Finite Group Databasebeta

← #10588306⋊C3 #13136763

Subgroups — #10588306⋊C3 #13136763

Order 1728.

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

normal subgroups only

This group has 148 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#11721353C211yesmaximal, characteristic
2C3×#4545—31nomaximal
3#10588306C311yesmaximal, characteristic, complemented
4#1094⋊C4—41nomaximal
5#88455C411yescharacteristic, complemented
6C32×#127D311yescharacteristic
7C3×#834—31no—
8C3×#834—31no—
9#127⋊C9—21no—
10#127⋊C9C2×C311yescharacteristic
11C2×#1094—41no—
12C3×#353—91no—
13#4545—31no—
14C2×C32×D4—31no—
15C2×C32×D4—31no—
16C32×C42Dic311yescharacteristic
17C3×#217—61no—
18C3×#217—61no—
19(C22⋊C9)⋊C4—41no—
20C42⋊C9—21no—
21C42⋊C9C3×C411yescharacteristic, complemented
22#1094—41no—
233−1+2⋊C4—161no—
24C3×#127—31no—
25C3×#127C3×D311yescharacteristic
26C3×#99—91no—
27C3×#99—91no—
28#834—31no—
29#834—31no—
30C23×C32—31no—
31C23×C32S411yescharacteristic
32C32×D4—121no—
33C32×D4—121no—
34C2×C32×C4—31no—
35C2×C3×Dic3—121no—
36C2×C3×Dic3—121no—
37C2×C22⋊C9—41no—
38C2×C22⋊C9—81no—
39#353—91noSylow
40C2×3−1+2—161no—
41C2×C3×D4—31no—
42C2×C3×D4—31no—
43C2×C3×D4—31no—
44C2×C3×D4—31no—
45C2×C3×D4—61no—
46C2×C3×D4—61no—
47C3×C22⋊C4—181no—
48C3×C22⋊C4—181no—
49C3×C42—31no—
50C3×C42C3×Dic311yescharacteristic
51#217—61no—
52#217—61no—
53C22×C32—61no—
54C22×C32—61no—
55C22×C32—121no—
56C22×C32SL(2,ℤ/4)11yescharacteristic
57C32×C4—61no—
58C3×Dic3—241no—
59C3×Dic3—241no—
60C22⋊C9—41no—
61C22⋊C9—81no—
62Dic9—161no—
63#127AGL(1,ℤ/9)11yescharacteristic
64#99—91no—
65#99—91no—
663−1+2—161noSylow
67C23×C3—31no—
68C23×C3—31no—
69C23×C3—31no—
70C23×C3—61no—
71C23×C3C3×S411yescharacteristic
72C3×D4—121no—
73C3×D4—121no—
74C3×D4—121no—
75C3×D4—121no—
76C3×D4—122no—
77C3×D4—122no—
78C2×C3×C4—31no—
79C2×C3×C4—31no—
80C2×C3×C4—61no—
81C2×C3×C4—361no—
82C2×C3×C4—361no—
83C2×Dic3—121no—
84C2×Dic3—121no—
85C2×C32—31no—
86C2×C32—41no—
87C2×C32—121no—
88C2×C9—161no—
89C2×C9—321no—
90C2×D4—31no—
91C2×D4—31no—
92C22⋊C4—181no—
93C22⋊C4—181no—
94C423−1+2⋊C411yescharacteristic, complemented
95C22×C3—31no—
96C22×C3—61no—
97C22×C3—61no—
98C22×C3—61no—
99C22×C3—61no—
100C22×C3—121no—
101C22×C3—121no—
102C22×C3—121no—
103C22×C3—121no—
104C22×C3—121no—
105C22×C3—121no—
106C22×C3C3×SL(2,ℤ/4)11yescharacteristic
107C3×C4—61no—
108C3×C4—61no—
109C3×C4—62no—
110C3×C4—721no—
111C3×C4—721no—
112Dic3—241no—
113Dic3—241no—
114C32#462611yescharacteristic
115C9—161no—
116C9—321no—
117C23—31no—
118C23((C22⋊C9)⋊C2)⋊C311yescharacteristic
119D4—121no—
120D4—121no—
121C2×C4—31no—
122C2×C4—361no—
123C2×C4—361no—
124C2×C3—31no—
125C2×C3—31no—
126C2×C3—41no—
127C2×C3—61no—
128C2×C3—121no—
129C2×C3—121no—
130C2×C3—121no—
131C2×C3—241no—
132C22—61no—
133C22—61no—
134C22—121no—
135C22#1094⋊C411yescharacteristic
136C4—61no—
137C4—721no—
138C4—721no—
139C3—31no—
140C3C3×#462611yescharacteristic
141C2—31no—
142C2—41no—
143C2—121no—
subgroupG/H
center, hypercenterC1G
commutator, hypocenterC42⋊C9C3×C4
FrattiniC23×C3C3×S4
Fitting, prime radicalC32×#127D3
socleC22×C3C3×SL(2,ℤ/4)
cosocle#127⋊C9C2×C3
LCSG ▷ C42⋊C9
UCSC1

Complements

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

NH ≅ G/Ntype
3. #10588306C3semidirect
5. #88455C4semidirect
21. C42⋊C9C3×C4semidirect
94. C423−1+2⋊C4semidirect

Maximal subgroups

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

subgroupindex
#117213532
C3×#45453
#105883063
#1094⋊C44