Finite Group Databasebeta

← #13165120

Subgroups — #13165120

Order 1728.

168 conjugacy classes of subgroups; only the 31 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 31 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#11724351C211yesmaximal, characteristic
2C2×C3⋊#3771C211yesmaximal, characteristic, complemented
3C3×#63748C211yesmaximal, characteristic, complemented
4#10593167—31nomaximal
5#89046—21no—
6C3×#3792—21no—
7C2×C3×#622C2211yescharacteristic
8#88943C411yescharacteristic
9#88942—21no—
10#88941C411yescharacteristic
11C3⋊#3771—21no—
12#64237—31no—
13C2×#3771—31no—
14#63748D311yescharacteristic, complemented
15C3×#638—41no—
16C2×C3×C3⋊Dic3D411yescharacteristic
17#6296—21no—
18#6295—21no—
19#6294—41no—
20C3×#622—41no—
21C3×#622C2×C411yescharacteristic
22C3×#622D411yescharacteristic
23#4497—91nomaximal
24#3805—61no—
25#3804—121no—
26#3792—21no—
27C2×#622C2×D311yescharacteristic
28#3776—31no—
29#3775—61no—
30#3774—31no—
31#3771—61no—
32C22×C33SD1611yescharacteristic
33C2×C3×C3⋊D3—41no—
34C3×C3⋊Dic3—41no—
35C3×C3⋊Dic3C22⋊C411yescharacteristic
36C3×C3⋊Dic3D811yescharacteristic
37#904—91no—
38C3×#108—91no—
39C2×C3⋊C16—91no—
40#638—41no—
41C2×C3⋊Dic3#6711yescharacteristic
42C3×#67—121no—
43C3×#67—121no—
44C3×#67—241no—
45C2×C3×Dic3—121no—
46#627—61no—
47#627—121no—
48#626—121no—
49#625—61no—
50#624—121no—
51D3×Dic3—121no—
52D3×Dic3—121no—
53#622—41no—
54#622C4×D311yescharacteristic
55#622D1211yescharacteristic
56#359—271noSylow
57C2×C33—21no—
58C2×C33#10211yescharacteristic
59C3×C3⋊D3—81no—
60#237—361no—
61#235—181no—
62C3×M16—181no—
63C2×C3×C8—91no—
64Dic12—91no—
65D24—91no—
66#204—181no—
67C3⋊C16—181no—
68C22×C32—41no—
69C22×C32—81no—
70C22×C32#21511yescharacteristic
71C2×C3⋊D3—41no—
72C2×C3×D3—121no—
73C2×C3×D3—121no—
74C2×C3×D3—241no—
75C3⋊Dic3—41no—
76C3⋊Dic3(C2×D3)⋊C411yescharacteristic
77C3⋊Dic3#21311yescharacteristic
78C3×Dic3—121no—
79C3×Dic3—121no—
80C3×Dic3—121no—
81C3×Dic3—121no—
82C3×Dic3—241no—
83#135—271no—
84C2×C16—271no—
85#108—91no—
86C33#35911yescharacteristic, complemented, Sylow
87C3×D4—361no—
88C2×C3×C4—91no—
89#67—121no—
90#67—121no—
91#67—241no—
92#67—361no—
93C2×Dic3—121no—
94C2×Dic3—361no—
95D12—181no—
96C4×D3—361no—
97C4×D3—361no—
98Dic6—181no—
99Dic6—361no—
100C3×C8—91no—
101C3×C8—91no—
102C3×C8—361no—
103C2×C32—21no—
104C2×C32—41no—
105C2×C32—81no—
106C2×C32—81no—
107C2×C32—161no—
108C2×C32#80911yescharacteristic
109C3⋊D3—81no—
110C3×D3—241no—
111C3×D3—241no—
112C3×D3—481no—
113#41—541no—
114Q16—271no—
115SD16—541no—
116D8—271no—
117M16—181no—
118C2×C8—91no—
119C16—541no—
120C22×C3—41no—
121C22×C3—81no—
122C22×C3—361no—
123C22×C3AΓL(1,𝔽9)11yescharacteristic
124C2×D3—121no—
125C2×D3—121no—
126C2×D3—241no—
127C3×C4—91no—
128C3×C4—91no—
129C3×C4—361no—
130Dic3—121no—
131Dic3—121no—
132Dic3—121no—
133Dic3—121no—
134Dic3—241no—
135C32—41no—
136C32—81no—
137C32#449711yescharacteristic, complemented
138Q8—541no—
139D4—361no—
140D4—541no—
141C2×C4—91no—
142C2×C4—1081no—
143C8—91no—
144C8—91no—
145C8—361no—
146C2×C3—21no—
147C2×C3—41no—
148C2×C3—81no—
149C2×C3—81no—
150C2×C3—161no—
151C2×C3—721no—
152C2×C3GO+(4,𝔽2)⋊C411yescharacteristic
153D3—241no—
154D3—241no—
155D3—481no—
156C22—361no—
157C22#8909211yescharacteristic
158C4—91no—
159C4—91no—
160C4—361no—
161C3—41no—
162C3—81no—
163C3#1059316711yescharacteristic, complemented
164C2—21no—
165C2—721no—
166C2#1172410111yescharacteristic
subgroupG/H
centerC2#11724101
commutatorC3×#622C2×C4
Frattini, hypercenterC22#89092
FittingC22×C33SD16
socleC2×C33#102
prime radical#11724351C2
cosocleC2×C3×#622C22
hypocenterC33#359
LCSG ▷ C3×#622 ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×C3⋊#3771C2semidirect
3. C3×#63748C2semidirect
14. #63748D3semidirect
86. C33#359semidirect
137. C32#4497semidirect
163. C3#10593167semidirect

Maximal subgroups

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

subgroupindex
#117243512
C2×C3⋊#37712
C3×#637482
#105931673
#44979