Finite Group Databasebeta

← C32×#5964 #13181723

Subgroups — C32×#5964 #13181723

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
2C32×#1010C214yesmaximal, complemented
3C32×#1009C214yesmaximal, complemented
4C32×D3×#41C218yesmaximal, complemented
5C32×#1007C212yesmaximal, complemented
6C32×#1004C216yesmaximal, complemented
7C2×C32×#235C216yesmaximal, complemented
8C32×#585—31nomaximal
9C3×#5964C314yesmaximal, direct product factor
10C33×#41C2214yescomplemented
11C33×#41C2214yescomplemented
12C2×C33×Q8C2211yescharacteristic, complemented
13C2×C33×D4C2213yescomplemented
14C22×C33×C4C2213yescomplemented
15C2×C32×#67C22112yescomplemented
16C32×#239C2218yescomplemented
17C32×D3×Q8C2218yescomplemented
18C32×#237C22124yescomplemented
19C32×D3×D4C22124yescomplemented
20C32×#235C2214yescomplemented
21C32×#235C22112yescomplemented
22C32×#235C22112yescomplemented
23C32×#235C22112yescomplemented
24C2×C32×D12C2216yescomplemented
25C2×C32×C4×D3C22112yescomplemented
26C2×C32×Dic6C2216yescomplemented
27C32×2−1+4—34no—
28C32×2−1+4—32no—
29C32×H(5,𝔽2)—34no—
30C32×H(5,𝔽2)—36no—
31C2×C32×#41—24no—
32C2×C32×#41—38no—
33C2×C32×#41—36no—
34C2×C32×#41D311yescharacteristic, complemented
35C2×C32×#41C2×C314yescomplemented
36C3×#1010C2×C3116yescomplemented
37C3×#1009C2×C3116yescomplemented
38C3×D3×#41C2×C3132yescomplemented
39C3×#1007C2×C318yescomplemented
40C3×#1004C2×C3124yescomplemented
41C2×C3×#235C2×C3124yescomplemented
42C23×C33C2313yes—
43C22×C32×D3C23112yes—
44C33×Q8C2314yes—
45C33×D4C23112yes—
46C2×C33×C4C2316yes—
47C2×C33×C4C2316yes—
48C2×C33×C4C2311yescharacteristic
49C2×C33×C4C2313yes—
50C32×#67C2318yes—
51C32×#67C23124yes—
52C2×C32×Dic3C23112yes—
53C32×D12C2314yes—
54C32×D12C23112yes—
55C32×C4×D3C2318yes—
56C32×C4×D3C23124yes—
57C32×Dic6C2314yes—
58C32×Dic6C23112yes—
59C3×#585—34no—
60#5964C3211yescharacteristic, direct product factor
61C32×#41—216no—
62C32×#41—216no—
63C32×#41—38no—
64C32×#41—324no—
65C32×#41—312no—
66C32×#41—34no—
67C32×#41—312no—
68C32×#41—312no—
69C32×#41C2×D314yescomplemented
70C32×#41C2×D314yescomplemented
71C32×#41C22×C3116yescomplemented
72C32×#41C22×C3116yescomplemented
73C2×C32×Q8—24no—
74C2×C32×Q8—38no—
75C2×C32×Q8—36no—
76C2×C32×Q8C2×D311yescharacteristic, complemented
77C2×C32×Q8C22×C314yescomplemented
78C2×C32×D4—212no—
79C2×C32×D4—312no—
80C2×C32×D4—324no—
81C2×C32×D4—36no—
82C2×C32×D4C2×D313yescomplemented
83C2×C32×D4C22×C3112yescomplemented
84C22×C32×C4—212no—
85C22×C32×C4—312no—
86C22×C32×C4C2×D313yescomplemented
87C22×C32×C4C22×C3112yescomplemented
88C2×C3×#67C22×C3148yescomplemented
89C3×#239C22×C3132yescomplemented
90C3×D3×Q8C22×C3132yescomplemented
91C3×#237C22×C3196yescomplemented
92C3×D3×D4C22×C3196yescomplemented
93C3×#235C22×C3116yescomplemented
94C3×#235C22×C3148yescomplemented
95C3×#235C22×C3148yescomplemented
96C3×#235C22×C3148yescomplemented
97C2×C3×D12C22×C3124yescomplemented
98C2×C3×C4×D3C22×C3148yescomplemented
99C2×C3×Dic6C22×C3124yescomplemented
100C22×C33—43no—
101C22×C33C2416yes—
102C22×C33C2411yescharacteristic
103C2×C32×D3—412no—
104C2×C32×D3C2418yes—
105C33×C4C2411yescharacteristic
106C33×C4C2416yes—
107C33×C4C2411yescharacteristic
108C32×Dic3C2418yes—
109C3×2−1+4—316no—
110C3×2−1+4—38no—
111C3×H(5,𝔽2)—316no—
112C3×H(5,𝔽2)—324no—
113C2×C3×#41—24no—
114C2×C3×#41—332no—
115C2×C3×#41—324no—
116C2×C3×#41C3×D314yescomplemented
117C2×C3×#41C2×C3211yescharacteristic, complemented
118#1010C2×C3214yescomplemented
119#1009C2×C3214yescomplemented
120D3×#41C2×C3218yescomplemented
121#1007C2×C3212yescomplemented
122#1004C2×C3216yescomplemented
123C2×#235C2×C3216yescomplemented
124C23×C32—212no—
125C23×C32—312no—
126C23×C32C22×D313yes—
127C23×C32C23×C3112yes—
128C22×C3×D3C23×C3148yes—
129C32×Q8—216no—
130C32×Q8—312no—
131C32×Q8—34no—
132C32×Q8C22×D314yes—
133C32×Q8C23×C3116yes—
134C32×D4—248no—
135C32×D4—324no—
136C32×D4—38no—
137C32×D4—312no—
138C32×D4—34no—
139C32×D4C22×D3112yes—
140C32×D4C23×C3148yes—
141C2×C32×C4—24no—
142C2×C32×C4—212no—
143C2×C32×C4—224no—
144C2×C32×C4—224no—
145C2×C32×C4—312no—
146C2×C32×C4—324no—
147C2×C32×C4—38no—
148C2×C32×C4C22×D316yes—
149C2×C32×C4C22×D313yes—
150C2×C32×C4C22×D316yes—
151C2×C32×C4C22×D311yescharacteristic
152C2×C32×C4C23×C3124yes—
153C2×C32×C4C23×C3124yes—
154C2×C32×C4C23×C314yes—
155C2×C32×C4C23×C3112yes—
156C3×#67C23×C3132yes—
157C3×#67C23×C3196yes—
158C2×C3×Dic3C23×C3148yes—
159C3×D12C23×C3116yes—
160C3×D12C23×C3148yes—
161C3×C4×D3C23×C3132yes—
162C3×C4×D3C23×C3196yes—
163C3×Dic6C23×C3116yes—
164C3×Dic6C23×C3148yes—
165#585—31noSylow
166C2×C33—21no—
167C2×C33—26no—
168C2×C33C2511yescharacteristic
169C32×D3—28no—
170C3×#41—216no—
171C3×#41—216no—
172C3×#41—332no—
173C3×#41—396no—
174C3×#41—348no—
175C3×#41—316no—
176C3×#41—348no—
177C3×#41—348no—
178C3×#41C2×C3×D3116yescomplemented
179C3×#41C2×C3×D3116yescomplemented
180C3×#41C22×C3214yescomplemented
181C3×#41C22×C3214yescomplemented
182C2×C3×Q8—24no—
183C2×C3×Q8—332no—
184C2×C3×Q8—324no—
185C2×C3×Q8C2×C3×D314yescomplemented
186C2×C3×Q8C22×C3211yescharacteristic, complemented
187C2×C3×D4—212no—
188C2×C3×D4—348no—
189C2×C3×D4—396no—
190C2×C3×D4—324no—
191C2×C3×D4C2×C3×D3112yescomplemented
192C2×C3×D4C22×C3213yescomplemented
193C22×C3×C4—212no—
194C22×C3×C4—348no—
195C22×C3×C4C2×C3×D3112yescomplemented
196C22×C3×C4C22×C3213yescomplemented
197C2×#67C22×C32112yescomplemented
198#239C22×C3218yescomplemented
199D3×Q8C22×C3218yescomplemented
200#237C22×C32124yescomplemented
201D3×D4C22×C32124yescomplemented
202#235C22×C3214yescomplemented
203#235C22×C32112yescomplemented
204#235C22×C32112yescomplemented
205#235C22×C32112yescomplemented
206C2×D12C22×C3216yescomplemented
207C2×C4×D3C22×C32112yescomplemented
208C2×Dic6C22×C3216yescomplemented
209C22×C32—24no—
210C22×C32—224no—
211C22×C32—38no—
212C22×C32—43no—
213C22×C32—412no—
214C22×C32—424no—
215C22×C32—1212no—
216C22×C32C23×D316yes—
217C22×C32C23×D311yescharacteristic
218C22×C32C24×C3124yes—
219C22×C32C24×C314yes—
220C2×C3×D3—448no—
221C2×C3×D3C24×C3132yes—
222C32×C4—24no—
223C32×C4—24no—
224C32×C4—224no—
225C32×C4—38no—
226C32×C4C23×D316yes—
227C32×C4C23×D311yescharacteristic
228C32×C4C23×D311yescharacteristic
229C32×C4C24×C314yes—
230C32×C4C24×C3124yes—
231C32×C4C24×C314yes—
232C3×Dic3C24×C3132yes—
2332−1+4—34no—
2342−1+4—32no—
235H(5,𝔽2)—34no—
236H(5,𝔽2)—36no—
237C2×#41—38no—
238C2×#41—36no—
239C2×#41C32×D311yescharacteristic, complemented
240C33#58511yescharacteristic, complemented, Sylow
241C23×C3—212no—
242C23×C3—348no—
243C23×C3C22×C3×D3112yes—
244C23×C3C23×C3213yes—
245C22×D3C23×C32112yes—
246C3×Q8—216no—
247C3×Q8—348no—
248C3×Q8—316no—
249C3×Q8C22×C3×D3116yes—
250C3×Q8C23×C3214yes—
251C3×D4—248no—
252C3×D4—396no—
253C3×D4—332no—
254C3×D4—348no—
255C3×D4—316no—
256C3×D4C22×C3×D3148yes—
257C3×D4C23×C32112yes—
258C2×C3×C4—24no—
259C2×C3×C4—212no—
260C2×C3×C4—224no—
261C2×C3×C4—224no—
262C2×C3×C4—348no—
263C2×C3×C4—332no—
264C2×C3×C4—396no—
265C2×C3×C4C22×C3×D3124yes—
266C2×C3×C4C22×C3×D314yes—
267C2×C3×C4C22×C3×D3124yes—
268C2×C3×C4C22×C3×D3112yes—
269C2×C3×C4C23×C3216yes—
270C2×C3×C4C23×C3216yes—
271C2×C3×C4C23×C3211yescharacteristic
272C2×C3×C4C23×C3213yes—
273#67C23×C3218yes—
274#67C23×C32124yes—
275C2×Dic3C23×C32112yes—
276D12C23×C3214yes—
277D12C23×C32112yes—
278C4×D3C23×C3218yes—
279C4×D3C23×C32124yes—
280Dic6C23×C3214yes—
281Dic6C23×C32112yes—
282C2×C32—28no—
283C2×C32—21no—
284C2×C32—26no—
285C2×C32—24no—
286C2×C32—24no—
287C2×C32—224no—
288C2×C32—424no—
289C2×C32—68no—
290C2×C32C24×D311yescharacteristic
291C2×C32C25×C314yes—
292C3×D3—232no—
293#41—38no—
294#41—324no—
295#41—312no—
296#41—34no—
297#41—312no—
298#41—312no—
299#41C2×C32×D314yescomplemented
300#41C2×C32×D314yescomplemented
301C2×Q8—38no—
302C2×Q8—36no—
303C2×Q8C2×C32×D311yescharacteristic, complemented
304C2×D4—312no—
305C2×D4—324no—
306C2×D4—36no—
307C2×D4C2×C32×D313yescomplemented
308C22×C4—312no—
309C22×C4C2×C32×D313yescomplemented
310C22×C3—24no—
311C22×C3—224no—
312C22×C3—332no—
313C22×C3—412no—
314C22×C3—43no—
315C22×C3—424no—
316C22×C3—1248no—
317C22×C3C23×C3×D3124yes—
318C22×C3C23×C3×D314yes—
319C22×C3C24×C3216yes—
320C22×C3C24×C3211yescharacteristic
321C2×D3—412no—
322C2×D3C24×C3218yes—
323C3×C4—24no—
324C3×C4—24no—
325C3×C4—224no—
326C3×C4—332no—
327C3×C4C23×C3×D314yes—
328C3×C4C23×C3×D3124yes—
329C3×C4C23×C3×D314yes—
330C3×C4C24×C3211yescharacteristic
331C3×C4C24×C3216yes—
332C3×C4C24×C3211yescharacteristic
333Dic3C24×C3218yes—
334C32—24no—
335C32#596411yescharacteristic, direct product factor
336C32C3×#58514yescomplemented
337C23—312no—
338C23C22×C32×D313yes—
339Q8—312no—
340Q8—34no—
341Q8C22×C32×D314yes—
342D4—324no—
343D4—38no—
344D4—312no—
345D4—34no—
346D4C22×C32×D3112yes—
347C2×C4—312no—
348C2×C4—38no—
349C2×C4—324no—
350C2×C4C22×C32×D316yes—
351C2×C4C22×C32×D311yescharacteristic
352C2×C4C22×C32×D316yes—
353C2×C4C22×C32×D313yes—
354C2×C3—28no—
355C2×C3—24no—
356C2×C3—224no—
357C2×C3—26no—
358C2×C3—21no—
359C2×C3—24no—
360C2×C3—424no—
361C2×C3—632no—
362C2×C3C24×C3×D314yes—
363C2×C3C25×C3211yescharacteristic
364D3—28no—
365C22—38no—
366C22—43no—
367C22—1212no—
368C22C23×C32×D316yes—
369C22C23×C32×D311yescharacteristic
370C4—38no—
371C4C23×C32×D311yescharacteristic
372C4C23×C32×D316yes—
373C4C23×C32×D311yescharacteristic
374C3—24no—
375C3C3×#596414yesdirect product factor
376C3C32×#58511yescharacteristic, complemented
377C2—21no—
378C2—26no—
379C2—68no—
380C2C24×C32×D311yescharacteristic
subgroupG/H
centerC32×C4C23×D3
commutator, cosocleC2×C3C25×C32
FrattiniC2C24×C32×D3
FittingC2×C33×#41C2
socleC2×C33C25
hypercenterC2×C32×#41D3
hypocenterC3C32×#585
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C32×C4 ◁ C2×C32×#41

Complements

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

NH ≅ G/Ntype
1. C2×C33×#41C2semidirect
2. C32×#1010C2semidirect
3. C32×#1009C2semidirect
4. C32×D3×#41C2semidirect
5. C32×#1007C2semidirect
6. C32×#1004C2semidirect
7. C2×C32×#235C2semidirect
9. C3×#5964C3direct (N × H)
10. C33×#41C22semidirect
11. C33×#41C22semidirect
12. C2×C33×Q8C22semidirect
13. C2×C33×D4C22semidirect
14. C22×C33×C4C22semidirect
15. C2×C32×#67C22semidirect
16. C32×#239C22semidirect
17. C32×D3×Q8C22semidirect
18. C32×#237C22semidirect
19. C32×D3×D4C22semidirect
20. C32×#235C22semidirect
21. C32×#235C22semidirect
22. C32×#235C22semidirect
23. C32×#235C22semidirect
24. C2×C32×D12C22semidirect
25. C2×C32×C4×D3C22semidirect
26. C2×C32×Dic6C22semidirect
34. C2×C32×#41D3semidirect
35. C2×C32×#41C2×C3semidirect
36. C3×#1010C2×C3semidirect
37. C3×#1009C2×C3semidirect
38. C3×D3×#41C2×C3semidirect
39. C3×#1007C2×C3semidirect
40. C3×#1004C2×C3semidirect
41. C2×C3×#235C2×C3semidirect
60. #5964C32direct (N × H)
69. C32×#41C2×D3semidirect
70. C32×#41C2×D3semidirect
71. C32×#41C22×C3semidirect
72. C32×#41C22×C3semidirect
76. C2×C32×Q8C2×D3semidirect
77. C2×C32×Q8C22×C3semidirect
82. C2×C32×D4C2×D3semidirect
83. C2×C32×D4C22×C3semidirect
86. C22×C32×C4C2×D3semidirect
87. C22×C32×C4C22×C3semidirect
88. C2×C3×#67C22×C3semidirect
89. C3×#239C22×C3semidirect
90. C3×D3×Q8C22×C3semidirect
91. C3×#237C22×C3semidirect
92. C3×D3×D4C22×C3semidirect
93. C3×#235C22×C3semidirect
94. C3×#235C22×C3semidirect
95. C3×#235C22×C3semidirect
96. C3×#235C22×C3semidirect
97. C2×C3×D12C22×C3semidirect
98. C2×C3×C4×D3C22×C3semidirect
99. C2×C3×Dic6C22×C3semidirect
116. C2×C3×#41C3×D3semidirect
117. C2×C3×#41C2×C32semidirect
118. #1010C2×C32semidirect
119. #1009C2×C32semidirect
120. D3×#41C2×C32semidirect
121. #1007C2×C32semidirect
122. #1004C2×C32semidirect
123. C2×#235C2×C32semidirect
178. C3×#41C2×C3×D3semidirect
179. C3×#41C2×C3×D3semidirect
180. C3×#41C22×C32semidirect
181. C3×#41C22×C32semidirect
185. C2×C3×Q8C2×C3×D3semidirect
186. C2×C3×Q8C22×C32semidirect
191. C2×C3×D4C2×C3×D3semidirect
192. C2×C3×D4C22×C32semidirect
195. C22×C3×C4C2×C3×D3semidirect
196. C22×C3×C4C22×C32semidirect
197. C2×#67C22×C32semidirect
198. #239C22×C32semidirect
199. D3×Q8C22×C32semidirect
200. #237C22×C32semidirect
201. D3×D4C22×C32semidirect
202. #235C22×C32semidirect
203. #235C22×C32semidirect
204. #235C22×C32semidirect
205. #235C22×C32semidirect
206. C2×D12C22×C32semidirect
207. C2×C4×D3C22×C32semidirect
208. C2×Dic6C22×C32semidirect
239. C2×#41C32×D3semidirect
240. C33#585semidirect
299. #41C2×C32×D3semidirect
300. #41C2×C32×D3semidirect
303. C2×Q8C2×C32×D3semidirect
307. C2×D4C2×C32×D3semidirect
309. C22×C4C2×C32×D3semidirect
335. C32#5964direct (N × H)
336. C32C3×#585semidirect
375. C3C3×#5964direct (N × H)
376. C3C32×#585semidirect

Maximal subgroups

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

subgroupindex
C2×C33×#412
C32×#10102
C32×#10092
C32×D3×#412
C32×#10072
C32×#10042
C2×C32×#2352
C32×#5853
C3×#59643