Finite Group Databasebeta

← A4×#3718 #13166133

Subgroups — A4×#3718 #13166133

Order 1728.

378 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 54 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
1C3×A4×Dic6C212yesmaximal
2A4×C3⋊(C3⋊C8)C211yesmaximal, characteristic
3C22×#3718C311yesmaximal, characteristic, complemented
4A4×#216—32nomaximal
5C32×C4×A4C2211yescharacteristic
6C3×Dic3×A4—22no—
7C3×#3718—41nomaximal
8C22×C3×Dic6C2×C312yes—
9C3×Q8×A4—32no—
10A4×Dic6—22no—
11A4×Dic6D312yes—
12C22×C3⋊(C3⋊C8)C2×C311yescharacteristic
13C2×#3718—31no—
14C2×#3718—32no—
15C2×#3718—31no—
16A4×C3⋊C8—32no—
17A4×C3⋊C8—61no—
18C2×C32×A4D411yescharacteristic
19C32×Dic6—42no—
20C3×C3⋊(C3⋊C8)—41no—
21C22×#216—32no—
22A4×Q16—91no—
23C22×C32×C4C22×C311yescharacteristic
24C22×C3×Dic3—22no—
25C2×C3×Dic6—32no—
26C2×C3×Dic6—32no—
27C2×C3×Dic6—62no—
28C3×C4×A4—22no—
29C3×C4×A4—21no—
30C3×C4×A4—41no—
31C3×C4×A4—62no—
32C3×C4×A4C2×D312yes—
33Dic3×A4—24no—
34Dic3×A4—22no—
35C2×C3⋊(C3⋊C8)—31no—
36C2×C3⋊(C3⋊C8)—31no—
37C3×#216—122no—
38#3718—32no—
39#3718—32no—
40#3718—31no—
41#3718A411yescharacteristic, direct product factor
42C32×A4Q1611yescharacteristic, complemented
43C33×C4—41no—
44C32×Dic3—82no—
45C22×C3×Q8—32no—
46C22×Dic6C3×D312yes—
47Q8×A4—32no—
48Q8×A4—62no—
49C2×#216—92no—
50C2×#216—92no—
51C2×#216—92no—
52C2×#216—92no—
53C22×C3⋊C8—32no—
54C22×C3⋊C8—61no—
55C8×A4—91no—
56C23×C32C3×D411yescharacteristic
57C2×C3×A4—22no—
58C2×C3×A4—21no—
59C2×C3×A4—41no—
60C2×C3×A4#6712yes—
61C32×Q8—122no—
62C2×C32×C4—31no—
63C2×C32×C4—31no—
64C2×C3×Dic3—62no—
65C2×C3×Dic3—62no—
66C3×Dic6—32no—
67C3×Dic6—42no—
68C3×Dic6—62no—
69C3×Dic6—62no—
70C3×Dic6—82no—
71C3×Dic6C2×A412yes—
72C3⋊(C3⋊C8)—31no—
73C3⋊(C3⋊C8)C2×A411yescharacteristic
74C3×C3⋊C8—122no—
75C3×C3⋊C8—241no—
76C22×Q16—91noSylow
77C2×C33—41no—
78C2×C3×Q8—92no—
79C2×C3×Q8—92no—
80C2×C3×Q8—182no—
81C22×C3×C4—21no—
82C22×C3×C4—62no—
83C22×C3×C4C2×C3×D312yes—
84C22×Dic3—22no—
85C2×Dic6—32no—
86C2×Dic6—32no—
87C2×Dic6—62no—
88C4×A4—22no—
89C4×A4—41no—
90C4×A4—64no—
91C4×A4—62no—
92C4×A4D3211yescharacteristic
93C3×Q16—361no—
94#216—32no—
95#216—92no—
96#216—92no—
97#216—94no—
98#216—92no—
99C2×C3⋊C8—92no—
100C2×C3⋊C8—92no—
101C2×C3⋊C8—181no—
102C2×C3⋊C8—181no—
103C22×C32—31no—
104C22×C32—31no—
105C22×C32C3×Q1611yescharacteristic, complemented
106C3×A4—22no—
107C3×A4—21no—
108C3×A4—41no—
109C3×A4#21612yescomplemented
110C32×C4—31no—
111C32×C4—42no—
112C32×C4—81no—
113C32×C4—82no—
114C32×C4—161no—
115C32×C4—242no—
116C32×C4C22×A411yescharacteristic
117C3×Dic3—22no—
118C3×Dic3—62no—
119C3×Dic3—82no—
120C3×Dic3—84no—
121C22×Q8—32no—
122C2×Q16—271no—
123C2×Q16—271no—
124C2×Q16—272no—
125C22×C8—91no—
126C33—41noSylow
127C23×C3—21no—
128C23×C3C3×#6712yes—
129C2×A4—22no—
130C2×A4—41no—
131C2×A4#62511yescharacteristic
132C3×Q8—32no—
133C3×Q8—92no—
134C3×Q8—122no—
135C3×Q8—182no—
136C3×Q8—182no—
137C3×Q8—242no—
138C2×C3×C4—32no—
139C2×C3×C4—32no—
140C2×C3×C4—61no—
141C2×C3×C4—61no—
142C2×C3×C4—182no—
143C2×C3×C4—182no—
144C2×Dic3—62no—
145C2×Dic3—62no—
146Dic6—32no—
147Dic6—62no—
148Dic6—62no—
149Dic6D3×A412yes—
150C3×C8—361no—
151C3⋊C8—32no—
152C3⋊C8—61no—
153C3⋊C8—92no—
154C3⋊C8—181no—
155C2×C32—31no—
156C2×C32—31no—
157C2×C32—42no—
158C2×C32—81no—
159C2×C32—82no—
160C2×C32—161no—
161C2×C32D4×A411yescharacteristic
162C2×Q8—92no—
163C2×Q8—92no—
164C2×Q8—182no—
165C22×C4—62no—
166C22×C4C3×D3211yescharacteristic
167Q16—91no—
168Q16—271no—
169Q16—272no—
170Q16—272no—
171C2×C8—271no—
172C2×C8—271no—
173C22×C3—21no—
174C22×C3—32no—
175C22×C3—32no—
176C22×C3—61no—
177C22×C3—61no—
178C22×C3C3×#21612yescomplemented
179A4—22no—
180A4—41no—
181A4#371811yescharacteristic, direct product factor
182C3×C4—21no—
183C3×C4—32no—
184C3×C4—41no—
185C3×C4—61no—
186C3×C4—62no—
187C3×C4—82no—
188C3×C4—161no—
189C3×C4—182no—
190C3×C4—242no—
191C3×C4—244no—
192C3×C4C2×D3×A412yes—
193Dic3—22no—
194Dic3—62no—
195C32—42no—
196C32—81no—
197C32—82no—
198C32—161no—
199C32A4×Q1611yescharacteristic, complemented
200C23C3×#62511yescharacteristic
201Q8—32no—
202Q8—92no—
203Q8—182no—
204Q8—182no—
205C2×C4—31no—
206C2×C4—31no—
207C2×C4—182no—
208C2×C4—182no—
209C8—91no—
210C8—271no—
211C2×C3—21no—
212C2×C3—32no—
213C2×C3—32no—
214C2×C3—41no—
215C2×C3—61no—
216C2×C3—61no—
217C2×C3—82no—
218C2×C3—161no—
219C2×C3A4×#6712yes—
220C22—31no—
221C22—31no—
222C22C3×#371811yescharacteristic, complemented
223C4—31no—
224C4—62no—
225C4—182no—
226C4D32×A411yescharacteristic
227C3—21no—
228C3—41no—
229C3—82no—
230C3—161no—
231C3A4×#21612yescomplemented
232C2—31no—
233C2—31no—
234C2A4×#62511yescharacteristic
subgroupG/H
centerC2A4×#625
commutator, Fitting, cosocleC22×C32×C4C22×C3
Frattini, hypercenterC4D32×A4
socleC23×C32C3×D4
prime radicalC2×C32×A4D4
hypocenterC22×C32C3×Q16
LCSG ▷ C22×C32×C4 ▷ C23×C32 ▷ C22×C32
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
3. C22×#3718C3semidirect
41. #3718A4direct (N × H)
42. C32×A4Q16semidirect
105. C22×C32C3×Q16semidirect
109. C3×A4#216semidirect
178. C22×C3C3×#216semidirect
181. A4#3718direct (N × H)
199. C32A4×Q16semidirect
222. C22C3×#3718semidirect
231. C3A4×#216semidirect

Maximal subgroups

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

subgroupindex
C3×A4×Dic62
A4×C3⋊(C3⋊C8)2
C22×#37183
A4×#2163
C3×#37184