Finite Group Databasebeta

← #13157229

Subgroups — #13157229

Order 1728.

388 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 33 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×#64145C211yesmaximal, characteristic
2#11723105C212yesmaximal, complemented
3#10590212—31nomaximal
4C2×C33×D4C411yescharacteristic
5C2×C3×#634C2211yescharacteristic
6#88993—22no—
7C3×#3757—22no—
8C3×C4×C3⋊Dic3C411yescharacteristic
9#64145D311yescharacteristic
10C3×#936—62no—
11#63780—32no—
12C23×C33—21no—
13C33×D4—41no—
14C2×C33×C4C2×C411yescharacteristic
15C2×C3×C3⋊Dic3—21no—
16C2×C3×C3⋊Dic3D412yes—
17C3×#634—41no—
18C3⋊#622—42no—
19#4542—91nomaximal
20C2×C32×D4—22no—
21C2×C32×D4—42no—
22C2×C32×D4Dic311yescharacteristic
23C2×#634C2×D311yescharacteristic
24C2×C3×Dic6—62no—
25#3792—62no—
26#3757—22no—
27C4×C3⋊Dic3Dic311yescharacteristic
28C3×#217—68no—
29C3×C4×Dic3—62no—
30C22×C33—42no—
31C22×C33—41no—
32C22×C33C22⋊C411yescharacteristic
33C33×C4—21no—
34C3×C3⋊Dic3—22no—
35C3×C3⋊Dic3—41no—
36C3×#124—91no—
37#936—62no—
38#836—92no—
39C23×C32—21no—
40C23×C32—24no—
41C23×C32—44no—
42C32×D4—41no—
43C32×D4—44no—
44C32×D4—48no—
45C2×C32×C4—22no—
46C2×C32×C4—42no—
47C2×C32×C4C2×Dic311yescharacteristic
48C2×C3⋊Dic3—21no—
49C2×C3⋊Dic3#6712yes—
50#634—41no—
51C2×C3×Dic3—64no—
52C2×C3×Dic3—64no—
53C3×Dic6—124no—
54#622—122no—
55#355—271noSylow
56C2×C33—21no—
57C2×C33—81no—
58C2×C33#9911yescharacteristic
59C2×C3×Q8—91no—
60C2×C3×D4—22no—
61C2×C3×D4—42no—
62C2×C3×D4#15711yescharacteristic
63C2×Dic6—62no—
64C3×C22⋊C4—182no—
65C3×C42—91no—
66#217—68no—
67C4×Dic3—62no—
68#208—182no—
69C22×C32—22no—
70C22×C32—42no—
71C22×C32—44no—
72C22×C32—42no—
73C22×C32—48no—
74C22×C32—41no—
75C22×C32—48no—
76C22×C32—88no—
77C22×C32#21711yescharacteristic
78C32×C4—21no—
79C32×C4—24no—
80C32×C4—44no—
81C3⋊Dic3—22no—
82C3⋊Dic3—41no—
83C3×Dic3—124no—
84C3×Dic3—124no—
85#124—91no—
86#101—272no—
87C33#35511yescharacteristic, complemented, Sylow
88C23×C3—24no—
89C23×C3—21no—
90C23×C3—44no—
91C3×Q8—361no—
92C3×D4—41no—
93C3×D4—44no—
94C3×D4—48no—
95C2×C3×C4—22no—
96C2×C3×C4—42no—
97C2×C3×C4—92no—
98C2×C3×C4—181no—
99C2×C3×C4C2×#15711yescharacteristic
100C2×Dic3—64no—
101C2×Dic3—64no—
102Dic6—124no—
103C3⋊C8—362no—
104C2×C32—21no—
105C2×C32—24no—
106C2×C32—22no—
107C2×C32—44no—
108C2×C32—42no—
109C2×C32—84no—
110C2×C32—88no—
111C2×C32—81no—
112C2×C32#83411yescharacteristic
113C2×Q8—91no—
114C2×D4C3⋊#15711yescharacteristic
115M16—542no—
116C22⋊C4—182no—
117C42—91no—
118C22×C3—22no—
119C22×C3—44no—
120C22×C3—42no—
121C22×C3—48no—
122C22×C3—48no—
123C22×C3—41no—
124C22×C3—42no—
125C22×C3—88no—
126C22×C3C22⋊#15711yescharacteristic
127C3×C4—24no—
128C3×C4—21no—
129C3×C4—44no—
130C3×C4—182no—
131C3×C4—361no—
132Dic3—124no—
133Dic3—124no—
134C32—22no—
135C32—42no—
136C32#454211yescharacteristic, complemented
137C23—21no—
138Q8—361no—
139D4—41no—
140C2×C4—92no—
141C2×C4—181no—
142C2×C4C2×C3⋊#15711yescharacteristic
143C8—1082no—
144C2×C3—24no—
145C2×C3—21no—
146C2×C3—22no—
147C2×C3—44no—
148C2×C3—42no—
149C2×C3—84no—
150C2×C3—88no—
151C2×C3—81no—
152C2×C3#6378611yescharacteristic
153C22—42no—
154C22—41no—
155C22#8899411yescharacteristic
156C4—21no—
157C4—182no—
158C4—361no—
159C3—22no—
160C3—42no—
161C3#1059021211yescharacteristic, complemented
162C2—21no—
163C2—81no—
164C2#1172311111yescharacteristic
subgroupG/H
centerC2#11723111
commutatorC2×C33×C4C2×C4
FrattiniC2×C4C2×C3⋊#157
Fitting, prime radicalC2×C33×D4C4
socleC2×C33#99
cosocleC2×C3×#634C22
hypercenterC2×D4C3⋊#157
hypocenterC33#355
LCSG ▷ C2×C33×C4 ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11723105C2semidirect
87. C33#355semidirect
136. C32#4542semidirect
161. C3#10590212semidirect

Maximal subgroups

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

subgroupindex
C3×#641452
#117231052
#105902123
#45429