Finite Group Databasebeta

← #3674⋊A4 #13147163

Subgroups — #3674⋊A4 #13147163

Order 1728.

231 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 231 conjugacy classes of subgroups — too many to draw. Only its 41 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
1Q8×C9⋊A4C211yesmaximal, characteristic
2Dic18⋊A4C211yesmaximal, characteristic
3(C9⋊C8)⋊A4C211yesmaximal, characteristic
4A4×#216—31nomaximal
5C22×#3674C311yesmaximal, characteristic, complemented
6C4×C9⋊A4—21no—
7C4×C9⋊A4C2211yescharacteristic
8Dic9⋊A4—21no—
9#3674⋊C3—41nomaximal
10C3×Q8×A4D311yescharacteristic
11A4×Dic6—31no—
12A4×C3⋊C8—31no—
13C22×Q8×C9C2×C311yescharacteristic
14C22×Dic18C2×C311yescharacteristic
15Q8×C22⋊C9—21no—
16C2×#3674—31no—
17C2×#3674—31no—
18C2×#3674—31no—
19C2×#3674—31no—
20C22×C9⋊C8C2×C311yescharacteristic
21C2×C9⋊A4D411yescharacteristic
22Q8×3−1+2—41no—
233−1+2⋊Q8—41no—
243−1+2⋊C8—41no—
25C22×#216—31no—
26A4×Q16—91no—
27C3×C4×A4—21no—
28C3×C4×A4C2×D311yescharacteristic
29Dic3×A4—61no—
30C3×#216—121no—
31C2×Q8×C9—31no—
32C2×Q8×C9—31no—
33C2×Q8×C9—61no—
34C22×C4×C9—21no—
35C22×C4×C9C22×C311yescharacteristic
36C22×Dic9—21no—
37C2×Dic18—31no—
38C2×Dic18—31no—
39C2×Dic18—61no—
40C4×C22⋊C9—22no—
41C4×C22⋊C9—21no—
42#3674—32no—
43#3674—31no—
44#3674—31no—
45#3674—31no—
46#3674A411yescharacteristic, complemented
47C2×C9⋊C8—31no—
48C2×C9⋊C8—31no—
49C9⋊A4Q1611yescharacteristic, complemented
50C4×3−1+2—41no—
51C4×3−1+2—81no—
523−1+2⋊C4—81no—
53C22×C3×Q8C3×D311yescharacteristic
54C22×Dic6—31no—
55Q8×A4—31no—
56Q8×A4—91no—
57C2×#216—91no—
58C2×#216—91no—
59C2×#216—91no—
60C2×#216—91no—
61C22×C3⋊C8—31no—
62C8×A4—91no—
63C2×C3×A4#6711yescharacteristic
64C32×Q8—41no—
65C3×Dic6—121no—
66C23×C9C3×D411yescharacteristic
67C2×C22⋊C9—21no—
68C3×C3⋊C8—121no—
69Q8×C9—31no—
70Q8×C9—61no—
71Q8×C9—61no—
72Q8×C9—81no—
73Q8×C9C2×A411yescharacteristic
74C2×C4×C9—31no—
75C2×C4×C9—31no—
76C2×C4×C9—61no—
77C2×C4×C9—61no—
78C2×Dic9—61no—
79C2×Dic9—61no—
80Dic18—31no—
81Dic18—61no—
82Dic18—61no—
83Dic18C2×A411yescharacteristic
84C9⋊C8—31no—
85C9⋊C8C2×A411yescharacteristic
86C22×Q16—91noSylow
87C2×3−1+2—41no—
88C2×C3×Q8—31no—
89C2×C3×Q8—31no—
90C2×C3×Q8—61no—
91C22×C3×C4—21no—
92C22×C3×C4C2×C3×D311yescharacteristic
93C22×Dic3—61no—
94C2×Dic6—91no—
95C2×Dic6—91no—
96C2×Dic6—181no—
97C4×A4—31no—
98C4×A4—61no—
99C4×A4—181no—
100C3×Q16—361no—
101#216—31no—
102#216—91no—
103#216—91no—
104#216—92no—
105#216—91no—
106C2×C3⋊C8—91no—
107C2×C3⋊C8—91no—
108C3×A4#21611yescharacteristic
109C32×C4—41no—
110C32×C4—81no—
111C3×Dic3—241no—
112C22×C9—31no—
113C22×C9—31no—
114C22×C9C3×Q1611yescharacteristic, complemented
115C22⋊C9—21no—
116C4×C9—21no—
117C4×C9—31no—
118C4×C9—61no—
119C4×C9—82no—
120C4×C9—81no—
121C4×C9C22×A411yescharacteristic
122Dic9—21no—
123Dic9—61no—
124C22×Q8—91no—
125C22×Q8AGL(1,ℤ/9)11yescharacteristic
126C2×Q16—271no—
127C2×Q16—271no—
128C2×Q16—271no—
129C2×Q16—271no—
130C22×C8—91no—
1313−1+2—41noSylow
132C23×C3C3×#6711yescharacteristic
133C2×A4—31no—
134C3×Q8—31no—
135C3×Q8—61no—
136C3×Q8—61no—
137C3×Q8—121no—
138C3×Q8—361no—
139C3×Q8D3×A411yescharacteristic
140C2×C3×C4—31no—
141C2×C3×C4—31no—
142C2×C3×C4—61no—
143C2×C3×C4—61no—
144C2×Dic3—181no—
145C2×Dic3—181no—
146Dic6—31no—
147Dic6—91no—
148Dic6—181no—
149Dic6—181no—
150C3×C8—361no—
151C3⋊C8—31no—
152C3⋊C8—91no—
153C2×C32—41no—
154C2×C9—31no—
155C2×C9—31no—
156C2×C9—81no—
157C2×C9D4×A411yescharacteristic
158C2×Q8—31no—
159C2×Q8—31no—
160C2×Q8—61no—
161C2×Q8—271no—
162C2×Q8—271no—
163C2×Q8—541no—
164C22×C4—21no—
165C22×C4—181no—
166C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
167Q16—91no—
168Q16—271no—
169Q16—271no—
170Q16—272no—
171Q16—271no—
172C2×C8—271no—
173C2×C8—271no—
174C22×C3—31no—
175C22×C3—31no—
176C22×C3C3×#21611yescharacteristic
177A4—31no—
178C3×C4—21no—
179C3×C4—31no—
180C3×C4—61no—
181C3×C4—121no—
182C3×C4—241no—
183C3×C4—721no—
184C3×C4C2×D3×A411yescharacteristic
185Dic3—61no—
186Dic3—181no—
187C32—41no—
188C9—81no—
189C9A4×Q1611yescharacteristic, complemented
190C23#611⋊C311yescharacteristic
191Q8—31no—
192Q8—61no—
193Q8—61no—
194Q8—91no—
195Q8—271no—
196Q8—541no—
197Q8—541no—
198Q8D9⋊A411yescharacteristic
199C2×C4—31no—
200C2×C4—31no—
201C2×C4—61no—
202C2×C4—61no—
203C2×C4—541no—
204C2×C4—541no—
205C8—91no—
206C8—271no—
207C2×C3—31no—
208C2×C3—31no—
209C2×C3—121no—
210C2×C3A4×#6711yescharacteristic
211C22—31no—
212C22—31no—
213C22#3674⋊C311yescharacteristic, complemented
214C4—21no—
215C4—31no—
216C4—61no—
217C4—181no—
218C4—541no—
219C4C2×D9⋊A411yescharacteristic
220C3—121no—
221C3A4×#21611yescharacteristic
222C2—31no—
223C2—31no—
224C2#611⋊A411yescharacteristic
subgroupG/H
centerC2#611⋊A4
commutator, cosocleC22×C4×C9C22×C3
FrattiniC3×C4C2×D3×A4
FittingC22×Q8×C9C2×C3
socleC23×C3C3×#67
prime radicalC2×C3×A4#67
hypercenterQ8D9⋊A4
hypocenterC22×C9C3×Q16
LCSG ▷ C22×C4×C9 ▷ C23×C9 ▷ C22×C9
UCSC1 ◁ C2 ◁ C4 ◁ Q8

Complements

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

NH ≅ G/Ntype
5. C22×#3674C3semidirect
46. #3674A4semidirect
49. C9⋊A4Q16semidirect
114. C22×C9C3×Q16semidirect
189. C9A4×Q16semidirect
213. C22#3674⋊C3semidirect

Maximal subgroups

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

subgroupindex
Q8×C9⋊A42
Dic18⋊A42
(C9⋊C8)⋊A42
A4×#2163
C22×#36743
#3674⋊C34