Finite Group Databasebeta

← D3×#63713 #13164759

Subgroups — D3×#63713 #13164759

Order 1728.

2528 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11722667C211yesmaximal, characteristic, complemented
2C2×D3×Q8×C9C211yesmaximal, characteristic, complemented
3C3×#63713C211yesmaximal, characteristic, complemented
4D3×#3701C214yesmaximal, complemented
5D3×Q8×D9C214yesmaximal, complemented
6#11722510C214yesmaximal, complemented
7#11722509C214yesmaximal, complemented
8D3×#3697C216yesmaximal, complemented
9#11722458C216yesmaximal, complemented
10D3×#1007—31nomaximal
11C2×#63713—31nomaximal
12C2×C3×Q8×C9C2211yescharacteristic, complemented
13#88746C2214yescomplemented
14Q8×C3⋊D9C2214yescomplemented
15#88736C2216yescomplemented
16D3×Q8×C9C2214yes—
17D3×Q8×C9C2214yescomplemented
18C3×#3701C2214yescomplemented
19C3×Q8×D9C2214yescomplemented
20C2×C4×D3×C9C2213yes—
21C3×#3697C2216yescomplemented
22C2×C9×Dic6C2213yescomplemented
23D3×#611C2214yes—
24#88658C22112yescomplemented
25D3×D36C2216yes—
26C4×D3×D9C22112yes—
27#88642C2216yescomplemented
28#88641C2216yescomplemented
29#88640C22112yescomplemented
30D3×Dic18C2216yes—
31(C3⋊D9)⋊Q8C22112yescomplemented
32#88635C22112yescomplemented
33#88634C22112yescomplemented
34D9×Dic6C22112yescomplemented
35#64364—31no—
36C3×#1007—31no—
37C2×C3×D3×Q8D311yescharacteristic
38D3×#239—34no—
39D32×Q8—34no—
40#64316—34no—
41#64315—34no—
42D3×#235—36no—
43#64297—36no—
44C22×Q8×C9—31no—
45#63713—31no—
46#63713—34no—
47#63713—34no—
48#63713—36no—
49#63713—61no—
50#63713D311yescharacteristic, direct product factor
51C2×#3701—34no—
52C2×Q8×D9—34no—
53C2×#3697—36no—
54C22×D3×C9C2311yescharacteristic
55C2×D3×D9C2314yes—
56C3×Q8×C9C2314yes—
57C2×C3×C4×C9C2313yes—
58#6233C2314yes—
59C4⋊(C3⋊D9)C2316yes—
60C4×C3⋊D9C23112yes—
61#6226C2316yes—
62C3×#611C2314yes—
63C2×C9×Dic3C2313yes—
64C4×D3×C9C2316yes—
65C4×D3×C9C2316yes—
66C3×D36C2316yes—
67C3×C4×D9C23112yes—
68C9×Dic6C23112yes—
69C3×Dic18C2316yes—
70#6195C2314yes—
71#6194C2314yes—
72D3×Dic9C2314yes—
73#6192C23112yes—
74(C3⋊D9)⋊C4C23112yes—
75Dic3×D9C23112yes—
76#6189C23112yes—
77D3×2−1+4—91no—
78C2×#1007—91no—
79C2×C32×Q8C2×D311yescharacteristic
80#3832—34no—
81Q8×C3⋊D3—34no—
82#3828—36no—
83C3×#239—34no—
84C3×D3×Q8—34no—
85C3×D3×Q8C2×D314yes—
86C3×D3×Q8C2×D314yes—
87C3×#235—36no—
88C2×C3×C4×D3C2×D313yes—
89C2×C3×Dic6C2×D313yes—
90D3×#67—34no—
91#3804—312no—
92D3×D12—36no—
93C4×D32—312no—
94#3799—36no—
95#3799—312no—
96#3798—312no—
97#3797—312no—
98#3796—312no—
99#3795—36no—
100D3×Dic6—36no—
101D3×Dic6—312no—
102C2×Q8×C9—21no—
103C2×Q8×C9—34no—
104C2×Q8×C9—34no—
105C2×Q8×C9—33no—
106C2×Q8×C9C2×D311yescharacteristic, complemented
107C22×C4×C9—33no—
108C2×#611—34no—
109#3701—34no—
110#3701—312no—
111#3701—312no—
112#3701—64no—
113#3701C2×D314yescomplemented
114Q8×D9—34no—
115Q8×D9—312no—
116Q8×D9—312no—
117Q8×D9—64no—
118Q8×D9C2×D314yescomplemented
119#3697—36no—
120#3697—312no—
121#3697—36no—
122#3697—312no—
123#3697—36no—
124#3697—66no—
125#3697C2×D316yescomplemented
126C2×D36—36no—
127C2×C4×D9—312no—
128C2×Dic18—36no—
129C22×C3×C9C2411yescharacteristic
130C2×C3⋊D9C2414yes—
131C2×D3×C9—22no—
132C2×D3×C9C2411yescharacteristic
133C2×D3×C9C2411yescharacteristic
134C2×C3×D9C2414yes—
135D3×D9—28no—
136C3×C4×C9C2416yes—
137C3⋊Dic9C2414yes—
138C9×Dic3C2416yes—
139C3×Dic9C2414yes—
140C3×2−1+4—91no—
141C22×C3×Q8—31no—
142#1010—94no—
143#1010—96no—
144D3×#41—94no—
145D3×#41—96no—
146#1007—31no—
147#1007—96no—
148#1007—94no—
149#1007—94no—
150#1007—91no—
151#1007—91no—
152#1007—94no—
153#1007—181no—
154C2×#239—94no—
155C2×D3×Q8—94no—
156C2×D3×Q8—94no—
157C2×D3×Q8D911yescharacteristic, complemented
158C2×#235—96no—
159C22×C3×D3C22×D311yescharacteristic
160C2×D32—34no—
161C32×Q8C22×D314yes—
162C2×C32×C4C22×D313yes—
163#638—34no—
164C4⋊(C3⋊D3)—36no—
165C4×C3⋊D3—312no—
166#634—36no—
167C3×#67—34no—
168C2×C3×Dic3C22×D313yes—
169C3×D12—36no—
170C3×C4×D3—312no—
171C3×C4×D3C22×D316yes—
172C3×C4×D3C22×D316yes—
173C3×Dic6—36no—
174C3×Dic6C22×D3112yes—
175#627—312no—
176#626—34no—
177#626—312no—
178#625—34no—
179#624—312no—
180D3×Dic3—34no—
181D3×Dic3—312no—
182C23×C9—31no—
183C22×D9—34no—
184Q8×C9—24no—
185Q8×C9—312no—
186Q8×C9C22×D314yes—
187C2×C4×C9—23no—
188C2×C4×C9—33no—
189C2×C4×C9—36no—
190C2×C4×C9—36no—
191C2×C4×C9C22×D313yes—
192#611—34no—
193#611—34no—
194#611—34no—
195#611—64no—
196#611C22×D314yes—
197C2×Dic9—34no—
198D36—312no—
199D36—36no—
200D36—66no—
201D36C22×D316yes—
202C4×D9—312no—
203C4×D9—312no—
204C4×D9—312no—
205C4×D9—612no—
206C4×D9C22×D3112yes—
207Dic18—36no—
208Dic18—312no—
209Dic18—66no—
210Dic18C22×D316yes—
211C2×2−1+4—271noSylow
212C2×C3×C9—21no—
213C2×C3×C9C2511yescharacteristic
214C3⋊D9—24no—
215D3×C9—21no—
216D3×C92−1+412yescomplemented
217C3×D9—24no—
218C3×#41—94no—
219C3×#41—96no—
220C2×C3×Q8—21no—
221C2×C3×Q8—34no—
222C2×C3×Q8—34no—
223C2×C3×Q8—33no—
224C2×C3×Q8—94no—
225C2×C3×Q8C2×D911yescharacteristic, complemented
226C2×C3×Q8D3211yescharacteristic
227C22×C3×C4—33no—
228C2×#67—94no—
229#239—34no—
230#239—912no—
231#239—912no—
232#239—94no—
233#239—94no—
234#239—96no—
235#239—184no—
236D3×Q8—34no—
237D3×Q8—912no—
238D3×Q8—912no—
239D3×Q8—94no—
240D3×Q8—94no—
241D3×Q8—96no—
242D3×Q8—912no—
243D3×Q8—184no—
244D3×Q8C2×D914yes—
245D3×Q8C2×D914yescomplemented
246#237—912no—
247#237—96no—
248#237—912no—
249D3×D4—94no—
250D3×D4—96no—
251#235—36no—
252#235—96no—
253#235—912no—
254#235—96no—
255#235—912no—
256#235—96no—
257#235—96no—
258#235—912no—
259#235—912no—
260#235—186no—
261C2×D12—96no—
262C2×C4×D3—912no—
263C2×C4×D3—912no—
264C2×C4×D3C2×D913yes—
265C2×Dic6—96no—
266C2×Dic6—912no—
267C2×Dic6C2×D913yescomplemented
268C22×C32C23×D311yescharacteristic
269C2×C3⋊D3—34no—
270C2×C3×D3—22no—
271C2×C3×D3—34no—
272C2×C3×D3C23×D311yescharacteristic
273C2×C3×D3C23×D311yescharacteristic
274D32—68no—
275C32×C4C23×D316yes—
276C3⋊Dic3—34no—
277C3×Dic3—34no—
278C3×Dic3C23×D316yes—
279C22×C9—21no—
280C22×C9—31no—
281C22×C9—31no—
282C22×C9—62no—
283C22×C9C23×D311yescharacteristic
284C2×D9—34no—
285C2×D9—64no—
286C2×D9—68no—
287C2×D9C23×D314yes—
288C4×C9—26no—
289C4×C9—36no—
290C4×C9C23×D316yes—
291Dic9—34no—
292Dic9—64no—
293Dic9C23×D314yes—
2942−1+4—91no—
2952−1+4—274no—
2962−1+4—274no—
2972−1+4—276no—
2982−1+4—271no—
299C2×#41—276no—
300C2×#41—274no—
301C22×Q8—31no—
302C22×Q8—274no—
303C3×C9C2×2−1+411yescharacteristic, complemented, Sylow
304C23×C3—31no—
305C22×D3—94no—
306C22×D3—94no—
307C22×D3C22×D911yescharacteristic
308C3×Q8—24no—
309C3×Q8—312no—
310C3×Q8—96no—
311C3×Q8C22×D914yes—
312C3×Q8C2×D3214yes—
313C3×D4—94no—
314C3×D4—96no—
315C2×C3×C4—23no—
316C2×C3×C4—33no—
317C2×C3×C4—36no—
318C2×C3×C4—36no—
319C2×C3×C4—912no—
320C2×C3×C4C22×D913yes—
321C2×C3×C4C2×D3213yes—
322#67—34no—
323#67—94no—
324#67—94no—
325#67—94no—
326#67—94no—
327#67—94no—
328#67—912no—
329#67—184no—
330C2×Dic3—94no—
331C2×Dic3—912no—
332C2×Dic3C22×D913yes—
333D12—36no—
334D12—912no—
335D12—94no—
336D12—96no—
337D12—96no—
338D12—186no—
339C4×D3—312no—
340C4×D3—912no—
341C4×D3—912no—
342C4×D3—94no—
343C4×D3—912no—
344C4×D3—912no—
345C4×D3—912no—
346C4×D3—1812no—
347C4×D3C22×D916yes—
348C4×D3C22×D916yes—
349Dic6—36no—
350Dic6—912no—
351Dic6—96no—
352Dic6—96no—
353Dic6—912no—
354Dic6—186no—
355Dic6C22×D9112yes—
356C2×C32—21no—
357C2×C32C24×D311yescharacteristic
358C3⋊D3—64no—
359C3×D3—21no—
360C3×D3—64no—
361C3×D3#100712yes—
362C2×C9—21no—
363C2×C9—21no—
364C2×C9—32no—
365C2×C9—41no—
366C2×C9—61no—
367C2×C9C24×D311yescharacteristic
368D9—24no—
369D9—64no—
370D9—124no—
371#41—94no—
372#41—96no—
373#41—2712no—
374#41—2712no—
375#41—276no—
376#41—276no—
377#41—2712no—
378#41—2712no—
379#41—274no—
380#41—276no—
381C2×Q8—34no—
382C2×Q8—34no—
383C2×Q8—33no—
384C2×Q8—94no—
385C2×Q8—276no—
386C2×Q8—2712no—
387C2×Q8—2712no—
388C2×Q8—274no—
389C2×Q8D3×D911yescharacteristic, complemented
390C2×D4—276no—
391C2×D4—274no—
392C22×C4—33no—
393C22×C4—2712no—
394C22×C3—21no—
395C22×C3—31no—
396C22×C3—31no—
397C22×C3—62no—
398C22×C3—94no—
399C22×C3C23×D911yescharacteristic
400C22×C3C22×D3211yescharacteristic
401C2×D3—22no—
402C2×D3—34no—
403C2×D3—94no—
404C2×D3—94no—
405C2×D3—184no—
406C2×D3—188no—
407C2×D3—188no—
408C2×D3C23×D911yescharacteristic
409C2×D3C23×D911yescharacteristic
410C3×C4—26no—
411C3×C4—36no—
412C3×C4—94no—
413C3×C4C23×D916yes—
414C3×C4C22×D3216yes—
415Dic3—34no—
416Dic3—94no—
417Dic3—94no—
418Dic3—184no—
419Dic3C23×D916yes—
420C32C2×#100711yescharacteristic
421C9—21no—
422C9D3×2−1+411yescharacteristic, complemented
423C23—31no—
424C23—274no—
425Q8—312no—
426Q8—96no—
427Q8—2712no—
428Q8—276no—
429Q8C2×D3×D914yes—
430D4—94no—
431D4—96no—
432D4—2712no—
433D4—274no—
434D4—274no—
435D4—274no—
436D4—276no—
437C2×C4—33no—
438C2×C4—36no—
439C2×C4—36no—
440C2×C4—912no—
441C2×C4—2712no—
442C2×C4—2712no—
443C2×C4—274no—
444C2×C4—2712no—
445C2×C4C2×D3×D913yes—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3—21no—
449C2×C3—32no—
450C2×C3—41no—
451C2×C3—61no—
452C2×C3—184no—
453C2×C3C24×D911yescharacteristic
454C2×C3C23×D3211yescharacteristic
455D3—21no—
456D3—64no—
457D3—184no—
458D3—184no—
459D3—364no—
460D3#6371312yesdirect product factor
461C22—31no—
462C22—31no—
463C22—62no—
464C22—94no—
465C22—274no—
466C22—548no—
467C22C22×D3×D911yescharacteristic
468C4—36no—
469C4—94no—
470C4—274no—
471C4C22×D3×D916yes—
472C3—21no—
473C3C2×#6371311yescharacteristic, complemented
474C3D3×#100711yescharacteristic
475C2—21no—
476C2—32no—
477C2—61no—
478C2—184no—
479C2—544no—
480C2C23×D3×D911yescharacteristic
subgroupG/H
centerC2C23×D3×D9
commutator, cosocleC2×C3×C9C25
FrattiniC2×C3C23×D32
FittingC2×C3×Q8×C9C22
socleC2×C32C24×D3
hypercenterC2×Q8D3×D9
hypocenterC3×C9C2×2−1+4
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. #11722667C2semidirect
2. C2×D3×Q8×C9C2semidirect
3. C3×#63713C2semidirect
4. D3×#3701C2semidirect
5. D3×Q8×D9C2semidirect
6. #11722510C2semidirect
7. #11722509C2semidirect
8. D3×#3697C2semidirect
9. #11722458C2semidirect
12. C2×C3×Q8×C9C22semidirect
13. #88746C22semidirect
14. Q8×C3⋊D9C22semidirect
15. #88736C22semidirect
17. D3×Q8×C9C22semidirect
18. C3×#3701C22semidirect
19. C3×Q8×D9C22semidirect
21. C3×#3697C22semidirect
22. C2×C9×Dic6C22semidirect
24. #88658C22semidirect
27. #88642C22semidirect
28. #88641C22semidirect
29. #88640C22semidirect
31. (C3⋊D9)⋊Q8C22semidirect
32. #88635C22semidirect
33. #88634C22semidirect
34. D9×Dic6C22semidirect
50. #63713D3direct (N × H)
106. C2×Q8×C9C2×D3semidirect
113. #3701C2×D3semidirect
118. Q8×D9C2×D3semidirect
125. #3697C2×D3semidirect
157. C2×D3×Q8D9semidirect
216. D3×C92−1+4semidirect
225. C2×C3×Q8C2×D9semidirect
245. D3×Q8C2×D9semidirect
267. C2×Dic6C2×D9semidirect
303. C3×C9C2×2−1+4semidirect
389. C2×Q8D3×D9semidirect
422. C9D3×2−1+4semidirect
460. D3#63713direct (N × H)
473. C3C2×#63713semidirect

Maximal subgroups

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

subgroupindex
#117226672
C2×D3×Q8×C92
C3×#637132
D3×#37012
D3×Q8×D92
#117225102
#117225092
D3×#36972
#117224582
D3×#10073
C2×#637133