Finite Group Databasebeta

← C3×#10590161 #13157087

Subgroups — C3×#10590161 #13157087

Order 1728.

332 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 60 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×#64096C211yesmaximal, characteristic
2C3×C4⋊#622C211yesmaximal, characteristic
3C3×C4×#622C211yesmaximal, characteristic
4#10590161C311yesmaximal, characteristic, direct product factor
5C2×C3×#634C411yescharacteristic
6C2×C3×#622—21no—
7C2×C3×#622—21no—
8C33×C4⋊C4C411yescharacteristic
9C3×#3750—21no—
10C3×C4×C3⋊Dic3—21no—
11C3×C4×C3⋊Dic3C2211yescharacteristic
12#64096C2×C311yescharacteristic
13C3×#887—62no—
14C4⋊#622C2×C311yescharacteristic
15C4×#622C2×C311yescharacteristic
16C2×C33×C4—21no—
17C2×C33×C4C2×C411yescharacteristic
18C2×C3×C3⋊Dic3—21no—
19C2×C3×C3⋊Dic3D411yescharacteristic
20C2×C3×C3⋊Dic3D411yescharacteristic
21C3×#634—21no—
22C3×#634C812yescomplemented
23C3×#622—22no—
24C3×#622—41no—
25C3×Q8⋊C8—91nomaximal
26C2×#634C3×C411yescharacteristic
27C2×C3×Dic6—62no—
28C2×#622—21no—
29C2×#622—21no—
30C32×C4⋊C4—22no—
31C32×C4⋊C4—42no—
32C32×C4⋊C4C3×C411yescharacteristic
33#3750—21no—
34C4×C3⋊Dic3—21no—
35C4×C3⋊Dic3C22×C311yescharacteristic
36C3×Dic3⋊C4—64no—
37C3×C4×Dic3—62no—
38C3×C4×Dic3—64no—
39C22×C33C22⋊C411yescharacteristic
40C33×C4—41no—
41C33×C4C2×C811yescharacteristic
42C33×C4M1611yescharacteristic
43C3×C3⋊Dic3—22no—
44C3×C3⋊Dic3—21no—
45C3×C3⋊Dic3SD1611yescharacteristic
46C3×C3⋊Dic3Q1611yescharacteristic
47C3×C4×Q8—91no—
48#887—62no—
49C3×C4⋊C8—91no—
50C3×C4×C8—91no—
51C2×C32×C4—21no—
52C2×C32×C4—24no—
53C2×C32×C4—22no—
54C2×C32×C4—44no—
55C2×C32×C4—42no—
56C2×C32×C4C2×C3×C411yescharacteristic
57C2×C3⋊Dic3—21no—
58C2×C3⋊Dic3C3×D411yescharacteristic
59C2×C3⋊Dic3C3×D411yescharacteristic
60#634—21no—
61#634C3×C812yescomplemented
62C2×C3×Dic3—62no—
63C2×C3×Dic3—62no—
64C2×C3×Dic3—64no—
65C3×Dic6—64no—
66C3×Dic6—64no—
67#622—22no—
68#622—41no—
69Q8⋊C8—91noSylow
70C2×C33C22⋊C811yescharacteristic
71C2×C33#10311yescharacteristic
72C2×C33#10411yescharacteristic
73C2×C3×Q8—91no—
74C2×Dic6—62no—
75C2×C3×C8—181no—
76C2×C3×C8—181no—
77C3×C4⋊C4—22no—
78C3×C4⋊C4—42no—
79C3×C4⋊C4—181no—
80C3×C4⋊C4#15711yescharacteristic
81C3×C42—91no—
82C3×C42—181no—
83Dic3⋊C4—64no—
84C4×Dic3—62no—
85C4×Dic3—64no—
86C22×C32—22no—
87C22×C32—42no—
88C22×C32C3×C22⋊C411yescharacteristic
89C32×C4—22no—
90C32×C4—22no—
91C32×C4—42no—
92C32×C4—42no—
93C32×C4—41no—
94C32×C4—44no—
95C32×C4—84no—
96C32×C4C2×C3×C811yescharacteristic
97C32×C4C3×M1611yescharacteristic
98C3⋊Dic3—22no—
99C3⋊Dic3—21no—
100C3⋊Dic3C3×SD1611yescharacteristic
101C3⋊Dic3C3×Q1611yescharacteristic
102C3×Dic3—62no—
103C3×Dic3—62no—
104C3×Dic3—68no—
105C3×Dic3—122no—
106C4×Q8—91no—
107C4⋊C8—91no—
108C4×C8—91no—
109C33Q8⋊C811yescharacteristic, complemented, Sylow
110C3×Q8—92no—
111C3×Q8—181no—
112C2×C3×C4—24no—
113C2×C3×C4—21no—
114C2×C3×C4—22no—
115C2×C3×C4—44no—
116C2×C3×C4—42no—
117C2×C3×C4—91no—
118C2×C3×C4—91no—
119C2×C3×C4—181no—
120C2×C3×C4C2×#15711yescharacteristic
121C2×Dic3—62no—
122C2×Dic3—62no—
123C2×Dic3—64no—
124Dic6—64no—
125Dic6—64no—
126C3×C8—182no—
127C3×C8—361no—
128C2×C32—22no—
129C2×C32—22no—
130C2×C32—22no—
131C2×C32—42no—
132C2×C32—42no—
133C2×C32—42no—
134C2×C32C3×C22⋊C811yescharacteristic
135C2×C32C3×#10311yescharacteristic
136C2×C32C3×#10411yescharacteristic
137C2×Q8—91no—
138C2×C8—181no—
139C2×C8—181no—
140C4⋊C4—181no—
141C4⋊C4C3×#15711yescharacteristic
142C42—91no—
143C42—181no—
144C22×C3—22no—
145C22×C3—42no—
146C22×C3C22⋊#15711yescharacteristic
147C3×C4—22no—
148C3×C4—22no—
149C3×C4—44no—
150C3×C4—41no—
151C3×C4—42no—
152C3×C4—42no—
153C3×C4—84no—
154C3×C4—91no—
155C3×C4—91no—
156C3×C4—182no—
157C3×C4—181no—
158C3×C4#378711yescharacteristic
159C3×C4#378811yescharacteristic
160Dic3—62no—
161Dic3—62no—
162Dic3—68no—
163Dic3—122no—
164C32—22no—
165C32—42no—
166C32C3×Q8⋊C811yescharacteristic, complemented
167Q8—92no—
168Q8—181no—
169C2×C4—21no—
170C2×C4—91no—
171C2×C4—91no—
172C2×C4—181no—
173C2×C4C2×C3×#15711yescharacteristic
174C8—182no—
175C8—361no—
176C2×C3—22no—
177C2×C3—22no—
178C2×C3—22no—
179C2×C3—42no—
180C2×C3—42no—
181C2×C3—42no—
182C2×C3#6377811yescharacteristic
183C2×C3#6378311yescharacteristic
184C2×C3#6378411yescharacteristic
185C22C3×C22⋊#15711yescharacteristic
186C4—41no—
187C4—91no—
188C4—91no—
189C4—182no—
190C4—181no—
191C4C3×#378711yescharacteristic
192C4C3×#378811yescharacteristic
193C3—22no—
194C3—42no—
195C3#1059016111yescharacteristic, direct product factor
196C2C3×#6377811yescharacteristic
197C2C3×#6378311yescharacteristic
198C2C3×#6378411yescharacteristic
subgroupG/H
centerC22×C3C22⋊#157
commutatorC32×C4C2×C3×C8
FrattiniC2×C4C2×C3×#157
FittingC33×C4⋊C4C4
socle, prime radicalC22×C33C22⋊C4
cosocleC4×C3⋊Dic3C22×C3
hypercenterC3×C4⋊C4#157
hypocenterC32C3×Q8⋊C8
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. #10590161C3direct (N × H)
22. C3×#634C8semidirect
61. #634C3×C8semidirect
109. C33Q8⋊C8semidirect
166. C32C3×Q8⋊C8semidirect
195. C3#10590161direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#640962
C3×C4⋊#6222
C3×C4×#6222
#105901613
C3×Q8⋊C89