Finite Group Databasebeta

← C3×D9×#131 #13153399

Subgroups — C3×D9×#131 #13153399

Order 1728.

860 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#41×D9C211yesmaximal, characteristic
2C3×C9×#131C211yesmaximal, characteristic, complemented
3C3×#63510C211yesmaximal, characteristic, complemented
4C3×#63469C213yesmaximal, complemented
5C3×M16×D9C213yesmaximal, complemented
6C3×#63464C213yesmaximal, complemented
7C2×C3×C8×D9C213yesmaximal, complemented
8C3×D3×#131—31nomaximal
9D9×#131C311yesmaximal, characteristic, direct product factor
10C3×C9×#41C2211yescharacteristic
11C3×#3701C411yescharacteristic
12C3×Q8×D9C411yescharacteristic
13C3×#3699C413yes—
14C3×D4×D9C413yes—
15C3×#3697C2213yes—
16C2×C3×C4×D9C2213yes—
17C3×C9×M16C2213yescomplemented
18C2×C3×C8×C9C2213yescomplemented
19C3×#3667C2213yescomplemented
20C2×C3×C9⋊C8C2213yescomplemented
21C3×#3663C2216yescomplemented
22C3×C8×D9C2211yescharacteristic
23C3×C8×D9C2213yes—
24C3×C8×D9C2213yescomplemented
25C3×C8×D9C2213yescomplemented
26C3×D3×#41—31no—
27C32×#131D311yescharacteristic
28C3×#948—31no—
29C3×#907—33no—
30C3×D3×M16—33no—
31C3×#901—33no—
32C2×C3×D3×C8—33no—
33#41×D9C2×C311yescharacteristic
34C9×#131—21no—
35C9×#131C2×C311yescharacteristic, complemented
36#63510C2×C311yescharacteristic, complemented
37#63469C2×C313yescomplemented
38M16×D9C2×C313yescomplemented
39#63464C2×C313yescomplemented
40C2×C8×D9C2×C313yescomplemented
41C22×C3×D9C2×C413yes—
42C3×Q8×C9C2×C411yescharacteristic
43C3×D4×C9C2×C413yes—
44C2×C3×C4×C9C2313yes—
45C3×#611C2×C416yes—
46C2×C3×Dic9C2×C413yes—
47C3×D36C2×C413yes—
48C3×C4×D9C2×C413yes—
49C3×C4×D9C2×C413yes—
50C3×C4×D9C2311yescharacteristic
51C3×C4×D9C2313yes—
52C3×Dic18C2×C413yes—
53C3×C8×C9C2311yescharacteristic
54C3×C8×C9C2313yes—
55C3×C9⋊C8C2311yescharacteristic
56C3×C9⋊C8C2313yes—
57C2×C3×#131—91no—
58D3×#131—31no—
59C32×#41C2×D311yescharacteristic
60C3×#239—31no—
61C3×D3×Q8—31no—
62C3×#237—33no—
63C3×D3×D4—33no—
64C3×#235—33no—
65C2×C3×C4×D3—33no—
66C32×M16C2×D313yes—
67C2×C32×C8C2×D313yes—
68C3×#208—33no—
69C2×C3×C3⋊C8—33no—
70C3×ΓL(1,𝔽25)—36no—
71C3×D3×C8—31no—
72C3×D3×C8—33no—
73C3×D3×C8—33no—
74C3×D3×C8—33no—
75C9×#41—21no—
76C9×#41C22×C311yescharacteristic
77#3701C3×C411yescharacteristic
78Q8×D9C3×C411yescharacteristic
79#3699C3×C413yes—
80D4×D9C3×C413yes—
81#3697C22×C313yes—
82C2×C4×D9C22×C313yes—
83C9×M16—23no—
84C9×M16C22×C313yescomplemented
85C2×C8×C9—23no—
86C2×C8×C9C22×C313yescomplemented
87#3667C22×C313yescomplemented
88C2×C9⋊C8C22×C313yescomplemented
89#3663C22×C316yescomplemented
90C8×D9C22×C311yescharacteristic
91C8×D9C22×C313yes—
92C8×D9C22×C313yescomplemented
93C8×D9C22×C313yescomplemented
94C22×C3×C9C22×C413yes—
95C2×C3×D9—26no—
96C2×C3×D9C22×C413yes—
97C2×C3×D9C22×C411yescharacteristic
98C3×C4×C9C22×C413yes—
99C3×C4×C9C2411yescharacteristic
100C3×Dic9C22×C413yes—
101C3×Dic9C22×C411yescharacteristic
102C2×C3×#41—91no—
103D3×#41—31no—
104C3×#131—21no—
105C3×#131—91no—
106C3×#131—93no—
107C3×#131—93no—
108C3×#131D911yescharacteristic, direct product factor
109C3×#131C3×D311yescharacteristic
110C2×C3×M16—93no—
111C22×C3×C8—93no—
112#948—31no—
113#907—33no—
114D3×M16—33no—
115#901—33no—
116C2×D3×C8—33no—
117C22×C3×D3—33no—
118C32×Q8C4×D311yescharacteristic
119C32×D4C4×D313yes—
120C2×C32×C4C22×D313yes—
121C3×#67—36no—
122C2×C3×Dic3—33no—
123C3×D12—33no—
124C3×C4×D3—33no—
125C3×C4×D3—33no—
126C3×C4×D3—31no—
127C3×C4×D3—33no—
128C3×Dic6—33no—
129C22×D9C2×C3×C413yes—
130C32×C8C22×D313yes—
131C32×C8C22×D311yescharacteristic
132C3×C3⋊C8—31no—
133C3×C3⋊C8—33no—
134Q8×C9—21no—
135Q8×C9C2×C3×C411yescharacteristic
136D4×C9—23no—
137D4×C9C2×C3×C413yes—
138C2×C4×C9—23no—
139C2×C4×C9C23×C313yes—
140#611C2×C3×C416yes—
141C2×Dic9C2×C3×C413yes—
142D36C2×C3×C413yes—
143C4×D9C2×C3×C413yes—
144C4×D9C2×C3×C413yes—
145C4×D9C23×C311yescharacteristic
146C4×D9C23×C313yes—
147Dic18C2×C3×C413yes—
148C8×C9—21no—
149C8×C9—23no—
150C8×C9C23×C311yescharacteristic
151C8×C9C23×C313yes—
152C9⋊C8C23×C311yescharacteristic
153C9⋊C8C23×C313yes—
154C2×#131—91noSylow
155C2×C3×C9—23no—
156C2×C3×C9C23×C411yescharacteristic
157C3×D9—23no—
158C3×D9#13112yesdirect product factor
159C3×#41—21no—
160C3×#41—91no—
161C3×#41—93no—
162C3×#41—93no—
163C3×#41C2×D911yescharacteristic
164C3×#41C2×C3×D311yescharacteristic
165C2×C3×Q8—91no—
166C2×C3×D4—93no—
167C22×C3×C4—93no—
168#239—31no—
169D3×Q8—31no—
170#237—33no—
171D3×D4—33no—
172#235—33no—
173C2×C4×D3—33no—
174C3×M16—23no—
175C3×M16—93no—
176C3×M16—96no—
177C3×M16C2×D913yescomplemented
178C3×M16C2×C3×D313yes—
179C2×C3×C8—23no—
180C2×C3×C8—93no—
181C2×C3×C8—93no—
182C2×C3×C8—93no—
183C2×C3×C8—91no—
184C2×C3×C8—93no—
185C2×C3×C8C2×D913yescomplemented
186C2×C3×C8C2×C3×D313yes—
187#208—33no—
188C2×C3⋊C8—33no—
189ΓL(1,𝔽25)—36no—
190D3×C8—31no—
191D3×C8—33no—
192D3×C8—33no—
193D3×C8—33no—
194C22×C32C2×C4×D313yes—
195C2×C3×D3—31no—
196C2×C3×D3—33no—
197C2×C3×D3—66no—
198C32×C4C2×C4×D313yes—
199C32×C4C23×D311yescharacteristic
200C3×Dic3—33no—
201C3×Dic3—31no—
202C22×C9—23no—
203C22×C9C22×C3×C413yes—
204C2×D9—26no—
205C2×D9C22×C3×C413yes—
206C2×D9C22×C3×C411yescharacteristic
207C4×C9—21no—
208C4×C9—23no—
209C4×C9C22×C3×C413yes—
210C4×C9C24×C311yescharacteristic
211Dic9C22×C3×C413yes—
212Dic9C22×C3×C411yescharacteristic
213C2×#41—91no—
214#131—91no—
215#131—93no—
216#131—93no—
217#131C3×D911yescharacteristic, direct product factor
218C2×M16—93no—
219C22×C8—93no—
220C3×C9C2×#13111yescharacteristic, complemented, Sylow
221C23×C3—93no—
222C22×D3—33no—
223C3×Q8—21no—
224C3×Q8—93no—
225C3×Q8C4×D911yescharacteristic
226C3×Q8C3×C4×D311yescharacteristic
227C3×D4—23no—
228C3×D4—96no—
229C3×D4—93no—
230C3×D4C4×D913yes—
231C3×D4C3×C4×D313yes—
232C2×C3×C4—23no—
233C2×C3×C4—93no—
234C2×C3×C4—93no—
235C2×C3×C4—93no—
236C2×C3×C4—91no—
237C2×C3×C4—93no—
238C2×C3×C4C22×D913yes—
239C2×C3×C4C22×C3×D313yes—
240#67—36no—
241C2×Dic3—33no—
242D12—33no—
243C4×D3—31no—
244C4×D3—33no—
245C4×D3—33no—
246C4×D3—33no—
247Dic6—33no—
248C3×C8—21no—
249C3×C8—23no—
250C3×C8—91no—
251C3×C8—93no—
252C3×C8C22×D913yes—
253C3×C8C22×D911yescharacteristic
254C3×C8C22×C3×D311yescharacteristic
255C3×C8C22×C3×D313yes—
256C3⋊C8—31no—
257C3⋊C8—33no—
258C2×C32—23no—
259C2×C32C22×C4×D311yescharacteristic
260C3×D3—32no—
261C3×D3—63no—
262C2×C9—23no—
263C2×C9—21no—
264C2×C9—43no—
265C2×C9C23×C3×C411yescharacteristic
266D9—23no—
267D9C3×#13112yesdirect product factor
268#41—91no—
269#41—93no—
270#41—93no—
271#41C2×C3×D911yescharacteristic
272C2×Q8—91no—
273C2×D4—93no—
274C22×C4—93no—
275M16—93no—
276M16—96no—
277M16C2×C3×D913yescomplemented
278C2×C8—93no—
279C2×C8—91no—
280C2×C8—93no—
281C2×C8—93no—
282C2×C8—93no—
283C2×C8C2×C3×D913yescomplemented
284C22×C3—23no—
285C22×C3—91no—
286C22×C3—93no—
287C22×C3—186no—
288C22×C3C2×C4×D913yes—
289C22×C3C2×C3×C4×D313yes—
290C2×D3—31no—
291C2×D3—33no—
292C2×D3—66no—
293C3×C4—21no—
294C3×C4—23no—
295C3×C4—91no—
296C3×C4—93no—
297C3×C4C2×C4×D913yes—
298C3×C4C23×D911yescharacteristic
299C3×C4C2×C3×C4×D313yes—
300C3×C4C23×C3×D311yescharacteristic
301Dic3—33no—
302Dic3—31no—
303C32D3×#13111yescharacteristic
304C9—21no—
305C9C2×C3×#13111yescharacteristic, complemented
306C23—93no—
307Q8—93no—
308Q8C3×C4×D911yescharacteristic
309D4—96no—
310D4—93no—
311D4C3×C4×D913yes—
312C2×C4—93no—
313C2×C4—91no—
314C2×C4—93no—
315C2×C4—93no—
316C2×C4—93no—
317C2×C4C22×C3×D913yes—
318C8—91no—
319C8—93no—
320C8C22×C3×D911yescharacteristic
321C8C22×C3×D913yes—
322C2×C3—23no—
323C2×C3—23no—
324C2×C3—21no—
325C2×C3—43no—
326C2×C3—92no—
327C2×C3—183no—
328C2×C3C22×C4×D911yescharacteristic
329C2×C3C22×C3×C4×D311yescharacteristic
330D3—32no—
331D3—63no—
332C22—91no—
333C22—93no—
334C22—186no—
335C22C2×C3×C4×D913yes—
336C4—91no—
337C4—93no—
338C4C2×C3×C4×D913yes—
339C4C23×C3×D911yescharacteristic
340C3—21no—
341C3D9×#13111yescharacteristic, direct product factor
342C3C3×D3×#13111yescharacteristic
343C2—23no—
344C2—92no—
345C2—183no—
346C2C22×C3×C4×D911yescharacteristic
subgroupG/H
centerC3×C8C22×D9
commutatorC2×C9C23×C3×C4
FrattiniC3×C4C23×C3×D3
FittingC3×C9×#131C2
socleC2×C32C22×C4×D3
prime radicalC3×#41×D9C2
cosocleC4×C9C24×C3
hypercenterC3×#131D9
hypocenterC9C2×C3×#131
LCSG ▷ C2×C9 ▷ C9
UCSC1 ◁ C3×C8 ◁ C3×#131

Complements

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

NH ≅ G/Ntype
2. C3×C9×#131C2semidirect
3. C3×#63510C2semidirect
4. C3×#63469C2semidirect
5. C3×M16×D9C2semidirect
6. C3×#63464C2semidirect
7. C2×C3×C8×D9C2semidirect
9. D9×#131C3direct (N × H)
17. C3×C9×M16C22semidirect
18. C2×C3×C8×C9C22semidirect
19. C3×#3667C22semidirect
20. C2×C3×C9⋊C8C22semidirect
21. C3×#3663C22semidirect
24. C3×C8×D9C22semidirect
25. C3×C8×D9C22semidirect
35. C9×#131C2×C3semidirect
36. #63510C2×C3semidirect
37. #63469C2×C3semidirect
38. M16×D9C2×C3semidirect
39. #63464C2×C3semidirect
40. C2×C8×D9C2×C3semidirect
84. C9×M16C22×C3semidirect
86. C2×C8×C9C22×C3semidirect
87. #3667C22×C3semidirect
88. C2×C9⋊C8C22×C3semidirect
89. #3663C22×C3semidirect
92. C8×D9C22×C3semidirect
93. C8×D9C22×C3semidirect
108. C3×#131D9direct (N × H)
158. C3×D9#131direct (N × H)
177. C3×M16C2×D9semidirect
185. C2×C3×C8C2×D9semidirect
217. #131C3×D9direct (N × H)
220. C3×C9C2×#131semidirect
267. D9C3×#131direct (N × H)
277. M16C2×C3×D9semidirect
283. C2×C8C2×C3×D9semidirect
305. C9C2×C3×#131semidirect
341. C3D9×#131direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#41×D92
C3×C9×#1312
C3×#635102
C3×#634692
C3×M16×D92
C3×#634642
C2×C3×C8×D92
C3×D3×#1313
D9×#1313