Finite Group Databasebeta

← C2×C3⋊D3×#239 #13181626

Subgroups — C2×C3⋊D3×#239 #13181626

Order 1728.

5996 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×#89080C211yesmaximal, characteristic, complemented
2C2×C3×Q8×C3⋊D3C211yesmaximal, characteristic, complemented
3C2×C32×#239C211yesmaximal, characteristic, complemented
4C3⋊D3×#239C2116yesmaximal, direct product factor
5C2×C3⋊D3×D12C213yesmaximal, complemented
6C2×C4×D3×C3⋊D3C213yesmaximal, complemented
7C2×#89022C213yesmaximal, complemented
8C2×#89015C213yesmaximal, complemented
9C2×#41×C3⋊D3—31nomaximal
10C2×D3×#239—34nomaximal
11C22×D3×C3⋊D3C2213yes—
12C2×C33×Q8C2211yescharacteristic, complemented
13#89080C2218yescomplemented
14C2×C4⋊(C3⋊(C3⋊D3))C2213yescomplemented
15C2×C4×C3⋊(C3⋊D3)C2213yescomplemented
16C3×Q8×C3⋊D3C2218yescomplemented
17C2×C3×C4×C3⋊D3C2213yes—
18C2×C3×#634C2213yescomplemented
19C32×#239C2218yescomplemented
20C2×C32×D12C2213yescomplemented
21C2×C32×C4×D3C2213yescomplemented
22C2×#6291C2216yescomplemented
23C2×#6289C2213yescomplemented
24C2×Dic3×C3⋊D3C2211yescharacteristic
25C2×D3×C3⋊Dic3C2213yescomplemented
26C3⋊D3×D12C22124yescomplemented
27C4×D3×C3⋊D3C22124yescomplemented
28#89022C22124yescomplemented
29#89015C22124yescomplemented
30C2×C32×#41—31no—
31#41×C3⋊D3—316no—
32C2×#3832—31no—
33C2×#3832—34no—
34C2×#3832—64no—
35C2×Q8×C3⋊D3D311yescharacteristic, complemented
36C2×#3830—33no—
37C2×D4×C3⋊D3—33no—
38C2×#3828—33no—
39C22×C4×C3⋊D3—33no—
40C2×C3×#239D314yescomplemented
41C2×C3×D3×Q8—34no—
42D3×#239—364no—
43C2×D3×D12—312no—
44C2×C4×D32—312no—
45C2×#3799—312no—
46C2×#3796—312no—
47C22×C3⋊(C3⋊D3)C2313yes—
48C22×C3×C3⋊D3C2311yescharacteristic
49C22×C32×D3C2313yes—
50C2×D3×C3⋊D3—212no—
51C2×D3×C3⋊D3C23112yes—
52C33×Q8C2314yescomplemented
53C2×C33×C4C2313yes—
54C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
55C4⋊(C3⋊(C3⋊D3))C23112yescomplemented
56C4×C3⋊(C3⋊D3)C23112yescomplemented
57C2×C3×C3⋊Dic3C2313yes—
58C3×C4×C3⋊D3C23112yes—
59C3×#634C23112yescomplemented
60C2×C32×Dic3C2311yescharacteristic
61C32×D12C23112yescomplemented
62C32×C4×D3C23112yescomplemented
63#6291C23124yescomplemented
64#6289C23112yescomplemented
65Dic3×C3⋊D3C2314yes—
66D3×C3⋊Dic3C23112yescomplemented
67C2×D3×#41—94no—
68C22×#239—91no—
69C23×C3⋊D3—33no—
70C22×D32—312no—
71C32×#41—38no—
72C2×C32×Q8—24no—
73C2×C32×Q8C2×D314yescomplemented
74C2×C32×Q8C2×D311yescharacteristic, complemented
75C2×C32×D4—33no—
76C22×C32×C4—33no—
77C2×#638—36no—
78C22×C3⋊Dic3—33no—
79#3832—38no—
80#3832—332no—
81#3832—632no—
82Q8×C3⋊D3C2×D318yescomplemented
83#3830—324no—
84D4×C3⋊D3—324no—
85#3828—324no—
86C2×C4⋊(C3⋊D3)—33no—
87C2×C4⋊(C3⋊D3)—312no—
88C2×C4⋊(C3⋊D3)—612no—
89C2×C4×C3⋊D3—31no—
90C2×C4×C3⋊D3—33no—
91C2×C4×C3⋊D3—312no—
92C2×C4×C3⋊D3—33no—
93C2×C4×C3⋊D3—324no—
94C2×C4×C3⋊D3—612no—
95C2×C4×C3⋊D3C2×D313yes—
96C2×#634C2×D313yescomplemented
97C3×#239C2×D3132yescomplemented
98C3×D3×Q8—332no—
99C2×C3×D12C2×D3112yescomplemented
100C2×C3×C4×D3—312no—
101C2×C3×C4×D3C2×D3112yescomplemented
102C2×C3×Dic6—312no—
103C2×#626—324no—
104C2×#624—312no—
105C2×D3×Dic3—312no—
106C2×D3×Dic3—34no—
107D3×D12—396no—
108C4×D32—396no—
109#3799—396no—
110#3796—396no—
111C22×C33C2411yescharacteristic
112C2×C3⋊(C3⋊D3)—26no—
113C2×C3⋊(C3⋊D3)C2416yes—
114C2×C3×C3⋊D3#4114yescomplemented
115C2×C3×C3⋊D3C2412yes—
116C2×C32×D3—26no—
117C2×C32×D3C2416yes—
118D3×C3⋊D3—224no—
119C33×C4C2416yes—
120C3⋊(C3⋊Dic3)C2412yes—
121C3×C3⋊Dic3C2416yes—
122C32×Dic3C2412yes—
123C2×C3×#41—34no—
124C22×C3×Q8—91no—
125D3×#41—964no—
126C2×#239—94no—
127C2×#239—91no—
128C2×#239—916no—
129C2×#239—93no—
130C2×#239—93no—
131C2×#239—184no—
132C2×#239C3⋊D311yescharacteristic, direct product factor
133C2×D3×Q8—34no—
134C2×#237—912no—
135C2×D3×D4—912no—
136C2×#235—912no—
137C22×D12—93no—
138C22×C4×D3—912no—
139C22×C4×D3—93no—
140C23×C32—33no—
141C22×C3⋊D3—33no—
142C22×C3⋊D3—312no—
143C22×C3⋊D3—312no—
144C22×C3⋊D3—612no—
145C22×C3⋊D3—612no—
146C22×C3⋊D3C22×D311yescharacteristic
147C22×C3×D3—34no—
148C22×C3×D3C22×D3112yes—
149C2×D32—348no—
150C2×D32—648no—
151C32×Q8—216no—
152C32×Q8C22×D3116yescomplemented
153C32×Q8C22×D314yescomplemented
154C32×D4—312no—
155C2×C32×C4—212no—
156C2×C32×C4—31no—
157C2×C32×C4—312no—
158C2×C32×C4C22×D3112yes—
159C2×C32×C4C22×D313yes—
160#638—324no—
161C2×C3⋊Dic3—31no—
162C2×C3⋊Dic3—312no—
163C2×C3⋊Dic3—34no—
164C2×C3⋊Dic3—64no—
165C2×C3⋊Dic3C22×D313yes—
166C4⋊(C3⋊D3)—312no—
167C4⋊(C3⋊D3)—348no—
168C4⋊(C3⋊D3)—648no—
169C4×C3⋊D3—34no—
170C4×C3⋊D3—312no—
171C4×C3⋊D3—312no—
172C4×C3⋊D3—348no—
173C4×C3⋊D3—648no—
174C4×C3⋊D3C22×D3112yes—
175#634C22×D3112yescomplemented
176C2×C3×Dic3—312no—
177C2×C3×Dic3C22×D314yes—
178C3×D12C22×D3148yescomplemented
179C3×C4×D3—348no—
180C3×C4×D3C22×D3148yescomplemented
181C3×Dic6—348no—
182#626—396no—
183#624—348no—
184D3×Dic3—348no—
185D3×Dic3—316no—
186C22×#41—271noSylow
187C2×C33C2×#4112yescomplemented
188C2×C33C2511yescharacteristic
189C3⋊(C3⋊D3)—26no—
190C3×C3⋊D3C2×#4114yescomplemented
191C32×D3—26no—
192C23×D3—912no—
193C23×D3—93no—
194C3×#41—332no—
195C2×C3×Q8—24no—
196C2×C3×Q8—98no—
197C2×C3×Q8—93no—
198C2×C3×Q8D3214yescomplemented
199C2×C3×Q8C2×C3⋊D311yescharacteristic, complemented
200C2×C3×D4—312no—
201C22×C3×C4—312no—
202C22×C3×C4—93no—
203C2×#67—924no—
204C22×Dic3—912no—
205C22×Dic3—91no—
206#239—932no—
207#239—98no—
208#239—924no—
209#239—924no—
210#239—1832no—
211#239C2×C3⋊D318yesdirect product factor
212D3×Q8—332no—
213#237—996no—
214D3×D4—996no—
215#235—996no—
216C2×D12—96no—
217C2×D12—912no—
218C2×D12—93no—
219C2×D12—924no—
220C2×D12—1812no—
221C2×D12C2×C3⋊D313yescomplemented
222C2×C4×D3—312no—
223C2×C4×D3—996no—
224C2×C4×D3—912no—
225C2×C4×D3—94no—
226C2×C4×D3—93no—
227C2×C4×D3—912no—
228C2×C4×D3—93no—
229C2×C4×D3—93no—
230C2×C4×D3—924no—
231C2×C4×D3—1812no—
232C2×C4×D3C2×C3⋊D313yescomplemented
233C2×Dic6—312no—
234C22×C32—24no—
235C22×C32—36no—
236C22×C32—66no—
237C22×C32C23×D314yes—
238C22×C32C23×D311yescharacteristic
239C2×C3⋊D3—36no—
240C2×C3⋊D3—324no—
241C2×C3⋊D3—624no—
242C2×C3⋊D3—66no—
243C2×C3⋊D3—624no—
244C2×C3⋊D3—624no—
245C2×C3⋊D3—1224no—
246C2×C3⋊D3#23914yesdirect product factor
247C2×C3⋊D3C23×D312yes—
248C2×C3×D3—224no—
249C2×C3×D3—316no—
250C2×C3×D3—38no—
251C2×C3×D3C23×D3124yes—
252D32—696no—
253C32×C4—224no—
254C32×C4—32no—
255C32×C4C23×D3124yes—
256C32×C4C23×D316yes—
257C3⋊Dic3—32no—
258C3⋊Dic3—38no—
259C3⋊Dic3—68no—
260C3⋊Dic3C23×D316yes—
261C3×Dic3—324no—
262C3×Dic3C23×D318yes—
263C2×#41—31no—
264C2×#41—273no—
265C2×#41—273no—
266C2×#41—2716no—
267C2×#41—271no—
268C22×Q8—91no—
269C22×D4—273no—
270C23×C4—273no—
271C33C22×#4111yescharacteristic, complemented, Sylow
272C23×C3—312no—
273C23×C3—91no—
274C22×D3—34no—
275C22×D3—948no—
276C22×D3—912no—
277C22×D3—93no—
278C22×D3—912no—
279C22×D3—1812no—
280C22×D3—1812no—
281C22×D3—1848no—
282C22×D3C22×C3⋊D313yes—
283C3×Q8—216no—
284C3×Q8—912no—
285C3×Q8C2×D32116yescomplemented
286C3×Q8C22×C3⋊D314yescomplemented
287C3×D4—348no—
288C2×C3×C4—212no—
289C2×C3×C4—34no—
290C2×C3×C4—348no—
291C2×C3×C4—93no—
292C2×C3×C4—912no—
293C2×C3×C4C2×D32112yes—
294C2×C3×C4C22×C3⋊D313yes—
295#67—996no—
296C2×Dic3—312no—
297C2×Dic3—948no—
298C2×Dic3—94no—
299C2×Dic3—91no—
300C2×Dic3—94no—
301C2×Dic3—184no—
302C2×Dic3C22×C3⋊D311yescharacteristic
303D12—924no—
304D12—948no—
305D12—912no—
306D12—1848no—
307D12C22×C3⋊D3112yescomplemented
308C4×D3—348no—
309C4×D3—948no—
310C4×D3—916no—
311C4×D3—912no—
312C4×D3—948no—
313C4×D3—912no—
314C4×D3—912no—
315C4×D3—1848no—
316C4×D3C22×C3⋊D3112yescomplemented
317Dic6—348no—
318C2×C32—28no—
319C2×C32—24no—
320C2×C32—66no—
321C2×C32C2×#23912yescomplemented
322C2×C32D3×#4118yescomplemented
323C2×C32C24×D314yes—
324C2×C32C24×D311yescharacteristic
325C3⋊D3—66no—
326C3⋊D3—624no—
327C3⋊D3—1224no—
328C3⋊D3C2×#23914yesdirect product factor
329C3×D3—224no—
330C3×D3—316no—
331C24—273no—
332#41—38no—
333#41—2724no—
334#41—2724no—
335#41—278no—
336C2×Q8—98no—
337C2×Q8—93no—
338C2×Q8D3×C3⋊D311yescharacteristic, complemented
339C2×D4—33no—
340C2×D4—2724no—
341C2×D4—276no—
342C2×D4—273no—
343C22×C4—33no—
344C22×C4—93no—
345C22×C4—273no—
346C22×C4—2724no—
347C22×C4—271no—
348C22×C4—273no—
349C22×C4—273no—
350C22×C3—24no—
351C22×C3—324no—
352C22×C3—624no—
353C22×C3—94no—
354C22×C3—92no—
355C22×C3C22×D3214yes—
356C22×C3C23×C3⋊D311yescharacteristic
357C2×D3—26no—
358C2×D3—316no—
359C2×D3—38no—
360C2×D3—924no—
361C2×D3—96no—
362C2×D3—1824no—
363C2×D3—1824no—
364C2×D3—1896no—
365C2×D3—1824no—
366C2×D3—186no—
367C2×D3—3624no—
368C2×D3C23×C3⋊D316yes—
369C3×C4—224no—
370C3×C4—38no—
371C3×C4—96no—
372C3×C4C22×D32124yes—
373C3×C4C23×C3⋊D316yes—
374Dic3—324no—
375Dic3—98no—
376Dic3—92no—
377Dic3—188no—
378Dic3C23×C3⋊D312yes—
379C32—24no—
380C32C22×#23911yescharacteristic, complemented
381C32C2×D3×#4114yescomplemented
382C23—33no—
383C23—91no—
384C23—2712no—
385C23—273no—
386C23—5412no—
387Q8—912no—
388Q8C2×D3×C3⋊D314yescomplemented
389D4—312no—
390D4—2724no—
391D4—2712no—
392C2×C4—31no—
393C2×C4—312no—
394C2×C4—93no—
395C2×C4—912no—
396C2×C4—2712no—
397C2×C4—274no—
398C2×C4—2712no—
399C2×C4—271no—
400C2×C4—2712no—
401C2×C4C2×D3×C3⋊D313yes—
402C2×C3—28no—
403C2×C3—24no—
404C2×C3—624no—
405C2×C3—94no—
406C2×C3D3×#23918yescomplemented
407C2×C3#41×C3⋊D312yescomplemented
408C2×C3C23×D3214yes—
409C2×C3C24×C3⋊D311yescharacteristic
410D3—26no—
411D3—316no—
412D3—1824no—
413D3—186no—
414D3—3624no—
415C22—36no—
416C22—66no—
417C22—94no—
418C22—92no—
419C22—276no—
420C22—5424no—
421C22—546no—
422C22C22×D3×C3⋊D311yescharacteristic
423C4—32no—
424C4—96no—
425C4—272no—
426C4C22×D3×C3⋊D316yes—
427C3—24no—
428C3C2×D3×#23914yescomplemented
429C3C2×#41×C3⋊D311yescharacteristic, complemented
430C2—66no—
431C2—94no—
432C2—546no—
433C2C3⋊D3×#23912yesdirect product factor
434C2C23×D3×C3⋊D311yescharacteristic
subgroupG/H
centerC22C22×D3×C3⋊D3
commutator, cosocleC2×C33C25
FrattiniC2C23×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C24
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33C22×#41
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#89080C2semidirect
2. C2×C3×Q8×C3⋊D3C2semidirect
3. C2×C32×#239C2semidirect
4. C3⋊D3×#239C2direct (N × H)
5. C2×C3⋊D3×D12C2semidirect
6. C2×C4×D3×C3⋊D3C2semidirect
7. C2×#89022C2semidirect
8. C2×#89015C2semidirect
12. C2×C33×Q8C22semidirect
13. #89080C22semidirect
14. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
15. C2×C4×C3⋊(C3⋊D3)C22semidirect
16. C3×Q8×C3⋊D3C22semidirect
18. C2×C3×#634C22semidirect
19. C32×#239C22semidirect
20. C2×C32×D12C22semidirect
21. C2×C32×C4×D3C22semidirect
22. C2×#6291C22semidirect
23. C2×#6289C22semidirect
25. C2×D3×C3⋊Dic3C22semidirect
26. C3⋊D3×D12C22semidirect
27. C4×D3×C3⋊D3C22semidirect
28. #89022C22semidirect
29. #89015C22semidirect
35. C2×Q8×C3⋊D3D3semidirect
40. C2×C3×#239D3semidirect
52. C33×Q8C23semidirect
55. C4⋊(C3⋊(C3⋊D3))C23semidirect
56. C4×C3⋊(C3⋊D3)C23semidirect
59. C3×#634C23semidirect
61. C32×D12C23semidirect
62. C32×C4×D3C23semidirect
63. #6291C23semidirect
64. #6289C23semidirect
66. D3×C3⋊Dic3C23semidirect
73. C2×C32×Q8C2×D3semidirect
74. C2×C32×Q8C2×D3semidirect
82. Q8×C3⋊D3C2×D3semidirect
96. C2×#634C2×D3semidirect
97. C3×#239C2×D3semidirect
99. C2×C3×D12C2×D3semidirect
101. C2×C3×C4×D3C2×D3semidirect
114. C2×C3×C3⋊D3#41semidirect
132. C2×#239C3⋊D3direct (N × H)
152. C32×Q8C22×D3semidirect
153. C32×Q8C22×D3semidirect
175. #634C22×D3semidirect
178. C3×D12C22×D3semidirect
180. C3×C4×D3C22×D3semidirect
187. C2×C33C2×#41semidirect
190. C3×C3⋊D3C2×#41semidirect
198. C2×C3×Q8D32semidirect
199. C2×C3×Q8C2×C3⋊D3semidirect
211. #239C2×C3⋊D3direct (N × H)
221. C2×D12C2×C3⋊D3semidirect
232. C2×C4×D3C2×C3⋊D3semidirect
246. C2×C3⋊D3#239direct (N × H)
271. C33C22×#41semidirect
285. C3×Q8C2×D32semidirect
286. C3×Q8C22×C3⋊D3semidirect
307. D12C22×C3⋊D3semidirect
316. C4×D3C22×C3⋊D3semidirect
321. C2×C32C2×#239semidirect
322. C2×C32D3×#41semidirect
328. C3⋊D3C2×#239direct (N × H)
338. C2×Q8D3×C3⋊D3semidirect
380. C32C22×#239semidirect
381. C32C2×D3×#41semidirect
388. Q8C2×D3×C3⋊D3semidirect
406. C2×C3D3×#239semidirect
407. C2×C3#41×C3⋊D3semidirect
428. C3C2×D3×#239semidirect
429. C3C2×#41×C3⋊D3semidirect
433. C2C3⋊D3×#239direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#890802
C2×C3×Q8×C3⋊D32
C2×C32×#2392
C3⋊D3×#2392
C2×C3⋊D3×D122
C2×C4×D3×C3⋊D32
C2×#890222
C2×#890152
C2×#41×C3⋊D33
C2×D3×#2393