Finite Group Databasebeta

← #13173064

Subgroups — #13173064

Order 1728.

800 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 139 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
1C32×D4×A4C211yesmaximal, characteristic
2#11724463C211yesmaximal, characteristic, complemented
3C3⋊(C3⋊(A4⋊C8))C211yesmaximal, characteristic, complemented
4#10593648—312nomaximal
5#10592917—31nomaximal
6C22×C32×A4—21no—
7C32×C4×A4C2211yescharacteristic
8C3⋊(C3⋊SL(2,ℤ/4))—21no—
9#88861—41nomaximal
10C22×C32×D4D311yescharacteristic
11C3×D4×A4D3112yes—
12#64259—312no—
13C2×#3754—31no—
14C2×#3754—31no—
15#64133—31no—
16C3⋊(A4⋊C8)—312no—
17#63659—31no—
18#63659—31no—
19#63657—31no—
20C2×C32×A4—41no—
21C2×C32×A4D411yescharacteristic
22C33×D4—41no—
23#6308—41no—
24C3⋊(C3⋊(C3⋊C8))—41no—
25#5414—99no—
26#5218—94no—
27C24×C32—21no—
28C22×C3×A4—212no—
29C2×C32×D4—31no—
30C2×C32×D4—31no—
31C2×C32×D4—61no—
32C22×C32×C4C2×D311yescharacteristic
33C2×#634—31no—
34C3×C4×A4C2×D3112yes—
35C3⋊SL(2,ℤ/4)—612no—
36#3757—61no—
37#3754—62no—
38#3754—61no—
39#3754—61no—
40#3754—1212no—
41C4⋊(C3⋊Dic3)—31no—
42#3750—61no—
43C2×C3⋊(C3⋊C8)—31no—
44C2×C3⋊(C3⋊C8)—31no—
45C22×C33—81no—
46C32×A4SD1611yescharacteristic, complemented
47C33×C4—41no—
48C3⋊(C3⋊Dic3)—81no—
49C22×C3×D4C3⋊D314yes—
50D4×A4C3⋊D319yes—
51A4⋊Q8—99no—
52C2×#214—94no—
53C2×#214—94no—
54#924—94no—
55A4⋊C8—99no—
56#832—94no—
57#832—94no—
58#830—94no—
59C23×C32—41no—
60C23×C32—61no—
61C23×C32—61no—
62C23×C32—121no—
63C23×C32#6711yescharacteristic
64C2×C3×A4—412no—
65C2×C3×A4#67112yes—
66C32×D4—32no—
67C32×D4—31no—
68C32×D4—412no—
69C32×D4—61no—
70C32×D4S411yescharacteristic
71C2×C32×C4—31no—
72C2×C32×C4—31no—
73C2×C3⋊Dic3—61no—
74C2×C3⋊Dic3—61no—
75#634—61no—
76#634—61no—
77#634—1212no—
78C3⋊(C3⋊C8)—61no—
79C3⋊(C3⋊C8)—61no—
80C3⋊(C3⋊C8)—1212no—
81#450—271noSylow
82C2×C33—41no—
83C2×C33—161no—
84C24×C3—24no—
85C22×A4—29no—
86C2×C3×D4—34no—
87C2×C3×D4—34no—
88C2×C3×D4—64no—
89C22×C3×C4C2×C3⋊D314yes—
90C2×Dic6—94no—
91C4×A4C2×C3⋊D319yes—
92SL(2,ℤ/4)—189no—
93#217—184no—
94#214—188no—
95#214—184no—
96#214—184no—
97#214—369no—
98C4⋊Dic3—94no—
99Dic3⋊C4—184no—
100C2×C3⋊C8—94no—
101C2×C3⋊C8—94no—
102C22×C32—21no—
103C22×C32—31no—
104C22×C32—31no—
105C22×C32—61no—
106C22×C32—812no—
107C22×C32—121no—
108C22×C32—121no—
109C22×C32—121no—
110C22×C32—121no—
111C22×C32#21411yescharacteristic, complemented
112C3×A4#214112yescomplemented
113C32×C4—31no—
114C32×C4—412no—
115C32×C4C2×S411yescharacteristic
116C3⋊Dic3—121no—
117C3⋊Dic3—121no—
118C3⋊Dic3—2412no—
119C22×D4C3⋊(C3⋊D3)11yescharacteristic
120C2×SD16—271no—
121C2×SD16—271no—
122C22⋊Q8—271no—
123#102—271no—
124#102—271no—
125C22⋊C8—271no—
126C33—41noSylow
127C23×C3—44no—
128C23×C3—64no—
129C23×C3—64no—
130C23×C3—124no—
131C23×C3#63814yes—
132C2×A4—49no—
133C2×A4#63819yes—
134C3×D4—38no—
135C3×D4—34no—
136C3×D4—49no—
137C3×D4—64no—
138C3×D4C3⋊S414yes—
139C2×C3×C4—34no—
140C2×C3×C4—34no—
141C2×Dic3—184no—
142C2×Dic3—184no—
143Dic6—184no—
144Dic6—184no—
145Dic6—369no—
146C3⋊C8—184no—
147C3⋊C8—184no—
148C3⋊C8—369no—
149C2×C32—31no—
150C2×C32—31no—
151C2×C32—41no—
152C2×C32—412no—
153C2×C32—121no—
154C2×C32—1612no—
155C2×C32GL(2,ℤ/4)11yescharacteristic
156C24—21no—
157C2×Q8—271no—
158C2×D4—31no—
159C2×D4—31no—
160C2×D4—61no—
161C22×C4C2×C3⋊(C3⋊D3)11yescharacteristic
162SD16—542no—
163SD16—541no—
164SD16—541no—
165C2×C8—271no—
166C2×C8—271no—
167C4⋊C4—271no—
168C4⋊C4—541no—
169C22⋊C4—541no—
170C22×C3—24no—
171C22×C3—34no—
172C22×C3—34no—
173C22×C3—64no—
174C22×C3—89no—
175C22×C3—124no—
176C22×C3—124no—
177C22×C3—124no—
178C22×C3—124no—
179C22×C3#375414yescomplemented
180A4#375419yescomplemented
181C3×C4—34no—
182C3×C4—49no—
183C3×C4C2×C3⋊S414yes—
184Dic3—364no—
185Dic3—364no—
186Dic3—729no—
187C32—412no—
188C32#541411yescharacteristic, complemented
189C23—41no—
190C23—61no—
191C23—61no—
192C23—121no—
193C23#631211yescharacteristic
194Q8—541no—
195Q8—541no—
196D4—32no—
197D4—31no—
198D4—61no—
199D4C3⋊(C3⋊S4)11yescharacteristic
200C2×C4—31no—
201C2×C4—31no—
202C2×C4—541no—
203C2×C4—541no—
204C8—541no—
205C8—541no—
206C2×C3—34no—
207C2×C3—34no—
208C2×C3—44no—
209C2×C3—49no—
210C2×C3—124no—
211C2×C3—169no—
212C2×C3#6426914yes—
213C22—21no—
214C22—31no—
215C22—31no—
216C22—61no—
217C22—121no—
218C22—121no—
219C22—121no—
220C22—121no—
221C22#8886111yescharacteristic, complemented
222C4—31no—
223C4—1081no—
224C4—1081no—
225C4C2×C3⋊(C3⋊S4)11yescharacteristic
226C3—49no—
227C3#1059364814yescomplemented
228C2—31no—
229C2—31no—
230C2—41no—
231C2—121no—
232C2#1172447311yescharacteristic
subgroupG/H
centerC2#11724473
commutator, cosocleC32×C4×A4C22
FrattiniC4C2×C3⋊(C3⋊S4)
FittingC22×C32×D4D3
socleC23×C32#67
prime radicalC32×D4×A4C2
hypercenterD4C3⋊(C3⋊S4)
hypocenterC32×A4SD16
LCSG ▷ C32×C4×A4 ▷ C2×C32×A4 ▷ C32×A4
UCSC1 ◁ C2 ◁ C4 ◁ D4

Complements

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

NH ≅ G/Ntype
2. #11724463C2semidirect
3. C3⋊(C3⋊(A4⋊C8))C2semidirect
46. C32×A4SD16semidirect
111. C22×C32#214semidirect
112. C3×A4#214semidirect
179. C22×C3#3754semidirect
180. A4#3754semidirect
188. C32#5414semidirect
221. C22#88861semidirect
227. C3#10593648semidirect

Maximal subgroups

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

subgroupindex
C32×D4×A42
#117244632
C3⋊(C3⋊(A4⋊C8))2
#105936483
#105929173
#888614