Finite Group Databasebeta

← #13163529

Subgroups — #13163529

Order 1728.

1680 conjugacy classes of subgroups; only the 413 normal subgroups are shown.

normal subgroups only

This group has 1680 conjugacy classes of subgroups — too many to draw. Only its 413 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
1C2×#88851C211yesmaximal, characteristic
2#11723934C211yesmaximal, characteristic
3#11723928C211yesmaximal, characteristic, complemented
4C33×#107C211yesmaximal, characteristic, complemented
5#11722185C211yesmaximal, characteristic, complemented
6#11722180C211yesmaximal, characteristic, complemented
7#11722179C211yesmaximal, characteristic
8#10592588—313nomaximal
9C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
10C2×C4×C3⋊(C3⋊D3)—21no—
11C2×#6308C2211yescharacteristic
12C2×C33×C8C2211yescharacteristic
13C33×C4⋊C4C2211yescharacteristic
14C33×C4⋊C4C2211yescharacteristic
15#88859—21no—
16#88857—21no—
17C4×C3⋊(C3⋊Dic3)—21no—
18C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20#88851C414yescomplemented
21C2×#3744—313no—
22#64102—313no—
23#64096—313no—
24C32×#107D3113yescomplemented
25#63642—313no—
26#63637—313no—
27#63636—313no—
28C22×C3⋊(C3⋊D3)—21no—
29C2×C33×C4—21no—
30C2×C33×C4—21no—
31C2×C33×C4C2311yescharacteristic
32C2×C3⋊(C3⋊Dic3)—21no—
33C2×C3⋊(C3⋊Dic3)D411yescharacteristic
34C2×C3⋊(C3⋊Dic3)D411yescharacteristic
35C4⋊(C3⋊(C3⋊D3))—21no—
36C4⋊(C3⋊(C3⋊D3))C2×C412yes—
37C4×C3⋊(C3⋊D3)—42no—
38#6308—21no—
39#6308C2×C412yescomplemented
40C33×C8C2×C412yescomplemented
41C3⋊(C3⋊(C3⋊C8))—21no—
42#4889—913no—
43C2×C4⋊(C3⋊D3)—313no—
44C2×C4×C3⋊D3—613no—
45C2×#634—313no—
46C2×C32×C8C2×D3113yes—
47C32×C4⋊C4C2×D3113yes—
48C32×C4⋊C4C2×D3113yes—
49#3752—613no—
50#3750—613no—
51C4×C3⋊Dic3—313no—
52C4×C3⋊Dic3—613no—
53C2×C3⋊(C3⋊C8)—313no—
54#3744—352no—
55C22×C33C2×D411yescharacteristic
56C2×C3⋊(C3⋊D3)—22no—
57C2×C3⋊(C3⋊D3)—41no—
58C2×C3⋊(C3⋊D3)—41no—
59C33×C4—41no—
60C33×C4—41no—
61C33×C4C22×C411yescharacteristic
62C33×C4#4111yescharacteristic
63C3⋊(C3⋊Dic3)—22no—
64C3⋊(C3⋊Dic3)—21no—
65C3⋊(C3⋊Dic3)—21no—
66C2×#204—913no—
67#893—913no—
68#887—913no—
69C3×#107C3⋊D3113yescomplemented
70#815—913no—
71#810—913no—
72#809—913no—
73C22×C3⋊D3—613no—
74C2×C32×C4—213no—
75C2×C32×C4—213no—
76C2×C32×C4C22×D3113yes—
77C2×C3⋊Dic3—313no—
78C2×C3⋊Dic3—313no—
79C2×C3⋊Dic3—613no—
80C4⋊(C3⋊D3)—326no—
81C4⋊(C3⋊D3)—613no—
82C4×C3⋊D3—1226no—
83#634—326no—
84#634—613no—
85C32×C8C4×D3126yescomplemented
86C3⋊(C3⋊C8)—613no—
87#440—271noSylow
88C2×C33C4×D411yescharacteristic
89C2×C33AGL(1,ℤ/8)11yescharacteristic
90C2×C33#13711yescharacteristic
91C3⋊(C3⋊D3)—42no—
92C2×D12—913no—
93C2×C4×D3—1813no—
94C2×Dic6—913no—
95C2×C3×C8C2×C3⋊D3113yes—
96C3×C4⋊C4C2×C3⋊D3113yes—
97C3×C4⋊C4C2×C3⋊D3113yes—
98(C2×D3)⋊C4—1813no—
99Dic3⋊C4—1813no—
100C4×Dic3—913no—
101C4×Dic3—1813no—
102C2×C3⋊C8—913no—
103#204—952no—
104C22×C32D3×D4113yes—
105C2×C3⋊D3—626no—
106C2×C3⋊D3—1213no—
107C2×C3⋊D3—1213no—
108C32×C4—413no—
109C32×C4—413no—
110C32×C4C2×C4×D3113yes—
111C32×C4#239113yes—
112C3⋊Dic3—626no—
113C3⋊Dic3—613no—
114C3⋊Dic3—613no—
115C2×SD16—271no—
116C4×Q8—271no—
117C4×D4—271no—
118#107C3⋊(C3⋊D3)11yescharacteristic, complemented
119#103—271no—
120#102—271no—
121#97—271no—
122C33#44011yescharacteristic, complemented, Sylow
123C22×D3—1813no—
124C2×C3×C4—213no—
125C2×C3×C4—213no—
126C2×C3×C4C22×C3⋊D3113yes—
127C2×Dic3—913no—
128C2×Dic3—913no—
129C2×Dic3—1813no—
130D12—926no—
131D12—1813no—
132C4×D3—3626no—
133Dic6—926no—
134Dic6—1813no—
135C3×C8C4×C3⋊D3126yescomplemented
136C3⋊C8—1813no—
137C2×C32#893113yes—
138C2×C32#911113yes—
139C2×C32#918113yes—
140C3⋊D3—1226no—
141C2×Q8—271no—
142C2×D4—271no—
143C22×C4—541no—
144SD16—274no—
145C2×C8—271no—
146C2×C8C2×C3⋊(C3⋊D3)11yescharacteristic
147C4⋊C4—541no—
148C4⋊C4C2×C3⋊(C3⋊D3)11yescharacteristic
149C4⋊C4C2×C3⋊(C3⋊D3)11yescharacteristic
150C22⋊C4—541no—
151C42—271no—
152C42—541no—
153C22×C3D4×C3⋊D3113yes—
154C2×D3—1826no—
155C2×D3—3613no—
156C2×D3—3613no—
157C3×C4—413no—
158C3×C4—413no—
159C3×C4C2×C4×C3⋊D3113yes—
160C3×C4#3832113yes—
161Dic3—1826no—
162Dic3—1813no—
163Dic3—1813no—
164C32#4889113yescomplemented
165C23—541no—
166Q8—272no—
167Q8—541no—
168D4—272no—
169D4—541no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—271no—
173C2×C4—271no—
174C2×C4—541no—
175C2×C4—1082no—
176C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
177C8—541no—
178C8C4×C3⋊(C3⋊D3)12yescomplemented
179C2×C3#64102113yes—
180C2×C3#64120113yes—
181C2×C3#64127113yes—
182D3—3626no—
183C22—542no—
184C22—1081no—
185C22—1081no—
186C22D4×C3⋊(C3⋊D3)11yescharacteristic
187C4—41no—
188C4—41no—
189C4—542no—
190C4—541no—
191C4—541no—
192C4C2×C4×C3⋊(C3⋊D3)11yescharacteristic
193C4#8908011yescharacteristic
194C3#10592588113yescomplemented
195C2—1082no—
196C2#1172393411yescharacteristic
197C2#1172395211yescharacteristic
198C2#1172395911yescharacteristic
subgroupG/H
centerC22D4×C3⋊(C3⋊D3)
commutatorC33×C4C22×C4
FrattiniC2×C4C22×C3⋊(C3⋊D3)
FittingC33×#107C2
socleC22×C33C2×D4
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
cosocleC2×C33×C4C23
hypercenter#107C3⋊(C3⋊D3)
hypocenterC33#440
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
3. #11723928C2semidirect
4. C33×#107C2semidirect
5. #11722185C2semidirect
6. #11722180C2semidirect
20. #88851C4semidirect
24. C32×#107D3semidirect
39. #6308C2×C4semidirect
40. C33×C8C2×C4semidirect
69. C3×#107C3⋊D3semidirect
85. C32×C8C4×D3semidirect
118. #107C3⋊(C3⋊D3)semidirect
122. C33#440semidirect
135. C3×C8C4×C3⋊D3semidirect
164. C32#4889semidirect
178. C8C4×C3⋊(C3⋊D3)semidirect
194. C3#10592588semidirect

Maximal subgroups

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

subgroupindex
C2×#888512
#117239342
#117239282
C33×#1072
#117221852
#117221802
#117221792
#105925883