Finite Group Databasebeta

← C3×#10592788 #13163129

Subgroups — C3×#10592788 #13163129

Order 1728.

872 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 872 conjugacy classes of subgroups — too many to draw. Only its 190 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, complemented
2C3×#64084C211yesmaximal, characteristic
3C3×#63637C212yesmaximal, complemented
4C3×#63636C212yesmaximal
5C3×#63630C211yesmaximal, characteristic, complemented
6#10592788C311yesmaximal, characteristic, direct product factor
7C3×#5089—34nomaximal
8C2×C33×Q8—21no—
9C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
10C2×C3×#634C2211yescharacteristic
11C33×C4⋊C4—21no—
12C33×C4⋊C4C2212yes—
13C33×C42C2211yescharacteristic
14C3×#3752—21no—
15C3×#3752—21no—
16C2×C3×C3⋊(C3⋊C8)C2212yes—
17C32×C4⋊Q8—24no—
18C32×C4⋊Q8D314yescomplemented
19C32×C4⋊Q8C2×C311yescharacteristic, complemented
20#64084C2×C311yescharacteristic
21C3×#875—34no—
22#63637C2×C312yescomplemented
23#63636C2×C312yes—
24#63630C2×C311yescharacteristic, complemented
25C3×#810—38no—
26C3×#809—38no—
27C3×#803—34no—
28C22×C3×C3⋊D3—21no—
29C33×Q8—42no—
30C2×C33×C4—22no—
31C2×C33×C4D411yescharacteristic
32C2×C33×C4D411yescharacteristic
33C2×C33×C4C2311yescharacteristic
34C2×C3×C3⋊Dic3—21no—
35C3×C4⋊(C3⋊D3)—22no—
36C3×#634—22no—
37C3×C3⋊(C3⋊C8)—22no—
38C3×#489—91no—
39#5089—34no—
40C2×C32×Q8—28no—
41C2×C32×Q8—24no—
42C2×C32×Q8—21no—
43C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
44C2×#634C22×C311yescharacteristic
45C2×C3×D12—34no—
46C2×C3×Dic6—34no—
47C32×C4⋊C4—28no—
48C32×C4⋊C4—28no—
49C32×C4⋊C4—24no—
50C32×C4⋊C4—21no—
51C32×C4⋊C4C2×D318yes—
52C32×C4⋊C4C22×C312yes—
53C32×C42—24no—
54C32×C42C2×D314yes—
55C32×C42C22×C311yescharacteristic
56#3752—21no—
57#3752—21no—
58C2×C3⋊(C3⋊C8)C22×C312yes—
59C3×(C2×D3)⋊C4—64no—
60C3×(C2×D3)⋊C4—64no—
61C2×C3×C3⋊C8—38no—
62C22×C33C2×D411yescharacteristic
63C2×C3×C3⋊D3—41no—
64C2×C3×C3⋊D3—41no—
65C2×C3×C3⋊D3—41no—
66C33×C4—21no—
67C33×C4—21no—
68C33×C4—42no—
69C33×C4#4112yes—
70C3×C3⋊Dic3—41no—
71C3×C4⋊Q8—24no—
72C3×C4⋊Q8C3×D314yescomplemented
73C3×C4⋊Q8C3⋊D311yescharacteristic, complemented
74C3×#124—91no—
75#875—34no—
76C3×#103—92no—
77C3×#102—92no—
78C3×#97—91no—
79#810—38no—
80#809—38no—
81#803—34no—
82C22×C3⋊D3—21no—
83C22×C3×D3—64no—
84C32×Q8—416no—
85C32×Q8—48no—
86C32×Q8—42no—
87C2×C32×C4—216no—
88C2×C32×C4—28no—
89C2×C32×C4—22no—
90C2×C32×C4—24no—
91C2×C32×C4—24no—
92C2×C32×C4—24no—
93C2×C32×C4#6714yes—
94C2×C32×C4#6714yes—
95C2×C32×C4C3×D411yescharacteristic
96C2×C32×C4C3×D411yescharacteristic
97C2×C32×C4C22×D314yes—
98C2×C32×C4C23×C311yescharacteristic
99C2×C3⋊Dic3—21no—
100C4⋊(C3⋊D3)—22no—
101#634—22no—
102C2×C3×Dic3—64no—
103C3×D12—68no—
104C3×Dic6—68no—
105C3⋊(C3⋊C8)—22no—
106C3×C3⋊C8—68no—
107#489—91noSylow
108C2×C33#12411yescharacteristic
109C2×C33AGL(1,ℤ/8)11yescharacteristic
110C2×C33#13711yescharacteristic
111C3×C3⋊D3—81no—
112C2×C3×Q8—28no—
113C2×C3×Q8—24no—
114C2×C3×Q8—21no—
115C2×C3×Q8—91no—
116C2×C3×D4—91no—
117C2×D12—34no—
118C2×Dic6—34no—
119C2×C3×C8—92no—
120C3×C4⋊C4—28no—
121C3×C4⋊C4—24no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4—28no—
124C3×C4⋊C4C2×C3×D318yes—
125C3×C4⋊C4C2×C3⋊D312yes—
126C3×C22⋊C4—181no—
127C3×C22⋊C4—181no—
128C3×C42—24no—
129C3×C42C2×C3×D314yes—
130C3×C42C2×C3⋊D311yescharacteristic
131(C2×D3)⋊C4—64no—
132(C2×D3)⋊C4—64no—
133C2×C3⋊C8—38no—
134C22×C32—24no—
135C22×C32C2×#6714yes—
136C22×C32C2×C3×D411yescharacteristic
137C2×C3⋊D3—41no—
138C2×C3⋊D3—41no—
139C2×C3⋊D3—41no—
140C2×C3×D3—124no—
141C2×C3×D3—124no—
142C2×C3×D3—124no—
143C32×C4—28no—
144C32×C4—28no—
145C32×C4—24no—
146C32×C4—24no—
147C32×C4—21no—
148C32×C4—21no—
149C32×C4—28no—
150C32×C4—416no—
151C32×C4—48no—
152C32×C4—42no—
153C32×C4#23918yes—
154C32×C4C3×#4112yes—
155C3⋊Dic3—41no—
156C3×Dic3—124no—
157C4⋊Q8C3×C3⋊D311yescharacteristic, complemented
158#124—91no—
159#103—92no—
160#102—92no—
161#97—91no—
162C33#48911yescharacteristic, complemented, Sylow
163C23×C3—181no—
164C22×D3—64no—
165C3×Q8—416no—
166C3×Q8—48no—
167C3×Q8—42no—
168C3×Q8—182no—
169C3×D4—182no—
170C2×C3×C4—216no—
171C2×C3×C4—28no—
172C2×C3×C4—22no—
173C2×C3×C4—24no—
174C2×C3×C4—24no—
175C2×C3×C4—24no—
176C2×C3×C4—181no—
177C2×C3×C4C3×#6714yes—
178C2×C3×C4C3×#6714yes—
179C2×C3×C4#63811yescharacteristic
180C2×C3×C4#63811yescharacteristic
181C2×C3×C4C22×C3×D314yes—
182C2×C3×C4C22×C3⋊D311yescharacteristic
183C2×Dic3—64no—
184D12—68no—
185Dic6—68no—
186C3×C8—182no—
187C3⋊C8—68no—
188C2×C32—24no—
189C2×C32—24no—
190C2×C32—24no—
191C2×C32#93214yes—
192C2×C32#94214yes—
193C2×C32#94714yes—
194C2×C32C3×#12411yescharacteristic
195C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
196C2×C32C3×#13711yescharacteristic
197C3⋊D3—81no—
198C3×D3—244no—
199C2×Q8—21no—
200C2×Q8—91no—
201C2×D4—91no—
202C2×C8—92no—
203C4⋊C4—21no—
204C4⋊C4C2×C3×C3⋊D312yes—
205C22⋊C4—181no—
206C22⋊C4—181no—
207C42C2×C3×C3⋊D311yescharacteristic
208C22×C3—24no—
209C22×C3—361no—
210C22×C3—361no—
211C22×C3—361no—
212C22×C3C2×C3×#6714yes—
213C22×C3C2×#63811yescharacteristic
214C2×D3—124no—
215C2×D3—124no—
216C2×D3—124no—
217C3×C4—28no—
218C3×C4—28no—
219C3×C4—24no—
220C3×C4—24no—
221C3×C4—21no—
222C3×C4—21no—
223C3×C4—28no—
224C3×C4—416no—
225C3×C4—48no—
226C3×C4—42no—
227C3×C4—361no—
228C3×C4C3×#23918yes—
229C3×C4#383212yes—
230Dic3—124no—
231C32—24no—
232C32#508914yescomplemented
233C32C3×#48911yescharacteristic, complemented
234C23—181no—
235Q8—42no—
236Q8—182no—
237D4—182no—
238C2×C4—22no—
239C2×C4—181no—
240C2×C4C3×#63811yescharacteristic
241C2×C4C3×#63811yescharacteristic
242C2×C4C22×C3×C3⋊D311yescharacteristic
243C8—182no—
244C2×C3—24no—
245C2×C3—24no—
246C2×C3—24no—
247C2×C3—721no—
248C2×C3C3×#93214yes—
249C2×C3C3×#94214yes—
250C2×C3C3×#94714yes—
251C2×C3#6414111yescharacteristic
252C2×C3#6415111yescharacteristic
253C2×C3#6415611yescharacteristic
254D3—244no—
255C22—361no—
256C22—361no—
257C22—361no—
258C22C2×C3×#63811yescharacteristic
259C4—21no—
260C4—21no—
261C4—42no—
262C4—361no—
263C4C3×#383212yes—
264C3—24no—
265C3C3×#508914yescomplemented
266C3#1059278811yescharacteristic, direct product factor
267C2—721no—
268C2C3×#6414111yescharacteristic
269C2C3×#6415111yescharacteristic
270C2C3×#6415611yescharacteristic
subgroupG/H
centerC22×C3C2×#638
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×C4⋊Q8C2
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC3×C4⋊Q8C3⋊D3
hypocenterC32C3×#489
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
1. C33×C4⋊Q8C2semidirect
3. C3×#63637C2semidirect
5. C3×#63630C2semidirect
6. #10592788C3direct (N × H)
18. C32×C4⋊Q8D3semidirect
19. C32×C4⋊Q8C2×C3semidirect
22. #63637C2×C3semidirect
24. #63630C2×C3semidirect
72. C3×C4⋊Q8C3×D3semidirect
73. C3×C4⋊Q8C3⋊D3semidirect
157. C4⋊Q8C3×C3⋊D3semidirect
162. C33#489semidirect
232. C32#5089semidirect
233. C32C3×#489semidirect
265. C3C3×#5089semidirect
266. C3#10592788direct (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×#640842
C3×#636372
C3×#636362
C3×#636302
#105927883
C3×#50893