Finite Group Databasebeta

← #13163402

Subgroups — #13163402

Order 1728.

1736 conjugacy classes of subgroups; only the 351 normal subgroups are shown.

normal subgroups only

This group has 1736 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#11723977C211yesmaximal, characteristic, complemented
2C2×#88860C211yesmaximal, characteristic
3C2×#88851C211yesmaximal, characteristic, complemented
4#11723934C211yesmaximal, characteristic, complemented
5C33×#102C211yesmaximal, characteristic, complemented
6#11722184C211yesmaximal, characteristic, complemented
7#11722180C211yesmaximal, characteristic, complemented
8#10592486—313nomaximal
9C2×C33×D4C2211yescharacteristic
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C4×C3⋊(C3⋊D3)—21no—
12C2×#6308C2211yescharacteristic
13C2×C33×C8C2211yescharacteristic
14C33×C4⋊C4C2211yescharacteristic
15#88864—21no—
16#88864—21no—
17#88860—22no—
18#88859—21no—
19C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
20C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
21#88851—22no—
22#64145—313no—
23C2×#3753—313no—
24C2×#3744—313no—
25#64102—313no—
26C32×#102D3113yescomplemented
27#63641—313no—
28#63637—313no—
29C23×C33—21no—
30C22×C3⋊(C3⋊D3)—21no—
31C33×D4—21no—
32C33×D4—21no—
33C2×C33×C4—21no—
34C2×C33×C4C2311yescharacteristic
35C2×C3⋊(C3⋊Dic3)—21no—
36C2×C3⋊(C3⋊Dic3)D411yescharacteristic
37C2×C3⋊(C3⋊Dic3)D411yescharacteristic
38C4⋊(C3⋊(C3⋊D3))—21no—
39C4⋊(C3⋊(C3⋊D3))D412yes—
40C4×C3⋊(C3⋊D3)—42no—
41#6308—21no—
42#6308—21no—
43C33×C8—21no—
44C3⋊(C3⋊(C3⋊C8))—21no—
45#4787—913no—
46C2×C32×D4C2×D3113yes—
47C2×C4⋊(C3⋊D3)—313no—
48C2×C4×C3⋊D3—613no—
49C2×#634—313no—
50C2×C32×C8C2×D3113yes—
51C32×C4⋊C4C2×D3113yes—
52#3757—613no—
53#3757—613no—
54#3753—626no—
55#3752—613no—
56C4×C3⋊Dic3—313no—
57C2×C3⋊(C3⋊C8)—313no—
58#3744—626no—
59C22×C33—41no—
60C22×C33—41no—
61C22×C33—41no—
62C22×C33C2×D411yescharacteristic
63C2×C3⋊(C3⋊D3)—22no—
64C2×C3⋊(C3⋊D3)—41no—
65C2×C3⋊(C3⋊D3)—41no—
66C33×C4—41no—
67C33×C4C2×D411yescharacteristic
68C33×C4#4111yescharacteristic
69C3⋊(C3⋊Dic3)—21no—
70C3⋊(C3⋊Dic3)—21no—
71C3⋊(C3⋊Dic3)—41no—
72#936—913no—
73C2×#213—913no—
74C2×#204—913no—
75#893—913no—
76C3×#102C3⋊D3113yescomplemented
77Dic3⋊C8—913no—
78#810—913no—
79C23×C32—213no—
80C22×C3⋊D3—613no—
81C32×D4—213no—
82C32×D4—213no—
83C2×C32×C4—213no—
84C2×C32×C4C22×D3113yes—
85C2×C3⋊Dic3—313no—
86C2×C3⋊Dic3—313no—
87C2×C3⋊Dic3—613no—
88C4⋊(C3⋊D3)—326no—
89C4⋊(C3⋊D3)—613no—
90C4×C3⋊D3—1226no—
91#634—613no—
92#634—613no—
93C32×C8—213no—
94C3⋊(C3⋊C8)—613no—
95#463—271noSylow
96C2×C33—81no—
97C2×C33#12111yescharacteristic
98C2×C33#13511yescharacteristic
99C2×C33AGL(1,ℤ/8)11yescharacteristic
100C3⋊(C3⋊D3)—42no—
101C2×C3×D4C2×C3⋊D3113yes—
102C2×D12—913no—
103C2×C4×D3—1813no—
104C2×Dic6—913no—
105C2×C3×C8C2×C3⋊D3113yes—
106C3×C4⋊C4C2×C3⋊D3113yes—
107#217—1813no—
108#217—1813no—
109#213—1826no—
110(C2×D3)⋊C4—1813no—
111C4×Dic3—913no—
112C2×C3⋊C8—913no—
113#204—1826no—
114C22×C32—413no—
115C22×C32—413no—
116C22×C32—413no—
117C22×C32D3×D4113yes—
118C2×C3⋊D3—626no—
119C2×C3⋊D3—1213no—
120C2×C3⋊D3—1213no—
121C32×C4—413no—
122C32×C4#235113yes—
123C32×C4D3×D4113yes—
124C3⋊Dic3—613no—
125C3⋊Dic3—613no—
126C3⋊Dic3—1213no—
127C2×SD16—271no—
128C2×D8—271no—
129#124—271no—
130C4×D4—271no—
131C4⋊C8—271no—
132#103—271no—
133#102C3⋊(C3⋊D3)11yescharacteristic, complemented
134C33#46311yescharacteristic, complemented, Sylow
135C23×C3—213no—
136C22×D3—1813no—
137C3×D4—213no—
138C3×D4—213no—
139C2×C3×C4—213no—
140C2×C3×C4C22×C3⋊D3113yes—
141C2×Dic3—913no—
142C2×Dic3—913no—
143C2×Dic3—1813no—
144D12—926no—
145D12—1813no—
146C4×D3—3626no—
147Dic6—1813no—
148Dic6—1813no—
149C3×C8—213no—
150C3⋊C8—1813no—
151C2×C32—813no—
152C2×C32#884113yes—
153C2×C32#911113yes—
154C2×C32#916113yes—
155C3⋊D3—1226no—
156C2×Q8—271no—
157C2×D4—271no—
158C2×D4C2×C3⋊(C3⋊D3)11yescharacteristic
159C22×C4—541no—
160SD16—542no—
161D8—542no—
162C2×C8—271no—
163C2×C8C2×C3⋊(C3⋊D3)11yescharacteristic
164C4⋊C4C2×C3⋊(C3⋊D3)11yescharacteristic
165C22⋊C4—541no—
166C22⋊C4—541no—
167C22⋊C4—541no—
168C42—271no—
169C22×C3—413no—
170C22×C3—413no—
171C22×C3—413no—
172C22×C3D4×C3⋊D3113yes—
173C2×D3—1826no—
174C2×D3—3613no—
175C2×D3—3613no—
176C3×C4—413no—
177C3×C4#3828113yes—
178C3×C4D4×C3⋊D3113yes—
179Dic3—1813no—
180Dic3—1813no—
181Dic3—3613no—
182C32#4787113yescomplemented
183C23—21no—
184C23—541no—
185Q8—541no—
186Q8—541no—
187D4—21no—
188D4—21no—
189D4—272no—
190D4—541no—
191C2×C4—21no—
192C2×C4—271no—
193C2×C4—271no—
194C2×C4—541no—
195C2×C4—1082no—
196C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
197C8—21no—
198C8—541no—
199C2×C3—813no—
200C2×C3#64093113yes—
201C2×C3#64120113yes—
202C2×C3#64125113yes—
203D3—3626no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—542no—
208C22—1081no—
209C22—1081no—
210C22D4×C3⋊(C3⋊D3)11yescharacteristic
211C4—41no—
212C4—541no—
213C4—541no—
214C4—1081no—
215C4#8907611yescharacteristic
216C4D4×C3⋊(C3⋊D3)11yescharacteristic
217C3#10592486113yescomplemented
218C2—81no—
219C2—1082no—
220C2#1172392511yescharacteristic
221C2#1172395211yescharacteristic
222C2#1172395711yescharacteristic
subgroupG/H
centerC22D4×C3⋊(C3⋊D3)
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C22×C3⋊(C3⋊D3)
FittingC33×#102C2
socleC22×C33C2×D4
prime radicalC2×#88860C2
hypercenter#102C3⋊(C3⋊D3)
hypocenterC33#463
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
1. #11723977C2semidirect
3. C2×#88851C2semidirect
4. #11723934C2semidirect
5. C33×#102C2semidirect
6. #11722184C2semidirect
7. #11722180C2semidirect
26. C32×#102D3semidirect
76. C3×#102C3⋊D3semidirect
133. #102C3⋊(C3⋊D3)semidirect
134. C33#463semidirect
182. C32#4787semidirect
217. C3#10592486semidirect

Maximal subgroups

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

subgroupindex
#117239772
C2×#888602
C2×#888512
#117239342
C33×#1022
#117221842
#117221802
#105924863