Finite Group Databasebeta

← C3×#64138 #11723884

Subgroups — C3×#64138 #11723884

Order 864.

684 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 154 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×C33×C4C211yesmaximal, characteristic, complemented
2C2×C3×#638C211yesmaximal, characteristic
3C3×#3757C211yesmaximal, characteristic, complemented
4C3×#3752C212yesmaximal, complemented
5C3×#3750C212yesmaximal, complemented
6#64138C311yesmaximal, characteristic, direct product factor
7C3×#929—34nomaximal
8C23×C33C2211yescharacteristic
9C22×C3×C3⋊D3C2211yescharacteristic
10C2×C33×C4—22no—
11C2×C33×C4C2212yes—
12C3×#638—22no—
13C2×C3×C3⋊Dic3C2211yescharacteristic
14C2×C3×C3⋊Dic3C2212yes—
15C22×C32×C4—24no—
16C22×C32×C4D314yescomplemented
17C22×C32×C4C2×C311yescharacteristic, complemented
18C2×#638C2×C311yescharacteristic
19C2×C3×#67—34no—
20#3757C2×C311yescharacteristic, complemented
21#3752C2×C312yescomplemented
22#3750C2×C312yescomplemented
23C3×#217—34no—
24C3×(C2×D3)⋊C4—38no—
25C3×Dic3⋊C4—38no—
26C22×C33—22no—
27C22×C33D412yes—
28C22×C33C2311yescharacteristic
29C2×C3×C3⋊D3—21no—
30C2×C3×C3⋊D3—22no—
31C33×C4—22no—
32C3×C3⋊Dic3—21no—
33C3×C3⋊Dic3—22no—
34C3×#123—91no—
35#929—34no—
36C23×C32—24no—
37C23×C32C2×D314yes—
38C23×C32C22×C311yescharacteristic
39C22×C3⋊D3C22×C311yescharacteristic
40C22×C3×D3—34no—
41C2×C32×C4—216no—
42C2×C32×C4—28no—
43C2×C32×C4—22no—
44C2×C32×C4—28no—
45C2×C32×C4C2×D318yes—
46C2×C32×C4C22×C312yes—
47#638—22no—
48C2×C3⋊Dic3C22×C311yescharacteristic
49C2×C3⋊Dic3C22×C312yes—
50C3×#67—68no—
51C2×C3×Dic3—34no—
52C2×C3×Dic3—38no—
53C2×C33—22no—
54C2×C33C2×D411yescharacteristic
55C2×C33#4112yes—
56C3×C3⋊D3—41no—
57C2×C3×D4—91no—
58C22×C3×C4—24no—
59C22×C3×C4C3×D314yescomplemented
60C22×C3×C4C3⋊D311yescharacteristic, complemented
61C2×#67—34no—
62C3×C4⋊C4—92no—
63C3×C22⋊C4—91no—
64C3×C22⋊C4—92no—
65#217—34no—
66(C2×D3)⋊C4—38no—
67Dic3⋊C4—38no—
68C22×C32—216no—
69C22×C32—28no—
70C22×C32—22no—
71C22×C32—28no—
72C22×C32—24no—
73C22×C32#6718yes—
74C22×C32C3×D412yes—
75C22×C32C22×D314yes—
76C22×C32C23×C311yescharacteristic
77C2×C3⋊D3—21no—
78C2×C3⋊D3—22no—
79C2×C3×D3—68no—
80C2×C3×D3—64no—
81C32×C4—216no—
82C32×C4—28no—
83C32×C4—22no—
84C3⋊Dic3—21no—
85C3⋊Dic3—22no—
86C3×Dic3—68no—
87C3×Dic3—64no—
88#123—91noSylow
89C33#12311yescharacteristic, complemented, Sylow
90C23×C3—24no—
91C23×C3—91no—
92C23×C3C2×C3×D314yes—
93C23×C3C2×C3⋊D311yescharacteristic
94C22×D3—34no—
95C3×D4—182no—
96C2×C3×C4—216no—
97C2×C3×C4—28no—
98C2×C3×C4—22no—
99C2×C3×C4—28no—
100C2×C3×C4—92no—
101C2×C3×C4—91no—
102C2×C3×C4C2×C3×D318yes—
103C2×C3×C4C2×C3⋊D312yes—
104#67—68no—
105C2×Dic3—34no—
106C2×Dic3—38no—
107C2×C32—216no—
108C2×C32—28no—
109C2×C32—22no—
110C2×C32—28no—
111C2×C32—24no—
112C2×C32#23518yes—
113C2×C32C2×#6714yes—
114C2×C32C2×C3×D411yescharacteristic
115C2×C32C3×#4112yes—
116C3⋊D3—41no—
117C3×D3—124no—
118C2×D4—91no—
119C22×C4C3×C3⋊D311yescharacteristic, complemented
120C4⋊C4—92no—
121C22⋊C4—91no—
122C22⋊C4—92no—
123C22×C3—216no—
124C22×C3—28no—
125C22×C3—22no—
126C22×C3—28no—
127C22×C3—24no—
128C22×C3—182no—
129C22×C3—181no—
130C22×C3C3×#6718yes—
131C22×C3#63812yes—
132C22×C3C22×C3×D314yes—
133C22×C3C22×C3⋊D311yescharacteristic
134C2×D3—68no—
135C2×D3—64no—
136C3×C4—216no—
137C3×C4—28no—
138C3×C4—22no—
139C3×C4—182no—
140C3×C4—181no—
141Dic3—68no—
142Dic3—64no—
143C32—24no—
144C32#92914yescomplemented
145C32C3×#12311yescharacteristic, complemented
146C23—91no—
147C23C2×C3×C3⋊D311yescharacteristic
148D4—182no—
149C2×C4—22no—
150C2×C4—92no—
151C2×C4—91no—
152C2×C4C2×C3×C3⋊D312yes—
153C2×C3—216no—
154C2×C3—28no—
155C2×C3—22no—
156C2×C3—28no—
157C2×C3—24no—
158C2×C3—361no—
159C2×C3C3×#23518yes—
160C2×C3C2×C3×#6714yes—
161C2×C3#382812yes—
162C2×C3C2×#63811yescharacteristic
163D3—124no—
164C22—22no—
165C22—182no—
166C22—181no—
167C22C3×#63812yes—
168C22C22×C3×C3⋊D311yescharacteristic
169C4—22no—
170C4—182no—
171C4—181no—
172C3—24no—
173C3C3×#92914yescomplemented
174C3#6413811yescharacteristic, direct product factor
175C2—22no—
176C2—361no—
177C2C3×#382812yes—
178C2C2×C3×#63811yescharacteristic
subgroupG/H
centerC22×C3C22×C3⋊D3
commutator, cosocleC22×C32C23×C3
FrattiniC22C22×C3×C3⋊D3
FittingC22×C33×C4C2
socleC22×C33C23
prime radicalC2×C3×#638C2
hypercenterC22×C3×C4C3⋊D3
hypocenterC32C3×#123
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
1. C22×C33×C4C2semidirect
3. C3×#3757C2semidirect
4. C3×#3752C2semidirect
5. C3×#3750C2semidirect
6. #64138C3direct (N × H)
16. C22×C32×C4D3semidirect
17. C22×C32×C4C2×C3semidirect
20. #3757C2×C3semidirect
21. #3752C2×C3semidirect
22. #3750C2×C3semidirect
59. C22×C3×C4C3×D3semidirect
60. C22×C3×C4C3⋊D3semidirect
89. C33#123semidirect
119. C22×C4C3×C3⋊D3semidirect
144. C32#929semidirect
145. C32C3×#123semidirect
173. C3C3×#929semidirect
174. C3#64138direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C33×C42
C2×C3×#6382
C3×#37572
C3×#37522
C3×#37502
#641383
C3×#9293