Finite Group Databasebeta

← A4×#3774 #13180228

Subgroups — A4×#3774 #13180228

Order 1728.

358 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 27 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
1A4×#625C212yesmaximal, complemented
2A4×#622C211yesmaximal, characteristic, complemented
3C22×#3774C311yesmaximal, characteristic, complemented
4C2×C3×D3×A4—22no—
5A4×C3⋊Dic3C2211yescharacteristic
6C3×#3774—41nomaximal
7C22×#625C2×C312yescomplemented
8C22×#622C2×C311yescharacteristic, complemented
9A4×#67—62no—
10C2×#3774—31no—
11C2×#3774—32no—
12C2×#3774—31no—
13C2×C32×A4D411yescharacteristic
14C3×D3×A4—42no—
15C3×#625—42no—
16C3×#622—41no—
17A4×D8—91nomaximal
18C23×C3×D3—22no—
19C22×C3×A4—62no—
20C2×D3×A4—22no—
21C2×D3×A4—42no—
22C22×C3⋊Dic3C22×C311yescharacteristic
23C2×#625—32no—
24C2×#625—32no—
25C2×#625—62no—
26C2×#622—31no—
27C2×#622—31no—
28Dic3×A4—62no—
29#3774—32no—
30#3774—32no—
31#3774—31no—
32#3774A411yescharacteristic, direct product factor
33C2×C32×D3—82no—
34C32×A4D811yescharacteristic, complemented
35C3×C3⋊Dic3—41no—
36C22×#67—62no—
37D4×A4—92no—
38C8×A4—91no—
39C23×C32C3×D411yescharacteristic
40C22×C3×D3—42no—
41C22×C3×D3—62no—
42C22×C3×D3—62no—
43C22×C3×D3—122no—
44C2×C3×A4—22no—
45C2×C3×A4—42no—
46C2×C3×A4—122no—
47D3×A4—42no—
48D3×A4—44no—
49C2×C3⋊Dic3—31no—
50C2×C3⋊Dic3—31no—
51C3×#67—242no—
52#625—32no—
53#625—62no—
54#625—62no—
55#625C2×A412yescomplemented
56#622—31no—
57#622C2×A411yescharacteristic, complemented
58C22×D8—91noSylow
59C2×C33—41no—
60C32×D3—162no—
61C24×C3—62no—
62C23×D3—22no—
63C22×A4—62no—
64C22×A4—122no—
65C2×#67—182no—
66C2×#67—182no—
67C2×#67—182no—
68C2×#67—182no—
69C22×Dic3—62no—
70C4×A4—91no—
71C3×D8—361no—
72C22×C32—31no—
73C22×C32—31no—
74C22×C32—242no—
75C22×C32C3×D811yescharacteristic, complemented
76C2×C3×D3—22no—
77C2×C3×D3—62no—
78C2×C3×D3—82no—
79C2×C3×D3—122no—
80C2×C3×D3—122no—
81C2×C3×D3—122no—
82C2×C3×D3—122no—
83C2×C3×D3—162no—
84C3×A4—22no—
85C3×A4—42no—
86C3⋊Dic3—31no—
87C3⋊Dic3C22×A411yescharacteristic
88C3×Dic3—242no—
89C22×D4—92no—
90C2×D8—271no—
91C2×D8—271no—
92C2×D8—272no—
93C22×C8—91no—
94C33—41noSylow
95C23×C3—22no—
96C23×C3—122no—
97C23×C3—182no—
98C23×C3—182no—
99C23×C3—362no—
100C22×D3—42no—
101C22×D3—62no—
102C22×D3—62no—
103C22×D3—122no—
104C2×A4—42no—
105C2×A4—124no—
106C2×A4—122no—
107C2×A4GO+(4,𝔽2)11yescharacteristic
108C3×D4—362no—
109#67—62no—
110#67—182no—
111#67—182no—
112#67—182no—
113#67—184no—
114C2×Dic3—182no—
115C2×Dic3—182no—
116C3×C8—361no—
117C2×C32—31no—
118C2×C32—31no—
119C2×C32—82no—
120C2×C32—162no—
121C2×C32—482no—
122C2×C32D4×A411yescharacteristic
123C3×D3—42no—
124C3×D3—122no—
125C3×D3—162no—
126C3×D3—164no—
127C24—62no—
128C2×D4—272no—
129C2×D4—272no—
130C2×D4—542no—
131C22×C4—91no—
132D8—91no—
133D8—271no—
134D8—272no—
135D8—272no—
136C2×C8—271no—
137C2×C8—271no—
138C22×C3—22no—
139C22×C3—62no—
140C22×C3—62no—
141C22×C3—62no—
142C22×C3—182no—
143C22×C3—242no—
144C22×C3—362no—
145C22×C3—362no—
146C22×C3—362no—
147C22×C3—362no—
148C22×C3—482no—
149C2×D3—22no—
150C2×D3—62no—
151C2×D3—122no—
152C2×D3—122no—
153C2×D3—122no—
154C2×D3—122no—
155A4—42no—
156A4#377411yescharacteristic, direct product factor
157C3×C4—361no—
158Dic3—62no—
159Dic3—182no—
160C32—82no—
161C32—162no—
162C32A4×D811yescharacteristic, complemented
163C23—122no—
164C23—182no—
165C23—182no—
166C23—362no—
167C23C3×GO+(4,𝔽2)11yescharacteristic
168D4—92no—
169D4—272no—
170D4—542no—
171D4—542no—
172C2×C4—271no—
173C2×C4—271no—
174C8—91no—
175C8—271no—
176C2×C3—22no—
177C2×C3—41no—
178C2×C3—62no—
179C2×C3—62no—
180C2×C3—122no—
181C2×C3—162no—
182C2×C3—362no—
183C2×C3—482no—
184C2×C3—484no—
185D3—42no—
186D3—122no—
187C22—31no—
188C22—31no—
189C22—62no—
190C22—182no—
191C22—362no—
192C22—362no—
193C22—362no—
194C22—362no—
195C22C3×#377411yescharacteristic, complemented
196C4—91no—
197C4—271no—
198C3—22no—
199C3—41no—
200C3—162no—
201C2—31no—
202C2—31no—
203C2—122no—
204C2—362no—
205C2A4×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2A4×GO+(4,𝔽2)
commutator, cosocleC22×C3⋊Dic3C22×C3
Fitting, socleC23×C32C3×D4
hypocenterC22×C32C3×D8
LCSG ▷ C22×C3⋊Dic3 ▷ C23×C32 ▷ C22×C32
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. A4×#625C2semidirect
2. A4×#622C2semidirect
3. C22×#3774C3semidirect
7. C22×#625C2×C3semidirect
8. C22×#622C2×C3semidirect
32. #3774A4direct (N × H)
34. C32×A4D8semidirect
55. #625C2×A4semidirect
57. #622C2×A4semidirect
75. C22×C32C3×D8semidirect
156. A4#3774direct (N × H)
162. C32A4×D8semidirect
195. C22C3×#3774semidirect

Maximal subgroups

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

subgroupindex
A4×#6252
A4×#6222
C22×#37743
C3×#37744
A4×D89