Finite Group Databasebeta

← C2×C3×#5166 #11865670

Subgroups — C2×C3×#5166 #11865670

Order 1152.

1149 conjugacy classes of subgroups.

The subgroup lattice has 520 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
3C22×C3×#216C211yesmaximal, characteristic, complemented
4C2×C3×#936C211yesmaximal, characteristic
5C2×C3×#832C211yesmaximal, characteristic
6C3×#5166C218yesmaximal, direct product factor
7C2×C3×#816C211yesmaximal, characteristic, complemented
8C2×C3×Dic3⋊C8C211yesmaximal, characteristic, complemented
9C2×C3×#464—31nomaximal
10C2×#5166C311yesmaximal, characteristic, direct product factor
11C22×C32×Q8C2211yescharacteristic
12C22×C32×D4C2211yescharacteristic
13C22×C3×Dic6C2211yescharacteristic
14C2×C32×SD16—24no—
15C2×C32×SD16C2214yes—
16C22×C32×C8C2211yescharacteristic
17C2×C3×#217—21no—
18C2×C3×#217—21no—
19C3×Q8×Dic3—22no—
20C3×Q8×Dic3C2214yescomplemented
21C2×C3×#216—24no—
22C2×C3×#216C2214yescomplemented
23C3×#936—22no—
24C3×#936C2214yes—
25C2×C3×C4⋊Dic3—21no—
26C2×C3×C4⋊Dic3C2211yescharacteristic
27C2×C3×C4×Dic3—21no—
28C2×C3×C4×Dic3C2211yescharacteristic
29C22×C3×C3⋊C8C2211yescharacteristic
30C3×#832C2214yes—
31C3×#816C2214yescomplemented
32C3×Dic3⋊C8C2214yescomplemented
33C22×C3×Q16—31no—
34C22×C3×SD16—21no—
35C22×C3×SD16D311yescharacteristic
36C22×C3×SD16C2×C311yescharacteristic
37C2×C3×#124—31no—
38C2×C3×C4×Q8—31no—
39C2×Q8×Dic3C2×C311yescharacteristic, complemented
40C22×#216C2×C311yescharacteristic, complemented
41C2×#936C2×C311yescharacteristic
42C3×#464—38no—
43C2×C3×C4⋊C8—31no—
44C2×C3×#103—31no—
45C2×C3×#102—31no—
46C2×#832C2×C311yescharacteristic
47#5166C2×C318yesdirect product factor
48C2×#816C2×C311yescharacteristic, complemented
49C2×Dic3⋊C8C2×C311yescharacteristic, complemented
50C24×C32—21no—
51C2×C32×Q8—21no—
52C2×C32×Q8—22no—
53C2×C32×Q8D414yes—
54C2×C32×Q8C2312yes—
55C2×C32×D4—22no—
56C2×C32×D4—21no—
57C2×C32×D4—22no—
58C2×C32×D4C2312yes—
59C22×C32×C4—21no—
60C22×C32×C4C2311yescharacteristic
61C22×C3×Dic3—21no—
62C22×C3×Dic3—21no—
63C22×C3×Dic3D411yescharacteristic
64C22×C3×Dic3D411yescharacteristic
65C2×C3×Dic6—22no—
66C2×C3×Dic6—22no—
67C2×C3×Dic6—21no—
68C2×C3×Dic6C2312yes—
69C32×SD16—28no—
70C2×C32×C8—22no—
71C2×C32×C8C2312yes—
72C3×#217—24no—
73C3×#217—24no—
74C3×#216—28no—
75C3×C4⋊Dic3—21no—
76C3×C4⋊Dic3—24no—
77C3×C4⋊Dic3C2312yes—
78C3×C4×Dic3—24no—
79C3×C4×Dic3D412yes—
80C3×C4×Dic3C2312yes—
81C2×C3×C3⋊C8—22no—
82C2×C3×C3⋊C8C2312yes—
83C2×#464—31noSylow
84C22×C3×Q8—21no—
85C22×C3×Q8—31no—
86C22×C3×Q8C2×D311yescharacteristic
87C22×C3×Q8C22×C311yescharacteristic
88C22×C3×D4—21no—
89C22×C3×D4C2×D311yescharacteristic
90C22×C3×D4C22×C311yescharacteristic
91C22×Dic6C22×C311yescharacteristic
92C2×C3×Q16—34no—
93C2×C3×Q16—64no—
94C2×C3×SD16—28no—
95C2×C3×SD16—24no—
96C2×C3×SD16—24no—
97C2×C3×SD16—24no—
98C2×C3×SD16C2×D314yes—
99C2×C3×SD16C22×C314yes—
100C22×C3×C8—21no—
101C22×C3×C8—31no—
102C22×C3×C8C2×D311yescharacteristic
103C22×C3×C8C22×C311yescharacteristic
104C3×#124—34no—
105C3×#124—62no—
106C3×C4×Q8—34no—
107C3×C4×Q8—62no—
108C2×C3×C4⋊C4—31no—
109C2×C3×C4⋊C4—61no—
110C2×C3×C22⋊C4—61no—
111C2×C3×C22⋊C4—61no—
112C2×C3×C42—31no—
113C2×C3×C42—61no—
114C2×#217—21no—
115C2×#217—21no—
116Q8×Dic3—22no—
117Q8×Dic3C22×C314yescomplemented
118C2×#216—24no—
119C2×#216C22×C314yescomplemented
120#936—22no—
121#936C22×C314yes—
122C2×C4⋊Dic3—21no—
123C2×C4⋊Dic3C22×C311yescharacteristic
124C2×C4×Dic3—21no—
125C2×C4×Dic3C22×C311yescharacteristic
126C22×C3⋊C8C22×C311yescharacteristic
127C3×C4⋊C8—34no—
128C3×#103—34no—
129C3×#102—34no—
130#832C22×C314yes—
131#816C22×C314yescomplemented
132Dic3⋊C8C22×C314yescomplemented
133C23×C32—22no—
134C23×C32—42no—
135C23×C32—42no—
136C23×C32—42no—
137C23×C32C2×D411yescharacteristic
138C32×Q8—24no—
139C32×Q8—22no—
140C32×Q8C2×D414yes—
141C32×D4—22no—
142C32×D4—22no—
143C32×D4—42no—
144C2×C32×C4—24no—
145C2×C32×C4—22no—
146C2×C32×C4C2×D411yescharacteristic
147C2×C32×C4C2×D412yes—
148C2×C32×C4#4112yes—
149C2×C32×C4C2411yescharacteristic
150C2×C3×Dic3—22no—
151C2×C3×Dic3—22no—
152C2×C3×Dic3—22no—
153C2×C3×Dic3—22no—
154C2×C3×Dic3—42no—
155C2×C3×Dic3—42no—
156C2×C3×Dic3C2×D412yes—
157C2×C3×Dic3C2×D412yes—
158C3×Dic6—22no—
159C3×Dic6—22no—
160C3×Dic6—42no—
161C32×C8—22no—
162C3×C3⋊C8—22no—
163C22×Q16—31no—
164C22×SD16C3×D311yescharacteristic
165C2×#124—31no—
166C2×C4×Q8—31no—
167#464—38no—
168C2×C4⋊C8—31no—
169C2×#103—31no—
170C2×#102—31no—
171C24×C3—21no—
172C24×C3—21no—
173C24×C3—41no—
174C2×C3×Q8—21no—
175C2×C3×Q8—22no—
176C2×C3×Q8—21no—
177C2×C3×Q8—22no—
178C2×C3×Q8—24no—
179C2×C3×Q8—22no—
180C2×C3×Q8—32no—
181C2×C3×Q8—41no—
182C2×C3×Q8—42no—
183C2×C3×Q8—61no—
184C2×C3×Q8—62no—
185C2×C3×Q8—62no—
186C2×C3×Q8#6714yes—
187C2×C3×Q8C3×D414yes—
188C2×C3×Q8C22×D312yes—
189C2×C3×Q8C23×C312yes—
190C2×C3×D4—24no—
191C2×C3×D4—21no—
192C2×C3×D4—22no—
193C2×C3×D4—22no—
194C2×C3×D4—22no—
195C2×C3×D4—21no—
196C2×C3×D4—22no—
197C2×C3×D4—22no—
198C2×C3×D4—41no—
199C2×C3×D4—42no—
200C2×C3×D4C22×D312yes—
201C2×C3×D4C23×C312yes—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—31no—
206C22×C3×C4—31no—
207C22×C3×C4—41no—
208C22×C3×C4—61no—
209C22×C3×C4—61no—
210C22×C3×C4C22×D311yescharacteristic
211C22×C3×C4C23×C311yescharacteristic
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3C3×D411yescharacteristic
215C22×Dic3C3×D411yescharacteristic
216C2×Dic6—22no—
217C2×Dic6—22no—
218C2×Dic6—21no—
219C2×Dic6C23×C312yes—
220C3×Q16—68no—
221C3×SD16—216no—
222C3×SD16—28no—
223C3×SD16—28no—
224C2×C3×C8—24no—
225C2×C3×C8—22no—
226C2×C3×C8—22no—
227C2×C3×C8—22no—
228C2×C3×C8—32no—
229C2×C3×C8—62no—
230C2×C3×C8C22×D312yes—
231C2×C3×C8C23×C312yes—
232C3×C4⋊C4—32no—
233C3×C4⋊C4—64no—
234C3×C4⋊C4—61no—
235C3×C22⋊C4—64no—
236C3×C22⋊C4—64no—
237C3×C42—32no—
238C3×C42—32no—
239C3×C42—64no—
240#217—24no—
241#217—24no—
242#216—28no—
243C4⋊Dic3—21no—
244C4⋊Dic3—24no—
245C4⋊Dic3C23×C312yes—
246C4×Dic3—24no—
247C4×Dic3C3×D412yes—
248C4×Dic3C23×C312yes—
249C2×C3⋊C8—22no—
250C2×C3⋊C8C23×C312yes—
251C22×C32—42no—
252C22×C32—42no—
253C22×C32—42no—
254C22×C32—84no—
255C22×C32#12112yes—
256C22×C32#13512yes—
257C22×C32#13712yes—
258C22×C32C22×D411yescharacteristic
259C32×C4—24no—
260C32×C4#12112yes—
261C32×C4C22×D411yescharacteristic
262C32×C4C2×#4111yescharacteristic
263C3×Dic3—22no—
264C3×Dic3—22no—
265C3×Dic3—42no—
266C3×Dic3—42no—
267C22×Q8—31no—
268C22×Q8C2×C3×D311yescharacteristic
269C22×D4C2×C3×D311yescharacteristic
270C2×Q16—34no—
271C2×Q16—64no—
272C2×SD16—24no—
273C2×SD16C2×C3×D314yes—
274C22×C8—31no—
275C22×C8C2×C3×D311yescharacteristic
276#124—34no—
277#124—62no—
278C4×Q8—34no—
279C4×Q8—62no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—61no—
282C2×C22⋊C4—61no—
283C2×C22⋊C4—61no—
284C2×C42—31no—
285C2×C42—61no—
286C4⋊C8—34no—
287#103—34no—
288#102—34no—
289C23×C3—22no—
290C23×C3—22no—
291C23×C3—21no—
292C23×C3—42no—
293C23×C3—44no—
294C23×C3—42no—
295C23×C3—42no—
296C23×C3—42no—
297C23×C3—42no—
298C23×C3—42no—
299C23×C3—42no—
300C23×C3—82no—
301C23×C3—82no—
302C23×C3D3×D411yescharacteristic
303C23×C3C2×C3×D411yescharacteristic
304C3×Q8—24no—
305C3×Q8—24no—
306C3×Q8—22no—
307C3×Q8—22no—
308C3×Q8—24no—
309C3×Q8—44no—
310C3×Q8—42no—
311C3×Q8—62no—
312C3×Q8—62no—
313C3×Q8—122no—
314C3×Q8C2×#6714yes—
315C3×Q8C2×C3×D414yes—
316C3×D4—24no—
317C3×D4—22no—
318C3×D4—22no—
319C3×D4—22no—
320C3×D4—22no—
321C3×D4—42no—
322C3×D4—44no—
323C3×D4—42no—
324C3×D4—42no—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—32no—
334C2×C3×C4—32no—
335C2×C3×C4—44no—
336C2×C3×C4—42no—
337C2×C3×C4—62no—
338C2×C3×C4—62no—
339C2×C3×C4—62no—
340C2×C3×C4—62no—
341C2×C3×C4—122no—
342C2×C3×C4—122no—
343C2×C3×C4D3×D411yescharacteristic
344C2×C3×C4#23712yes—
345C2×C3×C4C2×#6712yes—
346C2×C3×C4C2×C3×D411yescharacteristic
347C2×C3×C4C2×C3×D412yes—
348C2×C3×C4C3×#4112yes—
349C2×C3×C4C23×D311yescharacteristic
350C2×C3×C4C24×C311yescharacteristic
351C2×Dic3—22no—
352C2×Dic3—22no—
353C2×Dic3—22no—
354C2×Dic3—22no—
355C2×Dic3—42no—
356C2×Dic3—42no—
357C2×Dic3C2×C3×D412yes—
358C2×Dic3C2×C3×D412yes—
359Dic6—22no—
360Dic6—22no—
361Dic6—42no—
362C3×C8—24no—
363C3×C8—22no—
364C3×C8—22no—
365C3×C8—62no—
366C3⋊C8—22no—
367C2×C32—82no—
368C2×C32#46414yescomplemented
369C2×C32C2×#12111yescharacteristic
370C2×C32C2×#13511yescharacteristic
371C2×C32C2×#13711yescharacteristic
372C24—21no—
373C2×Q8—21no—
374C2×Q8—22no—
375C2×Q8—32no—
376C2×Q8—61no—
377C2×Q8—62no—
378C2×Q8—62no—
379C2×Q8C3×#6714yes—
380C2×Q8C22×C3×D312yes—
381C2×D4—21no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4C22×C3×D312yes—
385C22×C4—21no—
386C22×C4—31no—
387C22×C4—31no—
388C22×C4—61no—
389C22×C4—61no—
390C22×C4C22×C3×D311yescharacteristic
391Q16—68no—
392SD16—28no—
393C2×C8—22no—
394C2×C8—32no—
395C2×C8—62no—
396C2×C8C22×C3×D312yes—
397C4⋊C4—32no—
398C4⋊C4—64no—
399C4⋊C4—61no—
400C22⋊C4—64no—
401C22⋊C4—64no—
402C42—32no—
403C42—32no—
404C42—64no—
405C22×C3—22no—
406C22×C3—22no—
407C22×C3—22no—
408C22×C3—21no—
409C22×C3—44no—
410C22×C3—42no—
411C22×C3—42no—
412C22×C3—42no—
413C22×C3—42no—
414C22×C3—42no—
415C22×C3—42no—
416C22×C3—82no—
417C22×C3—82no—
418C22×C3—88no—
419C22×C3—84no—
420C22×C3—84no—
421C22×C3#91512yes—
422C22×C3#91612yes—
423C22×C3#93912yes—
424C22×C3C3×#12112yes—
425C22×C3C3×#13512yes—
426C22×C3C3×#13712yes—
427C22×C3C2×D3×D411yescharacteristic
428C22×C3C22×C3×D411yescharacteristic
429C3×C4—24no—
430C3×C4—24no—
431C3×C4—22no—
432C3×C4—21no—
433C3×C4—21no—
434C3×C4—44no—
435C3×C4—62no—
436C3×C4—62no—
437C3×C4—122no—
438C3×C4—122no—
439C3×C4#93912yes—
440C3×C4C3×#12112yes—
441C3×C4C2×#23711yescharacteristic
442C3×C4C22×#6711yescharacteristic
443C3×C4C22×C3×D411yescharacteristic
444C3×C4C2×C3×#4111yescharacteristic
445Dic3—22no—
446Dic3—22no—
447Dic3—42no—
448Dic3—42no—
449C32C2×#46411yescharacteristic, complemented, Sylow
450C23—22no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23C3×D3×D411yescharacteristic
455Q8—24no—
456Q8—22no—
457Q8—62no—
458Q8—62no—
459Q8—122no—
460Q8C2×C3×#6714yes—
461D4—22no—
462D4—22no—
463D4—42no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—32no—
467C2×C4—32no—
468C2×C4—62no—
469C2×C4—62no—
470C2×C4—62no—
471C2×C4—62no—
472C2×C4—122no—
473C2×C4—122no—
474C2×C4C3×D3×D411yescharacteristic
475C2×C4C3×#23712yes—
476C2×C4C2×C3×#6712yes—
477C2×C4C23×C3×D311yescharacteristic
478C8—22no—
479C8—62no—
480C2×C3—24no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3—21no—
484C2×C3—84no—
485C2×C3—82no—
486C2×C3—82no—
487C2×C3#516614yesdirect product factor
488C2×C3C3×#46414yescomplemented
489C2×C3C2×#91511yescharacteristic
490C2×C3C2×#91611yescharacteristic
491C2×C3C2×#93911yescharacteristic
492C2×C3C2×C3×#12111yescharacteristic
493C2×C3C2×C3×#13511yescharacteristic
494C2×C3C2×C3×#13711yescharacteristic
495C22—42no—
496C22—42no—
497C22—42no—
498C22—84no—
499C22C3×#91512yes—
500C22C3×#91612yes—
501C22C3×#93912yes—
502C22C2×C3×D3×D411yescharacteristic
503C4—24no—
504C4—62no—
505C4—62no—
506C4—122no—
507C4—122no—
508C4C3×#93912yes—
509C4C2×C3×#23711yescharacteristic
510C4C22×C3×#6711yescharacteristic
511C3—21no—
512C3C2×#516611yescharacteristic, direct product factor
513C3C2×C3×#46411yescharacteristic, complemented
514C2—82no—
515C2C3×#516614yesdirect product factor
516C2C2×C3×#91511yescharacteristic
517C2C2×C3×#91611yescharacteristic
518C2C2×C3×#93911yescharacteristic
subgroupG/H
centerC23×C3D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC22×C32×SD16C2
socleC23×C32C2×D4
prime radicalC22×C32×D4C22
hypercenterC22×C3×SD16D3
hypocenterC3C2×C3×#464
LCSG ▷ C2×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
3. C22×C3×#216C2semidirect
6. C3×#5166C2direct (N × H)
7. C2×C3×#816C2semidirect
8. C2×C3×Dic3⋊C8C2semidirect
10. C2×#5166C3direct (N × H)
20. C3×Q8×Dic3C22semidirect
22. C2×C3×#216C22semidirect
31. C3×#816C22semidirect
32. C3×Dic3⋊C8C22semidirect
39. C2×Q8×Dic3C2×C3semidirect
40. C22×#216C2×C3semidirect
47. #5166C2×C3direct (N × H)
48. C2×#816C2×C3semidirect
49. C2×Dic3⋊C8C2×C3semidirect
117. Q8×Dic3C22×C3semidirect
119. C2×#216C22×C3semidirect
131. #816C22×C3semidirect
132. Dic3⋊C8C22×C3semidirect
368. C2×C32#464semidirect
449. C32C2×#464semidirect
487. C2×C3#5166direct (N × H)
488. C2×C3C3×#464semidirect
512. C3C2×#5166direct (N × H)
513. C3C2×C3×#464semidirect
515. C2C3×#5166direct (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
C22×C3×#2162
C2×C3×#9362
C2×C3×#8322
C3×#51662
C2×C3×#8162
C2×C3×Dic3⋊C82
C2×C3×#4643
C2×#51663