Finite Group Databasebeta

← #10590796

Subgroups — #10590796

Order 576.

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

normal subgroups only

This group has 243 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
1C2×C3×#204C211yesmaximal, characteristic
2#63863C211yesmaximal, characteristic
3#63843C211yesmaximal, characteristic, complemented
4C2×#3718C211yesmaximal, characteristic, complemented
5#63641C211yesmaximal, characteristic, complemented
6C3×#816C211yesmaximal, characteristic, complemented
7#63562C211yesmaximal, characteristic
8#5166—31nomaximal
9#4798—31nomaximal
10C2×C3×D12C2211yescharacteristic
11C2×C3×Dic6C2211yescharacteristic
12C2×C3×Dic6C2211yescharacteristic
13C2×C32×C8C2211yescharacteristic
14C4×C3⋊Dic3C2211yescharacteristic
15C2×C3⋊(C3⋊C8)C2211yescharacteristic
16C3×C4⋊Dic3C2211yescharacteristic
17C3×#204—22no—
18Dic3⋊Dic3—21no—
19(C2×D3)⋊Dic3—21no—
20(C2×D3)⋊Dic3—21no—
21Dic32—21no—
22#3718—22no—
23C2×C3×SD16—31no—
24#947—31no—
25Q8×Dic3—31no—
26C2×#216—31no—
27C2×#216—31no—
28#936—31no—
29C2×#204D311yescharacteristic
30#887—31no—
31C3×#103—31no—
32#832—31no—
33#816D311yescharacteristic, complemented
34Dic3⋊C8—31no—
35Dic3⋊C8—31no—
36Dic3⋊C8—61no—
37#809—31no—
38C22×C3×D3—21no—
39C2×C32×C4C2311yescharacteristic
40C2×C3⋊Dic3D411yescharacteristic
41C2×C3⋊Dic3D411yescharacteristic
42C2×C3×Dic3—21no—
43C2×C3×Dic3—21no—
44C2×C3×Dic3—21no—
45C3×D12—21no—
46C3×D12—21no—
47C3×Dic6—21no—
48C3×Dic6—21no—
49C3×Dic6—21no—
50C3×Dic6D412yes—
51C32×C8—21no—
52C3⋊(C3⋊C8)—21no—
53#464—91noSylow
54C2×C3×Q8—31no—
55C2×C3×Q8—31no—
56C2×C3×D4—31no—
57C2×D12C2×D311yescharacteristic
58C2×Dic6C2×D311yescharacteristic
59C2×Dic6C2×D311yescharacteristic
60C3×SD16—62no—
61C2×C3×C8—21no—
62C2×C3×C8C2×D311yescharacteristic
63C2×C3×C8C2×D311yescharacteristic
64C3×C4⋊C4—31no—
65#217—61no—
66#217—61no—
67#216—62no—
68#216—62no—
69(C2×D3)⋊C4—61no—
70(C2×D3)⋊C4—61no—
71C4⋊Dic3—61no—
72C4⋊Dic3C2×D311yescharacteristic
73Dic3⋊C4—61no—
74C4×Dic3—31no—
75C4×Dic3—31no—
76C4×Dic3—61no—
77C4×Dic3—61no—
78C4×Dic3—61no—
79C2×C3⋊C8—31no—
80C2×C3⋊C8—31no—
81C2×C3⋊C8—61no—
82#204—22no—
83C22×C32C2×D411yescharacteristic
84C2×C3×D3—41no—
85C2×C3×D3—41no—
86C2×C3×D3—41no—
87C32×C4C2×D411yescharacteristic
88C32×C4#4111yescharacteristic
89C3⋊Dic3—21no—
90C3⋊Dic3—21no—
91C3×Dic3—22no—
92C3×Dic3—41no—
93C3×Dic3—41no—
94C2×Q16—91no—
95C2×SD16—31no—
96#124—91no—
97C4×Q8—91no—
98C4⋊C8—91no—
99#103—31no—
100#102—91no—
101C23×C3—61no—
102C22×D3—21no—
103C3×Q8—32no—
104C3×Q8—61no—
105C3×Q8—61no—
106C3×Q8—61no—
107C3×D4—61no—
108C3×D4—61no—
109C2×C3×C4—21no—
110C2×C3×C4—61no—
111C2×C3×C4—61no—
112C2×C3×C4—61no—
113C2×C3×C4C22×D311yescharacteristic
114C2×C3×C4C22×D311yescharacteristic
115C2×Dic3—21no—
116C2×Dic3—21no—
117C2×Dic3—21no—
118C2×Dic3—31no—
119C2×Dic3—31no—
120C2×Dic3—31no—
121C2×Dic3—31no—
122C2×Dic3—61no—
123C2×Dic3—61no—
124D12—21no—
125D12—21no—
126Dic6—21no—
127Dic6—21no—
128Dic6—21no—
129Dic6#6712yes—
130C3×C8—21no—
131C3×C8—21no—
132C3×C8—41no—
133C3⋊C8—61no—
134C3⋊C8—61no—
135C3⋊C8—121no—
136C2×C32#12111yescharacteristic
137C2×C32#13511yescharacteristic
138C2×C32#13711yescharacteristic
139C3×D3—81no—
140C2×Q8—31no—
141C2×Q8—31no—
142C2×D4—31no—
143Q16—182no—
144SD16—62no—
145C2×C8—91no—
146C2×C8D3211yescharacteristic
147C4⋊C4—31no—
148C4⋊C4—181no—
149C22⋊C4—181no—
150C22⋊C4—181no—
151C42—91no—
152C42—181no—
153C22×C3—21no—
154C22×C3—121no—
155C22×C3—121no—
156C22×C3—121no—
157C22×C3D3×D411yescharacteristic
158C22×C3D3×D411yescharacteristic
159C2×D3—41no—
160C2×D3—41no—
161C2×D3—41no—
162C3×C4—21no—
163C3×C4—21no—
164C3×C4—62no—
165C3×C4—121no—
166C3×C4—121no—
167C3×C4#23511yescharacteristic
168C3×C4D3×D411yescharacteristic
169C3×C4#23711yescharacteristic
170C3×C4C2×#6711yescharacteristic
171Dic3—22no—
172Dic3—41no—
173Dic3—41no—
174Dic3—62no—
175Dic3—61no—
176Dic3—61no—
177Dic3—61no—
178Dic3—61no—
179Dic3—121no—
180C32#46411yescharacteristic, complemented, Sylow
181C23—61no—
182Q8—32no—
183Q8—61no—
184Q8—61no—
185Q8—61no—
186D4—61no—
187D4—61no—
188C2×C4—61no—
189C2×C4—61no—
190C2×C4—61no—
191C2×C4—91no—
192C2×C4—91no—
193C2×C4C2×D3211yescharacteristic
194C8—21no—
195C8—181no—
196C2×C3—21no—
197C2×C3—21no—
198C2×C3—21no—
199C2×C3—241no—
200C2×C3#88411yescharacteristic
201C2×C3#91511yescharacteristic
202C2×C3#91611yescharacteristic
203C2×C3#91611yescharacteristic
204C2×C3#91811yescharacteristic
205C2×C3#93911yescharacteristic
206D3—81no—
207C22—121no—
208C22—121no—
209C22—121no—
210C22#380211yescharacteristic
211C4—62no—
212C4—121no—
213C4—121no—
214C4—181no—
215C4—181no—
216C4#379511yescharacteristic
217C4D3×#6711yescharacteristic
218C3—21no—
219C3#479811yescharacteristic, complemented
220C3#516611yescharacteristic, complemented
221C2—241no—
222C2#6380911yescharacteristic
223C2#6381111yescharacteristic
224C2#6388511yescharacteristic
subgroupG/H
centerC22#3802
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC2×C32×C8C22
socleC22×C32C2×D4
prime radicalC2×C3×D12C22
hypercenterC2×C8D32
hypocenterC32#464
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
3. #63843C2semidirect
4. C2×#3718C2semidirect
5. #63641C2semidirect
6. C3×#816C2semidirect
33. #816D3semidirect
180. C32#464semidirect
219. C3#4798semidirect
220. C3#5166semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#2042
#638632
#638432
C2×#37182
#636412
C3×#8162
#635622
#51663
#47983