Finite Group Databasebeta

← #13173333

Subgroups — #13173333

Order 1728.

584 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 109 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×C4⋊(C3⋊#157)C212yesmaximal
2C3×C42×C3⋊D3C211yesmaximal, characteristic
3#11723106C214yesmaximal
4#10593736—31nomaximal
5C22×C3⋊#157C2214yes—
6C2×C3×C4×C3⋊D3C2212yes—
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C4⋊(C3⋊#157)—24no—
9C33×C42C411yescharacteristic, complemented
10C3×C4×C3⋊Dic3C411yescharacteristic, complemented
11C3×C4×C3⋊Dic3C412yescomplemented
12C2×C4⋊#157—32no—
13C42×C3⋊D3D311yescharacteristic
14C3×C42×D3—62no—
15#63781—34no—
16C22×C3×C3⋊D3C2311yescharacteristic
17C2×C3⋊#157—24no—
18C2×C3⋊#157—28no—
19C2×C33×C4C2×C412yes—
20C2×C33×C4C2×C411yescharacteristic
21C2×C3×C3⋊Dic3C2×C412yes—
22C2×C3×C3⋊Dic3C2×C411yescharacteristic
23C3×C4×C3⋊D3—24no—
24C3×C4×C3⋊D3D412yes—
25C3×C4×C3⋊D3Q812yes—
26#4936—91nomaximal
27C22×#157—34no—
28C2×C4×C3⋊D3C2×D312yes—
29C2×C4×C3⋊D3C2×D311yescharacteristic
30C2×C3×C4×D3—62no—
31C2×C3×C4×D3—64no—
32C4⋊#157—64no—
33C32×C42—22no—
34C32×C42—42no—
35C32×C42Dic311yescharacteristic, complemented
36C4×C3⋊Dic3Dic311yescharacteristic, complemented
37C4×C3⋊Dic3Dic312yescomplemented
38C3×C4×Dic3—64no—
39C3×C4×Dic3—62no—
40C22×C33C22×C411yescharacteristic
41C2×C3×C3⋊D3C2×D411yescharacteristic
42C2×C3×C3⋊D3C2×Q811yescharacteristic
43C2×C3×C3⋊D3#4114yes—
44C3⋊#157—44no—
45C33×C4—22no—
46C33×C4C4⋊C412yes—
47C3×C3⋊Dic3—22no—
48C3×C3⋊Dic3C4⋊C412yes—
49C2×C3×C42—91no—
50C2×C4⋊Dic3—92no—
51C42×D3—62no—
52C3⋊#95—94no—
53C22×C3⋊D3C22×D311yescharacteristic
54C22×C3×D3—62no—
55C2×#157—64no—
56C2×#157—68no—
57C2×C32×C4—24no—
58C2×C32×C4—22no—
59C2×C32×C4—44no—
60C2×C32×C4—42no—
61C2×C32×C4C2×Dic312yes—
62C2×C32×C4C2×Dic311yescharacteristic
63C2×C3⋊Dic3C2×Dic312yes—
64C2×C3⋊Dic3C2×Dic311yescharacteristic
65C4×C3⋊D3—24no—
66C4×C3⋊D3Dic612yes—
67C4×C3⋊D3D1212yes—
68C2×C3×Dic3—62no—
69C2×C3×Dic3—64no—
70C3×C4×D3—64no—
71C3×C4×D3—64no—
72C3×C4×D3—616no—
73#382—271noSylow
74C2×C33C2×C4⋊C411yescharacteristic
75C2×C33#11712yes—
76C3×C3⋊D3#12412yes—
77C3×C3⋊D3#12512yes—
78C22×C3×C4—91no—
79C22×C3×C4—92no—
80C22×Dic3—94no—
81C2×C4×D3—62no—
82C2×C4×D3—64no—
83C3×C42—22no—
84C3×C42—42no—
85C3×C42—92no—
86C3×C42—91no—
87C3×C42#15711yescharacteristic, complemented
88C4⋊Dic3—184no—
89C4×Dic3—64no—
90C4×Dic3—62no—
91C22×C32—22no—
92C22×C32—42no—
93C22×C32C22×Dic311yescharacteristic
94C2×C3⋊D3C2×Dic611yescharacteristic
95C2×C3⋊D3C2×D1211yescharacteristic
96C2×C3⋊D3#23514yes—
97C2×C3×D3—62no—
98C2×C3×D3—62no—
99C2×C3×D3—68no—
100#157—124no—
101C32×C4—22no—
102C32×C4—28no—
103C32×C4—24no—
104C32×C4—48no—
105C32×C4—44no—
106C32×C4C4⋊Dic312yes—
107C3⋊Dic3—22no—
108C3⋊Dic3C4⋊Dic312yes—
109C3×Dic3—64no—
110C3×Dic3—68no—
111C2×C4⋊C4—272no—
112C2×C42—91no—
113#95—274no—
114C33#38211yescharacteristic, complemented, Sylow
115C23×C3—91no—
116C22×D3—62no—
117C2×C3×C4—24no—
118C2×C3×C4—22no—
119C2×C3×C4—44no—
120C2×C3×C4—42no—
121C2×C3×C4—91no—
122C2×C3×C4—92no—
123C2×C3×C4—92no—
124C2×C3×C4—92no—
125C2×C3×C4—184no—
126C2×C3×C4C2×#15712yes—
127C2×C3×C4C2×#15711yescharacteristic
128C2×Dic3—62no—
129C2×Dic3—64no—
130C2×Dic3—184no—
131C2×Dic3—188no—
132C4×D3—64no—
133C4×D3—64no—
134C4×D3—616no—
135C2×C32—22no—
136C2×C32—24no—
137C2×C32—42no—
138C2×C32—44no—
139C2×C32C2×C4⋊Dic311yescharacteristic
140C2×C32#92612yes—
141C3⋊D3#87012yes—
142C3⋊D3#87512yes—
143C3×D3—64no—
144C3×D3—64no—
145C22×C4—91no—
146C22×C4—92no—
147C22×C4—274no—
148C4⋊C4—544no—
149C42—92no—
150C42—91no—
151C42C3⋊#15711yescharacteristic, complemented
152C22×C3—22no—
153C22×C3—42no—
154C22×C3—91no—
155C22×C3—91no—
156C22×C3—94no—
157C22×C3C22×#15711yescharacteristic
158C2×D3—62no—
159C2×D3—62no—
160C2×D3—68no—
161C3×C4—28no—
162C3×C4—22no—
163C3×C4—24no—
164C3×C4—48no—
165C3×C4—44no—
166C3×C4—92no—
167C3×C4—182no—
168C3×C4C4⋊#15712yes—
169Dic3—64no—
170Dic3—68no—
171Dic3—364no—
172C32—22no—
173C32—42no—
174C32#493611yescharacteristic, complemented
175C23—91no—
176C2×C4—91no—
177C2×C4—92no—
178C2×C4—92no—
179C2×C4—92no—
180C2×C4—184no—
181C2×C4—544no—
182C2×C4—548no—
183C2×C4C2×C3⋊#15712yes—
184C2×C4C2×C3⋊#15711yescharacteristic
185C2×C3—22no—
186C2×C3—24no—
187C2×C3—42no—
188C2×C3—44no—
189C2×C3—92no—
190C2×C3—92no—
191C2×C3C2×C4⋊#15711yescharacteristic
192C2×C3#6428612yes—
193D3—64no—
194D3—64no—
195C22—91no—
196C22—91no—
197C22—94no—
198C22C22×C3⋊#15711yescharacteristic
199C4—22no—
200C4—92no—
201C4—182no—
202C4—1084no—
203C4C4⋊(C3⋊#157)12yes—
204C3—22no—
205C3—42no—
206C3#1059373611yescharacteristic, complemented
207C2—92no—
208C2—92no—
209C2C2×C4⋊(C3⋊#157)11yescharacteristic
210C2#1172451412yes—
subgroupG/H
center, FrattiniC22C22×C3⋊#157
commutator, socleC22×C33C22×C4
FittingC33×C42C4
prime radical, cosocleC22×C3×C3⋊D3C23
hypercenterC42C3⋊#157
hypocenterC33#382
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
9. C33×C42C4semidirect
10. C3×C4×C3⋊Dic3C4semidirect
11. C3×C4×C3⋊Dic3C4semidirect
35. C32×C42Dic3semidirect
36. C4×C3⋊Dic3Dic3semidirect
37. C4×C3⋊Dic3Dic3semidirect
87. C3×C42#157semidirect
114. C33#382semidirect
151. C42C3⋊#157semidirect
174. C32#4936semidirect
206. C3#10593736semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊(C3⋊#157)2
C3×C42×C3⋊D32
#117231062
#105937363
#49369