Finite Group Databasebeta

← #13138752

Subgroups — #13138752

Order 1728.

387 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 387 conjugacy classes of subgroups — too many to draw. Only its 83 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#11721117C212yesmaximal
2C8×#1084C211yesmaximal, characteristic
3#63644⋊C3C212yesmaximal
4#11720314C212yesmaximal
5#10590889—32nomaximal
6C2×#6196—22no—
7C2×C8×H(3,𝔽3)C2211yescharacteristic
8C4×#1084C2211yescharacteristic
9C2×#6180C2211yescharacteristic
10C4⋊((C3⋊Dic3)⋊C3)C2214yes—
11#88449—22no—
12#63918—34no—
13#63644D312yes—
14C3×#817—32no—
15C3×C8×Dic3—32no—
16C3×C8×Dic3—61no—
17#63573—34no—
18C2×C4×H(3,𝔽3)C2311yescharacteristic
19C2×#1084D412yes—
20C2×(C3⋊Dic3)⋊C3—24no—
21#6196—44no—
22C8×H(3,𝔽3)Q812yescomplemented
23#6180Q812yescomplemented
24#4855—92no—
25#4680—91no—
26C2×#627—64no—
27C2×C32×C8—21no—
28C2×C32×C8C2×D312yes—
29C4⋊(C3⋊Dic3)C2×D314yes—
30C3×C4⋊Dic3—34no—
31C3×C4×Dic3—32no—
32C3×C4×Dic3—61no—
33C2×C3×C3⋊C8—32no—
34C2×C3×C3⋊C8—61no—
35Dic3⋊Dic3—64no—
36C22×H(3,𝔽3)C2×D411yescharacteristic
37C4×H(3,𝔽3)C2×Q811yescharacteristic
38C4×H(3,𝔽3)C2×Q811yescharacteristic
39#1084SD1614yes—
40(C3⋊Dic3)⋊C3—44no—
41#888—94no—
42#869—92no—
43C3×#106—92no—
44C3×C4×C8—91no—
45#817—32no—
46#817—92no—
47#817—92no—
48C8×Dic3—32no—
49C8×Dic3—61no—
50#808—94no—
51C2×C32×C4—21no—
52C2×C32×C4C22×D312yes—
53C2×C3⋊Dic3—24no—
54C2×C3×Dic3—34no—
55C2×C3×Dic3—62no—
56C2×C3×Dic3—64no—
57#627—128no—
58C32×C8—22no—
59C32×C8Dic614yescomplemented
60C3×C3⋊C8—34no—
61C3×C3⋊C8—62no—
62#498—271noSylow
63C2×H(3,𝔽3)C4⋊Q811yescharacteristic
64C2×H(3,𝔽3)C2×SD1612yes—
65C2×Dic6—184no—
66C2×Dic6—182no—
67C2×C3×C8—32no—
68C2×C3×C8—61no—
69C2×C3×C8—91no—
70C2×C3×C8D3211yescharacteristic
71C3×C4⋊C4—94no—
72C3×C42—91no—
73C4⋊Dic3—34no—
74C4⋊Dic3—94no—
75C4⋊Dic3—182no—
76Dic3⋊C4—184no—
77C4×Dic3—32no—
78C4×Dic3—61no—
79C2×C3⋊C8—32no—
80C2×C3⋊C8—61no—
81C22×C32—21no—
82C22×C32D3×D412yes—
83C32×C4—21no—
84C32×C4—21no—
85C32×C4C2×Dic612yes—
86C32×C4D3×Q812yes—
87C3⋊Dic3—44no—
88C3×Dic3—38no—
89C3×Dic3—64no—
90C3×Dic3—124no—
91C4⋊Q8—272no—
92#106—92no—
93#106—272no—
94C4×C8—91no—
95H(3,𝔽3)#49811yescharacteristic, complemented, Sylow
96C2×C3×C4—32no—
97C2×C3×C4—61no—
98C2×C3×C4—92no—
99C2×C3×C4—184no—
100C2×C3×C4C2×D3211yescharacteristic
101C2×Dic3—34no—
102C2×Dic3—62no—
103C2×Dic3—64no—
104C2×Dic3—184no—
105Dic6—368no—
106Dic6—364no—
107C3×C8—34no—
108C3×C8—62no—
109C3×C8—92no—
110C3×C8#62712yes—
111C3⋊C8—34no—
112C3⋊C8—62no—
113C2×C32—22no—
114C2×C32—21no—
115C2×C32#88812yes—
116C2×C32D3×SD1614yes—
117C2×Q8—542no—
118C2×C8—91no—
119C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
120C4⋊C4—94no—
121C4⋊C4—542no—
122C42—91no—
123C22×C3—32no—
124C22×C3—61no—
125C22×C3#380211yescharacteristic
126C3×C4—32no—
127C3×C4—32no—
128C3×C4—61no—
129C3×C4—61no—
130C3×C4—94no—
131C3×C4—364no—
132C3×C4#379711yescharacteristic
133C3×C4C2×#62711yescharacteristic
134Dic3—38no—
135Dic3—64no—
136Dic3—124no—
137Dic3—364no—
138C32—21no—
139C32#485512yescomplemented
140Q8—1084no—
141C2×C4—92no—
142C2×C4—184no—
143C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
144C8—92no—
145C8#619612yescomplemented
146C2×C3—34no—
147C2×C3—32no—
148C2×C3—62no—
149C2×C3—61no—
150C2×C3#6379612yes—
151C2×C3#6391911yescharacteristic
152C22#8865511yescharacteristic
153C4—94no—
154C4—364no—
155C4#8865111yescharacteristic
156C4C2×#619611yescharacteristic
157C3—32no—
158C3—61no—
159C3#1059090011yescharacteristic
160C2#1172083712yes—
161C2#1172111711yescharacteristic
subgroupG/H
centerC22#88655
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#498
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
22. C8×H(3,𝔽3)Q8semidirect
23. #6180Q8semidirect
59. C32×C8Dic6semidirect
95. H(3,𝔽3)#498semidirect
139. C32#4855semidirect
145. C8#6196semidirect

Maximal subgroups

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

subgroupindex
#117211172
C8×#10842
#63644⋊C32
#117203142
#105908893