Finite Group Databasebeta

← C3×#10593267 #13166194

Subgroups — C3×#10593267 #13166194

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
1C22×C3×AGL(1,𝔽9)C212yesmaximal
2C2×C3×C4×#157C211yesmaximal, characteristic
3#10593267C311yesmaximal, characteristic, direct product factor
4C22×C3×#157C411yescharacteristic
5C22×C3×#157C2211yescharacteristic
6C2×C3×AGL(1,𝔽9)—22no—
7C2×C3×AGL(1,𝔽9)—22no—
8C2×C3×AGL(1,𝔽9)C414yescomplemented
9C2×C3×C4×C3⋊D3C411yescharacteristic
10C3×C4×#157—22no—
11C22×AGL(1,𝔽9)C2×C312yes—
12C2×C4×#157C2×C311yescharacteristic
13C22×C3×C3⋊D3C2×C411yescharacteristic
14C2×C3×#157—21no—
15C2×C3×#157—21no—
16C2×C3×#157C812yescomplemented
17C2×C3×#157C2×C412yes—
18C2×C3×#157D411yescharacteristic
19C2×C3×#157D412yes—
20C2×C3×#157Q811yescharacteristic
21C3×AGL(1,𝔽9)—24no—
22C2×C33×C4C811yescharacteristic, complemented
23C2×C3×C3⋊Dic3C811yescharacteristic, complemented
24C3×C4×C3⋊D3—21no—
25C3×C4×C3⋊D3—21no—
26C3×#336—91nomaximal
27C22×#157C3×C411yescharacteristic
28C22×#157C22×C311yescharacteristic
29C2×AGL(1,𝔽9)—22no—
30C2×AGL(1,𝔽9)—22no—
31C2×AGL(1,𝔽9)C3×C414yescomplemented
32C2×C4×C3⋊D3C3×C411yescharacteristic
33C2×C3×C4×D3—121no—
34C4×#157—22no—
35C22×C33C2×C811yescharacteristic
36C2×C3×C3⋊D3C4211yescharacteristic
37C2×C3×C3⋊D3C22⋊C411yescharacteristic
38C2×C3×C3⋊D3C4⋊C411yescharacteristic
39C2×C3×C3⋊D3C2×C811yescharacteristic
40C2×C3×C3⋊D3M1611yescharacteristic
41C2×C3×C3⋊D3M1611yescharacteristic
42C3×#157—22no—
43C3×#157C22⋊C412yes—
44C3×#157C4⋊C412yes—
45C33×C4—21no—
46C3×C3⋊Dic3—21no—
47C22×C3×C8—92no—
48C2×C3×C42—91no—
49C22×C3⋊D3C2×C3×C411yescharacteristic
50C22×C3×D3—121no—
51C2×#157—21no—
52C2×#157—21no—
53C2×#157C3×C812yescomplemented
54C2×#157C2×C3×C412yes—
55C2×#157C3×D411yescharacteristic
56C2×#157C3×D412yes—
57C2×#157C3×Q811yescharacteristic
58AGL(1,𝔽9)—24no—
59C2×C32×C4—41no—
60C2×C32×C4—81no—
61C2×C32×C4C3×C811yescharacteristic, complemented
62C2×C3⋊Dic3C3×C811yescharacteristic, complemented
63C4×C3⋊D3—21no—
64C4×C3⋊D3—21no—
65C2×C3×Dic3—121no—
66C3×C4×D3—122no—
67C3×C4×D3—122no—
68#336—91noSylow
69C2×C33C4×C811yescharacteristic
70C2×C33C22⋊C811yescharacteristic
71C2×C33C4⋊C811yescharacteristic
72C3×C3⋊D3#9511yescharacteristic
73C3×C3⋊D3#9711yescharacteristic
74C3×C3⋊D3C22⋊C811yescharacteristic
75C3×C3⋊D3C4⋊C811yescharacteristic
76C22×C3×C4—91no—
77C22×C3×C4—91no—
78C22×C3×C4—91no—
79C2×C4×D3—121no—
80C2×C3×C8—94no—
81C2×C3×C8—182no—
82C2×C3×C8—182no—
83C3×C42—182no—
84C22×C32—41no—
85C22×C32—81no—
86C22×C32C2×C3×C811yescharacteristic
87C2×C3⋊D3C3×C4211yescharacteristic
88C2×C3⋊D3C3×C22⋊C411yescharacteristic
89C2×C3⋊D3C3×C4⋊C411yescharacteristic
90C2×C3⋊D3C2×C3×C811yescharacteristic
91C2×C3⋊D3C3×M1611yescharacteristic
92C2×C3⋊D3C3×M1611yescharacteristic
93C2×C3×D3—121no—
94C2×C3×D3—121no—
95C2×C3×D3—121no—
96C2×C3×D3—121no—
97C2×C3×D3—121no—
98C2×C3×D3—121no—
99#157—22no—
100#157C3×C22⋊C412yes—
101#157C3×C4⋊C412yes—
102C32×C4—21no—
103C32×C4—42no—
104C32×C4—82no—
105C3⋊Dic3—21no—
106C3×Dic3—122no—
107C22×C8—92no—
108C2×C42—91no—
109C33#33611yescharacteristic, complemented, Sylow
110C23×C3—91no—
111C22×D3—121no—
112C2×C3×C4—41no—
113C2×C3×C4—81no—
114C2×C3×C4—92no—
115C2×C3×C4—92no—
116C2×C3×C4—91no—
117C2×C3×C4—91no—
118C2×C3×C4—92no—
119C2×C3×C4—91no—
120C2×C3×C4—181no—
121C2×C3×C4—181no—
122C2×C3×C4—181no—
123C2×C3×C4—181no—
124C2×C3×C4AGL(1,𝔽9)11yescharacteristic, complemented
125C2×Dic3—121no—
126C4×D3—122no—
127C4×D3—122no—
128C3×C8—184no—
129C2×C32—41no—
130C2×C32—41no—
131C2×C32—41no—
132C2×C32—81no—
133C2×C32—81no—
134C2×C32—81no—
135C2×C32C3×C4×C811yescharacteristic
136C2×C32C3×C22⋊C811yescharacteristic
137C2×C32C3×C4⋊C811yescharacteristic
138C3⋊D3C3×#9511yescharacteristic
139C3⋊D3C3×#9711yescharacteristic
140C3⋊D3C3×C22⋊C811yescharacteristic
141C3⋊D3C3×C4⋊C811yescharacteristic
142C3×D3—121no—
143C3×D3—121no—
144C3×D3—121no—
145C3×D3—121no—
146C22×C4—91no—
147C22×C4—91no—
148C22×C4—91no—
149C2×C8—94no—
150C2×C8—182no—
151C2×C8—182no—
152C42—182no—
153C22×C3—41no—
154C22×C3—81no—
155C22×C3—91no—
156C22×C3—91no—
157C22×C3—91no—
158C22×C3—91no—
159C22×C3—91no—
160C22×C3—91no—
161C22×C3C2×AGL(1,𝔽9)11yescharacteristic
162C2×D3—121no—
163C2×D3—121no—
164C2×D3—121no—
165C2×D3—121no—
166C2×D3—121no—
167C2×D3—121no—
168C3×C4—21no—
169C3×C4—42no—
170C3×C4—82no—
171C3×C4—92no—
172C3×C4—92no—
173C3×C4—182no—
174C3×C4—181no—
175Dic3—122no—
176C32—41no—
177C32—81no—
178C32C3×#33611yescharacteristic, complemented
179C23—91no—
180C2×C4—92no—
181C2×C4—92no—
182C2×C4—91no—
183C2×C4—91no—
184C2×C4—92no—
185C2×C4—91no—
186C2×C4—181no—
187C2×C4—181no—
188C2×C4—181no—
189C2×C4—181no—
190C2×C4C3×AGL(1,𝔽9)11yescharacteristic, complemented
191C8—184no—
192C2×C3—41no—
193C2×C3—41no—
194C2×C3—41no—
195C2×C3—81no—
196C2×C3—81no—
197C2×C3—81no—
198C2×C3—91no—
199C2×C3—91no—
200C2×C3—91no—
201C2×C3—91no—
202C2×C3C4×AGL(1,𝔽9)11yescharacteristic
203C2×C3C4⋊AGL(1,𝔽9)11yescharacteristic
204C2×C3C22⋊AGL(1,𝔽9)11yescharacteristic
205D3—121no—
206D3—121no—
207D3—121no—
208D3—121no—
209C22—91no—
210C22—91no—
211C22—91no—
212C22—91no—
213C22—91no—
214C22—91no—
215C22C2×C3×AGL(1,𝔽9)11yescharacteristic
216C4—21no—
217C4—92no—
218C4—92no—
219C4—182no—
220C4—181no—
221C3—41no—
222C3—81no—
223C3#1059326711yescharacteristic, direct product factor
224C2—91no—
225C2—91no—
226C2—91no—
227C2—91no—
228C2C3×C4×AGL(1,𝔽9)11yescharacteristic
229C2C3×C4⋊AGL(1,𝔽9)11yescharacteristic
230C2C3×C22⋊AGL(1,𝔽9)11yescharacteristic
subgroupG/H
centerC22×C3C2×AGL(1,𝔽9)
commutatorC2×C32C3×C4×C8
FrattiniC22C2×C3×AGL(1,𝔽9)
FittingC2×C33×C4C8
socleC22×C33C2×C8
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC22×#157C22×C3
hypercenterC2×C3×C4AGL(1,𝔽9)
hypocenterC32C3×#336
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
3. #10593267C3direct (N × H)
8. C2×C3×AGL(1,𝔽9)C4semidirect
16. C2×C3×#157C8semidirect
22. C2×C33×C4C8semidirect
23. C2×C3×C3⋊Dic3C8semidirect
31. C2×AGL(1,𝔽9)C3×C4semidirect
53. C2×#157C3×C8semidirect
61. C2×C32×C4C3×C8semidirect
62. C2×C3⋊Dic3C3×C8semidirect
109. C33#336semidirect
124. C2×C3×C4AGL(1,𝔽9)semidirect
178. C32C3×#336semidirect
190. C2×C4C3×AGL(1,𝔽9)semidirect
223. C3#10593267direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×AGL(1,𝔽9)2
C2×C3×C4×#1572
#105932673
C3×#3369