Finite Group Databasebeta

← C9×#5177 #13143361

Subgroups — C9×#5177 #13143361

Order 1728.

380 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C9×SD16C211yesmaximal, characteristic, complemented
2C9×#942C211yesmaximal, characteristic, complemented
3C9×#932C211yesmaximal, characteristic
4C9×#901C211yesmaximal, characteristic, complemented
5C9×#836C211yesmaximal, characteristic, complemented
6C9×#833C211yesmaximal, characteristic, complemented
7C9×#819C211yesmaximal, characteristic, complemented
8C3×#5177C311yesmaximal, characteristic
9C9×#471—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C3×D4×C9C2211yescharacteristic
12C9×#235C2211yescharacteristic
13C3×C9×SD16—21no—
14C3×C9×SD16—21no—
15C2×C3×C8×C9C2211yescharacteristic
16C9×#216—21no—
17C9×#215—21no—
18C9×#214—21no—
19C9×#213—21no—
20C9×#208C2211yescharacteristic
21C9×#208C2211yescharacteristic
22C9×#208C2211yescharacteristic
23C9×ΓL(1,𝔽25)—21no—
24D3×C8×C9—21no—
25C2×C32×SD16C2×C311yescharacteristic
26C3×#942C2×C311yescharacteristic
27C3×#932C2×C311yescharacteristic
28C3×#901C2×C311yescharacteristic
29C3×#836C2×C311yescharacteristic
30C3×#833C2×C311yescharacteristic
31C3×#819C2×C311yescharacteristic
32C9×#137—31no—
33C9×AGL(1,ℤ/8)—31no—
34C2×C9×SD16—21no—
35C2×C9×SD16D311yescharacteristic, complemented
36C9×#131—31no—
37C9×#108—31no—
38C9×#101—31no—
39C9×#100—31no—
40C23×C3×C9—21no—
41C3×Q8×C9—21no—
42C3×Q8×C9—21no—
43C3×D4×C9—21no—
44C3×D4×C9—21no—
45C2×C3×C4×C9—21no—
46C2×C3×C4×C9C2311yescharacteristic
47C9×#67—21no—
48C9×D12D411yescharacteristic
49C4×D3×C9—21no—
50C9×Dic6D411yescharacteristic
51C3×C8×C9D411yescharacteristic
52C3×C8×C9D411yescharacteristic
53C9×C3⋊C8—21no—
54C9×C3⋊C8—21no—
55C9×C3⋊C8—21no—
56C3×#471—31no—
57#5177C911yescharacteristic, direct product factor
58C2×C32×Q8C22×C311yescharacteristic
59C2×C32×D4C22×C311yescharacteristic
60C3×#235C22×C311yescharacteristic
61C32×SD16—21no—
62C32×SD16—21no—
63C2×C32×C8C22×C311yescharacteristic
64C3×#216—21no—
65C3×#215—21no—
66C3×#214—21no—
67C3×#213—21no—
68C3×#208C22×C311yescharacteristic
69C3×#208C22×C311yescharacteristic
70C3×#208C22×C311yescharacteristic
71C3×ΓL(1,𝔽25)—21no—
72C3×D3×C8—21no—
73C9×#41—31no—
74C2×Q8×C9—21no—
75C2×Q8×C9C2×D311yescharacteristic
76C2×D4×C9—21no—
77C2×D4×C9C2×D311yescharacteristic
78C9×Q16—61no—
79C9×SD16—22no—
80C9×SD16—22no—
81C9×SD16—21no—
82C9×SD16—21no—
83C9×SD16—61no—
84C9×SD16—61no—
85C9×D8—61no—
86C9×M16—31no—
87C9×M16—31no—
88C9×M16—31no—
89C9×M16—61no—
90C2×C8×C9—21no—
91C2×C8×C9—61no—
92C2×C8×C9C2×D311yescharacteristic
93C22×C3×C9—41no—
94C22×C3×C9—81no—
95C22×C3×C9#4111yescharacteristic
96C2×D3×C9—21no—
97C3×C4×C9—41no—
98C3×C4×C9C2×D411yescharacteristic
99C3×C4×C9C2×D411yescharacteristic
100C9×Dic3—21no—
101C3×#137—31no—
102C3×AGL(1,ℤ/8)—31no—
103C2×C3×SD16—21no—
104C2×C3×SD16C2×C911yescharacteristic, complemented
105C2×C3×SD16C3×D311yescharacteristic
106C3×#131—31no—
107#942C2×C911yescharacteristic, complemented
108#932C2×C911yescharacteristic
109#901C2×C911yescharacteristic, complemented
110C3×#108—31no—
111C3×#101—31no—
112C3×#100—31no—
113#836C2×C911yescharacteristic, complemented
114#833C2×C911yescharacteristic, complemented
115#819C2×C911yescharacteristic, complemented
116C23×C32—21no—
117C32×Q8—21no—
118C32×Q8—21no—
119C32×D4—21no—
120C32×D4—21no—
121C2×C32×C4—21no—
122C2×C32×C4C23×C311yescharacteristic
123C3×#67—21no—
124C3×D12C3×D411yescharacteristic
125C3×C4×D3—21no—
126C3×Dic6C3×D411yescharacteristic
127C23×C9—21no—
128C23×C9—41no—
129C32×C8C3×D411yescharacteristic
130C32×C8C3×D411yescharacteristic
131C3×C3⋊C8—21no—
132C3×C3⋊C8—21no—
133C3×C3⋊C8—21no—
134Q8×C9—22no—
135Q8×C9—21no—
136Q8×C9—21no—
137Q8×C9—31no—
138Q8×C9—41no—
139D4×C9—22no—
140D4×C9—21no—
141D4×C9—21no—
142D4×C9—31no—
143D4×C9—41no—
144D4×C9—61no—
145C2×C4×C9—21no—
146C2×C4×C9—21no—
147C2×C4×C9—41no—
148C2×C4×C9—61no—
149C2×C4×C9C22×D311yescharacteristic
150C8×C9—21no—
151C8×C9—21no—
152C8×C9—61no—
153C8×C9—61no—
154C8×C9—61no—
155C8×C9#6711yescharacteristic
156C8×C9#6711yescharacteristic
157#471—31noSylow
158C2×C3×C9—21no—
159C2×C3×C9—81no—
160C2×C3×C9#12111yescharacteristic
161D3×C9—41no—
162C3×#41—31no—
163C2×C3×Q8—21no—
164C2×C3×Q8C22×C911yescharacteristic
165C2×C3×Q8C2×C3×D311yescharacteristic
166C2×C3×D4—21no—
167C2×C3×D4C22×C911yescharacteristic
168C2×C3×D4C2×C3×D311yescharacteristic
169#235C22×C911yescharacteristic
170C3×Q16—61no—
171C3×SD16—22no—
172C3×SD16—22no—
173C3×SD16—21no—
174C3×SD16—21no—
175C3×SD16—21no—
176C3×SD16—21no—
177C3×SD16—61no—
178C3×SD16—61no—
179C3×D8—61no—
180C3×M16—31no—
181C3×M16—31no—
182C3×M16—31no—
183C3×M16—61no—
184C2×C3×C8—21no—
185C2×C3×C8—61no—
186C2×C3×C8C22×C911yescharacteristic
187C2×C3×C8C2×C3×D311yescharacteristic
188#216—21no—
189#215—21no—
190#214—21no—
191#213—21no—
192#208C22×C911yescharacteristic
193#208C22×C911yescharacteristic
194#208C22×C911yescharacteristic
195ΓL(1,𝔽25)—21no—
196D3×C8—21no—
197C22×C32—41no—
198C22×C32—81no—
199C22×C32C3×#4111yescharacteristic
200C2×C3×D3—21no—
201C32×C4—41no—
202C32×C4C2×C3×D411yescharacteristic
203C32×C4C2×C3×D411yescharacteristic
204C3×Dic3—21no—
205C22×C9—21no—
206C22×C9—42no—
207C22×C9—41no—
208C22×C9—61no—
209C22×C9—81no—
210C22×C9—82no—
211C22×C9#23711yescharacteristic
212C4×C9—21no—
213C4×C9—21no—
214C4×C9—42no—
215C4×C9—41no—
216C4×C9—61no—
217C4×C9D3×D411yescharacteristic
218C4×C9C2×#6711yescharacteristic
219#137—31no—
220AGL(1,ℤ/8)—31no—
221C2×SD16D3×C911yescharacteristic, complemented
222#131—31no—
223#108—31no—
224#101—31no—
225#100—31no—
226C3×C9#47111yescharacteristic, complemented, Sylow
227C23×C3—21no—
228C23×C3—21no—
229C23×C3—41no—
230C3×Q8—22no—
231C3×Q8—21no—
232C3×Q8—21no—
233C3×Q8—21no—
234C3×Q8—21no—
235C3×Q8—31no—
236C3×Q8—41no—
237C3×D4—22no—
238C3×D4—21no—
239C3×D4—21no—
240C3×D4—21no—
241C3×D4—21no—
242C3×D4—31no—
243C3×D4—41no—
244C3×D4—61no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—41no—
249C2×C3×C4—61no—
250C2×C3×C4C23×C911yescharacteristic
251C2×C3×C4C22×C3×D311yescharacteristic
252#67—21no—
253D12D4×C911yescharacteristic
254C4×D3—21no—
255Dic6D4×C911yescharacteristic
256C3×C8—21no—
257C3×C8—21no—
258C3×C8—61no—
259C3×C8—61no—
260C3×C8—61no—
261C3×C8D4×C911yescharacteristic
262C3×C8D4×C911yescharacteristic
263C3×C8C3×#6711yescharacteristic
264C3×C8C3×#6711yescharacteristic
265C3⋊C8—21no—
266C3⋊C8—21no—
267C3⋊C8—21no—
268C2×C32—21no—
269C2×C32—81no—
270C2×C32C3×#12111yescharacteristic
271C3×D3—41no—
272C2×C9—22no—
273C2×C9—21no—
274C2×C9—21no—
275C2×C9—82no—
276C2×C9—81no—
277C2×C9—121no—
278C2×C9#93811yescharacteristic
279#41—31no—
280C2×Q8C2×D3×C911yescharacteristic
281C2×D4C2×D3×C911yescharacteristic
282Q16—61no—
283SD16—21no—
284SD16—21no—
285SD16—61no—
286SD16—61no—
287D8—61no—
288M16—31no—
289M16—31no—
290M16—31no—
291M16—61no—
292C2×C8—61no—
293C2×C8C2×D3×C911yescharacteristic
294C22×C3—21no—
295C22×C3—42no—
296C22×C3—41no—
297C22×C3—41no—
298C22×C3—61no—
299C22×C3—81no—
300C22×C3—81no—
301C22×C3—82no—
302C22×C3C9×#4111yescharacteristic
303C22×C3C3×#23711yescharacteristic
304C2×D3—21no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—42no—
308C3×C4—41no—
309C3×C4—41no—
310C3×C4—61no—
311C3×C4C2×D4×C911yescharacteristic
312C3×C4C2×D4×C911yescharacteristic
313C3×C4C3×D3×D411yescharacteristic
314C3×C4C2×C3×#6711yescharacteristic
315Dic3—21no—
316C32C3×#47111yescharacteristic
317C9—21no—
318C9#517711yescharacteristic, direct product factor
319C23—21no—
320Q8—21no—
321Q8—21no—
322Q8—31no—
323D4—21no—
324D4—21no—
325D4—31no—
326D4—61no—
327C2×C4—21no—
328C2×C4—61no—
329C2×C4C22×D3×C911yescharacteristic
330C8—61no—
331C8—61no—
332C8—61no—
333C8C9×#6711yescharacteristic
334C8C9×#6711yescharacteristic
335C2×C3—22no—
336C2×C3—21no—
337C2×C3—21no—
338C2×C3—21no—
339C2×C3—82no—
340C2×C3—81no—
341C2×C3—81no—
342C2×C3—121no—
343C2×C3C9×#12111yescharacteristic
344C2×C3C3×#93811yescharacteristic
345D3—41no—
346C22—41no—
347C22—61no—
348C22—81no—
349C22C9×#23711yescharacteristic
350C4—41no—
351C4—61no—
352C4D3×D4×C911yescharacteristic
353C4C2×C9×#6711yescharacteristic
354C3—21no—
355C3C9×#47111yescharacteristic, complemented
356C3C3×#517711yescharacteristic
357C2—21no—
358C2—81no—
359C2—121no—
360C2C9×#93811yescharacteristic
subgroupG/H
centerC2×C9#938
commutatorC2×C3×C4C23×C9
FrattiniC2×C3×C4C22×C3×D3
FittingC2×C3×C9×SD16C2
socleC2×C32C3×#121
prime radicalC3×#932C2×C3
cosocleC2×C32×C4C23×C3
hypercenterC2×C9×SD16D3
hypocenterC3C9×#471
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C9 ◁ C2×C4×C9 ◁ C2×C9×SD16

Complements

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

NH ≅ G/Ntype
1. C2×C3×C9×SD16C2semidirect
2. C9×#942C2semidirect
4. C9×#901C2semidirect
5. C9×#836C2semidirect
6. C9×#833C2semidirect
7. C9×#819C2semidirect
35. C2×C9×SD16D3semidirect
57. #5177C9direct (N × H)
104. C2×C3×SD16C2×C9semidirect
107. #942C2×C9semidirect
109. #901C2×C9semidirect
113. #836C2×C9semidirect
114. #833C2×C9semidirect
115. #819C2×C9semidirect
221. C2×SD16D3×C9semidirect
226. C3×C9#471semidirect
318. C9#5177direct (N × H)
355. C3C9×#471semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C9×SD162
C9×#9422
C9×#9322
C9×#9012
C9×#8362
C9×#8332
C9×#8192
C3×#51773
C9×#4713