Finite Group Databasebeta

← #13163399

Subgroups — #13163399

Order 1728.

1512 conjugacy classes of subgroups; only the 407 normal subgroups are shown.

normal subgroups only

This group has 1512 conjugacy classes of subgroups — too many to draw. Only its 407 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
1D4×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic
2#11723929C211yesmaximal, characteristic, complemented
3C33×#102C211yesmaximal, characteristic
4#11722202C211yesmaximal, characteristic
5#11722187C211yesmaximal, characteristic, complemented
6#11722184C211yesmaximal, characteristic, complemented
7#11722178C211yesmaximal, characteristic, complemented
8#10592460—313nomaximal
9C2×C33×D4C2211yescharacteristic
10C22×C3⋊(C3⋊Dic3)—21no—
11C2×#6308—21no—
12C2×C33×C8C2211yescharacteristic
13C33×C4⋊C4C2211yescharacteristic
14#88864—21no—
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
17#88857—21no—
18C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20D4×C3⋊Dic3—313no—
21#64097—313no—
22C32×#102D3113yes—
23#63659—313no—
24#63644—313no—
25#63641—313no—
26#63635—313no—
27C23×C33—21no—
28C33×D4—21no—
29C33×D4Q812yescomplemented
30C2×C33×C4—21no—
31C2×C33×C4C2311yescharacteristic
32C2×C3⋊(C3⋊Dic3)—21no—
33C2×C3⋊(C3⋊Dic3)—21no—
34C2×C3⋊(C3⋊Dic3)—42no—
35C2×C3⋊(C3⋊Dic3)D411yescharacteristic
36C2×C3⋊(C3⋊Dic3)D411yescharacteristic
37#6308—42no—
38C33×C8—21no—
39C3⋊(C3⋊(C3⋊C8))—21no—
40#4761—913no—
41C2×C32×D4C2×D3113yes—
42C22×C3⋊Dic3—613no—
43C2×#634—613no—
44C2×C32×C8C2×D3113yes—
45C32×C4⋊C4C2×D3113yes—
46#3757—613no—
47C4⋊(C3⋊Dic3)—313no—
48C4⋊(C3⋊Dic3)—313no—
49#3750—613no—
50C4×C3⋊Dic3—313no—
51C2×C3⋊(C3⋊C8)—313no—
52C22×C33—22no—
53C22×C33—41no—
54C22×C33—41no—
55C22×C33C2×D411yescharacteristic
56C33×C4—41no—
57C33×C4C2×Q811yescharacteristic
58C33×C4#4111yescharacteristic
59C3⋊(C3⋊Dic3)—21no—
60C3⋊(C3⋊Dic3)—41no—
61C3⋊(C3⋊Dic3)—41no—
62C3⋊(C3⋊Dic3)SD1612yes—
63D4×Dic3—913no—
64#888—913no—
65C3×#102C3⋊D3113yes—
66#832—913no—
67#817—913no—
68Dic3⋊C8—913no—
69#808—913no—
70C23×C32—213no—
71C32×D4—213no—
72C32×D4Dic6126yescomplemented
73C2×C32×C4—213no—
74C2×C32×C4C22×D3113yes—
75C2×C3⋊Dic3—313no—
76C2×C3⋊Dic3—313no—
77C2×C3⋊Dic3—613no—
78C2×C3⋊Dic3—613no—
79C2×C3⋊Dic3—1226no—
80#634—1226no—
81C32×C8—213no—
82C3⋊(C3⋊C8)—613no—
83#476—271noSylow
84C2×C33—42no—
85C2×C33C22⋊Q811yescharacteristic
86C2×C33C2×SD1611yescharacteristic
87C2×C33AGL(1,ℤ/8)11yescharacteristic
88C2×C3×D4C2×C3⋊D3113yes—
89C22×Dic3—1813no—
90C2×Dic6—1813no—
91C2×C3×C8C2×C3⋊D3113yes—
92C3×C4⋊C4C2×C3⋊D3113yes—
93#217—1813no—
94C4⋊Dic3—913no—
95C4⋊Dic3—913no—
96Dic3⋊C4—1813no—
97C4×Dic3—913no—
98C2×C3⋊C8—913no—
99C22×C32—226no—
100C22×C32—413no—
101C22×C32—413no—
102C22×C32D3×D4113yes—
103C32×C4—413no—
104C32×C4C2×Dic6113yes—
105C32×C4#237113yes—
106C3⋊Dic3—326no—
107C3⋊Dic3—613no—
108C3⋊Dic3—1213no—
109C3⋊Dic3—1213no—
110C4⋊Q8—271no—
111C4×D4—271no—
112#106—271no—
113#106—271no—
114C4⋊C8—271no—
115#102—271no—
116#102C3⋊(C3⋊D3)11yescharacteristic
117C33#47611yescharacteristic, complemented, Sylow
118C23×C3—213no—
119C3×D4—213no—
120C3×D4#634126yescomplemented
121C2×C3×C4—213no—
122C2×C3×C4C22×C3⋊D3113yes—
123C2×Dic3—913no—
124C2×Dic3—913no—
125C2×Dic3—1813no—
126C2×Dic3—1813no—
127C2×Dic3—3626no—
128Dic6—3626no—
129C3×C8—213no—
130C3⋊C8—1813no—
131C2×C32—426no—
132C2×C32#878113yes—
133C2×C32#911113yes—
134C2×C32D3×SD16113yes—
135C2×Q8—541no—
136C2×D4C2×C3⋊(C3⋊D3)11yescharacteristic
137C22×C4—541no—
138C2×C8—271no—
139C2×C8C2×C3⋊(C3⋊D3)11yescharacteristic
140C4⋊C4—271no—
141C4⋊C4—271no—
142C4⋊C4—541no—
143C4⋊C4C2×C3⋊(C3⋊D3)11yescharacteristic
144C22⋊C4—541no—
145C42—271no—
146C22×C3—226no—
147C22×C3—413no—
148C22×C3—413no—
149C22×C3D4×C3⋊D3113yes—
150C3×C4—413no—
151C3×C4C2×#634113yes—
152C3×C4#3830113yes—
153Dic3—926no—
154Dic3—1813no—
155Dic3—3613no—
156Dic3—3613no—
157C32#4761113yescomplemented
158C23—21no—
159Q8—1082no—
160D4—21no—
161D4#630812yescomplemented
162C2×C4—21no—
163C2×C4—271no—
164C2×C4—271no—
165C2×C4—541no—
166C2×C4—541no—
167C2×C4—1082no—
168C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
169C8—21no—
170C8—541no—
171C2×C3—426no—
172C2×C3#64087113yes—
173C2×C3#64120113yes—
174C2×C3SD16×C3⋊D3113yes—
175C22—22no—
176C22—41no—
177C22—41no—
178C22D4×C3⋊(C3⋊D3)11yescharacteristic
179C4—41no—
180C4—272no—
181C4—541no—
182C4—1081no—
183C4—1081no—
184C4C2×#630811yescharacteristic
185C4#8907811yescharacteristic
186C3#10592460113yescomplemented
187C2—42no—
188C2#1172391911yescharacteristic
189C2#1172395211yescharacteristic
190C2SD16×C3⋊(C3⋊D3)11yescharacteristic
subgroupG/H
centerC22D4×C3⋊(C3⋊D3)
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C22×C3⋊(C3⋊D3)
FittingC33×#102C2
socleC22×C33C2×D4
prime radicalC2×C33×D4C22
hypercenter#102C3⋊(C3⋊D3)
hypocenterC33#476
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ #102

Complements

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

NH ≅ G/Ntype
2. #11723929C2semidirect
5. #11722187C2semidirect
6. #11722184C2semidirect
7. #11722178C2semidirect
29. C33×D4Q8semidirect
72. C32×D4Dic6semidirect
117. C33#476semidirect
120. C3×D4#634semidirect
157. C32#4761semidirect
161. D4#6308semidirect
186. C3#10592460semidirect

Maximal subgroups

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

subgroupindex
D4×C3⋊(C3⋊Dic3)2
#117239292
C33×#1022
#117222022
#117221872
#117221842
#117221782
#105924603