Finite Group Databasebeta

← #13136659

Subgroups — #13136659

Order 1728.

175 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 175 conjugacy classes of subgroups — too many to draw. Only its 25 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#11720833C211yesmaximal, characteristic
2#11720775C211yesmaximal, characteristic
3#11720251C211yesmaximal, characteristic
4#88651—21no—
5C4⋊#1088—21no—
6C8×#272C2211yescharacteristic
7#3746⋊C3—21no—
8#88433C412yescomplemented
9(C3⋊(C3⋊C16))⋊C3—21no—
10D3×Dic12—61no—
11D3×C3⋊C16—61no—
12C2×#1088—41no—
13C4×#272D411yescharacteristic
14#634⋊C3—22no—
15#6199—41no—
16#6196—41no—
17C8×H(3,𝔽3)D411yescharacteristic
18#6180C2×C411yescharacteristic
19#4492—91nomaximal
20D3×Dic6—62no—
21#3746—21no—
22C3×Dic12—61no—
23C3×D3×C8—61no—
24C3×D3×C8—61no—
25#3719—62no—
26C3⋊(C3⋊C16)—21no—
27C3×C3⋊C16—61no—
28C2×#272D811yescharacteristic
29#1088—81no—
30C4×H(3,𝔽3)C22⋊C411yescharacteristic
31#1084SD1611yescharacteristic
32(C3⋊Dic3)⋊C3—42no—
33D3×Q16—181no—
34C2×Dic12—91no—
35C3×#107—91no—
36C2×C3⋊C16—91no—
37D3×C16—181no—
38#634—22no—
39C3×C4×D3—61no—
40C3×C4×D3—61no—
41C3×Dic6—62no—
42#627—122no—
43D3×Dic3—122no—
44C32×C8—21no—
45C32×C8—21no—
46C3×C3⋊C8—61no—
47C3×C3⋊C8—61no—
48#358—271noSylow
49C2×H(3,𝔽3)#10211yescharacteristic
50#272SD3211yescharacteristic
51#272Q3211yescharacteristic
52D3×Q8—182no—
53C2×Dic6—181no—
54C3×Q16—181no—
55C2×C3×C8—91no—
56C3×C4⋊C4—181no—
57#216—182no—
58Dic12—61no—
59Dic12—92no—
60Dic12—181no—
61D3×C8—61no—
62D3×C8—61no—
63C3×C16—181no—
64C3⋊C16—61no—
65C3⋊C16—181no—
66C2×C3×D3—61no—
67C2×C3×D3—61no—
68C32×C4—21no—
69C32×C4—21no—
70C3⋊Dic3—42no—
71C3×Dic3—61no—
72C3×Dic3—61no—
73C3×Dic3—122no—
74C2×Q16—271no—
75C2×C16—271no—
76#107—91no—
77H(3,𝔽3)#35811yescharacteristic, complemented, Sylow
78C3×Q8—182no—
79C2×C3×C4—91no—
80C2×C3×C4—361no—
81C2×Dic3—361no—
82C4×D3—61no—
83C4×D3—61no—
84C4×D3—362no—
85Dic6—62no—
86Dic6—182no—
87Dic6—362no—
88Dic6—361no—
89C3×C8—61no—
90C3×C8—61no—
91C3×C8—91no—
92C3×C8GO+(4,𝔽2)11yescharacteristic
93C3⋊C8—61no—
94C3⋊C8—61no—
95C2×C32—21no—
96C2×C32—21no—
97C3×D3—61no—
98C3×D3—61no—
99C3×D3—61no—
100C3×D3—61no—
101C2×Q8—541no—
102Q16—181no—
103Q16—272no—
104C2×C8—91no—
105C4⋊C4—181no—
106C16—181no—
107C22×C3—91no—
108C2×D3—61no—
109C2×D3—61no—
110C3×C4—61no—
111C3×C4—61no—
112C3×C4—91no—
113C3×C4—362no—
114C3×C4—721no—
115C3×C4D32⋊C411yescharacteristic
116Dic3—61no—
117Dic3—61no—
118Dic3—122no—
119Dic3—362no—
120C32—21no—
121C32—21no—
122Q8—182no—
123Q8—1081no—
124C2×C4—91no—
125C2×C4—361no—
126C2×C4—1081no—
127C8—91no—
128C8#1088⋊C211yescharacteristic
129C2×C3—61no—
130C2×C3—61no—
131C2×C3—91no—
132C2×C3—91no—
133C2×C3#6372911yescharacteristic
134D3—61no—
135D3—61no—
136D3—61no—
137D3—61no—
138C22—91no—
139C4—91no—
140C4—362no—
141C4—721no—
142C4(((C3⋊D3)⋊C3)⋊C2)⋊C411yescharacteristic
143C3—61no—
144C3—61no—
145C3#1059001711yescharacteristic
146C2—91no—
147C2—91no—
148C2#1172069511yescharacteristic
subgroupG/H
centerC2#11720695
commutator#6180C2×C4
FrattiniC3×C8GO+(4,𝔽2)
FittingC8×H(3,𝔽3)D4
socleC2×C3#63729
prime radicalC2×#272D8
cosocleC8×#272C22
hypercenterC8#1088⋊C2
hypocenterH(3,𝔽3)#358
LCSG ▷ #6180 ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
8. #88433C4semidirect
77. H(3,𝔽3)#358semidirect

Maximal subgroups

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

subgroupindex
#117208332
#117207752
#117202512
#44929