Finite Group Databasebeta

← C3×#10592780 #13163119

Subgroups — C3×#10592780 #13163119

Order 1728.

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

normal subgroups only

This group has 812 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×C4⋊Q8C211yesmaximal, characteristic
2C3×#64079C211yesmaximal, characteristic
3C3×#63635C212yesmaximal
4C3×#63634C212yesmaximal
5C3×#63630C211yesmaximal, characteristic
6#10592780C311yesmaximal, characteristic, direct product factor
7C3×#5081—34nomaximal
8C2×C33×Q8—21no—
9C33×C4⋊C4—21no—
10C33×C4⋊C4C2212yes—
11C33×C42C2211yescharacteristic
12C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
13C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C3×#3750—21no—
15C3×#3750—21no—
16C2×C3×C3⋊(C3⋊C8)C2212yes—
17C32×C4⋊Q8—24no—
18C32×C4⋊Q8D314yes—
19C32×C4⋊Q8C2×C311yescharacteristic
20#64079C2×C311yescharacteristic
21C3×#870—34no—
22#63635C2×C312yes—
23#63634C2×C312yes—
24#63630C2×C311yescharacteristic
25C3×#808—38no—
26C3×#807—38no—
27C3×#803—34no—
28C33×Q8—42no—
29C2×C33×C4—22no—
30C2×C33×C4D411yescharacteristic
31C2×C33×C4D411yescharacteristic
32C2×C33×C4C2311yescharacteristic
33C2×C3×C3⋊Dic3—21no—
34C2×C3×C3⋊Dic3—21no—
35C3×C3⋊(C3⋊C8)Q814yescomplemented
36C3×#501—91no—
37#5081—34no—
38C2×C32×Q8—28no—
39C2×C32×Q8—24no—
40C2×C32×Q8—21no—
41C32×C4⋊C4—28no—
42C32×C4⋊C4—28no—
43C32×C4⋊C4—24no—
44C32×C4⋊C4—21no—
45C32×C4⋊C4C2×D318yes—
46C32×C4⋊C4C22×C312yes—
47C32×C42—24no—
48C32×C42C2×D314yes—
49C32×C42C22×C311yescharacteristic
50C4⋊(C3⋊Dic3)C22×C311yescharacteristic
51C4⋊(C3⋊Dic3)C22×C311yescharacteristic
52#3750—21no—
53#3750—21no—
54C2×C3⋊(C3⋊C8)C22×C312yes—
55C3×C4⋊Dic3—34no—
56C3×C4⋊Dic3—34no—
57C3×Dic3⋊C4—64no—
58C3×Dic3⋊C4—64no—
59C2×C3×C3⋊C8—38no—
60C22×C33C2×D411yescharacteristic
61C33×C4—21no—
62C33×C4—21no—
63C33×C4—42no—
64C33×C4C2×Q812yes—
65C3×C3⋊Dic3—41no—
66C3×C3⋊Dic3—41no—
67C3×C4⋊Q8—24no—
68C3×C4⋊Q8C3×D314yes—
69C3×C4⋊Q8C3⋊D311yescharacteristic
70C3×#125—91no—
71#870—34no—
72C3×#107—92no—
73C3×#106—92no—
74C3×#97—91no—
75#808—38no—
76#807—38no—
77#803—34no—
78C32×Q8—416no—
79C32×Q8—48no—
80C32×Q8—42no—
81C2×C32×C4—216no—
82C2×C32×C4—28no—
83C2×C32×C4—22no—
84C2×C32×C4—24no—
85C2×C32×C4—24no—
86C2×C32×C4—24no—
87C2×C32×C4#6714yes—
88C2×C32×C4#6714yes—
89C2×C32×C4C3×D411yescharacteristic
90C2×C32×C4C3×D411yescharacteristic
91C2×C32×C4C22×D314yes—
92C2×C32×C4C23×C311yescharacteristic
93C2×C3⋊Dic3—21no—
94C2×C3⋊Dic3—21no—
95C2×C3×Dic3—64no—
96C2×C3×Dic3—64no—
97C3⋊(C3⋊C8)C3×Q814yescomplemented
98C3×C3⋊C8—316no—
99#501—91noSylow
100C2×C33C4⋊Q811yescharacteristic
101C2×C33AGL(1,ℤ/8)11yescharacteristic
102C2×C33#13711yescharacteristic
103C2×C3×Q8—28no—
104C2×C3×Q8—24no—
105C2×C3×Q8—21no—
106C2×C3×C8—92no—
107C3×C4⋊C4—28no—
108C3×C4⋊C4—24no—
109C3×C4⋊C4—21no—
110C3×C4⋊C4—28no—
111C3×C4⋊C4—91no—
112C3×C4⋊C4—91no—
113C3×C4⋊C4—181no—
114C3×C4⋊C4—181no—
115C3×C4⋊C4C2×C3×D318yes—
116C3×C4⋊C4C2×C3⋊D312yes—
117C3×C42—24no—
118C3×C42C2×C3×D314yes—
119C3×C42C2×C3⋊D311yescharacteristic
120C4⋊Dic3—34no—
121C4⋊Dic3—34no—
122Dic3⋊C4—64no—
123Dic3⋊C4—64no—
124C2×C3⋊C8—38no—
125C22×C32—24no—
126C22×C32C2×#6714yes—
127C22×C32C2×C3×D411yescharacteristic
128C32×C4—28no—
129C32×C4—28no—
130C32×C4—24no—
131C32×C4—24no—
132C32×C4—21no—
133C32×C4—21no—
134C32×C4—28no—
135C32×C4—416no—
136C32×C4—48no—
137C32×C4—42no—
138C32×C4D3×Q818yes—
139C32×C4C2×C3×Q812yes—
140C3⋊Dic3—41no—
141C3⋊Dic3—41no—
142C3×Dic3—124no—
143C3×Dic3—124no—
144C4⋊Q8C3×C3⋊D311yescharacteristic
145#125—91no—
146#107—92no—
147#106—92no—
148#97—91no—
149C33#50111yescharacteristic, complemented, Sylow
150C3×Q8—416no—
151C3×Q8—48no—
152C3×Q8—42no—
153C2×C3×C4—216no—
154C2×C3×C4—28no—
155C2×C3×C4—22no—
156C2×C3×C4—24no—
157C2×C3×C4—24no—
158C2×C3×C4—24no—
159C2×C3×C4—181no—
160C2×C3×C4—181no—
161C2×C3×C4C3×#6714yes—
162C2×C3×C4C3×#6714yes—
163C2×C3×C4#63811yescharacteristic
164C2×C3×C4#63811yescharacteristic
165C2×C3×C4C22×C3×D314yes—
166C2×C3×C4C22×C3⋊D311yescharacteristic
167C2×Dic3—64no—
168C2×Dic3—64no—
169C3×C8—94no—
170C3⋊C8—316no—
171C2×C32—24no—
172C2×C32—24no—
173C2×C32—24no—
174C2×C32#93214yes—
175C2×C32#94214yes—
176C2×C32#94414yes—
177C2×C32C3×C4⋊Q811yescharacteristic
178C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
179C2×C32C3×#13711yescharacteristic
180C2×Q8—21no—
181C2×C8—92no—
182C4⋊C4—21no—
183C4⋊C4—91no—
184C4⋊C4—91no—
185C4⋊C4—181no—
186C4⋊C4—181no—
187C4⋊C4C2×C3×C3⋊D312yes—
188C42C2×C3×C3⋊D311yescharacteristic
189C22×C3—24no—
190C22×C3C2×C3×#6714yes—
191C22×C3C2×#63811yescharacteristic
192C3×C4—28no—
193C3×C4—28no—
194C3×C4—24no—
195C3×C4—24no—
196C3×C4—21no—
197C3×C4—21no—
198C3×C4—28no—
199C3×C4—416no—
200C3×C4—48no—
201C3×C4—42no—
202C3×C4—361no—
203C3×C4—361no—
204C3×C4C3×D3×Q818yes—
205C3×C4Q8×C3⋊D312yes—
206Dic3—124no—
207Dic3—124no—
208C32—24no—
209C32#508114yescomplemented
210C32C3×#50111yescharacteristic, complemented
211Q8—42no—
212C2×C4—22no—
213C2×C4—181no—
214C2×C4—181no—
215C2×C4C3×#63811yescharacteristic
216C2×C4C3×#63811yescharacteristic
217C2×C4C22×C3×C3⋊D311yescharacteristic
218C8—94no—
219C2×C3—24no—
220C2×C3—24no—
221C2×C3—24no—
222C2×C3C3×#93214yes—
223C2×C3C3×#94214yes—
224C2×C3C3×#94414yes—
225C2×C3#6414111yescharacteristic
226C2×C3#6415111yescharacteristic
227C2×C3#6415311yescharacteristic
228C22C2×C3×#63811yescharacteristic
229C4—21no—
230C4—21no—
231C4—42no—
232C4—361no—
233C4—361no—
234C4C3×Q8×C3⋊D312yes—
235C3—24no—
236C3C3×#508114yescomplemented
237C3#1059278011yescharacteristic, direct product factor
238C2C3×#6414111yescharacteristic
239C2C3×#6415111yescharacteristic
240C2C3×#6415311yescharacteristic
subgroupG/H
centerC22×C3C2×#638
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×C4⋊Q8C2
socle, prime radicalC22×C33C2×D4
hypercenterC3×C4⋊Q8C3⋊D3
hypocenterC32C3×#501
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C4⋊Q8

Complements

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

NH ≅ G/Ntype
6. #10592780C3direct (N × H)
35. C3×C3⋊(C3⋊C8)Q8semidirect
97. C3⋊(C3⋊C8)C3×Q8semidirect
149. C33#501semidirect
209. C32#5081semidirect
210. C32C3×#501semidirect
236. C3C3×#5081semidirect
237. C3#10592780direct (N × H)

Maximal subgroups

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

subgroupindex
C33×C4⋊Q82
C3×#640792
C3×#636352
C3×#636342
C3×#636302
#105927803
C3×#50813