Finite Group Databasebeta

← C2×C32×#915 #13179137

Subgroups — C2×C32×#915 #13179137

Order 1728.

1820 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×D3×Q8C211yesmaximal, characteristic, complemented
2C2×C32×#237C211yesmaximal, characteristic
3C2×C33×SD16C211yesmaximal, characteristic, complemented
4C2×C32×#216C211yesmaximal, characteristic, complemented
5C2×C32×#214C211yesmaximal, characteristic, complemented
6C32×#915C218yesmaximal, direct product factor
7C2×C32×Dic12C211yesmaximal, characteristic, complemented
8C2×C32×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
9C2×C32×#137—31nomaximal
10C2×C3×#915C314yesmaximal, direct product factor
11C2×C33×Q8C2211yescharacteristic
12C2×C33×D4C2211yescharacteristic
13C2×C32×#67—21no—
14C22×C32×Dic3—21no—
15C32×D3×Q8—22no—
16C32×D3×Q8C2214yescomplemented
17C32×#237—22no—
18C32×#237C2214yes—
19C2×C32×C4×D3—21no—
20C2×C32×C4×D3C2211yescharacteristic
21C2×C32×Dic6—21no—
22C2×C32×Dic6C2211yescharacteristic
23C2×C32×Dic6C2211yescharacteristic
24C33×SD16C2214yescomplemented
25C2×C33×C8C2211yescharacteristic
26C32×#216C2214yescomplemented
27C32×#214C2214yescomplemented
28C2×C32×C3⋊C8C2211yescharacteristic
29C32×Dic12C2214yescomplemented
30C32×ΓL(1,𝔽25)C2214yescomplemented
31C2×C32×#41—31no—
32C22×C32×Q8—31no—
33C2×C3×D3×Q8C2×C314yescomplemented
34C2×C3×#237C2×C314yes—
35C32×#137—38no—
36C2×C32×Q16—31no—
37C2×C32×Q16—31no—
38C2×C32×SD16—24no—
39C2×C32×SD16—31no—
40C2×C32×SD16D311yescharacteristic, complemented
41C2×C32×SD16C2×C314yescomplemented
42C2×C32×M16—31no—
43C2×C3×#216C2×C314yescomplemented
44C2×C3×#214C2×C314yescomplemented
45C3×#915C2×C3132yesdirect product factor
46C2×C3×Dic12C2×C314yescomplemented
47C2×C3×ΓL(1,𝔽25)C2×C314yescomplemented
48C23×C33—21no—
49C22×C32×D3D411yescharacteristic
50C33×Q8—21no—
51C33×Q8C2312yes—
52C33×D4—21no—
53C33×D4C2312yes—
54C2×C33×C4—21no—
55C2×C33×C4C2311yescharacteristic
56C32×#67—24no—
57C2×C32×Dic3—21no—
58C2×C32×Dic3—21no—
59C2×C32×Dic3—24no—
60C2×C32×Dic3D411yescharacteristic
61C32×C4×D3—24no—
62C32×C4×D3D412yes—
63C32×C4×D3C2312yes—
64C32×Dic6—21no—
65C32×Dic6—21no—
66C32×Dic6—24no—
67C32×Dic6C2312yes—
68C32×Dic6C2312yes—
69C33×C8C2312yes—
70C32×C3⋊C8C2312yes—
71C2×C3×#137—34no—
72C2×#915C3211yescharacteristic, direct product factor
73C32×#41—34no—
74C32×#41—62no—
75C2×C32×Q8—24no—
76C2×C32×Q8—34no—
77C2×C32×Q8—31no—
78C2×C32×Q8—31no—
79C2×C32×Q8—61no—
80C2×C32×Q8—62no—
81C2×C32×Q8C2×D311yescharacteristic
82C2×C32×Q8C22×C314yes—
83C2×C32×D4—24no—
84C2×C32×D4—61no—
85C2×C32×D4C2×D311yescharacteristic
86C2×C32×D4C22×C314yes—
87C22×C32×C4—31no—
88C22×C32×C4—61no—
89C22×C32×C4—61no—
90C2×C3×#67—24no—
91C22×C3×Dic3—24no—
92C3×D3×Q8—28no—
93C3×D3×Q8C22×C3116yescomplemented
94C3×#237—28no—
95C3×#237C22×C3116yes—
96C2×C3×C4×D3—24no—
97C2×C3×C4×D3C22×C314yes—
98C2×C3×Dic6—24no—
99C2×C3×Dic6C22×C314yes—
100C2×C3×Dic6C22×C314yes—
101C32×Q16—34no—
102C32×Q16—34no—
103C32×SD16—216no—
104C32×SD16—34no—
105C32×SD16C2×D314yescomplemented
106C32×SD16C22×C3116yescomplemented
107C32×M16—34no—
108C2×C32×C8—24no—
109C2×C32×C8—31no—
110C2×C32×C8C2×D311yescharacteristic
111C2×C32×C8C22×C314yes—
112C3×#216C22×C3116yescomplemented
113C3×#214C22×C3116yescomplemented
114C2×C3×C3⋊C8C22×C314yes—
115C3×Dic12C22×C3116yescomplemented
116C3×ΓL(1,𝔽25)C22×C3116yescomplemented
117C22×C33—22no—
118C22×C33—42no—
119C22×C33C2×D411yescharacteristic
120C2×C32×D3—22no—
121C2×C32×D3C2×D412yes—
122C33×C4—22no—
123C33×C4C2×D411yescharacteristic
124C33×C4C2411yescharacteristic
125C32×Dic3—22no—
126C32×Dic3—22no—
127C32×Dic3C2×D412yes—
128C2×C3×#41—34no—
129C22×C3×Q8—34no—
130C2×D3×Q8C2×C3211yescharacteristic, complemented
131C2×#237C2×C3211yescharacteristic
132C3×#137—332no—
133C2×C3×Q16—34no—
134C2×C3×Q16—34no—
135C2×C3×SD16—24no—
136C2×C3×SD16—34no—
137C2×C3×SD16C3×D314yescomplemented
138C2×C3×SD16C2×C3211yescharacteristic, complemented
139C2×C3×M16—34no—
140C2×#216C2×C3211yescharacteristic, complemented
141C2×#214C2×C3211yescharacteristic, complemented
142#915C2×C3218yesdirect product factor
143C2×Dic12C2×C3211yescharacteristic, complemented
144C2×ΓL(1,𝔽25)C2×C3211yescharacteristic, complemented
145C23×C32—21no—
146C23×C32—24no—
147C23×C32—31no—
148C23×C32—44no—
149C22×C3×D3C3×D414yes—
150C32×Q8—21no—
151C32×Q8—28no—
152C32×Q8—24no—
153C32×Q8—32no—
154C32×Q8—32no—
155C32×Q8—44no—
156C32×Q8—64no—
157C32×Q8—61no—
158C32×Q8—61no—
159C32×Q8C22×D312yes—
160C32×Q8C23×C318yes—
161C32×D4—21no—
162C32×D4—28no—
163C32×D4—24no—
164C32×D4—44no—
165C32×D4—64no—
166C32×D4C22×D312yes—
167C32×D4C23×C318yes—
168C2×C32×C4—21no—
169C2×C32×C4—24no—
170C2×C32×C4—24no—
171C2×C32×C4—31no—
172C2×C32×C4—32no—
173C2×C32×C4—32no—
174C2×C32×C4—44no—
175C2×C32×C4—64no—
176C2×C32×C4—64no—
177C2×C32×C4—61no—
178C2×C32×C4—61no—
179C2×C32×C4C22×D311yescharacteristic
180C2×C32×C4C23×C314yes—
181C3×#67—216no—
182C2×C3×Dic3—24no—
183C2×C3×Dic3—24no—
184C2×C3×Dic3—216no—
185C2×C3×Dic3C3×D414yes—
186C3×C4×D3—216no—
187C3×C4×D3C3×D418yes—
188C3×C4×D3C23×C318yes—
189C3×Dic6—24no—
190C3×Dic6—24no—
191C3×Dic6—216no—
192C3×Dic6C23×C318yes—
193C3×Dic6C23×C318yes—
194C32×C8—28no—
195C32×C8—32no—
196C32×C8C22×D312yes—
197C32×C8C23×C318yes—
198C3×C3⋊C8C23×C318yes—
199C2×#137—31noSylow
200C2×C33—42no—
201C2×C33#13712yescomplemented
202C2×C33C22×D411yescharacteristic
203C32×D3—22no—
204C3×#41—316no—
205C3×#41—68no—
206C2×C3×Q8—24no—
207C2×C3×Q8—316no—
208C2×C3×Q8—34no—
209C2×C3×Q8—34no—
210C2×C3×Q8—64no—
211C2×C3×Q8—68no—
212C2×C3×Q8C2×C3×D314yes—
213C2×C3×Q8C22×C3211yescharacteristic
214C2×C3×D4—24no—
215C2×C3×D4—64no—
216C2×C3×D4C2×C3×D314yes—
217C2×C3×D4C22×C3211yescharacteristic
218C22×C3×C4—34no—
219C22×C3×C4—64no—
220C22×C3×C4—64no—
221C2×#67—21no—
222C22×Dic3—21no—
223D3×Q8—22no—
224D3×Q8C22×C3214yescomplemented
225#237—22no—
226#237C22×C3214yes—
227C2×C4×D3—21no—
228C2×C4×D3C22×C3211yescharacteristic
229C2×Dic6—21no—
230C2×Dic6C22×C3211yescharacteristic
231C2×Dic6C22×C3211yescharacteristic
232C3×Q16—316no—
233C3×Q16—316no—
234C3×SD16—216no—
235C3×SD16—316no—
236C3×SD16C2×C3×D3116yescomplemented
237C3×SD16C22×C3214yescomplemented
238C3×M16—316no—
239C2×C3×C8—24no—
240C2×C3×C8—34no—
241C2×C3×C8C2×C3×D314yes—
242C2×C3×C8C22×C3211yescharacteristic
243#216C22×C3214yescomplemented
244#214C22×C3214yescomplemented
245C2×C3⋊C8C22×C3211yescharacteristic
246Dic12C22×C3214yescomplemented
247ΓL(1,𝔽25)C22×C3214yescomplemented
248C22×C32—22no—
249C22×C32—24no—
250C22×C32—28no—
251C22×C32—32no—
252C22×C32—48no—
253C22×C32—42no—
254C22×C32—48no—
255C22×C32—62no—
256C22×C32—88no—
257C22×C32D3×D411yescharacteristic
258C22×C32C2×C3×D414yes—
259C2×C3×D3—28no—
260C2×C3×D3C2×C3×D418yes—
261C32×C4—22no—
262C32×C4—24no—
263C32×C4—24no—
264C32×C4—28no—
265C32×C4—32no—
266C32×C4—48no—
267C32×C4—62no—
268C32×C4—62no—
269C32×C4D3×D411yescharacteristic
270C32×C4C2×C3×D414yes—
271C32×C4C23×D311yescharacteristic
272C32×C4C24×C314yes—
273C3×Dic3—28no—
274C3×Dic3—28no—
275C3×Dic3C2×C3×D418yes—
276C2×#41—31no—
277C22×Q8—31no—
278#137—38no—
279C2×Q16—31no—
280C2×Q16—31no—
281C2×SD16—31no—
282C2×SD16C32×D311yescharacteristic, complemented
283C2×M16—31no—
284C33C2×#13711yescharacteristic, complemented, Sylow
285C23×C3—24no—
286C23×C3—21no—
287C23×C3—34no—
288C23×C3—44no—
289C22×D3C32×D411yescharacteristic
290C3×Q8—24no—
291C3×Q8—21no—
292C3×Q8—28no—
293C3×Q8—38no—
294C3×Q8—38no—
295C3×Q8—44no—
296C3×Q8—616no—
297C3×Q8—64no—
298C3×Q8—64no—
299C3×Q8C22×C3×D318yes—
300C3×Q8C23×C3212yes—
301C3×D4—24no—
302C3×D4—21no—
303C3×D4—28no—
304C3×D4—44no—
305C3×D4—616no—
306C3×D4C22×C3×D318yes—
307C3×D4C23×C3212yes—
308C2×C3×C4—24no—
309C2×C3×C4—21no—
310C2×C3×C4—24no—
311C2×C3×C4—34no—
312C2×C3×C4—38no—
313C2×C3×C4—38no—
314C2×C3×C4—44no—
315C2×C3×C4—616no—
316C2×C3×C4—616no—
317C2×C3×C4—64no—
318C2×C3×C4—64no—
319C2×C3×C4C22×C3×D314yes—
320C2×C3×C4C23×C3211yescharacteristic
321#67—24no—
322C2×Dic3—21no—
323C2×Dic3—21no—
324C2×Dic3—24no—
325C2×Dic3C32×D411yescharacteristic
326C4×D3—24no—
327C4×D3C32×D412yes—
328C4×D3C23×C3212yes—
329Dic6—21no—
330Dic6—21no—
331Dic6—24no—
332Dic6C23×C3212yes—
333Dic6C23×C3212yes—
334C3×C8—28no—
335C3×C8—38no—
336C3×C8C22×C3×D318yes—
337C3×C8C23×C3212yes—
338C3⋊C8C23×C3212yes—
339C2×C32—28no—
340C2×C32—24no—
341C2×C32—42no—
342C2×C32—48no—
343C2×C32—62no—
344C2×C32—88no—
345C2×C32#91512yesdirect product factor
346C2×C32C3×#13718yescomplemented
347C2×C32C2×D3×D411yescharacteristic
348C2×C32C22×C3×D414yes—
349C3×D3—28no—
350#41—34no—
351#41—62no—
352C2×Q8—34no—
353C2×Q8—31no—
354C2×Q8—31no—
355C2×Q8—61no—
356C2×Q8—62no—
357C2×Q8C2×C32×D311yescharacteristic
358C2×D4—61no—
359C2×D4C2×C32×D311yescharacteristic
360C22×C4—31no—
361C22×C4—61no—
362C22×C4—61no—
363Q16—34no—
364Q16—34no—
365SD16—34no—
366SD16C2×C32×D314yescomplemented
367M16—34no—
368C2×C8—31no—
369C2×C8C2×C32×D311yescharacteristic
370C22×C3—28no—
371C22×C3—22no—
372C22×C3—24no—
373C22×C3—38no—
374C22×C3—48no—
375C22×C3—48no—
376C22×C3—42no—
377C22×C3—68no—
378C22×C3—88no—
379C22×C3C3×D3×D414yes—
380C22×C3C2×C32×D411yescharacteristic
381C2×D3—22no—
382C2×D3C2×C32×D412yes—
383C3×C4—28no—
384C3×C4—22no—
385C3×C4—24no—
386C3×C4—24no—
387C3×C4—38no—
388C3×C4—48no—
389C3×C4—68no—
390C3×C4—68no—
391C3×C4C3×D3×D414yes—
392C3×C4C2×C32×D411yescharacteristic
393C3×C4C23×C3×D314yes—
394C3×C4C24×C3211yescharacteristic
395Dic3—22no—
396Dic3—22no—
397Dic3C2×C32×D412yes—
398C32—24no—
399C32C2×#91511yescharacteristic, direct product factor
400C32C2×C3×#13714yescomplemented
401C23—21no—
402C23—31no—
403Q8—21no—
404Q8—32no—
405Q8—32no—
406Q8—64no—
407Q8—61no—
408Q8—61no—
409Q8C22×C32×D312yes—
410D4—21no—
411D4—64no—
412D4C22×C32×D312yes—
413C2×C4—21no—
414C2×C4—31no—
415C2×C4—32no—
416C2×C4—32no—
417C2×C4—64no—
418C2×C4—64no—
419C2×C4—61no—
420C2×C4—61no—
421C2×C4C22×C32×D311yescharacteristic
422C8—32no—
423C8C22×C32×D312yes—
424C2×C3—28no—
425C2×C3—24no—
426C2×C3—48no—
427C2×C3—42no—
428C2×C3—68no—
429C2×C3—88no—
430C2×C3C3×#91518yesdirect product factor
431C2×C3C32×#13712yescomplemented
432C2×C3C2×C3×D3×D414yes—
433C2×C3C22×C32×D411yescharacteristic
434D3—22no—
435C22—22no—
436C22—32no—
437C22—42no—
438C22—62no—
439C22C32×D3×D411yescharacteristic
440C4—22no—
441C4—32no—
442C4—62no—
443C4—62no—
444C4C32×D3×D411yescharacteristic
445C4C23×C32×D311yescharacteristic
446C3—24no—
447C3C2×C3×#91514yesdirect product factor
448C3C2×C32×#13711yescharacteristic, complemented
449C2—42no—
450C2—62no—
451C2C32×#91512yesdirect product factor
452C2C2×C32×D3×D411yescharacteristic
subgroupG/H
centerC22×C32D3×D4
commutator, cosocleC3×C4C24×C32
FrattiniC4C23×C32×D3
FittingC2×C33×SD16C2
socleC22×C33C2×D4
prime radicalC2×C32×#237C2
hypercenterC2×C32×SD16D3
hypocenterC3C2×C32×#137
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C32 ◁ C2×C32×C4 ◁ C2×C32×SD16

Complements

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

NH ≅ G/Ntype
1. C2×C32×D3×Q8C2semidirect
3. C2×C33×SD16C2semidirect
4. C2×C32×#216C2semidirect
5. C2×C32×#214C2semidirect
6. C32×#915C2direct (N × H)
7. C2×C32×Dic12C2semidirect
8. C2×C32×ΓL(1,𝔽25)C2semidirect
10. C2×C3×#915C3direct (N × H)
16. C32×D3×Q8C22semidirect
24. C33×SD16C22semidirect
26. C32×#216C22semidirect
27. C32×#214C22semidirect
29. C32×Dic12C22semidirect
30. C32×ΓL(1,𝔽25)C22semidirect
33. C2×C3×D3×Q8C2×C3semidirect
40. C2×C32×SD16D3semidirect
41. C2×C32×SD16C2×C3semidirect
43. C2×C3×#216C2×C3semidirect
44. C2×C3×#214C2×C3semidirect
45. C3×#915C2×C3direct (N × H)
46. C2×C3×Dic12C2×C3semidirect
47. C2×C3×ΓL(1,𝔽25)C2×C3semidirect
72. C2×#915C32direct (N × H)
93. C3×D3×Q8C22×C3semidirect
105. C32×SD16C2×D3semidirect
106. C32×SD16C22×C3semidirect
112. C3×#216C22×C3semidirect
113. C3×#214C22×C3semidirect
115. C3×Dic12C22×C3semidirect
116. C3×ΓL(1,𝔽25)C22×C3semidirect
130. C2×D3×Q8C2×C32semidirect
137. C2×C3×SD16C3×D3semidirect
138. C2×C3×SD16C2×C32semidirect
140. C2×#216C2×C32semidirect
141. C2×#214C2×C32semidirect
142. #915C2×C32direct (N × H)
143. C2×Dic12C2×C32semidirect
144. C2×ΓL(1,𝔽25)C2×C32semidirect
201. C2×C33#137semidirect
224. D3×Q8C22×C32semidirect
236. C3×SD16C2×C3×D3semidirect
237. C3×SD16C22×C32semidirect
243. #216C22×C32semidirect
244. #214C22×C32semidirect
246. Dic12C22×C32semidirect
247. ΓL(1,𝔽25)C22×C32semidirect
282. C2×SD16C32×D3semidirect
284. C33C2×#137semidirect
345. C2×C32#915direct (N × H)
346. C2×C32C3×#137semidirect
366. SD16C2×C32×D3semidirect
399. C32C2×#915direct (N × H)
400. C32C2×C3×#137semidirect
430. C2×C3C3×#915direct (N × H)
431. C2×C3C32×#137semidirect
447. C3C2×C3×#915direct (N × H)
448. C3C2×C32×#137semidirect
451. C2C32×#915direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×D3×Q82
C2×C32×#2372
C2×C33×SD162
C2×C32×#2162
C2×C32×#2142
C32×#9152
C2×C32×Dic122
C2×C32×ΓL(1,𝔽25)2
C2×C32×#1373
C2×C3×#9153