Finite Group Databasebeta

← #10590590

Subgroups — #10590590

Order 576.

261 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 261 conjugacy classes of subgroups — too many to draw. Only its 74 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×#893C211yesmaximal, characteristic
2#63844C211yesmaximal, characteristic, complemented
3C2×#3716C211yesmaximal, characteristic
4#63637C211yesmaximal, characteristic, complemented
5C3×#807C211yesmaximal, characteristic, complemented
6#63562C211yesmaximal, characteristic
7#63554C211yesmaximal, characteristic, complemented
8#5018—31nomaximal
9#4889—31nomaximal
10C2×#634C2211yescharacteristic
11C2×C3×D12C2211yescharacteristic
12C2×C3×C4×D3—21no—
13C32×C4⋊C4C2211yescharacteristic
14C2×C3⋊(C3⋊C8)C2211yescharacteristic
15C3×(C2×D3)⋊C4—21no—
16C3×C4⋊Dic3C2211yescharacteristic
17C3×C4×Dic3C2211yescharacteristic
18C2×C3×C3⋊C8C2211yescharacteristic
19#3724—21no—
20Dic32—21no—
21#3716C414yescomplemented
22C3×C4×D4—31no—
23C2×#214—31no—
24C2×#204—31no—
25#893D311yescharacteristic
26#887—31no—
27C4×Dic6—31no—
28C3×#107—31no—
29#832—31no—
30#815—31no—
31#810—31no—
32#810—31no—
33#810—61no—
34#809—31no—
35#807D311yescharacteristic, complemented
36#803—31no—
37C22×C3×D3—21no—
38C2×C32×C4—21no—
39C2×C32×C4C2311yescharacteristic
40C2×C3⋊Dic3—21no—
41#634—21no—
42#634C2×C412yescomplemented
43C2×C3×Dic3—21no—
44C2×C3×Dic3D411yescharacteristic
45C2×C3×Dic3D411yescharacteristic
46C3×D12—21no—
47C3×D12C2×C412yes—
48C3×C4×D3—42no—
49C3⋊(C3⋊C8)—21no—
50C3×C3⋊C8C2×C412yescomplemented
51#440—91noSylow
52C2×C3×D4—31no—
53C22×C3×C4—61no—
54C2×D12C2×D311yescharacteristic
55C2×C4×D3—21no—
56C2×Dic6—31no—
57C2×Dic6—31no—
58C2×Dic6—61no—
59C2×C3×C8—31no—
60C3×C4⋊C4—21no—
61C3×C4⋊C4—31no—
62C3×C4⋊C4C2×D311yescharacteristic
63C3×C4⋊C4C2×D311yescharacteristic
64C3×C22⋊C4—61no—
65C3×C42—31no—
66#214—34no—
67(C2×D3)⋊C4—21no—
68C4⋊Dic3C2×D311yescharacteristic
69Dic3⋊C4—61no—
70Dic3⋊C4—61no—
71C4×Dic3—61no—
72C4×Dic3—61no—
73C4×Dic3C2×D311yescharacteristic
74C2×C3⋊C8—31no—
75C2×C3⋊C8—31no—
76C2×C3⋊C8—61no—
77C2×C3⋊C8C2×D311yescharacteristic
78#204—34no—
79C22×C32C2×D411yescharacteristic
80C2×C3×D3—22no—
81C2×C3×D3—41no—
82C2×C3×D3—41no—
83C32×C4—41no—
84C32×C4C22×C411yescharacteristic
85C32×C4#4111yescharacteristic
86C3⋊Dic3—22no—
87C3×Dic3—21no—
88C3×Dic3—21no—
89C3×Dic3—41no—
90C2×SD16—91no—
91C4×Q8—91no—
92C4×D4—31no—
93#107—31no—
94#103—91no—
95#102—91no—
96#97—91no—
97C23×C3—61no—
98C22×D3—21no—
99C3×D4—32no—
100C3×D4—61no—
101C2×C3×C4—21no—
102C2×C3×C4—21no—
103C2×C3×C4—21no—
104C2×C3×C4—31no—
105C2×C3×C4—31no—
106C2×C3×C4—41no—
107C2×C3×C4—61no—
108C2×C3×C4—122no—
109C2×C3×C4C22×D311yescharacteristic
110C2×C3×C4C22×D311yescharacteristic
111C2×Dic3—21no—
112C2×Dic3—61no—
113C2×Dic3—61no—
114C2×Dic3—121no—
115C2×Dic3#6711yescharacteristic
116C2×Dic3#6711yescharacteristic
117D12—21no—
118D12C4×D312yes—
119C4×D3—42no—
120Dic6—32no—
121Dic6—32no—
122Dic6—62no—
123Dic6—61no—
124Dic6—61no—
125Dic6—121no—
126C3×C8—32no—
127C3⋊C8—61no—
128C3⋊C8—61no—
129C3⋊C8—121no—
130C3⋊C8C4×D312yescomplemented
131C2×C32C4×D411yescharacteristic
132C2×C32AGL(1,ℤ/8)11yescharacteristic
133C2×C32#13711yescharacteristic
134C3×D3—42no—
135C2×Q8—91no—
136C2×D4—31no—
137C22×C4—61no—
138SD16—94no—
139C2×C8—31no—
140C2×C8—91no—
141C4⋊C4—31no—
142C4⋊C4—181no—
143C4⋊C4D3211yescharacteristic
144C22⋊C4—61no—
145C42—31no—
146C42—181no—
147C22×C3—21no—
148C22×C3—62no—
149C22×C3—121no—
150C22×C3—121no—
151C22×C3D3×D411yescharacteristic
152C22×C3C2×#6711yescharacteristic
153C2×D3—22no—
154C2×D3—41no—
155C2×D3—41no—
156C3×C4—21no—
157C3×C4—21no—
158C3×C4—41no—
159C3×C4—41no—
160C3×C4—61no—
161C3×C4—61no—
162C3×C4—81no—
163C3×C4—121no—
164C3×C4C2×C4×D311yescharacteristic
165C3×C4C2×C4×D311yescharacteristic
166C3×C4#23511yescharacteristic
167C3×C4#23911yescharacteristic
168Dic3—21no—
169Dic3—21no—
170Dic3—41no—
171Dic3—62no—
172Dic3—62no—
173Dic3—122no—
174C32#44011yescharacteristic, complemented, Sylow
175C23—61no—
176Q8—92no—
177Q8—181no—
178D4—32no—
179D4—61no—
180C2×C4—21no—
181C2×C4—31no—
182C2×C4—31no—
183C2×C4—61no—
184C2×C4—122no—
185C2×C4—181no—
186C2×C4C2×D3211yescharacteristic
187C8—32no—
188C8—181no—
189C2×C3—21no—
190C2×C3—21no—
191C2×C3—21no—
192C2×C3—122no—
193C2×C3#89311yescharacteristic
194C2×C3#91111yescharacteristic
195C2×C3#91811yescharacteristic
196C2×C3C4×#6711yescharacteristic
197C2×C3#93211yescharacteristic
198C2×C3#95111yescharacteristic
199D3—42no—
200C22—62no—
201C22—121no—
202C22—121no—
203C22D3×#6711yescharacteristic
204C4—41no—
205C4—61no—
206C4—61no—
207C4—121no—
208C4—182no—
209C4#379911yescharacteristic
210C4C4×D3211yescharacteristic
211C3—21no—
212C3#488911yescharacteristic, complemented
213C3#501811yescharacteristic, complemented
214C2—122no—
215C2#6390411yescharacteristic
216C2#6393211yescharacteristic
217C2#6394311yescharacteristic
subgroupG/H
centerC22D3×#67
commutatorC32×C4C22×C4
FrattiniC2×C4C2×D32
FittingC32×C4⋊C4C22
socleC22×C32C2×D4
prime radicalC2×C3×D12C22
cosocleC2×C32×C4C23
hypercenterC4⋊C4D32
hypocenterC32#440
LCSG ▷ 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. #63844C2semidirect
4. #63637C2semidirect
5. C3×#807C2semidirect
7. #63554C2semidirect
21. #3716C4semidirect
35. #807D3semidirect
42. #634C2×C4semidirect
50. C3×C3⋊C8C2×C4semidirect
130. C3⋊C8C4×D3semidirect
174. C32#440semidirect
212. C3#4889semidirect
213. C3#5018semidirect

Maximal subgroups

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

subgroupindex
C3×#8932
#638442
C2×#37162
#636372
C3×#8072
#635622
#635542
#50183
#48893