Finite Group Databasebeta

← C9×#4748 #13141670

Subgroups — C9×#4748 #13141670

Order 1728.

358 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C9×#214C211yesmaximal, characteristic, complemented
2C9×#932C211yesmaximal, characteristic
3C9×#909C211yesmaximal, characteristic, complemented
4C9×#907C211yesmaximal, characteristic, complemented
5C3×C9×#100C211yesmaximal, characteristic, complemented
6C9×#824C211yesmaximal, characteristic, complemented
7C9×#822C211yesmaximal, characteristic, complemented
8C3×#4748C311yesmaximal, characteristic
9C9×#471—31nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C9×#235C2211yescharacteristic
12C2×C9×Dic6C2211yescharacteristic
13C3×C9×M16C2211yescharacteristic
14C3×C9×M16C2211yescharacteristic
15C9×#214—21no—
16C9×#214—21no—
17C9×#214—21no—
18C9×#213—21no—
19C9×#208C2211yescharacteristic
20C2×C9×C3⋊C8C2211yescharacteristic
21C9×Dic12—21no—
22C9×#204—21no—
23C9×ΓL(1,𝔽25)—21no—
24D3×C8×C9—21no—
25C2×C3×#214C2×C311yescharacteristic
26C3×#932C2×C311yescharacteristic
27C3×#909C2×C311yescharacteristic
28C3×#907C2×C311yescharacteristic
29C32×#100C2×C311yescharacteristic
30C3×#824C2×C311yescharacteristic
31C3×#822C2×C311yescharacteristic
32C9×#137—31no—
33C9×AGL(1,ℤ/8)—31no—
34C2×C9×SD16—31no—
35C9×#131—31no—
36C9×#108—31no—
37C9×#101—31no—
38C9×#100—21no—
39C9×#100D311yescharacteristic, complemented
40C23×C3×C9—21no—
41C3×D4×C9—21no—
42C3×D4×C9—21no—
43C2×C3×C4×C9C2311yescharacteristic
44C9×#67—21no—
45C2×C9×Dic3—21no—
46C9×D12D411yescharacteristic
47C4×D3×C9—21no—
48C9×Dic6—21no—
49C9×Dic6—21no—
50C9×Dic6D411yescharacteristic
51C3×C8×C9—21no—
52C3×C8×C9—21no—
53C9×C3⋊C8—21no—
54C9×C3⋊C8D411yescharacteristic
55C9×C3⋊C8D411yescharacteristic
56C3×#471—31no—
57#4748C911yescharacteristic, direct product factor
58C2×C32×D4C22×C311yescharacteristic
59C3×#235C22×C311yescharacteristic
60C2×C3×Dic6C22×C311yescharacteristic
61C32×M16C22×C311yescharacteristic
62C32×M16C22×C311yescharacteristic
63C3×#214—21no—
64C3×#214—21no—
65C3×#214—21no—
66C3×#213—21no—
67C3×#208C22×C311yescharacteristic
68C2×C3×C3⋊C8C22×C311yescharacteristic
69C3×Dic12—21no—
70C3×#204—21no—
71C3×ΓL(1,𝔽25)—21no—
72C3×D3×C8—21no—
73C9×#41—31no—
74C2×Q8×C9—31no—
75C2×D4×C9—21no—
76C2×D4×C9C2×D311yescharacteristic
77C9×Q16—61no—
78C9×SD16—61no—
79C9×SD16—61no—
80C9×SD16—61no—
81C9×SD16—61no—
82C9×D8—61no—
83C9×M16—21no—
84C9×M16—21no—
85C9×M16—31no—
86C9×M16—61no—
87C9×M16C2×D311yescharacteristic
88C9×M16C2×D311yescharacteristic
89C2×C8×C9—31no—
90C2×C8×C9—61no—
91C22×C3×C9—41no—
92C22×C3×C9—81no—
93C22×C3×C9#4111yescharacteristic
94C2×D3×C9—21no—
95C3×C4×C9C2×D411yescharacteristic
96C3×C4×C9C2×D411yescharacteristic
97C9×Dic3—21no—
98C9×Dic3—41no—
99C3×#137—31no—
100C3×AGL(1,ℤ/8)—31no—
101C2×C3×SD16—31no—
102C3×#131—31no—
103C2×#214C2×C911yescharacteristic, complemented
104#932C2×C911yescharacteristic
105#909C2×C911yescharacteristic, complemented
106#907C2×C911yescharacteristic, complemented
107C3×#108—31no—
108C3×#101—31no—
109C3×#100—21no—
110C3×#100C2×C911yescharacteristic, complemented
111C3×#100C3×D311yescharacteristic
112#824C2×C911yescharacteristic, complemented
113#822C2×C911yescharacteristic, complemented
114C23×C32—21no—
115C32×D4—21no—
116C32×D4—21no—
117C2×C32×C4C23×C311yescharacteristic
118C3×#67—21no—
119C2×C3×Dic3—21no—
120C3×D12C3×D411yescharacteristic
121C3×C4×D3—21no—
122C3×Dic6—21no—
123C3×Dic6—21no—
124C3×Dic6C3×D411yescharacteristic
125C23×C9—22no—
126C23×C9—21no—
127C32×C8—21no—
128C32×C8—21no—
129C3×C3⋊C8—21no—
130C3×C3⋊C8C3×D411yescharacteristic
131C3×C3⋊C8C3×D411yescharacteristic
132Q8×C9—31no—
133Q8×C9—61no—
134Q8×C9—61no—
135D4×C9—21no—
136D4×C9—21no—
137D4×C9—31no—
138D4×C9—41no—
139D4×C9—41no—
140D4×C9—61no—
141C2×C4×C9—21no—
142C2×C4×C9—61no—
143C2×C4×C9—61no—
144C2×C4×C9C22×D311yescharacteristic
145C8×C9—21no—
146C8×C9—21no—
147C8×C9—31no—
148C8×C9—31no—
149C8×C9—41no—
150C8×C9—41no—
151C8×C9—61no—
152#471—31noSylow
153C2×C3×C9—21no—
154C2×C3×C9—81no—
155C2×C3×C9#12111yescharacteristic
156D3×C9—41no—
157C3×#41—31no—
158C2×C3×Q8—31no—
159C2×C3×D4—21no—
160C2×C3×D4C22×C911yescharacteristic
161C2×C3×D4C2×C3×D311yescharacteristic
162#235C22×C911yescharacteristic
163C2×Dic6C22×C911yescharacteristic
164C3×Q16—61no—
165C3×SD16—61no—
166C3×SD16—61no—
167C3×SD16—61no—
168C3×SD16—61no—
169C3×D8—61no—
170C3×M16—21no—
171C3×M16—21no—
172C3×M16—31no—
173C3×M16—61no—
174C3×M16C22×C911yescharacteristic
175C3×M16C22×C911yescharacteristic
176C3×M16C2×C3×D311yescharacteristic
177C3×M16C2×C3×D311yescharacteristic
178C2×C3×C8—31no—
179C2×C3×C8—61no—
180#214—21no—
181#214—21no—
182#214—21no—
183#213—21no—
184#208C22×C911yescharacteristic
185C2×C3⋊C8C22×C911yescharacteristic
186Dic12—21no—
187#204—21no—
188ΓL(1,𝔽25)—21no—
189D3×C8—21no—
190C22×C32—41no—
191C22×C32—81no—
192C22×C32C3×#4111yescharacteristic
193C2×C3×D3—21no—
194C32×C4C2×C3×D411yescharacteristic
195C32×C4C2×C3×D411yescharacteristic
196C3×Dic3—21no—
197C3×Dic3—41no—
198C22×C9—21no—
199C22×C9—42no—
200C22×C9—41no—
201C22×C9—61no—
202C22×C9—81no—
203C22×C9—82no—
204C22×C9#23511yescharacteristic
205C4×C9—21no—
206C4×C9—21no—
207C4×C9—61no—
208C4×C9—121no—
209C4×C9D3×D411yescharacteristic
210C4×C9D3×D411yescharacteristic
211#137—31no—
212AGL(1,ℤ/8)—31no—
213C2×SD16—31no—
214#131—31no—
215#108—31no—
216#101—31no—
217#100D3×C911yescharacteristic, complemented
218C3×C9#47111yescharacteristic, complemented, Sylow
219C23×C3—22no—
220C23×C3—21no—
221C23×C3—21no—
222C3×Q8—31no—
223C3×Q8—61no—
224C3×Q8—61no—
225C3×D4—21no—
226C3×D4—21no—
227C3×D4—21no—
228C3×D4—21no—
229C3×D4—31no—
230C3×D4—41no—
231C3×D4—41no—
232C3×D4—61no—
233C2×C3×C4—21no—
234C2×C3×C4—61no—
235C2×C3×C4—61no—
236C2×C3×C4C23×C911yescharacteristic
237C2×C3×C4C22×C3×D311yescharacteristic
238#67—21no—
239C2×Dic3—21no—
240D12D4×C911yescharacteristic
241C4×D3—21no—
242Dic6—21no—
243Dic6—21no—
244Dic6D4×C911yescharacteristic
245C3×C8—21no—
246C3×C8—21no—
247C3×C8—21no—
248C3×C8—21no—
249C3×C8—31no—
250C3×C8—31no—
251C3×C8—41no—
252C3×C8—41no—
253C3×C8—61no—
254C3⋊C8—21no—
255C3⋊C8D4×C911yescharacteristic
256C3⋊C8D4×C911yescharacteristic
257C2×C32—21no—
258C2×C32—81no—
259C2×C32C3×#12111yescharacteristic
260C3×D3—41no—
261C2×C9—21no—
262C2×C9—21no—
263C2×C9—41no—
264C2×C9—82no—
265C2×C9—81no—
266C2×C9—121no—
267C2×C9#88411yescharacteristic
268#41—31no—
269C2×Q8—31no—
270C2×D4C2×D3×C911yescharacteristic
271Q16—61no—
272SD16—61no—
273SD16—61no—
274SD16—61no—
275SD16—61no—
276D8—61no—
277M16—31no—
278M16—61no—
279M16C2×D3×C911yescharacteristic
280M16C2×D3×C911yescharacteristic
281C2×C8—31no—
282C2×C8—61no—
283C22×C3—21no—
284C22×C3—42no—
285C22×C3—41no—
286C22×C3—41no—
287C22×C3—61no—
288C22×C3—81no—
289C22×C3—81no—
290C22×C3—82no—
291C22×C3C9×#4111yescharacteristic
292C22×C3C3×#23511yescharacteristic
293C2×D3—21no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—61no—
297C3×C4—121no—
298C3×C4C2×D4×C911yescharacteristic
299C3×C4C2×D4×C911yescharacteristic
300C3×C4C3×D3×D411yescharacteristic
301C3×C4C3×D3×D411yescharacteristic
302Dic3—21no—
303Dic3—41no—
304C32C3×#47111yescharacteristic
305C9—21no—
306C9#474811yescharacteristic, direct product factor
307C23—21no—
308Q8—31no—
309Q8—61no—
310Q8—61no—
311D4—21no—
312D4—21no—
313D4—31no—
314D4—61no—
315C2×C4—61no—
316C2×C4—61no—
317C2×C4C22×D3×C911yescharacteristic
318C8—21no—
319C8—21no—
320C8—31no—
321C8—31no—
322C8—61no—
323C2×C3—21no—
324C2×C3—21no—
325C2×C3—21no—
326C2×C3—41no—
327C2×C3—82no—
328C2×C3—81no—
329C2×C3—81no—
330C2×C3—121no—
331C2×C3C9×#12111yescharacteristic
332C2×C3C3×#88411yescharacteristic
333D3—41no—
334C22—41no—
335C22—61no—
336C22—81no—
337C22C9×#23511yescharacteristic
338C4—61no—
339C4—121no—
340C4D3×D4×C911yescharacteristic
341C4D3×D4×C911yescharacteristic
342C3—21no—
343C3C9×#47111yescharacteristic, complemented
344C3C3×#474811yescharacteristic
345C2—21no—
346C2—81no—
347C2—121no—
348C2C9×#88411yescharacteristic
subgroupG/H
centerC2×C9#884
commutatorC2×C3×C4C23×C9
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×#100C2
socleC2×C32C3×#121
prime radicalC3×#932C2×C3
cosocleC2×C32×C4C23×C3
hypercenterC9×#100D3
hypocenterC3C9×#471
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C9 ◁ C2×C4×C9 ◁ C9×#100

Complements

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

NH ≅ G/Ntype
1. C2×C9×#214C2semidirect
3. C9×#909C2semidirect
4. C9×#907C2semidirect
5. C3×C9×#100C2semidirect
6. C9×#824C2semidirect
7. C9×#822C2semidirect
39. C9×#100D3semidirect
57. #4748C9direct (N × H)
103. C2×#214C2×C9semidirect
105. #909C2×C9semidirect
106. #907C2×C9semidirect
110. C3×#100C2×C9semidirect
112. #824C2×C9semidirect
113. #822C2×C9semidirect
217. #100D3×C9semidirect
218. C3×C9#471semidirect
306. C9#4748direct (N × H)
343. C3C9×#471semidirect

Maximal subgroups

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

subgroupindex
C2×C9×#2142
C9×#9322
C9×#9092
C9×#9072
C3×C9×#1002
C9×#8242
C9×#8222
C3×#47483
C9×#4713