Finite Group Databasebeta

← #13135767

Subgroups — #13135767

Order 1728.

429 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 429 conjugacy classes of subgroups — too many to draw. Only its 91 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×#88540C211yesmaximal, characteristic
2H(3,𝔽3)×C22⋊C8C211yesmaximal, characteristic
3#11720573C211yesmaximal, characteristic
4C3×#4469—34nomaximal
5C22×C4×H(3,𝔽3)C2211yescharacteristic
6C22×#1084C411yescharacteristic
7C22×#1084C411yescharacteristic
8C2×C8×H(3,𝔽3)—21no—
9#88540—21no—
10#88540—21no—
11C2×#6180—21no—
12C2×C3×Dic3⋊C4—34no—
13C32×C22⋊C8D314yes—
14C3×#830—34no—
15C23×H(3,𝔽3)C2×C411yescharacteristic
16C2×C4×H(3,𝔽3)—41no—
17C2×C4×H(3,𝔽3)D411yescharacteristic
18C2×C4×H(3,𝔽3)D411yescharacteristic
19C2×#1084—21no—
20C2×#1084—21no—
21C2×#1084—21no—
22C2×#1084—41no—
23C2×#1084C812yescomplemented
24C8×H(3,𝔽3)—41no—
25#6180—41no—
26C3×#324—91no—
27#4469—34no—
28C22×C32×C4C2×D314yes—
29C22×C3×Dic3—34no—
30C22×C3×Dic3—34no—
31C2×C32×C8—24no—
32C3×Dic3⋊C4—64no—
33C3×Dic3⋊C4—64no—
34C2×C3×C3⋊C8—64no—
35C22×H(3,𝔽3)—21no—
36C22×H(3,𝔽3)—21no—
37C22×H(3,𝔽3)C22⋊C411yescharacteristic
38C22×H(3,𝔽3)C2×C811yescharacteristic
39C22×H(3,𝔽3)M1611yescharacteristic
40C4×H(3,𝔽3)—21no—
41C4×H(3,𝔽3)—21no—
42#1084—22no—
43#1084—41no—
44C2×C3×C4⋊C4—91no—
45C2×Dic3⋊C4—34no—
46C3×C22⋊C8—34no—
47C3×C22⋊C8—91no—
48C3×C22⋊C8C3⋊D311yescharacteristic
49#830—34no—
50C23×C32C4×D314yes—
51C2×C32×C4—44no—
52C2×C32×C4D1214yes—
53C2×C32×C4#6714yes—
54C2×C3×Dic3—38no—
55C2×C3×Dic3—64no—
56C2×C3×Dic3—64no—
57C2×C3×Dic3—64no—
58C2×C3×Dic3—124no—
59C32×C8—44no—
60C3×C3⋊C8—124no—
61#324—91noSylow
62C2×H(3,𝔽3)—21no—
63C2×H(3,𝔽3)—21no—
64C2×H(3,𝔽3)C22⋊C811yescharacteristic
65C2×H(3,𝔽3)#9911yescharacteristic
66C2×H(3,𝔽3)#10111yescharacteristic
67C22×C3×C4—34no—
68C22×C3×C4—91no—
69C22×C3×C4—91no—
70C22×C3×C4C2×C3⋊D311yescharacteristic
71C22×Dic3—34no—
72C22×Dic3—34no—
73C2×C3×C8—21no—
74C2×C3×C8—64no—
75C2×C3×C8—181no—
76C3×C4⋊C4—181no—
77C3×C4⋊C4—181no—
78Dic3⋊C4—64no—
79Dic3⋊C4—64no—
80C2×C3⋊C8—64no—
81C22×C32—24no—
82C22×C32—24no—
83C22×C32D3×C814yes—
84C22×C32ΓL(1,𝔽25)14yes—
85C22×C32(C2×D3)⋊C414yes—
86C32×C4—24no—
87C32×C4—24no—
88C3×Dic3—68no—
89C3×Dic3—124no—
90C2×C4⋊C4—91no—
91C22⋊C8—91no—
92C22⋊C8#27211yescharacteristic
93H(3,𝔽3)#32411yescharacteristic, complemented, Sylow
94C23×C3—34no—
95C23×C3C4×C3⋊D311yescharacteristic
96C2×C3×C4—34no—
97C2×C3×C4—34no—
98C2×C3×C4—41no—
99C2×C3×C4—92no—
100C2×C3×C4—124no—
101C2×C3×C4—181no—
102C2×C3×C4—181no—
103C2×C3×C4—181no—
104C2×C3×C4—361no—
105C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
106C2×C3×C4#63811yescharacteristic
107C2×Dic3—38no—
108C2×Dic3—64no—
109C2×Dic3—64no—
110C2×Dic3—64no—
111C2×Dic3—124no—
112C3×C8—41no—
113C3×C8—124no—
114C3×C8—361no—
115C3⋊C8—124no—
116C2×C32—24no—
117C2×C32—24no—
118C2×C32#80614yes—
119C2×C32(C2×D3)⋊C814yes—
120C2×C32#82414yes—
121C22×C4—91no—
122C22×C4—91no—
123C22×C4C2×#27211yescharacteristic
124C2×C8—21no—
125C2×C8—181no—
126C4⋊C4—181no—
127C4⋊C4—181no—
128C22×C3—21no—
129C22×C3—21no—
130C22×C3—34no—
131C22×C3—34no—
132C22×C3—34no—
133C22×C3—64no—
134C22×C3—64no—
135C22×C3C8×C3⋊D311yescharacteristic
136C22×C3#374311yescharacteristic
137C22×C3#375211yescharacteristic
138C3×C4—21no—
139C3×C4—21no—
140C3×C4—64no—
141C3×C4—64no—
142C3×C4—182no—
143C3×C4—361no—
144Dic3—68no—
145Dic3—124no—
146C32#446914yescomplemented
147C23C4×#27211yescharacteristic
148C2×C4—41no—
149C2×C4—92no—
150C2×C4—181no—
151C2×C4—181no—
152C2×C4—181no—
153C2×C4—361no—
154C2×C4C4⋊#27211yescharacteristic
155C2×C4#623511yescharacteristic
156C8—41no—
157C8—361no—
158C2×C3—21no—
159C2×C3—21no—
160C2×C3—34no—
161C2×C3—34no—
162C2×C3—34no—
163C2×C3—64no—
164C2×C3—64no—
165C2×C3#6363311yescharacteristic
166C2×C3#6364711yescharacteristic
167C2×C3#6365111yescharacteristic
168C22—21no—
169C22—21no—
170C22C8×#27211yescharacteristic
171C22#8852711yescharacteristic
172C22#8854211yescharacteristic
173C4—21no—
174C4—21no—
175C4—182no—
176C4—361no—
177C3—34no—
178C3#1058933511yescharacteristic
179C2—21no—
180C2—21no—
181C2#1172052411yescharacteristic
182C2#1172055311yescharacteristic
183C2#1172056111yescharacteristic
subgroupG/H
center, socleC22×C3#3752
commutatorC22×H(3,𝔽3)C2×C8
FrattiniC22×C3×C4C2×C3⋊D3
FittingH(3,𝔽3)×C22⋊C8C2
prime radicalC23×H(3,𝔽3)C2×C4
cosocleC22×C4×H(3,𝔽3)C22
hypercenterC3×C22⋊C8C3⋊D3
hypocenterH(3,𝔽3)#324
LCSG ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×C22⋊C8

Complements

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

NH ≅ G/Ntype
23. C2×#1084C8semidirect
93. H(3,𝔽3)#324semidirect
146. C32#4469semidirect

Maximal subgroups

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

subgroupindex
C2×#885402
H(3,𝔽3)×C22⋊C82
#117205732
C3×#44693