Finite Group Databasebeta

← #13163423

Subgroups — #13163423

Order 1728.

1484 conjugacy classes of subgroups; only the 349 normal subgroups are shown.

normal subgroups only

This group has 1484 conjugacy classes of subgroups — too many to draw. Only its 349 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#11723979C211yesmaximal, characteristic
2#11723933C211yesmaximal, characteristic, complemented
3C33×#102C211yesmaximal, characteristic, complemented
4#11722202C211yesmaximal, characteristic, complemented
5#11722190C211yesmaximal, characteristic, complemented
6#11722188C211yesmaximal, characteristic, complemented
7#11722178C211yesmaximal, characteristic, complemented
8#10592483—313nomaximal
9C2×C33×D4C2211yescharacteristic
10C2×#6312—21no—
11C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
12C2×C33×C8C2211yescharacteristic
13C33×C4⋊C4C2211yescharacteristic
14#88864—21no—
15#88859—21no—
16C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
17C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
18C4×C3⋊(C3⋊Dic3)—21no—
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20#64147—313no—
21#64101—313no—
22C32×#102D3113yescomplemented
23#63659—313no—
24#63647—313no—
25#63645—313no—
26#63635—313no—
27C23×C33—21no—
28C22×C3⋊(C3⋊D3)D411yescharacteristic
29C33×D4—21no—
30C33×D4—21no—
31C2×C33×C4—21no—
32C2×C33×C4C2311yescharacteristic
33#6312—42no—
34C2×C3⋊(C3⋊Dic3)—21no—
35C2×C3⋊(C3⋊Dic3)—21no—
36C2×C3⋊(C3⋊Dic3)D411yescharacteristic
37C4×C3⋊(C3⋊D3)—21no—
38C4×C3⋊(C3⋊D3)—21no—
39C33×C8—21no—
40C3⋊(C3⋊(C3⋊C8))—21no—
41#4784—913no—
42C2×C32×D4C2×D3113yes—
43C2×#638—613no—
44C2×C4×C3⋊D3—313no—
45C2×C32×C8C2×D3113yes—
46C32×C4⋊C4C2×D3113yes—
47#3757—613no—
48#3752—613no—
49C4⋊(C3⋊Dic3)—313no—
50C4⋊(C3⋊Dic3)—313no—
51C4×C3⋊Dic3—613no—
52C2×C3⋊(C3⋊C8)—313no—
53C22×C33—41no—
54C22×C33—41no—
55C22×C33—41no—
56C22×C33C2×D411yescharacteristic
57C2×C3⋊(C3⋊D3)—21no—
58C2×C3⋊(C3⋊D3)—21no—
59C2×C3⋊(C3⋊D3)—21no—
60C33×C4—41no—
61C33×C4#4111yescharacteristic
62C33×C4#4111yescharacteristic
63C3⋊(C3⋊Dic3)—21no—
64C3⋊(C3⋊Dic3)—41no—
65C3⋊(C3⋊Dic3)—41no—
66#938—913no—
67#892—913no—
68C3×#102C3⋊D3113yescomplemented
69#832—913no—
70(C2×D3)⋊C8—913no—
71#818—913no—
72#808—913no—
73C23×C32—213no—
74C22×C3⋊D3—313no—
75C32×D4—213no—
76C32×D4—213no—
77C2×C32×C4—213no—
78C2×C32×C4C22×D3113yes—
79#638—1226no—
80C2×C3⋊Dic3—313no—
81C2×C3⋊Dic3—613no—
82C2×C3⋊Dic3—613no—
83C4×C3⋊D3—613no—
84C4×C3⋊D3—613no—
85C32×C8—213no—
86C3⋊(C3⋊C8)—613no—
87#482—271noSylow
88C2×C33—81no—
89C2×C33#12311yescharacteristic
90C2×C33#13511yescharacteristic
91C2×C33AGL(1,ℤ/8)11yescharacteristic
92C3⋊(C3⋊D3)—41no—
93C2×C3×D4C2×C3⋊D3113yes—
94C2×#67—1813no—
95C2×C4×D3—913no—
96C2×C3×C8C2×C3⋊D3113yes—
97C3×C4⋊C4C2×C3⋊D3113yes—
98#217—1813no—
99(C2×D3)⋊C4—1813no—
100C4⋊Dic3—913no—
101C4⋊Dic3—913no—
102C4×Dic3—1813no—
103C2×C3⋊C8—913no—
104C22×C32—413no—
105C22×C32—413no—
106C22×C32—413no—
107C22×C32D3×D4113yes—
108C2×C3⋊D3—613no—
109C2×C3⋊D3—613no—
110C2×C3⋊D3—613no—
111C32×C4—413no—
112C32×C4#235113yes—
113C32×C4#237113yes—
114C3⋊Dic3—613no—
115C3⋊Dic3—1213no—
116C3⋊Dic3—1213no—
117#121—271no—
118#117—271no—
119#107—271no—
120#106—271no—
121#102—271no—
122#102C3⋊(C3⋊D3)11yescharacteristic, complemented
123C22⋊C8—271no—
124C33#48211yescharacteristic, complemented, Sylow
125C23×C3—213no—
126C22×D3—913no—
127C3×D4—213no—
128C3×D4—213no—
129C2×C3×C4—213no—
130C2×C3×C4C22×C3⋊D3113yes—
131#67—3626no—
132C2×Dic3—913no—
133C2×Dic3—1813no—
134C2×Dic3—1813no—
135C4×D3—1813no—
136C4×D3—1813no—
137C3×C8—213no—
138C3⋊C8—1813no—
139C2×C32—813no—
140C2×C32#883113yes—
141C2×C32#912113yes—
142C2×C32#914113yes—
143C3⋊D3—1213no—
144C2×D4—541no—
145C2×D4C2×C3⋊(C3⋊D3)11yescharacteristic
146C22×C4—271no—
147C2×C8—271no—
148C2×C8C2×C3⋊(C3⋊D3)11yescharacteristic
149C4⋊C4—271no—
150C4⋊C4—271no—
151C4⋊C4C2×C3⋊(C3⋊D3)11yescharacteristic
152C22⋊C4—541no—
153C22⋊C4—541no—
154C42—541no—
155C22×C3—413no—
156C22×C3—413no—
157C22×C3—413no—
158C22×C3D4×C3⋊D3113yes—
159C2×D3—1813no—
160C2×D3—1813no—
161C2×D3—1813no—
162C3×C4—413no—
163C3×C4#3828113yes—
164C3×C4#3830113yes—
165Dic3—1813no—
166Dic3—3613no—
167Dic3—3613no—
168C32#4784113yescomplemented
169C23—21no—
170C23—271no—
171D4—21no—
172D4—21no—
173D4—1082no—
174C2×C4—21no—
175C2×C4—271no—
176C2×C4—541no—
177C2×C4—541no—
178C2×C4—541no—
179C2×C4—541no—
180C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
181C8—21no—
182C8—541no—
183C2×C3—813no—
184C2×C3#64092113yes—
185C2×C3#64121113yes—
186C2×C3#64123113yes—
187D3—3613no—
188C22—41no—
189C22—41no—
190C22—41no—
191C22—541no—
192C22—541no—
193C22—541no—
194C22D4×C3⋊(C3⋊D3)11yescharacteristic
195C4—41no—
196C4—541no—
197C4—1081no—
198C4—1081no—
199C4#8907611yescharacteristic
200C4#8907811yescharacteristic
201C3#10592483113yescomplemented
202C2—81no—
203C2—1081no—
204C2#1172392411yescharacteristic
205C2#1172395311yescharacteristic
206C2#1172395511yescharacteristic
subgroupG/H
centerC22D4×C3⋊(C3⋊D3)
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C22×C3⋊(C3⋊D3)
FittingC33×#102C2
socleC22×C33C2×D4
prime radical#11723979C2
hypercenter#102C3⋊(C3⋊D3)
hypocenterC33#482
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. #11723933C2semidirect
3. C33×#102C2semidirect
4. #11722202C2semidirect
5. #11722190C2semidirect
6. #11722188C2semidirect
7. #11722178C2semidirect
22. C32×#102D3semidirect
68. C3×#102C3⋊D3semidirect
122. #102C3⋊(C3⋊D3)semidirect
124. C33#482semidirect
168. C32#4784semidirect
201. C3#10592483semidirect

Maximal subgroups

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

subgroupindex
#117239792
#117239332
C33×#1022
#117222022
#117221902
#117221882
#117221782
#105924833