Finite Group Databasebeta

← C3×#10593544 #13167621

Subgroups — C3×#10593544 #13167621

Order 1728.

306 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 94 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×#64286C211yesmaximal, characteristic
2C2×C3×C4⋊#157C211yesmaximal, characteristic
3C3×#64283C211yesmaximal, characteristic
4C3×#63746C212yesmaximal, complemented
5C3×#63745C212yesmaximal, complemented
6#10593544C311yesmaximal, characteristic, direct product factor
7C22×C3×#157—21no—
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C3×C22⋊#157—21no—
10C3×#3792C412yescomplemented
11C3×C4⋊#157—21no—
12C3×C4⋊#157C2211yescharacteristic
13C3×C4⋊#157C2212yes—
14C3×C4⋊#157C2211yescharacteristic
15C3×C4×#157—21no—
16C3×#3788C412yescomplemented
17C3×#3787C2212yes—
18#64286C2×C311yescharacteristic
19C2×C4⋊#157C2×C311yescharacteristic
20#64283C2×C311yescharacteristic
21#63746C2×C312yescomplemented
22#63745C2×C312yescomplemented
23C22×C3×C3⋊D3D411yescharacteristic
24C2×C3×#157—22no—
25C2×C3×#157—21no—
26C2×C3×#157—21no—
27C2×C3×#157—42no—
28C2×C33×C4Q811yescharacteristic
29C2×C3×C3⋊Dic3C2×C411yescharacteristic
30C3×C4×C3⋊D3C2×C411yescharacteristic
31C3×C4×C3⋊D3D411yescharacteristic
32C3×C4×C3⋊D3Q811yescharacteristic
33C3×C4×C3⋊D3C2311yescharacteristic
34C3×#622C2×C414yescomplemented
35C3×#428—91nomaximal
36C22×#157—21no—
37C2×C4×C3⋊D3C22×C311yescharacteristic
38C2×C3×C4×D3—121no—
39C22⋊#157—21no—
40#3792C3×C412yescomplemented
41C4⋊#157—21no—
42C4⋊#157C22×C311yescharacteristic
43C4⋊#157C22×C312yes—
44C4⋊#157C22×C311yescharacteristic
45C4×#157—21no—
46#3788C3×C412yescomplemented
47#3787C22×C312yes—
48C22×C33C4⋊C411yescharacteristic
49C2×C3×C3⋊D3—21no—
50C2×C3×C3⋊D3—21no—
51C2×C3×C3⋊D3C4⋊C411yescharacteristic
52C2×C3×C3⋊D3C2×D411yescharacteristic
53C3×#157—41no—
54C3×#157—42no—
55C3×#157—41no—
56C33×C4C4⋊C411yescharacteristic
57C33×C4C2×Q811yescharacteristic
58C3×C3⋊Dic3C4⋊C411yescharacteristic
59C3×C3⋊Dic3C22×C411yescharacteristic
60C2×C3×M16—91no—
61C3×#117—91no—
62C2×C3×C4⋊C4—91no—
63C3×#107—92no—
64C3×#106—92no—
65C22×C3⋊D3C3×D411yescharacteristic
66C22×C3×D3—121no—
67C2×#157—22no—
68C2×#157—21no—
69C2×#157—21no—
70C2×#157—42no—
71C2×C32×C4—41no—
72C2×C32×C4—81no—
73C2×C32×C4C3×Q811yescharacteristic
74C2×C3⋊Dic3C2×C3×C411yescharacteristic
75C4×C3⋊D3C2×C3×C411yescharacteristic
76C4×C3⋊D3C3×D411yescharacteristic
77C4×C3⋊D3C3×Q811yescharacteristic
78C4×C3⋊D3C23×C311yescharacteristic
79C2×C3×Dic3—121no—
80C3×C4×D3—121no—
81C3×C4×D3—121no—
82C3×C4×D3—121no—
83C3×C4×D3—121no—
84#622C2×C3×C414yescomplemented
85#428—91noSylow
86C2×C33—21no—
87C2×C33C2×C4⋊C411yescharacteristic
88C3×C3⋊D3—21no—
89C3×C3⋊D3AGL(1,ℤ/8)11yescharacteristic
90C3×C3⋊D3#13711yescharacteristic
91C22×C3×C4—91no—
92C22×C3×C4—181no—
93C2×C4×D3—121no—
94C3×M16—92no—
95C3×M16—92no—
96C2×C3×C8—92no—
97C3×C4⋊C4—91no—
98C3×C4⋊C4—92no—
99C3×C4⋊C4—91no—
100C3×C4⋊C4—181no—
101C3×C22⋊C4—181no—
102C3×C42—181no—
103C22×C32—41no—
104C22×C32—81no—
105C22×C32C3×C4⋊C411yescharacteristic
106C2×C3⋊D3—21no—
107C2×C3⋊D3—21no—
108C2×C3⋊D3C3×C4⋊C411yescharacteristic
109C2×C3⋊D3C2×C3×D411yescharacteristic
110C2×C3×D3—121no—
111C2×C3×D3—121no—
112C2×C3×D3—122no—
113C2×C3×D3—122no—
114#157—41no—
115#157—42no—
116#157—41no—
117C32×C4—41no—
118C32×C4—41no—
119C32×C4—81no—
120C32×C4—81no—
121C32×C4C3×C4⋊C411yescharacteristic
122C32×C4C2×C3×Q811yescharacteristic
123C3⋊Dic3C3×C4⋊C411yescharacteristic
124C3⋊Dic3C22×C3×C411yescharacteristic
125C3×Dic3—121no—
126C3×Dic3—121no—
127C2×M16—91no—
128#117—91no—
129C2×C4⋊C4—91no—
130#107—92no—
131#106—92no—
132C33#42811yescharacteristic, complemented, Sylow
133C23×C3—91no—
134C22×D3—121no—
135C2×C3×C4—41no—
136C2×C3×C4—81no—
137C2×C3×C4—91no—
138C2×C3×C4—91no—
139C2×C3×C4—91no—
140C2×C3×C4—91no—
141C2×C3×C4—91no—
142C2×C3×C4—181no—
143C2×C3×C4—181no—
144C2×C3×C4—182no—
145C2×C3×C4—362no—
146C2×C3×C4PSU(3,𝔽2)11yescharacteristic
147C2×Dic3—121no—
148C4×D3—121no—
149C4×D3—121no—
150C4×D3—121no—
151C4×D3—121no—
152C3×C8—94no—
153C2×C32—21no—
154C2×C32—42no—
155C2×C32—41no—
156C2×C32—82no—
157C2×C32—81no—
158C2×C32C2×C3×C4⋊C411yescharacteristic
159C3⋊D3—21no—
160C3⋊D3C3×AGL(1,ℤ/8)11yescharacteristic
161C3⋊D3C3×#13711yescharacteristic
162C3×D3—121no—
163C3×D3—121no—
164C3×D3—122no—
165C22×C4—91no—
166C22×C4—181no—
167M16—92no—
168M16—92no—
169C2×C8—92no—
170C4⋊C4—91no—
171C4⋊C4—92no—
172C4⋊C4—91no—
173C4⋊C4—181no—
174C22⋊C4—181no—
175C42—181no—
176C22×C3—41no—
177C22×C3—81no—
178C22×C3—91no—
179C22×C3—91no—
180C22×C3—181no—
181C22×C3—181no—
182C22×C3#377711yescharacteristic
183C2×D3—121no—
184C2×D3—121no—
185C2×D3—122no—
186C2×D3—122no—
187C3×C4—41no—
188C3×C4—41no—
189C3×C4—81no—
190C3×C4—81no—
191C3×C4—91no—
192C3×C4—91no—
193C3×C4—362no—
194C3×C4—361no—
195C3×C4—361no—
196C3×C4#377711yescharacteristic
197C3×C4C2×PSU(3,𝔽2)11yescharacteristic
198Dic3—121no—
199Dic3—121no—
200C32—41no—
201C32—81no—
202C32C3×#42811yescharacteristic, complemented
203C23—91no—
204C2×C4—91no—
205C2×C4—91no—
206C2×C4—91no—
207C2×C4—91no—
208C2×C4—91no—
209C2×C4—181no—
210C2×C4—181no—
211C2×C4—182no—
212C2×C4—362no—
213C2×C4C3×PSU(3,𝔽2)11yescharacteristic
214C8—94no—
215C2×C3—21no—
216C2×C3—42no—
217C2×C3—41no—
218C2×C3—82no—
219C2×C3—81no—
220C2×C3—91no—
221C2×C3—91no—
222C2×C3—181no—
223C2×C3C2×#377711yescharacteristic
224D3—121no—
225D3—121no—
226D3—122no—
227C22—91no—
228C22—91no—
229C22—181no—
230C22—181no—
231C22C3×#377711yescharacteristic
232C4—91no—
233C4—91no—
234C4—362no—
235C4—361no—
236C4—361no—
237C4C3×#377711yescharacteristic
238C4C2×C3×PSU(3,𝔽2)11yescharacteristic
239C3—41no—
240C3—81no—
241C3#1059354411yescharacteristic, direct product factor
242C2—21no—
243C2—91no—
244C2—91no—
245C2—181no—
246C2C2×C3×#377711yescharacteristic
subgroupG/H
centerC2×C3C2×#3777
commutatorC3⋊Dic3C22×C3×C4
FrattiniC4C2×C3×PSU(3,𝔽2)
FittingC2×C33×C4Q8
socleC2×C33C2×C4⋊C4
prime radicalC22×C3×C3⋊D3D4
cosocleC4×C3⋊D3C23×C3
hypercenterC2×C3×C4PSU(3,𝔽2)
hypocenterC32C3×#428
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
4. C3×#63746C2semidirect
5. C3×#63745C2semidirect
6. #10593544C3direct (N × H)
10. C3×#3792C4semidirect
16. C3×#3788C4semidirect
21. #63746C2×C3semidirect
22. #63745C2×C3semidirect
34. C3×#622C2×C4semidirect
40. #3792C3×C4semidirect
46. #3788C3×C4semidirect
84. #622C2×C3×C4semidirect
132. C33#428semidirect
202. C32C3×#428semidirect
241. C3#10593544direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#642862
C2×C3×C4⋊#1572
C3×#642832
C3×#637462
C3×#637452
#105935443
C3×#4289