Finite Group Databasebeta

← C3×D32×Q16 #13174355

Subgroups — C3×D32×Q16 #13174355

Order 1728.

1230 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×D32×Q8C212yesmaximal
2C3×Q16×C3⋊D3C211yesmaximal, characteristic, complemented
3C32×D3×Q16C212yesmaximal, complemented
4C3×#63944C212yesmaximal, complemented
5C3×D3×#216C214yesmaximal, complemented
6C3×#63802C211yesmaximal, characteristic, complemented
7C3×D3×Dic12C212yesmaximal, complemented
8C3×D32×C8C211yesmaximal, characteristic
9C2×C3×D3×Q16—32nomaximal
10D32×Q16C311yesmaximal, characteristic, direct product factor
11C3×Q8×C3⋊D3C2212yes—
12C32×D3×Q8C2214yes—
13C3×C4×D32—22no—
14C3×C4×D32C2211yescharacteristic
15C3×#3797—22no—
16C3×#3797C2212yes—
17C3×D3×Dic6—24no—
18C3×D3×Dic6C2214yes—
19C33×Q16C2211yescharacteristic, complemented
20C3×#3756C2212yescomplemented
21C3×#3746C2211yescharacteristic, complemented
22C3×C8×C3⋊D3C2211yescharacteristic
23C32×#216C2214yescomplemented
24C32×Dic12C2212yescomplemented
25C32×D3×C8C2212yes—
26C3×#3719C2214yescomplemented
27C3×#3718C2212yescomplemented
28C3×#3710C2211yescharacteristic
29C3×D3×C3⋊C8C2212yes—
30C2×C3×D3×Q8—34no—
31D32×Q8C2×C312yes—
32C2×C32×Q16—32no—
33Q16×C3⋊D3C2×C311yescharacteristic, complemented
34C2×C3×#216—34no—
35C3×D3×Q16—22no—
36C3×D3×Q16—34no—
37C3×D3×Q16—32no—
38C3×D3×Q16—34no—
39C3×D3×Q16—32no—
40C3×D3×Q16—32no—
41C3×D3×Q16—61no—
42C3×D3×Q16D312yesdirect product factor
43C3×D3×Q16C2×C312yescomplemented
44C2×C3×Dic12—32no—
45C2×C3×D3×C8—32no—
46#63944C2×C312yescomplemented
47D3×#216C2×C314yescomplemented
48#63802C2×C311yescharacteristic, complemented
49D3×Dic12C2×C312yescomplemented
50D32×C8C2×C311yescharacteristic
51C2×C3×D32D411yescharacteristic
52C33×Q8C2312yes—
53C3×C4×C3⋊D3—22no—
54C3×C4×C3⋊D3C2311yescharacteristic
55C3×#634—22no—
56C3×#634C2312yes—
57C32×C4×D3—24no—
58C32×C4×D3C2312yes—
59C32×Dic6—24no—
60C32×Dic6C2314yes—
61C3×#627—22no—
62C3×#627—24no—
63C3×#624—22no—
64C3×#624D411yescharacteristic
65C3×D3×Dic3—24no—
66C3×D3×Dic3D412yes—
67C33×C8C2311yescharacteristic
68C3×C3⋊(C3⋊C8)C2311yescharacteristic
69C32×C3⋊C8C2312yes—
70C22×C3×Q16—91no—
71C2×D3×Q16—32no—
72C2×C32×Q8—34no—
73Q8×C3⋊D3C22×C312yes—
74C3×D3×Q8—24no—
75C3×D3×Q8—34no—
76C3×D3×Q8—34no—
77C3×D3×Q8—34no—
78C3×D3×Q8—64no—
79C3×D3×Q8—64no—
80C3×D3×Q8—62no—
81C3×D3×Q8C2×D314yes—
82C3×D3×Q8C22×C314yes—
83C2×C3×C4×D3—32no—
84C2×C3×C4×D3—64no—
85C2×C3×Dic6—34no—
86C2×C3×Dic6—64no—
87C4×D32—22no—
88C4×D32C22×C311yescharacteristic
89#3797—22no—
90#3797C22×C312yes—
91D3×Dic6—24no—
92D3×Dic6C22×C314yes—
93C32×Q16—21no—
94C32×Q16—22no—
95C32×Q16—34no—
96C32×Q16—32no—
97C32×Q16—41no—
98C32×Q16C2×D312yescomplemented
99C32×Q16C22×C311yescharacteristic, complemented
100C2×C32×C8—32no—
101#3756C22×C312yescomplemented
102#3746C22×C311yescharacteristic, complemented
103C8×C3⋊D3C22×C311yescharacteristic
104C3×#216—24no—
105C3×#216—34no—
106C3×#216—34no—
107C3×#216—34no—
108C3×#216—62no—
109C3×#216C2×D314yescomplemented
110C3×#216C22×C314yescomplemented
111C2×C3×C3⋊C8—32no—
112C3×Dic12—22no—
113C3×Dic12—34no—
114C3×Dic12—32no—
115C3×Dic12—61no—
116C3×Dic12C2×D312yescomplemented
117C3×Dic12C22×C312yescomplemented
118C3×D3×C8—22no—
119C3×D3×C8—32no—
120C3×D3×C8—32no—
121C3×D3×C8—32no—
122C3×D3×C8—61no—
123C3×D3×C8C2×D312yes—
124C3×D3×C8C22×C312yes—
125#3719C22×C314yescomplemented
126#3718C22×C312yescomplemented
127#3710C22×C311yescharacteristic
128D3×C3⋊C8C22×C312yes—
129C2×C3×C3⋊D3C2×D411yescharacteristic
130C2×C32×D3C2×D412yes—
131C3×D32Q1614yesdirect product factor
132C33×C4—22no—
133C33×C4C2411yescharacteristic
134C3×C3⋊Dic3—22no—
135C3×C3⋊Dic3C2×D411yescharacteristic
136C32×Dic3—24no—
137C32×Dic3C2×D412yes—
138C22×C3×Q8—92no—
139C2×D3×Q8—34no—
140C2×C3×Q16—32no—
141C2×C3×Q16—32no—
142C2×C3×Q16—62no—
143C2×C3×Q16—92no—
144C2×C3×Q16—92no—
145C2×C3×Q16—91no—
146C2×C3×Q16—94no—
147C2×C3×Q16—91no—
148C22×C3×C8—91no—
149C2×#216—34no—
150D3×Q16—34no—
151D3×Q16—32no—
152D3×Q16—34no—
153D3×Q16—32no—
154D3×Q16—32no—
155D3×Q16—61no—
156D3×Q16C3×D312yesdirect product factor
157C2×Dic12—32no—
158C2×D3×C8—32no—
159C22×C3×D3—32no—
160C2×D32C3×D411yescharacteristic
161C32×Q8—22no—
162C32×Q8—24no—
163C32×Q8—34no—
164C32×Q8—42no—
165C32×Q8—64no—
166C32×Q8C22×D314yes—
167C32×Q8C23×C312yes—
168C2×C32×C4—32no—
169C2×C32×C4—64no—
170C4×C3⋊D3—22no—
171C4×C3⋊D3C23×C311yescharacteristic
172#634—22no—
173#634C23×C312yes—
174C2×C3×Dic3—32no—
175C2×C3×Dic3—64no—
176C3×C4×D3—24no—
177C3×C4×D3—24no—
178C3×C4×D3—22no—
179C3×C4×D3—32no—
180C3×C4×D3—32no—
181C3×C4×D3—32no—
182C3×C4×D3—44no—
183C3×C4×D3—61no—
184C3×C4×D3—64no—
185C3×C4×D3—64no—
186C3×C4×D3—64no—
187C3×C4×D3—122no—
188C3×C4×D3C22×D312yes—
189C3×C4×D3C23×C312yes—
190C3×Dic6—24no—
191C3×Dic6—24no—
192C3×Dic6—24no—
193C3×Dic6—34no—
194C3×Dic6—44no—
195C3×Dic6—62no—
196C3×Dic6—64no—
197C3×Dic6—64no—
198C3×Dic6—64no—
199C3×Dic6—64no—
200C3×Dic6—122no—
201C3×Dic6C22×D314yes—
202C3×Dic6C23×C314yes—
203#627—22no—
204#627—24no—
205#624—22no—
206#624C3×D411yescharacteristic
207D3×Dic3—24no—
208D3×Dic3C3×D412yes—
209C32×C8—21no—
210C32×C8—22no—
211C32×C8—32no—
212C32×C8—41no—
213C32×C8C22×D312yes—
214C32×C8C23×C311yescharacteristic
215C3⋊(C3⋊C8)C23×C311yescharacteristic
216C3×C3⋊C8—22no—
217C3×C3⋊C8—32no—
218C3×C3⋊C8—61no—
219C3×C3⋊C8C22×D312yes—
220C3×C3⋊C8C23×C312yes—
221C22×Q16—91noSylow
222C2×C33C22×D411yescharacteristic
223C3×C3⋊D3C2×Q1612yescomplemented
224C32×D3C2×Q1614yescomplemented
225C2×C3×Q8—34no—
226C2×C3×Q8—34no—
227C2×C3×Q8—64no—
228C2×C3×Q8—94no—
229C2×C3×Q8—92no—
230C2×C3×Q8—92no—
231C2×C3×Q8—184no—
232C2×C3×Q8—182no—
233C22×C3×C4—91no—
234C22×C3×C4—182no—
235D3×Q8—34no—
236D3×Q8—34no—
237D3×Q8—34no—
238D3×Q8—62no—
239D3×Q8—64no—
240D3×Q8—64no—
241D3×Q8C2×C3×D314yes—
242C2×C4×D3—32no—
243C2×C4×D3—64no—
244C2×Dic6—34no—
245C2×Dic6—64no—
246C3×Q16—21no—
247C3×Q16—22no—
248C3×Q16—34no—
249C3×Q16—34no—
250C3×Q16—32no—
251C3×Q16—32no—
252C3×Q16—41no—
253C3×Q16—62no—
254C3×Q16—64no—
255C3×Q16—94no—
256C3×Q16—92no—
257C3×Q16—92no—
258C3×Q16—91no—
259C3×Q16D3211yescharacteristic, direct product factor
260C3×Q16C2×C3×D312yescomplemented
261C2×C3×C8—32no—
262C2×C3×C8—32no—
263C2×C3×C8—62no—
264C2×C3×C8—91no—
265C2×C3×C8—92no—
266C2×C3×C8—91no—
267#216—34no—
268#216—34no—
269#216—34no—
270#216—62no—
271#216C2×C3×D314yescomplemented
272C2×C3⋊C8—32no—
273Dic12—34no—
274Dic12—32no—
275Dic12—61no—
276Dic12C2×C3×D312yescomplemented
277D3×C8—32no—
278D3×C8—32no—
279D3×C8—32no—
280D3×C8—61no—
281D3×C8C2×C3×D312yes—
282C22×C32—32no—
283C2×C3⋊D3C2×C3×D411yescharacteristic
284C2×C3×D3—22no—
285C2×C3×D3—32no—
286C2×C3×D3—38no—
287C2×C3×D3—61no—
288C2×C3×D3D3×D412yes—
289C2×C3×D3C2×C3×D412yes—
290D32C3×Q1614yesdirect product factor
291C32×C4—24no—
292C32×C4—22no—
293C32×C4—21no—
294C32×C4—22no—
295C32×C4—32no—
296C32×C4—41no—
297C32×C4—44no—
298C32×C4—42no—
299C32×C4—64no—
300C32×C4—82no—
301C32×C4C23×D312yes—
302C32×C4C24×C311yescharacteristic
303C3⋊Dic3—22no—
304C3⋊Dic3C2×C3×D411yescharacteristic
305C3×Dic3—24no—
306C3×Dic3—24no—
307C3×Dic3—22no—
308C3×Dic3—32no—
309C3×Dic3—44no—
310C3×Dic3—61no—
311C3×Dic3—64no—
312C3×Dic3—122no—
313C3×Dic3D3×D412yes—
314C3×Dic3C2×C3×D412yes—
315C22×Q8—92no—
316C2×Q16—32no—
317C2×Q16—92no—
318C2×Q16—92no—
319C2×Q16—91no—
320C2×Q16—94no—
321C2×Q16—91no—
322C22×C8—91no—
323C33C22×Q1611yescharacteristic, complemented, Sylow
324C23×C3—91no—
325C22×D3—32no—
326C3×Q8—24no—
327C3×Q8—22no—
328C3×Q8—34no—
329C3×Q8—34no—
330C3×Q8—42no—
331C3×Q8—64no—
332C3×Q8—64no—
333C3×Q8—64no—
334C3×Q8—92no—
335C3×Q8—124no—
336C3×Q8—182no—
337C3×Q8—184no—
338C3×Q8—182no—
339C3×Q8C2×D3212yes—
340C3×Q8C22×C3×D314yes—
341C2×C3×C4—32no—
342C2×C3×C4—32no—
343C2×C3×C4—62no—
344C2×C3×C4—64no—
345C2×C3×C4—64no—
346C2×C3×C4—91no—
347C2×C3×C4—92no—
348C2×C3×C4—91no—
349C2×C3×C4—124no—
350C2×C3×C4—182no—
351C2×C3×C4—182no—
352C2×C3×C4—184no—
353C2×Dic3—32no—
354C2×Dic3—64no—
355C4×D3—24no—
356C4×D3—32no—
357C4×D3—32no—
358C4×D3—32no—
359C4×D3—61no—
360C4×D3—64no—
361C4×D3—64no—
362C4×D3—64no—
363C4×D3—122no—
364C4×D3C22×C3×D312yes—
365Dic6—24no—
366Dic6—34no—
367Dic6—62no—
368Dic6—64no—
369Dic6—64no—
370Dic6—64no—
371Dic6—64no—
372Dic6—122no—
373Dic6C22×C3×D314yes—
374C3×C8—22no—
375C3×C8—21no—
376C3×C8—32no—
377C3×C8—32no—
378C3×C8—41no—
379C3×C8—62no—
380C3×C8—91no—
381C3×C8C2×D3211yescharacteristic
382C3×C8C22×C3×D312yes—
383C3⋊C8—32no—
384C3⋊C8—61no—
385C3⋊C8C22×C3×D312yes—
386C2×C32—21no—
387C2×C32—22no—
388C2×C32—34no—
389C2×C32—41no—
390C2×C32C2×D3×D412yes—
391C2×C32C22×C3×D411yescharacteristic
392C3⋊D3C2×C3×Q1612yescomplemented
393C3×D3—24no—
394C3×D3—34no—
395C3×D3—62no—
396C3×D3D3×Q1614yesdirect product factor
397C3×D3C2×C3×Q1614yescomplemented
398C2×Q8—34no—
399C2×Q8—94no—
400C2×Q8—92no—
401C2×Q8—92no—
402C2×Q8—184no—
403C2×Q8—182no—
404C22×C4—91no—
405C22×C4—182no—
406Q16—34no—
407Q16—32no—
408Q16—94no—
409Q16—92no—
410Q16—92no—
411Q16—91no—
412Q16C3×D3211yescharacteristic, direct product factor
413C2×C8—32no—
414C2×C8—91no—
415C2×C8—92no—
416C2×C8—91no—
417C22×C3—32no—
418C22×C3—32no—
419C22×C3—62no—
420C22×C3—94no—
421C22×C3—91no—
422C2×D3—32no—
423C2×D3—38no—
424C2×D3—61no—
425C2×D3C3×D3×D412yes—
426C3×C4—22no—
427C3×C4—24no—
428C3×C4—21no—
429C3×C4—22no—
430C3×C4—32no—
431C3×C4—32no—
432C3×C4—42no—
433C3×C4—44no—
434C3×C4—41no—
435C3×C4—62no—
436C3×C4—64no—
437C3×C4—64no—
438C3×C4—82no—
439C3×C4—91no—
440C3×C4—124no—
441C3×C4—182no—
442C3×C4C22×D3211yescharacteristic
443C3×C4C23×C3×D312yes—
444Dic3—24no—
445Dic3—32no—
446Dic3—61no—
447Dic3—64no—
448Dic3—122no—
449Dic3C3×D3×D412yes—
450C32—22no—
451C32—21no—
452C32—41no—
453C32C2×D3×Q1612yescomplemented
454C32C22×C3×Q1611yescharacteristic, complemented
455C23—91no—
456Q8—34no—
457Q8—64no—
458Q8—92no—
459Q8—182no—
460Q8—184no—
461Q8—182no—
462Q8C2×C3×D3212yes—
463C2×C4—32no—
464C2×C4—64no—
465C2×C4—91no—
466C2×C4—92no—
467C2×C4—91no—
468C2×C4—182no—
469C2×C4—182no—
470C2×C4—184no—
471C8—32no—
472C8—91no—
473C8C2×C3×D3211yescharacteristic
474C2×C3—21no—
475C2×C3—22no—
476C2×C3—34no—
477C2×C3—34no—
478C2×C3—41no—
479C2×C3—64no—
480C2×C3—92no—
481C2×C3D32×D411yescharacteristic
482C2×C3C2×C3×D3×D412yes—
483D3—34no—
484D3—62no—
485D3C3×D3×Q1614yesdirect product factor
486C22—32no—
487C22—94no—
488C22—91no—
489C4—22no—
490C4—32no—
491C4—64no—
492C4—91no—
493C4—182no—
494C4C22×C3×D3211yescharacteristic
495C3—21no—
496C3—22no—
497C3—41no—
498C3D32×Q1611yescharacteristic, direct product factor
499C3C2×C3×D3×Q1612yescomplemented
500C2—34no—
501C2—92no—
502C2C3×D32×D411yescharacteristic
subgroupG/H
centerC2×C3D32×D4
commutator, cosocleC32×C4C24×C3
FrattiniC4C22×C3×D32
FittingC33×Q16C22
socleC2×C33C22×D4
prime radicalC2×C3×D32D4
hypercenterC3×Q16D32
hypocenterC32C22×C3×Q16
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×Q16

Complements

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

NH ≅ G/Ntype
2. C3×Q16×C3⋊D3C2semidirect
3. C32×D3×Q16C2semidirect
4. C3×#63944C2semidirect
5. C3×D3×#216C2semidirect
6. C3×#63802C2semidirect
7. C3×D3×Dic12C2semidirect
10. D32×Q16C3direct (N × H)
19. C33×Q16C22semidirect
20. C3×#3756C22semidirect
21. C3×#3746C22semidirect
23. C32×#216C22semidirect
24. C32×Dic12C22semidirect
26. C3×#3719C22semidirect
27. C3×#3718C22semidirect
33. Q16×C3⋊D3C2×C3semidirect
42. C3×D3×Q16D3direct (N × H)
43. C3×D3×Q16C2×C3semidirect
46. #63944C2×C3semidirect
47. D3×#216C2×C3semidirect
48. #63802C2×C3semidirect
49. D3×Dic12C2×C3semidirect
98. C32×Q16C2×D3semidirect
99. C32×Q16C22×C3semidirect
101. #3756C22×C3semidirect
102. #3746C22×C3semidirect
109. C3×#216C2×D3semidirect
110. C3×#216C22×C3semidirect
116. C3×Dic12C2×D3semidirect
117. C3×Dic12C22×C3semidirect
125. #3719C22×C3semidirect
126. #3718C22×C3semidirect
131. C3×D32Q16direct (N × H)
156. D3×Q16C3×D3direct (N × H)
223. C3×C3⋊D3C2×Q16semidirect
224. C32×D3C2×Q16semidirect
259. C3×Q16D32direct (N × H)
260. C3×Q16C2×C3×D3semidirect
271. #216C2×C3×D3semidirect
276. Dic12C2×C3×D3semidirect
290. D32C3×Q16direct (N × H)
323. C33C22×Q16semidirect
392. C3⋊D3C2×C3×Q16semidirect
396. C3×D3D3×Q16direct (N × H)
397. C3×D3C2×C3×Q16semidirect
412. Q16C3×D32direct (N × H)
453. C32C2×D3×Q16semidirect
454. C32C22×C3×Q16semidirect
485. D3C3×D3×Q16direct (N × H)
498. C3D32×Q16direct (N × H)
499. C3C2×C3×D3×Q16semidirect

Maximal subgroups

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

subgroupindex
C3×D32×Q82
C3×Q16×C3⋊D32
C32×D3×Q162
C3×#639442
C3×D3×#2162
C3×#638022
C3×D3×Dic122
C3×D32×C82
C2×C3×D3×Q163
D32×Q163