Finite Group Databasebeta

← C2×#11724626 #13181681

Subgroups — C2×#11724626 #13181681

Order 1728.

4082 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3832C212yesmaximal, complemented
2C2×C3×Q8×C3⋊D3C211yesmaximal, characteristic, complemented
3#11724626C2116yesmaximal, direct product factor
4C2×C4⋊#1113C213yesmaximal, complemented
5C2×C4×#1113C213yesmaximal, complemented
6C2×#89041C216yesmaximal, complemented
7C2×#64319—31nomaximal
8C2×D3×#239—32nomaximal
9C22×#1113C2213yes—
10C2×C33×Q8C2211yescharacteristic, complemented
11C3×#3832C22116yescomplemented
12C3×Q8×C3⋊D3C2218yescomplemented
13C2×C3×C4⋊(C3⋊D3)C2216yescomplemented
14C2×C3×C4×C3⋊D3C2213yes—
15C2×C3×C4×C3⋊D3C2216yescomplemented
16C2×C3×#634C2213yescomplemented
17C2×#6295C2216yescomplemented
18C2×#6294C2211yescharacteristic
19C2×#6294C2216yescomplemented
20C4⋊#1113C22124yescomplemented
21C4×#1113C22124yescomplemented
22#89041C22148yescomplemented
23C2×#3832D312yescomplemented
24C2×Q8×C3⋊D3D311yescharacteristic, complemented
25C2×C3×#239—32no—
26C2×C3×#239—32no—
27C2×C3×#239—62no—
28C2×C3×D3×Q8—32no—
29C2×C3×D3×Q8—61no—
30#64319—316no—
31D3×#239—332no—
32C2×#3802—33no—
33C2×D3×D12—36no—
34C2×C4×D32—33no—
35C2×C4×D32—36no—
36C2×#3799—36no—
37C2×#3796—36no—
38C2×#3795—36no—
39C22×C3×C3⋊D3C2316yes—
40C22×C3×C3⋊D3C2311yescharacteristic
41C2×#1113—212no—
42C2×#1113C23112yes—
43C33×Q8C2314yescomplemented
44C2×C33×C4C2313yes—
45C2×C3×C3⋊Dic3C2312yes—
46C2×C3×C3⋊Dic3C2313yes—
47C3×C4⋊(C3⋊D3)C23124yescomplemented
48C3×C4×C3⋊D3C23112yes—
49C3×C4×C3⋊D3C23124yescomplemented
50C3×#634C23112yescomplemented
51#6295C23124yescomplemented
52#6294C2314yes—
53#6294C23124yescomplemented
54C2×D3×#41—92no—
55C22×#239—91no—
56C22×D32—36no—
57C22×D32—33no—
58C2×C32×Q8—21no—
59C2×C32×Q8—22no—
60C2×C32×Q8—41no—
61C2×C32×Q8C2×D312yescomplemented
62C2×C32×Q8C2×D311yescharacteristic, complemented
63#3832C2×D3116yescomplemented
64Q8×C3⋊D3C2×D318yescomplemented
65C2×C4⋊(C3⋊D3)C2×D316yescomplemented
66C2×C4×C3⋊D3C2×D313yes—
67C2×C4×C3⋊D3C2×D316yescomplemented
68C2×#634C2×D313yescomplemented
69C3×#239—316no—
70C3×#239—316no—
71C3×#239—616no—
72C3×D3×Q8—316no—
73C3×D3×Q8—68no—
74C2×C3×D12—36no—
75C2×C3×D12—36no—
76C2×C3×D12—66no—
77C2×C3×C4×D3—36no—
78C2×C3×C4×D3—36no—
79C2×C3×C4×D3—36no—
80C2×C3×C4×D3—63no—
81C2×C3×C4×D3—66no—
82C2×C3×Dic6—36no—
83C2×C3×Dic6—63no—
84C2×#626—312no—
85C2×#625—36no—
86C2×#624—36no—
87C2×#624—31no—
88C2×D3×Dic3—36no—
89C2×D3×Dic3—36no—
90C2×D3×Dic3—32no—
91#3802—324no—
92D3×D12—348no—
93C4×D32—324no—
94C4×D32—348no—
95#3799—348no—
96#3796—348no—
97#3795—348no—
98C22×C33C2411yescharacteristic
99C2×C3×C3⋊D3—212no—
100C2×C3×C3⋊D3#4114yescomplemented
101C2×C3×C3⋊D3C24112yes—
102C2×C3×C3⋊D3C2412yes—
103#1113—224no—
104C33×C4C2416yes—
105C3×C3⋊Dic3C2414yes—
106C3×C3⋊Dic3C2416yes—
107C2×C3×#41—92no—
108C22×C3×Q8—91no—
109D3×#41—932no—
110C2×#239—32no—
111C2×#239—32no—
112C2×#239—62no—
113C2×#239—96no—
114C2×#239—916no—
115C2×D3×Q8—32no—
116C2×D3×Q8—61no—
117C2×#237—96no—
118C2×D3×D4—96no—
119C2×#235—96no—
120C22×D12—93no—
121C22×C4×D3—93no—
122C22×C4×D3—96no—
123C22×C3⋊D3C22×D316yes—
124C22×C3⋊D3C22×D311yescharacteristic
125C22×C3×D3—32no—
126C22×C3×D3—36no—
127C22×C3×D3—36no—
128C22×C3×D3—61no—
129C22×C3×D3—66no—
130C2×D32—324no—
131C2×D32—312no—
132C2×D32—624no—
133C2×D32—612no—
134C32×Q8—24no—
135C32×Q8—28no—
136C32×Q8—44no—
137C32×Q8C22×D318yescomplemented
138C32×Q8C22×D314yescomplemented
139C2×C32×C4—23no—
140C2×C32×C4—26no—
141C2×C32×C4—43no—
142C2×C32×C4C22×D316yes—
143C2×C32×C4C22×D313yes—
144C2×C3⋊Dic3C22×D312yes—
145C2×C3⋊Dic3C22×D313yes—
146C4⋊(C3⋊D3)C22×D3124yescomplemented
147C4×C3⋊D3C22×D3112yes—
148C4×C3⋊D3C22×D3124yescomplemented
149#634C22×D3112yescomplemented
150C2×C3×Dic3—32no—
151C2×C3×Dic3—32no—
152C2×C3×Dic3—36no—
153C2×C3×Dic3—63no—
154C2×C3×Dic3—62no—
155C3×D12—324no—
156C3×D12—324no—
157C3×D12—624no—
158C3×C4×D3—324no—
159C3×C4×D3—324no—
160C3×C4×D3—324no—
161C3×C4×D3—612no—
162C3×C4×D3—624no—
163C3×Dic6—324no—
164C3×Dic6—612no—
165#626—348no—
166#625—324no—
167#624—34no—
168#624—324no—
169D3×Dic3—38no—
170D3×Dic3—324no—
171D3×Dic3—324no—
172C22×#41—271noSylow
173C2×C33C2×#4112yescomplemented
174C2×C33C2511yescharacteristic
175C3×C3⋊D3—212no—
176C3×C3⋊D3C2×#4114yescomplemented
177C23×D3—93no—
178C23×D3—96no—
179C3×#41—916no—
180C2×C3×Q8—21no—
181C2×C3×Q8—22no—
182C2×C3×Q8—41no—
183C2×C3×Q8—98no—
184C2×C3×Q8—93no—
185C2×C3×Q8D3212yescomplemented
186C2×C3×Q8D3211yescharacteristic, complemented
187C2×C3×D4—96no—
188C22×C3×C4—93no—
189C22×C3×C4—96no—
190C2×#67—912no—
191C22×Dic3—91no—
192C22×Dic3—96no—
193#239—316no—
194#239—316no—
195#239—616no—
196#239—948no—
197D3×Q8—316no—
198D3×Q8—68no—
199#237—948no—
200D3×D4—948no—
201#235—948no—
202C2×D12—36no—
203C2×D12—36no—
204C2×D12—66no—
205C2×D12—924no—
206C2×D12—96no—
207C2×C4×D3—36no—
208C2×C4×D3—36no—
209C2×C4×D3—36no—
210C2×C4×D3—63no—
211C2×C4×D3—66no—
212C2×C4×D3—924no—
213C2×C4×D3—948no—
214C2×C4×D3—92no—
215C2×C4×D3—96no—
216C2×C4×D3—96no—
217C2×Dic6—36no—
218C2×Dic6—63no—
219C22×C32—21no—
220C22×C32—22no—
221C22×C32—41no—
222C22×C32C23×D311yescharacteristic
223C22×C32C23×D312yes—
224C2×C3⋊D3—212no—
225C2×C3⋊D3#23914yescomplemented
226C2×C3⋊D3C23×D3112yes—
227C2×C3⋊D3C23×D312yes—
228C2×C3×D3—38no—
229C2×C3×D3—34no—
230C2×C3×D3—312no—
231C2×C3×D3—312no—
232C2×C3×D3—64no—
233C2×C3×D3—62no—
234C2×C3×D3—612no—
235C2×C3×D3—612no—
236C2×C3×D3—612no—
237C2×C3×D3—1212no—
238D32—648no—
239D32—624no—
240C32×C4—26no—
241C32×C4—212no—
242C32×C4—46no—
243C32×C4C23×D3112yes—
244C32×C4C23×D316yes—
245C3⋊Dic3C23×D314yes—
246C3⋊Dic3C23×D316yes—
247C3×Dic3—34no—
248C3×Dic3—34no—
249C3×Dic3—312no—
250C3×Dic3—66no—
251C3×Dic3—64no—
252C2×#41—92no—
253C2×#41—276no—
254C2×#41—2716no—
255C22×Q8—91no—
256C22×D4—273no—
257C23×C4—273no—
258C33C22×#4111yescharacteristic, complemented, Sylow
259C23×C3—91no—
260C23×C3—96no—
261C22×D3—32no—
262C22×D3—36no—
263C22×D3—36no—
264C22×D3—61no—
265C22×D3—66no—
266C22×D3—912no—
267C22×D3—924no—
268C22×D3—1824no—
269C22×D3—1812no—
270C3×Q8—24no—
271C3×Q8—28no—
272C3×Q8—44no—
273C3×Q8—912no—
274C3×Q8C2×D3218yescomplemented
275C3×Q8C2×D3214yescomplemented
276C3×D4—924no—
277C2×C3×C4—23no—
278C2×C3×C4—26no—
279C2×C3×C4—43no—
280C2×C3×C4—92no—
281C2×C3×C4—93no—
282C2×C3×C4—912no—
283C2×C3×C4—924no—
284C2×C3×C4C2×D3216yes—
285C2×C3×C4C2×D3213yes—
286#67—948no—
287C2×Dic3—36no—
288C2×Dic3—32no—
289C2×Dic3—32no—
290C2×Dic3—63no—
291C2×Dic3—62no—
292C2×Dic3—94no—
293C2×Dic3—924no—
294D12—324no—
295D12—324no—
296D12—624no—
297D12—924no—
298C4×D3—324no—
299C4×D3—324no—
300C4×D3—324no—
301C4×D3—612no—
302C4×D3—624no—
303C4×D3—98no—
304C4×D3—924no—
305C4×D3—924no—
306Dic6—324no—
307Dic6—612no—
308C2×C32—24no—
309C2×C32—22no—
310C2×C32—21no—
311C2×C32—22no—
312C2×C32—42no—
313C2×C32—41no—
314C2×C32C2×#23912yescomplemented
315C2×C32D3×#4114yescomplemented
316C2×C32C24×D311yescharacteristic
317C2×C32C24×D312yes—
318C3⋊D3—212no—
319C3⋊D3C2×#23914yescomplemented
320C3×D3—38no—
321C3×D3—64no—
322C3×D3—612no—
323C3×D3—612no—
324C3×D3—1212no—
325C24—273no—
326#41—916no—
327#41—2748no—
328C2×Q8—98no—
329C2×Q8—93no—
330C2×Q8#111311yescharacteristic, complemented
331C2×D4—96no—
332C2×D4—2724no—
333C2×D4—276no—
334C22×C4—93no—
335C22×C4—96no—
336C22×C4—2724no—
337C22×C4—271no—
338C22×C4—276no—
339C22×C3—21no—
340C22×C3—22no—
341C22×C3—41no—
342C22×C3—94no—
343C22×C3—92no—
344C22×C3—912no—
345C22×C3—1812no—
346C22×C3C22×D3212yes—
347C22×C3C22×D3211yescharacteristic
348C2×D3—38no—
349C2×D3—34no—
350C2×D3—312no—
351C2×D3—312no—
352C2×D3—62no—
353C2×D3—64no—
354C2×D3—612no—
355C2×D3—612no—
356C2×D3—612no—
357C2×D3—1212no—
358C2×D3—1848no—
359C2×D3—1824no—
360C3×C4—26no—
361C3×C4—212no—
362C3×C4—46no—
363C3×C4—94no—
364C3×C4—96no—
365C3×C4C22×D32112yes—
366C3×C4C22×D3216yes—
367Dic3—312no—
368Dic3—34no—
369Dic3—34no—
370Dic3—66no—
371Dic3—64no—
372C32—22no—
373C32—21no—
374C32—41no—
375C32C22×#23911yescharacteristic, complemented
376C32C2×D3×#4112yescomplemented
377C23—91no—
378C23—96no—
379C23—2712no—
380C23—5412no—
381Q8—912no—
382Q8C2×#111314yescomplemented
383D4—924no—
384D4—2724no—
385C2×C4—92no—
386C2×C4—93no—
387C2×C4—912no—
388C2×C4—924no—
389C2×C4—274no—
390C2×C4—2724no—
391C2×C4C2×#111313yes—
392C2×C3—22no—
393C2×C3—24no—
394C2×C3—21no—
395C2×C3—22no—
396C2×C3—42no—
397C2×C3—41no—
398C2×C3—94no—
399C2×C3—1812no—
400C2×C3D3×#23914yescomplemented
401C2×C3#6431912yescomplemented
402C2×C3C23×D3212yes—
403C2×C3C23×D3211yescharacteristic
404D3—38no—
405D3—64no—
406D3—612no—
407D3—612no—
408D3—1212no—
409C22—94no—
410C22—92no—
411C22—912no—
412C22—1812no—
413C22—5424no—
414C22C22×#111311yescharacteristic
415C4—94no—
416C4—96no—
417C4C22×#111316yes—
418C3—21no—
419C3—22no—
420C3—41no—
421C3C2×D3×#23912yescomplemented
422C3C2×#6431911yescharacteristic, complemented
423C2—94no—
424C2—1812no—
425C2#1172462612yesdirect product factor
426C2C23×#111311yescharacteristic
subgroupG/H
centerC22C22×#1113
commutator, cosocleC2×C33C25
FrattiniC2C23×#1113
FittingC2×C33×Q8C22
socleC22×C33C24
hypercenterC2×Q8#1113
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×C3×#3832C2semidirect
2. C2×C3×Q8×C3⋊D3C2semidirect
3. #11724626C2direct (N × H)
4. C2×C4⋊#1113C2semidirect
5. C2×C4×#1113C2semidirect
6. C2×#89041C2semidirect
10. C2×C33×Q8C22semidirect
11. C3×#3832C22semidirect
12. C3×Q8×C3⋊D3C22semidirect
13. C2×C3×C4⋊(C3⋊D3)C22semidirect
15. C2×C3×C4×C3⋊D3C22semidirect
16. C2×C3×#634C22semidirect
17. C2×#6295C22semidirect
19. C2×#6294C22semidirect
20. C4⋊#1113C22semidirect
21. C4×#1113C22semidirect
22. #89041C22semidirect
23. C2×#3832D3semidirect
24. C2×Q8×C3⋊D3D3semidirect
43. C33×Q8C23semidirect
47. C3×C4⋊(C3⋊D3)C23semidirect
49. C3×C4×C3⋊D3C23semidirect
50. C3×#634C23semidirect
51. #6295C23semidirect
53. #6294C23semidirect
61. C2×C32×Q8C2×D3semidirect
62. C2×C32×Q8C2×D3semidirect
63. #3832C2×D3semidirect
64. Q8×C3⋊D3C2×D3semidirect
65. C2×C4⋊(C3⋊D3)C2×D3semidirect
67. C2×C4×C3⋊D3C2×D3semidirect
68. C2×#634C2×D3semidirect
100. C2×C3×C3⋊D3#41semidirect
137. C32×Q8C22×D3semidirect
138. C32×Q8C22×D3semidirect
146. C4⋊(C3⋊D3)C22×D3semidirect
148. C4×C3⋊D3C22×D3semidirect
149. #634C22×D3semidirect
173. C2×C33C2×#41semidirect
176. C3×C3⋊D3C2×#41semidirect
185. C2×C3×Q8D32semidirect
186. C2×C3×Q8D32semidirect
225. C2×C3⋊D3#239semidirect
258. C33C22×#41semidirect
274. C3×Q8C2×D32semidirect
275. C3×Q8C2×D32semidirect
314. C2×C32C2×#239semidirect
315. C2×C32D3×#41semidirect
319. C3⋊D3C2×#239semidirect
330. C2×Q8#1113semidirect
375. C32C22×#239semidirect
376. C32C2×D3×#41semidirect
382. Q8C2×#1113semidirect
400. C2×C3D3×#239semidirect
401. C2×C3#64319semidirect
421. C3C2×D3×#239semidirect
422. C3C2×#64319semidirect
425. C2#11724626direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38322
C2×C3×Q8×C3⋊D32
#117246262
C2×C4⋊#11132
C2×C4×#11132
C2×#890412
C2×#643193
C2×D3×#2393