Finite Group Databasebeta

← C2×C3×#4768 #11873424

Subgroups — C2×C3×#4768 #11873424

Order 1152.

925 conjugacy classes of subgroups.

The subgroup lattice has 477 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#936C211yesmaximal, characteristic
2C2×C3×#890C211yesmaximal, characteristic, complemented
3C2×C32×#102C211yesmaximal, characteristic
4C2×C3×#832C211yesmaximal, characteristic
5C2×C3×#816C211yesmaximal, characteristic, complemented
6C2×C3×C8×Dic3C211yesmaximal, characteristic, complemented
7C2×C3×#810C211yesmaximal, characteristic, complemented
8C3×#4768C218yesmaximal, direct product factor
9C2×C3×#488—31nomaximal
10C2×#4768C311yesmaximal, characteristic, direct product factor
11C22×C32×D4C2211yescharacteristic
12C22×C3×Dic6C2211yescharacteristic
13C22×C32×C8C2211yescharacteristic
14C2×C32×C4⋊C4C2211yescharacteristic
15C2×C3×#217—21no—
16C2×C3×#217—21no—
17C3×#936—22no—
18C3×#936C2214yes—
19C2×C3×C4⋊Dic3—21no—
20C2×C3×C4⋊Dic3C2211yescharacteristic
21C2×C3×Dic3⋊C4—21no—
22C2×C3×C4×Dic3C2211yescharacteristic
23C22×C3×C3⋊C8C2211yescharacteristic
24C3×#890—22no—
25C3×#890C2214yescomplemented
26C32×#102C2214yes—
27C3×#832C2214yes—
28C3×#816C2214yescomplemented
29C3×C8×Dic3C2214yescomplemented
30C3×#810C2214yescomplemented
31C2×C3×#125—31no—
32C2×C3×#124—31no—
33C2×#936C2×C311yescharacteristic
34C2×#890C2×C311yescharacteristic, complemented
35C3×#488—38no—
36C2×C3×#103—31no—
37C2×C3×#103—31no—
38C2×C3×#102—21no—
39C2×C3×#102—31no—
40C2×C3×#102D311yescharacteristic
41C2×C3×#102C2×C311yescharacteristic
42C2×C3×C4×C8—31no—
43C2×#832C2×C311yescharacteristic
44C2×#816C2×C311yescharacteristic, complemented
45C2×C8×Dic3C2×C311yescharacteristic, complemented
46C2×#810C2×C311yescharacteristic, complemented
47#4768C2×C318yesdirect product factor
48C24×C32—21no—
49C2×C32×D4—21no—
50C2×C32×D4—22no—
51C2×C32×D4—22no—
52C2×C32×D4C2312yes—
53C22×C32×C4—21no—
54C22×C32×C4C2311yescharacteristic
55C22×C3×Dic3—21no—
56C22×C3×Dic3—21no—
57C22×C3×Dic3D411yescharacteristic
58C22×C3×Dic3D411yescharacteristic
59C2×C3×Dic6—22no—
60C2×C3×Dic6—22no—
61C2×C3×Dic6—21no—
62C2×C3×Dic6C2312yes—
63C2×C32×C8—22no—
64C2×C32×C8C2312yes—
65C32×C4⋊C4—21no—
66C32×C4⋊C4C2312yes—
67C3×#217—24no—
68C3×#217—24no—
69C3×C4⋊Dic3—21no—
70C3×C4⋊Dic3—24no—
71C3×C4⋊Dic3C2312yes—
72C3×Dic3⋊C4—24no—
73C3×C4×Dic3D412yes—
74C3×C4×Dic3C2312yes—
75C2×C3×C3⋊C8—22no—
76C2×C3×C3⋊C8C2312yes—
77C2×#488—31noSylow
78C22×C3×Q8—31no—
79C22×C3×D4—21no—
80C22×C3×D4C2×D311yescharacteristic
81C22×C3×D4C22×C311yescharacteristic
82C22×Dic6C22×C311yescharacteristic
83C22×C3×C8—21no—
84C22×C3×C8—31no—
85C22×C3×C8C2×D311yescharacteristic
86C22×C3×C8C22×C311yescharacteristic
87C3×#125—34no—
88C3×#125—62no—
89C3×#124—34no—
90C3×#124—62no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—31no—
93C2×C3×C4⋊C4—61no—
94C2×C3×C4⋊C4—61no—
95C2×C3×C4⋊C4C2×D311yescharacteristic
96C2×C3×C4⋊C4C22×C311yescharacteristic
97C2×C3×C22⋊C4—61no—
98C2×C3×C22⋊C4—61no—
99C2×C3×C42—31no—
100C2×#217—21no—
101C2×#217—21no—
102#936—22no—
103#936C22×C314yes—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3C22×C311yescharacteristic
106C2×Dic3⋊C4—21no—
107C2×C4×Dic3C22×C311yescharacteristic
108C22×C3⋊C8C22×C311yescharacteristic
109#890—22no—
110#890C22×C314yescomplemented
111C3×#103—34no—
112C3×#103—34no—
113C3×#102—24no—
114C3×#102—34no—
115C3×#102C2×D314yes—
116C3×#102C22×C314yes—
117C3×C4×C8—34no—
118#832C22×C314yes—
119#816C22×C314yescomplemented
120C8×Dic3C22×C314yescomplemented
121#810C22×C314yescomplemented
122C23×C32—22no—
123C23×C32—42no—
124C23×C32—42no—
125C23×C32—42no—
126C23×C32C2×D411yescharacteristic
127C32×D4—22no—
128C32×D4—22no—
129C32×D4—42no—
130C2×C32×C4—22no—
131C2×C32×C4—42no—
132C2×C32×C4C2×D411yescharacteristic
133C2×C32×C4#4112yes—
134C2×C32×C4#4112yes—
135C2×C32×C4C2411yescharacteristic
136C2×C3×Dic3—22no—
137C2×C3×Dic3—22no—
138C2×C3×Dic3—22no—
139C2×C3×Dic3—22no—
140C2×C3×Dic3—42no—
141C2×C3×Dic3—42no—
142C2×C3×Dic3C2×D412yes—
143C2×C3×Dic3C2×D412yes—
144C3×Dic6—22no—
145C3×Dic6—22no—
146C3×Dic6—42no—
147C32×C8—22no—
148C3×C3⋊C8—22no—
149C2×#125—31no—
150C2×#124—31no—
151#488—38no—
152C2×#103—31no—
153C2×#103—31no—
154C2×#102—31no—
155C2×#102C3×D311yescharacteristic
156C2×C4×C8—31no—
157C24×C3—21no—
158C24×C3—21no—
159C24×C3—41no—
160C2×C3×Q8—32no—
161C2×C3×Q8—61no—
162C2×C3×Q8—62no—
163C2×C3×Q8—62no—
164C2×C3×D4—24no—
165C2×C3×D4—22no—
166C2×C3×D4—21no—
167C2×C3×D4—22no—
168C2×C3×D4—21no—
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C2×C3×D4—22no—
172C2×C3×D4—41no—
173C2×C3×D4—42no—
174C2×C3×D4C22×D312yes—
175C2×C3×D4C23×C312yes—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—31no—
180C22×C3×C4—31no—
181C22×C3×C4—41no—
182C22×C3×C4—61no—
183C22×C3×C4—61no—
184C22×C3×C4C22×D311yescharacteristic
185C22×C3×C4C23×C311yescharacteristic
186C22×Dic3—21no—
187C22×Dic3—21no—
188C22×Dic3C3×D411yescharacteristic
189C22×Dic3C3×D411yescharacteristic
190C2×Dic6—22no—
191C2×Dic6—22no—
192C2×Dic6—21no—
193C2×Dic6C23×C312yes—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C2×C3×C8—22no—
197C2×C3×C8—32no—
198C2×C3×C8—42no—
199C2×C3×C8—62no—
200C2×C3×C8C22×D312yes—
201C2×C3×C8C23×C312yes—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—32no—
206C3×C4⋊C4—41no—
207C3×C4⋊C4—64no—
208C3×C4⋊C4—64no—
209C3×C4⋊C4—61no—
210C3×C4⋊C4C22×D312yes—
211C3×C4⋊C4C23×C312yes—
212C3×C22⋊C4—64no—
213C3×C22⋊C4—64no—
214C3×C42—32no—
215C3×C42—32no—
216#217—24no—
217#217—24no—
218C4⋊Dic3—21no—
219C4⋊Dic3—24no—
220C4⋊Dic3C23×C312yes—
221Dic3⋊C4—24no—
222C4×Dic3C3×D412yes—
223C4×Dic3C23×C312yes—
224C2×C3⋊C8—22no—
225C2×C3⋊C8C23×C312yes—
226C22×C32—42no—
227C22×C32—42no—
228C22×C32—42no—
229C22×C32—84no—
230C22×C32#12412yes—
231C22×C32#13512yes—
232C22×C32#13512yes—
233C22×C32C22×D411yescharacteristic
234C32×C4—42no—
235C32×C4#12412yes—
236C32×C4C2×#4111yescharacteristic
237C32×C4C2×#4111yescharacteristic
238C3×Dic3—22no—
239C3×Dic3—22no—
240C3×Dic3—42no—
241C3×Dic3—42no—
242C22×Q8—31no—
243C22×D4C2×C3×D311yescharacteristic
244C22×C8—31no—
245C22×C8C2×C3×D311yescharacteristic
246#125—34no—
247#125—62no—
248#124—34no—
249#124—62no—
250C2×C4⋊C4—31no—
251C2×C4⋊C4—61no—
252C2×C4⋊C4—61no—
253C2×C4⋊C4C2×C3×D311yescharacteristic
254C2×C22⋊C4—61no—
255C2×C22⋊C4—61no—
256C2×C42—31no—
257#103—34no—
258#103—34no—
259#102—34no—
260#102C2×C3×D314yes—
261C4×C8—34no—
262C23×C3—22no—
263C23×C3—22no—
264C23×C3—21no—
265C23×C3—42no—
266C23×C3—44no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3—42no—
271C23×C3—42no—
272C23×C3—42no—
273C23×C3—82no—
274C23×C3—82no—
275C23×C3D3×D411yescharacteristic
276C23×C3C2×C3×D411yescharacteristic
277C3×Q8—62no—
278C3×Q8—62no—
279C3×Q8—122no—
280C3×D4—24no—
281C3×D4—22no—
282C3×D4—22no—
283C3×D4—22no—
284C3×D4—22no—
285C3×D4—42no—
286C3×D4—44no—
287C3×D4—42no—
288C3×D4—42no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—21no—
292C2×C3×C4—22no—
293C2×C3×C4—21no—
294C2×C3×C4—22no—
295C2×C3×C4—32no—
296C2×C3×C4—32no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—62no—
301C2×C3×C4—62no—
302C2×C3×C4—62no—
303C2×C3×C4—62no—
304C2×C3×C4—82no—
305C2×C3×C4—122no—
306C2×C3×C4—122no—
307C2×C3×C4#23512yes—
308C2×C3×C4D3×D411yescharacteristic
309C2×C3×C4#23712yes—
310C2×C3×C4C2×C3×D411yescharacteristic
311C2×C3×C4C3×#4112yes—
312C2×C3×C4C3×#4112yes—
313C2×C3×C4C23×D311yescharacteristic
314C2×C3×C4C24×C311yescharacteristic
315C2×Dic3—22no—
316C2×Dic3—22no—
317C2×Dic3—22no—
318C2×Dic3—22no—
319C2×Dic3—42no—
320C2×Dic3—42no—
321C2×Dic3C2×C3×D412yes—
322C2×Dic3C2×C3×D412yes—
323Dic6—22no—
324Dic6—22no—
325Dic6—42no—
326C3×C8—22no—
327C3×C8—22no—
328C3×C8—42no—
329C3×C8—62no—
330C3⋊C8—22no—
331C2×C32—82no—
332C2×C32#48814yescomplemented
333C2×C32C2×#12411yescharacteristic
334C2×C32C2×#13511yescharacteristic
335C2×C32C2×#13511yescharacteristic
336C24—21no—
337C2×Q8—32no—
338C2×Q8—61no—
339C2×Q8—62no—
340C2×Q8—62no—
341C2×D4—22no—
342C2×D4—21no—
343C2×D4—22no—
344C2×D4C22×C3×D312yes—
345C22×C4—21no—
346C22×C4—31no—
347C22×C4—31no—
348C22×C4—61no—
349C22×C4—61no—
350C22×C4C22×C3×D311yescharacteristic
351C2×C8—22no—
352C2×C8—32no—
353C2×C8—62no—
354C2×C8C22×C3×D312yes—
355C4⋊C4—21no—
356C4⋊C4—32no—
357C4⋊C4—64no—
358C4⋊C4—64no—
359C4⋊C4—61no—
360C4⋊C4C22×C3×D312yes—
361C22⋊C4—64no—
362C22⋊C4—64no—
363C42—32no—
364C42—32no—
365C22×C3—22no—
366C22×C3—22no—
367C22×C3—22no—
368C22×C3—21no—
369C22×C3—44no—
370C22×C3—42no—
371C22×C3—42no—
372C22×C3—42no—
373C22×C3—42no—
374C22×C3—42no—
375C22×C3—42no—
376C22×C3—82no—
377C22×C3—82no—
378C22×C3—88no—
379C22×C3—84no—
380C22×C3—84no—
381C22×C3#88512yes—
382C22×C3#91212yes—
383C22×C3#91612yes—
384C22×C3C3×#12412yes—
385C22×C3C3×#13512yes—
386C22×C3C3×#13512yes—
387C22×C3C2×D3×D411yescharacteristic
388C22×C3C22×C3×D411yescharacteristic
389C3×C4—22no—
390C3×C4—21no—
391C3×C4—21no—
392C3×C4—42no—
393C3×C4—42no—
394C3×C4—62no—
395C3×C4—62no—
396C3×C4—82no—
397C3×C4—122no—
398C3×C4—122no—
399C3×C4#88512yes—
400C3×C4C3×#12412yes—
401C3×C4C2×#23511yescharacteristic
402C3×C4C2×#23711yescharacteristic
403C3×C4C2×C3×#4111yescharacteristic
404C3×C4C2×C3×#4111yescharacteristic
405Dic3—22no—
406Dic3—22no—
407Dic3—42no—
408Dic3—42no—
409C32C2×#48811yescharacteristic, complemented, Sylow
410C23—22no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23C3×D3×D411yescharacteristic
415Q8—62no—
416Q8—62no—
417Q8—122no—
418D4—22no—
419D4—22no—
420D4—42no—
421C2×C4—22no—
422C2×C4—32no—
423C2×C4—32no—
424C2×C4—42no—
425C2×C4—62no—
426C2×C4—62no—
427C2×C4—62no—
428C2×C4—62no—
429C2×C4—122no—
430C2×C4—122no—
431C2×C4C3×#23512yes—
432C2×C4C3×D3×D411yescharacteristic
433C2×C4C3×#23712yes—
434C2×C4C23×C3×D311yescharacteristic
435C8—22no—
436C8—62no—
437C2×C3—21no—
438C2×C3—21no—
439C2×C3—21no—
440C2×C3—24no—
441C2×C3—84no—
442C2×C3—82no—
443C2×C3—82no—
444C2×C3#476814yesdirect product factor
445C2×C3C3×#48814yescomplemented
446C2×C3C2×#88511yescharacteristic
447C2×C3C2×#91211yescharacteristic
448C2×C3C2×#91611yescharacteristic
449C2×C3C2×C3×#12411yescharacteristic
450C2×C3C2×C3×#13511yescharacteristic
451C2×C3C2×C3×#13511yescharacteristic
452C22—42no—
453C22—42no—
454C22—42no—
455C22—84no—
456C22C3×#88512yes—
457C22C3×#91212yes—
458C22C3×#91612yes—
459C22C2×C3×D3×D411yescharacteristic
460C4—42no—
461C4—62no—
462C4—62no—
463C4—122no—
464C4—122no—
465C4C3×#88512yes—
466C4C2×C3×#23511yescharacteristic
467C4C2×C3×#23711yescharacteristic
468C3—21no—
469C3C2×#476811yescharacteristic, direct product factor
470C3C2×C3×#48811yescharacteristic, complemented
471C2—82no—
472C2C3×#476814yesdirect product factor
473C2C2×C3×#88511yescharacteristic
474C2C2×C3×#91211yescharacteristic
475C2C2×C3×#91611yescharacteristic
subgroupG/H
centerC23×C3D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC2×C32×#102C2
socleC23×C32C2×D4
prime radicalC22×C32×D4C22
hypercenterC2×C3×#102D3
hypocenterC3C2×C3×#488
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C2×C3×#102

Complements

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

NH ≅ G/Ntype
2. C2×C3×#890C2semidirect
5. C2×C3×#816C2semidirect
6. C2×C3×C8×Dic3C2semidirect
7. C2×C3×#810C2semidirect
8. C3×#4768C2direct (N × H)
10. C2×#4768C3direct (N × H)
25. C3×#890C22semidirect
28. C3×#816C22semidirect
29. C3×C8×Dic3C22semidirect
30. C3×#810C22semidirect
34. C2×#890C2×C3semidirect
44. C2×#816C2×C3semidirect
45. C2×C8×Dic3C2×C3semidirect
46. C2×#810C2×C3semidirect
47. #4768C2×C3direct (N × H)
110. #890C22×C3semidirect
119. #816C22×C3semidirect
120. C8×Dic3C22×C3semidirect
121. #810C22×C3semidirect
332. C2×C32#488semidirect
409. C32C2×#488semidirect
444. C2×C3#4768direct (N × H)
445. C2×C3C3×#488semidirect
469. C3C2×#4768direct (N × H)
470. C3C2×C3×#488semidirect
472. C2C3×#4768direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#9362
C2×C3×#8902
C2×C32×#1022
C2×C3×#8322
C2×C3×#8162
C2×C3×C8×Dic32
C2×C3×#8102
C3×#47682
C2×C3×#4883
C2×#47683