Finite Group Databasebeta

← #13163505

Subgroups — #13163505

Order 1728.

1484 conjugacy classes of subgroups; only the 349 normal subgroups are shown.

normal subgroups only

This group has 1484 conjugacy classes of subgroups — too many to draw. Only its 349 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#11723936C211yesmaximal, characteristic
2#11723933C211yesmaximal, characteristic, complemented
3C33×#106C211yesmaximal, characteristic, complemented
4#11722191C211yesmaximal, characteristic, complemented
5#11722190C211yesmaximal, characteristic, complemented
6#11722179C211yesmaximal, characteristic, complemented
7#11722177C211yesmaximal, characteristic, complemented
8#10592565—313nomaximal
9C2×C4⋊(C3⋊(C3⋊D3))—21no—
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
12C2×C33×C8C2211yescharacteristic
13C33×C4⋊C4C2211yescharacteristic
14C33×C4⋊C4C2211yescharacteristic
15#88859—21no—
16#88859—21no—
17C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
18C4×C3⋊(C3⋊Dic3)—21no—
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20#64104—313no—
21#64101—313no—
22C32×#106D3113yescomplemented
23#63648—313no—
24#63647—313no—
25#63636—313no—
26#63634—313no—
27C22×C3⋊(C3⋊D3)—21no—
28C22×C3⋊(C3⋊D3)D411yescharacteristic
29C2×C33×C4—21no—
30C2×C33×C4—21no—
31C2×C33×C4C2311yescharacteristic
32C2×C3⋊(C3⋊Dic3)—21no—
33C2×C3⋊(C3⋊Dic3)D411yescharacteristic
34C4⋊(C3⋊(C3⋊D3))—21no—
35C4⋊(C3⋊(C3⋊D3))—21no—
36C4⋊(C3⋊(C3⋊D3))—42no—
37C4×C3⋊(C3⋊D3)—21no—
38C4×C3⋊(C3⋊D3)—21no—
39C33×C8—21no—
40C3⋊(C3⋊(C3⋊C8))—21no—
41#4866—913no—
42C2×C4⋊(C3⋊D3)—313no—
43C2×C4⋊(C3⋊D3)—613no—
44C2×C4×C3⋊D3—313no—
45C2×C32×C8C2×D3113yes—
46C32×C4⋊C4C2×D3113yes—
47C32×C4⋊C4C2×D3113yes—
48#3752—613no—
49#3752—613no—
50C4⋊(C3⋊Dic3)—313no—
51C4×C3⋊Dic3—613no—
52C2×C3⋊(C3⋊C8)—313no—
53C22×C33C2×D411yescharacteristic
54C2×C3⋊(C3⋊D3)—21no—
55C2×C3⋊(C3⋊D3)—21no—
56C2×C3⋊(C3⋊D3)—21no—
57C2×C3⋊(C3⋊D3)—41no—
58C2×C3⋊(C3⋊D3)—41no—
59C2×C3⋊(C3⋊D3)—41no—
60C33×C4—41no—
61C33×C4—41no—
62C33×C4#4111yescharacteristic
63C33×C4#4111yescharacteristic
64C3⋊(C3⋊Dic3)—21no—
65C3⋊(C3⋊Dic3)—41no—
66#895—913no—
67#892—913no—
68C3×#106C3⋊D3113yescomplemented
69#821—913no—
70(C2×D3)⋊C8—913no—
71#809—913no—
72#807—913no—
73C22×C3⋊D3—313no—
74C22×C3⋊D3—613no—
75C2×C32×C4—213no—
76C2×C32×C4—213no—
77C2×C32×C4C22×D3113yes—
78C2×C3⋊Dic3—313no—
79C2×C3⋊Dic3—613no—
80C4⋊(C3⋊D3)—613no—
81C4⋊(C3⋊D3)—613no—
82C4⋊(C3⋊D3)—1226no—
83C4×C3⋊D3—613no—
84C4×C3⋊D3—613no—
85C32×C8—213no—
86C3⋊(C3⋊C8)—613no—
87#482—271noSylow
88C2×C33#12311yescharacteristic
89C2×C33#13511yescharacteristic
90C2×C33AGL(1,ℤ/8)11yescharacteristic
91C3⋊(C3⋊D3)—41no—
92C3⋊(C3⋊D3)—81no—
93C2×D12—913no—
94C2×D12—1813no—
95C2×C4×D3—913no—
96C2×C3×C8C2×C3⋊D3113yes—
97C3×C4⋊C4C2×C3⋊D3113yes—
98C3×C4⋊C4C2×C3⋊D3113yes—
99(C2×D3)⋊C4—1813no—
100(C2×D3)⋊C4—1813no—
101C4⋊Dic3—913no—
102C4×Dic3—1813no—
103C2×C3⋊C8—913no—
104C22×C32D3×D4113yes—
105C2×C3⋊D3—613no—
106C2×C3⋊D3—613no—
107C2×C3⋊D3—613no—
108C2×C3⋊D3—1213no—
109C2×C3⋊D3—1213no—
110C2×C3⋊D3—1213no—
111C32×C4—413no—
112C32×C4—413no—
113C32×C4#235113yes—
114C32×C4#239113yes—
115C3⋊Dic3—613no—
116C3⋊Dic3—1213no—
117#121—271no—
118#117—271no—
119#107—271no—
120#106C3⋊(C3⋊D3)11yescharacteristic, complemented
121#102—271no—
122#102—271no—
123C22⋊C8—271no—
124C33#48211yescharacteristic, complemented, Sylow
125C22×D3—913no—
126C22×D3—1813no—
127C2×C3×C4—213no—
128C2×C3×C4—213no—
129C2×C3×C4C22×C3⋊D3113yes—
130C2×Dic3—913no—
131C2×Dic3—1813no—
132D12—1813no—
133D12—1813no—
134D12—3626no—
135C4×D3—1813no—
136C4×D3—1813no—
137C3×C8—213no—
138C3⋊C8—1813no—
139C2×C32#894113yes—
140C2×C32#914113yes—
141C2×C32#916113yes—
142C3⋊D3—1213no—
143C3⋊D3—2413no—
144C2×D4—271no—
145C2×D4—541no—
146C22×C4—271no—
147C2×C8—271no—
148C2×C8C2×C3⋊(C3⋊D3)11yescharacteristic
149C4⋊C4—271no—
150C4⋊C4C2×C3⋊(C3⋊D3)11yescharacteristic
151C4⋊C4C2×C3⋊(C3⋊D3)11yescharacteristic
152C22⋊C4—541no—
153C22⋊C4—541no—
154C42—541no—
155C22×C3D4×C3⋊D3113yes—
156C2×D3—1813no—
157C2×D3—1813no—
158C2×D3—1813no—
159C2×D3—3613no—
160C2×D3—3613no—
161C2×D3—3613no—
162C3×C4—413no—
163C3×C4—413no—
164C3×C4#3828113yes—
165C3×C4#3832113yes—
166Dic3—1813no—
167Dic3—3613no—
168C32#4866113yescomplemented
169C23—271no—
170C23—541no—
171D4—541no—
172D4—541no—
173D4—1082no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—271no—
177C2×C4—541no—
178C2×C4—541no—
179C2×C4—541no—
180C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
181C8—21no—
182C8—541no—
183C2×C3#64103113yes—
184C2×C3#64123113yes—
185C2×C3#64125113yes—
186D3—3613no—
187D3—7213no—
188C22—541no—
189C22—541no—
190C22—541no—
191C22—1081no—
192C22—1081no—
193C22—1081no—
194C22D4×C3⋊(C3⋊D3)11yescharacteristic
195C4—41no—
196C4—41no—
197C4—541no—
198C4—1081no—
199C4#8907611yescharacteristic
200C4#8908011yescharacteristic
201C3#10592565113yescomplemented
202C2—1081no—
203C2—2161no—
204C2#1172393511yescharacteristic
205C2#1172395511yescharacteristic
206C2#1172395711yescharacteristic
subgroupG/H
centerC22D4×C3⋊(C3⋊D3)
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C22×C3⋊(C3⋊D3)
FittingC33×#106C2
socleC22×C33C2×D4
prime radical#11723936C2
hypercenter#106C3⋊(C3⋊D3)
hypocenterC33#482
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ #106

Complements

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

NH ≅ G/Ntype
2. #11723933C2semidirect
3. C33×#106C2semidirect
4. #11722191C2semidirect
5. #11722190C2semidirect
6. #11722179C2semidirect
7. #11722177C2semidirect
22. C32×#106D3semidirect
68. C3×#106C3⋊D3semidirect
120. #106C3⋊(C3⋊D3)semidirect
124. C33#482semidirect
168. C32#4866semidirect
201. C3#10592565semidirect

Maximal subgroups

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

subgroupindex
#117239362
#117239332
C33×#1062
#117221912
#117221902
#117221792
#117221772
#105925653