Finite Group Databasebeta

← #10588563⋊C3 #13142244

Subgroups — #10588563⋊C3 #13142244

Order 1728.

420 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 100 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
1C2×D72⋊C3C211yesmaximal, characteristic, complemented
2C2×C8×AGL(1,ℤ/9)C211yesmaximal, characteristic, complemented
3#63457⋊C3C212yesmaximal, complemented
43−1+2×#107C211yesmaximal, characteristic, complemented
5#63369⋊C3C212yesmaximal, complemented
6C3×#4883—31nomaximal
7#10588563C311yesmaximal, characteristic, complemented
8C2×C4⋊AGL(1,ℤ/9)—22no—
9C2×C4⋊AGL(1,ℤ/9)C2212yes—
10C2×C4×AGL(1,ℤ/9)C2211yescharacteristic
11C2×C8×3−1+2C2211yescharacteristic
12C4⋊C4×3−1+2C2212yes—
13(C2×AGL(1,ℤ/9))⋊C4—22no—
14C2×3−1+2⋊C8C2211yescharacteristic
15D72⋊C3—22no—
16C8×AGL(1,ℤ/9)—22no—
17C2×C3×D24—31no—
18C2×C3×D3×C8—31no—
19C3×#895—32no—
20C32×#107D311yescharacteristic
21C3×#809—32no—
22C2×D72C2×C311yescharacteristic, complemented
23C2×C8×D9C2×C311yescharacteristic, complemented
24#63457C2×C312yescomplemented
25C9×#107—21no—
26C9×#107C2×C311yescharacteristic, complemented
27#63369C2×C312yescomplemented
28C22×AGL(1,ℤ/9)—22no—
29C22×AGL(1,ℤ/9)D411yescharacteristic
30C2×C4×3−1+2—22no—
31C2×C4×3−1+2C2311yescharacteristic
32C2×3−1+2⋊C4D411yescharacteristic
33C4⋊AGL(1,ℤ/9)—22no—
34C4⋊AGL(1,ℤ/9)—22no—
35C4⋊AGL(1,ℤ/9)—44no—
36C4×AGL(1,ℤ/9)—21no—
37C4×AGL(1,ℤ/9)—21no—
38C8×3−1+2D412yescomplemented
393−1+2⋊C8—21no—
40C3×#466—91no—
41#4883—31no—
42C2×C3×D12—32no—
43C2×C3×D12—62no—
44C2×C3×C4×D3—31no—
45C2×C32×C8C2×D311yescharacteristic
46C32×C4⋊C4C2×D312yes—
47C3×(C2×D3)⋊C4—62no—
48C2×C3×C3⋊C8—31no—
49C3×D24—62no—
50C3×D3×C8—62no—
51C2×D36—22no—
52C2×D36C22×C312yes—
53C2×C4×D9C22×C311yescharacteristic
54C2×C8×C9—21no—
55C2×C8×C9C22×C311yescharacteristic
56C9×C4⋊C4—22no—
57C9×C4⋊C4C22×C312yes—
58(C2×D9)⋊C4—22no—
59C2×C9⋊C8C22×C311yescharacteristic
60D72—22no—
61C8×D9—22no—
62C22×3−1+2C2×D411yescharacteristic
63C2×AGL(1,ℤ/9)—21no—
64C2×AGL(1,ℤ/9)—21no—
65C2×AGL(1,ℤ/9)—42no—
66C2×AGL(1,ℤ/9)—42no—
67C2×AGL(1,ℤ/9)—42no—
68C2×AGL(1,ℤ/9)D812yescomplemented
69C4×3−1+2—42no—
70C4×3−1+2C2×D411yescharacteristic
71C4×3−1+2#4111yescharacteristic
723−1+2⋊C4—21no—
73C2×C3×D8—91no—
74C22×C3×C8—91no—
75C3×#121—92no—
76C2×D24—31no—
77C2×D3×C8—31no—
78#895—32no—
79C3×#107—31no—
80C3×#107C3×D311yescharacteristic
81C3×#102—92no—
82#809—32no—
83C22×C3×D3—31no—
84C22×C3×D3—62no—
85C2×C32×C4—22no—
86C2×C32×C4C22×D311yescharacteristic
87C2×C3×Dic3—31no—
88C3×D12—62no—
89C3×D12—62no—
90C3×D12—124no—
91C3×C4×D3—61no—
92C3×C4×D3—61no—
93C22×D9—22no—
94C22×D9C3×D411yescharacteristic
95C32×C8D1212yes—
96C3×C3⋊C8—61no—
97C2×C4×C9—22no—
98C2×C4×C9—21no—
99C2×C4×C9—42no—
100C2×C4×C9C23×C311yescharacteristic
101C2×Dic9C3×D411yescharacteristic
102D36—22no—
103D36—22no—
104D36—44no—
105C4×D9—21no—
106C4×D9—21no—
107C8×C9—22no—
108C8×C9C3×D412yescomplemented
109C9⋊C8—21no—
110#466—91noSylow
111C2×3−1+2#12111yescharacteristic
112C2×3−1+2C2×D811yescharacteristic
113C2×3−1+2#13511yescharacteristic
114AGL(1,ℤ/9)—22no—
115AGL(1,ℤ/9)—82no—
116C2×C3×D4—92no—
117C2×C3×D4—182no—
118C22×C3×C4—91no—
119C2×D12—32no—
120C2×D12—62no—
121C2×C4×D3—31no—
122C3×D8—182no—
123C2×C3×C8—31no—
124C2×C3×C8—91no—
125C2×C3×C8—182no—
126C2×C3×C8C2×C3×D311yescharacteristic
127C3×C4⋊C4—32no—
128C3×C4⋊C4C2×C3×D312yes—
129C3×C22⋊C4—182no—
130(C2×D3)⋊C4—62no—
131C2×C3⋊C8—31no—
132D24—62no—
133D3×C8—62no—
134C22×C32D3×D411yescharacteristic
135C2×C3×D3—32no—
136C2×C3×D3—61no—
137C2×C3×D3—61no—
138C2×C3×D3—122no—
139C2×C3×D3—122no—
140C2×C3×D3—122no—
141C32×C4—42no—
142C32×C4C2×D1211yescharacteristic
143C32×C4#23911yescharacteristic
144C3×Dic3—61no—
145C22×C9—21no—
146C22×C9C2×C3×D411yescharacteristic
147C2×D9—21no—
148C2×D9—21no—
149C2×D9—42no—
150C2×D9—42no—
151C2×D9—42no—
152C2×D9C3×D812yescomplemented
153C4×C9—21no—
154C4×C9—21no—
155C4×C9—42no—
156C4×C9—82no—
157C4×C9C2×C3×D411yescharacteristic
158C4×C9C3×#4111yescharacteristic
159Dic9—21no—
160C2×D8—91no—
161C22×C8—91no—
162#121—92no—
163#107AGL(1,ℤ/9)11yescharacteristic, complemented
164#102—92no—
1653−1+2#46611yescharacteristic, complemented, Sylow
166C23×C3—91no—
167C23×C3—182no—
168C22×D3—31no—
169C22×D3—62no—
170C3×D4—182no—
171C3×D4—182no—
172C3×D4—364no—
173C2×C3×C4—22no—
174C2×C3×C4—31no—
175C2×C3×C4—62no—
176C2×C3×C4—91no—
177C2×C3×C4—181no—
178C2×C3×C4—181no—
179C2×C3×C4C22×C3×D311yescharacteristic
180C2×Dic3—31no—
181D12—62no—
182D12—62no—
183D12—124no—
184C4×D3—61no—
185C4×D3—61no—
186C3×C8—32no—
187C3×C8—181no—
188C3×C8C3×D1212yes—
189C3⋊C8—61no—
190C2×C32#89511yescharacteristic
191C2×C32D3×D811yescharacteristic
192C2×C32#91911yescharacteristic
193C3×D3—62no—
194C3×D3—242no—
195C2×C9—21no—
196C2×C9—21no—
197C2×C9—21no—
198C2×C9C3×#12111yescharacteristic
199C2×C9C2×C3×D811yescharacteristic
200C2×C9C3×#13511yescharacteristic
201D9—22no—
202D9—82no—
203C2×D4—92no—
204C2×D4—182no—
205C22×C4—91no—
206D8—182no—
207C2×C8—91no—
208C2×C8—182no—
209C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
210C4⋊C4C2×AGL(1,ℤ/9)12yes—
211C22⋊C4—182no—
212C22×C3—31no—
213C22×C3—92no—
214C22×C3—181no—
215C22×C3—181no—
216C22×C3—362no—
217C22×C3—362no—
218C22×C3—362no—
219C22×C3C3×D3×D411yescharacteristic
220C2×D3—32no—
221C2×D3—61no—
222C2×D3—61no—
223C2×D3—122no—
224C2×D3—122no—
225C2×D3—122no—
226C3×C4—31no—
227C3×C4—31no—
228C3×C4—42no—
229C3×C4—122no—
230C3×C4—181no—
231C3×C4C2×C3×D1211yescharacteristic
232C3×C4C3×#23911yescharacteristic
233Dic3—61no—
234C32#488311yescharacteristic
235C9—21no—
236C9C3×#46611yescharacteristic, complemented
237C23—91no—
238C23—182no—
239D4—182no—
240D4—182no—
241D4—364no—
242C2×C4—22no—
243C2×C4—91no—
244C2×C4—181no—
245C2×C4—181no—
246C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
247C8—181no—
248C8C4⋊AGL(1,ℤ/9)12yescomplemented
249C2×C3—31no—
250C2×C3—31no—
251C2×C3—31no—
252C2×C3—182no—
253C2×C3—722no—
254C2×C3C3×#89511yescharacteristic
255C2×C3C3×D3×D811yescharacteristic
256C2×C3C3×#91911yescharacteristic
257D3—62no—
258D3—242no—
259C22—92no—
260C22—181no—
261C22—181no—
262C22—362no—
263C22—362no—
264C22—362no—
265C22D4×AGL(1,ℤ/9)11yescharacteristic
266C4—42no—
267C4—181no—
268C4C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
269C4#8872411yescharacteristic
270C3—31no—
271C3C3×#488311yescharacteristic
272C2—182no—
273C2—722no—
274C2#63457⋊C311yescharacteristic
275C2D8×AGL(1,ℤ/9)11yescharacteristic
276C2#63481⋊C311yescharacteristic
subgroupG/H
centerC22D4×AGL(1,ℤ/9)
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#107C2
socleC22×C3C3×D3×D4
hypercenter#107AGL(1,ℤ/9)
hypocenterC9C3×#466
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
1. C2×D72⋊C3C2semidirect
2. C2×C8×AGL(1,ℤ/9)C2semidirect
3. #63457⋊C3C2semidirect
4. 3−1+2×#107C2semidirect
5. #63369⋊C3C2semidirect
7. #10588563C3semidirect
22. C2×D72C2×C3semidirect
23. C2×C8×D9C2×C3semidirect
24. #63457C2×C3semidirect
26. C9×#107C2×C3semidirect
27. #63369C2×C3semidirect
38. C8×3−1+2D4semidirect
68. C2×AGL(1,ℤ/9)D8semidirect
108. C8×C9C3×D4semidirect
152. C2×D9C3×D8semidirect
163. #107AGL(1,ℤ/9)semidirect
165. 3−1+2#466semidirect
236. C9C3×#466semidirect
248. C8C4⋊AGL(1,ℤ/9)semidirect

Maximal subgroups

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

subgroupindex
C2×D72⋊C32
C2×C8×AGL(1,ℤ/9)2
#63457⋊C32
3−1+2×#1072
#63369⋊C32
C3×#48833
#105885633