Finite Group Databasebeta

← C9×#5401 #13146765

Subgroups — C9×#5401 #13146765

Order 1728.

232 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 45 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
1C9×C4⋊S4C211yesmaximal, characteristic
2C9×A4⋊Q8C211yesmaximal, characteristic, complemented
3C8×C9×A4C211yesmaximal, characteristic, complemented
4C3×#5401C311yesmaximal, characteristic
5C9×#465—31nomaximal
6C2×C9×S4—21no—
7C4×C9×A4C2211yescharacteristic
8C9×SL(2,ℤ/4)—21no—
9C9×#204—41nomaximal
10C3×C4⋊S4C2×C311yescharacteristic
11C3×A4⋊Q8C2×C311yescharacteristic
12C3×C8×A4C2×C311yescharacteristic
13C2×C9×SD16—31no—
14C22×C8×C9D311yescharacteristic, complemented
15C9×C22⋊Q8—31no—
16C9×#121—31no—
17C8×C22⋊C9—21no—
18C9×#106—31no—
19C9×#103—31no—
20C9×#102—31no—
21C2×C9×A4D411yescharacteristic
22C9×S4—41no—
23C9×D12—41no—
24C9×Dic6—41no—
25C3×C8×C9—41no—
26#5401C911yescharacteristic, direct product factor
27C3×#465—31no—
28C2×C3×S4—21no—
29C3×C4×A4C22×C311yescharacteristic
30C3×SL(2,ℤ/4)—21no—
31C3×#204—41no—
32C2×Q8×C9—31no—
33C2×D4×C9—31no—
34C2×D4×C9—61no—
35C22×C4×C9C2×D311yescharacteristic
36C4×C22⋊C9—21no—
37C9×SD16—61no—
38C9×SD16—61no—
39C2×C8×C9—31no—
40C2×C8×C9—31no—
41C9×C4⋊C4—31no—
42C9×C4⋊C4—31no—
43C9×C4⋊C4—61no—
44C9×C22⋊C4—61no—
45C9×C22⋊C4—61no—
46C2×D3×C9—81no—
47C9×A4SD1611yescharacteristic, complemented
48C3×C4×C9—41no—
49C9×Dic3—81no—
50C4⋊S4C2×C911yescharacteristic
51A4⋊Q8C2×C911yescharacteristic, complemented
52C2×C3×SD16—31no—
53C22×C3×C8C3×D311yescharacteristic
54C3×C22⋊Q8—31no—
55C3×#121—31no—
56C8×A4—21no—
57C8×A4C2×C911yescharacteristic, complemented
58C3×#106—31no—
59C3×#103—31no—
60C3×#102—31no—
61C2×C3×A4C3×D411yescharacteristic
62C3×S4—41no—
63C3×D12—41no—
64C3×Dic6—41no—
65C23×C9—61no—
66C23×C9D1211yescharacteristic
67C2×C22⋊C9—21no—
68C32×C8—41no—
69Q8×C9—61no—
70Q8×C9—61no—
71D4×C9—61no—
72D4×C9—61no—
73D4×C9—121no—
74D4×C9—121no—
75C2×C4×C9—31no—
76C2×C4×C9—31no—
77C2×C4×C9—61no—
78C2×C4×C9—61no—
79C2×C4×C9—61no—
80C8×C9—31no—
81C8×C9—81no—
82C8×C9S411yescharacteristic, complemented
83#465—31noSylow
84C2×C3×C9—41no—
85D3×C9—161no—
86C2×S4—21no—
87C2×C3×Q8—31no—
88C2×C3×D4—31no—
89C2×C3×D4—61no—
90C22×C3×C4C2×C3×D311yescharacteristic
91C4×A4—21no—
92C4×A4C22×C911yescharacteristic
93SL(2,ℤ/4)—21no—
94C3×SD16—61no—
95C3×SD16—61no—
96C2×C3×C8—31no—
97C2×C3×C8—31no—
98C3×C4⋊C4—31no—
99C3×C4⋊C4—31no—
100C3×C4⋊C4—61no—
101C3×C22⋊C4—61no—
102C3×C22⋊C4—61no—
103#204—41no—
104C2×C3×D3—81no—
105C3×A4C3×SD1611yescharacteristic
106C32×C4—41no—
107C3×Dic3—81no—
108C22×C9—31no—
109C22×C9—31no—
110C22×C9—121no—
111C22×C9—121no—
112C22×C9—121no—
113C22×C9#20411yescharacteristic, complemented
114C22⋊C9—21no—
115C4×C9—31no—
116C4×C9—81no—
117C4×C9—121no—
118C4×C9—121no—
119C4×C9—121no—
120C4×C9C2×S411yescharacteristic
121C2×SD16—31no—
122C22×C8D3×C911yescharacteristic, complemented
123C22⋊Q8—31no—
124#121—31no—
125#106—31no—
126#103—31no—
127#102—31no—
128C3×C9—41noSylow
129C23×C3—61no—
130C23×C3C3×D1211yescharacteristic
131C2×A4—21no—
132C2×A4D4×C911yescharacteristic
133S4—41no—
134C3×Q8—61no—
135C3×Q8—61no—
136C3×D4—61no—
137C3×D4—61no—
138C3×D4—121no—
139C3×D4—121no—
140C2×C3×C4—31no—
141C2×C3×C4—31no—
142C2×C3×C4—61no—
143C2×C3×C4—61no—
144C2×C3×C4—61no—
145D12—41no—
146Dic6—41no—
147C3×C8—31no—
148C3×C8—41no—
149C3×C8—81no—
150C3×C8C3×S411yescharacteristic
151C2×C32—41no—
152C3×D3—161no—
153C2×C9—31no—
154C2×C9—31no—
155C2×C9—81no—
156C2×C9—241no—
157C2×C9C4⋊S411yescharacteristic
158C2×Q8—31no—
159C2×D4—31no—
160C2×D4—61no—
161C22×C4C2×D3×C911yescharacteristic
162SD16—61no—
163SD16—61no—
164C2×C8—31no—
165C2×C8—31no—
166C4⋊C4—31no—
167C4⋊C4—31no—
168C4⋊C4—61no—
169C22⋊C4—61no—
170C22⋊C4—61no—
171C22×C3—31no—
172C22×C3—31no—
173C22×C3—121no—
174C22×C3—121no—
175C22×C3—121no—
176C22×C3C3×#20411yescharacteristic
177C2×D3—81no—
178A4—21no—
179A4C9×SD1611yescharacteristic, complemented
180C3×C4—31no—
181C3×C4—41no—
182C3×C4—81no—
183C3×C4—121no—
184C3×C4—121no—
185C3×C4—121no—
186C3×C4C2×C3×S411yescharacteristic
187Dic3—81no—
188C32—41no—
189C9—81no—
190C9#540111yescharacteristic, direct product factor
191C23—61no—
192C23C9×D1211yescharacteristic
193Q8—61no—
194Q8—61no—
195D4—61no—
196D4—61no—
197D4—121no—
198D4—121no—
199C2×C4—31no—
200C2×C4—31no—
201C2×C4—61no—
202C2×C4—61no—
203C2×C4—61no—
204C8—31no—
205C8C9×S411yescharacteristic, complemented
206C2×C3—31no—
207C2×C3—31no—
208C2×C3—41no—
209C2×C3—81no—
210C2×C3—241no—
211C2×C3C3×C4⋊S411yescharacteristic
212D3—161no—
213C22—31no—
214C22—31no—
215C22—121no—
216C22—121no—
217C22—121no—
218C22C9×#20411yescharacteristic, complemented
219C4—31no—
220C4—121no—
221C4—121no—
222C4—121no—
223C4C2×C9×S411yescharacteristic
224C3—41no—
225C3—81no—
226C3C3×#540111yescharacteristic
227C2—31no—
228C2—31no—
229C2—241no—
230C2C9×C4⋊S411yescharacteristic
subgroupG/H
centerC2×C9C4⋊S4
commutatorC4×A4C22×C9
FrattiniC3×C4C2×C3×S4
FittingC22×C8×C9D3
socleC23×C3C3×D12
prime radicalC3×C4⋊S4C2×C3
cosocleC3×C4×A4C22×C3
hypercenterC8×C9S4
hypocenterA4C9×SD16
LCSG ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C2×C9 ◁ C4×C9 ◁ C8×C9

Complements

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

NH ≅ G/Ntype
2. C9×A4⋊Q8C2semidirect
3. C8×C9×A4C2semidirect
14. C22×C8×C9D3semidirect
26. #5401C9direct (N × H)
47. C9×A4SD16semidirect
51. A4⋊Q8C2×C9semidirect
57. C8×A4C2×C9semidirect
82. C8×C9S4semidirect
113. C22×C9#204semidirect
122. C22×C8D3×C9semidirect
179. A4C9×SD16semidirect
190. C9#5401direct (N × H)
205. C8C9×S4semidirect
218. C22C9×#204semidirect

Maximal subgroups

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

subgroupindex
C9×C4⋊S42
C9×A4⋊Q82
C8×C9×A42
C3×#54013
C9×#4653
C9×#2044