Finite Group Databasebeta

← #10588417⋊C3 #13141641

Subgroups — #10588417⋊C3 #13141641

Order 1728.

367 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 367 conjugacy classes of subgroups — too many to draw. Only its 102 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
1#63492⋊C3C211yesmaximal, characteristic
2C2×C4⋊(3−1+2⋊C4)C211yesmaximal, characteristic, complemented
33−1+2×C22⋊C8C211yesmaximal, characteristic, complemented
4#63380⋊C3C212yesmaximal, complemented
5#63377⋊C3C212yesmaximal, complemented
6C3×#4735—31nomaximal
7#10588417C311yesmaximal, characteristic, complemented
8C22×C4×3−1+2C2211yescharacteristic
9C2×#611⋊C3—21no—
10C22×3−1+2⋊C4—21no—
11C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
12C2×C8×3−1+2C2212yes—
13(C2×AGL(1,ℤ/9))⋊C4—21no—
14C4⋊(3−1+2⋊C4)—21no—
15C4⋊(3−1+2⋊C4)C2212yes—
16C4⋊(3−1+2⋊C4)C2211yescharacteristic
17C3×#930—31no—
18C2×C3×C4⋊Dic3—31no—
19C32×C22⋊C8D311yescharacteristic
20C3×#821—32no—
21C3×#817—32no—
22#63492C2×C311yescharacteristic
23C2×C4⋊Dic9C2×C311yescharacteristic, complemented
24C9×C22⋊C8—21no—
25C9×C22⋊C8C2×C311yescharacteristic, complemented
26#63380C2×C312yescomplemented
27#63377C2×C312yescomplemented
28C23×3−1+2D411yescharacteristic
29C22×AGL(1,ℤ/9)—21no—
30C2×C4×3−1+2—22no—
31C2×C4×3−1+2D411yescharacteristic
32C2×C4×3−1+2C2311yescharacteristic
33#611⋊C3—42no—
34C2×3−1+2⋊C4—22no—
35C2×3−1+2⋊C4—21no—
36C2×3−1+2⋊C4—42no—
37C4⋊AGL(1,ℤ/9)—22no—
38C8×3−1+2—22no—
39C3×#481—91no—
40#4735—31no—
41C22×C32×C4C2×D311yescharacteristic
42C2×C3×#67—61no—
43C22×C3×Dic3—61no—
44C2×C3×D12—31no—
45C2×C32×C8C2×D312yes—
46C3×(C2×D3)⋊C4—61no—
47C3×C4⋊Dic3—32no—
48C3×C4⋊Dic3—31no—
49C3×C4⋊Dic3—61no—
50C22×C4×C9—21no—
51C22×C4×C9C22×C311yescharacteristic
52C2×#611—21no—
53C22×Dic9—21no—
54C2×D36C22×C311yescharacteristic
55C2×C8×C9—22no—
56C2×C8×C9C22×C312yes—
57(C2×D9)⋊C4—21no—
58C4⋊Dic9—21no—
59C4⋊Dic9C22×C311yescharacteristic
60C4⋊Dic9C22×C312yes—
61C22×3−1+2—21no—
62C22×3−1+2—21no—
63C22×3−1+2SD1612yes—
64C22×3−1+2C2×D411yescharacteristic
65C2×AGL(1,ℤ/9)—41no—
66C2×AGL(1,ℤ/9)—41no—
67C2×AGL(1,ℤ/9)—41no—
68C4×3−1+2—21no—
69C4×3−1+2#4112yes—
703−1+2⋊C4—41no—
713−1+2⋊C4—42no—
72C3×#121—91no—
73C2×C3×C4⋊C4—91no—
74#930—31no—
75C2×C4⋊Dic3—31no—
76C3×#106—92no—
77C3×#102—92no—
78C3×C22⋊C8—31no—
79C3×C22⋊C8C3×D311yescharacteristic
80#821—32no—
81#817—32no—
82C23×C32D1211yescharacteristic
83C22×C3×D3—61no—
84C2×C32×C4—22no—
85C2×C32×C4D1211yescharacteristic
86C2×C32×C4C22×D311yescharacteristic
87C3×#67—122no—
88C2×C3×Dic3—62no—
89C2×C3×Dic3—61no—
90C2×C3×Dic3—122no—
91C3×D12—62no—
92C23×C9—21no—
93C23×C9C3×D411yescharacteristic
94C22×D9—21no—
95C32×C8—22no—
96C2×C4×C9—22no—
97C2×C4×C9—21no—
98C2×C4×C9—21no—
99C2×C4×C9—42no—
100C2×C4×C9C3×D411yescharacteristic
101C2×C4×C9C23×C311yescharacteristic
102#611—42no—
103C2×Dic9—22no—
104C2×Dic9—21no—
105C2×Dic9—42no—
106D36—22no—
107C8×C9—22no—
108C8×C9—42no—
109#481—91noSylow
110C2×3−1+2—22no—
111C2×3−1+2#12311yescharacteristic
112C2×3−1+2C2×SD1611yescharacteristic
113C2×3−1+2AGL(1,ℤ/8)11yescharacteristic
114AGL(1,ℤ/9)—81no—
115C2×C3×D4—91no—
116C2×C3×D4—181no—
117C22×C3×C4—31no—
118C22×C3×C4—181no—
119C22×C3×C4C2×C3×D311yescharacteristic
120C2×#67—61no—
121C22×Dic3—61no—
122C2×D12—31no—
123C2×C3×C8—32no—
124C2×C3×C8C2×C3×D312yes—
125C3×C4⋊C4—92no—
126C3×C4⋊C4—91no—
127C3×C4⋊C4—181no—
128C3×C22⋊C4—181no—
129(C2×D3)⋊C4—61no—
130C4⋊Dic3—31no—
131C4⋊Dic3—32no—
132C4⋊Dic3—61no—
133C22×C32—21no—
134C22×C32—21no—
135C22×C32#20412yes—
136C22×C32C2×D1211yescharacteristic
137C2×C3×D3—121no—
138C2×C3×D3—121no—
139C2×C3×D3—121no—
140C32×C4—21no—
141C32×C4#23712yes—
142C3×Dic3—121no—
143C3×Dic3—122no—
144C22×C9—21no—
145C22×C9—21no—
146C22×C9—22no—
147C22×C9—21no—
148C22×C9—41no—
149C22×C9—41no—
150C22×C9C3×SD1612yes—
151C22×C9C2×C3×D411yescharacteristic
152C2×D9—41no—
153C2×D9—41no—
154C2×D9—41no—
155C4×C9—21no—
156C4×C9—22no—
157C4×C9—41no—
158C4×C9C3×#4112yes—
159Dic9—41no—
160Dic9—42no—
161#121—91no—
162C2×C4⋊C4—91no—
163#106—92no—
164#102—92no—
165C22⋊C8AGL(1,ℤ/9)11yescharacteristic, complemented
1663−1+2#48111yescharacteristic, complemented, Sylow
167C23×C3—31no—
168C23×C3—181no—
169C23×C3C3×D1211yescharacteristic
170C22×D3—61no—
171C3×D4—182no—
172C3×D4—362no—
173C2×C3×C4—22no—
174C2×C3×C4—31no—
175C2×C3×C4—31no—
176C2×C3×C4—62no—
177C2×C3×C4—181no—
178C2×C3×C4—182no—
179C2×C3×C4—362no—
180C2×C3×C4C3×D1211yescharacteristic
181C2×C3×C4C22×C3×D311yescharacteristic
182#67—122no—
183C2×Dic3—62no—
184C2×Dic3—61no—
185C2×Dic3—122no—
186D12—62no—
187C3×C8—22no—
188C3×C8—62no—
189C2×C32—22no—
190C2×C32#88611yescharacteristic
191C2×C32C2×#20411yescharacteristic
192C2×C32#90811yescharacteristic
193C3×D3—241no—
194C2×C9—22no—
195C2×C9—21no—
196C2×C9—21no—
197C2×C9—21no—
198C2×C9—42no—
199C2×C9C3×#12311yescharacteristic
200C2×C9C2×C3×SD1611yescharacteristic
201C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
202D9—81no—
203C2×D4—91no—
204C2×D4—181no—
205C22×C4—181no—
206C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
207C2×C8C2×AGL(1,ℤ/9)12yes—
208C4⋊C4—92no—
209C4⋊C4—91no—
210C4⋊C4—181no—
211C22⋊C4—181no—
212C22×C3—21no—
213C22×C3—21no—
214C22×C3—31no—
215C22×C3—32no—
216C22×C3—61no—
217C22×C3—61no—
218C22×C3—361no—
219C22×C3—361no—
220C22×C3—361no—
221C22×C3C3×#20412yes—
222C22×C3C2×C3×D1211yescharacteristic
223C2×D3—121no—
224C2×D3—121no—
225C2×D3—121no—
226C3×C4—21no—
227C3×C4—32no—
228C3×C4—61no—
229C3×C4—362no—
230C3×C4—361no—
231C3×C4C3×#23712yes—
232Dic3—121no—
233Dic3—122no—
234C32#473511yescharacteristic
235C9—21no—
236C9C3×#48111yescharacteristic, complemented
237C23—181no—
238C23C4⋊AGL(1,ℤ/9)11yescharacteristic
239D4—182no—
240D4—362no—
241C2×C4—22no—
242C2×C4—181no—
243C2×C4—182no—
244C2×C4—362no—
245C2×C4C4⋊AGL(1,ℤ/9)11yescharacteristic
246C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
247C8—22no—
248C2×C3—22no—
249C2×C3—31no—
250C2×C3—31no—
251C2×C3—31no—
252C2×C3—62no—
253C2×C3—721no—
254C2×C3C3×#88611yescharacteristic
255C2×C3C2×C3×#20411yescharacteristic
256C2×C3C3×#90811yescharacteristic
257D3—241no—
258C22—21no—
259C22—21no—
260C22—361no—
261C22—361no—
262C22—361no—
263C22#3664⋊C312yes—
264C22C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
265C4—21no—
266C4—361no—
267C4—362no—
268C4#3699⋊C312yes—
269C3—31no—
270C3C3×#473511yescharacteristic
271C2—22no—
272C2—721no—
273C2#63448⋊C311yescharacteristic
274C2C2×#3664⋊C311yescharacteristic
275C2#63470⋊C311yescharacteristic
subgroupG/H
centerC22C2×C4⋊AGL(1,ℤ/9)
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×C22⋊C8C2
socleC22×C3C2×C3×D12
prime radical#63492⋊C3C2
hypercenterC22⋊C8AGL(1,ℤ/9)
hypocenterC9C3×#481
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊(3−1+2⋊C4)C2semidirect
3. 3−1+2×C22⋊C8C2semidirect
4. #63380⋊C3C2semidirect
5. #63377⋊C3C2semidirect
7. #10588417C3semidirect
23. C2×C4⋊Dic9C2×C3semidirect
25. C9×C22⋊C8C2×C3semidirect
26. #63380C2×C3semidirect
27. #63377C2×C3semidirect
165. C22⋊C8AGL(1,ℤ/9)semidirect
166. 3−1+2#481semidirect
236. C9C3×#481semidirect

Maximal subgroups

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

subgroupindex
#63492⋊C32
C2×C4⋊(3−1+2⋊C4)2
3−1+2×C22⋊C82
#63380⋊C32
#63377⋊C32
C3×#47353
#105884173