Finite Group Databasebeta

← D3×Q16×C3⋊D3 #13176763

Subgroups — D3×Q16×C3⋊D3 #13176763

Order 1728.

1788 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×Q8×C3⋊D3C212yesmaximal
2Q16×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3C3×Q16×C3⋊D3C211yesmaximal, characteristic, complemented
4C32×D3×Q16C211yesmaximal, characteristic, complemented
5#11723577C212yesmaximal, complemented
6C3⋊D3×#216C212yesmaximal, complemented
7D3×#3756C212yesmaximal, complemented
8#11723330C211yesmaximal, characteristic, complemented
9C3⋊D3×Dic12C211yesmaximal, characteristic, complemented
10D3×#3746C211yesmaximal, characteristic, complemented
11D3×C8×C3⋊D3C211yesmaximal, characteristic
12C2×Q16×C3⋊D3—31nomaximal
13D32×Q16—34nomaximal
14Q8×C3⋊(C3⋊D3)C2212yes—
15C3×Q8×C3⋊D3C2212yes—
16C32×D3×Q8C2212yes—
17C4×D3×C3⋊D3—22no—
18C4×D3×C3⋊D3C2211yescharacteristic
19#89019—22no—
20#89019C2212yes—
21C3⋊D3×Dic6—22no—
22C3⋊D3×Dic6C2212yes—
23D3×#634—22no—
24D3×#634C2212yes—
25C33×Q16C2211yescharacteristic, complemented
26#88863C2212yescomplemented
27#88853C2211yescharacteristic, complemented
28C8×C3⋊(C3⋊D3)C2211yescharacteristic
29C3×#3756C2212yescomplemented
30C3×#3746C2211yescharacteristic, complemented
31C3×C8×C3⋊D3C2211yescharacteristic
32C32×#216C2212yescomplemented
33C32×Dic12C2211yescharacteristic, complemented
34C32×D3×C8C2211yescharacteristic
35#88800C2212yescomplemented
36#88799C2212yescomplemented
37#88798C2212yescomplemented
38#88785C2211yescharacteristic
39C3⋊D3×C3⋊C8C2211yescharacteristic
40D3×C3⋊(C3⋊C8)C2211yescharacteristic
41C2×Q8×C3⋊D3—32no—
42D32×Q8—38no—
43C2×C32×Q16—31no—
44C2×#3756—32no—
45Q16×C3⋊D3—32no—
46Q16×C3⋊D3—31no—
47Q16×C3⋊D3—32no—
48Q16×C3⋊D3—31no—
49Q16×C3⋊D3—31no—
50Q16×C3⋊D3—34no—
51Q16×C3⋊D3—64no—
52Q16×C3⋊D3D311yescharacteristic, direct product factor
53C2×#3746—31no—
54C2×C8×C3⋊D3—31no—
55C3×D3×Q16—34no—
56C3×D3×Q16D314yescomplemented
57#63944—38no—
58D3×#216—38no—
59D3×#216—38no—
60#63802—34no—
61D3×Dic12—34no—
62D3×Dic12—34no—
63D32×C8—34no—
64C2×D3×C3⋊D3D411yescharacteristic
65C33×Q8C2312yes—
66C4×C3⋊(C3⋊D3)—22no—
67C4×C3⋊(C3⋊D3)C2311yescharacteristic
68#6308—22no—
69#6308C2312yes—
70C3×C4×C3⋊D3—22no—
71C3×C4×C3⋊D3C2311yescharacteristic
72C3×#634—22no—
73C3×#634C2312yes—
74C32×C4×D3—22no—
75C32×C4×D3C2311yescharacteristic
76C32×Dic6—22no—
77C32×Dic6C2312yes—
78#6293—22no—
79#6293—22no—
80#6293—22no—
81#6289—22no—
82#6289D411yescharacteristic
83Dic3×C3⋊D3—22no—
84Dic3×C3⋊D3D411yescharacteristic
85D3×C3⋊Dic3—22no—
86D3×C3⋊Dic3D411yescharacteristic
87C33×C8C2311yescharacteristic
88C3⋊(C3⋊(C3⋊C8))C2311yescharacteristic
89C3×C3⋊(C3⋊C8)C2311yescharacteristic
90C32×C3⋊C8C2311yescharacteristic
91C2×D3×Q16—94no—
92C2×D3×Q16—91no—
93C2×C32×Q8—32no—
94Q8×C3⋊D3—32no—
95Q8×C3⋊D3—38no—
96Q8×C3⋊D3—32no—
97Q8×C3⋊D3—32no—
98Q8×C3⋊D3—62no—
99Q8×C3⋊D3—62no—
100Q8×C3⋊D3—68no—
101Q8×C3⋊D3C2×D312yes—
102C2×C4×C3⋊D3—31no—
103C2×C4×C3⋊D3—62no—
104C2×#634—32no—
105C2×#634—62no—
106C3×D3×Q8—38no—
107C3×D3×Q8C2×D318yes—
108C4×D32—34no—
109C4×D32—68no—
110#3797—38no—
111#3797—68no—
112D3×Dic6—38no—
113D3×Dic6—38no—
114D3×Dic6—68no—
115D3×Dic6—68no—
116C32×Q16—24no—
117C32×Q16—32no—
118C32×Q16—31no—
119C32×Q16C2×D314yescomplemented
120C32×Q16C2×D311yescharacteristic, complemented
121C2×C32×C8—31no—
122#3756—32no—
123#3756—38no—
124#3756—32no—
125#3756—32no—
126#3756—68no—
127#3756C2×D312yescomplemented
128C2×C3⋊(C3⋊C8)—31no—
129#3746—32no—
130#3746—31no—
131#3746—34no—
132#3746—64no—
133#3746C2×D311yescharacteristic, complemented
134C8×C3⋊D3—31no—
135C8×C3⋊D3—31no—
136C8×C3⋊D3—34no—
137C8×C3⋊D3—31no—
138C8×C3⋊D3—64no—
139C8×C3⋊D3C2×D311yescharacteristic
140C3×#216—38no—
141C3×#216C2×D318yescomplemented
142C3×Dic12—34no—
143C3×Dic12C2×D314yescomplemented
144C3×D3×C8—34no—
145C3×D3×C8C2×D314yes—
146#3719—38no—
147#3719—38no—
148#3718—38no—
149#3710—34no—
150D3×C3⋊C8—34no—
151D3×C3⋊C8—34no—
152C2×C3⋊(C3⋊D3)C2×D411yescharacteristic
153C2×C3×C3⋊D3C2×D411yescharacteristic
154C2×C32×D3C2×D411yescharacteristic
155D3×C3⋊D3Q1614yesdirect product factor
156C33×C4—22no—
157C33×C4C2411yescharacteristic
158C3⋊(C3⋊Dic3)—22no—
159C3⋊(C3⋊Dic3)C2×D411yescharacteristic
160C3×C3⋊Dic3—22no—
161C3×C3⋊Dic3C2×D411yescharacteristic
162C32×Dic3—22no—
163C32×Dic3C2×D411yescharacteristic
164C2×D3×Q8—98no—
165C2×D3×Q8—92no—
166C2×C3×Q16—34no—
167C2×C3×Q16—91no—
168C2×#216—98no—
169C2×#216—92no—
170D3×Q16—34no—
171D3×Q16—94no—
172D3×Q16—94no—
173D3×Q16—98no—
174D3×Q16—98no—
175D3×Q16—92no—
176D3×Q16—94no—
177D3×Q16—91no—
178D3×Q16—92no—
179D3×Q16—91no—
180D3×Q16—91no—
181D3×Q16—184no—
182D3×Q16C3⋊D311yescharacteristic, direct product factor
183C2×Dic12—94no—
184C2×Dic12—91no—
185C2×D3×C8—94no—
186C2×D3×C8—91no—
187C22×C3⋊D3—31no—
188C2×D32—34no—
189C32×Q8—28no—
190C32×Q8—32no—
191C32×Q8—62no—
192C32×Q8C22×D318yes—
193C32×Q8C22×D312yes—
194C2×C32×C4—31no—
195C2×C32×C4—62no—
196C2×C3⋊Dic3—31no—
197C2×C3⋊Dic3—62no—
198C4×C3⋊D3—22no—
199C4×C3⋊D3—31no—
200C4×C3⋊D3—31no—
201C4×C3⋊D3—31no—
202C4×C3⋊D3—34no—
203C4×C3⋊D3—64no—
204C4×C3⋊D3—62no—
205C4×C3⋊D3—62no—
206C4×C3⋊D3—62no—
207C4×C3⋊D3—68no—
208C4×C3⋊D3—128no—
209C4×C3⋊D3C22×D311yescharacteristic
210#634—22no—
211#634—32no—
212#634—38no—
213#634—68no—
214#634—62no—
215#634—62no—
216#634—62no—
217#634—62no—
218#634—68no—
219#634—128no—
220#634C22×D312yes—
221C3×C4×D3—28no—
222C3×C4×D3—34no—
223C3×C4×D3—68no—
224C3×C4×D3C22×D314yes—
225C3×Dic6—28no—
226C3×Dic6—38no—
227C3×Dic6—68no—
228C3×Dic6C22×D318yes—
229#627—68no—
230#627—68no—
231#627—68no—
232#624—34no—
233#624—68no—
234D3×Dic3—34no—
235D3×Dic3—34no—
236D3×Dic3—68no—
237D3×Dic3—68no—
238C32×C8—24no—
239C32×C8—31no—
240C32×C8C22×D314yes—
241C32×C8C22×D311yescharacteristic
242C3⋊(C3⋊C8)—31no—
243C3⋊(C3⋊C8)—34no—
244C3⋊(C3⋊C8)—64no—
245C3⋊(C3⋊C8)C22×D311yescharacteristic
246C3×C3⋊C8—34no—
247C3×C3⋊C8C22×D314yes—
248C22×Q16—271noSylow
249C2×C33C22×D411yescharacteristic
250C3⋊(C3⋊D3)C2×Q1612yescomplemented
251C3×C3⋊D3C2×Q1612yescomplemented
252C32×D3C2×Q1612yescomplemented
253C2×C3×Q8—38no—
254C2×C3×Q8—92no—
255D3×Q8—38no—
256D3×Q8—98no—
257D3×Q8—98no—
258D3×Q8—98no—
259D3×Q8—92no—
260D3×Q8—92no—
261D3×Q8—92no—
262D3×Q8—188no—
263D3×Q8—182no—
264D3×Q8—188no—
265D3×Q8—182no—
266D3×Q8—188no—
267D3×Q8C2×C3⋊D312yes—
268C2×C4×D3—94no—
269C2×C4×D3—91no—
270C2×C4×D3—182no—
271C2×C4×D3—188no—
272C2×Dic6—98no—
273C2×Dic6—92no—
274C2×Dic6—182no—
275C2×Dic6—188no—
276C3×Q16—24no—
277C3×Q16—38no—
278C3×Q16—34no—
279C3×Q16—92no—
280C3×Q16—91no—
281C3×Q16D3214yescomplemented
282C3×Q16C2×C3⋊D311yescharacteristic, complemented
283C2×C3×C8—34no—
284C2×C3×C8—91no—
285#216—38no—
286#216—98no—
287#216—98no—
288#216—98no—
289#216—92no—
290#216—92no—
291#216—92no—
292#216—188no—
293#216C2×C3⋊D312yescomplemented
294C2×C3⋊C8—94no—
295C2×C3⋊C8—91no—
296Dic12—34no—
297Dic12—98no—
298Dic12—92no—
299Dic12—94no—
300Dic12—91no—
301Dic12—184no—
302Dic12C2×C3⋊D311yescharacteristic, complemented
303D3×C8—34no—
304D3×C8—94no—
305D3×C8—94no—
306D3×C8—91no—
307D3×C8—94no—
308D3×C8—91no—
309D3×C8—91no—
310D3×C8—184no—
311D3×C8C2×C3⋊D311yescharacteristic
312C22×C32—31no—
313C2×C3⋊D3—31no—
314C2×C3⋊D3—34no—
315C2×C3⋊D3—34no—
316C2×C3⋊D3—64no—
317C2×C3⋊D3D3×D411yescharacteristic
318C2×C3×D3—34no—
319C2×C3×D3D3×D414yes—
320D32—316no—
321C32×C4—22no—
322C32×C4—24no—
323C32×C4—28no—
324C32×C4—31no—
325C32×C4—48no—
326C32×C4—62no—
327C32×C4C23×D314yes—
328C32×C4C23×D311yescharacteristic
329C3⋊Dic3—22no—
330C3⋊Dic3—31no—
331C3⋊Dic3—34no—
332C3⋊Dic3—64no—
333C3⋊Dic3—62no—
334C3⋊Dic3—68no—
335C3⋊Dic3—128no—
336C3⋊Dic3D3×D411yescharacteristic
337C3×Dic3—28no—
338C3×Dic3—34no—
339C3×Dic3—68no—
340C3×Dic3D3×D414yes—
341C22×Q8—272no—
342C2×Q16—31no—
343C2×Q16—91no—
344C2×Q16—271no—
345C2×Q16—271no—
346C2×Q16—272no—
347C2×Q16—271no—
348C2×Q16—272no—
349C2×Q16—272no—
350C2×Q16—271no—
351C22×C8—271no—
352C33C22×Q1611yescharacteristic, complemented, Sylow
353C22×D3—94no—
354C22×D3—91no—
355C3×Q8—28no—
356C3×Q8—38no—
357C3×Q8—68no—
358C3×Q8—92no—
359C3×Q8—182no—
360C3×Q8C2×D3218yes—
361C3×Q8C22×C3⋊D312yes—
362C2×C3×C4—34no—
363C2×C3×C4—68no—
364C2×C3×C4—91no—
365C2×C3×C4—182no—
366C2×Dic3—94no—
367C2×Dic3—91no—
368C2×Dic3—182no—
369C2×Dic3—188no—
370C4×D3—22no—
371C4×D3—34no—
372C4×D3—68no—
373C4×D3—94no—
374C4×D3—94no—
375C4×D3—91no—
376C4×D3—94no—
377C4×D3—91no—
378C4×D3—91no—
379C4×D3—184no—
380C4×D3—182no—
381C4×D3—188no—
382C4×D3—182no—
383C4×D3—188no—
384C4×D3—188no—
385C4×D3—182no—
386C4×D3—368no—
387C4×D3C22×C3⋊D311yescharacteristic
388Dic6—22no—
389Dic6—38no—
390Dic6—68no—
391Dic6—98no—
392Dic6—92no—
393Dic6—188no—
394Dic6—182no—
395Dic6—182no—
396Dic6—188no—
397Dic6—188no—
398Dic6—188no—
399Dic6—182no—
400Dic6—188no—
401Dic6—182no—
402Dic6—368no—
403Dic6C22×C3⋊D312yes—
404C3×C8—24no—
405C3×C8—34no—
406C3×C8—91no—
407C3×C8C2×D3214yes—
408C3×C8C22×C3⋊D311yescharacteristic
409C3⋊C8—34no—
410C3⋊C8—94no—
411C3⋊C8—91no—
412C3⋊C8—184no—
413C3⋊C8C22×C3⋊D311yescharacteristic
414C2×C32—24no—
415C2×C32—32no—
416C2×C32C2×D3×D414yes—
417C2×C32C2×D3×D411yescharacteristic
418C3⋊D3—32no—
419C3⋊D3—38no—
420C3⋊D3—68no—
421C3⋊D3D3×Q1612yesdirect product factor
422C3×D3—38no—
423C3×D3D3×Q1618yescomplemented
424C2×Q8—32no—
425C2×Q8—92no—
426C2×Q8—272no—
427C2×Q8—272no—
428C2×Q8—272no—
429C2×Q8—272no—
430C2×Q8—542no—
431C2×Q8—542no—
432C2×Q8—542no—
433C22×C4—271no—
434C22×C4—542no—
435Q16—32no—
436Q16—31no—
437Q16—92no—
438Q16—91no—
439Q16—272no—
440Q16—272no—
441Q16—272no—
442Q16—272no—
443Q16—271no—
444Q16D3×C3⋊D311yescharacteristic, direct product factor
445C2×C8—31no—
446C2×C8—91no—
447C2×C8—271no—
448C2×C8—271no—
449C2×C8—271no—
450C2×C8—271no—
451C22×C3—34no—
452C22×C3—91no—
453C2×D3—34no—
454C2×D3—916no—
455C2×D3—94no—
456C2×D3—91no—
457C2×D3—94no—
458C2×D3—184no—
459C2×D3D4×C3⋊D311yescharacteristic
460C3×C4—28no—
461C3×C4—22no—
462C3×C4—24no—
463C3×C4—34no—
464C3×C4—48no—
465C3×C4—68no—
466C3×C4—91no—
467C3×C4—182no—
468C3×C4C22×D3214yes—
469C3×C4C23×C3⋊D311yescharacteristic
470Dic3—22no—
471Dic3—34no—
472Dic3—68no—
473Dic3—94no—
474Dic3—91no—
475Dic3—184no—
476Dic3—188no—
477Dic3—182no—
478Dic3—368no—
479Dic3D4×C3⋊D311yescharacteristic
480C32—24no—
481C32C2×D3×Q1614yescomplemented
482C32C2×D3×Q1611yescharacteristic, complemented
483C23—271no—
484Q8—32no—
485Q8—62no—
486Q8—92no—
487Q8—182no—
488Q8—272no—
489Q8—542no—
490Q8—542no—
491Q8—542no—
492Q8—542no—
493Q8C2×D3×C3⋊D312yes—
494C2×C4—31no—
495C2×C4—62no—
496C2×C4—91no—
497C2×C4—182no—
498C2×C4—271no—
499C2×C4—271no—
500C2×C4—271no—
501C2×C4—271no—
502C2×C4—542no—
503C2×C4—542no—
504C2×C4—542no—
505C2×C4—542no—
506C8—31no—
507C8—91no—
508C8—271no—
509C8C2×D3×C3⋊D311yescharacteristic
510C2×C3—24no—
511C2×C3—38no—
512C2×C3—92no—
513C2×C3D32×D414yes—
514C2×C3C2×D4×C3⋊D311yescharacteristic
515D3—38no—
516D3—92no—
517D3—98no—
518D3—188no—
519D3Q16×C3⋊D312yesdirect product factor
520C22—31no—
521C22—91no—
522C22—274no—
523C22—271no—
524C4—22no—
525C4—31no—
526C4—62no—
527C4—91no—
528C4—182no—
529C4—271no—
530C4—542no—
531C4C22×D3×C3⋊D311yescharacteristic
532C3—24no—
533C3D32×Q1614yescomplemented
534C3C2×Q16×C3⋊D311yescharacteristic, complemented
535C2—32no—
536C2—92no—
537C2—272no—
538C2D3×D4×C3⋊D311yescharacteristic
subgroupG/H
centerC2D3×D4×C3⋊D3
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC33×Q16C22
socleC2×C33C22×D4
prime radicalC2×D3×C3⋊D3D4
hypercenterQ16D3×C3⋊D3
hypocenterC33C22×Q16
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ Q16

Complements

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

NH ≅ G/Ntype
2. Q16×C3⋊(C3⋊D3)C2semidirect
3. C3×Q16×C3⋊D3C2semidirect
4. C32×D3×Q16C2semidirect
5. #11723577C2semidirect
6. C3⋊D3×#216C2semidirect
7. D3×#3756C2semidirect
8. #11723330C2semidirect
9. C3⋊D3×Dic12C2semidirect
10. D3×#3746C2semidirect
25. C33×Q16C22semidirect
26. #88863C22semidirect
27. #88853C22semidirect
29. C3×#3756C22semidirect
30. C3×#3746C22semidirect
32. C32×#216C22semidirect
33. C32×Dic12C22semidirect
35. #88800C22semidirect
36. #88799C22semidirect
37. #88798C22semidirect
52. Q16×C3⋊D3D3direct (N × H)
56. C3×D3×Q16D3semidirect
119. C32×Q16C2×D3semidirect
120. C32×Q16C2×D3semidirect
127. #3756C2×D3semidirect
133. #3746C2×D3semidirect
141. C3×#216C2×D3semidirect
143. C3×Dic12C2×D3semidirect
155. D3×C3⋊D3Q16direct (N × H)
182. D3×Q16C3⋊D3direct (N × H)
250. C3⋊(C3⋊D3)C2×Q16semidirect
251. C3×C3⋊D3C2×Q16semidirect
252. C32×D3C2×Q16semidirect
281. C3×Q16D32semidirect
282. C3×Q16C2×C3⋊D3semidirect
293. #216C2×C3⋊D3semidirect
302. Dic12C2×C3⋊D3semidirect
352. C33C22×Q16semidirect
421. C3⋊D3D3×Q16direct (N × H)
423. C3×D3D3×Q16semidirect
444. Q16D3×C3⋊D3direct (N × H)
481. C32C2×D3×Q16semidirect
482. C32C2×D3×Q16semidirect
519. D3Q16×C3⋊D3direct (N × H)
533. C3D32×Q16semidirect
534. C3C2×Q16×C3⋊D3semidirect

Maximal subgroups

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

subgroupindex
D3×Q8×C3⋊D32
Q16×C3⋊(C3⋊D3)2
C3×Q16×C3⋊D32
C32×D3×Q162
#117235772
C3⋊D3×#2162
D3×#37562
#117233302
C3⋊D3×Dic122
D3×#37462
D3×C8×C3⋊D32
C2×Q16×C3⋊D33
D32×Q163