Finite Group Databasebeta

← C3×#2018 #72733

Subgroups — C3×#2018 #72733

Order 384.

382 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 92 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×C3×AGL(1,ℤ/8)C212yesmaximal, complemented
2C22×C3×D8C211yesmaximal, characteristic, complemented
3C2×C3×#100C212yesmaximal, complemented
4C3×#408C211yesmaximal, characteristic, complemented
5C3×#341C211yesmaximal, characteristic, complemented
6#2018C311yesmaximal, characteristic, direct product factor, Sylow
7C2×C3×#41C2211yescharacteristic
8C22×C3×D4C2212yes—
9C3×AGL(1,ℤ/8)—24no—
10C3×AGL(1,ℤ/8)—24no—
11C2×C3×SD16—22no—
12C2×C3×D8—24no—
13C2×C3×D8—22no—
14C2×C3×D8—22no—
15C2×C3×M16C2211yescharacteristic
16C2×C3×M16C2212yes—
17C22×C3×C8C2211yescharacteristic
18C3×#100—24no—
19C3×C22⋊C8—22no—
20C2×AGL(1,ℤ/8)C2×C312yescomplemented
21C22×D8C2×C311yescharacteristic, complemented
22C2×#100C2×C312yescomplemented
23#408C2×C311yescharacteristic, complemented
24#341C2×C311yescharacteristic, complemented
25C24×C3—22no—
26C3×#41—22no—
27C3×#41—22no—
28C2×C3×Q8D411yescharacteristic
29C2×C3×D4—21no—
30C2×C3×D4—22no—
31C2×C3×D4—22no—
32C2×C3×D4—22no—
33C2×C3×D4—22no—
34C2×C3×D4—22no—
35C2×C3×D4D411yescharacteristic
36C2×C3×D4D414yes—
37C22×C3×C4—21no—
38C22×C3×C4C2311yescharacteristic
39C3×SD16—28no—
40C3×D8—28no—
41C3×D8—28no—
42C3×M16—22no—
43C3×M16—22no—
44C3×M16D414yes—
45C2×C3×C8—22no—
46C2×C3×C8—22no—
47C2×C3×C8—21no—
48C2×C3×C8D412yes—
49C2×#41C22×C311yescharacteristic
50C22×D4C22×C312yes—
51AGL(1,ℤ/8)—24no—
52AGL(1,ℤ/8)—24no—
53C2×SD16—22no—
54C2×D8—24no—
55C2×D8—22no—
56C2×D8—22no—
57C2×M16C22×C311yescharacteristic
58C2×M16C22×C312yes—
59C22×C8C22×C311yescharacteristic
60#100—24no—
61C22⋊C8—22no—
62C23×C3—21no—
63C23×C3—24no—
64C23×C3—42no—
65C23×C3—42no—
66C23×C3—84no—
67C23×C3#4111yescharacteristic
68C3×Q8—22no—
69C3×D4—24no—
70C3×D4—22no—
71C3×D4—24no—
72C3×D4—42no—
73C3×D4—42no—
74C3×D4—42no—
75C2×C3×C4—21no—
76C2×C3×C4—41no—
77C2×C3×C4—41no—
78C2×C3×C4C2×D412yes—
79C2×C3×C4C2×D411yescharacteristic
80C2×C3×C4C2×D412yes—
81C2×C3×C4C2×D411yescharacteristic
82C3×C8—22no—
83C3×C8—24no—
84C3×C8—41no—
85C24—22no—
86#41—22no—
87#41—22no—
88C2×Q8C3×D411yescharacteristic
89C2×D4—21no—
90C2×D4—22no—
91C2×D4—22no—
92C2×D4—22no—
93C2×D4—22no—
94C2×D4—22no—
95C2×D4C3×D411yescharacteristic
96C2×D4C3×D414yes—
97C22×C4—21no—
98C22×C4C23×C311yescharacteristic
99SD16—28no—
100D8—28no—
101D8—28no—
102M16—22no—
103M16—22no—
104M16C3×D414yes—
105C2×C8—22no—
106C2×C8—22no—
107C2×C8—21no—
108C2×C8C3×D412yes—
109C22×C3—22no—
110C22×C3—41no—
111C22×C3—44no—
112C22×C3—42no—
113C22×C3—84no—
114C22×C3—84no—
115C22×C3—84no—
116C22×C3#12112yes—
117C22×C3#12111yescharacteristic
118C3×C4—41no—
119C3×C4#12011yescharacteristic
120C3×C4#12012yes—
121C3×C4#12711yescharacteristic
122C23—21no—
123C23—24no—
124C23—42no—
125C23—42no—
126C23—84no—
127C23C3×#4111yescharacteristic
128Q8—22no—
129D4—24no—
130D4—22no—
131D4—24no—
132D4—42no—
133D4—42no—
134D4—42no—
135C2×C4—21no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4C2×C3×D411yescharacteristic
139C2×C4C2×C3×D412yes—
140C2×C4C2×C3×D411yescharacteristic
141C2×C4C2×C3×D412yes—
142C8—22no—
143C8—24no—
144C8—41no—
145C2×C3—22no—
146C2×C3—81no—
147C2×C3—84no—
148C2×C3#39211yescharacteristic
149C2×C3#47212yes—
150C22—22no—
151C22—41no—
152C22—44no—
153C22—42no—
154C22—84no—
155C22—84no—
156C22—84no—
157C22C3×#12112yes—
158C22C3×#12111yescharacteristic
159C4—41no—
160C4C3×#12011yescharacteristic
161C4C3×#12012yes—
162C4C3×#12711yescharacteristic
163C3#201811yescharacteristic, direct product factor, Sylow
164C2—22no—
165C2—81no—
166C2—84no—
167C2C3×#39211yescharacteristic
168C2C3×#47212yes—
subgroupG/H
center, socleC22×C3#121
commutator, Frattini, cosocleC22×C4C23×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×AGL(1,ℤ/8)C2semidirect
2. C22×C3×D8C2semidirect
3. C2×C3×#100C2semidirect
4. C3×#408C2semidirect
5. C3×#341C2semidirect
6. #2018C3direct (N × H)
20. C2×AGL(1,ℤ/8)C2×C3semidirect
21. C22×D8C2×C3semidirect
22. C2×#100C2×C3semidirect
23. #408C2×C3semidirect
24. #341C2×C3semidirect
163. C3#2018direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×AGL(1,ℤ/8)2
C22×C3×D82
C2×C3×#1002
C3×#4082
C3×#3412
#20183