Finite Group Databasebeta

← C3×#10589344 #13145902

Subgroups — C3×#10589344 #13145902

Order 1728.

512 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 106 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×#64153C211yesmaximal, characteristic
2C33×#101C211yesmaximal, characteristic
3C3×#63663C211yesmaximal, characteristic
4#10589344C311yesmaximal, characteristic, direct product factor
5C3×#4478—34nomaximal
6C2×C33×Q8C2211yescharacteristic
7C2×C3×#634C411yescharacteristic
8C33×M16—21no—
9C3×#3750—21no—
10C3×#3750—21no—
11C3×C4×C3⋊Dic3C411yescharacteristic
12C3×#3748—21no—
13#64153C2×C311yescharacteristic
14C3×#944—34no—
15C32×#101—24no—
16C32×#101D314yes—
17C32×#101C2×C311yescharacteristic
18#63663C2×C311yescharacteristic
19C3×#836—34no—
20C33×Q8—41no—
21C2×C33×C4C2×C411yescharacteristic
22C2×C33×C4D411yescharacteristic
23C2×C33×C4D411yescharacteristic
24C2×C3×C3⋊Dic3—21no—
25C2×C3×C3⋊Dic3—21no—
26C3×#634—41no—
27C33×C8—41no—
28C3×C3⋊(C3⋊C8)—41no—
29C3×#356—91no—
30#4478—34no—
31C2×C32×Q8—24no—
32C2×C32×Q8C2×D314yes—
33C2×C32×Q8C22×C311yescharacteristic
34C2×#634C3×C411yescharacteristic
35C2×C3×Dic6—34no—
36C32×M16—28no—
37C32×M16—24no—
38C32×M16—21no—
39#3750—21no—
40#3750—21no—
41C4×C3⋊Dic3C3×C411yescharacteristic
42#3748—21no—
43C3×Dic3⋊C4—64no—
44C3×Dic3⋊C4—64no—
45C3×C4×Dic3—34no—
46C3×#208—64no—
47C22×C33C22⋊C411yescharacteristic
48C33×C4—21no—
49C33×C4—21no—
50C33×C4—21no—
51C3×C3⋊Dic3—22no—
52C3×C3⋊Dic3—41no—
53C3×C4⋊Q8—91no—
54#944—34no—
55C3×#101—24no—
56C3×#101—91no—
57C3×#101C3×D314yes—
58C3×#101C3⋊D311yescharacteristic
59#836—34no—
60C32×Q8—44no—
61C32×Q8—41no—
62C32×Q8—48no—
63C2×C32×C4—24no—
64C2×C32×C4—24no—
65C2×C32×C4—24no—
66C2×C32×C4C4×D314yes—
67C2×C32×C4D1214yes—
68C2×C32×C4#6714yes—
69C2×C32×C4C2×C3×C411yescharacteristic
70C2×C32×C4C3×D411yescharacteristic
71C2×C32×C4C3×D411yescharacteristic
72C2×C3⋊Dic3—21no—
73C2×C3⋊Dic3—21no—
74#634—41no—
75C2×C3×Dic3—64no—
76C2×C3×Dic3—64no—
77C3×Dic6—124no—
78C32×C8—48no—
79C32×C8—44no—
80C32×C8—41no—
81C3⋊(C3⋊C8)—41no—
82C3×C3⋊C8—124no—
83#356—91noSylow
84C2×C33—21no—
85C2×C33#9911yescharacteristic
86C2×C3×Q8—24no—
87C2×C3×Q8—91no—
88C2×C3×Q8C2×C3×D314yes—
89C2×C3×Q8C2×C3⋊D311yescharacteristic
90C2×Dic6—34no—
91C3×M16—28no—
92C3×M16—24no—
93C3×M16—21no—
94C3×M16—181no—
95C3×C4⋊C4—181no—
96C3×C4⋊C4—181no—
97C3×C42—91no—
98Dic3⋊C4—64no—
99Dic3⋊C4—64no—
100C4×Dic3—34no—
101#208—64no—
102C22×C32—24no—
103C22×C32(C2×D3)⋊C414yes—
104C22×C32C3×C22⋊C411yescharacteristic
105C32×C4—28no—
106C32×C4—24no—
107C32×C4—21no—
108C32×C4—21no—
109C32×C4—21no—
110C32×C4—24no—
111C32×C4—24no—
112C32×C4—44no—
113C32×C4—44no—
114C3⋊Dic3—22no—
115C3⋊Dic3—41no—
116C3×Dic3—68no—
117C3×Dic3—124no—
118C4⋊Q8—91no—
119#101—91no—
120#101C3×C3⋊D311yescharacteristic
121C33#35611yescharacteristic, complemented, Sylow
122C3×Q8—44no—
123C3×Q8—41no—
124C3×Q8—48no—
125C3×Q8—361no—
126C2×C3×C4—24no—
127C2×C3×C4—24no—
128C2×C3×C4—24no—
129C2×C3×C4—181no—
130C2×C3×C4—181no—
131C2×C3×C4C3×C4×D314yes—
132C2×C3×C4C3×D1214yes—
133C2×C3×C4C3×#6714yes—
134C2×C3×C4C4×C3⋊D311yescharacteristic
135C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
136C2×C3×C4#63811yescharacteristic
137C2×Dic3—64no—
138C2×Dic3—64no—
139Dic6—124no—
140C3×C8—48no—
141C3×C8—44no—
142C3×C8—41no—
143C3×C8—361no—
144C3⋊C8—124no—
145C2×C32—21no—
146C2×C32—24no—
147C2×C32—24no—
148C2×C32—44no—
149C2×C32#80614yes—
150C2×C32C3×#9911yescharacteristic
151C2×Q8—91no—
152C2×Q8C2×C3×C3⋊D311yescharacteristic
153M16—21no—
154M16—181no—
155C4⋊C4—181no—
156C4⋊C4—181no—
157C42—91no—
158C22×C3—24no—
159C22×C3C3×(C2×D3)⋊C414yes—
160C22×C3#375211yescharacteristic
161C3×C4—28no—
162C3×C4—24no—
163C3×C4—24no—
164C3×C4—24no—
165C3×C4—21no—
166C3×C4—21no—
167C3×C4—21no—
168C3×C4—44no—
169C3×C4—44no—
170C3×C4—182no—
171C3×C4—361no—
172Dic3—68no—
173Dic3—124no—
174C32—24no—
175C32#447814yescomplemented
176C32C3×#35611yescharacteristic, complemented
177Q8—41no—
178Q8—361no—
179C2×C4—181no—
180C2×C4—181no—
181C2×C4C3×C4×C3⋊D311yescharacteristic
182C2×C4C3×C4⋊(C3⋊D3)11yescharacteristic
183C2×C4C3×#63811yescharacteristic
184C8—41no—
185C8—361no—
186C2×C3—24no—
187C2×C3—21no—
188C2×C3—24no—
189C2×C3—44no—
190C2×C3C3×#80614yes—
191C2×C3#6363311yescharacteristic
192C22C3×#375211yescharacteristic
193C4—21no—
194C4—21no—
195C4—21no—
196C4—182no—
197C4—361no—
198C3—24no—
199C3C3×#447814yescomplemented
200C3#1058934411yescharacteristic, direct product factor
201C2—21no—
202C2C3×#6363311yescharacteristic
subgroupG/H
centerC2×C3#63633
commutatorC2×C32×C4C2×C3×C4
FrattiniC2×Q8C2×C3×C3⋊D3
FittingC33×#101C2
socleC2×C33#99
prime radicalC22×C33C22⋊C4
cosocleC2×C32×Q8C22×C3
hypercenterC3×#101C3⋊D3
hypocenterC32C3×#356
LCSG ▷ C2×C32×C4 ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×Q8 ◁ C3×#101

Complements

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

NH ≅ G/Ntype
4. #10589344C3direct (N × H)
121. C33#356semidirect
175. C32#4478semidirect
176. C32C3×#356semidirect
199. C3C3×#4478semidirect
200. C3#10589344direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#641532
C33×#1012
C3×#636632
#105893443
C3×#44783