Finite Group Databasebeta

← #13163675

Subgroups — #13163675

Order 1728.

1484 conjugacy classes of subgroups; only the 403 normal subgroups are shown.

normal subgroups only

This group has 1484 conjugacy classes of subgroups — too many to draw. Only its 403 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
1C33×#121C211yesmaximal, characteristic
2#11723967C211yesmaximal, characteristic, complemented
3#11722202C211yesmaximal, characteristic, complemented
4#11722202C211yesmaximal, characteristic, complemented
5#11722200C211yesmaximal, characteristic, complemented
6#11722178C211yesmaximal, characteristic, complemented
7#11722177C211yesmaximal, characteristic, complemented
8#10592734—313nomaximal
9C2×C33×D4—21no—
10C2×C33×D4C2211yescharacteristic
11C22×C33×C4C2211yescharacteristic
12C33×C4⋊C4C2211yescharacteristic
13C33×C22⋊C4—21no—
14#88864—21no—
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
17C4×C3⋊(C3⋊Dic3)—21no—
18C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20C32×#121D3113yes—
21#64135—313no—
22#63659—313no—
23#63659—313no—
24#63657—313no—
25#63635—313no—
26#63634—313no—
27C23×C33—21no—
28C23×C33D411yescharacteristic
29C33×D4—21no—
30C33×D4—21no—
31C33×D4—42no—
32C2×C33×C4—21no—
33C2×C33×C4—21no—
34C2×C33×C4—21no—
35C2×C33×C4D411yescharacteristic
36C2×C33×C4C2311yescharacteristic
37C2×C3⋊(C3⋊Dic3)—21no—
38C2×C3⋊(C3⋊Dic3)—21no—
39C3⋊(C3⋊(C3⋊C8))—21no—
40C3⋊(C3⋊(C3⋊C8))—21no—
41#5035—913no—
42C2×C32×D4—213no—
43C2×C32×D4C2×D3113yes—
44C22×C32×C4C2×D3113yes—
45C32×C4⋊C4C2×D3113yes—
46C32×C22⋊C4—213no—
47#3757—613no—
48C4⋊(C3⋊Dic3)—313no—
49C4⋊(C3⋊Dic3)—313no—
50C4×C3⋊Dic3—613no—
51C2×C3⋊(C3⋊C8)—313no—
52C2×C3⋊(C3⋊C8)—313no—
53C22×C33—21no—
54C22×C33—21no—
55C22×C33—21no—
56C22×C33—41no—
57C22×C33—41no—
58C22×C33—41no—
59C22×C33C2×D411yescharacteristic
60C33×C4—21no—
61C33×C4—41no—
62C33×C4#4111yescharacteristic
63C33×C4#4111yescharacteristic
64C3⋊(C3⋊Dic3)—41no—
65C3⋊(C3⋊Dic3)—41no—
66C3×#121C3⋊D3113yes—
67#926—913no—
68#832—913no—
69#832—913no—
70#830—913no—
71#808—913no—
72#807—913no—
73C23×C32—213no—
74C23×C32#67113yes—
75C32×D4—213no—
76C32×D4—213no—
77C32×D4—426no—
78C2×C32×C4—213no—
79C2×C32×C4—213no—
80C2×C32×C4—213no—
81C2×C32×C4#67113yes—
82C2×C32×C4C22×D3113yes—
83C2×C3⋊Dic3—613no—
84C2×C3⋊Dic3—613no—
85C3⋊(C3⋊C8)—613no—
86C3⋊(C3⋊C8)—613no—
87#482—271noSylow
88C2×C33—41no—
89C2×C33—81no—
90C2×C33#12311yescharacteristic
91C2×C33#13511yescharacteristic
92C2×C33AGL(1,ℤ/8)11yescharacteristic
93C2×C3×D4—213no—
94C2×C3×D4C2×C3⋊D3113yes—
95C22×C3×C4C2×C3⋊D3113yes—
96C3×C4⋊C4C2×C3⋊D3113yes—
97C3×C22⋊C4—213no—
98#217—1813no—
99C4⋊Dic3—913no—
100C4⋊Dic3—913no—
101C4×Dic3—1813no—
102C2×C3⋊C8—913no—
103C2×C3⋊C8—913no—
104C22×C32—213no—
105C22×C32—213no—
106C22×C32—213no—
107C22×C32—413no—
108C22×C32—413no—
109C22×C32—413no—
110C22×C32C2×#67113yes—
111C32×C4—213no—
112C32×C4—413no—
113C32×C4#237113yes—
114C32×C4#237113yes—
115C3⋊Dic3—1213no—
116C3⋊Dic3—1213no—
117#121C3⋊(C3⋊D3)11yescharacteristic
118#117—271no—
119#107—271no—
120#106—271no—
121#102—271no—
122#102—271no—
123C22⋊C8—271no—
124C33#48211yescharacteristic, complemented, Sylow
125C23×C3—213no—
126C23×C3#638113yes—
127C3×D4—213no—
128C3×D4—213no—
129C3×D4—426no—
130C2×C3×C4—213no—
131C2×C3×C4—213no—
132C2×C3×C4—213no—
133C2×C3×C4#638113yes—
134C2×C3×C4C22×C3⋊D3113yes—
135C2×Dic3—1813no—
136C2×Dic3—1813no—
137C3⋊C8—1813no—
138C3⋊C8—1813no—
139C2×C32—413no—
140C2×C32—813no—
141C2×C32#932113yes—
142C2×C32#935113yes—
143C2×C32#950113yes—
144C2×D4—21no—
145C2×D4C2×C3⋊(C3⋊D3)11yescharacteristic
146C22×C4C2×C3⋊(C3⋊D3)11yescharacteristic
147C2×C8—271no—
148C2×C8—271no—
149C4⋊C4—271no—
150C4⋊C4—271no—
151C4⋊C4C2×C3⋊(C3⋊D3)11yescharacteristic
152C22⋊C4—21no—
153C22⋊C4—541no—
154C42—541no—
155C22×C3—213no—
156C22×C3—213no—
157C22×C3—213no—
158C22×C3—413no—
159C22×C3—413no—
160C22×C3—413no—
161C22×C3C2×#638113yes—
162C3×C4—213no—
163C3×C4—413no—
164C3×C4#3830113yes—
165C3×C4#3830113yes—
166Dic3—3613no—
167Dic3—3613no—
168C32#5035113yescomplemented
169C23—21no—
170C23#631211yescharacteristic
171D4—21no—
172D4—21no—
173D4—42no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—541no—
178C2×C4—541no—
179C2×C4#631211yescharacteristic
180C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
181C8—541no—
182C8—541no—
183C2×C3—413no—
184C2×C3—813no—
185C2×C3#64141113yes—
186C2×C3#64144113yes—
187C2×C3#64159113yes—
188C22—21no—
189C22—21no—
190C22—21no—
191C22—41no—
192C22—41no—
193C22—41no—
194C22C2×#631211yescharacteristic
195C4—21no—
196C4—41no—
197C4—1081no—
198C4—1081no—
199C4#8907811yescharacteristic
200C4#8907811yescharacteristic
201C3#10592734113yescomplemented
202C2—41no—
203C2—81no—
204C2#1172397311yescharacteristic
205C2#1172397611yescharacteristic
206C2#1172399111yescharacteristic
subgroupG/H
centerC22C2×#6312
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C22×C3⋊(C3⋊D3)
Fitting, prime radicalC33×#121C2
socleC22×C33C2×D4
hypercenter#121C3⋊(C3⋊D3)
hypocenterC33#482
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
2. #11723967C2semidirect
3. #11722202C2semidirect
4. #11722202C2semidirect
5. #11722200C2semidirect
6. #11722178C2semidirect
7. #11722177C2semidirect
124. C33#482semidirect
168. C32#5035semidirect
201. C3#10592734semidirect

Maximal subgroups

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

subgroupindex
C33×#1212
#117239672
#117222022
#117222002
#117221782
#117221772
#105927343