Finite Group Databasebeta

← C2×C3×D4×Dic6 #11902652

Subgroups — C2×C3×D4×Dic6 #11902652

Order 1152.

3349 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
1C23×C3×Dic6C212yesmaximal, complemented
2C2×C32×C4×D4C211yesmaximal, characteristic
3C2×C3×D4×Dic3C212yesmaximal
4C2×C3×#924C212yesmaximal, complemented
5C3×D4×Dic6C2116yesmaximal, direct product factor
6C2×C3×#888C212yesmaximal, complemented
7C2×C3×#878C214yesmaximal, complemented
8C2×C3×#869C211yesmaximal, characteristic, complemented
9C2×C3×C4×Dic6C211yesmaximal, characteristic, complemented
10C2×C3×D4×Q8—31nomaximal
11C2×D4×Dic6C311yesmaximal, characteristic, direct product factor
12C22×C32×D4C2211yescharacteristic
13C23×C32×C4C2212yes—
14C23×C3×Dic3C2214yes—
15C22×C3×Dic6—216no—
16C22×C3×Dic6C2214yes—
17C22×C3×Dic6C2212yes—
18C22×C3×Dic6C2211yescharacteristic
19C22×C3×Dic6C22116yescomplemented
20C32×C4×D4C2218yes—
21C2×C32×C4⋊C4C2211yescharacteristic
22C2×C32×C22⋊C4C2212yes—
23C2×C32×C42C2211yescharacteristic
24C2×C3×#217C2214yes—
25C3×D4×Dic3C22116yes—
26C2×C3×C4⋊Dic3C2212yes—
27C2×C3×C4⋊Dic3C2211yescharacteristic
28C2×C3×C4⋊Dic3C2212yes—
29C3×#924C22116yescomplemented
30C2×C3×Dic3⋊C4C2214yes—
31C2×C3×Dic3⋊C4C2212yes—
32C2×C3×C4×Dic3C2212yes—
33C3×#888C22116yescomplemented
34C3×#878C22132yescomplemented
35C3×#869C2218yescomplemented
36C3×C4×Dic6C2218yescomplemented
37C23×C3×Q8—32no—
38C23×Dic6C2×C312yescomplemented
39C3×D4×Q8—316no—
40C2×C3×C4⋊Q8—32no—
41C2×C3×C4⋊Q8—31no—
42C2×C3×C22⋊Q8—32no—
43C2×C3×C22⋊Q8—34no—
44C2×C3×C4×Q8—31no—
45C2×C3×C4×D4—21no—
46C2×C3×C4×D4—32no—
47C2×C3×C4×D4D311yescharacteristic
48C2×C3×C4×D4C2×C311yescharacteristic
49C2×D4×Dic3C2×C312yes—
50C2×#924C2×C312yescomplemented
51D4×Dic6C2×C3116yesdirect product factor
52C2×#888C2×C312yescomplemented
53C2×#878C2×C314yescomplemented
54C2×#869C2×C311yescharacteristic, complemented
55C2×C4×Dic6C2×C311yescharacteristic, complemented
56C24×C32C2312yes—
57C2×C32×D4Q818yescomplemented
58C2×C32×D4C2314yes—
59C22×C32×C4—28no—
60C22×C32×C4C2311yescharacteristic
61C22×C32×C4C2312yes—
62C22×C32×C4C2318yes—
63C22×C32×C4C2311yescharacteristic
64C22×C32×C4C2311yescharacteristic
65C22×C3×Dic3—216no—
66C22×C3×Dic3C23116yes—
67C22×C3×Dic3C2312yes—
68C22×C3×Dic3C2314yes—
69C22×C3×Dic3C2312yes—
70C2×C3×Dic6—28no—
71C2×C3×Dic6—216no—
72C2×C3×Dic6—264no—
73C2×C3×Dic6D418yesdirect product factor
74C2×C3×Dic6C23116yes—
75C2×C3×Dic6C2318yes—
76C2×C3×Dic6C2314yes—
77C32×C4⋊C4C2314yes—
78C32×C22⋊C4C2318yes—
79C32×C42C2314yes—
80C3×#217C23116yes—
81C3×C4⋊Dic3C2318yes—
82C3×C4⋊Dic3C2318yes—
83C3×C4⋊Dic3C2314yes—
84C3×Dic3⋊C4C2318yes—
85C3×Dic3⋊C4C23116yes—
86C3×C4×Dic3C2318yes—
87C2×D4×Q8—31noSylow
88C22×C3×Q8—31no—
89C22×C3×Q8—34no—
90C22×C3×Q8—32no—
91C22×C3×Q8—316no—
92C22×C3×Q8—616no—
93C22×C3×D4—21no—
94C22×C3×D4C2×D311yescharacteristic
95C22×C3×D4C22×C311yescharacteristic
96C23×C3×C4—22no—
97C23×C3×C4—34no—
98C23×C3×C4C2×D312yes—
99C23×C3×C4C22×C312yes—
100C23×Dic3C22×C314yes—
101C22×Dic6—216no—
102C22×Dic6C22×C312yes—
103C22×Dic6C22×C314yes—
104C22×Dic6C22×C311yescharacteristic
105C22×Dic6C22×C3116yescomplemented
106C3×C4⋊Q8—316no—
107C3×C4⋊Q8—38no—
108C3×C22⋊Q8—316no—
109C3×C22⋊Q8—332no—
110C3×C4×Q8—38no—
111C3×C4×D4—28no—
112C3×C4×D4—316no—
113C3×C4×D4C2×D318yes—
114C3×C4×D4C22×C318yes—
115C2×C3×C4⋊C4—21no—
116C2×C3×C4⋊C4—32no—
117C2×C3×C4⋊C4—34no—
118C2×C3×C4⋊C4—32no—
119C2×C3×C4⋊C4—31no—
120C2×C3×C4⋊C4—32no—
121C2×C3×C4⋊C4C2×D311yescharacteristic
122C2×C3×C4⋊C4C22×C311yescharacteristic
123C2×C3×C22⋊C4—22no—
124C2×C3×C22⋊C4—34no—
125C2×C3×C22⋊C4C2×D312yes—
126C2×C3×C22⋊C4C22×C312yes—
127C2×C3×C42—21no—
128C2×C3×C42—32no—
129C2×C3×C42C2×D311yescharacteristic
130C2×C3×C42C22×C311yescharacteristic
131C2×#217C22×C314yes—
132D4×Dic3C22×C3116yes—
133C2×C4⋊Dic3C22×C311yescharacteristic
134C2×C4⋊Dic3C22×C312yes—
135C2×C4⋊Dic3C22×C312yes—
136#924C22×C3116yescomplemented
137C2×Dic3⋊C4C22×C312yes—
138C2×Dic3⋊C4C22×C314yes—
139C2×C4×Dic3C22×C312yes—
140#888C22×C3116yescomplemented
141#878C22×C3132yescomplemented
142#869C22×C318yescomplemented
143C4×Dic6C22×C318yescomplemented
144C23×C32—24no—
145C23×C32—24no—
146C23×C32C2×Q818yescomplemented
147C23×C32C2414yes—
148C23×C32C2411yescharacteristic
149C32×D4C2×Q8116yescomplemented
150C2×C32×C4—24no—
151C2×C32×C4—22no—
152C2×C32×C4—216no—
153C2×C32×C4C2×D414yes—
154C2×C32×C4C2×Q814yescomplemented
155C2×C32×C4C2412yes—
156C2×C32×C4C2414yes—
157C2×C32×C4C2412yes—
158C2×C32×C4C2412yes—
159C2×C3×Dic3—28no—
160C2×C3×Dic3—24no—
161C2×C3×Dic3—232no—
162C2×C3×Dic3C2×D418yes—
163C2×C3×Dic3C2414yes—
164C2×C3×Dic3C2418yes—
165C2×C3×Dic3C2414yes—
166C3×Dic6—216no—
167C3×Dic6—232no—
168C3×Dic6C2×D4116yesdirect product factor
169C23×Q8—32no—
170D4×Q8—316no—
171C2×C4⋊Q8—32no—
172C2×C4⋊Q8—31no—
173C2×C22⋊Q8—32no—
174C2×C22⋊Q8—34no—
175C2×C4×Q8—31no—
176C2×C4×D4—32no—
177C2×C4×D4C3×D311yescharacteristic
178C24×C3—22no—
179C24×C3C22×D312yes—
180C24×C3C23×C312yes—
181C2×C3×Q8—38no—
182C2×C3×Q8—34no—
183C2×C3×Q8—316no—
184C2×C3×Q8—38no—
185C2×C3×Q8—616no—
186C2×C3×Q8—664no—
187C2×C3×Q8—68no—
188C2×C3×D4—28no—
189C2×C3×D4—24no—
190C2×C3×D4Dic618yesdirect product factor
191C2×C3×D4C3×Q818yescomplemented
192C2×C3×D4C22×D314yes—
193C2×C3×D4C23×C314yes—
194C22×C3×C4—28no—
195C22×C3×C4—28no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—22no—
200C22×C3×C4—28no—
201C22×C3×C4—32no—
202C22×C3×C4—316no—
203C22×C3×C4—32no—
204C22×C3×C4—34no—
205C22×C3×C4—48no—
206C22×C3×C4—616no—
207C22×C3×C4C22×D318yes—
208C22×C3×C4C22×D311yescharacteristic
209C22×C3×C4C22×D312yes—
210C22×C3×C4C22×D311yescharacteristic
211C22×C3×C4C22×D311yescharacteristic
212C22×C3×C4C23×C311yescharacteristic
213C22×C3×C4C23×C312yes—
214C22×C3×C4C23×C318yes—
215C22×C3×C4C23×C311yescharacteristic
216C22×C3×C4C23×C311yescharacteristic
217C22×Dic3—216no—
218C22×Dic3C23×C3116yes—
219C22×Dic3C23×C312yes—
220C22×Dic3C23×C314yes—
221C22×Dic3C23×C312yes—
222C2×Dic6—28no—
223C2×Dic6—216no—
224C2×Dic6—264no—
225C2×Dic6C3×D418yesdirect product factor
226C2×Dic6C23×C318yes—
227C2×Dic6C23×C3116yes—
228C2×Dic6C23×C314yes—
229C3×C4⋊C4—24no—
230C3×C4⋊C4—38no—
231C3×C4⋊C4—316no—
232C3×C4⋊C4—38no—
233C3×C4⋊C4—34no—
234C3×C4⋊C4—38no—
235C3×C4⋊C4C22×D314yes—
236C3×C4⋊C4C23×C314yes—
237C3×C22⋊C4—28no—
238C3×C22⋊C4—316no—
239C3×C22⋊C4C22×D318yes—
240C3×C22⋊C4C23×C318yes—
241C3×C42—24no—
242C3×C42—38no—
243C3×C42C22×D314yes—
244C3×C42C23×C314yes—
245#217C23×C3116yes—
246C4⋊Dic3C23×C314yes—
247C4⋊Dic3C23×C318yes—
248C4⋊Dic3C23×C318yes—
249Dic3⋊C4C23×C318yes—
250Dic3⋊C4C23×C3116yes—
251C4×Dic3C23×C318yes—
252C22×C32—216no—
253C22×C32—24no—
254C22×C32—24no—
255C22×C32C22×D412yes—
256C22×C32C22×Q812yes—
257C22×C32C22×Q818yescomplemented
258C22×C322−1+412yes—
259C22×C32C2511yescharacteristic
260C32×C4—24no—
261C32×C4—22no—
262C32×C4C22×D414yes—
263C32×C4C22×Q814yescomplemented
264C3×Dic3—28no—
265C3×Dic3—24no—
266C3×Dic3C22×D418yes—
267C22×Q8—31no—
268C22×Q8—34no—
269C22×Q8—32no—
270C22×Q8—316no—
271C22×Q8—616no—
272C22×D4C2×C3×D311yescharacteristic
273C23×C4—34no—
274C23×C4C2×C3×D312yes—
275C4⋊Q8—316no—
276C4⋊Q8—38no—
277C22⋊Q8—316no—
278C22⋊Q8—332no—
279C4×Q8—38no—
280C4×D4—316no—
281C4×D4C2×C3×D318yes—
282C2×C4⋊C4—32no—
283C2×C4⋊C4—32no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—34no—
286C2×C4⋊C4—32no—
287C2×C4⋊C4C2×C3×D311yescharacteristic
288C2×C22⋊C4—34no—
289C2×C22⋊C4C2×C3×D312yes—
290C2×C42—32no—
291C2×C42C2×C3×D311yescharacteristic
292C23×C3—24no—
293C23×C3—24no—
294C23×C3—24no—
295C23×C3—24no—
296C23×C3—21no—
297C23×C3—24no—
298C23×C3—28no—
299C23×C3—44no—
300C23×C3—44no—
301C23×C3C2×Dic618yescomplemented
302C23×C3C2×C3×Q818yescomplemented
303C23×C3C23×D314yes—
304C23×C3C23×D311yescharacteristic
305C23×C3C24×C314yes—
306C23×C3C24×C311yescharacteristic
307C3×Q8—316no—
308C3×Q8—632no—
309C3×Q8—616no—
310C3×D4—216no—
311C3×D4C2×Dic6116yesdirect product factor
312C3×D4C2×C3×Q8116yescomplemented
313C2×C3×C4—24no—
314C2×C3×C4—22no—
315C2×C3×C4—24no—
316C2×C3×C4—22no—
317C2×C3×C4—216no—
318C2×C3×C4—216no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—24no—
325C2×C3×C4—38no—
326C2×C3×C4—34no—
327C2×C3×C4—34no—
328C2×C3×C4—38no—
329C2×C3×C4—416no—
330C2×C3×C4—42no—
331C2×C3×C4—44no—
332C2×C3×C4—632no—
333C2×C3×C4—68no—
334C2×C3×C4—64no—
335C2×C3×C4C2×Dic614yescomplemented
336C2×C3×C4D3×D414yes—
337C2×C3×C4C2×C3×D414yes—
338C2×C3×C4C2×C3×Q814yescomplemented
339C2×C3×C4C23×D312yes—
340C2×C3×C4C23×D314yes—
341C2×C3×C4C23×D312yes—
342C2×C3×C4C23×D312yes—
343C2×C3×C4C24×C312yes—
344C2×C3×C4C24×C314yes—
345C2×C3×C4C24×C312yes—
346C2×C3×C4C24×C312yes—
347C2×Dic3—28no—
348C2×Dic3—24no—
349C2×Dic3—232no—
350C2×Dic3C2×C3×D418yes—
351C2×Dic3C24×C314yes—
352C2×Dic3C24×C318yes—
353C2×Dic3C24×C314yes—
354Dic6—216no—
355Dic6—232no—
356Dic6C2×C3×D4116yesdirect product factor
357C2×C32—28no—
358C2×C32D4×Q814yescomplemented
359C2×C32C23×D411yescharacteristic
360C2×C32C23×Q811yescharacteristic
361C2×C32C2×2−1+411yescharacteristic
362C24C22×C3×D312yes—
363C2×Q8—34no—
364C2×Q8—316no—
365C2×Q8—38no—
366C2×Q8—38no—
367C2×Q8—616no—
368C2×Q8—664no—
369C2×Q8—68no—
370C2×D4C3×Dic618yesdirect product factor
371C2×D4C22×C3×D314yes—
372C22×C4—28no—
373C22×C4—32no—
374C22×C4—32no—
375C22×C4—34no—
376C22×C4—316no—
377C22×C4—616no—
378C22×C4C22×C3×D311yescharacteristic
379C22×C4C22×C3×D312yes—
380C22×C4C22×C3×D318yes—
381C22×C4C22×C3×D311yescharacteristic
382C22×C4C22×C3×D311yescharacteristic
383C4⋊C4—38no—
384C4⋊C4—38no—
385C4⋊C4—34no—
386C4⋊C4—316no—
387C4⋊C4—38no—
388C4⋊C4C22×C3×D314yes—
389C22⋊C4—316no—
390C22⋊C4C22×C3×D318yes—
391C42—38no—
392C42C22×C3×D314yes—
393C22×C3—24no—
394C22×C3—24no—
395C22×C3—24no—
396C22×C3—24no—
397C22×C3—216no—
398C22×C3—216no—
399C22×C3—22no—
400C22×C3—28no—
401C22×C3—22no—
402C22×C3—22no—
403C22×C3—21no—
404C22×C3—44no—
405C22×C3—416no—
406C22×C3—44no—
407C22×C3C22×Dic612yes—
408C22×C3C22×Dic618yescomplemented
409C22×C3C2×D3×D412yes—
410C22×C3#101012yes—
411C22×C3C22×C3×D412yes—
412C22×C3C22×C3×Q812yes—
413C22×C3C22×C3×Q818yescomplemented
414C22×C3C3×2−1+412yes—
415C22×C3C24×D311yescharacteristic
416C22×C3C25×C311yescharacteristic
417C3×C4—24no—
418C3×C4—22no—
419C3×C4—24no—
420C3×C4—22no—
421C3×C4—24no—
422C3×C4—24no—
423C3×C4—38no—
424C3×C4—44no—
425C3×C4—42no—
426C3×C4—68no—
427C3×C4—64no—
428C3×C4C22×Dic614yescomplemented
429C3×C4C2×D3×D414yes—
430C3×C4C22×C3×D414yes—
431C3×C4C22×C3×Q814yescomplemented
432Dic3—28no—
433Dic3—24no—
434Dic3C22×C3×D418yes—
435C32C2×D4×Q811yescharacteristic, complemented, Sylow
436C23—24no—
437C23—24no—
438C23C2×C3×Dic618yescomplemented
439C23C23×C3×D314yes—
440C23C23×C3×D311yescharacteristic
441Q8—316no—
442Q8—632no—
443Q8—616no—
444D4C2×C3×Dic6116yesdirect product factor
445C2×C4—24no—
446C2×C4—22no—
447C2×C4—216no—
448C2×C4—38no—
449C2×C4—34no—
450C2×C4—34no—
451C2×C4—38no—
452C2×C4—632no—
453C2×C4—68no—
454C2×C4—64no—
455C2×C4C2×C3×Dic614yescomplemented
456C2×C4C3×D3×D414yes—
457C2×C4C23×C3×D312yes—
458C2×C4C23×C3×D314yes—
459C2×C4C23×C3×D312yes—
460C2×C4C23×C3×D312yes—
461C2×C3—28no—
462C2×C3—28no—
463C2×C3—24no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—21no—
467C2×C3—48no—
468C2×C3D4×Dic614yesdirect product factor
469C2×C3C3×D4×Q814yescomplemented
470C2×C3C23×Dic611yescharacteristic
471C2×C3C22×D3×D411yescharacteristic
472C2×C3C2×#101011yescharacteristic
473C2×C3C23×C3×D411yescharacteristic
474C2×C3C23×C3×Q811yescharacteristic
475C2×C3C2×C3×2−1+411yescharacteristic
476C22—216no—
477C22—24no—
478C22—24no—
479C22C22×C3×Dic612yes—
480C22C22×C3×Dic618yescomplemented
481C22C2×C3×D3×D412yes—
482C22C3×#101012yes—
483C22C24×C3×D311yescharacteristic
484C4—24no—
485C4—22no—
486C4—38no—
487C4—68no—
488C4—64no—
489C4C22×C3×Dic614yescomplemented
490C4C2×C3×D3×D414yes—
491C3—21no—
492C3C2×D4×Dic611yescharacteristic, direct product factor
493C3C2×C3×D4×Q811yescharacteristic, complemented
494C2—28no—
495C2C3×D4×Dic614yesdirect product factor
496C2C23×C3×Dic611yescharacteristic
497C2C22×C3×D3×D411yescharacteristic
498C2C2×C3×#101011yescharacteristic
subgroupG/H
centerC23×C3C23×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC2×C32×C4×D4C2
socleC23×C32C24
prime radicalC22×C32×D4C22
hypercenterC2×C3×C4×D4D3
hypocenterC3C2×C3×D4×Q8
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×D4

Complements

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

NH ≅ G/Ntype
1. C23×C3×Dic6C2semidirect
4. C2×C3×#924C2semidirect
5. C3×D4×Dic6C2direct (N × H)
6. C2×C3×#888C2semidirect
7. C2×C3×#878C2semidirect
8. C2×C3×#869C2semidirect
9. C2×C3×C4×Dic6C2semidirect
11. C2×D4×Dic6C3direct (N × H)
19. C22×C3×Dic6C22semidirect
29. C3×#924C22semidirect
33. C3×#888C22semidirect
34. C3×#878C22semidirect
35. C3×#869C22semidirect
36. C3×C4×Dic6C22semidirect
38. C23×Dic6C2×C3semidirect
50. C2×#924C2×C3semidirect
51. D4×Dic6C2×C3direct (N × H)
52. C2×#888C2×C3semidirect
53. C2×#878C2×C3semidirect
54. C2×#869C2×C3semidirect
55. C2×C4×Dic6C2×C3semidirect
57. C2×C32×D4Q8semidirect
73. C2×C3×Dic6D4direct (N × H)
105. C22×Dic6C22×C3semidirect
136. #924C22×C3semidirect
140. #888C22×C3semidirect
141. #878C22×C3semidirect
142. #869C22×C3semidirect
143. C4×Dic6C22×C3semidirect
146. C23×C32C2×Q8semidirect
149. C32×D4C2×Q8semidirect
154. C2×C32×C4C2×Q8semidirect
168. C3×Dic6C2×D4direct (N × H)
190. C2×C3×D4Dic6direct (N × H)
191. C2×C3×D4C3×Q8semidirect
225. C2×Dic6C3×D4direct (N × H)
257. C22×C32C22×Q8semidirect
263. C32×C4C22×Q8semidirect
301. C23×C3C2×Dic6semidirect
302. C23×C3C2×C3×Q8semidirect
311. C3×D4C2×Dic6direct (N × H)
312. C3×D4C2×C3×Q8semidirect
335. C2×C3×C4C2×Dic6semidirect
338. C2×C3×C4C2×C3×Q8semidirect
356. Dic6C2×C3×D4direct (N × H)
358. C2×C32D4×Q8semidirect
370. C2×D4C3×Dic6direct (N × H)
408. C22×C3C22×Dic6semidirect
413. C22×C3C22×C3×Q8semidirect
428. C3×C4C22×Dic6semidirect
431. C3×C4C22×C3×Q8semidirect
435. C32C2×D4×Q8semidirect
438. C23C2×C3×Dic6semidirect
444. D4C2×C3×Dic6direct (N × H)
455. C2×C4C2×C3×Dic6semidirect
468. C2×C3D4×Dic6direct (N × H)
469. C2×C3C3×D4×Q8semidirect
480. C22C22×C3×Dic6semidirect
489. C4C22×C3×Dic6semidirect
492. C3C2×D4×Dic6direct (N × H)
493. C3C2×C3×D4×Q8semidirect
495. C2C3×D4×Dic6direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×Dic62
C2×C32×C4×D42
C2×C3×D4×Dic32
C2×C3×#9242
C3×D4×Dic62
C2×C3×#8882
C2×C3×#8782
C2×C3×#8692
C2×C3×C4×Dic62
C2×C3×D4×Q83
C2×D4×Dic63