Finite Group Databasebeta

← C3×#10593543 #13167620

Subgroups — C3×#10593543 #13167620

Order 1728.

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

normal subgroups only

This group has 298 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×#64286C212yesmaximal
2C2×C3×#3787C211yesmaximal, characteristic
3C3×#63746C212yesmaximal, complemented
4C3×#63745C212yesmaximal, complemented
5#10593543C311yesmaximal, characteristic, direct product factor
6C2×C3×C4×C3⋊D3C2211yescharacteristic
7C3×C22⋊#157—22no—
8C2×C3×#622C412yescomplemented
9C3×C4⋊#157C2214yes—
10C3×C4×#157—22no—
11C3×#3787C412yescomplemented
12C3×#3787C2211yescharacteristic
13C3×#3787C2211yescharacteristic
14#64286C2×C312yes—
15C2×#3787C2×C311yescharacteristic
16#63746C2×C312yescomplemented
17#63745C2×C312yescomplemented
18C22×C3×C3⋊D3D411yescharacteristic
19C2×C3×#157—24no—
20C2×C33×C4Q811yescharacteristic
21C2×C3×C3⋊Dic3C2×C411yescharacteristic
22C3×C4×C3⋊D3C2×C411yescharacteristic
23C3×C4×C3⋊D3D411yescharacteristic
24C3×C4×C3⋊D3Q811yescharacteristic
25C3×C4×C3⋊D3C2311yescharacteristic
26C3×#622C2×C412yes—
27C3×#622C2×C412yes—
28C3×#427—91nomaximal
29C2×C4×C3⋊D3C22×C311yescharacteristic
30C2×C3×C4×D3—121no—
31C22⋊#157—22no—
32C2×#622C3×C412yescomplemented
33C4⋊#157C22×C314yes—
34C4×#157—22no—
35#3787C3×C412yescomplemented
36#3787C22×C311yescharacteristic
37#3787C22×C311yescharacteristic
38C22×C33C4⋊C411yescharacteristic
39C2×C3×C3⋊D3—22no—
40C2×C3×C3⋊D3C4⋊C411yescharacteristic
41C2×C3×C3⋊D3C2×D411yescharacteristic
42C3×#157—44no—
43C33×C4C4⋊C411yescharacteristic
44C33×C4C2×Q811yescharacteristic
45C3×C3⋊Dic3C4⋊C411yescharacteristic
46C3×C3⋊Dic3C22×C411yescharacteristic
47C22×C3×C8—91no—
48C3×#117—92no—
49C3×#107—92no—
50C3×#106—92no—
51C22×C3⋊D3C3×D411yescharacteristic
52C22×C3×D3—121no—
53C2×#157—24no—
54C2×C32×C4—41no—
55C2×C32×C4—81no—
56C2×C32×C4C3×Q811yescharacteristic
57C2×C3⋊Dic3C2×C3×C411yescharacteristic
58C4×C3⋊D3C2×C3×C411yescharacteristic
59C4×C3⋊D3C3×D411yescharacteristic
60C4×C3⋊D3C3×Q811yescharacteristic
61C4×C3⋊D3C23×C311yescharacteristic
62C2×C3×Dic3—121no—
63C3×C4×D3—121no—
64C3×C4×D3—121no—
65C3×C4×D3—121no—
66C3×C4×D3—121no—
67#622C2×C3×C412yes—
68#622C2×C3×C412yes—
69#427—91noSylow
70C2×C33—21no—
71C2×C33C2×C4⋊C411yescharacteristic
72C3×C3⋊D3—21no—
73C3×C3⋊D3#13512yes—
74C22×C3×C4—91no—
75C2×C4×D3—121no—
76C2×C3×C8—91no—
77C2×C3×C8—91no—
78C2×C3×C8—92no—
79C2×C3×C8—92no—
80C3×C4⋊C4—94no—
81C3×C22⋊C4—182no—
82C3×C42—182no—
83C22×C32—41no—
84C22×C32—81no—
85C22×C32C3×C4⋊C411yescharacteristic
86C2×C3⋊D3—22no—
87C2×C3⋊D3C3×C4⋊C411yescharacteristic
88C2×C3⋊D3C2×C3×D411yescharacteristic
89C2×C3×D3—121no—
90C2×C3×D3—121no—
91C2×C3×D3—124no—
92#157—44no—
93C32×C4—41no—
94C32×C4—41no—
95C32×C4—81no—
96C32×C4—81no—
97C32×C4C3×C4⋊C411yescharacteristic
98C32×C4C2×C3×Q811yescharacteristic
99C3⋊Dic3C3×C4⋊C411yescharacteristic
100C3⋊Dic3C22×C3×C411yescharacteristic
101C3×Dic3—121no—
102C3×Dic3—121no—
103C22×C8—91no—
104#117—92no—
105#107—92no—
106#106—92no—
107C33#42711yescharacteristic, complemented, Sylow
108C23×C3—91no—
109C22×D3—121no—
110C2×C3×C4—41no—
111C2×C3×C4—81no—
112C2×C3×C4—91no—
113C2×C3×C4—91no—
114C2×C3×C4—91no—
115C2×C3×C4—91no—
116C2×C3×C4—91no—
117C2×C3×C4—184no—
118C2×C3×C4PSU(3,𝔽2)11yescharacteristic
119C2×Dic3—121no—
120C4×D3—121no—
121C4×D3—121no—
122C4×D3—121no—
123C4×D3—121no—
124C3×C8—92no—
125C3×C8—92no—
126C2×C32—21no—
127C2×C32—42no—
128C2×C32—41no—
129C2×C32—82no—
130C2×C32—81no—
131C2×C32C2×C3×C4⋊C411yescharacteristic
132C3⋊D3—21no—
133C3⋊D3C3×#13512yes—
134C3×D3—122no—
135C3×D3—122no—
136C22×C4—91no—
137C2×C8—91no—
138C2×C8—91no—
139C2×C8—92no—
140C2×C8—92no—
141C4⋊C4—94no—
142C22⋊C4—182no—
143C42—182no—
144C22×C3—41no—
145C22×C3—81no—
146C22×C3—91no—
147C22×C3—91no—
148C22×C3—182no—
149C22×C3#377711yescharacteristic
150C2×D3—121no—
151C2×D3—121no—
152C2×D3—124no—
153C3×C4—41no—
154C3×C4—41no—
155C3×C4—81no—
156C3×C4—81no—
157C3×C4—91no—
158C3×C4—91no—
159C3×C4—364no—
160C3×C4#377711yescharacteristic
161C3×C4C2×PSU(3,𝔽2)11yescharacteristic
162Dic3—121no—
163Dic3—121no—
164C32—41no—
165C32—81no—
166C32C3×#42711yescharacteristic, complemented
167C23—91no—
168C2×C4—91no—
169C2×C4—91no—
170C2×C4—91no—
171C2×C4—91no—
172C2×C4—91no—
173C2×C4—184no—
174C2×C4C3×PSU(3,𝔽2)11yescharacteristic
175C8—92no—
176C8—92no—
177C2×C3—21no—
178C2×C3—42no—
179C2×C3—41no—
180C2×C3—82no—
181C2×C3—81no—
182C2×C3—92no—
183C2×C3—181no—
184C2×C3C2×#377711yescharacteristic
185D3—122no—
186D3—122no—
187C22—91no—
188C22—91no—
189C22—182no—
190C22C3×#377711yescharacteristic
191C4—91no—
192C4—91no—
193C4—364no—
194C4C3×#377711yescharacteristic
195C4C2×C3×PSU(3,𝔽2)11yescharacteristic
196C3—41no—
197C3—81no—
198C3#1059354311yescharacteristic, direct product factor
199C2—21no—
200C2—92no—
201C2—181no—
202C2C2×C3×#377711yescharacteristic
subgroupG/H
centerC3×C4#3777
commutatorC3⋊Dic3C22×C3×C4
FrattiniC4C2×C3×PSU(3,𝔽2)
FittingC2×C33×C4Q8
socleC2×C33C2×C4⋊C4
prime radicalC22×C3×C3⋊D3D4
cosocleC4×C3⋊D3C23×C3
hypercenterC2×C3×C4PSU(3,𝔽2)
hypocenterC32C3×#427
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×#63746C2semidirect
4. C3×#63745C2semidirect
5. #10593543C3direct (N × H)
8. C2×C3×#622C4semidirect
11. C3×#3787C4semidirect
16. #63746C2×C3semidirect
17. #63745C2×C3semidirect
32. C2×#622C3×C4semidirect
35. #3787C3×C4semidirect
107. C33#427semidirect
166. C32C3×#427semidirect
198. C3#10593543direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#642862
C2×C3×#37872
C3×#637462
C3×#637452
#105935433
C3×#4279