Finite Group Databasebeta

← C3×#1856 #72571

Subgroups — C3×#1856 #72571

Order 384.

324 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 112 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×#518C211yesmaximal, characteristic
2C3×#428C212yesmaximal, complemented
3C2×C3×#100C211yesmaximal, characteristic
4C3×#390C211yesmaximal, characteristic
5C3×#339C212yesmaximal, complemented
6#1856C311yesmaximal, characteristic, direct product factor, Sylow
7C22×C3×D4C2211yescharacteristic
8C2×C3×M16C2212yes—
9C3×C4×D4—22no—
10C3×C4×D4—22no—
11C3×#117C2212yes—
12C2×C3×C4⋊C4C2211yescharacteristic
13C2×C3×C22⋊C4—22no—
14C2×C3×C22⋊C4—22no—
15C2×C3×C42C2211yescharacteristic
16C3×#107—22no—
17C3×#106—22no—
18C3×#100C414yescomplemented
19#518C2×C311yescharacteristic
20#428C2×C312yescomplemented
21C2×#100C2×C311yescharacteristic
22#390C2×C311yescharacteristic
23#339C2×C312yescomplemented
24C24×C3D412yes—
25C2×C3×D4—22no—
26C2×C3×D4—22no—
27C2×C3×D4C2×C412yes—
28C2×C3×D4Q812yes—
29C22×C3×C4—21no—
30C22×C3×C4—22no—
31C22×C3×C4—22no—
32C22×C3×C4—21no—
33C22×C3×C4C2311yescharacteristic
34C3×M16—22no—
35C3×M16C2×C414yescomplemented
36C2×C3×C8—22no—
37C3×C4⋊C4—22no—
38C3×C4⋊C4D412yes—
39C3×C4⋊C4D412yes—
40C3×C22⋊C4—24no—
41C3×C22⋊C4—22no—
42C3×C22⋊C4—44no—
43C3×C42—22no—
44C3×C42—22no—
45C22×D4C22×C311yescharacteristic
46C2×M16C22×C312yes—
47C4×D4—22no—
48C4×D4—22no—
49#117C22×C312yes—
50C2×C4⋊C4C22×C311yescharacteristic
51C2×C22⋊C4—22no—
52C2×C22⋊C4—22no—
53C2×C42C22×C311yescharacteristic
54#107—22no—
55#106—22no—
56#100C3×C414yescomplemented
57C23×C3—22no—
58C23×C3—22no—
59C23×C3—42no—
60C23×C3—42no—
61C23×C3C4⋊C414yes—
62C23×C3C2×D411yescharacteristic
63C3×D4—24no—
64C3×D4—24no—
65C2×C3×C4—22no—
66C2×C3×C4—22no—
67C2×C3×C4—22no—
68C2×C3×C4—41no—
69C2×C3×C4—41no—
70C2×C3×C4—42no—
71C2×C3×C4—41no—
72C2×C3×C4—82no—
73C2×C3×C4—82no—
74C2×C3×C4C22×C411yescharacteristic
75C2×C3×C4C2×D412yes—
76C2×C3×C4C2×Q811yescharacteristic
77C2×C3×C4#4112yes—
78C3×C8—24no—
79C24C3×D412yes—
80C2×D4—22no—
81C2×D4—22no—
82C2×D4C2×C3×C412yes—
83C2×D4C3×Q812yes—
84C22×C4—21no—
85C22×C4—22no—
86C22×C4—22no—
87C22×C4—21no—
88C22×C4C23×C311yescharacteristic
89M16—22no—
90M16C2×C3×C414yescomplemented
91C2×C8—22no—
92C4⋊C4—22no—
93C4⋊C4C3×D412yes—
94C4⋊C4C3×D412yes—
95C22⋊C4—24no—
96C22⋊C4—22no—
97C22⋊C4—44no—
98C42—22no—
99C42—22no—
100C22×C3—21no—
101C22×C3—21no—
102C22×C3—24no—
103C22×C3—44no—
104C22×C3—42no—
105C22×C3—42no—
106C22×C3—82no—
107C22×C3C2×C4⋊C411yescharacteristic
108C22×C3#12011yescharacteristic
109C22×C3#12311yescharacteristic
110C3×C4—42no—
111C3×C4—42no—
112C3×C4—42no—
113C3×C4—81no—
114C3×C4C4×D412yes—
115C3×C4C22⋊Q812yes—
116C23—22no—
117C23—22no—
118C23—42no—
119C23—42no—
120C23C3×C4⋊C414yes—
121C23C2×C3×D411yescharacteristic
122D4—24no—
123D4—24no—
124C2×C4—22no—
125C2×C4—22no—
126C2×C4—22no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—42no—
130C2×C4—41no—
131C2×C4—82no—
132C2×C4—82no—
133C2×C4C22×C3×C411yescharacteristic
134C2×C4C2×C3×D412yes—
135C2×C4C2×C3×Q811yescharacteristic
136C2×C4C3×#4112yes—
137C8—24no—
138C2×C3—21no—
139C2×C3—21no—
140C2×C3—44no—
141C2×C3#38511yescharacteristic
142C2×C3#45311yescharacteristic
143C2×C3#45511yescharacteristic
144C22—21no—
145C22—21no—
146C22—24no—
147C22—44no—
148C22—42no—
149C22—42no—
150C22—82no—
151C22C2×C3×C4⋊C411yescharacteristic
152C22C3×#12011yescharacteristic
153C22C3×#12311yescharacteristic
154C4—42no—
155C4—42no—
156C4—42no—
157C4—81no—
158C4C3×C4×D412yes—
159C4C3×C22⋊Q812yes—
160C3#185611yescharacteristic, direct product factor, Sylow
161C2—21no—
162C2—21no—
163C2—44no—
164C2C3×#38511yescharacteristic
165C2C3×#45311yescharacteristic
166C2C3×#45511yescharacteristic
subgroupG/H
center, socleC22×C3#120
commutatorC2×C4C22×C3×C4
Frattini, cosocleC22×C4C23×C3
prime radicalC22×C3×D4C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#428C2semidirect
5. C3×#339C2semidirect
6. #1856C3direct (N × H)
18. C3×#100C4semidirect
20. #428C2×C3semidirect
23. #339C2×C3semidirect
35. C3×M16C2×C4semidirect
56. #100C3×C4semidirect
90. M16C2×C3×C4semidirect
160. C3#1856direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#5182
C3×#4282
C2×C3×#1002
C3×#3902
C3×#3392
#18563