Finite Group Databasebeta

← C3×#10596756 #13181743

Subgroups — C3×#10596756 #13181743

Order 1728.

5516 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C33×#41C211yesmaximal, characteristic, complemented
2C3×#64367C214yesmaximal, complemented
3C3×#64366C214yesmaximal, complemented
4C3×#41×C3⋊D3C218yesmaximal, complemented
5C3×#64364C212yesmaximal, complemented
6C3×#64361C216yesmaximal, complemented
7C2×C3×#3828C216yesmaximal, complemented
8#10596756C311yesmaximal, characteristic, direct product factor
9C3×#5964—34nomaximal
10C33×#41C2214yescomplemented
11C33×#41C2214yescomplemented
12C2×C33×Q8C2211yescharacteristic, complemented
13C2×C33×D4C2213yescomplemented
14C22×C33×C4C2213yescomplemented
15C2×C3×#638C22112yescomplemented
16C3×#3832C2218yescomplemented
17C3×Q8×C3⋊D3C2218yescomplemented
18C3×#3830C22124yescomplemented
19C3×D4×C3⋊D3C22124yescomplemented
20C3×#3828C2214yescomplemented
21C3×#3828C22112yescomplemented
22C3×#3828C22112yescomplemented
23C3×#3828C22112yescomplemented
24C2×C3×C4⋊(C3⋊D3)C2216yescomplemented
25C2×C3×C4×C3⋊D3C22112yescomplemented
26C2×C3×#634C2216yescomplemented
27C2×C32×#41—24no—
28C2×C32×#41D314yescomplemented
29C2×C32×#41C2×C311yescharacteristic, complemented
30#64367C2×C314yescomplemented
31#64366C2×C314yescomplemented
32#41×C3⋊D3C2×C318yescomplemented
33#64364C2×C312yescomplemented
34#64361C2×C316yescomplemented
35C2×#3828C2×C316yescomplemented
36C3×#1010—316no—
37C3×#1009—316no—
38C3×D3×#41—332no—
39C3×#1007—38no—
40C3×#1004—324no—
41C2×C3×#235—324no—
42C23×C33C2313yes—
43C22×C3×C3⋊D3C23112yes—
44C33×Q8C2314yes—
45C33×D4C23112yes—
46C2×C33×C4C2316yes—
47C2×C33×C4C2316yes—
48C2×C33×C4C2311yescharacteristic
49C2×C33×C4C2313yes—
50C3×#638C2318yes—
51C3×#638C23124yes—
52C2×C3×C3⋊Dic3C23112yes—
53C3×C4⋊(C3⋊D3)C2314yes—
54C3×C4⋊(C3⋊D3)C23112yes—
55C3×C4×C3⋊D3C2318yes—
56C3×C4×C3⋊D3C23124yes—
57C3×#634C2314yes—
58C3×#634C23112yes—
59C3×#585—91no—
60#5964—34no—
61C32×#41—216no—
62C32×#41—216no—
63C32×#41C2×D3116yescomplemented
64C32×#41C2×D3116yescomplemented
65C32×#41C22×C314yescomplemented
66C32×#41C22×C314yescomplemented
67C2×C32×Q8—24no—
68C2×C32×Q8C2×D314yescomplemented
69C2×C32×Q8C22×C311yescharacteristic, complemented
70C2×C32×D4—212no—
71C2×C32×D4C2×D3112yescomplemented
72C2×C32×D4C22×C313yescomplemented
73C22×C32×C4—212no—
74C22×C32×C4C2×D3112yescomplemented
75C22×C32×C4C22×C313yescomplemented
76C2×#638C22×C3112yescomplemented
77#3832C22×C318yescomplemented
78Q8×C3⋊D3C22×C318yescomplemented
79#3830C22×C3124yescomplemented
80D4×C3⋊D3C22×C3124yescomplemented
81#3828C22×C314yescomplemented
82#3828C22×C3112yescomplemented
83#3828C22×C3112yescomplemented
84#3828C22×C3112yescomplemented
85C2×C4⋊(C3⋊D3)C22×C316yescomplemented
86C2×C4×C3⋊D3C22×C3112yescomplemented
87C2×#634C22×C316yescomplemented
88C2×C3×#67—348no—
89C3×#239—332no—
90C3×D3×Q8—332no—
91C3×#237—396no—
92C3×D3×D4—396no—
93C3×#235—316no—
94C3×#235—348no—
95C3×#235—348no—
96C3×#235—348no—
97C2×C3×D12—324no—
98C2×C3×C4×D3—348no—
99C2×C3×Dic6—324no—
100C22×C33—43no—
101C22×C33C2416yes—
102C22×C33C2411yescharacteristic
103C2×C3×C3⋊D3—412no—
104C2×C3×C3⋊D3C2418yes—
105C33×C4C2411yescharacteristic
106C33×C4C2416yes—
107C33×C4C2411yescharacteristic
108C3×C3⋊Dic3C2418yes—
109C3×2−1+4—94no—
110C3×2−1+4—92no—
111C3×H(5,𝔽2)—94no—
112C3×H(5,𝔽2)—96no—
113C2×C3×#41—24no—
114C2×C3×#41—98no—
115C2×C3×#41—96no—
116C2×C3×#41C3×D314yescomplemented
117C2×C3×#41C3⋊D311yescharacteristic, complemented
118#1010—316no—
119#1009—316no—
120D3×#41—332no—
121#1007—38no—
122#1004—324no—
123C2×#235—324no—
124C23×C32—212no—
125C23×C32C22×D3112yes—
126C23×C32C23×C313yes—
127C22×C3⋊D3C23×C3112yes—
128C22×C3×D3—348no—
129C32×Q8—216no—
130C32×Q8C22×D3116yes—
131C32×Q8C23×C314yes—
132C32×D4—248no—
133C32×D4C22×D3148yes—
134C32×D4C23×C3112yes—
135C2×C32×C4—24no—
136C2×C32×C4—212no—
137C2×C32×C4—224no—
138C2×C32×C4—224no—
139C2×C32×C4C22×D3124yes—
140C2×C32×C4C22×D3124yes—
141C2×C32×C4C22×D314yes—
142C2×C32×C4C22×D3112yes—
143C2×C32×C4C23×C316yes—
144C2×C32×C4C23×C316yes—
145C2×C32×C4C23×C311yescharacteristic
146C2×C32×C4C23×C313yes—
147#638C23×C318yes—
148#638C23×C3124yes—
149C2×C3⋊Dic3C23×C3112yes—
150C4⋊(C3⋊D3)C23×C314yes—
151C4⋊(C3⋊D3)C23×C3112yes—
152C4×C3⋊D3C23×C318yes—
153C4×C3⋊D3C23×C3124yes—
154#634C23×C314yes—
155#634C23×C3112yes—
156C3×#67—332no—
157C3×#67—396no—
158C2×C3×Dic3—348no—
159C3×D12—316no—
160C3×D12—348no—
161C3×C4×D3—332no—
162C3×C4×D3—396no—
163C3×Dic6—316no—
164C3×Dic6—348no—
165#585—91noSylow
166C2×C33—21no—
167C2×C33—26no—
168C2×C33C2511yescharacteristic
169C3×C3⋊D3—28no—
170C3×#41—216no—
171C3×#41—216no—
172C3×#41—98no—
173C3×#41—924no—
174C3×#41—912no—
175C3×#41—94no—
176C3×#41—912no—
177C3×#41—912no—
178C3×#41C2×C3×D3116yescomplemented
179C3×#41C2×C3×D3116yescomplemented
180C3×#41C2×C3⋊D314yescomplemented
181C3×#41C2×C3⋊D314yescomplemented
182C2×C3×Q8—24no—
183C2×C3×Q8—98no—
184C2×C3×Q8—96no—
185C2×C3×Q8C2×C3×D314yescomplemented
186C2×C3×Q8C2×C3⋊D311yescharacteristic, complemented
187C2×C3×D4—212no—
188C2×C3×D4—912no—
189C2×C3×D4—924no—
190C2×C3×D4—96no—
191C2×C3×D4C2×C3×D3112yescomplemented
192C2×C3×D4C2×C3⋊D313yescomplemented
193C22×C3×C4—212no—
194C22×C3×C4—912no—
195C22×C3×C4C2×C3×D3112yescomplemented
196C22×C3×C4C2×C3⋊D313yescomplemented
197C2×#67—348no—
198#239—332no—
199D3×Q8—332no—
200#237—396no—
201D3×D4—396no—
202#235—316no—
203#235—348no—
204#235—348no—
205#235—348no—
206C2×D12—324no—
207C2×C4×D3—348no—
208C2×Dic6—324no—
209C22×C32—24no—
210C22×C32—224no—
211C22×C32—412no—
212C22×C32—43no—
213C22×C32—424no—
214C22×C32C23×D3124yes—
215C22×C32C23×D314yes—
216C22×C32C24×C316yes—
217C22×C32C24×C311yescharacteristic
218C2×C3⋊D3—412no—
219C2×C3⋊D3C24×C318yes—
220C2×C3×D3—332no—
221C2×C3×D3—1248no—
222C32×C4—24no—
223C32×C4—24no—
224C32×C4—224no—
225C32×C4C23×D314yes—
226C32×C4C23×D3124yes—
227C32×C4C23×D314yes—
228C32×C4C24×C311yescharacteristic
229C32×C4C24×C316yes—
230C32×C4C24×C311yescharacteristic
231C3⋊Dic3C24×C318yes—
232C3×Dic3—332no—
2332−1+4—94no—
2342−1+4—92no—
235H(5,𝔽2)—94no—
236H(5,𝔽2)—96no—
237C2×#41—98no—
238C2×#41—96no—
239C2×#41C3×C3⋊D311yescharacteristic, complemented
240C33#58511yescharacteristic, complemented, Sylow
241C23×C3—212no—
242C23×C3—912no—
243C23×C3C22×C3×D3112yes—
244C23×C3C22×C3⋊D313yes—
245C22×D3—348no—
246C3×Q8—216no—
247C3×Q8—912no—
248C3×Q8—94no—
249C3×Q8C22×C3×D3116yes—
250C3×Q8C22×C3⋊D314yes—
251C3×D4—248no—
252C3×D4—924no—
253C3×D4—98no—
254C3×D4—912no—
255C3×D4—94no—
256C3×D4C22×C3×D3148yes—
257C3×D4C22×C3⋊D3112yes—
258C2×C3×C4—24no—
259C2×C3×C4—212no—
260C2×C3×C4—224no—
261C2×C3×C4—224no—
262C2×C3×C4—912no—
263C2×C3×C4—924no—
264C2×C3×C4—98no—
265C2×C3×C4C22×C3×D3124yes—
266C2×C3×C4C22×C3×D3124yes—
267C2×C3×C4C22×C3×D314yes—
268C2×C3×C4C22×C3×D3112yes—
269C2×C3×C4C22×C3⋊D316yes—
270C2×C3×C4C22×C3⋊D313yes—
271C2×C3×C4C22×C3⋊D316yes—
272C2×C3×C4C22×C3⋊D311yescharacteristic
273#67—332no—
274#67—396no—
275C2×Dic3—348no—
276D12—316no—
277D12—348no—
278C4×D3—332no—
279C4×D3—396no—
280Dic6—316no—
281Dic6—348no—
282C2×C32—28no—
283C2×C32—24no—
284C2×C32—26no—
285C2×C32—21no—
286C2×C32—24no—
287C2×C32—224no—
288C2×C32—424no—
289C2×C32C24×D314yes—
290C2×C32C25×C311yescharacteristic
291C3⋊D3—28no—
292C3×D3—632no—
293#41—98no—
294#41—924no—
295#41—912no—
296#41—94no—
297#41—912no—
298#41—912no—
299#41C2×C3×C3⋊D314yescomplemented
300#41C2×C3×C3⋊D314yescomplemented
301C2×Q8—98no—
302C2×Q8—96no—
303C2×Q8C2×C3×C3⋊D311yescharacteristic, complemented
304C2×D4—912no—
305C2×D4—924no—
306C2×D4—96no—
307C2×D4C2×C3×C3⋊D313yescomplemented
308C22×C4—912no—
309C22×C4C2×C3×C3⋊D313yescomplemented
310C22×C3—24no—
311C22×C3—224no—
312C22×C3—412no—
313C22×C3—43no—
314C22×C3—424no—
315C22×C3—98no—
316C22×C3—3612no—
317C22×C3C23×C3×D3124yes—
318C22×C3C23×C3×D314yes—
319C22×C3C23×C3⋊D316yes—
320C22×C3C23×C3⋊D311yescharacteristic
321C2×D3—332no—
322C2×D3—1248no—
323C3×C4—24no—
324C3×C4—24no—
325C3×C4—224no—
326C3×C4—98no—
327C3×C4C23×C3×D314yes—
328C3×C4C23×C3×D3124yes—
329C3×C4C23×C3×D314yes—
330C3×C4C23×C3⋊D316yes—
331C3×C4C23×C3⋊D311yescharacteristic
332C3×C4C23×C3⋊D311yescharacteristic
333Dic3—332no—
334C32—24no—
335C32#596414yescomplemented
336C32C3×#58511yescharacteristic, complemented
337C23—912no—
338C23C22×C3×C3⋊D313yes—
339Q8—912no—
340Q8—94no—
341Q8C22×C3×C3⋊D314yes—
342D4—924no—
343D4—98no—
344D4—912no—
345D4—94no—
346D4C22×C3×C3⋊D3112yes—
347C2×C4—912no—
348C2×C4—98no—
349C2×C4—924no—
350C2×C4C22×C3×C3⋊D316yes—
351C2×C4C22×C3×C3⋊D311yescharacteristic
352C2×C4C22×C3×C3⋊D316yes—
353C2×C4C22×C3×C3⋊D313yes—
354C2×C3—28no—
355C2×C3—224no—
356C2×C3—24no—
357C2×C3—21no—
358C2×C3—26no—
359C2×C3—24no—
360C2×C3—424no—
361C2×C3—188no—
362C2×C3C24×C3×D314yes—
363C2×C3C24×C3⋊D311yescharacteristic
364D3—632no—
365C22—43no—
366C22—98no—
367C22—3612no—
368C22C23×C3×C3⋊D316yes—
369C22C23×C3×C3⋊D311yescharacteristic
370C4—98no—
371C4C23×C3×C3⋊D311yescharacteristic
372C4C23×C3×C3⋊D316yes—
373C4C23×C3×C3⋊D311yescharacteristic
374C3—24no—
375C3C3×#596414yescomplemented
376C3#1059675611yescharacteristic, direct product factor
377C2—21no—
378C2—26no—
379C2—188no—
380C2C24×C3×C3⋊D311yescharacteristic
subgroupG/H
centerC3×C4C23×C3⋊D3
commutator, cosocleC2×C32C25×C3
FrattiniC2C24×C3×C3⋊D3
FittingC2×C33×#41C2
socleC2×C33C25
hypercenterC2×C3×#41C3⋊D3
hypocenterC32C3×#585
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C3×C4 ◁ C2×C3×#41

Complements

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

NH ≅ G/Ntype
1. C2×C33×#41C2semidirect
2. C3×#64367C2semidirect
3. C3×#64366C2semidirect
4. C3×#41×C3⋊D3C2semidirect
5. C3×#64364C2semidirect
6. C3×#64361C2semidirect
7. C2×C3×#3828C2semidirect
8. #10596756C3direct (N × H)
10. C33×#41C22semidirect
11. C33×#41C22semidirect
12. C2×C33×Q8C22semidirect
13. C2×C33×D4C22semidirect
14. C22×C33×C4C22semidirect
15. C2×C3×#638C22semidirect
16. C3×#3832C22semidirect
17. C3×Q8×C3⋊D3C22semidirect
18. C3×#3830C22semidirect
19. C3×D4×C3⋊D3C22semidirect
20. C3×#3828C22semidirect
21. C3×#3828C22semidirect
22. C3×#3828C22semidirect
23. C3×#3828C22semidirect
24. C2×C3×C4⋊(C3⋊D3)C22semidirect
25. C2×C3×C4×C3⋊D3C22semidirect
26. C2×C3×#634C22semidirect
28. C2×C32×#41D3semidirect
29. C2×C32×#41C2×C3semidirect
30. #64367C2×C3semidirect
31. #64366C2×C3semidirect
32. #41×C3⋊D3C2×C3semidirect
33. #64364C2×C3semidirect
34. #64361C2×C3semidirect
35. C2×#3828C2×C3semidirect
63. C32×#41C2×D3semidirect
64. C32×#41C2×D3semidirect
65. C32×#41C22×C3semidirect
66. C32×#41C22×C3semidirect
68. C2×C32×Q8C2×D3semidirect
69. C2×C32×Q8C22×C3semidirect
71. C2×C32×D4C2×D3semidirect
72. C2×C32×D4C22×C3semidirect
74. C22×C32×C4C2×D3semidirect
75. C22×C32×C4C22×C3semidirect
76. C2×#638C22×C3semidirect
77. #3832C22×C3semidirect
78. Q8×C3⋊D3C22×C3semidirect
79. #3830C22×C3semidirect
80. D4×C3⋊D3C22×C3semidirect
81. #3828C22×C3semidirect
82. #3828C22×C3semidirect
83. #3828C22×C3semidirect
84. #3828C22×C3semidirect
85. C2×C4⋊(C3⋊D3)C22×C3semidirect
86. C2×C4×C3⋊D3C22×C3semidirect
87. C2×#634C22×C3semidirect
116. C2×C3×#41C3×D3semidirect
117. C2×C3×#41C3⋊D3semidirect
178. C3×#41C2×C3×D3semidirect
179. C3×#41C2×C3×D3semidirect
180. C3×#41C2×C3⋊D3semidirect
181. C3×#41C2×C3⋊D3semidirect
185. C2×C3×Q8C2×C3×D3semidirect
186. C2×C3×Q8C2×C3⋊D3semidirect
191. C2×C3×D4C2×C3×D3semidirect
192. C2×C3×D4C2×C3⋊D3semidirect
195. C22×C3×C4C2×C3×D3semidirect
196. C22×C3×C4C2×C3⋊D3semidirect
239. C2×#41C3×C3⋊D3semidirect
240. C33#585semidirect
299. #41C2×C3×C3⋊D3semidirect
300. #41C2×C3×C3⋊D3semidirect
303. C2×Q8C2×C3×C3⋊D3semidirect
307. C2×D4C2×C3×C3⋊D3semidirect
309. C22×C4C2×C3×C3⋊D3semidirect
335. C32#5964semidirect
336. C32C3×#585semidirect
375. C3C3×#5964semidirect
376. C3#10596756direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C33×#412
C3×#643672
C3×#643662
C3×#41×C3⋊D32
C3×#643642
C3×#643612
C2×C3×#38282
#105967563
C3×#59643