Finite Group Databasebeta

← #13167455

Subgroups — #13167455

Order 1728.

384 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 71 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
1C4⋊#6296C212yesmaximal
2C3×C4×#622C211yesmaximal, characteristic
3#11722797C214yesmaximal
4#10593395—31nomaximal
5C2×#6296—22no—
6C2×C3×#622C2212yes—
7C3×C4⋊(C3⋊Dic3)—22no—
8C3×C4×C3⋊Dic3C2211yescharacteristic
9#88815C2214yes—
10#63919—32no—
11#63918—62no—
12C4×#622D311yescharacteristic
13#63743—34no—
14C2×C33×C4D411yescharacteristic
15C2×C3×C3⋊Dic3—24no—
16C2×C3×C3⋊Dic3D411yescharacteristic
17C2×C3×C3⋊Dic3C2311yescharacteristic
18#6296—44no—
19C3×#622Q814yescomplemented
20#4682—91nomaximal
21C2×#627—64no—
22C2×#627—62no—
23C2×#622C2×D312yes—
24C4⋊(C3⋊Dic3)—22no—
25C4×C3⋊Dic3C2×D311yescharacteristic
26C3×C4⋊Dic3—62no—
27C3×C4⋊Dic3—62no—
28C3×C4⋊Dic3—122no—
29C3×C4×Dic3—62no—
30#3724—34no—
31Dic3⋊Dic3—64no—
32C22×C33C2×D411yescharacteristic
33C33×C4Q1612yes—
34C3×C3⋊Dic3—44no—
35C3×C3⋊Dic3D812yes—
36C3×C3⋊Dic3C2×Q812yes—
37#888—182no—
38#869—92no—
39C3×C4×C8—91no—
40#818—94no—
41C2×C32×C4—22no—
42C2×C32×C4—42no—
43C2×C32×C4D1211yescharacteristic
44C2×C3⋊Dic3—24no—
45C2×C3⋊Dic3D1211yescharacteristic
46C2×C3⋊Dic3C22×D311yescharacteristic
47C2×C3×Dic3—62no—
48C2×C3×Dic3—62no—
49C2×C3×Dic3—64no—
50C2×C3×Dic3—64no—
51C2×C3×Dic3—124no—
52#627—128no—
53#627—124no—
54#622Dic614yescomplemented
55#500—271noSylow
56C2×C33C4⋊Q811yescharacteristic
57C2×C33C2×D811yescharacteristic
58C2×C33C2×Q1611yescharacteristic
59C2×Dic6—182no—
60C2×Dic6—184no—
61C2×C3×C8—92no—
62C3×C4⋊C4—182no—
63C3×C42—91no—
64C4⋊Dic3—62no—
65C4⋊Dic3—62no—
66C4⋊Dic3—94no—
67C4⋊Dic3—122no—
68Dic3⋊C4—184no—
69C4×Dic3—62no—
70C22×C32—22no—
71C22×C32—42no—
72C22×C32C2×D1211yescharacteristic
73C32×C4—24no—
74C32×C4—44no—
75C32×C4Dic1212yes—
76C3⋊Dic3—44no—
77C3⋊Dic3D2412yes—
78C3⋊Dic3C2×Dic612yes—
79C3×Dic3—64no—
80C3×Dic3—64no—
81C3×Dic3—124no—
82C3×Dic3—124no—
83C3×Dic3—244no—
84C4⋊Q8—272no—
85#107—274no—
86C4×C8—91no—
87C33#50011yescharacteristic, complemented, Sylow
88C2×C3×C4—22no—
89C2×C3×C4—42no—
90C2×C3×C4—91no—
91C2×C3×C4—91no—
92C2×C3×C4—184no—
93C2×C3×C4GO+(4,𝔽2)11yescharacteristic
94C2×Dic3—62no—
95C2×Dic3—62no—
96C2×Dic3—64no—
97C2×Dic3—64no—
98C2×Dic3—124no—
99Dic6—368no—
100Dic6—364no—
101C3×C8—94no—
102C2×C32—22no—
103C2×C32—22no—
104C2×C32—22no—
105C2×C32—42no—
106C2×C32—42no—
107C2×C32—42no—
108C2×C32#86911yescharacteristic
109C2×C32C2×D2411yescharacteristic
110C2×C32C2×Dic1211yescharacteristic
111C2×Q8—542no—
112C2×C8—92no—
113C4⋊C4—182no—
114C4⋊C4—274no—
115C42—91no—
116C22×C3—22no—
117C22×C3—42no—
118C22×C3C2×GO+(4,𝔽2)11yescharacteristic
119C3×C4—24no—
120C3×C4—44no—
121C3×C4—92no—
122C3×C4—92no—
123C3×C4—364no—
124C3×C4#377612yes—
125Dic3—64no—
126Dic3—64no—
127Dic3—124no—
128Dic3—124no—
129Dic3—244no—
130C32—22no—
131C32—42no—
132C32#468211yescharacteristic, complemented
133Q8—1084no—
134C2×C4—91no—
135C2×C4—91no—
136C2×C4—184no—
137C2×C4#157⋊D311yescharacteristic
138C8—94no—
139C2×C3—22no—
140C2×C3—22no—
141C2×C3—22no—
142C2×C3—42no—
143C2×C3—42no—
144C2×C3—42no—
145C2×C3#6422211yescharacteristic
146C2×C3#6422511yescharacteristic
147C2×C3C2×#377611yescharacteristic
148C22C2×#157⋊D311yescharacteristic
149C4—92no—
150C4—92no—
151C4—364no—
152C4#8894312yes—
153C3—22no—
154C3—42no—
155C3#1059339511yescharacteristic, complemented
156C2#1172433511yescharacteristic
157C2#1172433611yescharacteristic
158C2C2×#8894311yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC2×C33×C4D4
socle, prime radicalC22×C33C2×D4
hypercenterC2×C4#157⋊D3
hypocenterC33#500
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
19. C3×#622Q8semidirect
54. #622Dic6semidirect
87. C33#500semidirect
132. C32#4682semidirect
155. C3#10593395semidirect

Maximal subgroups

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

subgroupindex
C4⋊#62962
C3×C4×#6222
#117227972
#105933953
#46829