Finite Group Databasebeta

← #13163492

Subgroups — #13163492

Order 1728.

1512 conjugacy classes of subgroups; only the 353 normal subgroups are shown.

normal subgroups only

This group has 1512 conjugacy classes of subgroups — too many to draw. Only its 353 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#11723934C211yesmaximal, characteristic
2#11723929C211yesmaximal, characteristic, complemented
3C33×#106C211yesmaximal, characteristic, complemented
4#11722191C211yesmaximal, characteristic
5#11722184C211yesmaximal, characteristic, complemented
6#11722179C211yesmaximal, characteristic
7#11722178C211yesmaximal, characteristic, complemented
8#10592569—313nomaximal
9C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
10C2×C4×C3⋊(C3⋊D3)—21no—
11C2×#6308—21no—
12C2×C33×C8C2211yescharacteristic
13C33×C4⋊C4C2211yescharacteristic
14C33×C4⋊C4C2211yescharacteristic
15#88859—21no—
16C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
17#88857—21no—
18C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20#64102—313no—
21#64097—313no—
22C32×#106D3113yescomplemented
23#63648—313no—
24#63641—313no—
25#63636—313no—
26#63635—313no—
27C22×C3⋊(C3⋊D3)—21no—
28C2×C33×C4—21no—
29C2×C33×C4—21no—
30C2×C33×C4C2311yescharacteristic
31C2×C3⋊(C3⋊Dic3)—21no—
32C2×C3⋊(C3⋊Dic3)D411yescharacteristic
33C2×C3⋊(C3⋊Dic3)D411yescharacteristic
34C4⋊(C3⋊(C3⋊D3))—21no—
35C4⋊(C3⋊(C3⋊D3))Q812yescomplemented
36C4×C3⋊(C3⋊D3)—42no—
37#6308—42no—
38C33×C8—21no—
39C3⋊(C3⋊(C3⋊C8))—21no—
40#4870—913no—
41C2×C4⋊(C3⋊D3)—313no—
42C2×C4×C3⋊D3—613no—
43C2×#634—613no—
44C2×C32×C8C2×D3113yes—
45C32×C4⋊C4C2×D3113yes—
46C32×C4⋊C4C2×D3113yes—
47#3752—613no—
48C4⋊(C3⋊Dic3)—313no—
49#3750—613no—
50C4×C3⋊Dic3—313no—
51C2×C3⋊(C3⋊C8)—313no—
52C22×C33C2×D411yescharacteristic
53C2×C3⋊(C3⋊D3)—22no—
54C2×C3⋊(C3⋊D3)—41no—
55C2×C3⋊(C3⋊D3)—41no—
56C33×C4—41no—
57C33×C4—41no—
58C33×C4C2×Q811yescharacteristic
59C33×C4#4111yescharacteristic
60C3⋊(C3⋊Dic3)—21no—
61C3⋊(C3⋊Dic3)—41no—
62C3⋊(C3⋊Dic3)SD1612yes—
63#893—913no—
64#888—913no—
65C3×#106C3⋊D3113yescomplemented
66#821—913no—
67Dic3⋊C8—913no—
68#809—913no—
69#808—913no—
70C22×C3⋊D3—613no—
71C2×C32×C4—213no—
72C2×C32×C4—213no—
73C2×C32×C4C22×D3113yes—
74C2×C3⋊Dic3—313no—
75C2×C3⋊Dic3—313no—
76C2×C3⋊Dic3—613no—
77C4⋊(C3⋊D3)—326no—
78C4⋊(C3⋊D3)—613no—
79C4×C3⋊D3—1226no—
80#634—1226no—
81C32×C8—213no—
82C3⋊(C3⋊C8)—613no—
83#476—271noSylow
84C2×C33C22⋊Q811yescharacteristic
85C2×C33C2×SD1611yescharacteristic
86C2×C33AGL(1,ℤ/8)11yescharacteristic
87C3⋊(C3⋊D3)—42no—
88C2×D12—913no—
89C2×C4×D3—1813no—
90C2×Dic6—1813no—
91C2×C3×C8C2×C3⋊D3113yes—
92C3×C4⋊C4C2×C3⋊D3113yes—
93C3×C4⋊C4C2×C3⋊D3113yes—
94(C2×D3)⋊C4—1813no—
95C4⋊Dic3—913no—
96Dic3⋊C4—1813no—
97C4×Dic3—913no—
98C2×C3⋊C8—913no—
99C22×C32D3×D4113yes—
100C2×C3⋊D3—626no—
101C2×C3⋊D3—1213no—
102C2×C3⋊D3—1213no—
103C32×C4—413no—
104C32×C4—413no—
105C32×C4#235113yes—
106C32×C4D3×Q8113yes—
107C3⋊Dic3—326no—
108C3⋊Dic3—613no—
109C3⋊Dic3—1213no—
110C4⋊Q8—271no—
111C4×D4—271no—
112#106—271no—
113#106C3⋊(C3⋊D3)11yescharacteristic, complemented
114C4⋊C8—271no—
115#102—271no—
116#102—271no—
117C33#47611yescharacteristic, complemented, Sylow
118C22×D3—1813no—
119C2×C3×C4—213no—
120C2×C3×C4—213no—
121C2×C3×C4C22×C3⋊D3113yes—
122C2×Dic3—913no—
123C2×Dic3—913no—
124C2×Dic3—1813no—
125D12—926no—
126D12—1813no—
127C4×D3—3626no—
128Dic6—3626no—
129C3×C8—213no—
130C3⋊C8—1813no—
131C2×C32#896113yes—
132C2×C32D3×SD16113yes—
133C2×C32#914113yes—
134C3⋊D3—1226no—
135C2×Q8—541no—
136C2×D4—271no—
137C22×C4—541no—
138C2×C8—271no—
139C2×C8C2×C3⋊(C3⋊D3)11yescharacteristic
140C4⋊C4—271no—
141C4⋊C4—541no—
142C4⋊C4C2×C3⋊(C3⋊D3)11yescharacteristic
143C4⋊C4C2×C3⋊(C3⋊D3)11yescharacteristic
144C22⋊C4—541no—
145C42—271no—
146C22×C3D4×C3⋊D3113yes—
147C2×D3—1826no—
148C2×D3—3613no—
149C2×D3—3613no—
150C3×C4—413no—
151C3×C4—413no—
152C3×C4#3828113yes—
153C3×C4Q8×C3⋊D3113yes—
154Dic3—926no—
155Dic3—1813no—
156Dic3—3613no—
157C32#4870113yescomplemented
158C23—541no—
159Q8—1082no—
160D4—272no—
161D4—541no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—271no—
165C2×C4—271no—
166C2×C4—541no—
167C2×C4—1082no—
168C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
169C8—21no—
170C8—541no—
171C2×C3#64105113yes—
172C2×C3SD16×C3⋊D3113yes—
173C2×C3#64123113yes—
174D3—3626no—
175C22—542no—
176C22—1081no—
177C22—1081no—
178C22D4×C3⋊(C3⋊D3)11yescharacteristic
179C4—41no—
180C4—41no—
181C4—272no—
182C4—541no—
183C4—1081no—
184C4#8907611yescharacteristic
185C4Q8×C3⋊(C3⋊D3)11yescharacteristic
186C3#10592569113yescomplemented
187C2—1082no—
188C2#1172393711yescharacteristic
189C2SD16×C3⋊(C3⋊D3)11yescharacteristic
190C2#1172395511yescharacteristic
subgroupG/H
centerC22D4×C3⋊(C3⋊D3)
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C22×C3⋊(C3⋊D3)
FittingC33×#106C2
socleC22×C33C2×D4
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
hypercenter#106C3⋊(C3⋊D3)
hypocenterC33#476
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. #11723929C2semidirect
3. C33×#106C2semidirect
5. #11722184C2semidirect
7. #11722178C2semidirect
22. C32×#106D3semidirect
35. C4⋊(C3⋊(C3⋊D3))Q8semidirect
65. C3×#106C3⋊D3semidirect
113. #106C3⋊(C3⋊D3)semidirect
117. C33#476semidirect
157. C32#4870semidirect
186. C3#10592569semidirect

Maximal subgroups

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

subgroupindex
#117239342
#117239292
C33×#1062
#117221912
#117221842
#117221792
#117221782
#105925693