Finite Group Databasebeta

← C3×#10592763 #13163101

Subgroups — C3×#10592763 #13163101

Order 1728.

764 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 198 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
1C33×#125C211yesmaximal, characteristic
2C3×#64078C211yesmaximal, characteristic
3C3×#63635C212yesmaximal
4C3×#63634C212yesmaximal
5C3×#63630C211yesmaximal, characteristic
6#10592763C311yesmaximal, characteristic, direct product factor
7C3×#5064—34nomaximal
8C2×C3×#634—21no—
9C33×C4⋊C4—21no—
10C33×C4⋊C4—21no—
11C33×C4⋊C4C2211yescharacteristic
12C33×C4⋊C4C2211yescharacteristic
13C33×C42C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)—21no—
15C3×C4⋊(C3⋊Dic3)C2212yes—
16C2×C3×C3⋊(C3⋊C8)C2212yes—
17C32×#125—24no—
18C32×#125D314yes—
19C32×#125C2×C311yescharacteristic
20#64078C2×C311yescharacteristic
21C3×#869—34no—
22#63635C2×C312yes—
23#63634C2×C312yes—
24#63630C2×C311yescharacteristic
25C3×#808—38no—
26C3×#807—38no—
27C3×#803—34no—
28C2×C33×C4—21no—
29C2×C33×C4—21no—
30C2×C33×C4D411yescharacteristic
31C2×C33×C4D411yescharacteristic
32C2×C33×C4C2311yescharacteristic
33C2×C3×C3⋊Dic3—22no—
34C3×#634—42no—
35C3×C3⋊(C3⋊C8)Q814yescomplemented
36C3×#501—91no—
37#5064—34no—
38C2×#634—21no—
39C2×C3×Dic6—64no—
40C32×C4⋊C4—28no—
41C32×C4⋊C4—28no—
42C32×C4⋊C4—24no—
43C32×C4⋊C4—24no—
44C32×C4⋊C4—24no—
45C32×C4⋊C4—24no—
46C32×C4⋊C4—21no—
47C32×C4⋊C4—21no—
48C32×C4⋊C4C2×D314yes—
49C32×C4⋊C4C2×D314yes—
50C32×C4⋊C4C22×C311yescharacteristic
51C32×C4⋊C4C22×C311yescharacteristic
52C32×C42—24no—
53C32×C42C2×D314yes—
54C32×C42C22×C311yescharacteristic
55C4⋊(C3⋊Dic3)—21no—
56C4⋊(C3⋊Dic3)C22×C312yes—
57C2×C3⋊(C3⋊C8)C22×C312yes—
58C3×C4⋊Dic3—38no—
59C3×C4⋊Dic3—64no—
60C2×C3×C3⋊C8—38no—
61C22×C33C2×D411yescharacteristic
62C33×C4—21no—
63C33×C4—21no—
64C33×C4—41no—
65C33×C4—41no—
66C33×C4C2×Q812yes—
67C3×C3⋊Dic3—42no—
68C3×C4⋊Q8—91no—
69C3×#125—24no—
70C3×#125C3×D314yes—
71C3×#125C3⋊D311yescharacteristic
72#869—34no—
73C3×#107—92no—
74C3×#106—92no—
75C3×#97—91no—
76#808—38no—
77#807—38no—
78#803—34no—
79C2×C32×C4—28no—
80C2×C32×C4—28no—
81C2×C32×C4—24no—
82C2×C32×C4—24no—
83C2×C32×C4—21no—
84C2×C32×C4—21no—
85C2×C32×C4—24no—
86C2×C32×C4—24no—
87C2×C32×C4—24no—
88C2×C32×C4#6714yes—
89C2×C32×C4#6714yes—
90C2×C32×C4C3×D411yescharacteristic
91C2×C32×C4C3×D411yescharacteristic
92C2×C32×C4C22×D314yes—
93C2×C32×C4C23×C311yescharacteristic
94C2×C3⋊Dic3—22no—
95#634—42no—
96C2×C3×Dic3—68no—
97C3×Dic6—128no—
98C3⋊(C3⋊C8)C3×Q814yescomplemented
99C3×C3⋊C8—316no—
100#501—91noSylow
101C2×C33C4⋊Q811yescharacteristic
102C2×C33AGL(1,ℤ/8)11yescharacteristic
103C2×C33#13711yescharacteristic
104C2×C3×Q8—181no—
105C2×Dic6—64no—
106C2×C3×C8—92no—
107C3×C4⋊C4—28no—
108C3×C4⋊C4—28no—
109C3×C4⋊C4—24no—
110C3×C4⋊C4—24no—
111C3×C4⋊C4—21no—
112C3×C4⋊C4—21no—
113C3×C4⋊C4—24no—
114C3×C4⋊C4—24no—
115C3×C4⋊C4—92no—
116C3×C4⋊C4—181no—
117C3×C4⋊C4C2×C3×D314yes—
118C3×C4⋊C4C2×C3×D314yes—
119C3×C4⋊C4C2×C3⋊D311yescharacteristic
120C3×C4⋊C4C2×C3⋊D311yescharacteristic
121C3×C42—24no—
122C3×C42C2×C3×D314yes—
123C3×C42C2×C3⋊D311yescharacteristic
124C4⋊Dic3—38no—
125C4⋊Dic3—64no—
126C2×C3⋊C8—38no—
127C22×C32—24no—
128C22×C32C2×#6714yes—
129C22×C32C2×C3×D411yescharacteristic
130C32×C4—21no—
131C32×C4—21no—
132C32×C4—28no—
133C32×C4—24no—
134C32×C4—24no—
135C32×C4—44no—
136C32×C4—44no—
137C32×C4—48no—
138C32×C4—48no—
139C32×C4—44no—
140C32×C4—44no—
141C32×C4—41no—
142C32×C4—41no—
143C32×C4D3×Q818yes—
144C32×C4C2×C3×Q812yes—
145C3⋊Dic3—42no—
146C3×Dic3—128no—
147C4⋊Q8—91no—
148#125C3×C3⋊D311yescharacteristic
149#107—92no—
150#106—92no—
151#97—91no—
152C33#50111yescharacteristic, complemented, Sylow
153C3×Q8—362no—
154C2×C3×C4—28no—
155C2×C3×C4—28no—
156C2×C3×C4—24no—
157C2×C3×C4—24no—
158C2×C3×C4—21no—
159C2×C3×C4—21no—
160C2×C3×C4—24no—
161C2×C3×C4—24no—
162C2×C3×C4—24no—
163C2×C3×C4—182no—
164C2×C3×C4C3×#6714yes—
165C2×C3×C4C3×#6714yes—
166C2×C3×C4#63811yescharacteristic
167C2×C3×C4#63811yescharacteristic
168C2×C3×C4C22×C3×D314yes—
169C2×C3×C4C22×C3⋊D311yescharacteristic
170C2×Dic3—68no—
171Dic6—128no—
172C3×C8—94no—
173C3⋊C8—316no—
174C2×C32—24no—
175C2×C32—24no—
176C2×C32—24no—
177C2×C32#94414yes—
178C2×C32#94914yes—
179C2×C32#95114yes—
180C2×C32C3×C4⋊Q811yescharacteristic
181C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
182C2×C32C3×#13711yescharacteristic
183C2×Q8—181no—
184C2×C8—92no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—92no—
188C4⋊C4—181no—
189C4⋊C4C2×C3×C3⋊D311yescharacteristic
190C4⋊C4C2×C3×C3⋊D311yescharacteristic
191C42C2×C3×C3⋊D311yescharacteristic
192C22×C3—24no—
193C22×C3C2×C3×#6714yes—
194C22×C3C2×#63811yescharacteristic
195C3×C4—24no—
196C3×C4—24no—
197C3×C4—21no—
198C3×C4—21no—
199C3×C4—28no—
200C3×C4—44no—
201C3×C4—44no—
202C3×C4—48no—
203C3×C4—48no—
204C3×C4—44no—
205C3×C4—44no—
206C3×C4—41no—
207C3×C4—41no—
208C3×C4—362no—
209C3×C4C3×D3×Q818yes—
210C3×C4Q8×C3⋊D312yes—
211Dic3—128no—
212C32—24no—
213C32#506414yescomplemented
214C32C3×#50111yescharacteristic, complemented
215Q8—362no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—182no—
219C2×C4C3×#63811yescharacteristic
220C2×C4C3×#63811yescharacteristic
221C2×C4C22×C3×C3⋊D311yescharacteristic
222C8—94no—
223C2×C3—24no—
224C2×C3—24no—
225C2×C3—24no—
226C2×C3C3×#94414yes—
227C2×C3C3×#94914yes—
228C2×C3C3×#95114yes—
229C2×C3#6415311yescharacteristic
230C2×C3#6415811yescharacteristic
231C2×C3#6416011yescharacteristic
232C22C2×C3×#63811yescharacteristic
233C4—21no—
234C4—21no—
235C4—41no—
236C4—41no—
237C4—362no—
238C4C3×Q8×C3⋊D312yes—
239C3—24no—
240C3C3×#506414yescomplemented
241C3#1059276311yescharacteristic, direct product factor
242C2C3×#6415311yescharacteristic
243C2C3×#6415811yescharacteristic
244C2C3×#6416011yescharacteristic
subgroupG/H
centerC22×C3C2×#638
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#125C2
socle, prime radicalC22×C33C2×D4
hypercenterC3×#125C3⋊D3
hypocenterC32C3×#501
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#125

Complements

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

NH ≅ G/Ntype
6. #10592763C3direct (N × H)
35. C3×C3⋊(C3⋊C8)Q8semidirect
98. C3⋊(C3⋊C8)C3×Q8semidirect
152. C33#501semidirect
213. C32#5064semidirect
214. C32C3×#501semidirect
240. C3C3×#5064semidirect
241. C3#10592763direct (N × H)

Maximal subgroups

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

subgroupindex
C33×#1252
C3×#640782
C3×#636352
C3×#636342
C3×#636302
#105927633
C3×#50643