Finite Group Databasebeta

← #11724473

Subgroups — #11724473

Order 864.

628 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C22×C32×A4C211yesmaximal, characteristic, complemented
2C2×C3⋊(C3⋊S4)C211yesmaximal, characteristic, complemented
3C3⋊(C3⋊SL(2,ℤ/4))C211yesmaximal, characteristic, complemented
4#64269—312nomaximal
5#64162—31nomaximal
6C2×C32×A4—21no—
7C2×C32×A4C2211yescharacteristic
8C3⋊(C3⋊S4)—21no—
9#6312—41nomaximal
10C24×C32D311yescharacteristic, complemented
11C22×C3×A4D3112yescomplemented
12C2×C3⋊S4—312no—
13C2×#638—31no—
14C2×#638—31no—
15C2×#638—31no—
16C3⋊SL(2,ℤ/4)—312no—
17#3757—31no—
18#3757—31no—
19#3757—31no—
20C22×C33—41no—
21C2×C3⋊(C3⋊D3)—41no—
22C32×A4D411yescharacteristic, complemented
23C3⋊(C3⋊Dic3)—41no—
24GL(2,ℤ/4)—99no—
25#953—94no—
26C23×C32—21no—
27C23×C32—31no—
28C23×C32—31no—
29C23×C32—61no—
30C23×C32C2×D311yescharacteristic
31C22×C3⋊D3—31no—
32C2×C3×A4—212no—
33C2×C3×A4C2×D3112yes—
34C3⋊S4—612no—
35#638—61no—
36#638—62no—
37#638—61no—
38#638—61no—
39#638—61no—
40#638—1212no—
41C2×C3⋊Dic3—31no—
42C2×C3⋊Dic3—31no—
43C2×C3⋊Dic3—31no—
44C2×C33—41no—
45C2×C33—81no—
46C3⋊(C3⋊D3)—81no—
47C24×C3C3⋊D314yescomplemented
48C22×A4C3⋊D319yescomplemented
49C2×S4—99no—
50C2×#67—94no—
51C2×#67—94no—
52C2×#67—94no—
53SL(2,ℤ/4)—99no—
54#217—94no—
55#217—94no—
56#217—94no—
57C22×C32—31no—
58C22×C32—31no—
59C22×C32—32no—
60C22×C32—31no—
61C22×C32—412no—
62C22×C32—61no—
63C22×C32—61no—
64C22×C32—61no—
65C22×C32#6711yescharacteristic, complemented
66C22×C32S411yescharacteristic, complemented
67C2×C3⋊D3—61no—
68C2×C3⋊D3—61no—
69C2×C3⋊D3—61no—
70C2×C3⋊D3—1212no—
71C3×A4#67112yescomplemented
72C3⋊Dic3—61no—
73C3⋊Dic3—61no—
74C3⋊Dic3—61no—
75C3⋊Dic3—1212no—
76#120—271noSylow
77C33—41noSylow
78C23×C3—24no—
79C23×C3—34no—
80C23×C3—34no—
81C23×C3—64no—
82C23×C3C2×C3⋊D314yes—
83C22×D3—94no—
84C2×A4—29no—
85C2×A4C2×C3⋊D319yes—
86S4—189no—
87#67—184no—
88#67—188no—
89#67—184no—
90#67—184no—
91#67—184no—
92#67—369no—
93C2×Dic3—94no—
94C2×Dic3—94no—
95C2×Dic3—94no—
96C2×C32—21no—
97C2×C32—31no—
98C2×C32—31no—
99C2×C32—412no—
100C2×C32—61no—
101C2×C32—812no—
102C2×C32C2×S411yescharacteristic
103C3⋊D3—121no—
104C3⋊D3—2412no—
105C24C3⋊(C3⋊D3)11yescharacteristic, complemented
106C2×D4—271no—
107C2×D4—271no—
108C2×D4—271no—
109C22⋊C4—271no—
110C22⋊C4—271no—
111C22⋊C4—271no—
112C22×C3—34no—
113C22×C3—34no—
114C22×C3—38no—
115C22×C3—34no—
116C22×C3—49no—
117C22×C3—64no—
118C22×C3—64no—
119C22×C3—64no—
120C22×C3#63814yescomplemented
121C22×C3C3⋊S414yescomplemented
122C2×D3—184no—
123C2×D3—184no—
124C2×D3—184no—
125C2×D3—369no—
126A4#63819yescomplemented
127Dic3—184no—
128Dic3—184no—
129Dic3—184no—
130Dic3—369no—
131C32—412no—
132C32GL(2,ℤ/4)11yescharacteristic, complemented
133C23—21no—
134C23—31no—
135C23—31no—
136C23—61no—
137C23—271no—
138C23C2×C3⋊(C3⋊D3)11yescharacteristic
139D4—541no—
140D4—542no—
141D4—541no—
142D4—541no—
143D4—541no—
144C2×C4—271no—
145C2×C4—271no—
146C2×C4—271no—
147C2×C3—24no—
148C2×C3—34no—
149C2×C3—34no—
150C2×C3—49no—
151C2×C3—64no—
152C2×C3—89no—
153C2×C3C2×C3⋊S414yes—
154D3—364no—
155D3—729no—
156C22—31no—
157C22—31no—
158C22—32no—
159C22—31no—
160C22—61no—
161C22—61no—
162C22—61no—
163C22—541no—
164C22—541no—
165C22—541no—
166C22#631211yescharacteristic, complemented
167C22C3⋊(C3⋊S4)11yescharacteristic, complemented
168C4—541no—
169C4—541no—
170C4—541no—
171C3—49no—
172C3#6426914yescomplemented
173C2—21no—
174C2—31no—
175C2—31no—
176C2—61no—
177C2—1081no—
178C2C2×C3⋊(C3⋊S4)11yescharacteristic
subgroupG/H
center, FrattiniC2C2×C3⋊(C3⋊S4)
commutator, cosocleC2×C32×A4C22
FittingC24×C32D3
socleC23×C32C2×D3
hypercenterC22C3⋊(C3⋊S4)
hypocenterC32×A4D4
LCSG ▷ C2×C32×A4 ▷ C32×A4
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×C32×A4C2semidirect
2. C2×C3⋊(C3⋊S4)C2semidirect
3. C3⋊(C3⋊SL(2,ℤ/4))C2semidirect
10. C24×C32D3semidirect
11. C22×C3×A4D3semidirect
22. C32×A4D4semidirect
47. C24×C3C3⋊D3semidirect
48. C22×A4C3⋊D3semidirect
65. C22×C32#67semidirect
66. C22×C32S4semidirect
71. C3×A4#67semidirect
105. C24C3⋊(C3⋊D3)semidirect
120. C22×C3#638semidirect
121. C22×C3C3⋊S4semidirect
126. A4#638semidirect
132. C32GL(2,ℤ/4)semidirect
166. C22#6312semidirect
167. C22C3⋊(C3⋊S4)semidirect
172. C3#64269semidirect

Maximal subgroups

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

subgroupindex
C22×C32×A42
C2×C3⋊(C3⋊S4)2
C3⋊(C3⋊SL(2,ℤ/4))2
#642693
#641623
#63124