Finite Group Databasebeta

← C9×#4754 #13141695

Subgroups — C9×#4754 #13141695

Order 1728.

350 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C9×#942C211yesmaximal, characteristic, complemented
2C2×C9×#215C211yesmaximal, characteristic, complemented
3C9×#908C211yesmaximal, characteristic
4C9×#907C211yesmaximal, characteristic, complemented
5C3×C9×#101C211yesmaximal, characteristic, complemented
6C9×#823C211yesmaximal, characteristic, complemented
7C9×#822C211yesmaximal, characteristic, complemented
8C3×#4754C311yesmaximal, characteristic
9C9×#471—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C9×#235C2211yescharacteristic
12C2×C9×D12C2211yescharacteristic
13C3×C9×M16C2211yescharacteristic
14C3×C9×M16C2211yescharacteristic
15C9×#216—21no—
16C9×#215—21no—
17C9×#215—21no—
18C9×#215—21no—
19C9×#208C2211yescharacteristic
20C2×C9×C3⋊C8C2211yescharacteristic
21C9×D24—21no—
22C9×#204—21no—
23C9×ΓL(1,𝔽25)—21no—
24D3×C8×C9—21no—
25C3×#942C2×C311yescharacteristic
26C2×C3×#215C2×C311yescharacteristic
27C3×#908C2×C311yescharacteristic
28C3×#907C2×C311yescharacteristic
29C32×#101C2×C311yescharacteristic
30C3×#823C2×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—21no—
38C9×#101D311yescharacteristic, complemented
39C9×#100—31no—
40C22×D3×C9—21no—
41C3×Q8×C9—21no—
42C3×Q8×C9—21no—
43C2×C3×C4×C9—21no—
44C2×C3×C4×C9C2311yescharacteristic
45C9×#67—21no—
46C9×D12—21no—
47C9×D12—21no—
48C9×D12D411yescharacteristic
49C4×D3×C9—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#4754C911yescharacteristic, direct product factor
58C2×C32×Q8C22×C311yescharacteristic
59C3×#235C22×C311yescharacteristic
60C2×C3×D12C22×C311yescharacteristic
61C32×M16C22×C311yescharacteristic
62C32×M16C22×C311yescharacteristic
63C3×#216—21no—
64C3×#215—21no—
65C3×#215—21no—
66C3×#215—21no—
67C3×#208C22×C311yescharacteristic
68C2×C3×C3⋊C8C22×C311yescharacteristic
69C3×D24—21no—
70C3×#204—21no—
71C3×ΓL(1,𝔽25)—21no—
72C3×D3×C8—21no—
73C9×#41—31no—
74C2×Q8×C9—21no—
75C2×Q8×C9C2×D311yescharacteristic
76C2×D4×C9—31no—
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#4111yescharacteristic
92C2×D3×C9—21no—
93C2×D3×C9—41no—
94C2×D3×C9—81no—
95C3×C4×C9—41no—
96C3×C4×C9C2×D411yescharacteristic
97C3×C4×C9C2×D411yescharacteristic
98C9×Dic3—21no—
99C3×#137—31no—
100C3×AGL(1,ℤ/8)—31no—
101C2×C3×SD16—31no—
102C3×#131—31no—
103#942C2×C911yescharacteristic, complemented
104C2×#215C2×C911yescharacteristic, complemented
105#908C2×C911yescharacteristic
106#907C2×C911yescharacteristic, complemented
107C3×#108—31no—
108C3×#101—21no—
109C3×#101C2×C911yescharacteristic, complemented
110C3×#101C3×D311yescharacteristic
111C3×#100—31no—
112#823C2×C911yescharacteristic, complemented
113#822C2×C911yescharacteristic, complemented
114C22×C3×D3—21no—
115C32×Q8—21no—
116C32×Q8—21no—
117C2×C32×C4—21no—
118C2×C32×C4C23×C311yescharacteristic
119C3×#67—21no—
120C3×D12—21no—
121C3×D12—21no—
122C3×D12C3×D411yescharacteristic
123C3×C4×D3—21no—
124C3×Dic6C3×D411yescharacteristic
125C23×C9—61no—
126C32×C8—21no—
127C32×C8—21no—
128C3×C3⋊C8—21no—
129C3×C3⋊C8C3×D411yescharacteristic
130C3×C3⋊C8C3×D411yescharacteristic
131Q8×C9—21no—
132Q8×C9—21no—
133Q8×C9—31no—
134Q8×C9—41no—
135Q8×C9—41no—
136D4×C9—31no—
137D4×C9—61no—
138D4×C9—61no—
139D4×C9—61no—
140C2×C4×C9—22no—
141C2×C4×C9—21no—
142C2×C4×C9—21no—
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#12111yescharacteristic
155D3×C9—41no—
156D3×C9—81no—
157C3×#41—31no—
158C2×C3×Q8—21no—
159C2×C3×Q8C22×C911yescharacteristic
160C2×C3×Q8C2×C3×D311yescharacteristic
161C2×C3×D4—31no—
162#235C22×C911yescharacteristic
163C2×D12C22×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#216—21no—
181#215—21no—
182#215—21no—
183#215—21no—
184#208C22×C911yescharacteristic
185C2×C3⋊C8C22×C911yescharacteristic
186D24—21no—
187#204—21no—
188ΓL(1,𝔽25)—21no—
189D3×C8—21no—
190C22×C32C3×#4111yescharacteristic
191C2×C3×D3—21no—
192C2×C3×D3—41no—
193C2×C3×D3—81no—
194C32×C4—41no—
195C32×C4C2×C3×D411yescharacteristic
196C32×C4C2×C3×D411yescharacteristic
197C3×Dic3—21no—
198C22×C9—21no—
199C22×C9—61no—
200C22×C9—121no—
201C22×C9—241no—
202C22×C9#23511yescharacteristic
203C4×C9—21no—
204C4×C9—21no—
205C4×C9—42no—
206C4×C9—41no—
207C4×C9—61no—
208C4×C9D3×D411yescharacteristic
209C4×C9D3×D411yescharacteristic
210#137—31no—
211AGL(1,ℤ/8)—31no—
212C2×SD16—31no—
213#131—31no—
214#108—31no—
215#101D3×C911yescharacteristic, complemented
216#100—31no—
217C3×C9#47111yescharacteristic, complemented, Sylow
218C23×C3—61no—
219C22×D3—21no—
220C3×Q8—21no—
221C3×Q8—21no—
222C3×Q8—21no—
223C3×Q8—21no—
224C3×Q8—31no—
225C3×Q8—41no—
226C3×Q8—41no—
227C3×D4—31no—
228C3×D4—61no—
229C3×D4—61no—
230C3×D4—61no—
231C2×C3×C4—22no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—61no—
236C2×C3×C4C23×C911yescharacteristic
237C2×C3×C4C22×C3×D311yescharacteristic
238#67—21no—
239D12—21no—
240D12—21no—
241D12D4×C911yescharacteristic
242C4×D3—21no—
243Dic6D4×C911yescharacteristic
244C3×C8—21no—
245C3×C8—21no—
246C3×C8—21no—
247C3×C8—21no—
248C3×C8—31no—
249C3×C8—31no—
250C3×C8—41no—
251C3×C8—41no—
252C3×C8—61no—
253C3⋊C8—21no—
254C3⋊C8D4×C911yescharacteristic
255C3⋊C8D4×C911yescharacteristic
256C2×C32—21no—
257C2×C32C3×#12111yescharacteristic
258C3×D3—41no—
259C3×D3—81no—
260C2×C9—21no—
261C2×C9—21no—
262C2×C9—41no—
263C2×C9—121no—
264C2×C9—241no—
265C2×C9#88411yescharacteristic
266#41—31no—
267C2×Q8C2×D3×C911yescharacteristic
268C2×D4—31no—
269Q16—61no—
270SD16—61no—
271SD16—61no—
272SD16—61no—
273SD16—61no—
274D8—61no—
275M16—31no—
276M16—61no—
277M16C2×D3×C911yescharacteristic
278M16C2×D3×C911yescharacteristic
279C2×C8—31no—
280C2×C8—61no—
281C22×C3—21no—
282C22×C3—61no—
283C22×C3—121no—
284C22×C3—241no—
285C22×C3C9×#4111yescharacteristic
286C22×C3C3×#23511yescharacteristic
287C2×D3—21no—
288C2×D3—41no—
289C2×D3—81no—
290C3×C4—21no—
291C3×C4—21no—
292C3×C4—42no—
293C3×C4—41no—
294C3×C4—41no—
295C3×C4—61no—
296C3×C4C2×D4×C911yescharacteristic
297C3×C4C2×D4×C911yescharacteristic
298C3×C4C3×D3×D411yescharacteristic
299C3×C4C3×D3×D411yescharacteristic
300Dic3—21no—
301C32C3×#47111yescharacteristic
302C9—21no—
303C9#475411yescharacteristic, direct product factor
304C23—61no—
305Q8—21no—
306Q8—21no—
307Q8—31no—
308D4—31no—
309D4—61no—
310D4—61no—
311D4—61no—
312C2×C4—21no—
313C2×C4—61no—
314C2×C4C22×D3×C911yescharacteristic
315C8—21no—
316C8—21no—
317C8—31no—
318C8—31no—
319C8—61no—
320C2×C3—21no—
321C2×C3—21no—
322C2×C3—21no—
323C2×C3—41no—
324C2×C3—121no—
325C2×C3—241no—
326C2×C3C9×#12111yescharacteristic
327C2×C3C3×#88411yescharacteristic
328D3—41no—
329D3—81no—
330C22—61no—
331C22—121no—
332C22—241no—
333C22C9×#23511yescharacteristic
334C4—41no—
335C4—61no—
336C4D3×D4×C911yescharacteristic
337C4D3×D4×C911yescharacteristic
338C3—21no—
339C3C9×#47111yescharacteristic, complemented
340C3C3×#475411yescharacteristic
341C2—21no—
342C2—121no—
343C2—241no—
344C2C9×#88411yescharacteristic
subgroupG/H
centerC2×C9#884
commutatorC2×C3×C4C23×C9
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×#101C2
socleC2×C32C3×#121
prime radicalC3×#908C2×C3
cosocleC2×C32×C4C23×C3
hypercenterC9×#101D3
hypocenterC3C9×#471
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C9 ◁ C2×C4×C9 ◁ C9×#101

Complements

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

NH ≅ G/Ntype
1. C9×#942C2semidirect
2. C2×C9×#215C2semidirect
4. C9×#907C2semidirect
5. C3×C9×#101C2semidirect
6. C9×#823C2semidirect
7. C9×#822C2semidirect
38. C9×#101D3semidirect
57. #4754C9direct (N × H)
103. #942C2×C9semidirect
104. C2×#215C2×C9semidirect
106. #907C2×C9semidirect
109. C3×#101C2×C9semidirect
112. #823C2×C9semidirect
113. #822C2×C9semidirect
215. #101D3×C9semidirect
217. C3×C9#471semidirect
303. C9#4754direct (N × H)
339. C3C9×#471semidirect

Maximal subgroups

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

subgroupindex
C9×#9422
C2×C9×#2152
C9×#9082
C9×#9072
C3×C9×#1012
C9×#8232
C9×#8222
C3×#47543
C9×#4713