Finite Group Databasebeta

← C3×#2201 #72916

Subgroups — C3×#2201 #72916

Order 384.

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

normal subgroups only

This group has 226 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×#458C212yesmaximal
3C3×#455C211yesmaximal, characteristic
4C3×#354C211yesmaximal, characteristic, complemented
5C3×#352C212yesmaximal, complemented
6#2201C311yesmaximal, characteristic, direct product factor, Sylow
7C3×H(5,𝔽2)C2211yescharacteristic
8C3×#137—21no—
9C3×C4⋊Q8—21no—
10C3×#126—22no—
11C3×#125—21no—
12C3×#124C2211yescharacteristic
13C3×#123—22no—
14C3×#123C2212yes—
15C3×C22⋊Q8—22no—
16C3×#104—21no—
17C3×#100C2211yescharacteristic
18C3×#99—22no—
19C3×#99C2212yes—
20#563C2×C311yescharacteristic, complemented
21#458C2×C312yes—
22#455C2×C311yescharacteristic
23#354C2×C311yescharacteristic, complemented
24#352C2×C312yescomplemented
25C3×#41—21no—
26C3×#41—41no—
27C2×C3×Q8—21no—
28C2×C3×Q8D411yescharacteristic
29C2×C3×D4—22no—
30C2×C3×D4—41no—
31C2×C3×D4C2311yescharacteristic
32C22×C3×C4D412yes—
33C3×Q16—41no—
34C3×SD16—41no—
35C3×M16—21no—
36C3×C4⋊C4—21no—
37C3×C4⋊C4—22no—
38C3×C4⋊C4—22no—
39C3×C4⋊C4—21no—
40C3×C4⋊C4—22no—
41C3×C4⋊C4—42no—
42C3×C22⋊C4—22no—
43C3×C22⋊C4—22no—
44C3×C22⋊C4—22no—
45C3×C22⋊C4—42no—
46C3×C22⋊C4D412yes—
47C3×C42—21no—
48C3×C42D411yescharacteristic
49H(5,𝔽2)C22×C311yescharacteristic
50#137—21no—
51C4⋊Q8—21no—
52#126—22no—
53#125—21no—
54#124C22×C311yescharacteristic
55#123—22no—
56#123C22×C312yes—
57C22⋊Q8—22no—
58#104—21no—
59#100C22×C311yescharacteristic
60#99—22no—
61#99C22×C312yes—
62C23×C3—22no—
63C23×C3C2×D412yes—
64C3×Q8—21no—
65C3×Q8—41no—
66C3×Q8—81no—
67C3×D4—21no—
68C3×D4—41no—
69C3×D4—42no—
70C3×D4—41no—
71C2×C3×C4—22no—
72C2×C3×C4—21no—
73C2×C3×C4—22no—
74C2×C3×C4—21no—
75C2×C3×C4—22no—
76C2×C3×C4—41no—
77C2×C3×C4—42no—
78C2×C3×C4—42no—
79C2×C3×C4C2×D411yescharacteristic
80C3×C8—41no—
81#41—21no—
82#41—41no—
83C2×Q8—21no—
84C2×Q8C3×D411yescharacteristic
85C2×D4—22no—
86C2×D4—41no—
87C2×D4C23×C311yescharacteristic
88C22×C4C3×D412yes—
89Q16—41no—
90SD16—41no—
91M16—21no—
92C4⋊C4—21no—
93C4⋊C4—22no—
94C4⋊C4—22no—
95C4⋊C4—21no—
96C4⋊C4—22no—
97C4⋊C4—42no—
98C22⋊C4—22no—
99C22⋊C4—22no—
100C22⋊C4—22no—
101C22⋊C4—42no—
102C22⋊C4C3×D412yes—
103C42—21no—
104C42C3×D411yescharacteristic
105C22×C3—22no—
106C22×C3—42no—
107C22×C3—41no—
108C22×C3—82no—
109C22×C3#12011yescharacteristic
110C3×C4—21no—
111C3×C4—42no—
112C3×C4—41no—
113C3×C4—41no—
114C3×C4—42no—
115C3×C4—41no—
116C3×C4—82no—
117C23—22no—
118C23C2×C3×D412yes—
119Q8—21no—
120Q8—41no—
121Q8—81no—
122D4—21no—
123D4—41no—
124D4—42no—
125D4—41no—
126C2×C4—22no—
127C2×C4—21no—
128C2×C4—22no—
129C2×C4—21no—
130C2×C4—22no—
131C2×C4—41no—
132C2×C4—42no—
133C2×C4—42no—
134C2×C4C2×C3×D411yescharacteristic
135C8—41no—
136C2×C3—21no—
137C2×C3—42no—
138C2×C3—81no—
139C2×C3B(4,𝔽2)11yescharacteristic
140C22—22no—
141C22—42no—
142C22—41no—
143C22—82no—
144C22C3×#12011yescharacteristic
145C4—21no—
146C4—42no—
147C4—41no—
148C4—41no—
149C4—42no—
150C4—41no—
151C4—82no—
152C3#220111yescharacteristic, direct product factor, Sylow
153C2—21no—
154C2—42no—
155C2—81no—
156C2C3×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC2×C3B(4,𝔽2)
commutator, Frattini, cosocleC2×D4C23×C3
prime radicalC3×H(5,𝔽2)C22
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×D4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#563C2semidirect
4. C3×#354C2semidirect
5. C3×#352C2semidirect
6. #2201C3direct (N × H)
20. #563C2×C3semidirect
23. #354C2×C3semidirect
24. #352C2×C3semidirect
152. C3#2201direct (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×#4582
C3×#4552
C3×#3542
C3×#3522
#22013