Finite Group Databasebeta

← C2×#11724597 #13181629

Subgroups — C2×#11724597 #13181629

Order 1728.

6044 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×#3832C211yesmaximal, characteristic, complemented
3C2×C32×#239C211yesmaximal, characteristic, complemented
4#11724597C2116yesmaximal, direct product factor
5C2×#89026C213yesmaximal, complemented
6C2×C3⋊D3×D12C213yesmaximal, complemented
7C2×D3×C4⋊(C3⋊D3)C213yesmaximal, complemented
8C2×#89020C213yesmaximal, complemented
9C2×#64366—31nomaximal
10C2×#64320—34nomaximal
11C22×D3×C3⋊D3C2216yescomplemented
12C2×C33×Q8C2211yescharacteristic, complemented
13#89080C2218yescomplemented
14C2×C4⋊(C3⋊(C3⋊D3))C2213yescomplemented
15C2×C4×C3⋊(C3⋊D3)C2213yescomplemented
16C3×#3832C2218yescomplemented
17C2×C3×C4⋊(C3⋊D3)C2213yescomplemented
18C2×C3×C4×C3⋊D3C2213yescomplemented
19C32×#239C2218yescomplemented
20C2×C32×D12C2213yescomplemented
21C2×C32×C4×D3C2213yescomplemented
22C2×#6293C2211yescharacteristic, complemented
23C2×#6292C2213yescomplemented
24C2×#6291C2213yescomplemented
25C2×#6290C2213yescomplemented
26#89026C22124yescomplemented
27C3⋊D3×D12C22124yescomplemented
28D3×C4⋊(C3⋊D3)C22124yescomplemented
29#89020C22124yescomplemented
30C2×C32×#41—31no—
31#64366—316no—
32C2×#3832—31no—
33C2×#3832—34no—
34C2×#3832—64no—
35C2×#3832D311yescharacteristic, complemented
36C2×D4×C3⋊D3—33no—
37C2×D4×C3⋊D3—33no—
38C2×#3828—33no—
39C22×C4⋊(C3⋊D3)—33no—
40C2×C3×#239—34no—
41C2×C3×#239D314yescomplemented
42#64320—364no—
43C2×#3802—312no—
44C2×D3×D12—312no—
45C2×D3×D12—312no—
46C2×#3798—312no—
47C22×C3⋊(C3⋊D3)C2313yes—
48C22×C3×C3⋊D3C2313yes—
49C22×C32×D3C2313yes—
50C2×D3×C3⋊D3—412no—
51C2×D3×C3⋊D3C23124yescomplemented
52C33×Q8C2314yescomplemented
53C2×C33×C4C2313yes—
54C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
55C4⋊(C3⋊(C3⋊D3))C23112yescomplemented
56C4×C3⋊(C3⋊D3)C23112yescomplemented
57C2×C3×C3⋊Dic3C2311yescharacteristic
58C3×C4⋊(C3⋊D3)C23112yescomplemented
59C3×C4×C3⋊D3C23112yescomplemented
60C2×C32×Dic3C2311yescharacteristic
61C32×D12C23112yescomplemented
62C32×C4×D3C23112yescomplemented
63#6293C2314yescomplemented
64#6292C23112yescomplemented
65#6291C23112yescomplemented
66#6290C23112yescomplemented
67C2×#1009—94no—
68C2×#1009—91no—
69C23×C3⋊D3—36no—
70C22×D32—324no—
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—33no—
78C2×#638—33no—
79#3832—38no—
80#3832—332no—
81#3832—632no—
82#3832C2×D318yescomplemented
83D4×C3⋊D3—324no—
84D4×C3⋊D3—324no—
85#3828—324no—
86C2×C4⋊(C3⋊D3)—33no—
87C2×C4⋊(C3⋊D3)—33no—
88C2×C4⋊(C3⋊D3)—312no—
89C2×C4⋊(C3⋊D3)—324no—
90C2×C4⋊(C3⋊D3)—612no—
91C2×C4⋊(C3⋊D3)C2×D313yescomplemented
92C2×C4×C3⋊D3—33no—
93C2×C4×C3⋊D3—312no—
94C2×C4×C3⋊D3—612no—
95C2×C4×C3⋊D3C2×D313yescomplemented
96C2×#634—31no—
97C3×#239—332no—
98C3×#239C2×D3132yescomplemented
99C2×C3×D12—312no—
100C2×C3×D12C2×D3112yescomplemented
101C2×C3×C4×D3—312no—
102C2×C3×C4×D3C2×D3112yescomplemented
103C2×#627—34no—
104C2×#626—312no—
105C2×#626—312no—
106C2×#625—312no—
107#3802—396no—
108D3×D12—396no—
109D3×D12—396no—
110#3798—396no—
111C22×C33C2411yescharacteristic
112C2×C3⋊(C3⋊D3)—26no—
113C2×C3⋊(C3⋊D3)C2416yes—
114C2×C3×C3⋊D3—26no—
115C2×C3×C3⋊D3C2416yes—
116C2×C32×D3—26no—
117C2×C32×D3C2416yes—
118D3×C3⋊D3—424no—
119C33×C4C2416yes—
120C3⋊(C3⋊Dic3)C2412yes—
121C3×C3⋊Dic3C2412yes—
122C32×Dic3C2412yes—
123C2×C3×#41—34no—
124C2×C3×#41—91no—
125#1009—964no—
126#1009—916no—
127C2×#239—34no—
128C2×#239—94no—
129C2×#239—91no—
130C2×#239—184no—
131C2×#239C3⋊D311yescharacteristic, complemented
132C2×D3×D4—912no—
133C2×D3×D4—912no—
134C2×D3×D4—93no—
135C2×D3×D4—93no—
136C2×#235—912no—
137C2×#235—93no—
138C22×D12—912no—
139C22×D12—93no—
140C23×C32—33no—
141C22×C3⋊D3—33no—
142C22×C3⋊D3—324no—
143C22×C3⋊D3—312no—
144C22×C3⋊D3—612no—
145C22×C3⋊D3—1212no—
146C22×C3⋊D3C22×D313yes—
147C22×C3×D3—312no—
148C22×C3×D3C22×D3112yes—
149C2×D32—396no—
150C2×D32—1248no—
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—312no—
161#638—312no—
162C2×C3⋊Dic3—31no—
163C2×C3⋊Dic3—34no—
164C2×C3⋊Dic3—64no—
165C2×C3⋊Dic3C22×D311yescharacteristic
166C4⋊(C3⋊D3)—312no—
167C4⋊(C3⋊D3)—312no—
168C4⋊(C3⋊D3)—348no—
169C4⋊(C3⋊D3)—648no—
170C4⋊(C3⋊D3)C22×D3112yescomplemented
171C4×C3⋊D3—312no—
172C4×C3⋊D3—348no—
173C4×C3⋊D3—648no—
174C4×C3⋊D3C22×D3112yescomplemented
175#634—34no—
176C2×C3×Dic3—34no—
177C2×C3×Dic3C22×D314yes—
178C3×D12—348no—
179C3×D12C22×D3148yescomplemented
180C3×C4×D3—348no—
181C3×C4×D3C22×D3148yescomplemented
182#627—316no—
183#626—348no—
184#626—348no—
185#625—348no—
186C2×H(5,𝔽2)—271noSylow
187C2×C33H(5,𝔽2)12yescomplemented
188C2×C33C2511yescharacteristic
189C3⋊(C3⋊D3)—26no—
190C3×C3⋊D3—26no—
191C32×D3—26no—
192C23×D3—924no—
193C23×D3—96no—
194C3×#41—332no—
195C3×#41—98no—
196C2×C3×Q8—24no—
197C2×C3×Q8D3214yescomplemented
198C2×C3×Q8C2×C3⋊D311yescharacteristic, complemented
199C2×C3×D4—312no—
200C2×C3×D4—93no—
201C22×C3×C4—312no—
202C22×C3×C4—93no—
203C2×#67—912no—
204C2×#67—912no—
205C2×#67—93no—
206C2×#67—93no—
207#239—332no—
208#239—932no—
209#239—98no—
210#239—1832no—
211#239C2×C3⋊D318yescomplemented
212D3×D4—996no—
213D3×D4—996no—
214D3×D4—924no—
215D3×D4—924no—
216#235—996no—
217#235—924no—
218C2×D12—312no—
219C2×D12—996no—
220C2×D12—912no—
221C2×D12—93no—
222C2×D12—912no—
223C2×D12—93no—
224C2×D12—924no—
225C2×D12—1812no—
226C2×D12C2×C3⋊D313yescomplemented
227C2×C4×D3—312no—
228C2×C4×D3—912no—
229C2×C4×D3—93no—
230C2×C4×D3—1812no—
231C2×C4×D3C2×C3⋊D313yescomplemented
232C2×Dic6—94no—
233C2×Dic6—91no—
234C22×C32—24no—
235C22×C32—36no—
236C22×C32—66no—
237C22×C32C23×D314yes—
238C22×C32C23×D311yescharacteristic
239C2×C3⋊D3—26no—
240C2×C3⋊D3—36no—
241C2×C3⋊D3—324no—
242C2×C3⋊D3—624no—
243C2×C3⋊D3—66no—
244C2×C3⋊D3—624no—
245C2×C3⋊D3—1224no—
246C2×C3⋊D3—1224no—
247C2×C3⋊D3C23×D316yes—
248C2×C3×D3—224no—
249C2×C3×D3—324no—
250C2×C3×D3—624no—
251C2×C3×D3C23×D3124yes—
252D32—1296no—
253C32×C4—224no—
254C32×C4—32no—
255C32×C4C23×D3124yes—
256C32×C4C23×D316yes—
257C3⋊Dic3—32no—
258C3⋊Dic3—38no—
259C3⋊Dic3—68no—
260C3⋊Dic3C23×D312yes—
261C3×Dic3—38no—
262C3×Dic3C23×D318yes—
263H(5,𝔽2)—2716no—
264C2×#41—31no—
265C2×#41—91no—
266C2×#41—273no—
267C2×#41—271no—
268C22×D4—273no—
269C22×D4—273no—
270C22×D4—273no—
271C33C2×H(5,𝔽2)11yescharacteristic, complemented, Sylow
272C23×C3—312no—
273C23×C3—93no—
274C22×D3—312no—
275C22×D3—996no—
276C22×D3—912no—
277C22×D3—93no—
278C22×D3—924no—
279C22×D3—1812no—
280C22×D3—3612no—
281C22×D3—3648no—
282C22×D3C22×C3⋊D313yes—
283C3×Q8—216no—
284C3×Q8C2×D32116yescomplemented
285C3×Q8C22×C3⋊D314yescomplemented
286C3×D4—348no—
287C3×D4—912no—
288C2×C3×C4—212no—
289C2×C3×C4—34no—
290C2×C3×C4—348no—
291C2×C3×C4—91no—
292C2×C3×C4—912no—
293C2×C3×C4C2×D32112yes—
294C2×C3×C4C22×C3⋊D313yes—
295#67—948no—
296#67—948no—
297#67—912no—
298#67—912no—
299C2×Dic3—34no—
300C2×Dic3—94no—
301C2×Dic3—91no—
302C2×Dic3—184no—
303C2×Dic3C22×C3⋊D311yescharacteristic
304D12—348no—
305D12—948no—
306D12—912no—
307D12—948no—
308D12—912no—
309D12—1848no—
310D12C22×C3⋊D3112yescomplemented
311C4×D3—348no—
312C4×D3—948no—
313C4×D3—912no—
314C4×D3—1848no—
315C4×D3C22×C3⋊D3112yescomplemented
316Dic6—916no—
317Dic6—94no—
318C2×C32—28no—
319C2×C32—24no—
320C2×C32—66no—
321C2×C32#100918yescomplemented
322C2×C32#100912yescomplemented
323C2×C32C24×D314yes—
324C2×C32C24×D311yescharacteristic
325C3⋊D3—26no—
326C3⋊D3—66no—
327C3⋊D3—624no—
328C3⋊D3—1224no—
329C3×D3—224no—
330C3×D3—624no—
331C24—276no—
332#41—38no—
333#41—98no—
334#41—2724no—
335#41—278no—
336C2×Q8—271no—
337C2×Q8D3×C3⋊D311yescharacteristic, complemented
338C2×D4—33no—
339C2×D4—93no—
340C2×D4—2724no—
341C2×D4—273no—
342C2×D4—2724no—
343C2×D4—273no—
344C2×D4—2724no—
345C2×D4—273no—
346C2×D4—273no—
347C22×C4—33no—
348C22×C4—93no—
349C22×C4—273no—
350C22×C3—24no—
351C22×C3—324no—
352C22×C3—624no—
353C22×C3—96no—
354C22×C3—186no—
355C22×C3C22×D3214yes—
356C22×C3C23×C3⋊D311yescharacteristic
357C2×D3—26no—
358C2×D3—324no—
359C2×D3—624no—
360C2×D3—924no—
361C2×D3—96no—
362C2×D3—1824no—
363C2×D3—1824no—
364C2×D3—186no—
365C2×D3—3624no—
366C2×D3—3624no—
367C2×D3—3696no—
368C2×D3C23×C3⋊D316yes—
369C3×C4—224no—
370C3×C4—38no—
371C3×C4—92no—
372C3×C4C22×D32124yes—
373C3×C4C23×C3⋊D316yes—
374Dic3—38no—
375Dic3—98no—
376Dic3—92no—
377Dic3—188no—
378Dic3C23×C3⋊D312yes—
379C32—24no—
380C32C2×#100914yescomplemented
381C32C2×#100911yescharacteristic, complemented
382C23—33no—
383C23—93no—
384C23—2724no—
385C23—273no—
386C23—10812no—
387Q8—274no—
388Q8C2×D3×C3⋊D314yescomplemented
389D4—312no—
390D4—912no—
391D4—2712no—
392D4—2712no—
393D4—2712no—
394D4—2712no—
395C2×C4—31no—
396C2×C4—312no—
397C2×C4—91no—
398C2×C4—912no—
399C2×C4—271no—
400C2×C4—2712no—
401C2×C4C2×D3×C3⋊D313yes—
402C2×C3—28no—
403C2×C3—24no—
404C2×C3—624no—
405C2×C3—186no—
406C2×C3#6432018yescomplemented
407C2×C3#6436612yescomplemented
408C2×C3C23×D3214yes—
409C2×C3C24×C3⋊D311yescharacteristic
410D3—26no—
411D3—624no—
412D3—1824no—
413D3—186no—
414D3—3624no—
415C22—36no—
416C22—66no—
417C22—96no—
418C22—186no—
419C22—276no—
420C22—546no—
421C22—10824no—
422C22C22×D3×C3⋊D311yescharacteristic
423C4—32no—
424C4—92no—
425C4—272no—
426C4C22×D3×C3⋊D316yes—
427C3—24no—
428C3C2×#6432014yescomplemented
429C3C2×#6436611yescharacteristic, complemented
430C2—66no—
431C2—186no—
432C2—546no—
433C2#1172459712yesdirect 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
hypocenterC33C2×H(5,𝔽2)
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×#3832C2semidirect
3. C2×C32×#239C2semidirect
4. #11724597C2direct (N × H)
5. C2×#89026C2semidirect
6. C2×C3⋊D3×D12C2semidirect
7. C2×D3×C4⋊(C3⋊D3)C2semidirect
8. C2×#89020C2semidirect
11. C22×D3×C3⋊D3C22semidirect
12. C2×C33×Q8C22semidirect
13. #89080C22semidirect
14. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
15. C2×C4×C3⋊(C3⋊D3)C22semidirect
16. C3×#3832C22semidirect
17. C2×C3×C4⋊(C3⋊D3)C22semidirect
18. C2×C3×C4×C3⋊D3C22semidirect
19. C32×#239C22semidirect
20. C2×C32×D12C22semidirect
21. C2×C32×C4×D3C22semidirect
22. C2×#6293C22semidirect
23. C2×#6292C22semidirect
24. C2×#6291C22semidirect
25. C2×#6290C22semidirect
26. #89026C22semidirect
27. C3⋊D3×D12C22semidirect
28. D3×C4⋊(C3⋊D3)C22semidirect
29. #89020C22semidirect
35. C2×#3832D3semidirect
41. C2×C3×#239D3semidirect
51. C2×D3×C3⋊D3C23semidirect
52. C33×Q8C23semidirect
55. C4⋊(C3⋊(C3⋊D3))C23semidirect
56. C4×C3⋊(C3⋊D3)C23semidirect
58. C3×C4⋊(C3⋊D3)C23semidirect
59. C3×C4×C3⋊D3C23semidirect
61. C32×D12C23semidirect
62. C32×C4×D3C23semidirect
63. #6293C23semidirect
64. #6292C23semidirect
65. #6291C23semidirect
66. #6290C23semidirect
73. C2×C32×Q8C2×D3semidirect
74. C2×C32×Q8C2×D3semidirect
82. #3832C2×D3semidirect
91. C2×C4⋊(C3⋊D3)C2×D3semidirect
95. C2×C4×C3⋊D3C2×D3semidirect
98. C3×#239C2×D3semidirect
100. C2×C3×D12C2×D3semidirect
102. C2×C3×C4×D3C2×D3semidirect
131. C2×#239C3⋊D3semidirect
152. C32×Q8C22×D3semidirect
153. C32×Q8C22×D3semidirect
170. C4⋊(C3⋊D3)C22×D3semidirect
174. C4×C3⋊D3C22×D3semidirect
179. C3×D12C22×D3semidirect
181. C3×C4×D3C22×D3semidirect
187. C2×C33H(5,𝔽2)semidirect
197. C2×C3×Q8D32semidirect
198. C2×C3×Q8C2×C3⋊D3semidirect
211. #239C2×C3⋊D3semidirect
226. C2×D12C2×C3⋊D3semidirect
231. C2×C4×D3C2×C3⋊D3semidirect
271. C33C2×H(5,𝔽2)semidirect
284. C3×Q8C2×D32semidirect
285. C3×Q8C22×C3⋊D3semidirect
310. D12C22×C3⋊D3semidirect
315. C4×D3C22×C3⋊D3semidirect
321. C2×C32#1009semidirect
322. C2×C32#1009semidirect
337. C2×Q8D3×C3⋊D3semidirect
380. C32C2×#1009semidirect
381. C32C2×#1009semidirect
388. Q8C2×D3×C3⋊D3semidirect
406. C2×C3#64320semidirect
407. C2×C3#64366semidirect
428. C3C2×#64320semidirect
429. C3C2×#64366semidirect
433. C2#11724597direct (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×#38322
C2×C32×#2392
#117245972
C2×#890262
C2×C3⋊D3×D122
C2×D3×C4⋊(C3⋊D3)2
C2×#890202
C2×#643663
C2×#643203