Finite Group Databasebeta

← C3×C4×#3777 #13167624

Subgroups — C3×C4×#3777 #13167624

Order 1728.

458 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 162 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×C4×#157C211yesmaximal, characteristic
3C2×C3×#3777C212yesmaximal
4C3×#63747C212yesmaximal
5C4×#3777C311yesmaximal, characteristic, direct product factor
6C22×C3×#157C2211yescharacteristic
7C22×C3×#157C2214yes—
8C22×C3×#157C2211yescharacteristic
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C3×C4×#157—24no—
11C3×C4×#157C414yescomplemented
12C3×#3777C418yesdirect product factor
13C2×C4×#157C2×C312yes—
14C2×C4×#157C2×C311yescharacteristic
15C2×#3777C2×C312yes—
16#63747C2×C312yes—
17C22×C3×C3⋊D3C2311yescharacteristic
18C2×C3×#157—24no—
19C2×C3×#157—22no—
20C2×C3×#157—24no—
21C2×C3×#157C2×C418yes—
22C2×C3×#157C2×C412yes—
23C2×C3×#157C2×C414yes—
24C2×C3×#157C2×C412yes—
25C2×C33×C4Q811yescharacteristic
26C2×C3×C3⋊Dic3Q811yescharacteristic
27C3×C4×C3⋊D3C2×C412yes—
28C3×C4×C3⋊D3D412yes—
29C3×C4×C4⋊C4—91nomaximal
30C22×#157C22×C311yescharacteristic
31C22×#157C22×C314yes—
32C22×#157C22×C311yescharacteristic
33C2×C4×C3⋊D3C22×C311yescharacteristic
34C2×C3×C4×D3—121no—
35C4×#157—24no—
36C4×#157C3×C414yescomplemented
37#3777C3×C418yesdirect product factor
38C22×C33C2×Q811yescharacteristic
39C2×C3×C3⋊D3C22×C411yescharacteristic
40C2×C3×C3⋊D3C22×C412yes—
41C2×C3×C3⋊D3C2×D411yescharacteristic
42C2×C3×C3⋊D3#4112yes—
43C3×#157—22no—
44C3×#157—28no—
45C3×#157C4214yescomplemented
46C33×C4C4⋊C412yescomplemented
47C3×C3⋊Dic3C4⋊C412yescomplemented
48C2×C3×C4⋊C4—92no—
49C2×C3×C42—91no—
50C2×C3×C42—92no—
51C3×#95—92no—
52C22×C3⋊D3C23×C311yescharacteristic
53C22×C3×D3—121no—
54C2×#157—24no—
55C2×#157—22no—
56C2×#157—24no—
57C2×#157C2×C3×C418yes—
58C2×#157C2×C3×C412yes—
59C2×#157C2×C3×C414yes—
60C2×#157C2×C3×C412yes—
61C2×C32×C4—41no—
62C2×C32×C4—81no—
63C2×C32×C4C3×Q811yescharacteristic
64C2×C3⋊Dic3C3×Q811yescharacteristic
65C4×C3⋊D3C2×C3×C412yes—
66C4×C3⋊D3C3×D412yes—
67C2×C3×Dic3—121no—
68C3×C4×D3—122no—
69C3×C4×D3—122no—
70C4×C4⋊C4—91noSylow
71C2×C33C2×C4⋊C411yescharacteristic
72C2×C33C4×Q812yes—
73C3×C3⋊D3C2×C4211yescharacteristic
74C3×C3⋊D3#11711yescharacteristic
75C3×C3⋊D3C4×D412yes—
76C22×C3×C4—91no—
77C22×C3×C4—91no—
78C22×C3×C4—91no—
79C22×C3×C4—94no—
80C2×C4×D3—121no—
81C3×C4⋊C4—98no—
82C3×C42—94no—
83C3×C42—184no—
84C22×C32—41no—
85C22×C32—81no—
86C22×C32C2×C3×Q811yescharacteristic
87C2×C3⋊D3C22×C3×C411yescharacteristic
88C2×C3⋊D3C22×C3×C412yes—
89C2×C3⋊D3C2×C3×D411yescharacteristic
90C2×C3⋊D3C3×#4112yes—
91C2×C3×D3—121no—
92C2×C3×D3—122no—
93C2×C3×D3—121no—
94C2×C3×D3—122no—
95#157—22no—
96#157—28no—
97#157C3×C4214yescomplemented
98C32×C4—42no—
99C32×C4—82no—
100C32×C4C3×C4⋊C412yescomplemented
101C3⋊Dic3C3×C4⋊C412yescomplemented
102C3×Dic3—122no—
103C2×C4⋊C4—92no—
104C2×C42—91no—
105C2×C42—92no—
106#95—92no—
107C33C4×C4⋊C411yescharacteristic, complemented, Sylow
108C23×C3—91no—
109C22×D3—121no—
110C2×C3×C4—41no—
111C2×C3×C4—81no—
112C2×C3×C4—92no—
113C2×C3×C4—94no—
114C2×C3×C4—91no—
115C2×C3×C4—92no—
116C2×C3×C4—92no—
117C2×C3×C4—92no—
118C2×C3×C4—98no—
119C2×C3×C4—184no—
120C2×C3×C4—182no—
121C2×C3×C4—184no—
122C2×C3×C4PSU(3,𝔽2)11yescharacteristic
123C2×Dic3—121no—
124C4×D3—122no—
125C4×D3—122no—
126C2×C32—42no—
127C2×C32—41no—
128C2×C32—81no—
129C2×C32—82no—
130C2×C32C2×C3×C4⋊C411yescharacteristic
131C2×C32C3×C4×Q812yes—
132C3⋊D3C2×C3×C4211yescharacteristic
133C3⋊D3C3×#11711yescharacteristic
134C3⋊D3C3×C4×D412yes—
135C3×D3—121no—
136C3×D3—121no—
137C3×D3—122no—
138C22×C4—91no—
139C22×C4—91no—
140C22×C4—91no—
141C22×C4—94no—
142C4⋊C4—98no—
143C42—94no—
144C42—184no—
145C22×C3—41no—
146C22×C3—81no—
147C22×C3—92no—
148C22×C3—92no—
149C22×C3—91no—
150C22×C3—91no—
151C22×C3C2×PSU(3,𝔽2)11yescharacteristic
152C2×D3—121no—
153C2×D3—121no—
154C2×D3—122no—
155C2×D3—122no—
156C3×C4—42no—
157C3×C4—82no—
158C3×C4—94no—
159C3×C4—92no—
160C3×C4—188no—
161C3×C4—182no—
162C3×C4#377712yesdirect product factor
163Dic3—122no—
164C32—41no—
165C32—81no—
166C32C3×C4×C4⋊C411yescharacteristic, complemented
167C23—91no—
168C2×C4—92no—
169C2×C4—94no—
170C2×C4—91no—
171C2×C4—92no—
172C2×C4—92no—
173C2×C4—92no—
174C2×C4—98no—
175C2×C4—184no—
176C2×C4—182no—
177C2×C4—184no—
178C2×C4C3×PSU(3,𝔽2)11yescharacteristic
179C2×C3—42no—
180C2×C3—41no—
181C2×C3—82no—
182C2×C3—81no—
183C2×C3—91no—
184C2×C3—92no—
185C2×C3—91no—
186C2×C3C4×PSU(3,𝔽2)12yes—
187C2×C3C2×#377711yescharacteristic
188D3—121no—
189D3—121no—
190D3—122no—
191C22—92no—
192C22—92no—
193C22—91no—
194C22—91no—
195C22C2×C3×PSU(3,𝔽2)11yescharacteristic
196C4—94no—
197C4—92no—
198C4—188no—
199C4—182no—
200C4C3×#377712yesdirect product factor
201C3—41no—
202C3—81no—
203C3C4×#377711yescharacteristic, direct product factor
204C2—91no—
205C2—92no—
206C2—91no—
207C2C3×C4×PSU(3,𝔽2)12yes—
208C2C2×C3×#377711yescharacteristic
subgroupG/H
center, hypercenterC2×C3×C4PSU(3,𝔽2)
commutatorC3⋊D3C2×C3×C42
FrattiniC22C2×C3×PSU(3,𝔽2)
FittingC2×C33×C4Q8
socleC22×C33C2×Q8
prime radicalC22×C3×C3⋊D3C23
cosocleC22×C3⋊D3C23×C3
hypocenterC32C3×C4×C4⋊C4
LCSG ▷ C3⋊D3 ▷ C32
UCSC1 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
5. C4×#3777C3direct (N × H)
11. C3×C4×#157C4semidirect
12. C3×#3777C4direct (N × H)
36. C4×#157C3×C4semidirect
37. #3777C3×C4direct (N × H)
45. C3×#157C42semidirect
46. C33×C4C4⋊C4semidirect
47. C3×C3⋊Dic3C4⋊C4semidirect
97. #157C3×C42semidirect
100. C32×C4C3×C4⋊C4semidirect
101. C3⋊Dic3C3×C4⋊C4semidirect
107. C33C4×C4⋊C4semidirect
162. C3×C4#3777direct (N × H)
166. C32C3×C4×C4⋊C4semidirect
200. C4C3×#3777direct (N × H)
203. C3C4×#3777direct (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×#37772
C3×#637472
C4×#37773
C3×C4×C4⋊C49