Finite Group Databasebeta

← #11724199

Subgroups — #11724199

Order 864.

271 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 271 conjugacy classes of subgroups — too many to draw. Only its 44 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×Dic3×A4C211yesmaximal, characteristic
2C3⋊(C3⋊SL(2,ℤ/4))C211yesmaximal, characteristic
3C3×C3⋊SL(2,ℤ/4)C211yesmaximal, characteristic
4#64259—31nomaximal
5#64204—33nomaximal
6#63964—31nomaximal
7C2×C32×A4C2211yescharacteristic
8#6293—41nomaximal
9C22×C3×Dic3D311yescharacteristic
10C3×C4×A4—31no—
11C2×#627—31no—
12Dic3×A4D313yes—
13C3⋊SL(2,ℤ/4)—31no—
14C3⋊SL(2,ℤ/4)—33no—
15C3⋊SL(2,ℤ/4)—61no—
16C3⋊SL(2,ℤ/4)—63no—
17C3⋊SL(2,ℤ/4)D311yescharacteristic
18C3×SL(2,ℤ/4)—33no—
19#3757—31no—
20C3×#217—31no—
21#3724—31no—
22Dic3⋊Dic3—31no—
23Dic3⋊Dic3—31no—
24C32×A4Q811yescharacteristic, complemented
25C3⋊(C3⋊Dic3)—41no—
26C3×C3⋊Dic3—41no—
27C32×Dic3—41no—
28A4⋊Q8—93no—
29#924—91no—
30#878—91no—
31C23×C32C2×D311yescharacteristic
32C2×C3×A4—21no—
33C2×C3×A4—23no—
34C2×C3×A4C2×D313yes—
35C2×C3×A4C2×D311yescharacteristic
36C2×C3⋊Dic3—31no—
37C2×C3⋊Dic3—31no—
38#634—121no—
39C2×C3×Dic3—31no—
40C2×C3×Dic3—31no—
41C2×C3×Dic3—31no—
42C2×C3×Dic3—31no—
43#627—61no—
44#627—61no—
45#627—123no—
46C2×C33—41no—
47C22×C3×C4—31no—
48C22×Dic3C3⋊D311yescharacteristic
49C2×Dic6—91no—
50C2×Dic6—91no—
51C4×A4—33no—
52SL(2,ℤ/4)—33no—
53SL(2,ℤ/4)—93no—
54SL(2,ℤ/4)—183no—
55C3×C22⋊C4—91no—
56#217—31no—
57#217—91no—
58#217—91no—
59#217—181no—
60C4⋊Dic3—91no—
61C4⋊Dic3—91no—
62Dic3⋊C4—91no—
63Dic3⋊C4—91no—
64Dic3⋊C4—91no—
65Dic3⋊C4—91no—
66C22×C32—31no—
67C22×C32—31no—
68C22×C32Dic611yescharacteristic, complemented
69C3×A4—21no—
70C3×A4—23no—
71C3×A4Dic613yescomplemented
72C3×A4Dic611yescharacteristic, complemented
73C32×C4—121no—
74C3⋊Dic3—41no—
75C3⋊Dic3—61no—
76C3⋊Dic3—61no—
77C3⋊Dic3—121no—
78C3⋊Dic3—123no—
79C3⋊Dic3—241no—
80C3⋊Dic3—243no—
81C3×Dic3—31no—
82C3×Dic3—43no—
83C3×Dic3—61no—
84C3×Dic3—61no—
85C3×Dic3—123no—
86C3×Dic3S411yescharacteristic
87C22⋊Q8—271noSylow
88C33—41noSylow
89C23×C3—21no—
90C23×C3D3211yescharacteristic
91C23×C3C2×C3⋊D311yescharacteristic
92C2×A4—23no—
93C2×A4D3213yes—
94C2×C3×C4—91no—
95C2×C3×C4—91no—
96C2×C3×C4—91no—
97C2×C3×C4—91no—
98C2×Dic3—31no—
99C2×Dic3—31no—
100C2×Dic3—31no—
101C2×Dic3—31no—
102C2×Dic3—91no—
103C2×Dic3—91no—
104C2×Dic3—91no—
105C2×Dic3—91no—
106C2×Dic3—181no—
107C2×Dic3—181no—
108Dic6—181no—
109Dic6—181no—
110Dic6—181no—
111Dic6—181no—
112Dic6—363no—
113C2×C32—31no—
114C2×C32—31no—
115C2×C32—41no—
116C2×C32—43no—
117C2×C32—83no—
118C2×C32—81no—
119C2×C32C2×S411yescharacteristic
120C2×Q8—271no—
121C22×C4—31no—
122C4⋊C4—271no—
123C4⋊C4—271no—
124C4⋊C4—271no—
125C22⋊C4—91no—
126C22⋊C4—271no—
127C22×C3—21no—
128C22×C3—31no—
129C22×C3—31no—
130C22×C3—31no—
131C22×C3—31no—
132C22×C3—61no—
133C22×C3—61no—
134C22×C3#62711yescharacteristic, complemented
135C22×C3#63411yescharacteristic, complemented
136A4—23no—
137A4#62713yescomplemented
138C3×C4—31no—
139C3×C4—91no—
140C3×C4—123no—
141C3×C4—181no—
142C3×C4—181no—
143Dic3—31no—
144Dic3—61no—
145Dic3—61no—
146Dic3—123no—
147Dic3—181no—
148Dic3—181no—
149Dic3—181no—
150Dic3—181no—
151Dic3—361no—
152Dic3—363no—
153Dic3—361no—
154Dic3—723no—
155Dic3C3⋊S411yescharacteristic
156C32—41no—
157C32—43no—
158C32—83no—
159C32—81no—
160C32A4⋊Q811yescharacteristic, complemented
161C23D3×C3⋊D311yescharacteristic
162Q8—541no—
163Q8—541no—
164C2×C4—91no—
165C2×C4—91no—
166C2×C4—91no—
167C2×C4—91no—
168C2×C4—271no—
169C2×C4—271no—
170C2×C3—21no—
171C2×C3—31no—
172C2×C3—31no—
173C2×C3—31no—
174C2×C3—31no—
175C2×C3—43no—
176C2×C3—61no—
177C2×C3—61no—
178C2×C3—83no—
179C2×C3D3×S411yescharacteristic
180C2×C3C2×C3⋊S411yescharacteristic
181C22—31no—
182C22—31no—
183C22#629311yescharacteristic, complemented
184C4—31no—
185C4—91no—
186C4—181no—
187C4—181no—
188C4—541no—
189C4—541no—
190C3—21no—
191C3—43no—
192C3—83no—
193C3#6420411yescharacteristic, complemented
194C3#6425911yescharacteristic, complemented
195C2—31no—
196C2—31no—
197C2D3×C3⋊S411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2D3×C3⋊S4
commutator, prime radical, cosocleC2×C32×A4C22
Fitting, socleC23×C32C2×D3
hypocenterC32×A4Q8
LCSG ▷ C2×C32×A4 ▷ C32×A4
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
24. C32×A4Q8semidirect
68. C22×C32Dic6semidirect
71. C3×A4Dic6semidirect
72. C3×A4Dic6semidirect
134. C22×C3#627semidirect
135. C22×C3#634semidirect
137. A4#627semidirect
160. C32A4⋊Q8semidirect
183. C22#6293semidirect
193. C3#64204semidirect
194. C3#64259semidirect

Maximal subgroups

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

subgroupindex
C3×Dic3×A42
C3⋊(C3⋊SL(2,ℤ/4))2
C3×C3⋊SL(2,ℤ/4)2
#642593
#642043
#639643
#62934