Finite Group Databasebeta

← #10590508

Subgroups — #10590508

Order 576.

307 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 307 conjugacy classes of subgroups — too many to draw. Only its 62 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#64104C211yesmaximal, characteristic, complemented
2#63916C211yesmaximal, characteristic, complemented
3C2×#3714C211yesmaximal, characteristic, complemented
4C2×#3712C211yesmaximal, characteristic, complemented
5C3×#809C211yesmaximal, characteristic, complemented
6C3×#808C211yesmaximal, characteristic, complemented
7#63564C211yesmaximal, characteristic, complemented
8#4865—31nomaximal
9#4782—31nomaximal
10C2×C4⋊(C3⋊D3)—21no—
11C2×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C4×C3⋊D3C2211yescharacteristic
13C2×C3×D12C2211yescharacteristic
14C2×#626—21no—
15C32×C4⋊C4C2211yescharacteristic
16#3752—21no—
17C3×C4⋊Dic3C2211yescharacteristic
18C2×C3×C3⋊C8C2211yescharacteristic
19C2×C3×C3⋊C8C2211yescharacteristic
20(C2×D3)⋊Dic3—21no—
21#3714—22no—
22#3712—22no—
23#938—31no—
24C2×#213—31no—
25C2×D24—31no—
26C2×ΓL(1,𝔽25)—31no—
27C2×ΓL(1,𝔽25)—31no—
28#895—31no—
29#895—31no—
30#895—31no—
31#895—61no—
32C3×#106—31no—
33C3×#102—31no—
34#821—31no—
35#809—31no—
36#809D311yescharacteristic, complemented
37#808D311yescharacteristic, complemented
38C22×C3⋊D3—21no—
39C22×C3⋊D3D411yescharacteristic
40C22×C3×D3—21no—
41C2×C32×C4—21no—
42C2×C32×C4C2311yescharacteristic
43C2×C3⋊Dic3D411yescharacteristic
44C4⋊(C3⋊D3)—21no—
45C4⋊(C3⋊D3)—21no—
46C4⋊(C3⋊D3)—42no—
47C4×C3⋊D3—21no—
48C4×C3⋊D3—21no—
49C2×C3×Dic3—21no—
50C3×D12—21no—
51C3×D12—21no—
52#626—42no—
53C3×C3⋊C8—21no—
54C3×C3⋊C8D412yescomplemented
55#469—91noSylow
56C2×C3×D4—31no—
57C2×#67—61no—
58C2×D12—31no—
59C2×D12—31no—
60C2×D12—61no—
61C2×D12—61no—
62C2×D12—61no—
63C2×D12—62no—
64C2×D12—61no—
65C2×D12C2×D311yescharacteristic
66C2×C4×D3—31no—
67C2×C4×D3—31no—
68C2×C4×D3—61no—
69C2×C3×C8—31no—
70C2×C3×C8—31no—
71C3×C4⋊C4—21no—
72C3×C4⋊C4—31no—
73C3×C4⋊C4C2×D311yescharacteristic
74C3×C4⋊C4C2×D311yescharacteristic
75#217—61no—
76#213—62no—
77(C2×D3)⋊C4—61no—
78(C2×D3)⋊C4—61no—
79(C2×D3)⋊C4—61no—
80(C2×D3)⋊C4—62no—
81C4⋊Dic3C2×D311yescharacteristic
82C2×C3⋊C8C2×D311yescharacteristic
83C2×C3⋊C8C2×D311yescharacteristic
84D24—62no—
85ΓL(1,𝔽25)—62no—
86ΓL(1,𝔽25)—62no—
87C22×C32C2×D411yescharacteristic
88C2×C3⋊D3—21no—
89C2×C3⋊D3—21no—
90C2×C3⋊D3—21no—
91C2×C3⋊D3—41no—
92C2×C3⋊D3—41no—
93C2×C3⋊D3—41no—
94C2×C3×D3—41no—
95C2×C3×D3—41no—
96C2×C3×D3—41no—
97C32×C4—41no—
98C32×C4C2×D411yescharacteristic
99C32×C4#4111yescharacteristic
100C3⋊Dic3—21no—
101C3×Dic3—41no—
102C2×D8—91no—
103C2×M16—91no—
104#121—91no—
105#121—91no—
106#106—31no—
107#102—31no—
108#102—91no—
109C23×C3—61no—
110C22×D3—21no—
111C22×D3—31no—
112C22×D3—31no—
113C22×D3—61no—
114C22×D3—62no—
115C22×D3—61no—
116C22×D3—61no—
117C3×D4—61no—
118C3×D4—61no—
119C2×C3×C4—22no—
120C2×C3×C4—21no—
121C2×C3×C4—21no—
122C2×C3×C4—21no—
123C2×C3×C4—61no—
124C2×C3×C4C22×D311yescharacteristic
125C2×C3×C4C22×D311yescharacteristic
126#67—122no—
127C2×Dic3—21no—
128C2×Dic3—31no—
129C2×Dic3—31no—
130C2×Dic3—61no—
131D12—21no—
132D12—21no—
133D12—61no—
134D12—61no—
135D12—61no—
136D12—61no—
137D12—121no—
138D12—121no—
139D12—124no—
140D12—122no—
141D12—122no—
142D12—122no—
143C4×D3—61no—
144C4×D3—61no—
145C4×D3—61no—
146C4×D3—61no—
147C4×D3—121no—
148C4×D3—121no—
149C3×C8—32no—
150C3×C8—61no—
151C3⋊C8—21no—
152C3⋊C8D1212yescomplemented
153C2×C32#12111yescharacteristic
154C2×C32AGL(1,ℤ/8)11yescharacteristic
155C2×C32AGL(1,ℤ/8)11yescharacteristic
156C3⋊D3—41no—
157C3⋊D3—81no—
158C3×D3—81no—
159C2×D4—31no—
160C2×D4—91no—
161C2×D4—181no—
162C2×D4—181no—
163C22×C4—91no—
164D8—182no—
165M16—182no—
166C2×C8—31no—
167C2×C8—31no—
168C4⋊C4—31no—
169C4⋊C4D3211yescharacteristic
170C22⋊C4—181no—
171C22⋊C4—181no—
172C22×C3—21no—
173C22×C3—121no—
174C22×C3—121no—
175C22×C3—121no—
176C22×C3D3×D411yescharacteristic
177C22×C3D3×D411yescharacteristic
178C2×D3—41no—
179C2×D3—41no—
180C2×D3—41no—
181C2×D3—62no—
182C2×D3—61no—
183C2×D3—61no—
184C2×D3—61no—
185C2×D3—61no—
186C2×D3—61no—
187C2×D3—61no—
188C2×D3—121no—
189C2×D3—121no—
190C2×D3—122no—
191C2×D3—122no—
192C2×D3—122no—
193C2×D3—121no—
194C2×D3—121no—
195C2×D3—121no—
196C2×D3—121no—
197C2×D3—121no—
198C2×D3—121no—
199C3×C4—21no—
200C3×C4—21no—
201C3×C4—42no—
202C3×C4—41no—
203C3×C4—41no—
204C3×C4—121no—
205C3×C4C2×D1211yescharacteristic
206C3×C4#23511yescharacteristic
207C3×C4D3×D411yescharacteristic
208C3×C4#23911yescharacteristic
209Dic3—41no—
210Dic3—61no—
211Dic3—61no—
212Dic3—121no—
213C32#46911yescharacteristic, complemented, Sylow
214C23—61no—
215C23—91no—
216C23—181no—
217D4—61no—
218D4—61no—
219D4—181no—
220D4—181no—
221D4—362no—
222D4—362no—
223C2×C4—21no—
224C2×C4—61no—
225C2×C4—91no—
226C2×C4—181no—
227C2×C4—181no—
228C2×C4C2×D3211yescharacteristic
229C8—32no—
230C8—61no—
231C2×C3—21no—
232C2×C3—21no—
233C2×C3—21no—
234C2×C3—241no—
235C2×C3#88411yescharacteristic
236C2×C3#89511yescharacteristic
237C2×C3#91111yescharacteristic
238C2×C3#91411yescharacteristic
239C2×C3#91411yescharacteristic
240C2×C3#91411yescharacteristic
241D3—81no—
242D3—122no—
243D3—121no—
244D3—121no—
245D3—242no—
246D3—241no—
247D3—241no—
248C22—121no—
249C22—121no—
250C22—121no—
251C22—181no—
252C22—181no—
253C22—181no—
254C22—361no—
255C22—361no—
256C22—361no—
257C22#381111yescharacteristic
258C4—41no—
259C4—121no—
260C4—181no—
261C4#379911yescharacteristic
262C4D3×D1211yescharacteristic
263C3—21no—
264C3#478211yescharacteristic, complemented
265C3#486511yescharacteristic, complemented
266C2—241no—
267C2—361no—
268C2—721no—
269C2#6391011yescharacteristic
270C2#6393711yescharacteristic
271C2#6394111yescharacteristic
subgroupG/H
centerC22#3811
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. #64104C2semidirect
2. #63916C2semidirect
3. C2×#3714C2semidirect
4. C2×#3712C2semidirect
5. C3×#809C2semidirect
6. C3×#808C2semidirect
7. #63564C2semidirect
36. #809D3semidirect
37. #808D3semidirect
54. C3×C3⋊C8D4semidirect
152. C3⋊C8D12semidirect
213. C32#469semidirect
264. C3#4782semidirect
265. C3#4865semidirect

Maximal subgroups

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

subgroupindex
#641042
#639162
C2×#37142
C2×#37122
C3×#8092
C3×#8082
#635642
#48653
#47823