Finite Group Databasebeta

← #13163501

Subgroups — #13163501

Order 1728.

1652 conjugacy classes of subgroups; only the 419 normal subgroups are shown.

normal subgroups only

This group has 1652 conjugacy classes of subgroups — too many to draw. Only its 419 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
1C2×#88850C211yesmaximal, characteristic
2#11723933C211yesmaximal, characteristic
3C4⋊C4×C3⋊(C3⋊D3)C211yesmaximal, characteristic
4C33×#106C211yesmaximal, characteristic, complemented
5#11722188C211yesmaximal, characteristic, complemented
6#11722178C211yesmaximal, characteristic, complemented
7#11722177C211yesmaximal, characteristic, complemented
8#10592560—313nomaximal
9C2×C4×C3⋊(C3⋊D3)—21no—
10C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
11C2×C33×C8C2211yescharacteristic
12C33×C4⋊C4C2211yescharacteristic
13C33×C4⋊C4C2211yescharacteristic
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#88850C414yescomplemented
21C2×#3743—313no—
22#64101—313no—
23C4⋊C4×C3⋊D3—313no—
24C32×#106D3113yescomplemented
25#63645—313no—
26#63635—313no—
27#63634—313no—
28C22×C3⋊(C3⋊D3)D411yescharacteristic
29C2×C33×C4—21no—
30C2×C33×C4—21no—
31C2×C33×C4C2311yescharacteristic
32C2×C3⋊(C3⋊Dic3)—21no—
33C2×C3⋊(C3⋊Dic3)—21no—
34C2×C3⋊(C3⋊Dic3)D411yescharacteristic
35C4×C3⋊(C3⋊D3)—42no—
36C4×C3⋊(C3⋊D3)C2×C412yes—
37C4×C3⋊(C3⋊D3)Q812yes—
38C33×C8C2×C412yescomplemented
39C3⋊(C3⋊(C3⋊C8))C2×C412yescomplemented
40#4861—913no—
41C2×C4×C3⋊D3—313no—
42C2×C4×C3⋊D3—613no—
43C2×C32×C8C2×D3113yes—
44C32×C4⋊C4C2×D3113yes—
45C32×C4⋊C4C2×D3113yes—
46#3752—613no—
47C4⋊(C3⋊Dic3)—313no—
48C4⋊(C3⋊Dic3)—313no—
49#3750—613no—
50C4×C3⋊Dic3—613no—
51C2×C3⋊(C3⋊C8)—313no—
52#3743—352no—
53C22×C33C2×D411yescharacteristic
54C2×C3⋊(C3⋊D3)—21no—
55C2×C3⋊(C3⋊D3)—21no—
56C2×C3⋊(C3⋊D3)C4⋊C412yes—
57C33×C4—41no—
58C33×C4—41no—
59C33×C4C22×C411yescharacteristic
60C33×C4C2×Q811yescharacteristic
61C3⋊(C3⋊Dic3)—41no—
62C3⋊(C3⋊Dic3)—41no—
63C3⋊(C3⋊Dic3)C4⋊C412yes—
64C2×ΓL(1,𝔽25)—913no—
65#892—913no—
66D3×C4⋊C4—913no—
67C3×#106C3⋊D3113yescomplemented
68#818—913no—
69#808—913no—
70#807—913no—
71C22×C3⋊D3—313no—
72C2×C32×C4—213no—
73C2×C32×C4—213no—
74C2×C32×C4C22×D3113yes—
75C2×C3⋊Dic3—313no—
76C2×C3⋊Dic3—613no—
77C2×C3⋊Dic3—613no—
78C4×C3⋊D3—326no—
79C4×C3⋊D3—326no—
80C4×C3⋊D3—1226no—
81C32×C8C4×D3126yescomplemented
82C3⋊(C3⋊C8)—326no—
83#428—271noSylow
84C2×C33C2×C4⋊C411yescharacteristic
85C2×C33AGL(1,ℤ/8)11yescharacteristic
86C2×C33#13711yescharacteristic
87C3⋊(C3⋊D3)—22no—
88C2×C4×D3—913no—
89C2×C4×D3—1813no—
90C2×C3×C8C2×C3⋊D3113yes—
91C3×C4⋊C4C2×C3⋊D3113yes—
92C3×C4⋊C4C2×C3⋊D3113yes—
93(C2×D3)⋊C4—1813no—
94C4⋊Dic3—913no—
95C4⋊Dic3—913no—
96Dic3⋊C4—1813no—
97C4×Dic3—1813no—
98C2×C3⋊C8—913no—
99ΓL(1,𝔽25)—952no—
100C22×C32D3×D4113yes—
101C2×C3⋊D3—326no—
102C2×C3⋊D3—613no—
103C2×C3⋊D3—613no—
104C32×C4—413no—
105C32×C4—413no—
106C32×C4C2×C4×D3113yes—
107C32×C4D3×Q8113yes—
108C3⋊Dic3—326no—
109C3⋊Dic3—1213no—
110C3⋊Dic3—1213no—
111C2×M16—271no—
112#117—271no—
113C2×C4⋊C4—271no—
114#107—271no—
115#107—271no—
116#106—271no—
117#106C3⋊(C3⋊D3)11yescharacteristic, complemented
118C33#42811yescharacteristic, complemented, Sylow
119C22×D3—913no—
120C2×C3×C4—213no—
121C2×C3×C4—213no—
122C2×C3×C4C22×C3⋊D3113yes—
123C2×Dic3—913no—
124C2×Dic3—1813no—
125C2×Dic3—1813no—
126C4×D3—926no—
127C4×D3—926no—
128C4×D3—3626no—
129C3×C8C4×C3⋊D3126yescomplemented
130C3⋊C8—926no—
131C2×C32D3×C4⋊C4113yes—
132C2×C32#914113yes—
133C2×C32#915113yes—
134C3⋊D3—626no—
135C22×C4—271no—
136C22×C4—541no—
137M16—274no—
138C2×C8—271no—
139C2×C8C2×C3⋊(C3⋊D3)11yescharacteristic
140C4⋊C4—271no—
141C4⋊C4—271no—
142C4⋊C4—541no—
143C4⋊C4C2×C3⋊(C3⋊D3)11yescharacteristic
144C4⋊C4C2×C3⋊(C3⋊D3)11yescharacteristic
145C22⋊C4—541no—
146C42—541no—
147C22×C3D4×C3⋊D3113yes—
148C2×D3—926no—
149C2×D3—1813no—
150C2×D3—1813no—
151C3×C4—413no—
152C3×C4—413no—
153C3×C4C2×C4×C3⋊D3113yes—
154C3×C4Q8×C3⋊D3113yes—
155Dic3—926no—
156Dic3—3613no—
157Dic3—3613no—
158C32#4861113yescomplemented
159C23—271no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—271no—
163C2×C4—272no—
164C2×C4—272no—
165C2×C4—541no—
166C2×C4—541no—
167C2×C4—1082no—
168C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
169C8—272no—
170C8C4×C3⋊(C3⋊D3)12yescomplemented
171C2×C3C4⋊C4×C3⋊D3113yes—
172C2×C3#64123113yes—
173C2×C3#64124113yes—
174D3—1826no—
175C22—272no—
176C22—541no—
177C22—541no—
178C22D4×C3⋊(C3⋊D3)11yescharacteristic
179C4—41no—
180C4—41no—
181C4—272no—
182C4—1081no—
183C4—1081no—
184C4C2×C4×C3⋊(C3⋊D3)11yescharacteristic
185C4Q8×C3⋊(C3⋊D3)11yescharacteristic
186C3#10592560113yescomplemented
187C2—542no—
188C2C4⋊C4×C3⋊(C3⋊D3)11yescharacteristic
189C2#1172395511yescharacteristic
190C2#1172395611yescharacteristic
subgroupG/H
centerC22D4×C3⋊(C3⋊D3)
commutatorC33×C4C22×C4
FrattiniC2×C4C22×C3⋊(C3⋊D3)
FittingC33×#106C2
socleC22×C33C2×D4
prime radicalC22×C3⋊(C3⋊D3)D4
cosocleC2×C33×C4C23
hypercenter#106C3⋊(C3⋊D3)
hypocenterC33#428
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ #106

Complements

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

NH ≅ G/Ntype
4. C33×#106C2semidirect
5. #11722188C2semidirect
6. #11722178C2semidirect
7. #11722177C2semidirect
20. #88850C4semidirect
24. C32×#106D3semidirect
38. C33×C8C2×C4semidirect
39. C3⋊(C3⋊(C3⋊C8))C2×C4semidirect
67. C3×#106C3⋊D3semidirect
81. C32×C8C4×D3semidirect
117. #106C3⋊(C3⋊D3)semidirect
118. C33#428semidirect
129. C3×C8C4×C3⋊D3semidirect
158. C32#4861semidirect
170. C8C4×C3⋊(C3⋊D3)semidirect
186. C3#10592560semidirect

Maximal subgroups

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

subgroupindex
C2×#888502
#117239332
C4⋊C4×C3⋊(C3⋊D3)2
C33×#1062
#117221882
#117221782
#117221772
#105925603