Finite Group Databasebeta

← C2×A4⋊PSU(3,𝔽2) #13181805

Subgroups — C2×A4⋊PSU(3,𝔽2) #13181805

Order 1728.

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

normal subgroups only

This group has 377 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
1C2×A4×#157C211yesmaximal, characteristic
2C2×A4⋊#157C212yesmaximal
3A4⋊PSU(3,𝔽2)C214yesmaximal, direct product factor
4C2×C22⋊PSU(3,𝔽2)—31nomaximal
5C2×A4×C3⋊D3C2211yescharacteristic
6C2×C3⋊PSU(3,𝔽2)—41nomaximal
7A4×#157C2212yes—
8A4⋊#157C2214yes—
9C23×#157D311yescharacteristic
10C22×PSU(3,𝔽2)—31no—
11C2×C22⋊#157—32no—
12C22⋊PSU(3,𝔽2)—34no—
13C22⋊PSU(3,𝔽2)—34no—
14C2×#3777—31no—
15C2×#3777—32no—
16C2×C32×A4Q811yescharacteristic, complemented
17C2×C3⋊#157—42no—
18C2×C3×#157—41no—
19A4×C3⋊D3Q811yescharacteristic, complemented
20A4×C3⋊D3C2311yescharacteristic
21C3⋊PSU(3,𝔽2)—44no—
22C2×A4⋊Q8—91nomaximal
23C23×C3⋊D3C2×D311yescharacteristic
24C22×#157—31no—
25C22×#157—31no—
26C22×#157—32no—
27C22×#157—32no—
28C22×#157—32no—
29C22×#157C2×D312yes—
30C2×D3×A4—121no—
31C2×PSU(3,𝔽2)—34no—
32C2×PSU(3,𝔽2)—61no—
33C2×PSU(3,𝔽2)—62no—
34C2×PSU(3,𝔽2)—61no—
35C22⋊#157—34no—
36C22⋊#157—34no—
37#3777—34no—
38#3777—38no—
39C2×C3×C3⋊D3—41no—
40C32×A4C2×Q811yescharacteristic, complemented
41C3⋊#157—44no—
42C3×#157—42no—
43C2×C4×A4—91no—
44C2×SL(2,ℤ/4)—92no—
45A4⋊Q8—94no—
46C23×C32Dic611yescharacteristic, complemented
47C22×C3⋊D3—31no—
48C22×C3⋊D3—31no—
49C22×C3⋊D3—31no—
50C22×C3⋊D3—31no—
51C22×C3⋊D3Dic611yescharacteristic, complemented
52C22×C3⋊D3C22×D311yescharacteristic
53C2×C3×A4—41no—
54C2×C3×A4—81no—
55C2×#157—31no—
56C2×#157—32no—
57C2×#157—32no—
58C2×#157—32no—
59C2×#157—34no—
60C2×#157—34no—
61C2×#157—61no—
62C2×#157—62no—
63C2×#157—62no—
64C2×#157—62no—
65C2×#157—62no—
66C2×#157S411yescharacteristic
67D3×A4—121no—
68D3×A4—121no—
69PSU(3,𝔽2)—64no—
70PSU(3,𝔽2)—64no—
71C2×C22⋊Q8—271noSylow
72C2×C33—41no—
73C3×C3⋊D3—41no—
74C3×C3⋊D3—41no—
75C23×D3—121no—
76C22×A4—91no—
77C2×Dic6—361no—
78C4×A4—92no—
79SL(2,ℤ/4)—94no—
80C22×C32—31no—
81C22×C32—31no—
82C22×C32C2×Dic611yescharacteristic, complemented
83C2×C3⋊D3—31no—
84C2×C3⋊D3—31no—
85C2×C3⋊D3—31no—
86C2×C3⋊D3—31no—
87C2×C3⋊D3—31no—
88C2×C3⋊D3—31no—
89C2×C3⋊D3—31no—
90C2×C3⋊D3—61no—
91C2×C3⋊D3C2×S411yescharacteristic
92C2×C3×D3—481no—
93C3×A4—41no—
94C3×A4—81no—
95#157—32no—
96#157—64no—
97#157—64no—
98#157C2×S412yes—
99C22×Q8—271no—
100C23×C4—91no—
101C22⋊Q8—274no—
102C22⋊Q8—274no—
103C2×C4⋊C4—271no—
104C2×C4⋊C4—272no—
105C2×C22⋊C4—272no—
106C33—41noSylow
107C23×C3—41no—
108C22×D3—121no—
109C22×D3—121no—
110C22×D3—361no—
111C22×D3—361no—
112C22×D3—361no—
113C22×D3—361no—
114C2×A4—81no—
115C2×A4—91no—
116C2×A4—91no—
117C2×A4PSU(3,𝔽2)11yescharacteristic, complemented
118C2×C3×C4—361no—
119C2×Dic3—362no—
120Dic6—364no—
121C2×C32—31no—
122C2×C32—31no—
123C2×C32—161no—
124C2×C32—321no—
125C2×C32A4⋊Q811yescharacteristic, complemented
126C3⋊D3—31no—
127C3⋊D3—31no—
128C3⋊D3A4⋊Q811yescharacteristic, complemented
129C3⋊D3C22×S411yescharacteristic
130C3×D3—481no—
131C3×D3—481no—
132C24—91no—
133C2×Q8—274no—
134C2×Q8—541no—
135C2×Q8—542no—
136C2×Q8—541no—
137C22×C4—92no—
138C22×C4—271no—
139C22×C4—271no—
140C22×C4—272no—
141C22×C4—272no—
142C22×C4—272no—
143C4⋊C4—274no—
144C4⋊C4—278no—
145C22⋊C4—274no—
146C22⋊C4—274no—
147C22×C3—41no—
148C22×C3—121no—
149C22×C3—121no—
150C22×C3—361no—
151C2×D3—121no—
152C2×D3—361no—
153C2×D3—361no—
154C2×D3—361no—
155C2×D3—361no—
156C2×D3—361no—
157C2×D3—361no—
158C2×D3—361no—
159C2×D3—362no—
160A4—81no—
161A4C2×PSU(3,𝔽2)11yescharacteristic, complemented
162C3×C4—362no—
163Dic3—364no—
164C32—161no—
165C32—321no—
166C32C2×A4⋊Q811yescharacteristic, complemented
167C23—91no—
168C23—91no—
169C23—271no—
170C23—271no—
171C23—271no—
172C23—271no—
173C23C3⋊PSU(3,𝔽2)11yescharacteristic, complemented
174Q8—544no—
175Q8—544no—
176C2×C4—91no—
177C2×C4—272no—
178C2×C4—272no—
179C2×C4—271no—
180C2×C4—272no—
181C2×C4—274no—
182C2×C4—274no—
183C2×C4—541no—
184C2×C4—542no—
185C2×C4—542no—
186C2×C4—542no—
187C2×C4—542no—
188C2×C3—41no—
189C2×C3—41no—
190C2×C3—121no—
191C2×C3—121no—
192C2×C3—321no—
193C2×C3—361no—
194C2×C3—361no—
195D3—121no—
196D3—121no—
197D3—361no—
198D3—361no—
199C22—31no—
200C22—31no—
201C22—91no—
202C22—271no—
203C22—271no—
204C22—271no—
205C22—271no—
206C22—271no—
207C22—271no—
208C22—271no—
209C22—541no—
210C22C2×C3⋊PSU(3,𝔽2)11yescharacteristic, complemented
211C4—92no—
212C4—272no—
213C4—544no—
214C4—544no—
215C3—41no—
216C3—41no—
217C3—321no—
218C2—31no—
219C2—31no—
220C2—91no—
221C2—91no—
222C2—271no—
223C2—271no—
224C2A4⋊PSU(3,𝔽2)11yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2A4⋊PSU(3,𝔽2)
commutator, cosocleA4×C3⋊D3C23
FrattiniC1G
Fitting, socleC23×C32Dic6
prime radicalC2×A4×C3⋊D3C22
hypocenterC32×A4C2×Q8
LCSG ▷ A4×C3⋊D3 ▷ C32×A4
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
3. A4⋊PSU(3,𝔽2)C2direct (N × H)
16. C2×C32×A4Q8semidirect
19. A4×C3⋊D3Q8semidirect
40. C32×A4C2×Q8semidirect
46. C23×C32Dic6semidirect
51. C22×C3⋊D3Dic6semidirect
82. C22×C32C2×Dic6semidirect
117. C2×A4PSU(3,𝔽2)semidirect
125. C2×C32A4⋊Q8semidirect
128. C3⋊D3A4⋊Q8semidirect
161. A4C2×PSU(3,𝔽2)semidirect
166. C32C2×A4⋊Q8semidirect
173. C23C3⋊PSU(3,𝔽2)semidirect
210. C22C2×C3⋊PSU(3,𝔽2)semidirect
224. C2A4⋊PSU(3,𝔽2)direct (N × H)

Maximal subgroups

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

subgroupindex
C2×A4×#1572
C2×A4⋊#1572
A4⋊PSU(3,𝔽2)2
C2×C22⋊PSU(3,𝔽2)3
C2×C3⋊PSU(3,𝔽2)4
C2×A4⋊Q89