Finite Group Databasebeta

← C2×C3×#5164 #11860046

Subgroups — C2×C3×#5164 #11860046

Order 1152.

1237 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×SD16C211yesmaximal, characteristic
2C2×C3×Q8×Dic3C211yesmaximal, characteristic, complemented
3C2×C3×D4×Dic3C211yesmaximal, characteristic
4C2×C3×#835C211yesmaximal, characteristic, complemented
5C2×C3×#832C211yesmaximal, characteristic
6C3×#5164C218yesmaximal, direct product factor
7C2×C3×#818C211yesmaximal, characteristic, complemented
8C2×C3×#815C211yesmaximal, characteristic, complemented
9C2×C3×#440—31nomaximal
10C2×#5164C311yesmaximal, characteristic, direct product factor
11C22×C32×Q8C2211yescharacteristic
12C22×C32×D4C2211yescharacteristic
13C23×C3×Dic3—21no—
14C2×C32×SD16C418yescomplemented
15C2×C32×SD16C2214yes—
16C22×C32×C8C2211yescharacteristic
17C2×C3×#217—21no—
18C3×Q8×Dic3—22no—
19C3×Q8×Dic3C2214yescomplemented
20C3×D4×Dic3—22no—
21C3×D4×Dic3C2214yes—
22C2×C3×C4⋊Dic3—21no—
23C2×C3×C4⋊Dic3C2211yescharacteristic
24C2×C3×C4⋊Dic3C2211yescharacteristic
25C2×C3×C4×Dic3—21no—
26C2×C3×C4×Dic3C2211yescharacteristic
27C22×C3×C3⋊C8C2211yescharacteristic
28C3×#835C2214yescomplemented
29C3×#832C2214yes—
30C3×#818C2214yescomplemented
31C3×#815C2214yescomplemented
32C22×C3×SD16—21no—
33C22×C3×SD16D311yescharacteristic
34C22×C3×SD16C2×C311yescharacteristic
35C2×C3×C4×Q8—31no—
36C2×C3×C4×D4—31no—
37C2×Q8×Dic3C2×C311yescharacteristic, complemented
38C2×D4×Dic3C2×C311yescharacteristic
39C3×#440—38no—
40C2×C3×#107—31no—
41C2×C3×#103—31no—
42C2×C3×#102—31no—
43C2×C3×#97—31no—
44C2×#835C2×C311yescharacteristic, complemented
45C2×#832C2×C311yescharacteristic
46#5164C2×C318yesdirect product factor
47C2×#818C2×C311yescharacteristic, complemented
48C2×#815C2×C311yescharacteristic, complemented
49C24×C32—21no—
50C2×C32×Q8—21no—
51C2×C32×Q8—22no—
52C2×C32×Q8C2×C414yescomplemented
53C2×C32×Q8C2312yes—
54C2×C32×D4—21no—
55C2×C32×D4—22no—
56C2×C32×D4C2×C414yes—
57C2×C32×D4C2312yes—
58C22×C32×C4—21no—
59C22×C32×C4C2311yescharacteristic
60C22×C3×Dic3—21no—
61C22×C3×Dic3—21no—
62C22×C3×Dic3—24no—
63C22×C3×Dic3—44no—
64C22×C3×Dic3D411yescharacteristic
65C22×C3×Dic3D411yescharacteristic
66C32×SD16C2×C4116yescomplemented
67C2×C32×C8C2×C414yescomplemented
68C2×C32×C8C2312yes—
69C3×#217—24no—
70C3×C4⋊Dic3—21no—
71C3×C4⋊Dic3—21no—
72C3×C4⋊Dic3—24no—
73C3×C4⋊Dic3C2312yes—
74C3×C4⋊Dic3C2312yes—
75C3×C4×Dic3—24no—
76C3×C4×Dic3D412yes—
77C3×C4×Dic3C2312yes—
78C2×C3×C3⋊C8—22no—
79C2×C3×C3⋊C8C2312yes—
80C2×#440—31noSylow
81C22×C3×Q8—21no—
82C22×C3×Q8C2×D311yescharacteristic
83C22×C3×Q8C22×C311yescharacteristic
84C22×C3×D4—21no—
85C22×C3×D4C2×D311yescharacteristic
86C22×C3×D4C22×C311yescharacteristic
87C23×C3×C4—61no—
88C23×Dic3—21no—
89C2×C3×SD16—28no—
90C2×C3×SD16—24no—
91C2×C3×SD16Dic318yescomplemented
92C2×C3×SD16C3×C418yescomplemented
93C2×C3×SD16C2×D314yes—
94C2×C3×SD16C22×C314yes—
95C22×C3×C8—21no—
96C22×C3×C8—31no—
97C22×C3×C8C2×D311yescharacteristic
98C22×C3×C8C22×C311yescharacteristic
99C3×C4×Q8—34no—
100C3×C4×Q8—62no—
101C3×C4×D4—34no—
102C3×C4×D4—62no—
103C2×C3×C4⋊C4—31no—
104C2×C3×C4⋊C4—31no—
105C2×C3×C4⋊C4—61no—
106C2×C3×C22⋊C4—61no—
107C2×C3×C42—31no—
108C2×C3×C42—61no—
109C2×#217—21no—
110Q8×Dic3—22no—
111Q8×Dic3C22×C314yescomplemented
112D4×Dic3—22no—
113D4×Dic3C22×C314yes—
114C2×C4⋊Dic3—21no—
115C2×C4⋊Dic3C22×C311yescharacteristic
116C2×C4⋊Dic3C22×C311yescharacteristic
117C2×C4×Dic3—21no—
118C2×C4×Dic3C22×C311yescharacteristic
119C22×C3⋊C8C22×C311yescharacteristic
120C3×#107—34no—
121C3×#103—34no—
122C3×#102—34no—
123C3×#97—34no—
124#835C22×C314yescomplemented
125#832C22×C314yes—
126#818C22×C314yescomplemented
127#815C22×C314yescomplemented
128C23×C32—24no—
129C23×C32—22no—
130C23×C32—42no—
131C23×C32—42no—
132C23×C32C2×D411yescharacteristic
133C32×Q8—24no—
134C32×Q8—22no—
135C32×Q8C22×C414yescomplemented
136C32×D4—24no—
137C32×D4—22no—
138C32×D4C22×C414yes—
139C2×C32×C4—24no—
140C2×C32×C4—22no—
141C2×C32×C4C22×C412yes—
142C2×C32×C4C2×D411yescharacteristic
143C2×C32×C4#4112yes—
144C2×C32×C4C2411yescharacteristic
145C2×C3×Dic3—22no—
146C2×C3×Dic3—22no—
147C2×C3×Dic3—22no—
148C2×C3×Dic3—22no—
149C2×C3×Dic3—48no—
150C2×C3×Dic3—42no—
151C2×C3×Dic3—42no—
152C2×C3×Dic3C2×D412yes—
153C2×C3×Dic3C2×D412yes—
154C32×C8C22×C414yescomplemented
155C3×C3⋊C8—22no—
156C22×SD16C3×D311yescharacteristic
157C2×C4×Q8—31no—
158C2×C4×D4—31no—
159#440—38no—
160C2×#107—31no—
161C2×#103—31no—
162C2×#102—31no—
163C2×#97—31no—
164C24×C3—21no—
165C24×C3—21no—
166C24×C3—41no—
167C2×C3×Q8—21no—
168C2×C3×Q8—22no—
169C2×C3×Q8—21no—
170C2×C3×Q8—22no—
171C2×C3×Q8—24no—
172C2×C3×Q8—22no—
173C2×C3×Q8—41no—
174C2×C3×Q8—42no—
175C2×C3×Q8C2×Dic314yescomplemented
176C2×C3×Q8C2×C3×C414yescomplemented
177C2×C3×Q8C22×D312yes—
178C2×C3×Q8C23×C312yes—
179C2×C3×D4—21no—
180C2×C3×D4—22no—
181C2×C3×D4—21no—
182C2×C3×D4—22no—
183C2×C3×D4—24no—
184C2×C3×D4—22no—
185C2×C3×D4—41no—
186C2×C3×D4—42no—
187C2×C3×D4C2×Dic314yes—
188C2×C3×D4C2×C3×C414yes—
189C2×C3×D4C22×D312yes—
190C2×C3×D4C23×C312yes—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—31no—
195C22×C3×C4—31no—
196C22×C3×C4—41no—
197C22×C3×C4—64no—
198C22×C3×C4—61no—
199C22×C3×C4—61no—
200C22×C3×C4—124no—
201C22×C3×C4C22×D311yescharacteristic
202C22×C3×C4C23×C311yescharacteristic
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—24no—
206C22×Dic3—44no—
207C22×Dic3C3×D411yescharacteristic
208C22×Dic3C3×D411yescharacteristic
209C3×SD16—216no—
210C3×SD16C2×Dic3116yescomplemented
211C3×SD16C2×C3×C4116yescomplemented
212C2×C3×C8—24no—
213C2×C3×C8—22no—
214C2×C3×C8—32no—
215C2×C3×C8—62no—
216C2×C3×C8C2×Dic314yescomplemented
217C2×C3×C8C2×C3×C414yescomplemented
218C2×C3×C8C22×D312yes—
219C2×C3×C8C23×C312yes—
220C3×C4⋊C4—32no—
221C3×C4⋊C4—32no—
222C3×C4⋊C4—64no—
223C3×C4⋊C4—61no—
224C3×C4⋊C4—61no—
225C3×C22⋊C4—64no—
226C3×C42—32no—
227C3×C42—32no—
228C3×C42—64no—
229#217—24no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—24no—
233C4⋊Dic3C23×C312yes—
234C4⋊Dic3C23×C312yes—
235C4×Dic3—24no—
236C4×Dic3C3×D412yes—
237C4×Dic3C23×C312yes—
238C2×C3⋊C8—22no—
239C2×C3⋊C8C23×C312yes—
240C22×C32—24no—
241C22×C32—42no—
242C22×C32—42no—
243C22×C32—48no—
244C22×C32C4×D412yes—
245C22×C32AGL(1,ℤ/8)12yes—
246C22×C32#13712yes—
247C22×C32C22×D411yescharacteristic
248C32×C4—24no—
249C32×C4C4×D412yes—
250C32×C4C23×C411yescharacteristic
251C32×C4C2×#4111yescharacteristic
252C3×Dic3—22no—
253C3×Dic3—22no—
254C3×Dic3—42no—
255C3×Dic3—42no—
256C22×Q8C2×C3×D311yescharacteristic
257C22×D4C2×C3×D311yescharacteristic
258C23×C4—61no—
259C2×SD16C3×Dic318yescomplemented
260C2×SD16C2×C3×D314yes—
261C22×C8—31no—
262C22×C8C2×C3×D311yescharacteristic
263C4×Q8—34no—
264C4×Q8—62no—
265C4×D4—34no—
266C4×D4—62no—
267C2×C4⋊C4—31no—
268C2×C4⋊C4—31no—
269C2×C4⋊C4—61no—
270C2×C22⋊C4—61no—
271C2×C42—31no—
272C2×C42—61no—
273#107—34no—
274#103—34no—
275#102—34no—
276#97—34no—
277C23×C3—24no—
278C23×C3—24no—
279C23×C3—22no—
280C23×C3—22no—
281C23×C3—21no—
282C23×C3—44no—
283C23×C3—42no—
284C23×C3—42no—
285C23×C3—42no—
286C23×C3—42no—
287C23×C3—42no—
288C23×C3—82no—
289C23×C3—82no—
290C23×C3D3×D411yescharacteristic
291C23×C3C2×C3×D411yescharacteristic
292C3×Q8—24no—
293C3×Q8—24no—
294C3×Q8—22no—
295C3×Q8—22no—
296C3×Q8—24no—
297C3×Q8—44no—
298C3×Q8—42no—
299C3×Q8C22×Dic314yescomplemented
300C3×Q8C22×C3×C414yescomplemented
301C3×D4—24no—
302C3×D4—24no—
303C3×D4—22no—
304C3×D4—22no—
305C3×D4—24no—
306C3×D4—44no—
307C3×D4—42no—
308C3×D4C22×Dic314yes—
309C3×D4C22×C3×C414yes—
310C2×C3×C4—24no—
311C2×C3×C4—24no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—32no—
319C2×C3×C4—32no—
320C2×C3×C4—44no—
321C2×C3×C4—42no—
322C2×C3×C4—62no—
323C2×C3×C4—62no—
324C2×C3×C4—62no—
325C2×C3×C4—62no—
326C2×C3×C4—128no—
327C2×C3×C4—122no—
328C2×C3×C4—122no—
329C2×C3×C4D3×D411yescharacteristic
330C2×C3×C4#23712yes—
331C2×C3×C4C22×Dic312yes—
332C2×C3×C4C22×C3×C412yes—
333C2×C3×C4C2×C3×D411yescharacteristic
334C2×C3×C4C3×#4112yes—
335C2×C3×C4C23×D311yescharacteristic
336C2×C3×C4C24×C311yescharacteristic
337C2×Dic3—22no—
338C2×Dic3—22no—
339C2×Dic3—22no—
340C2×Dic3—22no—
341C2×Dic3—48no—
342C2×Dic3—42no—
343C2×Dic3—42no—
344C2×Dic3C2×C3×D412yes—
345C2×Dic3C2×C3×D412yes—
346C3×C8—24no—
347C3×C8—62no—
348C3×C8C22×Dic314yescomplemented
349C3×C8C22×C3×C414yescomplemented
350C3⋊C8—22no—
351C2×C32—44no—
352C2×C32#44014yescomplemented
353C2×C32C2×C4×D411yescharacteristic
354C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
355C2×C32C2×#13711yescharacteristic
356C24—21no—
357C2×Q8—21no—
358C2×Q8—22no—
359C2×Q8C2×C3×Dic314yescomplemented
360C2×Q8C22×C3×D312yes—
361C2×D4—21no—
362C2×D4—22no—
363C2×D4C2×C3×Dic314yes—
364C2×D4C22×C3×D312yes—
365C22×C4—21no—
366C22×C4—31no—
367C22×C4—31no—
368C22×C4—64no—
369C22×C4—61no—
370C22×C4—61no—
371C22×C4—124no—
372C22×C4C22×C3×D311yescharacteristic
373SD16C2×C3×Dic3116yescomplemented
374C2×C8—32no—
375C2×C8—62no—
376C2×C8C2×C3×Dic314yescomplemented
377C2×C8C22×C3×D312yes—
378C4⋊C4—32no—
379C4⋊C4—32no—
380C4⋊C4—64no—
381C4⋊C4—61no—
382C4⋊C4—61no—
383C22⋊C4—64no—
384C42—32no—
385C42—32no—
386C42—64no—
387C22×C3—24no—
388C22×C3—24no—
389C22×C3—22no—
390C22×C3—22no—
391C22×C3—22no—
392C22×C3—21no—
393C22×C3—44no—
394C22×C3—42no—
395C22×C3—42no—
396C22×C3—48no—
397C22×C3—42no—
398C22×C3—42no—
399C22×C3—48no—
400C22×C3—88no—
401C22×C3—82no—
402C22×C3—82no—
403C22×C3#91412yes—
404C22×C3#91512yes—
405C22×C3D4×Dic312yes—
406C22×C3C3×C4×D412yes—
407C22×C3C3×AGL(1,ℤ/8)12yes—
408C22×C3C3×#13712yes—
409C22×C3C2×D3×D411yescharacteristic
410C22×C3C22×C3×D411yescharacteristic
411C3×C4—24no—
412C3×C4—24no—
413C3×C4—22no—
414C3×C4—21no—
415C3×C4—21no—
416C3×C4—44no—
417C3×C4—62no—
418C3×C4—62no—
419C3×C4—122no—
420C3×C4—122no—
421C3×C4D4×Dic312yes—
422C3×C4C3×C4×D412yes—
423C3×C4C2×#23711yescharacteristic
424C3×C4C23×Dic311yescharacteristic
425C3×C4C23×C3×C411yescharacteristic
426C3×C4C2×C3×#4111yescharacteristic
427Dic3—22no—
428Dic3—22no—
429Dic3—42no—
430Dic3—42no—
431C32C2×#44011yescharacteristic, complemented, Sylow
432C23—24no—
433C23—22no—
434C23—42no—
435C23—42no—
436C23C3×D3×D411yescharacteristic
437Q8—24no—
438Q8—22no—
439Q8C22×C3×Dic314yescomplemented
440D4—24no—
441D4—22no—
442D4C22×C3×Dic314yes—
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—32no—
446C2×C4—32no—
447C2×C4—62no—
448C2×C4—62no—
449C2×C4—62no—
450C2×C4—62no—
451C2×C4—128no—
452C2×C4—122no—
453C2×C4—122no—
454C2×C4C3×D3×D411yescharacteristic
455C2×C4C3×#23712yes—
456C2×C4C22×C3×Dic312yes—
457C2×C4C23×C3×D311yescharacteristic
458C8—62no—
459C8C22×C3×Dic314yescomplemented
460C2×C3—24no—
461C2×C3—21no—
462C2×C3—21no—
463C2×C3—21no—
464C2×C3—44no—
465C2×C3—44no—
466C2×C3—84no—
467C2×C3#516414yesdirect product factor
468C2×C3C3×#44014yescomplemented
469C2×C3C2×#91411yescharacteristic
470C2×C3C2×#91511yescharacteristic
471C2×C3C2×D4×Dic311yescharacteristic
472C2×C3C2×C3×C4×D411yescharacteristic
473C2×C3C2×C3×AGL(1,ℤ/8)11yescharacteristic
474C2×C3C2×C3×#13711yescharacteristic
475C22—24no—
476C22—48no—
477C22—42no—
478C22—42no—
479C22C3×#91412yes—
480C22C3×#91512yes—
481C22C3×D4×Dic312yes—
482C22C2×C3×D3×D411yescharacteristic
483C4—24no—
484C4—62no—
485C4—62no—
486C4—122no—
487C4—122no—
488C4C3×D4×Dic312yes—
489C4C2×C3×#23711yescharacteristic
490C4C23×C3×Dic311yescharacteristic
491C3—21no—
492C3C2×#516411yescharacteristic, direct product factor
493C3C2×C3×#44011yescharacteristic, complemented
494C2—44no—
495C2C3×#516414yesdirect product factor
496C2C2×C3×#91411yescharacteristic
497C2C2×C3×#91511yescharacteristic
498C2C2×C3×D4×Dic311yescharacteristic
subgroupG/H
centerC23×C3D3×D4
commutatorC3×C4C23×C3×C4
FrattiniC2×C4C23×C3×D3
FittingC22×C32×SD16C2
socleC23×C32C2×D4
prime radicalC22×C32×D4C22
cosocleC2×C3×C4C24×C3
hypercenterC22×C3×SD16D3
hypocenterC3C2×C3×#440
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C22×C3×SD16

Complements

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

NH ≅ G/Ntype
2. C2×C3×Q8×Dic3C2semidirect
4. C2×C3×#835C2semidirect
6. C3×#5164C2direct (N × H)
7. C2×C3×#818C2semidirect
8. C2×C3×#815C2semidirect
10. C2×#5164C3direct (N × H)
14. C2×C32×SD16C4semidirect
19. C3×Q8×Dic3C22semidirect
28. C3×#835C22semidirect
30. C3×#818C22semidirect
31. C3×#815C22semidirect
37. C2×Q8×Dic3C2×C3semidirect
44. C2×#835C2×C3semidirect
46. #5164C2×C3direct (N × H)
47. C2×#818C2×C3semidirect
48. C2×#815C2×C3semidirect
52. C2×C32×Q8C2×C4semidirect
66. C32×SD16C2×C4semidirect
67. C2×C32×C8C2×C4semidirect
91. C2×C3×SD16Dic3semidirect
92. C2×C3×SD16C3×C4semidirect
111. Q8×Dic3C22×C3semidirect
124. #835C22×C3semidirect
126. #818C22×C3semidirect
127. #815C22×C3semidirect
135. C32×Q8C22×C4semidirect
154. C32×C8C22×C4semidirect
175. C2×C3×Q8C2×Dic3semidirect
176. C2×C3×Q8C2×C3×C4semidirect
210. C3×SD16C2×Dic3semidirect
211. C3×SD16C2×C3×C4semidirect
216. C2×C3×C8C2×Dic3semidirect
217. C2×C3×C8C2×C3×C4semidirect
259. C2×SD16C3×Dic3semidirect
299. C3×Q8C22×Dic3semidirect
300. C3×Q8C22×C3×C4semidirect
348. C3×C8C22×Dic3semidirect
349. C3×C8C22×C3×C4semidirect
352. C2×C32#440semidirect
359. C2×Q8C2×C3×Dic3semidirect
373. SD16C2×C3×Dic3semidirect
376. C2×C8C2×C3×Dic3semidirect
431. C32C2×#440semidirect
439. Q8C22×C3×Dic3semidirect
459. C8C22×C3×Dic3semidirect
467. C2×C3#5164direct (N × H)
468. C2×C3C3×#440semidirect
492. C3C2×#5164direct (N × H)
493. C3C2×C3×#440semidirect
495. C2C3×#5164direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×SD162
C2×C3×Q8×Dic32
C2×C3×D4×Dic32
C2×C3×#8352
C2×C3×#8322
C3×#51642
C2×C3×#8182
C2×C3×#8152
C2×C3×#4403
C2×#51643