Finite Group Databasebeta

← C9×#5203 #13143495

Subgroups — C9×#5203 #13143495

Order 1728.

372 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
1C3×C9×#137C211yesmaximal, characteristic, complemented
2C9×#949C211yesmaximal, characteristic
3C9×#948C211yesmaximal, characteristic, complemented
4C2×C9×#215C211yesmaximal, characteristic, complemented
5C9×#836C211yesmaximal, characteristic, complemented
6C9×#823C211yesmaximal, characteristic, complemented
7C9×#822C211yesmaximal, characteristic, complemented
8C3×#5203C311yesmaximal, characteristic
9C9×#471—31nomaximal
10C3×C9×#41C2211yescharacteristic
11C2×C3×Q8×C9C2211yescharacteristic
12C2×C9×D12C2211yescharacteristic
13C3×C9×Q16—21no—
14C3×C9×SD16—21no—
15C3×C9×M16C2211yescharacteristic
16C9×#215—21no—
17C9×#215—21no—
18C9×#215—21no—
19C9×#213—21no—
20C9×#208—21no—
21C9×#208C2211yescharacteristic
22C9×#208C2211yescharacteristic
23C2×C9×C3⋊C8—21no—
24C2×C9×C3⋊C8C2211yescharacteristic
25C32×#137C2×C311yescharacteristic
26C3×#949C2×C311yescharacteristic
27C3×#948C2×C311yescharacteristic
28C2×C3×#215C2×C311yescharacteristic
29C3×#836C2×C311yescharacteristic
30C3×#823C2×C311yescharacteristic
31C3×#822C2×C311yescharacteristic
32C9×#137—21no—
33C9×#137D311yescharacteristic, complemented
34C9×AGL(1,ℤ/8)—31no—
35C2×C9×SD16—31no—
36C9×#131—31no—
37C9×#108—31no—
38C9×#101—31no—
39C9×#100—31no—
40C22×D3×C9—21no—
41C3×Q8×C9—21no—
42C3×Q8×C9—21no—
43C3×Q8×C9D411yescharacteristic
44C3×D4×C9—21no—
45C3×D4×C9D411yescharacteristic
46C2×C3×C4×C9—21no—
47C2×C3×C4×C9—21no—
48C2×C3×C4×C9C2311yescharacteristic
49C9×D12—21no—
50C9×D12—21no—
51C3×C8×C9—21no—
52C9×C3⋊C8—21no—
53C9×C3⋊C8—21no—
54C9×C3⋊C8D411yescharacteristic
55C9×C3⋊C8D411yescharacteristic
56C3×#471—31no—
57#5203C911yescharacteristic, direct product factor
58C32×#41C22×C311yescharacteristic
59C2×C32×Q8C22×C311yescharacteristic
60C2×C3×D12C22×C311yescharacteristic
61C32×Q16—21no—
62C32×SD16—21no—
63C32×M16C22×C311yescharacteristic
64C3×#215—21no—
65C3×#215—21no—
66C3×#215—21no—
67C3×#213—21no—
68C3×#208—21no—
69C3×#208C22×C311yescharacteristic
70C3×#208C22×C311yescharacteristic
71C2×C3×C3⋊C8—21no—
72C2×C3×C3⋊C8C22×C311yescharacteristic
73C9×#41—21no—
74C9×#41C2×D311yescharacteristic
75C2×Q8×C9—21no—
76C2×Q8×C9C2×D311yescharacteristic
77C2×D4×C9—31no—
78C9×Q16—22no—
79C9×Q16—21no—
80C9×SD16—22no—
81C9×SD16—21no—
82C9×SD16—61no—
83C9×SD16—61no—
84C9×SD16—61no—
85C9×D8—61no—
86C9×M16—21no—
87C9×M16—31no—
88C9×M16—31no—
89C9×M16—61no—
90C9×M16C2×D311yescharacteristic
91C2×C8×C9—31no—
92C2×C8×C9—61no—
93C22×C3×C9—21no—
94C22×C3×C9#4111yescharacteristic
95C2×D3×C9—41no—
96C2×D3×C9—81no—
97C3×C4×C9—21no—
98C3×C4×C9—41no—
99C3×C4×C9C2×D411yescharacteristic
100C3×C4×C9C2×D411yescharacteristic
101C3×#137—21no—
102C3×#137C2×C911yescharacteristic, complemented
103C3×#137C3×D311yescharacteristic
104C3×AGL(1,ℤ/8)—31no—
105C2×C3×SD16—31no—
106C3×#131—31no—
107#949C2×C911yescharacteristic
108#948C2×C911yescharacteristic, complemented
109C2×#215C2×C911yescharacteristic, complemented
110C3×#108—31no—
111C3×#101—31no—
112C3×#100—31no—
113#836C2×C911yescharacteristic, complemented
114#823C2×C911yescharacteristic, complemented
115#822C2×C911yescharacteristic, complemented
116C22×C3×D3—21no—
117C32×Q8—21no—
118C32×Q8—21no—
119C32×Q8C3×D411yescharacteristic
120C32×D4—21no—
121C32×D4C3×D411yescharacteristic
122C2×C32×C4—21no—
123C2×C32×C4—21no—
124C2×C32×C4C23×C311yescharacteristic
125C3×D12—21no—
126C3×D12—21no—
127C23×C9—61no—
128C32×C8—21no—
129C3×C3⋊C8—21no—
130C3×C3⋊C8—21no—
131C3×C3⋊C8C3×D411yescharacteristic
132C3×C3⋊C8C3×D411yescharacteristic
133Q8×C9—22no—
134Q8×C9—21no—
135Q8×C9—21no—
136Q8×C9—21no—
137Q8×C9—41no—
138Q8×C9#6711yescharacteristic
139D4×C9—21no—
140D4×C9—21no—
141D4×C9—41no—
142D4×C9—61no—
143D4×C9—61no—
144D4×C9#6711yescharacteristic
145C2×C4×C9—21no—
146C2×C4×C9—21no—
147C2×C4×C9—21no—
148C2×C4×C9—41no—
149C2×C4×C9—41no—
150C2×C4×C9C22×D311yescharacteristic
151C8×C9—22no—
152C8×C9—21no—
153C8×C9—31no—
154C8×C9—31no—
155C8×C9—61no—
156C8×C9—61no—
157#471—31noSylow
158C2×C3×C9—21no—
159C2×C3×C9—41no—
160C2×C3×C9#12111yescharacteristic
161D3×C9—81no—
162C3×#41—21no—
163C3×#41C22×C911yescharacteristic
164C3×#41C2×C3×D311yescharacteristic
165C2×C3×Q8—21no—
166C2×C3×Q8C22×C911yescharacteristic
167C2×C3×Q8C2×C3×D311yescharacteristic
168C2×C3×D4—31no—
169C2×D12C22×C911yescharacteristic
170C3×Q16—22no—
171C3×Q16—21no—
172C3×Q16—21no—
173C3×SD16—22no—
174C3×SD16—21no—
175C3×SD16—21no—
176C3×SD16—61no—
177C3×SD16—61no—
178C3×SD16—61no—
179C3×D8—61no—
180C3×M16—21no—
181C3×M16—31no—
182C3×M16—31no—
183C3×M16—61no—
184C3×M16C22×C911yescharacteristic
185C3×M16C2×C3×D311yescharacteristic
186C2×C3×C8—31no—
187C2×C3×C8—61no—
188#215—21no—
189#215—21no—
190#215—21no—
191#213—21no—
192#208—21no—
193#208C22×C911yescharacteristic
194#208C22×C911yescharacteristic
195C2×C3⋊C8—21no—
196C2×C3⋊C8C22×C911yescharacteristic
197C22×C32—21no—
198C22×C32C3×#4111yescharacteristic
199C2×C3×D3—41no—
200C2×C3×D3—81no—
201C32×C4—21no—
202C32×C4—41no—
203C32×C4C2×C3×D411yescharacteristic
204C32×C4C2×C3×D411yescharacteristic
205C22×C9—21no—
206C22×C9—21no—
207C22×C9—41no—
208C22×C9—121no—
209C22×C9—241no—
210C22×C9#23711yescharacteristic
211C4×C9—21no—
212C4×C9—21no—
213C4×C9—21no—
214C4×C9—41no—
215C4×C9—42no—
216C4×C9—41no—
217C4×C9D3×D411yescharacteristic
218C4×C9C2×#6711yescharacteristic
219#137D3×C911yescharacteristic, complemented
220AGL(1,ℤ/8)—31no—
221C2×SD16—31no—
222#131—31no—
223#108—31no—
224#101—31no—
225#100—31no—
226C3×C9#47111yescharacteristic, complemented, Sylow
227C23×C3—61no—
228C22×D3—21no—
229C3×Q8—22no—
230C3×Q8—21no—
231C3×Q8—21no—
232C3×Q8—21no—
233C3×Q8—21no—
234C3×Q8—21no—
235C3×Q8—41no—
236C3×Q8D4×C911yescharacteristic
237C3×Q8C3×#6711yescharacteristic
238C3×D4—21no—
239C3×D4—21no—
240C3×D4—21no—
241C3×D4—41no—
242C3×D4—61no—
243C3×D4—61no—
244C3×D4D4×C911yescharacteristic
245C3×D4C3×#6711yescharacteristic
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4C23×C911yescharacteristic
254C2×C3×C4C22×C3×D311yescharacteristic
255D12—21no—
256D12—21no—
257C3×C8—22no—
258C3×C8—21no—
259C3×C8—21no—
260C3×C8—31no—
261C3×C8—31no—
262C3×C8—61no—
263C3×C8—61no—
264C3⋊C8—21no—
265C3⋊C8—21no—
266C3⋊C8D4×C911yescharacteristic
267C3⋊C8D4×C911yescharacteristic
268C2×C32—21no—
269C2×C32—41no—
270C2×C32C3×#12111yescharacteristic
271C3×D3—81no—
272C2×C9—21no—
273C2×C9—21no—
274C2×C9—41no—
275C2×C9—41no—
276C2×C9—81no—
277C2×C9—241no—
278C2×C9#93911yescharacteristic
279#41C2×D3×C911yescharacteristic
280C2×Q8C2×D3×C911yescharacteristic
281C2×D4—31no—
282Q16—21no—
283SD16—21no—
284SD16—61no—
285SD16—61no—
286SD16—61no—
287D8—61no—
288M16—31no—
289M16—31no—
290M16—61no—
291M16C2×D3×C911yescharacteristic
292C2×C8—31no—
293C2×C8—61no—
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—41no—
298C22×C3—121no—
299C22×C3—241no—
300C22×C3C9×#4111yescharacteristic
301C22×C3C3×#23711yescharacteristic
302C2×D3—41no—
303C2×D3—81no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—21no—
308C3×C4—41no—
309C3×C4—42no—
310C3×C4—41no—
311C3×C4—41no—
312C3×C4C2×D4×C911yescharacteristic
313C3×C4C2×D4×C911yescharacteristic
314C3×C4C3×D3×D411yescharacteristic
315C3×C4C2×C3×#6711yescharacteristic
316C32C3×#47111yescharacteristic
317C9—21no—
318C9#520311yescharacteristic, direct product factor
319C23—61no—
320Q8—21no—
321Q8—21no—
322Q8C9×#6711yescharacteristic
323D4—21no—
324D4—61no—
325D4—61no—
326D4C9×#6711yescharacteristic
327C2×C4—21no—
328C2×C4—21no—
329C2×C4C22×D3×C911yescharacteristic
330C8—21no—
331C8—31no—
332C8—31no—
333C8—61no—
334C8—61no—
335C2×C3—21no—
336C2×C3—21no—
337C2×C3—21no—
338C2×C3—41no—
339C2×C3—41no—
340C2×C3—41no—
341C2×C3—81no—
342C2×C3—241no—
343C2×C3C9×#12111yescharacteristic
344C2×C3C3×#93911yescharacteristic
345D3—81no—
346C22—21no—
347C22—121no—
348C22—241no—
349C22C9×#23711yescharacteristic
350C4—21no—
351C4—41no—
352C4D3×D4×C911yescharacteristic
353C4C2×C9×#6711yescharacteristic
354C3—21no—
355C3C9×#47111yescharacteristic, complemented
356C3C3×#520311yescharacteristic
357C2—21no—
358C2—41no—
359C2—241no—
360C2C9×#93911yescharacteristic
subgroupG/H
centerC2×C9#939
commutatorC2×C3×C4C23×C9
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×#137C2
socleC2×C32C3×#121
prime radicalC3×#949C2×C3
cosocleC2×C32×C4C23×C3
hypercenterC9×#137D3
hypocenterC3C9×#471
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C9 ◁ C2×C4×C9 ◁ C9×#137

Complements

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

NH ≅ G/Ntype
1. C3×C9×#137C2semidirect
3. C9×#948C2semidirect
4. C2×C9×#215C2semidirect
5. C9×#836C2semidirect
6. C9×#823C2semidirect
7. C9×#822C2semidirect
33. C9×#137D3semidirect
57. #5203C9direct (N × H)
102. C3×#137C2×C9semidirect
108. #948C2×C9semidirect
109. C2×#215C2×C9semidirect
113. #836C2×C9semidirect
114. #823C2×C9semidirect
115. #822C2×C9semidirect
219. #137D3×C9semidirect
226. C3×C9#471semidirect
318. C9#5203direct (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
C3×C9×#1372
C9×#9492
C9×#9482
C2×C9×#2152
C9×#8362
C9×#8232
C9×#8222
C3×#52033
C9×#4713