Finite Group Databasebeta

← #13163635

Subgroups — #13163635

Order 1728.

1344 conjugacy classes of subgroups; only the 403 normal subgroups are shown.

normal subgroups only

This group has 1344 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×#117C211yesmaximal, characteristic
2#11723965C211yesmaximal, characteristic
3#11722200C211yesmaximal, characteristic
4#11722180C211yesmaximal, characteristic, complemented
5#11722180C211yesmaximal, characteristic, complemented
6#11722178C211yesmaximal, characteristic, complemented
7#11722177C211yesmaximal, characteristic, complemented
8#10592695—313nomaximal
9C22×C33×C4C2211yescharacteristic
10C2×#6308C2211yescharacteristic
11C33×C4⋊C4C2211yescharacteristic
12C33×C4⋊C4C2211yescharacteristic
13C33×C22⋊C4—21no—
14C33×C42—21no—
15#88864—21no—
16C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
17#88857—21no—
18C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20C32×#117D3113yes—
21#64133—313no—
22#63657—313no—
23#63637—313no—
24#63637—313no—
25#63635—313no—
26#63634—313no—
27C23×C33D411yescharacteristic
28C2×C33×C4—21no—
29C2×C33×C4—21no—
30C2×C33×C4—21no—
31C2×C33×C4—21no—
32C2×C33×C4D411yescharacteristic
33C2×C33×C4C2311yescharacteristic
34C2×C3⋊(C3⋊Dic3)—21no—
35C2×C3⋊(C3⋊Dic3)—21no—
36#6308—21no—
37#6308—21no—
38C3⋊(C3⋊(C3⋊C8))—21no—
39C3⋊(C3⋊(C3⋊C8))—21no—
40#4996—913no—
41C22×C32×C4C2×D3113yes—
42C2×#634—313no—
43C32×C4⋊C4C2×D3113yes—
44C32×C4⋊C4C2×D3113yes—
45C32×C22⋊C4—213no—
46C32×C42—213no—
47#3757—613no—
48C4⋊(C3⋊Dic3)—313no—
49#3750—613no—
50C2×C3⋊(C3⋊C8)—313no—
51C2×C3⋊(C3⋊C8)—313no—
52C22×C33—21no—
53C22×C33—21no—
54C22×C33—21no—
55C22×C33C2×D411yescharacteristic
56C33×C4—21no—
57C33×C4—41no—
58C33×C4—41no—
59C33×C4#4111yescharacteristic
60C33×C4#4111yescharacteristic
61C3⋊(C3⋊Dic3)—41no—
62C3⋊(C3⋊Dic3)—41no—
63C3×#117C3⋊D3113yes—
64#924—913no—
65#830—913no—
66#810—913no—
67#810—913no—
68#808—913no—
69#807—913no—
70C23×C32#67113yes—
71C2×C32×C4—213no—
72C2×C32×C4—213no—
73C2×C32×C4—213no—
74C2×C32×C4—213no—
75C2×C32×C4#67113yes—
76C2×C32×C4C22×D3113yes—
77C2×C3⋊Dic3—613no—
78C2×C3⋊Dic3—613no—
79#634—613no—
80#634—613no—
81C3⋊(C3⋊C8)—613no—
82C3⋊(C3⋊C8)—613no—
83#485—271noSylow
84C2×C33—41no—
85C2×C33#12311yescharacteristic
86C2×C33#13511yescharacteristic
87C2×C33#13711yescharacteristic
88C22×C3×C4C2×C3⋊D3113yes—
89C2×Dic6—913no—
90C3×C4⋊C4C2×C3⋊D3113yes—
91C3×C4⋊C4C2×C3⋊D3113yes—
92C3×C22⋊C4—213no—
93C3×C42—213no—
94#217—1813no—
95C4⋊Dic3—913no—
96Dic3⋊C4—1813no—
97C2×C3⋊C8—913no—
98C2×C3⋊C8—913no—
99C22×C32—213no—
100C22×C32—213no—
101C22×C32—213no—
102C22×C32C2×#67113yes—
103C32×C4—213no—
104C32×C4—413no—
105C32×C4—413no—
106C32×C4#235113yes—
107C32×C4#235113yes—
108C3⋊Dic3—1213no—
109C3⋊Dic3—1213no—
110C22⋊Q8—271no—
111#117C3⋊(C3⋊D3)11yescharacteristic
112#107—271no—
113#106—271no—
114#103—271no—
115#103—271no—
116C22⋊C8—271no—
117C33#48511yescharacteristic, complemented, Sylow
118C23×C3#638113yes—
119C2×C3×C4—213no—
120C2×C3×C4—213no—
121C2×C3×C4—213no—
122C2×C3×C4—213no—
123C2×C3×C4#638113yes—
124C2×C3×C4C22×C3⋊D3113yes—
125C2×Dic3—1813no—
126C2×Dic3—1813no—
127Dic6—1813no—
128Dic6—1813no—
129C3⋊C8—1813no—
130C3⋊C8—1813no—
131C2×C32—413no—
132C2×C32#929113yes—
133C2×C32#950113yes—
134C2×C32#951113yes—
135C2×Q8—271no—
136C22×C4C2×C3⋊(C3⋊D3)11yescharacteristic
137C2×C8—271no—
138C2×C8—271no—
139C4⋊C4—271no—
140C4⋊C4—541no—
141C4⋊C4C2×C3⋊(C3⋊D3)11yescharacteristic
142C4⋊C4C2×C3⋊(C3⋊D3)11yescharacteristic
143C22⋊C4—21no—
144C22⋊C4—541no—
145C42—21no—
146C22×C3—213no—
147C22×C3—213no—
148C22×C3—213no—
149C22×C3C2×#638113yes—
150C3×C4—213no—
151C3×C4—413no—
152C3×C4—413no—
153C3×C4#3828113yes—
154C3×C4#3828113yes—
155Dic3—3613no—
156Dic3—3613no—
157C32#4996113yescomplemented
158C23#631211yescharacteristic
159Q8—541no—
160Q8—541no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—541no—
166C2×C4—541no—
167C2×C4#631211yescharacteristic
168C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
169C8—541no—
170C8—541no—
171C2×C3—413no—
172C2×C3#64138113yes—
173C2×C3#64159113yes—
174C2×C3#64160113yes—
175C22—21no—
176C22—21no—
177C22—21no—
178C22C2×#631211yescharacteristic
179C4—21no—
180C4—41no—
181C4—41no—
182C4—1081no—
183C4—1081no—
184C4#8907611yescharacteristic
185C4#8907611yescharacteristic
186C3#10592695113yescomplemented
187C2—41no—
188C2#1172397011yescharacteristic
189C2#1172399111yescharacteristic
190C2#1172399211yescharacteristic
subgroupG/H
centerC22C2×#6312
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C22×C3⋊(C3⋊D3)
FittingC33×#117C2
socleC22×C33C2×D4
prime radicalC23×C33D4
hypercenter#117C3⋊(C3⋊D3)
hypocenterC33#485
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
4. #11722180C2semidirect
5. #11722180C2semidirect
6. #11722178C2semidirect
7. #11722177C2semidirect
117. C33#485semidirect
157. C32#4996semidirect
186. C3#10592695semidirect

Maximal subgroups

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

subgroupindex
C33×#1172
#117239652
#117222002
#117221802
#117221782
#117221772
#105926953