Finite Group Databasebeta

← #10590572

Subgroups — #10590572

Order 576.

246 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 86 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×#892C211yesmaximal, characteristic
2#63841C211yesmaximal, characteristic
3C2×D3×C3⋊C8C211yesmaximal, characteristic
4#63634C211yesmaximal, characteristic, complemented
5C3×#808C211yesmaximal, characteristic, complemented
6#63576C211yesmaximal, characteristic, complemented
7#63573C211yesmaximal, characteristic, complemented
8#4998—31nomaximal
9#4860—31nomaximal
10C2×C3×C4×D3C2211yescharacteristic
11C32×C4⋊C4C2211yescharacteristic
12C4⋊(C3⋊Dic3)C2211yescharacteristic
13C2×C3⋊(C3⋊C8)C2211yescharacteristic
14C3×(C2×D3)⋊C4—21no—
15C3×C4⋊Dic3C2211yescharacteristic
16C3×C4⋊Dic3C2211yescharacteristic
17C3×C4×Dic3—21no—
18C2×C3×C3⋊C8C2211yescharacteristic
19(C2×D3)⋊Dic3—21no—
20Dic32—21no—
21D3×C3⋊C8C414yescomplemented
22C3×#117—31no—
23#926—31no—
24C22×C3⋊C8—31no—
25C2×D3×C8—31no—
26#892—31no—
27#892D311yescharacteristic
28C3×#106—31no—
29#817—31no—
30#808—31no—
31#808D311yescharacteristic, complemented
32#807—31no—
33#807—31no—
34#807—31no—
35#807—31no—
36#807—61no—
37C22×C3×D3D411yescharacteristic
38C2×C32×C4—21no—
39C2×C32×C4C2311yescharacteristic
40C2×C3⋊Dic3—21no—
41C2×C3×Dic3—21no—
42C2×C3×Dic3—21no—
43C2×C3×Dic3D411yescharacteristic
44C3×C4×D3C2×C412yes—
45C3×C4×D3Q812yes—
46C3⋊(C3⋊C8)C2×C412yes—
47C3×C3⋊C8C2×C412yes—
48#427—91noSylow
49C22×C3×C4—31no—
50C2×C4×D3C2×D311yescharacteristic
51C2×C3×C8—31no—
52C3×C4⋊C4—21no—
53C3×C4⋊C4—31no—
54C3×C4⋊C4—31no—
55C3×C4⋊C4C2×D311yescharacteristic
56C3×C4⋊C4C2×D311yescharacteristic
57C3×C22⋊C4—61no—
58C3×C42—61no—
59#217—61no—
60(C2×D3)⋊C4—21no—
61(C2×D3)⋊C4—61no—
62C4⋊Dic3—31no—
63C4⋊Dic3—31no—
64C4⋊Dic3—61no—
65C4⋊Dic3C2×D311yescharacteristic
66C4⋊Dic3C2×D311yescharacteristic
67C4×Dic3—21no—
68C4×Dic3—61no—
69C4×Dic3—61no—
70C2×C3⋊C8—31no—
71C2×C3⋊C8—31no—
72C2×C3⋊C8—34no—
73C2×C3⋊C8—61no—
74C2×C3⋊C8C2×D311yescharacteristic
75D3×C8—34no—
76C22×C32C2×D411yescharacteristic
77C2×C3×D3—21no—
78C2×C3×D3—21no—
79C2×C3×D3C4⋊C412yes—
80C32×C4—41no—
81C32×C4C22×C411yescharacteristic
82C32×C4C2×Q811yescharacteristic
83C3⋊Dic3—41no—
84C3×Dic3—41no—
85C3×Dic3—41no—
86C3×Dic3C4⋊C412yes—
87C22×C8—91no—
88#117—31no—
89#117—91no—
90#107—91no—
91#107—91no—
92#106—31no—
93#106—91no—
94C23×C3—31no—
95C22×D3#6711yescharacteristic
96C2×C3×C4—21no—
97C2×C3×C4—21no—
98C2×C3×C4—21no—
99C2×C3×C4—31no—
100C2×C3×C4—32no—
101C2×C3×C4—32no—
102C2×C3×C4—41no—
103C2×C3×C4—61no—
104C2×C3×C4—61no—
105C2×C3×C4C22×D311yescharacteristic
106C2×C3×C4C22×D311yescharacteristic
107C2×Dic3—21no—
108C2×Dic3—21no—
109C2×Dic3—61no—
110C2×Dic3—61no—
111C2×Dic3—121no—
112C2×Dic3#6711yescharacteristic
113C4×D3Dic612yes—
114C4×D3C4×D312yes—
115C3×C8—32no—
116C3⋊C8—32no—
117C3⋊C8—32no—
118C3⋊C8—62no—
119C3⋊C8C4×D312yes—
120C2×C32C2×C4⋊C411yescharacteristic
121C2×C32#13511yescharacteristic
122C2×C32#13511yescharacteristic
123C3×D3—22no—
124C22×C4—31no—
125C2×C8—31no—
126C2×C8—94no—
127C2×C8—91no—
128C4⋊C4—31no—
129C4⋊C4—31no—
130C4⋊C4—91no—
131C4⋊C4D3211yescharacteristic
132C22⋊C4—61no—
133C22⋊C4—181no—
134C42—61no—
135C42—181no—
136C22×C3—21no—
137C22×C3—32no—
138C22×C3—61no—
139C22×C3—61no—
140C22×C3D3×D411yescharacteristic
141C22×C3C2×#6711yescharacteristic
142C2×D3—21no—
143C2×D3—21no—
144C2×D3Dic3⋊C412yes—
145C3×C4—21no—
146C3×C4—21no—
147C3×C4—32no—
148C3×C4—41no—
149C3×C4—41no—
150C3×C4—81no—
151C3×C4—121no—
152C3×C4—121no—
153C3×C4C2×Dic611yescharacteristic
154C3×C4C2×C4×D311yescharacteristic
155C3×C4C2×C4×D311yescharacteristic
156C3×C4D3×Q811yescharacteristic
157Dic3—41no—
158Dic3—41no—
159Dic3—121no—
160Dic3—121no—
161Dic3—241no—
162Dic3Dic3⋊C412yes—
163C32#42711yescharacteristic, complemented, Sylow
164C23—31no—
165C2×C4—21no—
166C2×C4—31no—
167C2×C4—32no—
168C2×C4—32no—
169C2×C4—61no—
170C2×C4—61no—
171C2×C4—181no—
172C2×C4C2×D3211yescharacteristic
173C8—32no—
174C8—92no—
175C2×C3—21no—
176C2×C3—21no—
177C2×C3—21no—
178C2×C3—62no—
179C2×C3D3×C4⋊C411yescharacteristic
180C2×C3#91611yescharacteristic
181C2×C3#91611yescharacteristic
182C2×C3C2×Dic3⋊C411yescharacteristic
183C2×C3#95011yescharacteristic
184C2×C3#95011yescharacteristic
185D3—22no—
186C22—32no—
187C22—61no—
188C22—61no—
189C22D3×#6711yescharacteristic
190C4—32no—
191C4—41no—
192C4—121no—
193C4—121no—
194C4—361no—
195C4D3×Dic611yescharacteristic
196C4C4×D3211yescharacteristic
197C3—21no—
198C3#486011yescharacteristic, complemented
199C3#499811yescharacteristic, complemented
200C2—62no—
201C2D3×Dic3⋊C411yescharacteristic
202C2#6393511yescharacteristic
203C2#6394711yescharacteristic
subgroupG/H
centerC22D3×#67
commutatorC32×C4C22×C4
FrattiniC2×C4C2×D32
FittingC32×C4⋊C4C22
socleC22×C32C2×D4
prime radicalC22×C3×D3D4
cosocleC2×C32×C4C23
hypercenterC4⋊C4D32
hypocenterC32#427
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
4. #63634C2semidirect
5. C3×#808C2semidirect
6. #63576C2semidirect
7. #63573C2semidirect
21. D3×C3⋊C8C4semidirect
31. #808D3semidirect
163. C32#427semidirect
198. C3#4860semidirect
199. C3#4998semidirect

Maximal subgroups

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

subgroupindex
C3×#8922
#638412
C2×D3×C3⋊C82
#636342
C3×#8082
#635762
#635732
#49983
#48603