Finite Group Databasebeta

← #13167454

Subgroups — #13167454

Order 1728.

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

normal subgroups only

This group has 496 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×#3787C211yesmaximal, characteristic
2#11723689C212yesmaximal
3#11722797C214yesmaximal, complemented
4#10593396—31nomaximal
5C2×C3×C4×C3⋊D3C2211yescharacteristic
6C2×#6294—22no—
7C2×C3×#622C2212yes—
8C3×#3787C414yescomplemented
9C3×C4⋊(C3⋊Dic3)—22no—
10#88815C2214yes—
11C2×#3787D311yescharacteristic
12#63900—32no—
13D3×C4⋊Dic3—62no—
14#63743—34no—
15C22×C3×C3⋊D3D411yescharacteristic
16C2×C33×C4D411yescharacteristic
17C2×C3×C3⋊Dic3—24no—
18C2×C3×C3⋊Dic3C2311yescharacteristic
19C3×C4×C3⋊D3C2×C412yes—
20C3×C4×C3⋊D3Q812yes—
21#6294—44no—
22C3×#622C2×C414yescomplemented
23C2×#818—91nomaximal
24C2×C4×C3⋊D3C2×D311yescharacteristic
25C2×C3×C4×D3—62no—
26C2×#624—62no—
27C2×D3×Dic3—64no—
28C2×#622C2×D312yes—
29#3787Dic314yescomplemented
30C4⋊(C3⋊Dic3)—22no—
31C3×C4⋊Dic3—62no—
32C3×C4⋊Dic3—62no—
33C3×C4⋊Dic3—122no—
34#3724—34no—
35Dic3⋊Dic3—64no—
36C22×C33C2×D411yescharacteristic
37C2×C3×C3⋊D3C4⋊C412yes—
38C2×C3×C3⋊D3D812yes—
39C2×C3×C3⋊D3Q1612yes—
40C33×C4C4⋊C412yes—
41C3×C3⋊Dic3—44no—
42C3×C3⋊Dic3C22×C411yescharacteristic
43C3×C3⋊Dic3C2×Q811yescharacteristic
44C22×C3×C8—91no—
45C2×C4⋊Dic3—92no—
46D3×C4⋊C4—182no—
47#818—94no—
48C22×C3⋊D3D1211yescharacteristic
49C22×C3×D3—62no—
50C2×C32×C4—22no—
51C2×C32×C4—42no—
52C2×C32×C4D1211yescharacteristic
53C2×C3⋊Dic3—24no—
54C2×C3⋊Dic3C22×D311yescharacteristic
55C4×C3⋊D3Dic612yes—
56C4×C3⋊D3C2×Dic312yes—
57C2×C3×Dic3—62no—
58C2×C3×Dic3—64no—
59C2×C3×Dic3—64no—
60C2×C3×Dic3—124no—
61C3×C4×D3—64no—
62C3×C4×D3—64no—
63#624—124no—
64D3×Dic3—128no—
65#622C2×Dic314yescomplemented
66C2×#107—271noSylow
67C2×C33C2×C4⋊C411yescharacteristic
68C2×C33C2×D811yescharacteristic
69C2×C33C2×Q1611yescharacteristic
70C3×C3⋊D3#10714yescomplemented
71C22×C3×C4—91no—
72C22×Dic3—182no—
73C2×C4×D3—62no—
74C2×C4×D3—184no—
75C2×C3×C8—92no—
76C2×C3×C8—94no—
77C3×C4⋊C4—182no—
78C4⋊Dic3—62no—
79C4⋊Dic3—62no—
80C4⋊Dic3—94no—
81C4⋊Dic3—122no—
82Dic3⋊C4—184no—
83C22×C32—22no—
84C22×C32—42no—
85C22×C32C2×D1211yescharacteristic
86C2×C3⋊D3D2412yes—
87C2×C3⋊D3Dic1212yes—
88C2×C3⋊D3C4⋊Dic312yes—
89C2×C3×D3—64no—
90C2×C3×D3—64no—
91C2×C3×D3—64no—
92C32×C4—24no—
93C32×C4—44no—
94C32×C4C4⋊Dic312yes—
95C3⋊Dic3—44no—
96C3⋊Dic3C2×Dic611yescharacteristic
97C3⋊Dic3C22×Dic311yescharacteristic
98C3×Dic3—62no—
99C3×Dic3—62no—
100C3×Dic3—124no—
101C3×Dic3—124no—
102C3×Dic3—244no—
103C22×C8—91no—
104C2×C4⋊C4—272no—
105#107—274no—
106C33C2×#10711yescharacteristic, complemented, Sylow
107C23×C3—91no—
108C22×D3—62no—
109C2×C3×C4—22no—
110C2×C3×C4—42no—
111C2×C3×C4—91no—
112C2×C3×C4—92no—
113C2×C3×C4—92no—
114C2×C3×C4—184no—
115C2×C3×C4GO+(4,𝔽2)11yescharacteristic
116C2×Dic3—62no—
117C2×Dic3—64no—
118C2×Dic3—64no—
119C2×Dic3—124no—
120C2×Dic3—364no—
121C4×D3—64no—
122C4×D3—64no—
123C4×D3—368no—
124C3×C8—94no—
125C2×C32—22no—
126C2×C32—22no—
127C2×C32—22no—
128C2×C32—42no—
129C2×C32—42no—
130C2×C32—42no—
131C2×C32C2×D2411yescharacteristic
132C2×C32C2×Dic1211yescharacteristic
133C2×C32C2×C4⋊Dic311yescharacteristic
134C3⋊D3#81814yescomplemented
135C3×D3—68no—
136C22×C4—91no—
137C22×C4—542no—
138C2×C8—92no—
139C2×C8—94no—
140C4⋊C4—182no—
141C4⋊C4—274no—
142C22×C3—22no—
143C22×C3—42no—
144C22×C3—92no—
145C22×C3—92no—
146C22×C3—92no—
147C22×C3C2×GO+(4,𝔽2)11yescharacteristic
148C2×D3—64no—
149C2×D3—64no—
150C2×D3—64no—
151C3×C4—24no—
152C3×C4—44no—
153C3×C4—91no—
154C3×C4—91no—
155C3×C4—364no—
156C3×C4#377312yes—
157Dic3—62no—
158Dic3—62no—
159Dic3—124no—
160Dic3—124no—
161Dic3—244no—
162C32—22no—
163C32—42no—
164C32C2×#81811yescharacteristic, complemented
165C23—91no—
166C2×C4—91no—
167C2×C4—92no—
168C2×C4—92no—
169C2×C4—184no—
170C2×C4—1084no—
171C2×C4#157⋊D311yescharacteristic
172C8—94no—
173C2×C3—22no—
174C2×C3—22no—
175C2×C3—22no—
176C2×C3—42no—
177C2×C3—42no—
178C2×C3—42no—
179C2×C3—94no—
180C2×C3#6422511yescharacteristic
181C2×C3#6422811yescharacteristic
182C2×C3C2×#377311yescharacteristic
183D3—68no—
184C22—92no—
185C22—92no—
186C22—92no—
187C22C2×#157⋊D311yescharacteristic
188C4—91no—
189C4—91no—
190C4—364no—
191C4#8894012yes—
192C3—22no—
193C3—42no—
194C3#1059339611yescharacteristic, complemented
195C2—94no—
196C2#1172433611yescharacteristic
197C2#1172434111yescharacteristic
198C2C2×#8894011yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutatorC3×C3⋊Dic3C22×C4
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4#157⋊D3
hypocenterC33C2×#107
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. #11722797C2semidirect
8. C3×#3787C4semidirect
22. C3×#622C2×C4semidirect
29. #3787Dic3semidirect
65. #622C2×Dic3semidirect
70. C3×C3⋊D3#107semidirect
106. C33C2×#107semidirect
134. C3⋊D3#818semidirect
164. C32C2×#818semidirect
194. C3#10593396semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37872
#117236892
#117227972
#105933963
C2×#8189