Finite Group Databasebeta

← C3×#10593546 #13167622

Subgroups — C3×#10593546 #13167622

Order 1728.

282 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 94 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
1C3×#64283C212yesmaximal
2C2×C3×#3787C211yesmaximal, characteristic
3C3×#63748C214yesmaximal, complemented
4#10593546C311yesmaximal, characteristic, direct product factor
5C2×C3×C4×C3⋊D3C2211yescharacteristic
6C3×#3792C2214yes—
7C2×C3×#622C2212yes—
8C3×#3788—22no—
9C3×#3787—22no—
10C3×#3787C412yes—
11C3×#3787C412yes—
12#64283C2×C312yes—
13C2×#3787C2×C311yescharacteristic
14#63748C2×C314yescomplemented
15C22×C3×C3⋊D3Q811yescharacteristic
16C2×C33×C4Q811yescharacteristic
17C2×C3×C3⋊Dic3C2311yescharacteristic
18C3×C4×C3⋊D3C2×C411yescharacteristic
19C3×C4×C3⋊D3C2×C411yescharacteristic
20C3×C4×C3⋊D3D411yescharacteristic
21C3×C4×C3⋊D3D411yescharacteristic
22C3×#622—24no—
23C3×#622C2×C414yes—
24C2×C3×#108—91nomaximal
25C2×C4×C3⋊D3C22×C311yescharacteristic
26C2×C3×C4×D3—121no—
27#3792C22×C314yes—
28C2×#622C22×C312yes—
29#3788—22no—
30#3787—22no—
31#3787C3×C412yes—
32#3787C3×C412yes—
33C22×C33C2×Q811yescharacteristic
34C2×C3×C3⋊D3—22no—
35C2×C3×C3⋊D3C4⋊C411yescharacteristic
36C2×C3×C3⋊D3C4⋊C411yescharacteristic
37C33×C4C4⋊C411yescharacteristic
38C33×C4C4⋊C411yescharacteristic
39C3×C3⋊Dic3C22×C411yescharacteristic
40C3×C3⋊Dic3C2×D411yescharacteristic
41C2×C3×M16—92no—
42C22×C3×C8—91no—
43C3×#108—94no—
44C22×C3⋊D3C3×Q811yescharacteristic
45C22×C3×D3—121no—
46C2×C32×C4—41no—
47C2×C32×C4—81no—
48C2×C32×C4C3×Q811yescharacteristic
49C2×C3⋊Dic3C23×C311yescharacteristic
50C4×C3⋊D3C2×C3×C411yescharacteristic
51C4×C3⋊D3C2×C3×C411yescharacteristic
52C4×C3⋊D3C3×D411yescharacteristic
53C4×C3⋊D3C3×D411yescharacteristic
54C2×C3×Dic3—121no—
55C3×C4×D3—121no—
56C3×C4×D3—121no—
57C3×C4×D3—121no—
58C3×C4×D3—121no—
59#622—24no—
60#622C2×C3×C414yes—
61C2×#108—91noSylow
62C2×C33—21no—
63C2×C33C2×C4⋊C411yescharacteristic
64C3×C3⋊D3—21no—
65C3×C3⋊D3#10812yescomplemented
66C22×C3×C4—91no—
67C2×C4×D3—121no—
68C3×M16—94no—
69C3×M16—182no—
70C2×C3×C8—92no—
71C2×C3×C8—92no—
72C2×C3×C8—92no—
73C2×C3×C8—182no—
74C22×C32—41no—
75C22×C32—81no—
76C22×C32C2×C3×Q811yescharacteristic
77C2×C3⋊D3—22no—
78C2×C3⋊D3C3×C4⋊C411yescharacteristic
79C2×C3⋊D3C3×C4⋊C411yescharacteristic
80C2×C3×D3—121no—
81C2×C3×D3—121no—
82C2×C3×D3—124no—
83C32×C4—41no—
84C32×C4—41no—
85C32×C4—81no—
86C32×C4—81no—
87C32×C4C3×C4⋊C411yescharacteristic
88C32×C4C3×C4⋊C411yescharacteristic
89C3⋊Dic3C22×C3×C411yescharacteristic
90C3⋊Dic3C2×C3×D411yescharacteristic
91C3×Dic3—121no—
92C3×Dic3—121no—
93C2×M16—92no—
94C22×C8—91no—
95#108—94no—
96C33C2×#10811yescharacteristic, complemented, Sylow
97C23×C3—91no—
98C22×D3—121no—
99C2×C3×C4—41no—
100C2×C3×C4—81no—
101C2×C3×C4—91no—
102C2×C3×C4—91no—
103C2×C3×C4—91no—
104C2×C3×C4—91no—
105C2×C3×C4—91no—
106C2×C3×C4PSU(3,𝔽2)11yescharacteristic
107C2×Dic3—121no—
108C4×D3—121no—
109C4×D3—121no—
110C4×D3—121no—
111C4×D3—121no—
112C3×C8—94no—
113C3×C8—184no—
114C2×C32—21no—
115C2×C32—42no—
116C2×C32—41no—
117C2×C32—82no—
118C2×C32—81no—
119C2×C32C2×C3×C4⋊C411yescharacteristic
120C3⋊D3—21no—
121C3⋊D3C3×#10812yescomplemented
122C3×D3—122no—
123C3×D3—122no—
124C22×C4—91no—
125M16—94no—
126M16—182no—
127C2×C8—92no—
128C2×C8—92no—
129C2×C8—92no—
130C2×C8—182no—
131C22×C3—41no—
132C22×C3—81no—
133C22×C3—91no—
134C22×C3—91no—
135C22×C3—182no—
136C22×C3C2×PSU(3,𝔽2)11yescharacteristic
137C2×D3—121no—
138C2×D3—121no—
139C2×D3—124no—
140C3×C4—41no—
141C3×C4—41no—
142C3×C4—81no—
143C3×C4—81no—
144C3×C4—91no—
145C3×C4—91no—
146C3×C4#377711yescharacteristic
147C3×C4#377711yescharacteristic
148Dic3—121no—
149Dic3—121no—
150C32—41no—
151C32—81no—
152C32C2×C3×#10811yescharacteristic, complemented
153C23—91no—
154C2×C4—91no—
155C2×C4—91no—
156C2×C4—91no—
157C2×C4—91no—
158C2×C4—91no—
159C2×C4C3×PSU(3,𝔽2)11yescharacteristic
160C8—94no—
161C8—184no—
162C2×C3—21no—
163C2×C3—42no—
164C2×C3—41no—
165C2×C3—82no—
166C2×C3—81no—
167C2×C3—92no—
168C2×C3—181no—
169C2×C3C2×#377711yescharacteristic
170D3—122no—
171D3—122no—
172C22—91no—
173C22—91no—
174C22—182no—
175C22C2×C3×PSU(3,𝔽2)11yescharacteristic
176C4—91no—
177C4—91no—
178C4C3×#377711yescharacteristic
179C4C3×#377711yescharacteristic
180C3—41no—
181C3—81no—
182C3#1059354611yescharacteristic, direct product factor
183C2—21no—
184C2—92no—
185C2—181no—
186C2C2×C3×#377711yescharacteristic
subgroupG/H
centerC3×C4#3777
commutatorC3⋊Dic3C22×C3×C4
FrattiniC22C2×C3×PSU(3,𝔽2)
FittingC2×C33×C4Q8
socleC2×C33C2×C4⋊C4
prime radicalC22×C3×C3⋊D3Q8
cosocleC2×C3⋊Dic3C23×C3
hypercenterC2×C3×C4PSU(3,𝔽2)
hypocenterC32C2×C3×#108
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C3×C4 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
3. C3×#63748C2semidirect
4. #10593546C3direct (N × H)
14. #63748C2×C3semidirect
65. C3×C3⋊D3#108semidirect
96. C33C2×#108semidirect
121. C3⋊D3C3×#108semidirect
152. C32C2×C3×#108semidirect
182. C3#10593546direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#642832
C2×C3×#37872
C3×#637482
#105935463
C2×C3×#1089