Finite Group Databasebeta

← #13135216

Subgroups — #13135216

Order 1728.

278 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 43 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#11721861C211yesmaximal, characteristic
2#63663⋊C3C212yesmaximal
3#10589184—32nomaximal
4C2×Q8×H(3,𝔽3)C2211yescharacteristic
5C2×#6229C411yescharacteristic
6#88540—22no—
7C4×#1084C411yescharacteristic
8#3748⋊C3—22no—
9C3×#944—32no—
10C3×#944—61no—
11#63663D312yes—
12C3×#836—32no—
13Q8×H(3,𝔽3)—41no—
14C2×C4×H(3,𝔽3)C2×C411yescharacteristic
15C2×C4×H(3,𝔽3)D412yes—
16C2×#1084—21no—
17C2×#1084—21no—
18#6229—41no—
19(C3⋊(C3⋊C8))⋊C3—42no—
20#4548—91no—
21#4478—92no—
22C2×C32×Q8—21no—
23C2×C32×Q8C2×D312yes—
24C2×C3×Dic6—32no—
25C2×C3×Dic6—61no—
26#3748—22no—
27C3×Dic3⋊C4—64no—
28C3×Dic3⋊C4—62no—
29C3×Dic3⋊C4—62no—
30C3×C4×Dic3—32no—
31C3×C4×Dic3—61no—
32C3×#208—62no—
33C22×H(3,𝔽3)C22⋊C411yescharacteristic
34C4×H(3,𝔽3)—21no—
35C4×H(3,𝔽3)—22no—
36#1084—22no—
37#1084—41no—
38C3×C4⋊Q8—91no—
39#944—32no—
40#944—61no—
41C3×#101—92no—
42#836—32no—
43#836—92no—
44C32×Q8—42no—
45C32×Q8—42no—
46C2×C32×C4—21no—
47C2×C32×C4—22no—
48C2×C32×C4C4×D312yes—
49C2×C32×C4D1212yes—
50C2×C32×C4#6712yes—
51C2×C3×Dic3—62no—
52C2×C3×Dic3—62no—
53C2×C3×Dic3—62no—
54C2×C3×Dic3—62no—
55C3×Dic6—122no—
56C3×Dic6—122no—
57C3⋊(C3⋊C8)—42no—
58C3×C3⋊C8—122no—
59#356—271noSylow
60C2×H(3,𝔽3)—21no—
61C2×H(3,𝔽3)#9911yescharacteristic
62C2×C3×Q8—32no—
63C2×C3×Q8—61no—
64C2×C3×Q8—91no—
65C2×C3×Q8D3211yescharacteristic
66C2×Dic6—32no—
67C2×Dic6—61no—
68C3×M16—182no—
69C3×C4⋊C4—182no—
70C3×C42—91no—
71Dic3⋊C4—62no—
72Dic3⋊C4—62no—
73Dic3⋊C4—64no—
74C4×Dic3—32no—
75C4×Dic3—61no—
76#208—62no—
77#208—182no—
78C22×C32—21no—
79C22×C32(C2×D3)⋊C412yes—
80C32×C4—22no—
81C32×C4—22no—
82C32×C4—22no—
83C32×C4—22no—
84C32×C4—42no—
85C3×Dic3—64no—
86C3×Dic3—64no—
87C3×Dic3—122no—
88C3×Dic3—122no—
89C4⋊Q8—91no—
90#101—92no—
91H(3,𝔽3)#35611yescharacteristic, complemented, Sylow
92C3×Q8—41no—
93C3×Q8—122no—
94C3×Q8—122no—
95C3×Q8—361no—
96C2×C3×C4—32no—
97C2×C3×C4—32no—
98C2×C3×C4—32no—
99C2×C3×C4—61no—
100C2×C3×C4—62no—
101C2×C3×C4—181no—
102C2×C3×C4—181no—
103C2×C3×C4#62411yescharacteristic
104C2×C3×C4#62612yes—
105C2×Dic3—62no—
106C2×Dic3—62no—
107C2×Dic3—62no—
108C2×Dic3—62no—
109Dic6—122no—
110Dic6—122no—
111C3×C8—362no—
112C3⋊C8—122no—
113C3⋊C8—362no—
114C2×C32—22no—
115C2×C32—22no—
116C2×C32—21no—
117C2×C32#80612yes—
118C2×Q8—91no—
119C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
120M16—182no—
121C4⋊C4—182no—
122C42—91no—
123C22×C3—32no—
124C22×C3—61no—
125C22×C3#372211yescharacteristic
126C3×C4—22no—
127C3×C4—21no—
128C3×C4—62no—
129C3×C4—62no—
130C3×C4—62no—
131C3×C4—62no—
132C3×C4—122no—
133C3×C4—182no—
134C3×C4—361no—
135Dic3—64no—
136Dic3—64no—
137Dic3—122no—
138Dic3—122no—
139C32—21no—
140C32#447812yescomplemented
141Q8—41no—
142Q8—361no—
143C2×C4—181no—
144C2×C4—181no—
145C2×C4#619911yescharacteristic
146C2×C4#620012yes—
147C8—362no—
148C2×C3—21no—
149C2×C3—32no—
150C2×C3—62no—
151C2×C3—62no—
152C2×C3—61no—
153C2×C3#6358111yescharacteristic
154C22#8845111yescharacteristic
155C4—22no—
156C4—21no—
157C4—182no—
158C4—361no—
159C3—32no—
160C3—61no—
161C3#1058918511yescharacteristic
162C2—21no—
163C2#1172033511yescharacteristic
subgroupG/H
centerC2#11720335
commutatorC2×C4×H(3,𝔽3)C2×C4
FrattiniC2×C3×Q8D32
Fitting, cosocleC2×Q8×H(3,𝔽3)C22
socleC2×C3#63581
prime radicalC22×H(3,𝔽3)C22⋊C4
hypercenterC2×Q8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#356
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
91. H(3,𝔽3)#356semidirect
140. C32#4478semidirect

Maximal subgroups

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

subgroupindex
#117218612
#63663⋊C32
#105891843