Finite Group Databasebeta

← C3×#2203 #72918

Subgroups — C3×#2203 #72918

Order 384.

216 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 56 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×#563C211yesmaximal, characteristic, complemented
2C3×#456C212yesmaximal, complemented
3C3×#454C211yesmaximal, characteristic
4C3×#356C212yesmaximal, complemented
5C3×#355C211yesmaximal, characteristic, complemented
6#2203C311yesmaximal, characteristic, direct product factor, Sylow
7C3×2−1+4C2211yescharacteristic
8C3×#137—22no—
9C3×AGL(1,ℤ/8)—21no—
10C3×C4⋊Q8C2212yes—
11C3×#126—22no—
12C3×#125—21no—
13C3×#124C2211yescharacteristic
14C3×#123—21no—
15C3×C22⋊Q8—22no—
16C3×#104—21no—
17C3×#104—22no—
18C3×#101C2211yescharacteristic
19C3×#101C2212yes—
20#563C2×C311yescharacteristic, complemented
21#456C2×C312yescomplemented
22#454C2×C311yescharacteristic
23#356C2×C312yescomplemented
24#355C2×C311yescharacteristic, complemented
25C3×#41—21no—
26C3×#41—22no—
27C3×#41—41no—
28C2×C3×Q8—41no—
29C2×C3×Q8D412yes—
30C2×C3×Q8C2311yescharacteristic
31C2×C3×D4D411yescharacteristic
32C22×C3×C4—21no—
33C3×Q16—42no—
34C3×SD16—42no—
35C3×SD16—41no—
36C3×D8—41no—
37C3×M16—21no—
38C3×M16—22no—
39C3×C4⋊C4—21no—
40C3×C4⋊C4—22no—
41C3×C4⋊C4—21no—
42C3×C4⋊C4—22no—
43C3×C4⋊C4—22no—
44C3×C22⋊C4—22no—
45C3×C22⋊C4—21no—
46C3×C22⋊C4—22no—
47C3×C42D411yescharacteristic
48C3×C42D412yes—
492−1+4C22×C311yescharacteristic
50#137—22no—
51AGL(1,ℤ/8)—21no—
52C4⋊Q8C22×C312yes—
53#126—22no—
54#125—21no—
55#124C22×C311yescharacteristic
56#123—21no—
57C22⋊Q8—22no—
58#104—21no—
59#104—22no—
60#101C22×C312yes—
61#101C22×C311yescharacteristic
62C23×C3—21no—
63C3×Q8—21no—
64C3×Q8—22no—
65C3×Q8—41no—
66C3×Q8—42no—
67C3×D4—21no—
68C3×D4—22no—
69C3×D4—41no—
70C3×D4—41no—
71C2×C3×C4—21no—
72C2×C3×C4—22no—
73C2×C3×C4—22no—
74C2×C3×C4—41no—
75C2×C3×C4—42no—
76C2×C3×C4—81no—
77C2×C3×C4C2×D412yes—
78C2×C3×C4C2×D411yescharacteristic
79C3×C8—41no—
80C3×C8—42no—
81#41—21no—
82#41—22no—
83#41—41no—
84C2×Q8—41no—
85C2×Q8C3×D412yes—
86C2×Q8C23×C311yescharacteristic
87C2×D4C3×D411yescharacteristic
88C22×C4—21no—
89Q16—42no—
90SD16—42no—
91SD16—41no—
92D8—41no—
93M16—21no—
94M16—22no—
95C4⋊C4—21no—
96C4⋊C4—22no—
97C4⋊C4—21no—
98C4⋊C4—22no—
99C4⋊C4—22no—
100C22⋊C4—22no—
101C22⋊C4—21no—
102C22⋊C4—22no—
103C42C3×D411yescharacteristic
104C42C3×D412yes—
105C22×C3—42no—
106C22×C3—41no—
107C22×C3—41no—
108C22×C3#12011yescharacteristic
109C3×C4—21no—
110C3×C4—22no—
111C3×C4—41no—
112C3×C4—42no—
113C3×C4—42no—
114C3×C4—41no—
115C23—21no—
116Q8—21no—
117Q8—22no—
118Q8—41no—
119Q8—42no—
120D4—21no—
121D4—22no—
122D4—41no—
123D4—41no—
124C2×C4—21no—
125C2×C4—22no—
126C2×C4—22no—
127C2×C4—41no—
128C2×C4—42no—
129C2×C4—81no—
130C2×C4C2×C3×D412yes—
131C2×C4C2×C3×D411yescharacteristic
132C8—41no—
133C8—42no—
134C2×C3—21no—
135C2×C3—81no—
136C2×C3—81no—
137C2×C3B(4,𝔽2)11yescharacteristic
138C22—42no—
139C22—41no—
140C22—41no—
141C22C3×#12011yescharacteristic
142C4—21no—
143C4—22no—
144C4—41no—
145C4—42no—
146C4—42no—
147C4—41no—
148C3#220311yescharacteristic, direct product factor, Sylow
149C2—21no—
150C2—81no—
151C2—81no—
152C2C3×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC2×C3B(4,𝔽2)
commutator, Frattini, cosocleC2×Q8C23×C3
prime radicalC3×#454C2
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#563C2semidirect
2. C3×#456C2semidirect
4. C3×#356C2semidirect
5. C3×#355C2semidirect
6. #2203C3direct (N × H)
20. #563C2×C3semidirect
21. #456C2×C3semidirect
23. #356C2×C3semidirect
24. #355C2×C3semidirect
148. C3#2203direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#5632
C3×#4562
C3×#4542
C3×#3562
C3×#3552
#22033