Finite Group Databasebeta

← #13167385

Subgroups — #13167385

Order 1728.

544 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C2×C3×C4×#157C211yesmaximal, characteristic
2#11723689C212yesmaximal
3#11722791C214yesmaximal
4#10593344—31nomaximal
5C22×C3×#157C2212yes—
6C2×C3×C4×C3⋊D3C2211yescharacteristic
7C2×#6294C2214yes—
8C3×C4×#157C414yescomplemented
9C3×C4⋊(C3⋊Dic3)—22no—
10#88815—22no—
11C2×C4×#157D311yescharacteristic
12#63900—32no—
13D3×C4⋊Dic3—62no—
14#63737—34no—
15C22×C3×C3⋊D3C2311yescharacteristic
16C2×C3×#157—24no—
17C2×C3×#157C2×C414yes—
18C2×C33×C4D411yescharacteristic
19C2×C3×C3⋊Dic3—24no—
20C2×C3×C3⋊Dic3D411yescharacteristic
21C3×C4×C3⋊D3C2×C412yes—
22C3×C4×C3⋊D3Q812yes—
23#6294—28no—
24#4936—91nomaximal
25C22×#157C2×D312yes—
26C2×C4×C3⋊D3C2×D311yescharacteristic
27C2×C3×C4×D3—62no—
28C2×#624—34no—
29C2×D3×Dic3—64no—
30C4×#157Dic314yescomplemented
31C4⋊(C3⋊Dic3)—22no—
32C3×C4⋊Dic3—62no—
33C3×C4⋊Dic3—62no—
34C3×C4⋊Dic3—122no—
35#3724—62no—
36Dic3⋊Dic3—64no—
37C22×C33C2×D411yescharacteristic
38C2×C3×C3⋊D3C22×C411yescharacteristic
39C2×C3×C3⋊D3C2×Q811yescharacteristic
40C2×C3×C3⋊D3#4114yes—
41C3×#157—24no—
42C33×C4C4⋊C412yes—
43C3×C3⋊Dic3—44no—
44C3×C3⋊Dic3C4⋊C412yes—
45C2×C3×C42—91no—
46C2×C4⋊Dic3—92no—
47D3×C4⋊C4—182no—
48C3⋊#95—94no—
49C22×C3⋊D3C22×D311yescharacteristic
50C22×C3×D3—62no—
51C2×#157—24no—
52C2×#157C2×Dic314yes—
53C2×C32×C4—22no—
54C2×C32×C4—42no—
55C2×C32×C4D1211yescharacteristic
56C2×C3⋊Dic3—24no—
57C2×C3⋊Dic3D1211yescharacteristic
58C4×C3⋊D3Dic612yes—
59C4×C3⋊D3C2×Dic312yes—
60C2×C3×Dic3—62no—
61C2×C3×Dic3—64no—
62C2×C3×Dic3—64no—
63C2×C3×Dic3—124no—
64C3×C4×D3—64no—
65C3×C4×D3—64no—
66#624—68no—
67D3×Dic3—128no—
68#382—271noSylow
69C2×C33C2×C4⋊C411yescharacteristic
70C2×C33#12412yes—
71C3×C3⋊D3#11712yes—
72C3×C3⋊D3#12512yes—
73C22×C3×C4—91no—
74C22×C3×C4—92no—
75C22×Dic3—94no—
76C2×C4×D3—62no—
77C2×C4×D3—184no—
78C3×C4⋊C4—182no—
79C3×C42—94no—
80C4⋊Dic3—62no—
81C4⋊Dic3—62no—
82C4⋊Dic3—122no—
83C4⋊Dic3—182no—
84Dic3⋊C4—184no—
85C22×C32—22no—
86C22×C32—42no—
87C22×C32C2×D1211yescharacteristic
88C2×C3⋊D3C2×Dic611yescharacteristic
89C2×C3⋊D3#23514yes—
90C2×C3⋊D3C22×Dic311yescharacteristic
91C2×C3×D3—62no—
92C2×C3×D3—62no—
93C2×C3×D3—64no—
94C2×C3×D3—64no—
95#157—24no—
96C32×C4—24no—
97C32×C4—44no—
98C32×C4C4⋊Dic312yes—
99C3⋊Dic3—44no—
100C3⋊Dic3C4⋊Dic312yes—
101C3×Dic3—64no—
102C3×Dic3—124no—
103C3×Dic3—124no—
104C3×Dic3—244no—
105C2×C4⋊C4—272no—
106C2×C42—91no—
107#95—274no—
108C33#38211yescharacteristic, complemented, Sylow
109C23×C3—91no—
110C22×D3—62no—
111C2×C3×C4—22no—
112C2×C3×C4—42no—
113C2×C3×C4—91no—
114C2×C3×C4—92no—
115C2×C3×C4—92no—
116C2×C3×C4—94no—
117C2×C3×C4—184no—
118C2×C3×C4—184no—
119C2×C3×C4GO+(4,𝔽2)11yescharacteristic
120C2×Dic3—62no—
121C2×Dic3—64no—
122C2×Dic3—64no—
123C2×Dic3—124no—
124C2×Dic3—188no—
125C4×D3—64no—
126C4×D3—64no—
127C4×D3—368no—
128C2×C32—22no—
129C2×C32—22no—
130C2×C32—22no—
131C2×C32—42no—
132C2×C32—42no—
133C2×C32—42no—
134C2×C32#87512yes—
135C2×C32C2×C4⋊Dic311yescharacteristic
136C3⋊D3#87012yes—
137C3⋊D3#92612yes—
138C3×D3—64no—
139C3×D3—64no—
140C22×C4—91no—
141C22×C4—92no—
142C22×C4—274no—
143C4⋊C4—182no—
144C4⋊C4—542no—
145C42—94no—
146C22×C3—22no—
147C22×C3—42no—
148C22×C3—94no—
149C22×C3—91no—
150C22×C3—91no—
151C22×C3C2×GO+(4,𝔽2)11yescharacteristic
152C2×D3—62no—
153C2×D3—62no—
154C2×D3—64no—
155C2×D3—64no—
156C3×C4—24no—
157C3×C4—44no—
158C3×C4—92no—
159C3×C4—184no—
160C3×C4—364no—
161C3×C4#377312yes—
162Dic3—64no—
163Dic3—124no—
164Dic3—124no—
165Dic3—244no—
166C32—22no—
167C32—42no—
168C32#493611yescharacteristic, complemented
169C23—91no—
170C2×C4—91no—
171C2×C4—92no—
172C2×C4—92no—
173C2×C4—94no—
174C2×C4—184no—
175C2×C4—184no—
176C2×C4—548no—
177C2×C4#157⋊D311yescharacteristic
178C2×C3—22no—
179C2×C3—22no—
180C2×C3—22no—
181C2×C3—42no—
182C2×C3—42no—
183C2×C3—42no—
184C2×C3—92no—
185C2×C3—92no—
186C2×C3#6422112yes—
187C2×C3C2×#377311yescharacteristic
188D3—64no—
189D3—64no—
190C22—94no—
191C22—91no—
192C22—91no—
193C22C2×#157⋊D311yescharacteristic
194C4—92no—
195C4—184no—
196C4—364no—
197C4#8894012yes—
198C3—22no—
199C3—42no—
200C3#1059334411yescharacteristic, complemented
201C2—92no—
202C2—92no—
203C2#1172433412yes—
204C2C2×#8894011yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutatorC2×C3×C3⋊D3C22×C4
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radical, cosocleC22×C3×C3⋊D3C23
hypercenterC2×C4#157⋊D3
hypocenterC33#382
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
8. C3×C4×#157C4semidirect
30. C4×#157Dic3semidirect
108. C33#382semidirect
168. C32#4936semidirect
200. C3#10593344semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×#1572
#117236892
#117227912
#105933443
#49369