Finite Group Databasebeta

← #13163507

Subgroups — #13163507

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#11723938C211yesmaximal, characteristic
2#11723933C211yesmaximal, characteristic
3C33×#106C211yesmaximal, characteristic, complemented
4#11722190C211yesmaximal, characteristic
5#11722186C211yesmaximal, characteristic, complemented
6#11722180C211yesmaximal, characteristic, complemented
7#11722177C211yesmaximal, characteristic, complemented
8#10592567—313nomaximal
9C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
10C2×#6308C2211yescharacteristic
11C2×C33×C8C2211yescharacteristic
12C33×C4⋊C4C2211yescharacteristic
13C33×C4⋊C4C2211yescharacteristic
14#88859—21no—
15#88859—21no—
16C4⋊(C3⋊(C3⋊Dic3))—21no—
17C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
18C4×C3⋊(C3⋊Dic3)—21no—
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20#64106—313no—
21#64101—313no—
22C32×#106D3113yescomplemented
23#63647—313no—
24#63643—313no—
25#63637—313no—
26#63634—313no—
27C22×C3⋊(C3⋊D3)D411yescharacteristic
28C2×C33×C4—21no—
29C2×C33×C4—21no—
30C2×C33×C4C2311yescharacteristic
31C2×C3⋊(C3⋊Dic3)—21no—
32C2×C3⋊(C3⋊Dic3)—21no—
33C2×C3⋊(C3⋊Dic3)D411yescharacteristic
34C4×C3⋊(C3⋊D3)—21no—
35C4×C3⋊(C3⋊D3)—21no—
36#6308—21no—
37#6308—21no—
38C33×C8—21no—
39C3⋊(C3⋊(C3⋊C8))—21no—
40#4868—913no—
41C2×C4×C3⋊D3—313no—
42C2×#634—313no—
43C2×C32×C8C2×D3113yes—
44C32×C4⋊C4C2×D3113yes—
45C32×C4⋊C4C2×D3113yes—
46#3752—613no—
47#3752—613no—
48C4⋊(C3⋊Dic3)—313no—
49C4⋊(C3⋊Dic3)—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#897—913no—
64#892—913no—
65C3×#106C3⋊D3113yescomplemented
66(C2×D3)⋊C8—913no—
67#816—913no—
68#810—913no—
69#807—913no—
70C22×C3⋊D3—313no—
71C2×C32×C4—213no—
72C2×C32×C4—213no—
73C2×C32×C4C22×D3113yes—
74C2×C3⋊Dic3—313no—
75C2×C3⋊Dic3—613no—
76C2×C3⋊Dic3—613no—
77C4×C3⋊D3—613no—
78C4×C3⋊D3—613no—
79#634—613no—
80#634—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×C4×D3—913no—
89C2×Dic6—913no—
90C2×C3×C8C2×C3⋊D3113yes—
91C3×C4⋊C4C2×C3⋊D3113yes—
92C3×C4⋊C4C2×C3⋊D3113yes—
93(C2×D3)⋊C4—1813no—
94(C2×D3)⋊C4—1813no—
95C4⋊Dic3—913no—
96C4⋊Dic3—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#239113yes—
107C3⋊Dic3—613no—
108C3⋊Dic3—1213no—
109C3⋊Dic3—1213no—
110C22⋊Q8—271no—
111#117—271no—
112#107—271no—
113#106C3⋊(C3⋊D3)11yescharacteristic, complemented
114#103—271no—
115#103—271no—
116C22⋊C8—271no—
117C33#48511yescharacteristic, complemented, Sylow
118C22×D3—913no—
119C2×C3×C4—213no—
120C2×C3×C4—213no—
121C2×C3×C4C22×C3⋊D3113yes—
122C2×Dic3—913no—
123C2×Dic3—1813no—
124C2×Dic3—1813no—
125C4×D3—1813no—
126C4×D3—1813no—
127Dic6—1813no—
128Dic6—1813no—
129C3×C8—213no—
130C3⋊C8—1813no—
131C2×C32#894113yes—
132C2×C32#915113yes—
133C2×C32#916113yes—
134C3⋊D3—1213no—
135C2×Q8—271no—
136C22×C4—271no—
137C2×C8—271no—
138C2×C8C2×C3⋊(C3⋊D3)11yescharacteristic
139C4⋊C4—271no—
140C4⋊C4—541no—
141C4⋊C4C2×C3⋊(C3⋊D3)11yescharacteristic
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#3832113yes—
154Dic3—1813no—
155Dic3—3613no—
156Dic3—3613no—
157C32#4868113yescomplemented
158C23—271no—
159Q8—541no—
160Q8—541no—
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#64103113yes—
172C2×C3#64124113yes—
173C2×C3#64125113yes—
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#8908011yescharacteristic
186C3#10592567113yescomplemented
187C2—1081no—
188C2#1172393511yescharacteristic
189C2#1172395611yescharacteristic
190C2#1172395711yescharacteristic
subgroupG/H
centerC22D4×C3⋊(C3⋊D3)
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C22×C3⋊(C3⋊D3)
FittingC33×#106C2
socleC22×C33C2×D4
prime radicalC22×C3⋊(C3⋊D3)D4
hypercenter#106C3⋊(C3⋊D3)
hypocenterC33#485
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ #106

Complements

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

NH ≅ G/Ntype
3. C33×#106C2semidirect
5. #11722186C2semidirect
6. #11722180C2semidirect
7. #11722177C2semidirect
22. C32×#106D3semidirect
65. C3×#106C3⋊D3semidirect
113. #106C3⋊(C3⋊D3)semidirect
117. C33#485semidirect
157. C32#4868semidirect
186. C3#10592567semidirect

Maximal subgroups

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

subgroupindex
#117239382
#117239332
C33×#1062
#117221902
#117221862
#117221802
#117221772
#105925673