Finite Group Databasebeta

← A4×C3⋊#226 #13180321

Subgroups — A4×C3⋊#226 #13180321

Order 1728.

306 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 33 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×SL(2,𝔽3)C211yesmaximal, characteristic
2C22×C3⋊#226C311yesmaximal, characteristic, complemented
3A4×#226—33nomaximal
4A4×#216—31nomaximal
5A4×C3⋊Dic3—41nomaximal
6C3×C3⋊#226—41nomaximal
7C3×C22⋊SL(2,𝔽3)—21no—
8C3×Q8×A4D311yescharacteristic
9C22×C3×SL(2,𝔽3)C2×C311yescharacteristic
10A4×Dic6—31no—
11C2×C3⋊#226—31no—
12C2×C3⋊#226—31no—
13A4×SL(2,𝔽3)—23no—
14A4×SL(2,𝔽3)D313yes—
15A4×C3⋊C8—31no—
16C2×C32×A4—41no—
17C32×SL(2,𝔽3)—41no—
18C22×#226—33no—
19C22×#216—31no—
20A4×Q16—91no—
21C22×C3⋊Dic3—41no—
22C2×C3×SL(2,𝔽3)—31no—
23C3×C4×A4—31no—
24Dic3×A4—61no—
25Dic3×A4—123no—
26C3⋊#226—31no—
27C3⋊#226A411yescharacteristic, direct product factor
28C3×#226—123no—
29C3×#216—121no—
30C32×A4—41no—
31C3×C3⋊Dic3—161no—
32C22×C3×Q8C3×D311yescharacteristic
33C22×Dic6—31no—
34C22⋊SL(2,𝔽3)—23no—
35Q8×A4—21no—
36Q8×A4—91no—
37Q8×A4C3⋊D311yescharacteristic
38C22×SL(2,𝔽3)C3×D313yes—
39C2×#226—93no—
40C2×#226—93no—
41C2×#216—91no—
42C2×#216—91no—
43C2×#216—91no—
44C2×#216—91no—
45C22×C3⋊C8—31no—
46C8×A4—91no—
47C23×C32—41no—
48C2×C3×A4—43no—
49C2×C3×A4—83no—
50C2×C3×A4—81no—
51C2×C3×A4S411yescharacteristic
52C32×Q8—41no—
53C2×C3⋊Dic3—121no—
54C2×C3⋊Dic3—121no—
55C3×Dic6—121no—
56C3×SL(2,𝔽3)—43no—
57C3×SL(2,𝔽3)—81no—
58C3×SL(2,𝔽3)—83no—
59C3×SL(2,𝔽3)—241no—
60C3×SL(2,𝔽3)C2×A411yescharacteristic
61C3×C3⋊C8—121no—
62C22×Q16—91noSylow
63C2×C33—161no—
64C2×C3×Q8—31no—
65C2×C3×Q8—91no—
66C22×C3×C4—31no—
67C22×Dic3—61no—
68C22×Dic3—123no—
69C2×Dic6—91no—
70C2×Dic6—91no—
71C2×Dic6—181no—
72C2×SL(2,𝔽3)—33no—
73C4×A4—31no—
74C4×A4—61no—
75C4×A4—181no—
76#226—33no—
77#226—93no—
78C3×Q16—361no—
79#216—31no—
80#216—91no—
81#216—91no—
82#216—92no—
83#216—91no—
84C2×C3⋊C8—91no—
85C2×C3⋊C8—91no—
86C22×C32—41no—
87C22×C32—121no—
88C22×C32—121no—
89C3×A4—43no—
90C3×A4—83no—
91C3×A4—81no—
92C3×A4#22611yescharacteristic, complemented
93C32×C4—121no—
94C3⋊Dic3—41no—
95C3⋊Dic3—121no—
96C3×Dic3—241no—
97C3×Dic3—483no—
98C22×Q8—91no—
99C22×Q8C3×C3⋊D311yescharacteristic
100C2×Q16—271no—
101C2×Q16—271no—
102C2×Q16—271no—
103C2×Q16—271no—
104C22×C8—91no—
105C33—161noSylow
106C23×C3—43no—
107C23×C3C3×S411yescharacteristic
108C2×A4—21no—
109C2×A4—83no—
110C2×A4C3⋊S411yescharacteristic
111C3×Q8—41no—
112C3×Q8—61no—
113C3×Q8—81no—
114C3×Q8—91no—
115C3×Q8—361no—
116C3×Q8D3×A411yescharacteristic
117C2×C3×C4—91no—
118C2×C3×C4—91no—
119C2×Dic3—181no—
120C2×Dic3—181no—
121C2×Dic3—363no—
122C2×Dic3—363no—
123Dic6—31no—
124Dic6—91no—
125Dic6—181no—
126Dic6—181no—
127SL(2,𝔽3)—83no—
128SL(2,𝔽3)—243no—
129SL(2,𝔽3)D3×A413yes—
130C3×C8—361no—
131C3⋊C8—31no—
132C3⋊C8—91no—
133C2×C32—41no—
134C2×C32—41no—
135C2×C32—121no—
136C2×C32—121no—
137C2×C32—163no—
138C2×C32—323no—
139C2×C32—321no—
140C2×Q8—31no—
141C2×Q8—91no—
142C2×Q8—271no—
143C2×Q8—271no—
144C2×Q8—541no—
145C22×C4—31no—
146C22×C4—181no—
147Q16—91no—
148Q16—271no—
149Q16—271no—
150Q16—272no—
151Q16—271no—
152C2×C8—271no—
153C2×C8—271no—
154C22×C3—31no—
155C22×C3—31no—
156C22×C3—43no—
157C22×C3—123no—
158C22×C3—123no—
159C22×C3C3×#22611yescharacteristic, complemented
160A4—21no—
161A4—83no—
162A4C3⋊#22611yescharacteristic, direct product factor
163C3×C4—31no—
164C3×C4—91no—
165C3×C4—121no—
166C3×C4—241no—
167C3×C4—721no—
168Dic3—61no—
169Dic3—123no—
170Dic3—181no—
171Dic3—363no—
172C32—41no—
173C32—41no—
174C32—163no—
175C32—323no—
176C32—321no—
177C23C3×C3⋊S411yescharacteristic
178Q8—61no—
179Q8—91no—
180Q8—91no—
181Q8—271no—
182Q8—541no—
183Q8—541no—
184Q8A4×C3⋊D311yescharacteristic
185C2×C4—91no—
186C2×C4—91no—
187C2×C4—541no—
188C2×C4—541no—
189C8—91no—
190C8—271no—
191C2×C3—31no—
192C2×C3—31no—
193C2×C3—43no—
194C2×C3—41no—
195C2×C3—81no—
196C2×C3—123no—
197C2×C3—123no—
198C2×C3—323no—
199C2×C3A4×S411yescharacteristic
200C22—31no—
201C22—31no—
202C22C3×C3⋊#22611yescharacteristic, complemented
203C4—31no—
204C4—91no—
205C4—181no—
206C4—541no—
207C3—43no—
208C3—41no—
209C3—81no—
210C3—323no—
211C3A4×#22611yescharacteristic, complemented
212C2—31no—
213C2—31no—
214C2A4×C3⋊S411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2A4×C3⋊S4
commutator, cosocle, hypocenterC22×C3×SL(2,𝔽3)C2×C3
FittingC22×C3×Q8C3×D3
socleC23×C3C3×S4
prime radicalC3×A4×SL(2,𝔽3)C2
LCSG ▷ C22×C3×SL(2,𝔽3)
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
2. C22×C3⋊#226C3semidirect
27. C3⋊#226A4direct (N × H)
92. C3×A4#226semidirect
159. C22×C3C3×#226semidirect
162. A4C3⋊#226direct (N × H)
202. C22C3×C3⋊#226semidirect
211. C3A4×#226semidirect

Maximal subgroups

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

subgroupindex
C3×A4×SL(2,𝔽3)2
C22×C3⋊#2263
A4×#2263
A4×#2163
A4×C3⋊Dic34
C3×C3⋊#2264