Finite Group Databasebeta

← (C3⋊#4626)⋊C3 #13136764

Subgroups — (C3⋊#4626)⋊C3 #13136764

Order 1728.

163 conjugacy classes of subgroups; only the 21 normal subgroups are shown.

normal subgroups only

This group has 163 conjugacy classes of subgroups — too many to draw. Only its 21 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
1#127⋊H(3,𝔽3)C211yesmaximal, characteristic
2C3⋊#4626C311yesmaximal, characteristic, complemented
3C3×#4545—31nomaximal
4(C3⋊SL(2,ℤ/4))⋊C3—41nomaximal
5C32⋊(C42⋊C3)C411yescharacteristic, complemented
6C32×#127D311yescharacteristic
7C3×#127⋊C3—21no—
8C3×#127⋊C3C2×C311yescharacteristic
9C3×#834—31no—
10C3×#834—31no—
11C2×C22⋊H(3,𝔽3)—41no—
12#4626—91no—
13C3×#353—91no—
14#4545—31no—
15C2×C32×D4—31no—
16C2×C32×D4—31no—
17C3⋊SL(2,ℤ/4)—41no—
18C32×C42Dic311yescharacteristic, complemented
19C3×#217—61no—
20C3×#217—61no—
21C3×C42⋊C3—21no—
22C3×C42⋊C3C3×C411yescharacteristic, complemented
23C22⋊H(3,𝔽3)—41no—
24(C3⋊Dic3)⋊C3—161no—
25C3×#127—31no—
26C3×#127C3×D311yescharacteristic
27#127⋊C3—31no—
28#127⋊C3—61no—
29C3×#99—91no—
30C3×#99—91no—
31#834—31no—
32#834—31no—
33C23×C32—31no—
34C23×C32S411yescharacteristic
35C2×C3×A4—41no—
36C2×C3×A4—81no—
37C32×D4—121no—
38C32×D4—121no—
39C2×C32×C4—31no—
40C2×C3×Dic3—121no—
41C2×C3×Dic3—121no—
42#353—91noSylow
43C2×H(3,𝔽3)—161no—
44C2×C3×D4—31no—
45C2×C3×D4—31no—
46C2×C3×D4—31no—
47C2×C3×D4—31no—
48C2×C3×D4—61no—
49C2×C3×D4—61no—
50SL(2,ℤ/4)—361no—
51C3×C22⋊C4—181no—
52C3×C22⋊C4—181no—
53C3×C42—31no—
54C3×C42C3×Dic311yescharacteristic
55#217—61no—
56#217—61no—
57C42⋊C3—31no—
58C42⋊C3—61no—
59C22×C32—61no—
60C22×C32—61no—
61C22×C32—121no—
62C22×C32SL(2,ℤ/4)11yescharacteristic
63C3×A4—41no—
64C3×A4—81no—
65C32×C4—61no—
66C3⋊Dic3—161no—
67C3×Dic3—241no—
68C3×Dic3—241no—
69#127(C3⋊D3)⋊C311yescharacteristic
70#99—91no—
71#99—91no—
72H(3,𝔽3)—161noSylow
73C23×C3—31no—
74C23×C3—31no—
75C23×C3—31no—
76C23×C3—61no—
77C23×C3C3×S411yescharacteristic
78C2×A4—121no—
79C2×A4—241no—
80C3×D4—121no—
81C3×D4—121no—
82C3×D4—121no—
83C3×D4—121no—
84C3×D4—122no—
85C3×D4—122no—
86C2×C3×C4—31no—
87C2×C3×C4—31no—
88C2×C3×C4—61no—
89C2×C3×C4—361no—
90C2×C3×C4—361no—
91C2×Dic3—121no—
92C2×Dic3—121no—
93C2×C32—31no—
94C2×C32—41no—
95C2×C32—121no—
96C2×C32—161no—
97C2×C32—321no—
98C2×D4—31no—
99C2×D4—31no—
100C22⋊C4—181no—
101C22⋊C4—181no—
102C42(C3⋊Dic3)⋊C311yescharacteristic, complemented
103C22×C3—31no—
104C22×C3—61no—
105C22×C3—61no—
106C22×C3—61no—
107C22×C3—61no—
108C22×C3—121no—
109C22×C3—121no—
110C22×C3—121no—
111C22×C3—121no—
112C22×C3—121no—
113C22×C3—121no—
114C22×C3C3×SL(2,ℤ/4)11yescharacteristic
115A4—121no—
116A4—241no—
117C3×C4—61no—
118C3×C4—61no—
119C3×C4—62no—
120C3×C4—721no—
121C3×C4—721no—
122Dic3—241no—
123Dic3—241no—
124Dic3—1441no—
125C32—161no—
126C32—321no—
127C32#462611yescharacteristic, complemented
128C23—31no—
129C23(C3⋊S4)⋊C311yescharacteristic
130D4—121no—
131D4—121no—
132C2×C4—31no—
133C2×C4—361no—
134C2×C4—361no—
135C2×C3—31no—
136C2×C3—31no—
137C2×C3—41no—
138C2×C3—61no—
139C2×C3—121no—
140C2×C3—121no—
141C2×C3—121no—
142C2×C3—241no—
143C2×C3—481no—
144C2×C3—961no—
145C22—61no—
146C22—61no—
147C22—121no—
148C22(C3⋊SL(2,ℤ/4))⋊C311yescharacteristic
149C4—61no—
150C4—721no—
151C4—721no—
152C3—31no—
153C3—481no—
154C3—961no—
155C3C3×#462611yescharacteristic
156C2—31no—
157C2—41no—
158C2—121no—
subgroupG/H
center, hypercenterC1G
commutator, hypocenterC3×C42⋊C3C3×C4
FrattiniC23×C3C3×S4
FittingC32×#127D3
socleC22×C3C3×SL(2,ℤ/4)
prime radical#127⋊H(3,𝔽3)C2
cosocleC3×#127⋊C3C2×C3
LCSG ▷ C3×C42⋊C3
UCSC1

Complements

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

NH ≅ G/Ntype
2. C3⋊#4626C3semidirect
5. C32⋊(C42⋊C3)C4semidirect
18. C32×C42Dic3semidirect
22. C3×C42⋊C3C3×C4semidirect
102. C42(C3⋊Dic3)⋊C3semidirect
127. C32#4626semidirect

Maximal subgroups

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

subgroupindex
#127⋊H(3,𝔽3)2
C3⋊#46263
C3×#45453
(C3⋊SL(2,ℤ/4))⋊C34