Finite Group Databasebeta

← #13163503

Subgroups — #13163503

Order 1728.

1596 conjugacy classes of subgroups; only the 351 normal subgroups are shown.

normal subgroups only

This group has 1596 conjugacy classes of subgroups — too many to draw. Only its 351 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
2C4⋊C4×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3C33×#106C211yesmaximal, characteristic, complemented
4#11722191C211yesmaximal, characteristic, complemented
5#11722190C211yesmaximal, characteristic, complemented
6#11722179C211yesmaximal, characteristic, complemented
7#11722178C211yesmaximal, characteristic, complemented
8#10592562—313nomaximal
9C2×C4⋊(C3⋊(C3⋊D3))—21no—
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C4×C3⋊(C3⋊D3)—21no—
12C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
13C2×C33×C8C2211yescharacteristic
14C33×C4⋊C4C2211yescharacteristic
15C33×C4⋊C4C2211yescharacteristic
16#88859—21no—
17C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
18#88857—21no—
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20#64104—313no—
21C4⋊C4×C3⋊D3—313no—
22C32×#106D3113yescomplemented
23#63648—313no—
24#63647—313no—
25#63636—313no—
26#63635—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—
39C4×C3⋊(C3⋊D3)—42no—
40C33×C8—21no—
41C3⋊(C3⋊(C3⋊C8))—21no—
42#4863—913no—
43C2×C4⋊(C3⋊D3)—313no—
44C2×C4⋊(C3⋊D3)—613no—
45C2×C4×C3⋊D3—313no—
46C2×C4×C3⋊D3—613no—
47C2×C32×C8C2×D3113yes—
48C32×C4⋊C4C2×D3113yes—
49C32×C4⋊C4C2×D3113yes—
50#3752—613no—
51C4⋊(C3⋊Dic3)—313no—
52#3750—613no—
53C2×C3⋊(C3⋊C8)—313no—
54C22×C33C2×D411yescharacteristic
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—
60C2×C3⋊(C3⋊D3)SD1612yes—
61C33×C4—41no—
62C33×C4—41no—
63C33×C4#4111yescharacteristic
64C33×C4#4111yescharacteristic
65C3⋊(C3⋊Dic3)—21no—
66C3⋊(C3⋊Dic3)—41no—
67#895—913no—
68D3×C4⋊C4—913no—
69C3×#106C3⋊D3113yescomplemented
70#821—913no—
71(C2×D3)⋊C8—913no—
72#809—913no—
73#808—913no—
74C22×C3⋊D3—313no—
75C22×C3⋊D3—613no—
76C2×C32×C4—213no—
77C2×C32×C4—213no—
78C2×C32×C4C22×D3113yes—
79C2×C3⋊Dic3—313no—
80C2×C3⋊Dic3—613no—
81C4⋊(C3⋊D3)—613no—
82C4⋊(C3⋊D3)—613no—
83C4⋊(C3⋊D3)—1226no—
84C4×C3⋊D3—613no—
85C4×C3⋊D3—613no—
86C4×C3⋊D3—1226no—
87C32×C8—213no—
88C3⋊(C3⋊C8)—613no—
89#481—271noSylow
90C2×C33#12311yescharacteristic
91C2×C33C2×SD1611yescharacteristic
92C2×C33AGL(1,ℤ/8)11yescharacteristic
93C3⋊(C3⋊D3)—22no—
94C3⋊(C3⋊D3)—81no—
95C2×D12—913no—
96C2×D12—1813no—
97C2×C4×D3—913no—
98C2×C4×D3—1813no—
99C2×C3×C8C2×C3⋊D3113yes—
100C3×C4⋊C4C2×C3⋊D3113yes—
101C3×C4⋊C4C2×C3⋊D3113yes—
102(C2×D3)⋊C4—1813no—
103C4⋊Dic3—913no—
104Dic3⋊C4—1813no—
105C2×C3⋊C8—913no—
106C22×C32D3×D4113yes—
107C2×C3⋊D3—326no—
108C2×C3⋊D3—613no—
109C2×C3⋊D3—613no—
110C2×C3⋊D3—1213no—
111C2×C3⋊D3—1213no—
112C2×C3⋊D3—1213no—
113C32×C4—413no—
114C32×C4—413no—
115C32×C4#235113yes—
116C32×C4#239113yes—
117C3⋊Dic3—613no—
118C3⋊Dic3—1213no—
119#121—271no—
120C2×C4⋊C4—271no—
121#106—271no—
122#106C3⋊(C3⋊D3)11yescharacteristic, complemented
123#102—271no—
124#102—271no—
125C22⋊C8—271no—
126C33#48111yescharacteristic, complemented, Sylow
127C22×D3—913no—
128C22×D3—1813no—
129C2×C3×C4—213no—
130C2×C3×C4—213no—
131C2×C3×C4C22×C3⋊D3113yes—
132C2×Dic3—913no—
133C2×Dic3—1813no—
134D12—1813no—
135D12—1813no—
136D12—3626no—
137C4×D3—1813no—
138C4×D3—1813no—
139C4×D3—3626no—
140C3×C8—213no—
141C3⋊C8—1813no—
142C2×C32#894113yes—
143C2×C32D3×SD16113yes—
144C2×C32#914113yes—
145C3⋊D3—626no—
146C3⋊D3—2413no—
147C2×D4—271no—
148C2×D4—541no—
149C22×C4—271no—
150C22×C4—541no—
151C2×C8—271no—
152C2×C8C2×C3⋊(C3⋊D3)11yescharacteristic
153C4⋊C4—271no—
154C4⋊C4—541no—
155C4⋊C4C2×C3⋊(C3⋊D3)11yescharacteristic
156C4⋊C4C2×C3⋊(C3⋊D3)11yescharacteristic
157C22⋊C4—541no—
158C22×C3D4×C3⋊D3113yes—
159C2×D3—926no—
160C2×D3—1813no—
161C2×D3—1813no—
162C2×D3—3613no—
163C2×D3—3613no—
164C2×D3—3613no—
165C3×C4—413no—
166C3×C4—413no—
167C3×C4#3828113yes—
168C3×C4#3832113yes—
169Dic3—1813no—
170Dic3—3613no—
171C32#4863113yescomplemented
172C23—271no—
173C23—541no—
174D4—541no—
175D4—541no—
176D4—1082no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—271no—
180C2×C4—541no—
181C2×C4—541no—
182C2×C4—541no—
183C2×C4—1082no—
184C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
185C8—21no—
186C8—541no—
187C2×C3#64103113yes—
188C2×C3SD16×C3⋊D3113yes—
189C2×C3#64123113yes—
190D3—1826no—
191D3—7213no—
192C22—272no—
193C22—541no—
194C22—541no—
195C22—1081no—
196C22—1081no—
197C22—1081no—
198C22D4×C3⋊(C3⋊D3)11yescharacteristic
199C4—41no—
200C4—41no—
201C4—541no—
202C4—1081no—
203C4#8907611yescharacteristic
204C4#8908011yescharacteristic
205C3#10592562113yescomplemented
206C2—542no—
207C2—2161no—
208C2#1172393511yescharacteristic
209C2SD16×C3⋊(C3⋊D3)11yescharacteristic
210C2#1172395511yescharacteristic
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#481
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. C4⋊C4×C3⋊(C3⋊D3)C2semidirect
3. C33×#106C2semidirect
4. #11722191C2semidirect
5. #11722190C2semidirect
6. #11722179C2semidirect
7. #11722178C2semidirect
22. C32×#106D3semidirect
69. C3×#106C3⋊D3semidirect
122. #106C3⋊(C3⋊D3)semidirect
126. C33#481semidirect
171. C32#4863semidirect
205. C3#10592562semidirect

Maximal subgroups

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

subgroupindex
#117239362
C4⋊C4×C3⋊(C3⋊D3)2
C33×#1062
#117221912
#117221902
#117221792
#117221782
#105925623