Finite Group Databasebeta

← C2×C3×#63745 #13167618

Subgroups — C2×C3×#63745 #13167618

Order 1728.

330 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 114 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×C3×C4⋊#157C212yesmaximal
2C2×C3×#3787C211yesmaximal, characteristic
3C3×#63745C214yesmaximal, direct product factor
4C2×#63745C311yesmaximal, characteristic, direct product factor
5C22×C3×#157—22no—
6C2×C3×C4×C3⋊D3C2211yescharacteristic
7C2×C3×#622C412yescomplemented
8C3×C4⋊#157—22no—
9C3×C4⋊#157C2214yes—
10C3×#3787C412yescomplemented
11C3×#3787C2212yes—
12C2×C4⋊#157C2×C312yes—
13C2×#3787C2×C311yescharacteristic
14#63745C2×C314yesdirect product factor
15C22×C3×C3⋊D3D411yescharacteristic
16C2×C3×#157—24no—
17C2×C3×#157—44no—
18C2×C33×C4Q811yescharacteristic
19C2×C3×C3⋊Dic3C2×C411yescharacteristic
20C3×C4×C3⋊D3C2×C411yescharacteristic
21C3×C4×C3⋊D3D411yescharacteristic
22C3×C4×C3⋊D3Q811yescharacteristic
23C3×C4×C3⋊D3C2311yescharacteristic
24C3×#622C2×C414yescomplemented
25C2×C3×#106—91nomaximal
26C22×#157—22no—
27C2×C4×C3⋊D3C22×C311yescharacteristic
28C2×C3×C4×D3—121no—
29C2×#622C3×C412yescomplemented
30C4⋊#157—22no—
31C4⋊#157C22×C314yes—
32#3787C3×C412yescomplemented
33#3787C22×C312yes—
34C22×C33C4⋊C411yescharacteristic
35C2×C3×C3⋊D3C4⋊C411yescharacteristic
36C2×C3×C3⋊D3SD1614yes—
37C2×C3×C3⋊D3C2×D411yescharacteristic
38C3×#157—44no—
39C33×C4C4⋊C411yescharacteristic
40C33×C4C2×Q811yescharacteristic
41C3×C3⋊Dic3C4⋊C411yescharacteristic
42C3×C3⋊Dic3C22×C411yescharacteristic
43C22×C3×C8—91no—
44C2×C3×C4⋊C4—92no—
45C3×#106—94no—
46C22×C3⋊D3C3×D411yescharacteristic
47C22×C3×D3—121no—
48C2×#157—24no—
49C2×#157—44no—
50C2×C32×C4—41no—
51C2×C32×C4—81no—
52C2×C32×C4C3×Q811yescharacteristic
53C2×C3⋊Dic3C2×C3×C411yescharacteristic
54C4×C3⋊D3C2×C3×C411yescharacteristic
55C4×C3⋊D3C3×D411yescharacteristic
56C4×C3⋊D3C3×Q811yescharacteristic
57C4×C3⋊D3C23×C311yescharacteristic
58C2×C3×Dic3—121no—
59C3×C4×D3—121no—
60C3×C4×D3—121no—
61C3×C4×D3—121no—
62C3×C4×D3—121no—
63#622C2×C3×C414yescomplemented
64C2×#106—91noSylow
65C2×C33#10612yescomplemented
66C2×C33C2×C4⋊C411yescharacteristic
67C3×C3⋊D3#10612yescomplemented
68C3×C3⋊D3C2×SD1612yes—
69C22×C3×C4—91no—
70C22×C3×C4—182no—
71C2×C4×D3—121no—
72C2×C3×C8—92no—
73C2×C3×C8—92no—
74C2×C3×C8—92no—
75C3×C4⋊C4—94no—
76C3×C4⋊C4—182no—
77C22×C32—41no—
78C22×C32—81no—
79C22×C32C3×C4⋊C411yescharacteristic
80C2×C3⋊D3C3×C4⋊C411yescharacteristic
81C2×C3⋊D3C3×SD1614yes—
82C2×C3⋊D3C2×C3×D411yescharacteristic
83C2×C3×D3—121no—
84C2×C3×D3—124no—
85C2×C3×D3—121no—
86#157—44no—
87C32×C4—41no—
88C32×C4—41no—
89C32×C4—81no—
90C32×C4—81no—
91C32×C4C3×C4⋊C411yescharacteristic
92C32×C4C2×C3×Q811yescharacteristic
93C3⋊Dic3C3×C4⋊C411yescharacteristic
94C3⋊Dic3C22×C3×C411yescharacteristic
95C3×Dic3—121no—
96C3×Dic3—121no—
97C22×C8—91no—
98C2×C4⋊C4—92no—
99#106—94no—
100C33C2×#10611yescharacteristic, complemented, Sylow
101C23×C3—91no—
102C22×D3—121no—
103C2×C3×C4—41no—
104C2×C3×C4—81no—
105C2×C3×C4—91no—
106C2×C3×C4—91no—
107C2×C3×C4—91no—
108C2×C3×C4—91no—
109C2×C3×C4—91no—
110C2×C3×C4—184no—
111C2×C3×C4—364no—
112C2×C3×C4PSU(3,𝔽2)11yescharacteristic
113C2×Dic3—121no—
114C4×D3—121no—
115C4×D3—121no—
116C4×D3—121no—
117C4×D3—121no—
118C3×C8—94no—
119C2×C32—42no—
120C2×C32—41no—
121C2×C32—82no—
122C2×C32—81no—
123C2×C32C3×#10612yescomplemented
124C2×C32C2×C3×C4⋊C411yescharacteristic
125C3⋊D3C3×#10612yescomplemented
126C3⋊D3C2×C3×SD1612yes—
127C3×D3—122no—
128C3×D3—122no—
129C22×C4—91no—
130C22×C4—182no—
131C2×C8—92no—
132C2×C8—92no—
133C2×C8—92no—
134C4⋊C4—94no—
135C4⋊C4—182no—
136C22×C3—41no—
137C22×C3—81no—
138C22×C3—94no—
139C22×C3—91no—
140C22×C3—91no—
141C22×C3#377711yescharacteristic
142C2×D3—121no—
143C2×D3—121no—
144C2×D3—124no—
145C3×C4—41no—
146C3×C4—41no—
147C3×C4—81no—
148C3×C4—81no—
149C3×C4—91no—
150C3×C4—91no—
151C3×C4—364no—
152C3×C4#377711yescharacteristic
153C3×C4C2×PSU(3,𝔽2)11yescharacteristic
154Dic3—121no—
155Dic3—121no—
156C32—41no—
157C32—81no—
158C32C2×C3×#10611yescharacteristic, complemented
159C23—91no—
160C2×C4—91no—
161C2×C4—91no—
162C2×C4—91no—
163C2×C4—91no—
164C2×C4—91no—
165C2×C4—184no—
166C2×C4—364no—
167C2×C4C3×PSU(3,𝔽2)11yescharacteristic
168C8—94no—
169C2×C3—42no—
170C2×C3—41no—
171C2×C3—82no—
172C2×C3—81no—
173C2×C3—92no—
174C2×C3—92no—
175C2×C3#6374512yesdirect product factor
176C2×C3C2×#377711yescharacteristic
177D3—122no—
178D3—122no—
179C22—94no—
180C22—91no—
181C22—91no—
182C22C3×#377711yescharacteristic
183C4—91no—
184C4—91no—
185C4—364no—
186C4C3×#377711yescharacteristic
187C4C2×C3×PSU(3,𝔽2)11yescharacteristic
188C3—41no—
189C3—81no—
190C3C2×#6374511yescharacteristic, direct product factor
191C2—92no—
192C2—92no—
193C2C3×#6374512yesdirect product factor
194C2C2×C3×#377711yescharacteristic
subgroupG/H
centerC22×C3#3777
commutatorC3⋊Dic3C22×C3×C4
FrattiniC4C2×C3×PSU(3,𝔽2)
FittingC2×C33×C4Q8
socleC22×C33C4⋊C4
prime radicalC22×C3×C3⋊D3D4
cosocleC4×C3⋊D3C23×C3
hypercenterC2×C3×C4PSU(3,𝔽2)
hypocenterC32C2×C3×#106
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
3. C3×#63745C2direct (N × H)
4. C2×#63745C3direct (N × H)
7. C2×C3×#622C4semidirect
10. C3×#3787C4semidirect
14. #63745C2×C3direct (N × H)
24. C3×#622C2×C4semidirect
29. C2×#622C3×C4semidirect
32. #3787C3×C4semidirect
63. #622C2×C3×C4semidirect
65. C2×C33#106semidirect
67. C3×C3⋊D3#106semidirect
100. C33C2×#106semidirect
123. C2×C32C3×#106semidirect
125. C3⋊D3C3×#106semidirect
158. C32C2×C3×#106semidirect
175. C2×C3#63745direct (N × H)
190. C3C2×#63745direct (N × H)
193. C2C3×#63745direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4⋊#1572
C2×C3×#37872
C3×#637452
C2×#637453
C2×C3×#1069