescharacteristic
77C23×C4C22×C411yescharacteristic
78C23×C4C22×C412yes—
79C23×C4C22×C411yescharacteristic
80C23×C4C22×C411yescharacteristic
81C23×C4C22×C412yes—
82C23×C4C22×C412yes—
83C23×C4C2411yescharacteristic
84C22×C8—28no—
85#126—416no—
86#125—416no—
87C2×C4⋊C4—24no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—24no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—28no—
97C2×C4⋊C4—24no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—44no—
100C2×C4⋊C4—44no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—44no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—82no—
107C2×C4⋊C4C22⋊C418yes—
108C2×C22⋊C4—22no—
109C2×C22⋊C4—22no—
110C2×C22⋊C4—24no—
111C2×C22⋊C4—22no—
112C2×C22⋊C4—24no—
113C2×C22⋊C4—24no—
114C2×C22⋊C4—44no—
115C2×C22⋊C4—42no—
116C2×C22⋊C4—42no—
117C2×C22⋊C4—82no—
118C2×C42—24no—
119C2×C42—82no—
120C2×C42C2×D414yes—
121C2×C42C2×D414yes—
122C2×C42C2×D414yes—
123C4⋊C8—84no—
124C22⋊C8—28no—
125C22⋊C8—28no—
126#97—84no—
127#95—24no—
128#95—24no—
129#95—24no—
130#95—24no—
131#95C22⋊C414yes—
132#95C22⋊C414yes—
133C24—21no—
134C24—22no—
135C24—24no—
136C24—41no—
137C24—41no—
138C24—42no—
139C24C23×C411yescharacteristic
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—24no—
143C22×C4—24no—
144C22×C4—24no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—24no—
152C22×C4—24no—
153C22×C4—24no—
154C22×C4—24no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—24no—
158C22×C4—41no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—41no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4C2×C22⋊C414yes—
171C22×C4C2×C22⋊C414yes—
172C22×C4C2×C22⋊C412yes—
173C22×C4C2×C22⋊C412yes—
174C22×C4C22×D412yes—
175C22×C4H(5,𝔽2)14yes—
176C2×C8—48no—
177C2×C8—48no—
178C2×C8—48no—
179C4⋊C4—48no—
180C4⋊C4—48no—
181C4⋊C4—48no—
182C4⋊C4—44no—
183C4⋊C4—44no—
184C4⋊C4—44no—
185C4⋊C4—44no—
186C4⋊C4—44no—
187C4⋊C4—44no—
188C4⋊C4—44no—
189C4⋊C4—84no—
190C22⋊C4—48no—
191C22⋊C4—44no—
192C22⋊C4—44no—
193C22⋊C4—44no—
194C22⋊C4—84no—
195C42—28no—
196C42—44no—
197C42—44no—
198C23—21no—
199C23—21no—
200C23—22no—
201C23—41no—
202C23—42no—
203C23—41no—
204C23—41no—
205C23—42no—
206C23—44no—
207C23—84no—
208C23—82no—
209C23—82no—
210C23—84no—
211C23C22×C22⋊C411yescharacteristic
212C23#51812yes—
213C23#56814yes—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—24no—
219C2×C4—24no—
220C2×C4—24no—
221C2×C4—24no—
222C2×C4—24no—
223C2×C4—24no—
224C2×C4—24no—
225C2×C4—24no—
226C2×C4—44no—
227C2×C4—44no—
228C2×C4—48no—
229C2×C4—44no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—44no—
244C2×C4—48no—
245C2×C4—84no—
246C2×C4—82no—
247C2×C4—84no—
248C2×C4—82no—
249C2×C4#53514yes—
250C8—88no—
251C22—24no—
252C22—22no—
253C22—22no—
254C22—24no—
255C22—82no—
256C22—81no—
257C22—81no—
258C22—81no—
259C22—82no—
260C22—164no—
261C22#230911yescharacteristic
262C22#287912yes—
263C22#288311yescharacteristic
264C22#288411yescharacteristic
265C22#288912yes—
266C22#298014yes—
267C4—24no—
268C4—42no—
269C4—42no—
270C4—44no—
271C4—44no—
272C4—81no—
273C4—82no—
274C4—81no—
275C4—82no—
276C4—81no—
277C2—24no—
278C2—161no—
279C2#1740612yes—
280C2#1741012yes—
281C2#2179111yescharacteristic
282C2#2179611yescharacteristic
283C2#2180211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21916C2semidirect
3. #9654C2semidirect
6. #8306C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26642
#219162
#96542
#96522
#83092
#83062