Finite Group Databasebeta

← #10590588

Subgroups — #10590588

Order 576.

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

normal subgroups only

This group has 261 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
2#63857C211yesmaximal, characteristic, complemented
3C2×#3716C211yesmaximal, characteristic, complemented
4C2×#3711C211yesmaximal, characteristic, complemented
5#63637C211yesmaximal, characteristic, complemented
6C3×#807C211yesmaximal, characteristic, complemented
7#63562C211yesmaximal, characteristic, complemented
8#5042—31nomaximal
9#4886—31nomaximal
10C2×#634C2211yescharacteristic
11C2×C3×D12—21no—
12C2×C3×D12C2211yescharacteristic
13C2×C3×C4×D3C2211yescharacteristic
14C32×C4⋊C4C2211yescharacteristic
15C2×C3⋊(C3⋊C8)C2211yescharacteristic
16C3×(C2×D3)⋊C4—21no—
17C3×C4⋊Dic3C2211yescharacteristic
18C2×C3×C3⋊C8C2211yescharacteristic
19Dic3⋊Dic3—21no—
20(C2×D3)⋊Dic3—21no—
21#3716—22no—
22#3711—22no—
23C3×#121—31no—
24C2×#214—31no—
25#924—31no—
26C2×#208—31no—
27C2×#204—31no—
28C2×ΓL(1,𝔽25)—31no—
29#897—31no—
30#895D311yescharacteristic
31C3×#107—31no—
32#832—31no—
33#810—31no—
34#810—31no—
35#810—61no—
36#809—31no—
37#807D311yescharacteristic, complemented
38C22×C3×D3—21no—
39C22×C3×D3D411yescharacteristic
40C2×C32×C4—21no—
41C2×C32×C4C2311yescharacteristic
42C2×C3⋊Dic3—21no—
43#634—21no—
44#634—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—
52C3⋊(C3⋊C8)—21no—
53C3×C3⋊C8D412yescomplemented
54#468—91noSylow
55C2×C3×D4—31no—
56C2×C3×D4—61no—
57C22×C3×C4—31no—
58C2×D12—21no—
59C2×D12C2×D311yescharacteristic
60C2×C4×D3C2×D311yescharacteristic
61C2×Dic6—31no—
62C2×Dic6—31no—
63C2×Dic6—61no—
64C2×C3×C8—31no—
65C3×C4⋊C4—21no—
66C3×C4⋊C4—31no—
67C3×C4⋊C4C2×D311yescharacteristic
68C3×C4⋊C4C2×D311yescharacteristic
69C3×C22⋊C4—61no—
70#217—61no—
71#214—62no—
72(C2×D3)⋊C4—21no—
73(C2×D3)⋊C4—61no—
74C4⋊Dic3—61no—
75C4⋊Dic3C2×D311yescharacteristic
76Dic3⋊C4—61no—
77#208—62no—
78C2×C3⋊C8—31no—
79C2×C3⋊C8—31no—
80C2×C3⋊C8—61no—
81C2×C3⋊C8C2×D311yescharacteristic
82#204—62no—
83ΓL(1,𝔽25)—62no—
84C22×C32C2×D411yescharacteristic
85C2×C3×D3—21no—
86C2×C3×D3—21no—
87C2×C3×D3—21no—
88C2×C3×D3—41no—
89C2×C3×D3—41no—
90C2×C3×D3—41no—
91C32×C4—41no—
92C32×C4C2×D411yescharacteristic
93C32×C4#4111yescharacteristic
94C3⋊Dic3—41no—
95C3×Dic3—21no—
96C3×Dic3—41no—
97C2×SD16—91no—
98C2×M16—91no—
99C22⋊Q8—91no—
100#121—31no—
101#107—31no—
102#103—91no—
103#102—91no—
104C23×C3—31no—
105C23×C3—61no—
106C22×D3—21no—
107C22×D3#6711yescharacteristic
108C3×D4—61no—
109C3×D4—61no—
110C3×D4—122no—
111C2×C3×C4—21no—
112C2×C3×C4—21no—
113C2×C3×C4—21no—
114C2×C3×C4—31no—
115C2×C3×C4—41no—
116C2×C3×C4—61no—
117C2×C3×C4—61no—
118C2×C3×C4—61no—
119C2×C3×C4C22×D311yescharacteristic
120C2×C3×C4C22×D311yescharacteristic
121C2×Dic3—21no—
122C2×Dic3—61no—
123C2×Dic3—61no—
124C2×Dic3—121no—
125C2×Dic3#6711yescharacteristic
126D12—21no—
127D12—21no—
128D12—42no—
129C4×D3—21no—
130C4×D3—21no—
131Dic6—62no—
132Dic6—61no—
133Dic6—61no—
134Dic6—61no—
135Dic6—61no—
136Dic6—121no—
137C3×C8—32no—
138C3⋊C8—61no—
139C3⋊C8—61no—
140C3⋊C8—121no—
141C3⋊C8D1212yescomplemented
142C2×C32#12111yescharacteristic
143C2×C32AGL(1,ℤ/8)11yescharacteristic
144C2×C32#13711yescharacteristic
145C3×D3—41no—
146C3×D3—81no—
147C2×Q8—91no—
148C2×D4—31no—
149C2×D4—61no—
150C22×C4—31no—
151SD16—182no—
152M16—182no—
153C2×C8—31no—
154C2×C8—91no—
155C4⋊C4—31no—
156C4⋊C4—181no—
157C4⋊C4D3211yescharacteristic
158C22⋊C4—61no—
159C22⋊C4—181no—
160C22×C3—21no—
161C22×C3—61no—
162C22×C3—61no—
163C22×C3—61no—
164C22×C3—121no—
165C22×C3—121no—
166C22×C3—121no—
167C22×C3D3×D411yescharacteristic
168C22×C3C2×#6711yescharacteristic
169C2×D3—21no—
170C2×D3—21no—
171C2×D3—21no—
172C2×D3—41no—
173C2×D3—41no—
174C2×D3—41no—
175C3×C4—21no—
176C3×C4—21no—
177C3×C4—41no—
178C3×C4—41no—
179C3×C4—61no—
180C3×C4—81no—
181C3×C4—121no—
182C3×C4C2×D1211yescharacteristic
183C3×C4D3×D411yescharacteristic
184C3×C4#23711yescharacteristic
185C3×C4#23911yescharacteristic
186Dic3—21no—
187Dic3—41no—
188Dic3—122no—
189Dic3—121no—
190Dic3—121no—
191C32#46811yescharacteristic, complemented, Sylow
192C23—31no—
193C23—61no—
194Q8—181no—
195Q8—181no—
196D4—61no—
197D4—61no—
198D4—122no—
199C2×C4—21no—
200C2×C4—31no—
201C2×C4—61no—
202C2×C4—61no—
203C2×C4—61no—
204C2×C4—181no—
205C2×C4C2×D3211yescharacteristic
206C8—32no—
207C8—181no—
208C2×C3—21no—
209C2×C3—21no—
210C2×C3—21no—
211C2×C3—121no—
212C2×C3—241no—
213C2×C3#89511yescharacteristic
214C2×C3#91111yescharacteristic
215C2×C3#91811yescharacteristic
216C2×C3#93211yescharacteristic
217C2×C3#93911yescharacteristic
218C2×C3#95111yescharacteristic
219D3—41no—
220D3—81no—
221C22—61no—
222C22—61no—
223C22—61no—
224C22—121no—
225C22—121no—
226C22—121no—
227C22D3×#6711yescharacteristic
228C4—41no—
229C4—61no—
230C4—121no—
231C4—361no—
232C4#379611yescharacteristic
233C4D3×D1211yescharacteristic
234C3—21no—
235C3#488611yescharacteristic, complemented
236C3#504211yescharacteristic, complemented
237C2—121no—
238C2—241no—
239C2#6388611yescharacteristic
240C2#6393211yescharacteristic
241C2#6394311yescharacteristic
subgroupG/H
centerC22D3×#67
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC32×C4⋊C4C22
socleC22×C32C2×D4
prime radicalC3×#895C2
hypercenterC4⋊C4D32
hypocenterC32#468
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
2. #63857C2semidirect
3. C2×#3716C2semidirect
4. C2×#3711C2semidirect
5. #63637C2semidirect
6. C3×#807C2semidirect
7. #63562C2semidirect
37. #807D3semidirect
53. C3×C3⋊C8D4semidirect
141. C3⋊C8D12semidirect
191. C32#468semidirect
235. C3#4886semidirect
236. C3#5042semidirect

Maximal subgroups

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

subgroupindex
C3×#8952
#638572
C2×#37162
C2×#37112
#636372
C3×#8072
#635622
#50423
#48863