Finite Group Databasebeta

← C3×#10589678 #13152682

Subgroups — C3×#10589678 #13152682

Order 1728.

940 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×C3×#3701C211yesmaximal, characteristic, complemented
2C2×C3×Q8×D9C211yesmaximal, characteristic, complemented
3C3×C4×Q8×C9C211yesmaximal, characteristic, complemented
4C3×#63459C213yesmaximal, complemented
5C3×#63457C213yesmaximal, complemented
6C3×#63437C213yesmaximal, complemented
7C3×C4×D36C213yesmaximal, complemented
8C3×#5576—31nomaximal
9#10589678C311yesmaximal, characteristic, direct product factor
10C2×C3×Q8×C9C2211yescharacteristic
11C3×#3701—24no—
12C3×Q8×D9—24no—
13C2×C3×D36C2213yes—
14C2×C3×D36C2213yes—
15C2×C3×C4×D9C2213yes—
16C2×C3×C4×D9C2213yes—
17C2×C3×Dic18C2213yes—
18C3×C9×C4⋊C4C2213yes—
19C3×C42×C9C2213yes—
20C3×(C2×D9)⋊C4C2211yescharacteristic
21C3×(C2×D9)⋊C4C2213yes—
22C3×(C2×D9)⋊C4C2216yes—
23C3×C4⋊Dic9C2213yes—
24C2×C3×#239—31no—
25C2×C3×D3×Q8—31no—
26C32×C4×Q8D311yescharacteristic
27C3×#897—33no—
28C3×#895—33no—
29C3×#875—33no—
30C3×C4×D12—33no—
31C2×#3701C2×C311yescharacteristic, complemented
32C2×Q8×D9C2×C311yescharacteristic, complemented
33C4×Q8×C9—21no—
34C4×Q8×C9C2×C311yescharacteristic, complemented
35#63459C2×C313yescomplemented
36#63457C2×C313yescomplemented
37#63437C2×C313yescomplemented
38C4×D36C2×C313yescomplemented
39C22×C3×D9C2313yes—
40C22×C3×D9C2311yescharacteristic
41C3×Q8×C9D414yescomplemented
42C2×C3×C4×C9C2313yes—
43C2×C3×C4×C9C2311yescharacteristic
44C2×C3×C4×C9C2313yes—
45C2×C3×Dic9C2313yes—
46C2×C3×Dic9C2311yescharacteristic
47C3×D36—26no—
48C3×D36—26no—
49C3×C4×D9—26no—
50C3×C4×D9—26no—
51C3×Dic18—26no—
52C3×#548—91no—
53#5576—31no—
54C2×C32×Q8C2×D311yescharacteristic
55C3×#239—64no—
56C3×D3×Q8—64no—
57C2×C3×D12—33no—
58C2×C3×D12—33no—
59C2×C3×C4×D3—33no—
60C2×C3×C4×D3—33no—
61C2×C3×Dic6—33no—
62C32×C4⋊C4C2×D313yes—
63C32×C42C2×D313yes—
64C3×(C2×D3)⋊C4—36no—
65C3×(C2×D3)⋊C4—31no—
66C3×(C2×D3)⋊C4—33no—
67C3×C4⋊Dic3—33no—
68C2×Q8×C9—21no—
69C2×Q8×C9C22×C311yescharacteristic
70#3701—24no—
71Q8×D9—24no—
72C2×D36C22×C313yes—
73C2×D36C22×C313yes—
74C2×C4×D9C22×C313yes—
75C2×C4×D9C22×C313yes—
76C2×Dic18C22×C313yes—
77C9×C4⋊C4—23no—
78C9×C4⋊C4C22×C313yes—
79C42×C9—23no—
80C42×C9C22×C313yes—
81(C2×D9)⋊C4C22×C311yescharacteristic
82(C2×D9)⋊C4C22×C313yes—
83(C2×D9)⋊C4C22×C316yes—
84C4⋊Dic9C22×C313yes—
85C22×C3×C9C2411yescharacteristic
86C2×C3×D9—21no—
87C2×C3×D9—21no—
88C2×C3×D9—23no—
89C2×C3×D9—23no—
90C2×C3×D9—23no—
91C2×C3×D9#4112yescomplemented
92C3×C4×C9—23no—
93C3×C4×C9—21no—
94C3×C4×C9C2×D416yes—
95C3×Dic9—23no—
96C3×Dic9—21no—
97C2×C3×#41—91no—
98C22×C3×Q8—91no—
99C2×#239—31no—
100C2×D3×Q8—31no—
101C3×#124—93no—
102C3×C22⋊Q8—93no—
103C3×#121—93no—
104C3×C4×Q8—21no—
105C3×C4×Q8D911yescharacteristic, complemented
106C3×C4×Q8C3×D311yescharacteristic
107C3×C4×D4—93no—
108#897—33no—
109#895—33no—
110#875—33no—
111C4×D12—33no—
112C22×C3×D3—31no—
113C22×C3×D3—33no—
114C32×Q8D1214yes—
115C2×C32×C4C22×D313yes—
116C2×C32×C4C22×D311yescharacteristic
117C2×C32×C4C22×D313yes—
118C2×C3×Dic3—31no—
119C2×C3×Dic3—33no—
120C3×D12—66no—
121C3×D12—66no—
122C3×C4×D3—66no—
123C3×C4×D3—66no—
124C3×Dic6—66no—
125C22×D9C23×C313yes—
126C22×D9C23×C311yescharacteristic
127Q8×C9—24no—
128Q8×C9C3×D414yescomplemented
129C2×C4×C9—21no—
130C2×C4×C9—23no—
131C2×C4×C9—23no—
132C2×C4×C9C23×C313yes—
133C2×C4×C9C23×C311yescharacteristic
134C2×C4×C9C23×C313yes—
135C2×Dic9C23×C311yescharacteristic
136C2×Dic9C23×C313yes—
137D36—26no—
138D36—26no—
139C4×D9—26no—
140C4×D9—26no—
141Dic18—26no—
142#548—91noSylow
143C2×C3×C9C22×D411yescharacteristic
144C2×C3×C9C2×#4111yescharacteristic
145C2×C3×C92−1+411yescharacteristic
146C3×D9—22no—
147C3×D9—43no—
148C3×#41—184no—
149C2×C3×Q8—21no—
150C2×C3×Q8—93no—
151C2×C3×Q8—184no—
152C2×C3×Q8C2×D911yescharacteristic
153C2×C3×Q8C2×C3×D311yescharacteristic
154C2×C3×D4—93no—
155C2×C3×D4—93no—
156C22×C3×C4—93no—
157C22×C3×C4—93no—
158#239—64no—
159D3×Q8—64no—
160C2×D12—33no—
161C2×D12—33no—
162C2×C4×D3—33no—
163C2×C4×D3—33no—
164C2×Dic6—33no—
165C3×C4⋊C4—23no—
166C3×C4⋊C4—93no—
167C3×C4⋊C4C2×D913yes—
168C3×C4⋊C4C2×C3×D313yes—
169C3×C22⋊C4—91no—
170C3×C22⋊C4—93no—
171C3×C22⋊C4—96no—
172C3×C42—23no—
173C3×C42C2×D913yes—
174C3×C42C2×C3×D313yes—
175(C2×D3)⋊C4—31no—
176(C2×D3)⋊C4—33no—
177(C2×D3)⋊C4—36no—
178C4⋊Dic3—33no—
179C22×C32C23×D311yescharacteristic
180C2×C3×D3—32no—
181C2×C3×D3—61no—
182C2×C3×D3—61no—
183C2×C3×D3—63no—
184C2×C3×D3—63no—
185C2×C3×D3—63no—
186C32×C4—21no—
187C32×C4—23no—
188C32×C4C2×D1216yes—
189C3×Dic3—63no—
190C3×Dic3—61no—
191C22×C9—21no—
192C22×C9C24×C311yescharacteristic
193C2×D9—21no—
194C2×D9—21no—
195C2×D9—23no—
196C2×D9—23no—
197C2×D9—23no—
198C2×D9C3×#4112yescomplemented
199C4×C9—22no—
200C4×C9—23no—
201C4×C9—21no—
202C4×C9—26no—
203C4×C9—43no—
204C4×C9C2×C3×D416yes—
205Dic9—23no—
206Dic9—21no—
207C2×#41—91no—
208C22×Q8—91no—
209#124—93no—
210C22⋊Q8—93no—
211#121—93no—
212C4×Q8C3×D911yescharacteristic, complemented
213C4×D4—93no—
214C3×C9#54811yescharacteristic, complemented, Sylow
215C23×C3—91no—
216C23×C3—93no—
217C22×D3—31no—
218C22×D3—33no—
219C3×Q8—24no—
220C3×Q8—186no—
221C3×Q8D3614yescomplemented
222C3×Q8C3×D1214yes—
223C3×D4—186no—
224C3×D4—186no—
225C2×C3×C4—21no—
226C2×C3×C4—23no—
227C2×C3×C4—23no—
228C2×C3×C4—93no—
229C2×C3×C4—91no—
230C2×C3×C4—186no—
231C2×C3×C4—186no—
232C2×C3×C4C22×D911yescharacteristic
233C2×C3×C4C22×D913yes—
234C2×C3×C4C22×D913yes—
235C2×C3×C4C22×C3×D313yes—
236C2×C3×C4C22×C3×D311yescharacteristic
237C2×C3×C4C22×C3×D313yes—
238C2×Dic3—31no—
239C2×Dic3—33no—
240D12—66no—
241D12—66no—
242C4×D3—66no—
243C4×D3—66no—
244Dic6—66no—
245C2×C32C22×D1211yescharacteristic
246C2×C32#100711yescharacteristic
247C2×C32D3×#4111yescharacteristic
248C3×D3—62no—
249C3×D3—123no—
250C2×C9—21no—
251C2×C9—21no—
252C2×C9—21no—
253C2×C9C22×C3×D411yescharacteristic
254C2×C9C2×C3×#4111yescharacteristic
255C2×C9C3×2−1+411yescharacteristic
256D9—22no—
257D9—43no—
258#41—184no—
259C2×Q8—93no—
260C2×Q8—184no—
261C2×Q8C2×C3×D911yescharacteristic
262C2×D4—93no—
263C2×D4—93no—
264C22×C4—93no—
265C22×C4—93no—
266C4⋊C4—93no—
267C4⋊C4C2×C3×D913yes—
268C22⋊C4—91no—
269C22⋊C4—93no—
270C22⋊C4—96no—
271C42C2×C3×D913yes—
272C22×C3—21no—
273C22×C3—92no—
274C22×C3—181no—
275C22×C3—181no—
276C22×C3—183no—
277C22×C3—183no—
278C22×C3—183no—
279C22×C3C23×D911yescharacteristic
280C22×C3C23×C3×D311yescharacteristic
281C2×D3—32no—
282C2×D3—61no—
283C2×D3—61no—
284C2×D3—63no—
285C2×D3—63no—
286C2×D3—63no—
287C3×C4—22no—
288C3×C4—21no—
289C3×C4—23no—
290C3×C4—21no—
291C3×C4—23no—
292C3×C4—26no—
293C3×C4—43no—
294C3×C4—181no—
295C3×C4—183no—
296C3×C4C2×D3616yes—
297C3×C4C2×C3×D1216yes—
298Dic3—63no—
299Dic3—61no—
300C32#557611yescharacteristic
301C9—21no—
302C9C3×#54811yescharacteristic, complemented
303C23—91no—
304C23—93no—
305Q8—186no—
306Q8C3×D3614yescomplemented
307D4—186no—
308D4—186no—
309C2×C4—93no—
310C2×C4—91no—
311C2×C4—186no—
312C2×C4—186no—
313C2×C4C22×C3×D911yescharacteristic
314C2×C4C22×C3×D913yes—
315C2×C4C22×C3×D913yes—
316C2×C3—21no—
317C2×C3—21no—
318C2×C3—21no—
319C2×C3—182no—
320C2×C3—363no—
321C2×C3C22×D3611yescharacteristic
322C2×C3#6371311yescharacteristic
323C2×C3#41×D911yescharacteristic
324C2×C3C22×C3×D1211yescharacteristic
325C2×C3C3×#100711yescharacteristic
326C2×C3C3×D3×#4111yescharacteristic
327D3—62no—
328D3—123no—
329C22—92no—
330C22—181no—
331C22—181no—
332C22—183no—
333C22—183no—
334C22—183no—
335C22C23×C3×D911yescharacteristic
336C4—21no—
337C4—23no—
338C4—181no—
339C4—183no—
340C4C2×C3×D3616yes—
341C3—21no—
342C3#1058967811yescharacteristic, direct product factor
343C3C3×#557611yescharacteristic
344C2—182no—
345C2—363no—
346C2C22×C3×D3611yescharacteristic
347C2C3×#6371311yescharacteristic
348C2C3×#41×D911yescharacteristic
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC3×C4×Q8×C9C2
socleC22×C32C23×D3
hypercenterC3×C4×Q8D9
hypocenterC9C3×#548
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C3×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3701C2semidirect
2. C2×C3×Q8×D9C2semidirect
3. C3×C4×Q8×C9C2semidirect
4. C3×#63459C2semidirect
5. C3×#63457C2semidirect
6. C3×#63437C2semidirect
7. C3×C4×D36C2semidirect
9. #10589678C3direct (N × H)
31. C2×#3701C2×C3semidirect
32. C2×Q8×D9C2×C3semidirect
34. C4×Q8×C9C2×C3semidirect
35. #63459C2×C3semidirect
36. #63457C2×C3semidirect
37. #63437C2×C3semidirect
38. C4×D36C2×C3semidirect
41. C3×Q8×C9D4semidirect
91. C2×C3×D9#41semidirect
105. C3×C4×Q8D9semidirect
128. Q8×C9C3×D4semidirect
198. C2×D9C3×#41semidirect
212. C4×Q8C3×D9semidirect
214. C3×C9#548semidirect
221. C3×Q8D36semidirect
302. C9C3×#548semidirect
306. Q8C3×D36semidirect
342. C3#10589678direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37012
C2×C3×Q8×D92
C3×C4×Q8×C92
C3×#634592
C3×#634572
C3×#634372
C3×C4×D362
C3×#55763
#105896783