Finite Group Databasebeta

← C2×D3×#3701 #13164755

Subgroups — C2×D3×#3701 #13164755

Order 1728.

2834 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88746C211yesmaximal, characteristic, complemented
2C2×D3×Q8×C9C211yesmaximal, characteristic, complemented
3C2×C3×#3701C211yesmaximal, characteristic, complemented
4D3×#3701C2116yesmaximal, direct product factor
5C2×D3×D36C213yesmaximal, complemented
6C2×C4×D3×D9C213yesmaximal, complemented
7C2×#88635C213yesmaximal, complemented
8C2×#88634C213yesmaximal, complemented
9C2×D3×#239—31nomaximal
10C22×#3701—31nomaximal
11C22×D3×D9C2213yes—
12C2×C3×Q8×C9C2211yescharacteristic, complemented
13#88746C2218yescomplemented
14C2×C4⋊(C3⋊D9)C2213yescomplemented
15C2×C4×C3⋊D9C2213yescomplemented
16D3×Q8×C9C2218yescomplemented
17C3×#3701C2218yescomplemented
18C2×C4×D3×C9C2213yes—
19C2×C3×D36C2213yescomplemented
20C2×C3×C4×D9C2213yescomplemented
21C2×C9×Dic6C2213yescomplemented
22C2×D3×Dic9C2211yescharacteristic
23C2×#6192C2216yescomplemented
24C2×(C3⋊D9)⋊C4C2213yescomplemented
25C2×Dic3×D9C2213yescomplemented
26D3×D36C22124yescomplemented
27C4×D3×D9C22124yescomplemented
28#88635C22124yescomplemented
29#88634C22124yescomplemented
30C2×#3832—31no—
31C2×C3×#239—31no—
32C2×C3×D3×Q8D311yescharacteristic
33D3×#239—316no—
34C2×D3×D12—33no—
35C2×C4×D32—33no—
36C2×#3799—33no—
37C2×#3796—33no—
38C22×Q8×C9—31no—
39C2×#3701—31no—
40C2×#3701—316no—
41C2×#3701—33no—
42C2×#3701—33no—
43C2×#3701—61no—
44C2×#3701D311yescharacteristic, direct product factor
45C22×D36—33no—
46C22×C4×D9—33no—
47C22×C3⋊D9C2313yes—
48C22×D3×C9C2311yescharacteristic
49C22×C3×D9C2313yes—
50C2×D3×D9—212no—
51C2×D3×D9C23112yes—
52C3×Q8×C9C2314yescomplemented
53C2×C3×C4×C9C2313yes—
54C2×C3⋊Dic9C2311yescharacteristic
55C4⋊(C3⋊D9)C23112yescomplemented
56C4×C3⋊D9C23112yescomplemented
57C2×C9×Dic3C2313yes—
58C2×C3×Dic9C2311yescharacteristic
59C4×D3×C9C23112yes—
60C3×D36C23112yescomplemented
61C3×C4×D9C23112yescomplemented
62C9×Dic6C23112yescomplemented
63D3×Dic9C2314yes—
64#6192C23124yescomplemented
65(C3⋊D9)⋊C4C23112yescomplemented
66Dic3×D9C23112yescomplemented
67C2×D3×#41—91no—
68C22×#239—91no—
69C22×D32—33no—
70C2×C32×Q8C2×D311yescharacteristic
71#3832—38no—
72C2×C4⋊(C3⋊D3)—33no—
73C2×C4×C3⋊D3—33no—
74C3×#239—38no—
75C3×D3×Q8C2×D318yes—
76C2×C3×D12—33no—
77C2×C3×C4×D3—33no—
78C2×C3×C4×D3C2×D313yes—
79C2×C3×Dic6C2×D313yes—
80C2×#626—36no—
81C2×#624—33no—
82C2×D3×Dic3—31no—
83C2×D3×Dic3—33no—
84D3×D12—324no—
85C4×D32—324no—
86#3799—324no—
87#3796—324no—
88C23×D9—33no—
89C2×Q8×C9—21no—
90C2×Q8×C9—38no—
91C2×Q8×C9—33no—
92C2×Q8×C9C2×D311yescharacteristic, complemented
93C22×C4×C9—33no—
94C22×Dic9—31no—
95#3701—38no—
96#3701—324no—
97#3701—324no—
98#3701—68no—
99#3701C2×D318yesdirect product factor
100C2×D36—36no—
101C2×D36—33no—
102C2×D36—324no—
103C2×D36—63no—
104C2×D36C2×D313yescomplemented
105C2×C4×D9—33no—
106C2×C4×D9—33no—
107C2×C4×D9—33no—
108C2×C4×D9—324no—
109C2×C4×D9—63no—
110C2×C4×D9C2×D313yescomplemented
111C22×C3×C9C2411yescharacteristic
112C2×C3⋊D9—26no—
113C2×C3⋊D9C2416yes—
114C2×D3×C9#4114yescomplemented
115C2×D3×C9C2412yes—
116C2×C3×D9—26no—
117C2×C3×D9C2416yes—
118D3×D9—224no—
119C3×C4×C9C2416yes—
120C3⋊Dic9C2412yes—
121C9×Dic3C2416yes—
122C3×Dic9C2412yes—
123C2×C3×#41—91no—
124C22×C3×Q8—31no—
125D3×#41—916no—
126C2×#239—31no—
127C2×#239—93no—
128C2×#239—93no—
129C2×#239—916no—
130C2×#239—91no—
131C2×#239—91no—
132C2×#239—181no—
133C2×D3×Q8D911yescharacteristic, complemented
134C2×#237—93no—
135C2×D3×D4—93no—
136C2×#235—93no—
137C22×D12—93no—
138C22×C4×D3—93no—
139C22×C4×D3—93no—
140C22×C3⋊D3—33no—
141C22×C3×D3—33no—
142C22×C3×D3C22×D311yescharacteristic
143C2×D32—312no—
144C2×D32—612no—
145C32×Q8C22×D314yes—
146C2×C32×C4C22×D313yes—
147C2×C3⋊Dic3—31no—
148C4⋊(C3⋊D3)—312no—
149C4×C3⋊D3—312no—
150C2×C3×Dic3—31no—
151C2×C3×Dic3C22×D313yes—
152C3×D12—312no—
153C3×C4×D3—312no—
154C3×C4×D3C22×D3112yes—
155C3×Dic6C22×D3112yes—
156#626—324no—
157#624—312no—
158D3×Dic3—34no—
159D3×Dic3—312no—
160C23×C9—31no—
161C22×D9—33no—
162C22×D9—312no—
163C22×D9—63no—
164C22×D9—612no—
165C22×D9C22×D313yes—
166Q8×C9—24no—
167Q8×C9—312no—
168Q8×C9C22×D314yescomplemented
169C2×C4×C9—23no—
170C2×C4×C9—33no—
171C2×C4×C9—312no—
172C2×C4×C9C22×D313yes—
173C2×Dic9—31no—
174C2×Dic9—34no—
175C2×Dic9—61no—
176C2×Dic9C22×D311yescharacteristic
177D36—324no—
178D36—312no—
179D36—612no—
180D36C22×D3112yescomplemented
181C4×D9—312no—
182C4×D9—312no—
183C4×D9—312no—
184C4×D9—612no—
185C4×D9C22×D3112yescomplemented
186C22×#41—271noSylow
187C2×C3×C9C2×#4112yescomplemented
188C2×C3×C9C2511yescharacteristic
189C3⋊D9—26no—
190D3×C9C2×#4114yescomplemented
191C3×D9—26no—
192C23×D3—93no—
193C23×D3—93no—
194C3×#41—98no—
195C2×C3×Q8—21no—
196C2×C3×Q8—38no—
197C2×C3×Q8—33no—
198C2×C3×Q8C2×D911yescharacteristic, complemented
199C2×C3×Q8D3211yescharacteristic
200C2×C3×D4—93no—
201C22×C3×C4—33no—
202C22×C3×C4—93no—
203C2×#67—96no—
204C22×Dic3—91no—
205C22×Dic3—93no—
206#239—38no—
207#239—924no—
208#239—924no—
209#239—98no—
210#239—98no—
211#239—188no—
212D3×Q8C2×D918yescomplemented
213#237—924no—
214D3×D4—924no—
215#235—924no—
216C2×D12—33no—
217C2×D12—924no—
218C2×D12—96no—
219C2×D12—93no—
220C2×D12—93no—
221C2×D12—183no—
222C2×C4×D3—33no—
223C2×C4×D3—924no—
224C2×C4×D3—93no—
225C2×C4×D3—93no—
226C2×C4×D3—91no—
227C2×C4×D3—93no—
228C2×C4×D3—93no—
229C2×C4×D3—93no—
230C2×C4×D3—924no—
231C2×C4×D3—183no—
232C2×C4×D3C2×D913yes—
233C2×Dic6C2×D913yescomplemented
234C22×C32C23×D311yescharacteristic
235C2×C3⋊D3—36no—
236C2×C3⋊D3—66no—
237C2×C3×D3—36no—
238C2×C3×D3—66no—
239C2×C3×D3#23914yes—
240C2×C3×D3C23×D312yes—
241D32—624no—
242C32×C4C23×D316yes—
243C3⋊Dic3—32no—
244C3×Dic3—32no—
245C3×Dic3C23×D316yes—
246C22×C9—21no—
247C22×C9—34no—
248C22×C9—32no—
249C22×C9C23×D311yescharacteristic
250C2×D9—26no—
251C2×D9—36no—
252C2×D9—66no—
253C2×D9—624no—
254C2×D9—66no—
255C2×D9—126no—
256C2×D9C23×D316yes—
257C4×C9—26no—
258C4×C9—36no—
259C4×C9C23×D316yes—
260Dic9—32no—
261Dic9—62no—
262Dic9C23×D312yes—
263C2×#41—91no—
264C2×#41—273no—
265C2×#41—273no—
266C2×#41—2716no—
267C2×#41—271no—
268C22×Q8—31no—
269C22×D4—273no—
270C23×C4—273no—
271C3×C9C22×#4111yescharacteristic, complemented, Sylow
272C23×C3—31no—
273C23×C3—93no—
274C22×D3—33no—
275C22×D3—912no—
276C22×D3—93no—
277C22×D3—93no—
278C22×D3—912no—
279C22×D3—183no—
280C22×D3—1812no—
281C22×D3—1812no—
282C22×D3C22×D911yescharacteristic
283C3×Q8—24no—
284C3×Q8—312no—
285C3×Q8C22×D914yescomplemented
286C3×Q8C2×D3214yes—
287C3×D4—912no—
288C2×C3×C4—23no—
289C2×C3×C4—33no—
290C2×C3×C4—312no—
291C2×C3×C4—91no—
292C2×C3×C4—912no—
293C2×C3×C4C22×D913yes—
294C2×C3×C4C2×D3213yes—
295#67—924no—
296C2×Dic3—31no—
297C2×Dic3—94no—
298C2×Dic3—91no—
299C2×Dic3—91no—
300C2×Dic3—912no—
301C2×Dic3—181no—
302C2×Dic3C22×D913yes—
303D12—312no—
304D12—924no—
305D12—912no—
306D12—912no—
307D12—1812no—
308C4×D3—312no—
309C4×D3—912no—
310C4×D3—912no—
311C4×D3—94no—
312C4×D3—912no—
313C4×D3—912no—
314C4×D3—912no—
315C4×D3—1812no—
316C4×D3C22×D9112yes—
317Dic6C22×D9112yescomplemented
318C2×C32C2×#23912yes—
319C2×C32C24×D311yescharacteristic
320C3⋊D3—66no—
321C3×D3—66no—
322C3×D3C2×#23914yes—
323C2×C9—22no—
324C2×C9—21no—
325C2×C9—34no—
326C2×C9D3×#4112yescomplemented
327C2×C9C24×D311yescharacteristic
328D9—26no—
329D9—66no—
330D9—126no—
331C24—273no—
332#41—98no—
333#41—2724no—
334#41—2724no—
335#41—278no—
336C2×Q8—38no—
337C2×Q8—33no—
338C2×Q8D3×D911yescharacteristic, complemented
339C2×D4—93no—
340C2×D4—2724no—
341C2×D4—276no—
342C2×D4—273no—
343C22×C4—33no—
344C22×C4—93no—
345C22×C4—273no—
346C22×C4—2724no—
347C22×C4—273no—
348C22×C4—271no—
349C22×C4—273no—
350C22×C3—21no—
351C22×C3—34no—
352C22×C3—32no—
353C22×C3—96no—
354C22×C3—186no—
355C22×C3C23×D911yescharacteristic
356C22×C3C22×D3211yescharacteristic
357C2×D3—36no—
358C2×D3—66no—
359C2×D3—96no—
360C2×D3—96no—
361C2×D3—186no—
362C2×D3—1824no—
363C2×D3—186no—
364C2×D3—186no—
365C2×D3—1824no—
366C2×D3—366no—
367C2×D3#370114yesdirect product factor
368C2×D3C23×D912yes—
369C3×C4—26no—
370C3×C4—36no—
371C3×C4—92no—
372C3×C4C23×D916yes—
373C3×C4C22×D3216yes—
374Dic3—32no—
375Dic3—92no—
376Dic3—92no—
377Dic3—182no—
378Dic3C23×D916yes—
379C32C22×#23911yescharacteristic
380C9—21no—
381C9C2×D3×#4111yescharacteristic, complemented
382C23—31no—
383C23—93no—
384C23—2712no—
385C23—273no—
386C23—5412no—
387Q8—312no—
388Q8C2×D3×D914yescomplemented
389D4—912no—
390D4—2724no—
391D4—2712no—
392C2×C4—33no—
393C2×C4—312no—
394C2×C4—91no—
395C2×C4—912no—
396C2×C4—2712no—
397C2×C4—2712no—
398C2×C4—274no—
399C2×C4—271no—
400C2×C4—2712no—
401C2×C4C2×D3×D913yes—
402C2×C3—22no—
403C2×C3—21no—
404C2×C3—34no—
405C2×C3—186no—
406C2×C3C2×#370112yescomplemented
407C2×C3C24×D911yescharacteristic
408C2×C3D3×#23912yes—
409C2×C3C23×D3211yescharacteristic
410D3—66no—
411D3—186no—
412D3—186no—
413D3—366no—
414D3C2×#370114yesdirect product factor
415C22—34no—
416C22—32no—
417C22—96no—
418C22—186no—
419C22—276no—
420C22—5424no—
421C22—546no—
422C22C22×D3×D911yescharacteristic
423C4—36no—
424C4—92no—
425C4—272no—
426C4C22×D3×D916yes—
427C3—21no—
428C3C22×#370111yescharacteristic, complemented
429C3C2×D3×#23911yescharacteristic
430C2—34no—
431C2—186no—
432C2—546no—
433C2D3×#370112yesdirect product factor
434C2C23×D3×D911yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC2×C3×C9C25
FrattiniC2×C3C23×D32
FittingC2×C3×Q8×C9C22
socleC22×C32C23×D3
hypercenterC2×Q8D3×D9
hypocenterC3×C9C22×#41
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#88746C2semidirect
2. C2×D3×Q8×C9C2semidirect
3. C2×C3×#3701C2semidirect
4. D3×#3701C2direct (N × H)
5. C2×D3×D36C2semidirect
6. C2×C4×D3×D9C2semidirect
7. C2×#88635C2semidirect
8. C2×#88634C2semidirect
12. C2×C3×Q8×C9C22semidirect
13. #88746C22semidirect
14. C2×C4⋊(C3⋊D9)C22semidirect
15. C2×C4×C3⋊D9C22semidirect
16. D3×Q8×C9C22semidirect
17. C3×#3701C22semidirect
19. C2×C3×D36C22semidirect
20. C2×C3×C4×D9C22semidirect
21. C2×C9×Dic6C22semidirect
23. C2×#6192C22semidirect
24. C2×(C3⋊D9)⋊C4C22semidirect
25. C2×Dic3×D9C22semidirect
26. D3×D36C22semidirect
27. C4×D3×D9C22semidirect
28. #88635C22semidirect
29. #88634C22semidirect
44. C2×#3701D3direct (N × H)
52. C3×Q8×C9C23semidirect
55. C4⋊(C3⋊D9)C23semidirect
56. C4×C3⋊D9C23semidirect
60. C3×D36C23semidirect
61. C3×C4×D9C23semidirect
62. C9×Dic6C23semidirect
64. #6192C23semidirect
65. (C3⋊D9)⋊C4C23semidirect
66. Dic3×D9C23semidirect
92. C2×Q8×C9C2×D3semidirect
99. #3701C2×D3direct (N × H)
104. C2×D36C2×D3semidirect
110. C2×C4×D9C2×D3semidirect
114. C2×D3×C9#41semidirect
133. C2×D3×Q8D9semidirect
168. Q8×C9C22×D3semidirect
180. D36C22×D3semidirect
185. C4×D9C22×D3semidirect
187. C2×C3×C9C2×#41semidirect
190. D3×C9C2×#41semidirect
198. C2×C3×Q8C2×D9semidirect
212. D3×Q8C2×D9semidirect
233. C2×Dic6C2×D9semidirect
271. C3×C9C22×#41semidirect
285. C3×Q8C22×D9semidirect
317. Dic6C22×D9semidirect
326. C2×C9D3×#41semidirect
338. C2×Q8D3×D9semidirect
367. C2×D3#3701direct (N × H)
381. C9C2×D3×#41semidirect
388. Q8C2×D3×D9semidirect
406. C2×C3C2×#3701semidirect
414. D3C2×#3701direct (N × H)
428. C3C22×#3701semidirect
433. C2D3×#3701direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#887462
C2×D3×Q8×C92
C2×C3×#37012
D3×#37012
C2×D3×D362
C2×C4×D3×D92
C2×#886352
C2×#886342
C2×D3×#2393
C22×#37013