Finite Group Databasebeta

← C3×#10596417 #13180394

Subgroups — C3×#10596417 #13180394

Order 1728.

542 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 68 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×GO+(4,𝔽2)C211yesmaximal, characteristic
2C2×C3×AΓL(1,𝔽9)C212yesmaximal, complemented
3C3×C22⋊PSU(3,𝔽2)C211yesmaximal, characteristic, complemented
4C3×C22⋊AGL(1,𝔽9)C211yesmaximal, characteristic, complemented
5C3×GO+(4,𝔽2)⋊C4C212yesmaximal, complemented
6#10596417C311yesmaximal, characteristic, direct product factor
7C22×C3×D32—21no—
8C22×C3×#157C2211yescharacteristic
9C2×C3×PSU(3,𝔽2)C2211yescharacteristic, complemented
10C2×C3×GO+(4,𝔽2)—21no—
11C2×C3×GO+(4,𝔽2)—24no—
12C2×C3×GO+(4,𝔽2)C2212yes—
13C2×C3×AGL(1,𝔽9)C2212yescomplemented
14C3×AΓL(1,𝔽9)—24no—
15C3×C22⋊#157—21no—
16C3×#3777—21no—
17C3×#3777C2211yescharacteristic, complemented
18C22×GO+(4,𝔽2)C2×C311yescharacteristic
19C2×AΓL(1,𝔽9)C2×C312yescomplemented
20C22⋊PSU(3,𝔽2)C2×C311yescharacteristic, complemented
21C22⋊AGL(1,𝔽9)C2×C311yescharacteristic, complemented
22GO+(4,𝔽2)⋊C4C2×C312yescomplemented
23C22×C3×C3⋊D3D411yescharacteristic
24C22×C32×D3—41no—
25C2×C3×D32—24no—
26C2×C3×D32—22no—
27C2×C3×D32—42no—
28C2×C3×D32—41no—
29C2×C3×#157—22no—
30C2×C3×#157—21no—
31C2×C3×#157—21no—
32C2×C3×#157D411yescharacteristic
33C2×C3×#157C2311yescharacteristic
34C3×PSU(3,𝔽2)—22no—
35C3×GO+(4,𝔽2)—24no—
36C3×GO+(4,𝔽2)—22no—
37C3×GO+(4,𝔽2)D414yes—
38C3×AGL(1,𝔽9)—22no—
39C3×#450—91nomaximal
40C23×C3×D3—121no—
41C22×D32—21no—
42C22×#157C22×C311yescharacteristic
43C2×PSU(3,𝔽2)C22×C311yescharacteristic, complemented
44C2×GO+(4,𝔽2)—21no—
45C2×GO+(4,𝔽2)—24no—
46C2×GO+(4,𝔽2)C22×C312yes—
47C2×AGL(1,𝔽9)C22×C312yescomplemented
48AΓL(1,𝔽9)—24no—
49C22⋊#157—21no—
50#3777—21no—
51#3777C22×C311yescharacteristic, complemented
52C22×C33SD1611yescharacteristic, complemented
53C2×C3×C3⋊D3—21no—
54C2×C3×C3⋊D3—21no—
55C2×C3×C3⋊D3SD1611yescharacteristic, complemented
56C2×C3×C3⋊D3C2×D411yescharacteristic
57C2×C32×D3—44no—
58C2×C32×D3—42no—
59C3×D32—24no—
60C3×D32—44no—
61C3×D32—42no—
62C3×#157—21no—
63C3×#157—41no—
64C3×#157—41no—
65C3×#157C2×D412yes—
66C22×C3×D4—91no—
67C2×C3×SD16—92no—
68C3×C22⋊Q8—91no—
69C3×#102—92no—
70C3×C22⋊C8—91no—
71C23×C32—121no—
72C22×C3⋊D3C3×D411yescharacteristic
73C22×C3×D3—41no—
74C22×C3×D3—41no—
75C22×C3×D3—81no—
76C22×C3×D3—121no—
77C22×C3×D3—122no—
78C22×C3×D3—124no—
79C22×C3×D3—122no—
80C22×C3×D3—124no—
81C2×D32—24no—
82C2×D32—22no—
83C2×D32—42no—
84C2×D32—41no—
85C2×#157—22no—
86C2×#157—21no—
87C2×#157—21no—
88C2×#157C3×D411yescharacteristic
89C2×#157C23×C311yescharacteristic
90PSU(3,𝔽2)—22no—
91GO+(4,𝔽2)—24no—
92GO+(4,𝔽2)—22no—
93GO+(4,𝔽2)C3×D414yes—
94AGL(1,𝔽9)—22no—
95#450—91noSylow
96C2×C33—21no—
97C2×C33C2×SD1611yescharacteristic
98C3×C3⋊D3—21no—
99C3×C3⋊D3#12011yescharacteristic
100C3×C3⋊D3AGL(1,ℤ/8)11yescharacteristic
101C32×D3—44no—
102C24×C3—181no—
103C23×D3—121no—
104C2×C3×Q8—91no—
105C2×C3×D4—92no—
106C2×C3×D4—181no—
107C2×C3×D4—184no—
108C22×C3×C4—91no—
109C3×SD16—184no—
110C2×C3×C8—92no—
111C3×C4⋊C4—91no—
112C3×C4⋊C4—181no—
113C3×C22⋊C4—181no—
114C22×C32—41no—
115C22×C32—81no—
116C22×C32—124no—
117C22×C32—122no—
118C22×C32C3×SD1611yescharacteristic, complemented
119C2×C3⋊D3—21no—
120C2×C3⋊D3—21no—
121C2×C3⋊D3C3×SD1611yescharacteristic, complemented
122C2×C3⋊D3C2×C3×D411yescharacteristic
123C2×C3×D3—44no—
124C2×C3×D3—44no—
125C2×C3×D3—42no—
126C2×C3×D3—42no—
127C2×C3×D3—84no—
128C2×C3×D3—82no—
129C2×C3×D3—121no—
130C2×C3×D3—121no—
131C2×C3×D3—124no—
132C2×C3×D3—122no—
133C2×C3×D3—122no—
134C2×C3×D3—124no—
135C2×C3×D3—128no—
136D32—24no—
137D32—44no—
138D32—42no—
139#157—21no—
140#157—41no—
141#157—41no—
142#157C2×C3×D412yes—
143C22×D4—91no—
144C2×SD16—92no—
145C22⋊Q8—91no—
146#102—92no—
147C22⋊C8—91no—
148C33#45011yescharacteristic, complemented, Sylow
149C23×C3—91no—
150C23×C3—121no—
151C23×C3—121no—
152C23×C3—182no—
153C23×C3—184no—
154C23×C3—241no—
155C23×C3—361no—
156C23×C3—362no—
157C22×D3—41no—
158C22×D3—121no—
159C22×D3—122no—
160C22×D3—124no—
161C22×D3—122no—
162C22×D3—124no—
163C3×Q8—182no—
164C3×D4—94no—
165C3×D4—182no—
166C3×D4—184no—
167C2×C3×C4—91no—
168C2×C3×C4—91no—
169C2×C3×C4—182no—
170C2×C3×C4—181no—
171C2×C3×C4—181no—
172C3×C8—182no—
173C2×C32—21no—
174C2×C32—42no—
175C2×C32—41no—
176C2×C32—82no—
177C2×C32—81no—
178C2×C32—124no—
179C2×C32C2×C3×SD1611yescharacteristic
180C3⋊D3—21no—
181C3⋊D3C3×#12011yescharacteristic
182C3⋊D3C3×AGL(1,ℤ/8)11yescharacteristic
183C3×D3—44no—
184C3×D3—44no—
185C3×D3—84no—
186C3×D3—121no—
187C3×D3—121no—
188C3×D3—122no—
189C24—181no—
190C2×Q8—91no—
191C2×D4—92no—
192C2×D4—181no—
193C2×D4—184no—
194C22×C4—91no—
195SD16—184no—
196C2×C8—92no—
197C4⋊C4—91no—
198C4⋊C4—181no—
199C22⋊C4—181no—
200C22×C3—41no—
201C22×C3—81no—
202C22×C3—91no—
203C22×C3—91no—
204C22×C3—124no—
205C22×C3—124no—
206C22×C3—122no—
207C22×C3—122no—
208C22×C3—181no—
209C22×C3—181no—
210C22×C3—184no—
211C22×C3—244no—
212C22×C3—242no—
213C22×C3—362no—
214C22×C3—364no—
215C22×C3AΓL(1,𝔽9)11yescharacteristic, complemented
216C2×D3—44no—
217C2×D3—42no—
218C2×D3—121no—
219C2×D3—121no—
220C2×D3—124no—
221C2×D3—122no—
222C2×D3—122no—
223C2×D3—124no—
224C2×D3—128no—
225C3×C4—92no—
226C3×C4—181no—
227C3×C4—361no—
228C3×C4—361no—
229C32—41no—
230C32—81no—
231C32C3×#45011yescharacteristic, complemented
232C23—91no—
233C23—121no—
234C23—182no—
235C23—184no—
236C23—361no—
237C23—362no—
238Q8—182no—
239D4—94no—
240D4—182no—
241D4—184no—
242C2×C4—91no—
243C2×C4—91no—
244C2×C4—182no—
245C2×C4—181no—
246C2×C4—181no—
247C8—182no—
248C2×C3—21no—
249C2×C3—42no—
250C2×C3—41no—
251C2×C3—82no—
252C2×C3—81no—
253C2×C3—91no—
254C2×C3—91no—
255C2×C3—124no—
256C2×C3—124no—
257C2×C3—181no—
258C2×C3—244no—
259C2×C3C2×AΓL(1,𝔽9)11yescharacteristic
260D3—44no—
261D3—121no—
262D3—121no—
263D3—122no—
264C22—91no—
265C22—91no—
266C22—124no—
267C22—122no—
268C22—181no—
269C22—181no—
270C22—184no—
271C22—362no—
272C22—364no—
273C22C3×AΓL(1,𝔽9)11yescharacteristic, complemented
274C4—92no—
275C4—181no—
276C4—361no—
277C4—361no—
278C3—41no—
279C3—81no—
280C3#1059641711yescharacteristic, direct product factor
281C2—21no—
282C2—91no—
283C2—91no—
284C2—124no—
285C2—181no—
286C2C2×C3×AΓL(1,𝔽9)11yescharacteristic
subgroupG/H
centerC2×C3C2×AΓL(1,𝔽9)
commutator, cosocleC2×#157C23×C3
FrattiniC2C2×C3×AΓL(1,𝔽9)
FittingC22×C33SD16
socleC2×C33C2×SD16
prime radicalC22×C3×GO+(4,𝔽2)C2
hypercenterC22×C3AΓL(1,𝔽9)
hypocenterC32C3×#450
LCSG ▷ C2×#157 ▷ C3⋊D3 ▷ C32
UCSC1 ◁ C2×C3 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
2. C2×C3×AΓL(1,𝔽9)C2semidirect
3. C3×C22⋊PSU(3,𝔽2)C2semidirect
4. C3×C22⋊AGL(1,𝔽9)C2semidirect
5. C3×GO+(4,𝔽2)⋊C4C2semidirect
6. #10596417C3direct (N × H)
9. C2×C3×PSU(3,𝔽2)C22semidirect
13. C2×C3×AGL(1,𝔽9)C22semidirect
17. C3×#3777C22semidirect
19. C2×AΓL(1,𝔽9)C2×C3semidirect
20. C22⋊PSU(3,𝔽2)C2×C3semidirect
21. C22⋊AGL(1,𝔽9)C2×C3semidirect
22. GO+(4,𝔽2)⋊C4C2×C3semidirect
43. C2×PSU(3,𝔽2)C22×C3semidirect
47. C2×AGL(1,𝔽9)C22×C3semidirect
51. #3777C22×C3semidirect
52. C22×C33SD16semidirect
55. C2×C3×C3⋊D3SD16semidirect
118. C22×C32C3×SD16semidirect
121. C2×C3⋊D3C3×SD16semidirect
148. C33#450semidirect
215. C22×C3AΓL(1,𝔽9)semidirect
231. C32C3×#450semidirect
273. C22C3×AΓL(1,𝔽9)semidirect
280. C3#10596417direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×GO+(4,𝔽2)2
C2×C3×AΓL(1,𝔽9)2
C3×C22⋊PSU(3,𝔽2)2
C3×C22⋊AGL(1,𝔽9)2
C3×GO+(4,𝔽2)⋊C42
#105964173
C3×#4509