Finite Group Databasebeta

← C3⋊D3×#537 #11901119

Subgroups — C3⋊D3×#537 #11901119

Order 1152.

4920 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×C3⋊D3C212yesmaximal, complemented
2C32×#537C211yesmaximal, characteristic, complemented
3#10596304C212yesmaximal, complemented
4#10596100C214yesmaximal, complemented
5C3⋊D3×C22⋊Q8C214yesmaximal, complemented
6#10596079C214yesmaximal, complemented
7C3⋊D3×#121C214yesmaximal, complemented
8C3⋊D3×#121C214yesmaximal, complemented
9#10596068C214yesmaximal, complemented
10#10595981C211yesmaximal, characteristic, complemented
11C2×C4⋊C4×C3⋊D3C211yesmaximal, characteristic, complemented
12D3×#537—34nomaximal
13C2×C32×#41C2212yescomplemented
14#41×C3⋊D3—216no—
15C2×#3832C2212yescomplemented
16C2×Q8×C3⋊D3C2212yes—
17C2×#3830C2214yescomplemented
18C2×#3830C2212yescomplemented
19C2×D4×C3⋊D3C2212yes—
20C2×D4×C3⋊D3C2214yes—
21C2×D4×C3⋊D3C2214yes—
22C2×#3828C2214yescomplemented
23C2×#3828C2212yescomplemented
24C22×C4×C3⋊D3C2214yes—
25C22×C4×C3⋊D3C2212yes—
26C22×C4×C3⋊D3C2211yescharacteristic
27C32×C22⋊Q8C2214yescomplemented
28C32×#121C2214yescomplemented
29C32×#121C2214yescomplemented
30C2×C32×C4⋊C4C2211yescharacteristic, complemented
31#64155C2214yescomplemented
32#64148C2218yescomplemented
33#64148C2218yescomplemented
34#64147C2214yescomplemented
35#64147C2214yescomplemented
36#64139C2214yescomplemented
37#64139C2214yescomplemented
38C2×C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
39#64133C2214yescomplemented
40C2×#3750C2212yescomplemented
41#64106C2214yescomplemented
42#64105C2218yescomplemented
43#64104C2214yescomplemented
44#64104C2214yescomplemented
45C4⋊C4×C3⋊D3C2214yes—
46C4⋊C4×C3⋊D3C2214yes—
47C4⋊C4×C3⋊D3C2214yescomplemented
48#64093C2218yescomplemented
49#64093C2218yescomplemented
50C22⋊C4×C3⋊D3C2218yes—
51#64087C2218yescomplemented
52C2×D3×#41—38no—
53C3×#537D314yescomplemented
54#5833—38no—
55#5629—316no—
56D3×C22⋊Q8—316no—
57#5608—316no—
58D3×#121—316no—
59D3×#121—316no—
60#5597—316no—
61#5510—34no—
62C2×D3×C4⋊C4—34no—
63C23×C3⋊D3C2314yes—
64C23×C3⋊D3C2311yescharacteristic
65C32×#41—28no—
66C2×C32×Q8C2312yes—
67C2×C32×D4C2312yes—
68C2×C32×D4C2314yes—
69C2×C32×D4C2314yes—
70C22×C32×C4C2314yes—
71C22×C32×C4C2312yes—
72C22×C32×C4C2311yescharacteristic
73C2×#638C2314yes—
74C2×#638C2314yes—
75C2×#638C2314yes—
76C2×#638C2318yes—
77C22×C3⋊Dic3C2314yes—
78C22×C3⋊Dic3C2312yes—
79C22×C3⋊Dic3C2311yescharacteristic
80#3832—28no—
81Q8×C3⋊D3—28no—
82#3830—216no—
83#3830—28no—
84D4×C3⋊D3—216no—
85D4×C3⋊D3—28no—
86D4×C3⋊D3—48no—
87#3828—28no—
88#3828—216no—
89C2×C4⋊(C3⋊D3)C2314yes—
90C2×C4⋊(C3⋊D3)C2312yes—
91C2×C4⋊(C3⋊D3)C2314yes—
92C2×C4×C3⋊D3—216no—
93C2×C4×C3⋊D3—44no—
94C2×C4×C3⋊D3D418yescomplemented
95C2×C4×C3⋊D3C2314yes—
96C2×C4×C3⋊D3C2314yes—
97C2×C4×C3⋊D3C2314yes—
98C2×C4×C3⋊D3C2314yes—
99C2×C4×C3⋊D3C2314yes—
100C2×C4×C3⋊D3C2312yes—
101C2×C4×C3⋊D3C2312yes—
102C2×#634C2314yes—
103C2×#634C2312yes—
104C32×C4⋊C4C2314yes—
105C32×C4⋊C4C2314yes—
106C32×C22⋊C4C2318yes—
107#3757C2318yes—
108#3752C2318yes—
109#3752C2318yes—
110C4⋊(C3⋊Dic3)C2314yes—
111C4⋊(C3⋊Dic3)C2314yes—
112#3750C2318yes—
113#3750C2318yes—
114C2×#537—91noSylow
115C2×C3×#41C2×D318yescomplemented
116D3×#41—664no—
117C2×#239—38no—
118C2×D3×Q8—38no—
119C2×#237—316no—
120C2×#237—38no—
121C2×D3×D4—38no—
122C2×D3×D4—316no—
123C2×D3×D4—316no—
124C2×#235—38no—
125C2×#235—316no—
126C22×C4×D3—34no—
127C22×C4×D3—316no—
128C22×C4×D3—38no—
129C3×C22⋊Q8C2×D3116yescomplemented
130C3×#121C2×D3116yescomplemented
131C3×#121C2×D3116yescomplemented
132C2×C3×C4⋊C4C2×D314yescomplemented
133#946—316no—
134#939—332no—
135#939—332no—
136#938—316no—
137#938—316no—
138#930—316no—
139#930—316no—
140C2×C4⋊Dic3—34no—
141#924—316no—
142C2×Dic3⋊C4—38no—
143#897—316no—
144#896—332no—
145#895—316no—
146#895—316no—
147D3×C4⋊C4—316no—
148D3×C4⋊C4—316no—
149D3×C4⋊C4—316no—
150#884—332no—
151#884—332no—
152D3×C22⋊C4—332no—
153#878—332no—
154C23×C32C2414yes—
155C23×C32C2411yescharacteristic
156C22×C3⋊D3—22no—
157C22×C3⋊D3—22no—
158C22×C3⋊D3—28no—
159C22×C3⋊D3—28no—
160C22×C3⋊D3—28no—
161C22×C3⋊D3C2×D414yes—
162C22×C3⋊D3C2414yes—
163C22×C3⋊D3C2411yescharacteristic
164C22×C3⋊D3C2411yescharacteristic
165C32×Q8—24no—
166C32×D4—28no—
167C32×D4—24no—
168C32×D4—44no—
169C2×C32×C4—28no—
170C2×C32×C4—42no—
171C2×C32×C4C2×D414yescomplemented
172C2×C32×C4C2414yes—
173C2×C32×C4C2414yes—
174C2×C32×C4C2412yes—
175#638—28no—
176#638—28no—
177#638—28no—
178#638—48no—
179C2×C3⋊Dic3—28no—
180C2×C3⋊Dic3—42no—
181C2×C3⋊Dic3C2×D414yescomplemented
182C2×C3⋊Dic3C2414yes—
183C2×C3⋊Dic3C2414yes—
184C2×C3⋊Dic3C2412yes—
185C4⋊(C3⋊D3)—28no—
186C4⋊(C3⋊D3)—24no—
187C4⋊(C3⋊D3)—44no—
188C4×C3⋊D3—28no—
189C4×C3⋊D3—28no—
190C4×C3⋊D3—28no—
191C4×C3⋊D3—28no—
192C4×C3⋊D3—44no—
193C4×C3⋊D3C2×D418yes—
194C4×C3⋊D3C2×D418yescomplemented
195#634—24no—
196#634—28no—
197C22×#41—92no—
198#537—92no—
199#537—94no—
200#537—94no—
201#537—94no—
202#537—91no—
203#537C3⋊D311yescharacteristic, direct product factor
204C2×C22⋊Q8—94no—
205C2×#121—94no—
206C2×#121—94no—
207C22×C4⋊C4—91no—
208C23×D3—34no—
209C23×D3—316no—
210C3×#41—232no—
211C2×C3×Q8C22×D318yes—
212C2×C3×D4C22×D318yes—
213C2×C3×D4C22×D3116yes—
214C2×C3×D4C22×D3116yes—
215C22×C3×C4C22×D3116yes—
216C22×C3×C4C22×D318yes—
217C22×C3×C4C22×D314yes—
218C2×#67—316no—
219C2×#67—316no—
220C2×#67—316no—
221C2×#67—332no—
222C22×Dic3—34no—
223C22×Dic3—316no—
224C22×Dic3—38no—
225#239—632no—
226D3×Q8—632no—
227#237—632no—
228#237—664no—
229D3×D4—632no—
230D3×D4—664no—
231D3×D4—1232no—
232#235—664no—
233#235—632no—
234C2×D12—316no—
235C2×D12—38no—
236C2×D12—316no—
237C2×C4×D3—332no—
238C2×C4×D3—38no—
239C2×C4×D3—38no—
240C2×C4×D3—316no—
241C2×C4×D3—316no—
242C2×C4×D3—316no—
243C2×C4×D3—316no—
244C2×C4×D3—316no—
245C2×C4×D3—664no—
246C2×C4×D3—1216no—
247C2×Dic6—316no—
248C2×Dic6—38no—
249C3×C4⋊C4C22×D3116yes—
250C3×C4⋊C4C22×D3116yes—
251C3×C22⋊C4C22×D3132yes—
252#217—332no—
253(C2×D3)⋊C4—332no—
254(C2×D3)⋊C4—332no—
255C4⋊Dic3—316no—
256C4⋊Dic3—316no—
257Dic3⋊C4—332no—
258Dic3⋊C4—332no—
259C22×C32—21no—
260C22×C32—21no—
261C22×C32—24no—
262C22×C32—24no—
263C22×C32—24no—
264C22×C32C22×D412yes—
265C22×C32C2511yescharacteristic
266C2×C3⋊D3—21no—
267C2×C3⋊D3—21no—
268C2×C3⋊D3—28no—
269C2×C3⋊D3—24no—
270C2×C3⋊D3—24no—
271C2×C3⋊D3—24no—
272C2×C3⋊D3—416no—
273C2×C3⋊D3C22×D412yes—
274C2×C3⋊D3C22×D412yes—
275C2×C3⋊D3H(5,𝔽2)12yes—
276C2×C3⋊D32−1+412yes—
277C32×C4—24no—
278C32×C4—24no—
279C32×C4C22×D414yes—
280C3⋊Dic3—24no—
281C3⋊Dic3—24no—
282C3⋊Dic3C22×D414yes—
283C2×#41—92no—
284C2×#41—94no—
285C2×#41—92no—
286C2×#41—94no—
287C2×#41—92no—
288C2×#41—1816no—
289C2×#41C2×C3⋊D312yescomplemented
290C22×Q8—92no—
291C22×D4—92no—
292C22×D4—94no—
293C22×D4—94no—
294C23×C4—91no—
295C23×C4—94no—
296C23×C4—92no—
297C22⋊Q8—94no—
298C22⋊Q8—94no—
299C22⋊Q8—98no—
300C22⋊Q8—94no—
301C22⋊Q8—98no—
302C22⋊Q8C2×C3⋊D314yescomplemented
303#121—94no—
304#121—94no—
305#121—98no—
306#121—98no—
307#121—94no—
308#121—94no—
309#121—94no—
310#121—94no—
311#121—98no—
312#121—98no—
313#121C2×C3⋊D314yescomplemented
314#121C2×C3⋊D314yescomplemented
315C2×C4⋊C4—92no—
316C2×C4⋊C4—91no—
317C2×C4⋊C4—94no—
318C2×C4⋊C4—94no—
319C2×C4⋊C4—94no—
320C2×C4⋊C4C2×C3⋊D311yescharacteristic, complemented
321C2×C22⋊C4—98no—
322C23×C3C23×D3116yes—
323C23×C3C23×D314yes—
324C22×D3—34no—
325C22×D3—316no—
326C22×D3—34no—
327C22×D3—316no—
328C22×D3—68no—
329C22×D3—68no—
330C22×D3—632no—
331C22×D3—632no—
332C22×D3—632no—
333C3×Q8—216no—
334C3×D4—232no—
335C3×D4—216no—
336C3×D4—416no—
337C2×C3×C4—232no—
338C2×C3×C4—48no—
339C2×C3×C4D3×D4116yescomplemented
340C2×C3×C4C23×D3116yes—
341C2×C3×C4C23×D3116yes—
342C2×C3×C4C23×D318yes—
343#67—632no—
344#67—632no—
345#67—632no—
346#67—1232no—
347C2×Dic3—316no—
348C2×Dic3—38no—
349C2×Dic3—316no—
350C2×Dic3—316no—
351C2×Dic3—632no—
352C2×Dic3—128no—
353D12—632no—
354D12—616no—
355D12—1216no—
356C4×D3—332no—
357C4×D3—332no—
358C4×D3—632no—
359C4×D3—632no—
360C4×D3—632no—
361C4×D3—632no—
362C4×D3—1216no—
363Dic6—632no—
364Dic6—616no—
365C2×C32—22no—
366C2×C32—44no—
367C2×C32C23×D411yescharacteristic
368C2×C32C2×H(5,𝔽2)11yescharacteristic
369C2×C32C2×2−1+411yescharacteristic
370C3⋊D3—22no—
371C3⋊D3—44no—
372C3⋊D3#53714yesdirect product factor
373C24—91no—
374C24—94no—
375#41—28no—
376#41—1816no—
377#41—1816no—
378#41—188no—
379#41—188no—
380#41—188no—
381C2×Q8—94no—
382C2×Q8—92no—
383C2×Q8—188no—
384C2×Q8C22×C3⋊D312yes—
385C2×D4—94no—
386C2×D4—94no—
387C2×D4—94no—
388C2×D4—98no—
389C2×D4—94no—
390C2×D4—92no—
391C2×D4—94no—
392C2×D4—188no—
393C2×D4—1816no—
394C2×D4—368no—
395C2×D4C22×C3⋊D314yes—
396C2×D4C22×C3⋊D312yes—
397C2×D4C22×C3⋊D314yes—
398C22×C4—91no—
399C22×C4—92no—
400C22×C4—92no—
401C22×C4—98no—
402C22×C4—94no—
403C22×C4—92no—
404C22×C4—94no—
405C22×C4—94no—
406C22×C4—94no—
407C22×C4—94no—
408C22×C4—94no—
409C22×C4—1816no—
410C22×C4—364no—
411C22×C4C22×C3⋊D314yes—
412C22×C4C22×C3⋊D312yes—
413C22×C4C22×C3⋊D311yescharacteristic
414C4⋊C4—98no—
415C4⋊C4—94no—
416C4⋊C4—98no—
417C4⋊C4—94no—
418C4⋊C4C22×C3⋊D314yes—
419C4⋊C4C22×C3⋊D314yes—
420C22⋊C4—98no—
421C22⋊C4—98no—
422C22⋊C4—98no—
423C22⋊C4C22×C3⋊D318yes—
424C22×C3—24no—
425C22×C3—24no—
426C22×C3—216no—
427C22×C3—216no—
428C22×C3—216no—
429C22×C3C2×D3×D418yes—
430C22×C3C24×D314yes—
431C2×D3—38no—
432C2×D3—38no—
433C2×D3—38no—
434C2×D3—38no—
435C2×D3—632no—
436C2×D3—64no—
437C2×D3—64no—
438C2×D3—616no—
439C2×D3—616no—
440C2×D3—616no—
441C2×D3—1264no—
442C3×C4—216no—
443C3×C4—216no—
444C3×C4C2×D3×D4116yes—
445Dic3—316no—
446Dic3—616no—
447Dic3—616no—
448C32C2×#53711yescharacteristic, complemented, Sylow
449C23—91no—
450C23—94no—
451C23—91no—
452C23—94no—
453C23—182no—
454C23—182no—
455C23—188no—
456C23—188no—
457C23—188no—
458C23C23×C3⋊D314yes—
459C23C23×C3⋊D311yescharacteristic
460Q8—24no—
461Q8—188no—
462Q8—184no—
463D4—28no—
464D4—24no—
465D4—44no—
466D4—188no—
467D4—188no—
468D4—188no—
469D4—188no—
470D4—184no—
471D4—364no—
472D4—368no—
473C2×C4—28no—
474C2×C4—42no—
475C2×C4—94no—
476C2×C4—92no—
477C2×C4—98no—
478C2×C4—98no—
479C2×C4—94no—
480C2×C4—94no—
481C2×C4—188no—
482C2×C4—188no—
483C2×C4—188no—
484C2×C4—188no—
485C2×C4—188no—
486C2×C4—364no—
487C2×C4—362no—
488C2×C4D4×C3⋊D314yescomplemented
489C2×C4C23×C3⋊D314yes—
490C2×C4C23×C3⋊D314yes—
491C2×C4C23×C3⋊D312yes—
492C2×C3—28no—
493C2×C3—416no—
494C2×C3C22×D3×D414yes—
495C2×C3D3×H(5,𝔽2)14yes—
496C2×C3D3×2−1+414yes—
497D3—316no—
498D3—68no—
499D3—1216no—
500C22—21no—
501C22—21no—
502C22—24no—
503C22—24no—
504C22—24no—
505C22—92no—
506C22—92no—
507C22—92no—
508C22—92no—
509C22—188no—
510C22—181no—
511C22—181no—
512C22—184no—
513C22—184no—
514C22—184no—
515C22—3616no—
516C22C2×D4×C3⋊D312yes—
517C22C24×C3⋊D311yescharacteristic
518C4—24no—
519C4—24no—
520C4—94no—
521C4—184no—
522C4—184no—
523C4C2×D4×C3⋊D314yes—
524C3D3×#53714yescomplemented
525C2—22no—
526C2—44no—
527C2—94no—
528C2—182no—
529C2—364no—
530C2C22×D4×C3⋊D311yescharacteristic
531C2C3⋊D3×H(5,𝔽2)11yescharacteristic
532C2C3⋊D3×2−1+411yescharacteristic
subgroupG/H
center, FrattiniC22C24×C3⋊D3
commutator, socle, cosocleC22×C32C25
FittingC32×#537C2
hypercenter#537C3⋊D3
hypocenterC32C2×#537
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #537

Complements

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

NH ≅ G/Ntype
1. C2×#41×C3⋊D3C2semidirect
2. C32×#537C2semidirect
3. #10596304C2semidirect
4. #10596100C2semidirect
5. C3⋊D3×C22⋊Q8C2semidirect
6. #10596079C2semidirect
7. C3⋊D3×#121C2semidirect
8. C3⋊D3×#121C2semidirect
9. #10596068C2semidirect
10. #10595981C2semidirect
11. C2×C4⋊C4×C3⋊D3C2semidirect
13. C2×C32×#41C22semidirect
15. C2×#3832C22semidirect
17. C2×#3830C22semidirect
18. C2×#3830C22semidirect
22. C2×#3828C22semidirect
23. C2×#3828C22semidirect
27. C32×C22⋊Q8C22semidirect
28. C32×#121C22semidirect
29. C32×#121C22semidirect
30. C2×C32×C4⋊C4C22semidirect
31. #64155C22semidirect
32. #64148C22semidirect
33. #64148C22semidirect
34. #64147C22semidirect
35. #64147C22semidirect
36. #64139C22semidirect
37. #64139C22semidirect
38. C2×C4⋊(C3⋊Dic3)C22semidirect
39. #64133C22semidirect
40. C2×#3750C22semidirect
41. #64106C22semidirect
42. #64105C22semidirect
43. #64104C22semidirect
44. #64104C22semidirect
47. C4⋊C4×C3⋊D3C22semidirect
48. #64093C22semidirect
49. #64093C22semidirect
51. #64087C22semidirect
53. C3×#537D3semidirect
94. C2×C4×C3⋊D3D4semidirect
115. C2×C3×#41C2×D3semidirect
129. C3×C22⋊Q8C2×D3semidirect
130. C3×#121C2×D3semidirect
131. C3×#121C2×D3semidirect
132. C2×C3×C4⋊C4C2×D3semidirect
171. C2×C32×C4C2×D4semidirect
181. C2×C3⋊Dic3C2×D4semidirect
194. C4×C3⋊D3C2×D4semidirect
203. #537C3⋊D3direct (N × H)
289. C2×#41C2×C3⋊D3semidirect
302. C22⋊Q8C2×C3⋊D3semidirect
313. #121C2×C3⋊D3semidirect
314. #121C2×C3⋊D3semidirect
320. C2×C4⋊C4C2×C3⋊D3semidirect
339. C2×C3×C4D3×D4semidirect
372. C3⋊D3#537direct (N × H)
448. C32C2×#537semidirect
488. C2×C4D4×C3⋊D3semidirect
524. C3D3×#537semidirect

Maximal subgroups

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

subgroupindex
C2×#41×C3⋊D32
C32×#5372
#105963042
#105961002
C3⋊D3×C22⋊Q82
#105960792
C3⋊D3×#1212
#105960682
#105959812
C2×C4⋊C4×C3⋊D32
D3×#5373