Finite Group Databasebeta

← C2×C3×#63744 #13167615

Subgroups — C2×C3×#63744 #13167615

Order 1728.

346 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 130 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
2C2×C3×#3787C212yesmaximal
3C3×#63744C214yesmaximal, direct product factor
4C2×#63744C311yesmaximal, characteristic, direct product factor
5C22×C3×#157C411yescharacteristic
6C22×C3×#157C411yescharacteristic
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×C3×#622—22no—
9C3×C4×#157C412yes—
10C3×C4×#157C2212yes—
11C3×#3787—22no—
12C3×#3787C2214yes—
13C2×C4×#157C2×C311yescharacteristic
14C2×#3787C2×C312yes—
15#63744C2×C314yesdirect product factor
16C22×C3×C3⋊D3C2×C411yescharacteristic
17C2×C3×#157—22no—
18C2×C3×#157C814yescomplemented
19C2×C3×#157C2×C412yes—
20C2×C3×#157C2×C412yes—
21C2×C33×C4Q811yescharacteristic
22C2×C3×C3⋊Dic3D411yescharacteristic
23C3×C4×C3⋊D3C2×C411yescharacteristic
24C3×C4×C3⋊D3D411yescharacteristic
25C3×C4×C3⋊D3Q811yescharacteristic
26C3×C4×C3⋊D3C2311yescharacteristic
27C3×#622—24no—
28C2×C3×C4⋊C8—91nomaximal
29C22×#157C3×C411yescharacteristic
30C22×#157C3×C411yescharacteristic
31C2×C4×C3⋊D3C22×C311yescharacteristic
32C2×C3×C4×D3—121no—
33C2×#622—22no—
34C4×#157C3×C412yes—
35C4×#157C22×C312yes—
36#3787—22no—
37#3787C22×C314yes—
38C22×C33C4⋊C411yescharacteristic
39C2×C3×C3⋊D3C4⋊C411yescharacteristic
40C2×C3×C3⋊D3C2×C812yes—
41C2×C3×C3⋊D3M1612yes—
42C2×C3×C3⋊D3C22×C411yescharacteristic
43C3×#157—22no—
44C3×#157C2×C814yescomplemented
45C33×C4C4⋊C411yescharacteristic
46C33×C4C2×Q811yescharacteristic
47C3×C3⋊Dic3C4⋊C411yescharacteristic
48C3×C3⋊Dic3C2×D411yescharacteristic
49C22×C3×C8—92no—
50C2×C3×C42—91no—
51C3×C4⋊C8—94no—
52C22×C3⋊D3C2×C3×C411yescharacteristic
53C22×C3×D3—121no—
54C2×#157—22no—
55C2×#157C3×C814yescomplemented
56C2×#157C2×C3×C412yes—
57C2×#157C2×C3×C412yes—
58C2×C32×C4—41no—
59C2×C32×C4—81no—
60C2×C32×C4C3×Q811yescharacteristic
61C2×C3⋊Dic3C3×D411yescharacteristic
62C4×C3⋊D3C2×C3×C411yescharacteristic
63C4×C3⋊D3C3×D411yescharacteristic
64C4×C3⋊D3C3×Q811yescharacteristic
65C4×C3⋊D3C23×C311yescharacteristic
66C2×C3×Dic3—121no—
67C3×C4×D3—121no—
68C3×C4×D3—121no—
69C3×C4×D3—121no—
70C3×C4×D3—121no—
71#622—24no—
72C2×C4⋊C8—91noSylow
73C2×C33C4⋊C812yescomplemented
74C2×C33C2×C4⋊C411yescharacteristic
75C3×C3⋊D3C4⋊C812yescomplemented
76C3×C3⋊D3C22×C811yescharacteristic
77C3×C3⋊D3C2×M1611yescharacteristic
78C22×C3×C4—91no—
79C22×C3×C4—91no—
80C22×C3×C4—91no—
81C2×C4×D3—121no—
82C2×C3×C8—94no—
83C2×C3×C8—182no—
84C2×C3×C8—182no—
85C3×C42—92no—
86C3×C42—92no—
87C22×C32—41no—
88C22×C32—81no—
89C22×C32C3×C4⋊C411yescharacteristic
90C2×C3⋊D3C3×C4⋊C411yescharacteristic
91C2×C3⋊D3C2×C3×C812yes—
92C2×C3⋊D3C3×M1612yes—
93C2×C3⋊D3C22×C3×C411yescharacteristic
94C2×C3×D3—121no—
95C2×C3×D3—122no—
96C2×C3×D3—122no—
97C2×C3×D3—121no—
98#157—22no—
99#157C2×C3×C814yescomplemented
100C32×C4—41no—
101C32×C4—41no—
102C32×C4—81no—
103C32×C4—81no—
104C32×C4C3×C4⋊C411yescharacteristic
105C32×C4C2×C3×Q811yescharacteristic
106C3⋊Dic3C3×C4⋊C411yescharacteristic
107C3⋊Dic3C2×C3×D411yescharacteristic
108C3×Dic3—121no—
109C3×Dic3—121no—
110C22×C8—92no—
111C2×C42—91no—
112C4⋊C8—94no—
113C33C2×C4⋊C811yescharacteristic, complemented, Sylow
114C23×C3—91no—
115C22×D3—121no—
116C2×C3×C4—41no—
117C2×C3×C4—81no—
118C2×C3×C4—94no—
119C2×C3×C4—92no—
120C2×C3×C4—91no—
121C2×C3×C4—91no—
122C2×C3×C4—91no—
123C2×C3×C4—91no—
124C2×C3×C4—91no—
125C2×C3×C4—92no—
126C2×C3×C4—182no—
127C2×C3×C4PSU(3,𝔽2)11yescharacteristic
128C2×Dic3—121no—
129C4×D3—121no—
130C4×D3—121no—
131C4×D3—121no—
132C4×D3—121no—
133C3×C8—184no—
134C2×C32—42no—
135C2×C32—41no—
136C2×C32—82no—
137C2×C32—81no—
138C2×C32C3×C4⋊C812yescomplemented
139C2×C32C2×C3×C4⋊C411yescharacteristic
140C3⋊D3C3×C4⋊C812yescomplemented
141C3⋊D3C22×C3×C811yescharacteristic
142C3⋊D3C2×C3×M1611yescharacteristic
143C3×D3—122no—
144C3×D3—121no—
145C3×D3—121no—
146C22×C4—91no—
147C22×C4—91no—
148C22×C4—91no—
149C2×C8—94no—
150C2×C8—182no—
151C2×C8—182no—
152C42—92no—
153C42—92no—
154C22×C3—41no—
155C22×C3—81no—
156C22×C3—92no—
157C22×C3—92no—
158C22×C3—91no—
159C22×C3—91no—
160C22×C3#377711yescharacteristic
161C2×D3—121no—
162C2×D3—121no—
163C2×D3—122no—
164C2×D3—122no—
165C3×C4—41no—
166C3×C4—41no—
167C3×C4—81no—
168C3×C4—81no—
169C3×C4—94no—
170C3×C4—91no—
171C3×C4—91no—
172C3×C4—182no—
173C3×C4#377711yescharacteristic
174C3×C4C2×PSU(3,𝔽2)11yescharacteristic
175Dic3—121no—
176Dic3—121no—
177C32—41no—
178C32—81no—
179C32C2×C3×C4⋊C811yescharacteristic, complemented
180C23—91no—
181C2×C4—92no—
182C2×C4—94no—
183C2×C4—91no—
184C2×C4—91no—
185C2×C4—91no—
186C2×C4—91no—
187C2×C4—91no—
188C2×C4—92no—
189C2×C4—182no—
190C2×C4C3×PSU(3,𝔽2)11yescharacteristic
191C8—184no—
192C2×C3—42no—
193C2×C3—41no—
194C2×C3—82no—
195C2×C3—81no—
196C2×C3—91no—
197C2×C3—91no—
198C2×C3—92no—
199C2×C3#6374412yesdirect product factor
200C2×C3C2×#377711yescharacteristic
201D3—122no—
202D3—121no—
203D3—121no—
204C22—92no—
205C22—92no—
206C22—91no—
207C22—91no—
208C22C3×#377711yescharacteristic
209C4—94no—
210C4—91no—
211C4—91no—
212C4—182no—
213C4C3×#377711yescharacteristic
214C4C2×C3×PSU(3,𝔽2)11yescharacteristic
215C3—41no—
216C3—81no—
217C3C2×#6374411yescharacteristic, direct product factor
218C2—91no—
219C2—91no—
220C2—92no—
221C2C3×#6374412yesdirect product factor
222C2C2×C3×#377711yescharacteristic
subgroupG/H
center, hypercenterC2×C3×C4PSU(3,𝔽2)
commutatorC3⋊D3C22×C3×C8
FrattiniC4C2×C3×PSU(3,𝔽2)
FittingC2×C33×C4Q8
socleC22×C33C4⋊C4
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC4×C3⋊D3C23×C3
hypocenterC32C2×C3×C4⋊C8
LCSG ▷ C3⋊D3 ▷ C32
UCSC1 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
3. C3×#63744C2direct (N × H)
4. C2×#63744C3direct (N × H)
15. #63744C2×C3direct (N × H)
18. C2×C3×#157C8semidirect
44. C3×#157C2×C8semidirect
55. C2×#157C3×C8semidirect
73. C2×C33C4⋊C8semidirect
75. C3×C3⋊D3C4⋊C8semidirect
99. #157C2×C3×C8semidirect
113. C33C2×C4⋊C8semidirect
138. C2×C32C3×C4⋊C8semidirect
140. C3⋊D3C3×C4⋊C8semidirect
179. C32C2×C3×C4⋊C8semidirect
199. C2×C3#63744direct (N × H)
217. C3C2×#63744direct (N × H)
221. C2C3×#63744direct (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×#637442
C2×#637443
C2×C3×C4⋊C89