Finite Group Databasebeta

← #13163451

Subgroups — #13163451

Order 1728.

1344 conjugacy classes of subgroups; only the 349 normal subgroups are shown.

normal subgroups only

This group has 1344 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#11723987C211yesmaximal, characteristic
2#11723933C211yesmaximal, characteristic
3C33×#103C211yesmaximal, characteristic, complemented
4#11722205C211yesmaximal, characteristic, complemented
5#11722190C211yesmaximal, characteristic
6#11722188C211yesmaximal, characteristic, complemented
7#11722178C211yesmaximal, characteristic, complemented
8#10592513—313nomaximal
9C2×C33×Q8C2211yescharacteristic
10C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
11C2×C33×C8C2211yescharacteristic
12C33×C4⋊C4C2211yescharacteristic
13#88859—21no—
14#88859—21no—
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
17#88857—21no—
18C4×C3⋊(C3⋊Dic3)—21no—
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20#64155—313no—
21#64101—313no—
22C32×#103D3113yescomplemented
23#63662—313no—
24#63647—313no—
25#63645—313no—
26#63635—313no—
27C22×C3⋊(C3⋊D3)D411yescharacteristic
28C33×Q8—21no—
29C33×Q8—21no—
30C2×C33×C4—21no—
31C2×C33×C4—21no—
32C2×C33×C4C2311yescharacteristic
33C2×C3⋊(C3⋊Dic3)—21no—
34C2×C3⋊(C3⋊Dic3)—21no—
35C2×C3⋊(C3⋊Dic3)D411yescharacteristic
36C4×C3⋊(C3⋊D3)—21no—
37C4×C3⋊(C3⋊D3)—21no—
38C33×C8—21no—
39C3⋊(C3⋊(C3⋊C8))—21no—
40#4814—913no—
41C2×C32×Q8C2×D3113yes—
42C2×C4×C3⋊D3—313no—
43C2×C32×C8C2×D3113yes—
44C32×C4⋊C4C2×D3113yes—
45#3752—613no—
46#3752—613no—
47C4⋊(C3⋊Dic3)—313no—
48C4⋊(C3⋊Dic3)—313no—
49#3750—613no—
50C4×C3⋊Dic3—613no—
51C2×C3⋊(C3⋊C8)—313no—
52C22×C33C2×D411yescharacteristic
53C2×C3⋊(C3⋊D3)—21no—
54C2×C3⋊(C3⋊D3)—21no—
55C2×C3⋊(C3⋊D3)—21no—
56C33×C4—41no—
57C33×C4—41no—
58C33×C4#4111yescharacteristic
59C33×C4#4111yescharacteristic
60C3⋊(C3⋊Dic3)—21no—
61C3⋊(C3⋊Dic3)—41no—
62C3⋊(C3⋊Dic3)—41no—
63#946—913no—
64#892—913no—
65C3×#103C3⋊D3113yescomplemented
66#835—913no—
67(C2×D3)⋊C8—913no—
68#818—913no—
69#808—913no—
70C22×C3⋊D3—313no—
71C32×Q8—213no—
72C32×Q8—213no—
73C2×C32×C4—213no—
74C2×C32×C4—213no—
75C2×C32×C4C22×D3113yes—
76C2×C3⋊Dic3—313no—
77C2×C3⋊Dic3—613no—
78C2×C3⋊Dic3—613no—
79C4×C3⋊D3—613no—
80C4×C3⋊D3—613no—
81C32×C8—213no—
82C3⋊(C3⋊C8)—613no—
83#485—271noSylow
84C2×C33#12311yescharacteristic
85C2×C33#13511yescharacteristic
86C2×C33#13711yescharacteristic
87C3⋊(C3⋊D3)—41no—
88C2×C3×Q8C2×C3⋊D3113yes—
89C2×C4×D3—913no—
90C2×C3×C8C2×C3⋊D3113yes—
91C3×C4⋊C4C2×C3⋊D3113yes—
92(C2×D3)⋊C4—1813no—
93(C2×D3)⋊C4—1813no—
94C4⋊Dic3—913no—
95C4⋊Dic3—913no—
96Dic3⋊C4—1813no—
97C4×Dic3—1813no—
98C2×C3⋊C8—913no—
99C22×C32D3×D4113yes—
100C2×C3⋊D3—613no—
101C2×C3⋊D3—613no—
102C2×C3⋊D3—613no—
103C32×C4—413no—
104C32×C4—413no—
105C32×C4#235113yes—
106C32×C4#237113yes—
107C3⋊Dic3—613no—
108C3⋊Dic3—1213no—
109C3⋊Dic3—1213no—
110C22⋊Q8—271no—
111#117—271no—
112#107—271no—
113#106—271no—
114#103—271no—
115#103C3⋊(C3⋊D3)11yescharacteristic, complemented
116C22⋊C8—271no—
117C33#48511yescharacteristic, complemented, Sylow
118C22×D3—913no—
119C3×Q8—213no—
120C3×Q8—213no—
121C2×C3×C4—213no—
122C2×C3×C4—213no—
123C2×C3×C4C22×C3⋊D3113yes—
124C2×Dic3—913no—
125C2×Dic3—1813no—
126C2×Dic3—1813no—
127C4×D3—1813no—
128C4×D3—1813no—
129C3×C8—213no—
130C3⋊C8—1813no—
131C2×C32#883113yes—
132C2×C32#915113yes—
133C2×C32#919113yes—
134C3⋊D3—1213no—
135C2×Q8C2×C3⋊(C3⋊D3)11yescharacteristic
136C22×C4—271no—
137C2×C8—271no—
138C2×C8C2×C3⋊(C3⋊D3)11yescharacteristic
139C4⋊C4—271no—
140C4⋊C4—271no—
141C4⋊C4—541no—
142C4⋊C4C2×C3⋊(C3⋊D3)11yescharacteristic
143C22⋊C4—541no—
144C22⋊C4—541no—
145C42—541no—
146C22×C3D4×C3⋊D3113yes—
147C2×D3—1813no—
148C2×D3—1813no—
149C2×D3—1813no—
150C3×C4—413no—
151C3×C4—413no—
152C3×C4#3828113yes—
153C3×C4#3830113yes—
154Dic3—1813no—
155Dic3—3613no—
156Dic3—3613no—
157C32#4814113yescomplemented
158C23—271no—
159Q8—21no—
160Q8—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—271no—
164C2×C4—541no—
165C2×C4—541no—
166C2×C4—541no—
167C2×C4—541no—
168C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
169C8—21no—
170C8—541no—
171C2×C3#64092113yes—
172C2×C3#64124113yes—
173C2×C3#64128113yes—
174D3—3613no—
175C22—541no—
176C22—541no—
177C22—541no—
178C22D4×C3⋊(C3⋊D3)11yescharacteristic
179C4—41no—
180C4—41no—
181C4—541no—
182C4—1081no—
183C4—1081no—
184C4#8907611yescharacteristic
185C4#8907811yescharacteristic
186C3#10592513113yescomplemented
187C2—1081no—
188C2#1172392411yescharacteristic
189C2#1172395611yescharacteristic
190C2#1172396011yescharacteristic
subgroupG/H
centerC22D4×C3⋊(C3⋊D3)
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C22×C3⋊(C3⋊D3)
FittingC33×#103C2
socleC22×C33C2×D4
prime radicalC22×C3⋊(C3⋊D3)D4
hypercenter#103C3⋊(C3⋊D3)
hypocenterC33#485
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
3. C33×#103C2semidirect
4. #11722205C2semidirect
6. #11722188C2semidirect
7. #11722178C2semidirect
22. C32×#103D3semidirect
65. C3×#103C3⋊D3semidirect
115. #103C3⋊(C3⋊D3)semidirect
117. C33#485semidirect
157. C32#4814semidirect
186. C3#10592513semidirect

Maximal subgroups

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

subgroupindex
#117239872
#117239332
C33×#1032
#117222052
#117221902
#117221882
#117221782
#105925133