Finite Group Databasebeta

← C2×C3×#63746 #13167619

Subgroups — C2×C3×#63746 #13167619

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×#63746C214yesmaximal, direct product factor
4C2×#63746C311yesmaximal, 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#63746C2×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×#107—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⋊D3D812yes—
37C2×C3×C3⋊D3Q1612yes—
38C2×C3×C3⋊D3C2×D411yescharacteristic
39C3×#157—44no—
40C33×C4C4⋊C411yescharacteristic
41C33×C4C2×Q811yescharacteristic
42C3×C3⋊Dic3C4⋊C411yescharacteristic
43C3×C3⋊Dic3C22×C411yescharacteristic
44C22×C3×C8—91no—
45C2×C3×C4⋊C4—92no—
46C3×#107—94no—
47C22×C3⋊D3C3×D411yescharacteristic
48C22×C3×D3—121no—
49C2×#157—24no—
50C2×#157—44no—
51C2×C32×C4—41no—
52C2×C32×C4—81no—
53C2×C32×C4C3×Q811yescharacteristic
54C2×C3⋊Dic3C2×C3×C411yescharacteristic
55C4×C3⋊D3C2×C3×C411yescharacteristic
56C4×C3⋊D3C3×D411yescharacteristic
57C4×C3⋊D3C3×Q811yescharacteristic
58C4×C3⋊D3C23×C311yescharacteristic
59C2×C3×Dic3—121no—
60C3×C4×D3—121no—
61C3×C4×D3—121no—
62C3×C4×D3—121no—
63C3×C4×D3—121no—
64#622C2×C3×C414yescomplemented
65C2×#107—91noSylow
66C2×C33#10712yescomplemented
67C2×C33C2×C4⋊C411yescharacteristic
68C3×C3⋊D3#10712yescomplemented
69C3×C3⋊D3C2×D811yescharacteristic
70C3×C3⋊D3C2×Q1611yescharacteristic
71C22×C3×C4—91no—
72C22×C3×C4—182no—
73C2×C4×D3—121no—
74C2×C3×C8—92no—
75C2×C3×C8—92no—
76C2×C3×C8—92no—
77C3×C4⋊C4—94no—
78C3×C4⋊C4—182no—
79C22×C32—41no—
80C22×C32—81no—
81C22×C32C3×C4⋊C411yescharacteristic
82C2×C3⋊D3C3×C4⋊C411yescharacteristic
83C2×C3⋊D3C3×D812yes—
84C2×C3⋊D3C3×Q1612yes—
85C2×C3⋊D3C2×C3×D411yescharacteristic
86C2×C3×D3—121no—
87C2×C3×D3—122no—
88C2×C3×D3—122no—
89C2×C3×D3—121no—
90#157—44no—
91C32×C4—41no—
92C32×C4—41no—
93C32×C4—81no—
94C32×C4—81no—
95C32×C4C3×C4⋊C411yescharacteristic
96C32×C4C2×C3×Q811yescharacteristic
97C3⋊Dic3C3×C4⋊C411yescharacteristic
98C3⋊Dic3C22×C3×C411yescharacteristic
99C3×Dic3—121no—
100C3×Dic3—121no—
101C22×C8—91no—
102C2×C4⋊C4—92no—
103#107—94no—
104C33C2×#10711yescharacteristic, complemented, Sylow
105C23×C3—91no—
106C22×D3—121no—
107C2×C3×C4—41no—
108C2×C3×C4—81no—
109C2×C3×C4—91no—
110C2×C3×C4—91no—
111C2×C3×C4—91no—
112C2×C3×C4—91no—
113C2×C3×C4—91no—
114C2×C3×C4—184no—
115C2×C3×C4—364no—
116C2×C3×C4PSU(3,𝔽2)11yescharacteristic
117C2×Dic3—121no—
118C4×D3—121no—
119C4×D3—121no—
120C4×D3—121no—
121C4×D3—121no—
122C3×C8—94no—
123C2×C32—42no—
124C2×C32—41no—
125C2×C32—82no—
126C2×C32—81no—
127C2×C32C3×#10712yescomplemented
128C2×C32C2×C3×C4⋊C411yescharacteristic
129C3⋊D3C3×#10712yescomplemented
130C3⋊D3C2×C3×D811yescharacteristic
131C3⋊D3C2×C3×Q1611yescharacteristic
132C3×D3—122no—
133C3×D3—121no—
134C3×D3—121no—
135C22×C4—91no—
136C22×C4—182no—
137C2×C8—92no—
138C2×C8—92no—
139C2×C8—92no—
140C4⋊C4—94no—
141C4⋊C4—182no—
142C22×C3—41no—
143C22×C3—81no—
144C22×C3—92no—
145C22×C3—92no—
146C22×C3—91no—
147C22×C3—91no—
148C22×C3#377711yescharacteristic
149C2×D3—121no—
150C2×D3—121no—
151C2×D3—122no—
152C2×D3—122no—
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—
166C32C2×C3×#10711yescharacteristic, complemented
167C23—91no—
168C2×C4—91no—
169C2×C4—91no—
170C2×C4—91no—
171C2×C4—91no—
172C2×C4—91no—
173C2×C4—184no—
174C2×C4—364no—
175C2×C4C3×PSU(3,𝔽2)11yescharacteristic
176C8—94no—
177C2×C3—42no—
178C2×C3—41no—
179C2×C3—82no—
180C2×C3—81no—
181C2×C3—91no—
182C2×C3—92no—
183C2×C3—91no—
184C2×C3#6374612yesdirect product factor
185C2×C3C2×#377711yescharacteristic
186D3—122no—
187D3—121no—
188D3—121no—
189C22—92no—
190C22—92no—
191C22—91no—
192C22—91no—
193C22C3×#377711yescharacteristic
194C4—91no—
195C4—91no—
196C4—364no—
197C4C3×#377711yescharacteristic
198C4C2×C3×PSU(3,𝔽2)11yescharacteristic
199C3—41no—
200C3—81no—
201C3C2×#6374611yescharacteristic, direct product factor
202C2—91no—
203C2—92no—
204C2—91no—
205C2C3×#6374612yesdirect product factor
206C2C2×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×#107
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×#63746C2direct (N × H)
4. C2×#63746C3direct (N × H)
7. C2×C3×#622C4semidirect
10. C3×#3787C4semidirect
14. #63746C2×C3direct (N × H)
24. C3×#622C2×C4semidirect
29. C2×#622C3×C4semidirect
32. #3787C3×C4semidirect
64. #622C2×C3×C4semidirect
66. C2×C33#107semidirect
68. C3×C3⋊D3#107semidirect
104. C33C2×#107semidirect
127. C2×C32C3×#107semidirect
129. C3⋊D3C3×#107semidirect
166. C32C2×C3×#107semidirect
184. C2×C3#63746direct (N × H)
201. C3C2×#63746direct (N × H)
205. C2C3×#63746direct (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×#637462
C2×#637463
C2×C3×#1079