Finite Group Databasebeta

← #10590586

Subgroups — #10590586

Order 576.

285 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 285 conjugacy classes of subgroups — too many to draw. Only its 64 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
1C3×#895C211yesmaximal, characteristic, complemented
2#63909C211yesmaximal, characteristic, complemented
3C2×#3714C211yesmaximal, characteristic, complemented
4C2×#3711C211yesmaximal, characteristic, complemented
5#63636C211yesmaximal, characteristic, complemented
6C3×#808C211yesmaximal, characteristic, complemented
7#63562C211yesmaximal, characteristic, complemented
8#5039—31nomaximal
9#4865—31nomaximal
10C2×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×D12—21no—
12C2×C3×D12C2211yescharacteristic
13C2×C3×C4×D3C2211yescharacteristic
14C2×#626—21no—
15C32×C4⋊C4C2211yescharacteristic
16C2×C3⋊(C3⋊C8)C2211yescharacteristic
17C3×(C2×D3)⋊C4—21no—
18C3×C4⋊Dic3C2211yescharacteristic
19C2×C3×C3⋊C8C2211yescharacteristic
20#3722—21no—
21#3714—22no—
22#3711—22no—
23C3×#121—31no—
24C2×#213—31no—
25#930—31no—
26C2×#208—31no—
27C2×D24—31no—
28C2×ΓL(1,𝔽25)—31no—
29#895—31no—
30#895D311yescharacteristic, complemented
31C3×#106—31no—
32#832—31no—
33#809—31no—
34#809—31no—
35#809—31no—
36#809—61no—
37#808D311yescharacteristic, complemented
38C22×C3⋊D3—21no—
39C22×C3×D3—21no—
40C22×C3×D3D411yescharacteristic
41C2×C32×C4—21no—
42C2×C32×C4C2311yescharacteristic
43C4⋊(C3⋊D3)—21no—
44C4⋊(C3⋊D3)—21no—
45C2×C3×Dic3—21no—
46C2×C3×Dic3D411yescharacteristic
47C3×D12—21no—
48C3×D12—21no—
49C3×D12—42no—
50C3×C4×D3—21no—
51C3×C4×D3—21no—
52#626—42no—
53C3⋊(C3⋊C8)—21no—
54C3×C3⋊C8D412yescomplemented
55#469—91noSylow
56C2×C3×D4—31no—
57C2×C3×D4—61no—
58C22×C3×C4—31no—
59C2×#67—61no—
60C2×D12—21no—
61C2×D12—31no—
62C2×D12—31no—
63C2×D12—61no—
64C2×D12—61no—
65C2×D12C2×D311yescharacteristic
66C2×C4×D3C2×D311yescharacteristic
67C2×C3×C8—31no—
68C3×C4⋊C4—21no—
69C3×C4⋊C4—31no—
70C3×C4⋊C4C2×D311yescharacteristic
71C3×C4⋊C4C2×D311yescharacteristic
72C3×C22⋊C4—61no—
73#213—62no—
74(C2×D3)⋊C4—21no—
75(C2×D3)⋊C4—61no—
76(C2×D3)⋊C4—61no—
77C4⋊Dic3C2×D311yescharacteristic
78#208—62no—
79C2×C3⋊C8—31no—
80C2×C3⋊C8—31no—
81C2×C3⋊C8—61no—
82C2×C3⋊C8C2×D311yescharacteristic
83D24—62no—
84ΓL(1,𝔽25)—62no—
85C22×C32C2×D411yescharacteristic
86C2×C3⋊D3—41no—
87C2×C3⋊D3—41no—
88C2×C3⋊D3—41no—
89C2×C3×D3—21no—
90C2×C3×D3—21no—
91C2×C3×D3—21no—
92C2×C3×D3—41no—
93C2×C3×D3—41no—
94C2×C3×D3—41no—
95C32×C4—41no—
96C32×C4C2×D411yescharacteristic
97C32×C4#4111yescharacteristic
98C3×Dic3—21no—
99C3×Dic3—41no—
100C2×D8—91no—
101C2×M16—91no—
102#121—31no—
103#121—91no—
104#106—31no—
105#102—91no—
106#102—91no—
107C23×C3—31no—
108C23×C3—61no—
109C22×D3—21no—
110C22×D3—61no—
111C22×D3—61no—
112C22×D3—121no—
113C22×D3#6711yescharacteristic
114C3×D4—61no—
115C3×D4—61no—
116C3×D4—122no—
117C2×C3×C4—21no—
118C2×C3×C4—21no—
119C2×C3×C4—21no—
120C2×C3×C4—31no—
121C2×C3×C4—41no—
122C2×C3×C4—61no—
123C2×C3×C4—61no—
124C2×C3×C4—61no—
125C2×C3×C4C22×D311yescharacteristic
126C2×C3×C4C22×D311yescharacteristic
127#67—122no—
128C2×Dic3—21no—
129C2×Dic3#6711yescharacteristic
130D12—21no—
131D12—21no—
132D12—42no—
133D12—62no—
134D12—61no—
135D12—61no—
136D12—61no—
137D12—61no—
138D12—121no—
139D12—122no—
140C4×D3—21no—
141C4×D3—21no—
142C3×C8—32no—
143C3⋊C8—61no—
144C3⋊C8—61no—
145C3⋊C8—121no—
146C3⋊C8D1212yescomplemented
147C2×C32#12111yescharacteristic
148C2×C32AGL(1,ℤ/8)11yescharacteristic
149C2×C32AGL(1,ℤ/8)11yescharacteristic
150C3⋊D3—81no—
151C3×D3—41no—
152C3×D3—81no—
153C2×D4—31no—
154C2×D4—61no—
155C2×D4—91no—
156C2×D4—181no—
157C22×C4—31no—
158D8—182no—
159M16—182no—
160C2×C8—31no—
161C2×C8—91no—
162C4⋊C4—31no—
163C4⋊C4D3211yescharacteristic
164C22⋊C4—61no—
165C22⋊C4—181no—
166C22×C3—21no—
167C22×C3—61no—
168C22×C3—61no—
169C22×C3—61no—
170C22×C3—121no—
171C22×C3—121no—
172C22×C3—121no—
173C22×C3D3×D411yescharacteristic
174C22×C3C2×#6711yescharacteristic
175C2×D3—21no—
176C2×D3—21no—
177C2×D3—21no—
178C2×D3—41no—
179C2×D3—41no—
180C2×D3—41no—
181C2×D3—122no—
182C2×D3—121no—
183C2×D3—121no—
184C2×D3—121no—
185C2×D3—121no—
186C2×D3—121no—
187C2×D3—121no—
188C2×D3—241no—
189C2×D3—241no—
190C3×C4—21no—
191C3×C4—21no—
192C3×C4—41no—
193C3×C4—41no—
194C3×C4—61no—
195C3×C4—81no—
196C3×C4—121no—
197C3×C4C2×D1211yescharacteristic
198C3×C4D3×D411yescharacteristic
199C3×C4#23711yescharacteristic
200C3×C4#23911yescharacteristic
201Dic3—21no—
202Dic3—41no—
203C32#46911yescharacteristic, complemented, Sylow
204C23—31no—
205C23—61no—
206C23—181no—
207D4—61no—
208D4—61no—
209D4—122no—
210D4—181no—
211D4—181no—
212D4—362no—
213C2×C4—21no—
214C2×C4—31no—
215C2×C4—61no—
216C2×C4—61no—
217C2×C4—61no—
218C2×C4C2×D3211yescharacteristic
219C8—32no—
220C8—181no—
221C2×C3—21no—
222C2×C3—21no—
223C2×C3—21no—
224C2×C3—121no—
225C2×C3—241no—
226C2×C3#89511yescharacteristic
227C2×C3#91411yescharacteristic
228C2×C3#91411yescharacteristic
229C2×C3#93211yescharacteristic
230C2×C3#93911yescharacteristic
231C2×C3#94911yescharacteristic
232D3—41no—
233D3—81no—
234D3—242no—
235D3—241no—
236D3—241no—
237C22—61no—
238C22—61no—
239C22—61no—
240C22—121no—
241C22—121no—
242C22—121no—
243C22—361no—
244C22—361no—
245C22—361no—
246C22D3×#6711yescharacteristic
247C4—41no—
248C4—61no—
249C4—121no—
250C4#379611yescharacteristic
251C4D3×D1211yescharacteristic
252C3—21no—
253C3#486511yescharacteristic, complemented
254C3#503911yescharacteristic, complemented
255C2—121no—
256C2—241no—
257C2—721no—
258C2#6388611yescharacteristic
259C2#6393411yescharacteristic
260C2#6393911yescharacteristic
subgroupG/H
centerC22D3×#67
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC32×C4⋊C4C22
socleC22×C32C2×D4
hypercenterC4⋊C4D32
hypocenterC32#469
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×#895C2semidirect
2. #63909C2semidirect
3. C2×#3714C2semidirect
4. C2×#3711C2semidirect
5. #63636C2semidirect
6. C3×#808C2semidirect
7. #63562C2semidirect
30. #895D3semidirect
37. #808D3semidirect
54. C3×C3⋊C8D4semidirect
146. C3⋊C8D12semidirect
203. C32#469semidirect
253. C3#4865semidirect
254. C3#5039semidirect

Maximal subgroups

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

subgroupindex
C3×#8952
#639092
C2×#37142
C2×#37112
#636362
C3×#8082
#635622
#50393
#48653