Finite Group Databasebeta

← C2×Dic6×D12 #11902676

Subgroups — C2×Dic6×D12 #11902676

Order 1152.

3281 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D3×Dic6C212yesmaximal, complemented
2C2×#64078C211yesmaximal, characteristic, complemented
3C2×C3×C4×D12C211yesmaximal, characteristic
4C2×C3×C4×Dic6C211yesmaximal, characteristic, complemented
5C2×#63918C212yesmaximal, complemented
6C2×#63893C214yesmaximal, complemented
7C2×Dic3×D12C212yesmaximal
8C2×#63857C212yesmaximal, complemented
9Dic6×D12C2116yesmaximal, direct product factor
10C2×Q8×D12—31nomaximal
11C2×D4×Dic6—31nomaximal
12C22×#634C2212yes—
13C22×C3×D12C2211yescharacteristic
14C22×C3×C4×D3C2212yes—
15C22×C3×Dic6C2211yescharacteristic
16C22×#627C2214yes—
17C22×D3×Dic3C2214yes—
18C2×D3×Dic6—216no—
19C2×D3×Dic6C22116yescomplemented
20C2×C32×C42C2211yescharacteristic
21C2×C4⋊(C3⋊Dic3)C2212yes—
22C2×C4⋊(C3⋊Dic3)C2212yes—
23#64078C2218yescomplemented
24C2×C3×(C2×D3)⋊C4C2212yes—
25C2×C3×C4⋊Dic3C2211yescharacteristic
26C2×C3×C4⋊Dic3C2211yescharacteristic
27C2×C3×Dic3⋊C4C2212yes—
28C2×C3×C4×Dic3C2212yes—
29C3×C4×D12C2218yes—
30C3×C4×Dic6C2218yescomplemented
31C2×Dic3⋊Dic3C2214yes—
32C2×(C2×D3)⋊Dic3C2214yes—
33#63918C22116yescomplemented
34#63893C22132yescomplemented
35Dic3×D12C22116yes—
36#63857C22116yescomplemented
37C22×D3×Q8—32no—
38C23×Dic6—32no—
39C2×C3×C4×Q8—31no—
40C2×C3×C4×D4—31no—
41C2×D4×Dic3—32no—
42C2×#924—32no—
43Q8×D12—316no—
44D4×Dic6—316no—
45C2×#897—32no—
46C2×#897—34no—
47C2×#888—32no—
48C2×#878—34no—
49C2×C4×D12—32no—
50C2×C4×D12D311yescharacteristic
51C2×#869—31no—
52C2×#869—31no—
53C2×#869—32no—
54C2×#869—61no—
55C2×C4×Dic6D311yescharacteristic, complemented
56C23×C3×D3C2312yes—
57C22×C32×C4C2311yescharacteristic
58C22×C32×C4C2311yescharacteristic
59C22×C32×C4C2311yescharacteristic
60C22×C3⋊Dic3C2314yes—
61C2×#634—28no—
62C2×#634C2318yes—
63C22×C3×Dic3C2312yes—
64C22×C3×Dic3C2312yes—
65C22×C3×Dic3C2312yes—
66C2×C3×D12Q818yescomplemented
67C2×C3×D12C2314yes—
68C2×C3×C4×D3—28no—
69C2×C3×C4×D3C2318yes—
70C2×C3×Dic6D418yescomplemented
71C2×C3×Dic6C2314yes—
72C2×#627—216no—
73C2×#627C23116yes—
74C2×D3×Dic3—216no—
75C2×D3×Dic3C23116yes—
76D3×Dic6—264no—
77C32×C42C2314yes—
78C4⋊(C3⋊Dic3)C2318yes—
79C4⋊(C3⋊Dic3)C2318yes—
80C3×(C2×D3)⋊C4C2318yes—
81C3×C4⋊Dic3C2314yes—
82C3×C4⋊Dic3C2314yes—
83C3×Dic3⋊C4C2318yes—
84C3×C4×Dic3C2318yes—
85Dic3⋊Dic3C23116yes—
86(C2×D3)⋊Dic3C23116yes—
87C2×D4×Q8—91noSylow
88C22×C3×Q8—31no—
89C22×C3×D4—31no—
90C23×C3×C4—32no—
91C23×Dic3—34no—
92C2×D3×Q8—316no—
93C2×D3×Q8—616no—
94C22×D12C2×D311yescharacteristic
95C22×C4×D3—34no—
96C22×C4×D3C2×D312yes—
97C22×Dic6—32no—
98C22×Dic6—32no—
99C22×Dic6—34no—
100C22×Dic6—34no—
101C22×Dic6—316no—
102C22×Dic6—62no—
103C22×Dic6—616no—
104C22×Dic6C2×D311yescharacteristic
105C3×C4×Q8—38no—
106C3×C4×D4—38no—
107C2×C3×C4⋊C4—32no—
108C2×C3×C4⋊C4—31no—
109C2×C3×C4⋊C4—31no—
110C2×C3×C22⋊C4—32no—
111C2×C3×C42—21no—
112C2×C3×C42—32no—
113C2×C3×C42C2×D311yescharacteristic
114C2×C3×C42C2×D311yescharacteristic
115C2×#217—34no—
116D4×Dic3—316no—
117C2×(C2×D3)⋊C4—34no—
118C2×(C2×D3)⋊C4C2×D312yes—
119C2×C4⋊Dic3—32no—
120C2×C4⋊Dic3—32no—
121C2×C4⋊Dic3—32no—
122C2×C4⋊Dic3—32no—
123C2×C4⋊Dic3—34no—
124C2×C4⋊Dic3—62no—
125C2×C4⋊Dic3—62no—
126C2×C4⋊Dic3C2×D311yescharacteristic
127C2×C4⋊Dic3C2×D311yescharacteristic
128#924—316no—
129C2×Dic3⋊C4—34no—
130C2×Dic3⋊C4C2×D312yes—
131C2×C4×Dic3C2×D312yes—
132#897—316no—
133#897—332no—
134#888—316no—
135#878—332no—
136C4×D12—316no—
137C4×D12C2×D318yes—
138#869—38no—
139#869—38no—
140#869—316no—
141#869—68no—
142C4×Dic6C2×D318yescomplemented
143C23×C32C2411yescharacteristic
144C22×C3×D3—24no—
145C22×C3×D3—24no—
146C22×C3×D3C2×Q818yescomplemented
147C22×C3×D3C2414yes—
148C2×C32×C4—22no—
149C2×C32×C4C2×D414yes—
150C2×C32×C4C2×Q814yescomplemented
151C2×C32×C4C2412yes—
152C2×C32×C4C2412yes—
153C2×C32×C4C2412yes—
154C2×C3⋊Dic3—28no—
155C2×C3⋊Dic3C2418yes—
156#634—216no—
157C2×C3×Dic3—24no—
158C2×C3×Dic3—24no—
159C2×C3×Dic3C2×D418yes—
160C2×C3×Dic3C2414yes—
161C2×C3×Dic3C2414yes—
162C2×C3×Dic3C2414yes—
163C3×D12C2×Q8116yescomplemented
164C3×C4×D3—216no—
165C3×Dic6C2×D4116yescomplemented
166#627—232no—
167D3×Dic3—232no—
168C23×Q8—92no—
169D4×Q8—916no—
170C2×C4⋊Q8—92no—
171C2×C4⋊Q8—91no—
172C2×C22⋊Q8—94no—
173C2×C22⋊Q8—92no—
174C2×C4×Q8—31no—
175C2×C4×D4—31no—
176C2×C4×D4—92no—
177C24×C3—32no—
178C23×D3C22×D312yes—
179C2×C3×Q8—38no—
180C2×C3×Q8—34no—
181C2×C3×D4—38no—
182C2×C3×D4—34no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—32no—
187C22×C3×C4—32no—
188C22×C3×C4—32no—
189C22×C3×C4—38no—
190C22×C3×C4—68no—
191C22×C3×C4C22×D311yescharacteristic
192C22×C3×C4C22×D311yescharacteristic
193C22×C3×C4C22×D311yescharacteristic
194C22×C3×C4C22×D311yescharacteristic
195C22×C3×C4C22×D311yescharacteristic
196C22×C3×C4C22×D311yescharacteristic
197C22×Dic3—34no—
198C22×Dic3—34no—
199C22×Dic3—316no—
200C22×Dic3—64no—
201C22×Dic3—616no—
202C22×Dic3C22×D312yes—
203C22×Dic3C22×D312yes—
204C22×Dic3C22×D312yes—
205D3×Q8—664no—
206C2×D12Dic618yesdirect product factor
207C2×D12C22×D314yes—
208C2×C4×D3—28no—
209C2×C4×D3—316no—
210C2×C4×D3—616no—
211C2×C4×D3C22×D318yes—
212C2×Dic6—38no—
213C2×Dic6—38no—
214C2×Dic6—316no—
215C2×Dic6—316no—
216C2×Dic6—68no—
217C2×Dic6—616no—
218C2×Dic6—68no—
219C2×Dic6—68no—
220C2×Dic6—616no—
221C2×Dic6—664no—
222C2×Dic6—128no—
223C2×Dic6D1218yesdirect product factor
224C2×Dic6C22×D314yes—
225C3×C4⋊C4—34no—
226C3×C4⋊C4—38no—
227C3×C4⋊C4—34no—
228C3×C22⋊C4—38no—
229C3×C42—24no—
230C3×C42—38no—
231C3×C42C22×D314yes—
232C3×C42C22×D314yes—
233#217—316no—
234(C2×D3)⋊C4—316no—
235(C2×D3)⋊C4C22×D318yes—
236C4⋊Dic3—38no—
237C4⋊Dic3—38no—
238C4⋊Dic3—38no—
239C4⋊Dic3—38no—
240C4⋊Dic3—316no—
241C4⋊Dic3—68no—
242C4⋊Dic3—68no—
243C4⋊Dic3C22×D314yes—
244C4⋊Dic3C22×D314yes—
245Dic3⋊C4—316no—
246Dic3⋊C4C22×D318yes—
247C4×Dic3C22×D318yes—
248C22×C32C22×D412yes—
249C22×C32C22×Q812yes—
250C22×C322−1+412yes—
251C22×C32C2511yescharacteristic
252C2×C3×D3—216no—
253C2×C3×D3—24no—
254C2×C3×D3—24no—
255C2×C3×D3C22×Q818yescomplemented
256C32×C4—22no—
257C32×C4C22×D414yes—
258C32×C4C22×Q814yescomplemented
259C3⋊Dic3—28no—
260C3×Dic3—24no—
261C3×Dic3—24no—
262C3×Dic3C22×D418yes—
263C22×Q8—31no—
264C22×Q8—94no—
265C22×Q8—916no—
266C22×Q8—92no—
267C22×Q8—1816no—
268C22×D4—31no—
269C23×C4—32no—
270C23×C4—94no—
271C4⋊Q8—916no—
272C4⋊Q8—98no—
273C22⋊Q8—932no—
274C22⋊Q8—916no—
275C4×Q8—38no—
276C4×D4—38no—
277C4×D4—916no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—32no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—94no—
282C2×C4⋊C4—92no—
283C2×C4⋊C4—92no—
284C2×C22⋊C4—32no—
285C2×C22⋊C4—94no—
286C2×C42—32no—
287C2×C42D3211yescharacteristic
288C23×C3—21no—
289C23×C3—38no—
290C23×C3—34no—
291C23×C3—64no—
292C23×C3—64no—
293C23×C3C23×D311yescharacteristic
294C23×C3C23×D311yescharacteristic
295C22×D3—24no—
296C22×D3—24no—
297C22×D3C2×Dic618yescomplemented
298C22×D3C23×D314yes—
299C3×Q8—316no—
300C3×D4—316no—
301C2×C3×C4—24no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—24no—
308C2×C3×C4—24no—
309C2×C3×C4—38no—
310C2×C3×C4—34no—
311C2×C3×C4—34no—
312C2×C3×C4—34no—
313C2×C3×C4—64no—
314C2×C3×C4—64no—
315C2×C3×C4—616no—
316C2×C3×C4C2×Dic614yescomplemented
317C2×C3×C4C2×D1214yes—
318C2×C3×C4D3×D414yes—
319C2×C3×C4D3×Q814yescomplemented
320C2×C3×C4C23×D312yes—
321C2×C3×C4C23×D312yes—
322C2×C3×C4C23×D312yes—
323C2×C3×C4C23×D312yes—
324C2×C3×C4C23×D312yes—
325C2×C3×C4C23×D312yes—
326C2×Dic3—24no—
327C2×Dic3—24no—
328C2×Dic3—38no—
329C2×Dic3—38no—
330C2×Dic3—68no—
331C2×Dic3—68no—
332C2×Dic3—68no—
333C2×Dic3—632no—
334C2×Dic3—128no—
335C2×Dic3C2×D1218yes—
336C2×Dic3C23×D314yes—
337C2×Dic3C23×D314yes—
338C2×Dic3C23×D314yes—
339D12C2×Dic6116yesdirect product factor
340C4×D3—216no—
341C4×D3—632no—
342Dic6—632no—
343Dic6—616no—
344Dic6—616no—
345Dic6—632no—
346Dic6—1216no—
347Dic6C2×D12116yesdirect product factor
348C2×C32D4×Q814yescomplemented
349C2×C32C23×D411yescharacteristic
350C2×C32C23×Q811yescharacteristic
351C2×C32C2×2−1+411yescharacteristic
352C3×D3—28no—
353C24—32no—
354C2×Q8—34no—
355C2×Q8—38no—
356C2×Q8—916no—
357C2×Q8—98no—
358C2×Q8—1816no—
359C2×Q8—1864no—
360C2×Q8—188no—
361C2×D4—34no—
362C2×D4—38no—
363C22×C4—32no—
364C22×C4—32no—
365C22×C4—32no—
366C22×C4—38no—
367C22×C4—68no—
368C22×C4—916no—
369C22×C4—94no—
370C22×C4—1816no—
371C22×C4C2×D3211yescharacteristic
372C22×C4C2×D3211yescharacteristic
373C22×C4C2×D3211yescharacteristic
374C4⋊C4—34no—
375C4⋊C4—38no—
376C4⋊C4—34no—
377C4⋊C4—916no—
378C4⋊C4—98no—
379C4⋊C4—98no—
380C22⋊C4—38no—
381C22⋊C4—916no—
382C42—38no—
383C42C2×D3214yes—
384C22×C3—22no—
385C22×C3—22no—
386C22×C3—22no—
387C22×C3—21no—
388C22×C3—38no—
389C22×C3—64no—
390C22×C3—64no—
391C22×C3—616no—
392C22×C3C22×Dic612yes—
393C22×C3C22×D1212yes—
394C22×C3C2×D3×D412yes—
395C22×C3C2×D3×Q812yes—
396C22×C3#101012yes—
397C22×C3#101012yes—
398C22×C3C24×D311yescharacteristic
399C22×C3C24×D311yescharacteristic
400C2×D3—24no—
401C2×D3—24no—
402C2×D3—216no—
403C2×D3C22×Dic618yescomplemented
404C3×C4—24no—
405C3×C4—22no—
406C3×C4—22no—
407C3×C4—24no—
408C3×C4—24no—
409C3×C4—38no—
410C3×C4—64no—
411C3×C4—64no—
412C3×C4C22×Dic614yescomplemented
413C3×C4C22×D1214yes—
414C3×C4C2×D3×D414yes—
415C3×C4C2×D3×Q814yescomplemented
416Dic3—24no—
417Dic3—24no—
418Dic3—68no—
419Dic3—68no—
420Dic3—128no—
421Dic3C22×D1218yes—
422C32C2×D4×Q811yescharacteristic, complemented, Sylow
423C23—38no—
424C23—34no—
425C23—64no—
426C23—64no—
427C23C22×D3211yescharacteristic
428Q8—316no—
429Q8—1832no—
430Q8—1816no—
431D4—316no—
432C2×C4—22no—
433C2×C4—38no—
434C2×C4—34no—
435C2×C4—34no—
436C2×C4—34no—
437C2×C4—64no—
438C2×C4—64no—
439C2×C4—616no—
440C2×C4—98no—
441C2×C4—1832no—
442C2×C4—188no—
443C2×C4D3×Dic614yescomplemented
444C2×C4D3×D1214yes—
445C2×C4C22×D3212yes—
446C2×C4C22×D3212yes—
447C2×C4C22×D3212yes—
448C2×C3—21no—
449C2×C3—21no—
450C2×C3—21no—
451C2×C3—24no—
452C2×C3—68no—
453C2×C3D4×Dic614yescomplemented
454C2×C3Q8×D1214yescomplemented
455C2×C3C23×Dic611yescharacteristic
456C2×C3C23×D1211yescharacteristic
457C2×C3C22×D3×D411yescharacteristic
458C2×C3C22×D3×Q811yescharacteristic
459C2×C3C2×#101011yescharacteristic
460C2×C3C2×#101011yescharacteristic
461D3—28no—
462C22—38no—
463C22—64no—
464C22—64no—
465C22—616no—
466C22C2×D3×Dic612yes—
467C22#6429812yes—
468C22C2×D3×D1212yes—
469C22C23×D3211yescharacteristic
470C4—22no—
471C4—38no—
472C4—64no—
473C4—64no—
474C4—188no—
475C4C2×D3×Dic614yescomplemented
476C4C2×D3×D1214yes—
477C3—21no—
478C3C2×D4×Dic611yescharacteristic, complemented
479C3C2×Q8×D1211yescharacteristic, complemented
480C2—68no—
481C2Dic6×D1214yesdirect product factor
482C2C22×D3×Dic611yescharacteristic
483C2C2×#6429811yescharacteristic
484C2C22×D3×D1211yescharacteristic
subgroupG/H
centerC23C22×D32
commutator, cosocleC22×C32C25
FrattiniC22C23×D32
FittingC2×C32×C42C22
socleC23×C32C24
prime radicalC22×C3×D12C22
hypercenterC2×C42D32
hypocenterC32C2×D4×Q8
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
1. C22×D3×Dic6C2semidirect
2. C2×#64078C2semidirect
4. C2×C3×C4×Dic6C2semidirect
5. C2×#63918C2semidirect
6. C2×#63893C2semidirect
8. C2×#63857C2semidirect
9. Dic6×D12C2direct (N × H)
19. C2×D3×Dic6C22semidirect
23. #64078C22semidirect
30. C3×C4×Dic6C22semidirect
33. #63918C22semidirect
34. #63893C22semidirect
36. #63857C22semidirect
55. C2×C4×Dic6D3semidirect
66. C2×C3×D12Q8semidirect
70. C2×C3×Dic6D4semidirect
142. C4×Dic6C2×D3semidirect
146. C22×C3×D3C2×Q8semidirect
150. C2×C32×C4C2×Q8semidirect
163. C3×D12C2×Q8semidirect
165. C3×Dic6C2×D4semidirect
206. C2×D12Dic6direct (N × H)
223. C2×Dic6D12direct (N × H)
255. C2×C3×D3C22×Q8semidirect
258. C32×C4C22×Q8semidirect
297. C22×D3C2×Dic6semidirect
316. C2×C3×C4C2×Dic6semidirect
319. C2×C3×C4D3×Q8semidirect
339. D12C2×Dic6direct (N × H)
347. Dic6C2×D12direct (N × H)
348. C2×C32D4×Q8semidirect
403. C2×D3C22×Dic6semidirect
412. C3×C4C22×Dic6semidirect
415. C3×C4C2×D3×Q8semidirect
422. C32C2×D4×Q8semidirect
443. C2×C4D3×Dic6semidirect
453. C2×C3D4×Dic6semidirect
454. C2×C3Q8×D12semidirect
475. C4C2×D3×Dic6semidirect
478. C3C2×D4×Dic6semidirect
479. C3C2×Q8×D12semidirect
481. C2Dic6×D12direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D3×Dic62
C2×#640782
C2×C3×C4×D122
C2×C3×C4×Dic62
C2×#639182
C2×#638932
C2×Dic3×D122
C2×#638572
Dic6×D122
C2×Q8×D123
C2×D4×Dic63