Finite Group Databasebeta

← C3×#10593539 #13167617

Subgroups — C3×#10593539 #13167617

Order 1728.

322 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 102 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×#157C211yesmaximal, characteristic
2C3×#64283C212yesmaximal
3C3×#63744C214yesmaximal, complemented
4#10593539C311yesmaximal, characteristic, direct product factor
5C22×C3×#157C412yes—
6C2×C3×C4×C3⋊D3C2211yescharacteristic
7C3×#3792—22no—
8C3×C4×#157C411yescharacteristic
9C3×C4×#157C411yescharacteristic
10C3×C4×#157C2212yes—
11C3×#3788—22no—
12C3×#3787C2214yes—
13C2×C4×#157C2×C311yescharacteristic
14#64283C2×C312yes—
15#63744C2×C314yescomplemented
16C22×C3×C3⋊D3C2×C411yescharacteristic
17C2×C3×#157—24no—
18C2×C3×#157C2×C412yes—
19C2×C3×#157C2×C412yes—
20C2×C33×C4Q811yescharacteristic
21C2×C3×C3⋊Dic3D411yescharacteristic
22C3×C4×C3⋊D3C2×C411yescharacteristic
23C3×C4×C3⋊D3D411yescharacteristic
24C3×C4×C3⋊D3Q811yescharacteristic
25C3×C4×C3⋊D3C2311yescharacteristic
26C3×#622—24no—
27C3×#423—91nomaximal
28C22×#157C3×C412yes—
29C2×C4×C3⋊D3C22×C311yescharacteristic
30C2×C3×C4×D3—121no—
31#3792—22no—
32C4×#157C3×C411yescharacteristic
33C4×#157C3×C411yescharacteristic
34C4×#157C22×C312yes—
35#3788—22no—
36#3787C22×C314yes—
37C22×C33C4⋊C411yescharacteristic
38C2×C3×C3⋊D3—22no—
39C2×C3×C3⋊D3C4⋊C411yescharacteristic
40C2×C3×C3⋊D3C22×C411yescharacteristic
41C3×#157—22no—
42C3×#157M1614yescomplemented
43C33×C4C4⋊C411yescharacteristic
44C33×C4C2×Q811yescharacteristic
45C3×C3⋊Dic3C4⋊C411yescharacteristic
46C3×C3⋊Dic3C2×D411yescharacteristic
47C2×C3×M16—92no—
48C2×C3×C42—91no—
49C3×C4⋊C8—94no—
50C22×C3⋊D3C2×C3×C411yescharacteristic
51C22×C3×D3—121no—
52C2×#157—24no—
53C2×#157C2×C3×C412yes—
54C2×#157C2×C3×C412yes—
55C2×C32×C4—41no—
56C2×C32×C4—81no—
57C2×C32×C4C3×Q811yescharacteristic
58C2×C3⋊Dic3C3×D411yescharacteristic
59C4×C3⋊D3C2×C3×C411yescharacteristic
60C4×C3⋊D3C3×D411yescharacteristic
61C4×C3⋊D3C3×Q811yescharacteristic
62C4×C3⋊D3C23×C311yescharacteristic
63C2×C3×Dic3—121no—
64C3×C4×D3—121no—
65C3×C4×D3—121no—
66C3×C4×D3—121no—
67C3×C4×D3—121no—
68#622—24no—
69#423—91noSylow
70C2×C33—21no—
71C2×C33C2×C4⋊C411yescharacteristic
72C3×C3⋊D3—21no—
73C3×C3⋊D3C2×M1612yes—
74C22×C3×C4—91no—
75C22×C3×C4—92no—
76C2×C4×D3—121no—
77C3×M16—182no—
78C3×M16—182no—
79C2×C3×C8—94no—
80C3×C42—92no—
81C3×C42—91no—
82C3×C42—91no—
83C22×C32—41no—
84C22×C32—81no—
85C22×C32C3×C4⋊C411yescharacteristic
86C2×C3⋊D3—22no—
87C2×C3⋊D3C3×C4⋊C411yescharacteristic
88C2×C3⋊D3C22×C3×C411yescharacteristic
89C2×C3×D3—121no—
90C2×C3×D3—121no—
91C2×C3×D3—124no—
92#157—22no—
93#157C3×M1614yescomplemented
94C32×C4—41no—
95C32×C4—41no—
96C32×C4—81no—
97C32×C4—81no—
98C32×C4C3×C4⋊C411yescharacteristic
99C32×C4C2×C3×Q811yescharacteristic
100C3⋊Dic3C3×C4⋊C411yescharacteristic
101C3⋊Dic3C2×C3×D411yescharacteristic
102C3×Dic3—121no—
103C3×Dic3—121no—
104C2×M16—92no—
105C2×C42—91no—
106C4⋊C8—94no—
107C33#42311yescharacteristic, complemented, Sylow
108C23×C3—91no—
109C22×D3—121no—
110C2×C3×C4—41no—
111C2×C3×C4—81no—
112C2×C3×C4—92no—
113C2×C3×C4—91no—
114C2×C3×C4—91no—
115C2×C3×C4—91no—
116C2×C3×C4—91no—
117C2×C3×C4—91no—
118C2×C3×C4—92no—
119C2×C3×C4—184no—
120C2×C3×C4PSU(3,𝔽2)11yescharacteristic
121C2×Dic3—121no—
122C4×D3—121no—
123C4×D3—121no—
124C4×D3—121no—
125C4×D3—121no—
126C3×C8—184no—
127C2×C32—21no—
128C2×C32—42no—
129C2×C32—41no—
130C2×C32—82no—
131C2×C32—81no—
132C2×C32C2×C3×C4⋊C411yescharacteristic
133C3⋊D3—21no—
134C3⋊D3C2×C3×M1612yes—
135C3×D3—122no—
136C3×D3—122no—
137C22×C4—91no—
138C22×C4—92no—
139M16—182no—
140M16—182no—
141C2×C8—94no—
142C42—92no—
143C42—91no—
144C42—91no—
145C22×C3—41no—
146C22×C3—81no—
147C22×C3—91no—
148C22×C3—91no—
149C22×C3—182no—
150C22×C3#377711yescharacteristic
151C2×D3—121no—
152C2×D3—121no—
153C2×D3—124no—
154C3×C4—41no—
155C3×C4—41no—
156C3×C4—81no—
157C3×C4—81no—
158C3×C4—94no—
159C3×C4—91no—
160C3×C4—91no—
161C3×C4—182no—
162C3×C4#377711yescharacteristic
163C3×C4C2×PSU(3,𝔽2)11yescharacteristic
164Dic3—121no—
165Dic3—121no—
166C32—41no—
167C32—81no—
168C32C3×#42311yescharacteristic, complemented
169C23—91no—
170C2×C4—92no—
171C2×C4—91no—
172C2×C4—91no—
173C2×C4—91no—
174C2×C4—91no—
175C2×C4—91no—
176C2×C4—92no—
177C2×C4—184no—
178C2×C4C3×PSU(3,𝔽2)11yescharacteristic
179C8—184no—
180C2×C3—21no—
181C2×C3—42no—
182C2×C3—41no—
183C2×C3—82no—
184C2×C3—81no—
185C2×C3—92no—
186C2×C3—181no—
187C2×C3C2×#377711yescharacteristic
188D3—122no—
189D3—122no—
190C22—91no—
191C22—91no—
192C22—182no—
193C22C3×#377711yescharacteristic
194C4—94no—
195C4—91no—
196C4—91no—
197C4—182no—
198C4C3×#377711yescharacteristic
199C4C2×C3×PSU(3,𝔽2)11yescharacteristic
200C3—41no—
201C3—81no—
202C3#1059353911yescharacteristic, direct product factor
203C2—21no—
204C2—92no—
205C2—181no—
206C2C2×C3×#377711yescharacteristic
subgroupG/H
centerC3×C4C2×PSU(3,𝔽2)
commutatorC2×C3⋊D3C22×C3×C4
FrattiniC4C2×C3×PSU(3,𝔽2)
FittingC2×C33×C4Q8
socleC2×C33C2×C4⋊C4
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC4×C3⋊D3C23×C3
hypercenterC2×C3×C4PSU(3,𝔽2)
hypocenterC32C3×#423
LCSG ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C3×C4 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
3. C3×#63744C2semidirect
4. #10593539C3direct (N × H)
15. #63744C2×C3semidirect
42. C3×#157M16semidirect
93. #157C3×M16semidirect
107. C33#423semidirect
168. C32C3×#423semidirect
202. C3#10593539direct (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
C3×#642832
C3×#637442
#105935393
C3×#4239