Finite Group Databasebeta

← #10589376⋊C3 #13135477

Subgroups — #10589376⋊C3 #13135477

Order 1728.

224 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 60 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#64113⋊C3C211yesmaximal, characteristic
2C2×C16×H(3,𝔽3)C211yesmaximal, characteristic, complemented
3#63646⋊C3C211yesmaximal, characteristic, complemented
4#10589376C311yesmaximal, characteristic, complemented
5C3×#4510—31nomaximal
6#3828⋊C3—21no—
7C2×C8×H(3,𝔽3)C2211yescharacteristic
8#3748⋊C3—21no—
9#3745⋊C3C411yescharacteristic
10#3744⋊C3—21no—
11#3746⋊C3C411yescharacteristic
12C16×H(3,𝔽3)—21no—
13#64113C2×C311yescharacteristic
14C3×#904—31no—
15C2×C32×C16—21no—
16C2×C32×C16D311yescharacteristic, complemented
17C2×C32×C16C2×C311yescharacteristic, complemented
18#63646C2×C311yescharacteristic, complemented
19C3×#819—31no—
20C2×C4×H(3,𝔽3)D411yescharacteristic
21#638⋊C3—41no—
22C4⋊((C3⋊D3)⋊C3)—21no—
23C4×(C3⋊D3)⋊C3—41no—
24#634⋊C3—21no—
25C8×H(3,𝔽3)C2×C411yescharacteristic
26C8×H(3,𝔽3)D411yescharacteristic
27(C3⋊(C3⋊C8))⋊C3—41no—
28C3×#359—91no—
29#4510—31no—
30#4510—91no—
31#3828—21no—
32C3×#235—61no—
33C2×C32×C8—21no—
34C2×C32×C8C2×D311yescharacteristic
35C2×C32×C8C22×C311yescharacteristic
36#3748—21no—
37#3746C3×C411yescharacteristic
38#3745C3×C411yescharacteristic
39#3744—21no—
40C3×#208—61no—
41C3×Dic12—31no—
42C3×D24—31no—
43C3×#204—61no—
44C32×C16—22no—
45C32×C16—21no—
46C32×C16—21no—
47C22×H(3,𝔽3)SD1611yescharacteristic
48C2×(C3⋊D3)⋊C3—41no—
49C4×H(3,𝔽3)C22⋊C411yescharacteristic
50C4×H(3,𝔽3)D811yescharacteristic
51(C3⋊Dic3)⋊C3—41no—
52C3×#135—91no—
53#904—31no—
54#904—91no—
55C2×C3×C16—31no—
56C2×C3×C16—31no—
57C2×C3×C16—61no—
58C2×C3×C16C3×D311yescharacteristic
59C3×#108—91no—
60#819—31no—
61#819—91no—
62C2×C32×C4—21no—
63C2×C32×C4D1211yescharacteristic
64C2×C32×C4C3×D411yescharacteristic
65#638—41no—
66C4⋊(C3⋊D3)—21no—
67C4×C3⋊D3—41no—
68#634—21no—
69C3×#67—121no—
70C3×D12—61no—
71C3×C4×D3—121no—
72C3×Dic6—61no—
73C32×C8—21no—
74C32×C8—21no—
75C32×C8C4×D311yescharacteristic
76C32×C8#6711yescharacteristic
77C32×C8C2×C3×C411yescharacteristic
78C32×C8C3×D411yescharacteristic
79C3⋊(C3⋊C8)—41no—
80C3×C3⋊C8—121no—
81#359—91noSylow
82C2×H(3,𝔽3)—21no—
83C2×H(3,𝔽3)#10211yescharacteristic
84(C3⋊D3)⋊C3—81no—
85C3×#41—181no—
86#235—61no—
87#235—181no—
88C3×Q16—91no—
89C3×SD16—181no—
90C3×D8—91no—
91C3×M16—181no—
92C2×C3×C8—31no—
93C2×C3×C8—31no—
94C2×C3×C8—61no—
95C2×C3×C8C2×C3×D311yescharacteristic
96#208—61no—
97#208—181no—
98Dic12—31no—
99Dic12—91no—
100D24—31no—
101D24—91no—
102#204—61no—
103#204—181no—
104C3×C16—21no—
105C3×C16—61no—
106C3×C16—61no—
107C3×C16—62no—
108C22×C32—21no—
109C22×C32#20411yescharacteristic
110C22×C32C3×SD1611yescharacteristic
111C2×C3⋊D3—41no—
112C2×C3×D3—121no—
113C32×C4—21no—
114C32×C4—21no—
115C32×C4D2411yescharacteristic
116C32×C4(C2×D3)⋊C411yescharacteristic
117C32×C4C3×C22⋊C411yescharacteristic
118C32×C4C3×D811yescharacteristic
119C3⋊Dic3—41no—
120C3×Dic3—121no—
121#135—91no—
122C2×C16(C3⋊D3)⋊C311yescharacteristic, complemented
123#108—91no—
124H(3,𝔽3)#35911yescharacteristic, complemented, Sylow
125C3×Q8—181no—
126C3×D4—181no—
127C3×D4—361no—
128C2×C3×C4—31no—
129C2×C3×C4—31no—
130C2×C3×C4—61no—
131C2×C3×C4—361no—
132C2×C3×C4C3×D1211yescharacteristic
133#67—121no—
134#67—361no—
135D12—61no—
136D12—181no—
137C4×D3—121no—
138C4×D3—361no—
139Dic6—61no—
140Dic6—181no—
141C3×C8—31no—
142C3×C8—31no—
143C3×C8—31no—
144C3×C8—31no—
145C3×C8—61no—
146C3×C8—61no—
147C3×C8—361no—
148C3×C8C3×C4×D311yescharacteristic
149C3×C8C3×#6711yescharacteristic
150C3⋊C8—121no—
151C3⋊C8—361no—
152C2×C32—22no—
153C2×C32—21no—
154C2×C32—21no—
155C2×C32—21no—
156C2×C32#82111yescharacteristic
157C2×C32C3×#10211yescharacteristic
158C3⋊D3—81no—
159C3×D3—241no—
160#41—181no—
161Q16—91no—
162SD16—181no—
163D8—91no—
164M16—181no—
165C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
166C16—21no—
167C22×C3—31no—
168C22×C3—31no—
169C22×C3—61no—
170C22×C3—361no—
171C22×C3C3×#20411yescharacteristic
172C2×D3—121no—
173C2×D3—361no—
174C3×C4—31no—
175C3×C4—31no—
176C3×C4—31no—
177C3×C4—31no—
178C3×C4—61no—
179C3×C4—61no—
180C3×C4—361no—
181C3×C4C3×D2411yescharacteristic
182C3×C4C3×(C2×D3)⋊C411yescharacteristic
183Dic3—121no—
184Dic3—361no—
185C32—21no—
186C32#451011yescharacteristic, complemented
187C32C3×#35911yescharacteristic, complemented
188Q8—181no—
189D4—181no—
190D4—361no—
191C2×C4—361no—
192C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
193C8—361no—
194C8C4×(C3⋊D3)⋊C311yescharacteristic
195C8#638⋊C311yescharacteristic
196C2×C3—21no—
197C2×C3—31no—
198C2×C3—31no—
199C2×C3—61no—
200C2×C3—61no—
201C2×C3—62no—
202C2×C3—61no—
203C2×C3—721no—
204C2×C3C3×#82111yescharacteristic
205D3—241no—
206D3—721no—
207C22—361no—
208C22#3744⋊C311yescharacteristic
209C4—361no—
210C4#3745⋊C311yescharacteristic
211C4#3752⋊C311yescharacteristic
212C3—31no—
213C3—31no—
214C3—61no—
215C3C3×#451011yescharacteristic
216C2—21no—
217C2—721no—
218C2#63648⋊C311yescharacteristic
subgroupG/H
centerC4#3745⋊C3
commutatorC32×C8C2×C3×C4
FrattiniC2×C3×C8C2×C3×D3
FittingC2×C16×H(3,𝔽3)C2
socleC2×C3C3×#821
prime radical#64113⋊C3C2
cosocleC2×C32×C8C22×C3
hypercenterC2×C16(C3⋊D3)⋊C3
hypocenterC32C3×#359
LCSG ▷ C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16

Complements

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

NH ≅ G/Ntype
2. C2×C16×H(3,𝔽3)C2semidirect
3. #63646⋊C3C2semidirect
4. #10589376C3semidirect
16. C2×C32×C16D3semidirect
17. C2×C32×C16C2×C3semidirect
18. #63646C2×C3semidirect
122. C2×C16(C3⋊D3)⋊C3semidirect
124. H(3,𝔽3)#359semidirect
186. C32#4510semidirect
187. C32C3×#359semidirect

Maximal subgroups

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

subgroupindex
#64113⋊C32
C2×C16×H(3,𝔽3)2
#63646⋊C32
#105893763
C3×#45103