Finite Group Databasebeta

← C2×Dic3×#213 #11859836

Subgroups — C2×Dic3×#213 #11859836

Order 1152.

1333 conjugacy classes of subgroups.

The subgroup lattice has 525 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×Dic3C211yesmaximal, characteristic, complemented
2C22×C3×#213C211yesmaximal, characteristic
3C2×Dic3×D12C211yesmaximal, characteristic, complemented
4C2×#63659C211yesmaximal, characteristic, complemented
5Dic3×#213C218yesmaximal, direct product factor
6C2×#63575C211yesmaximal, characteristic, complemented
7C2×#63562C211yesmaximal, characteristic, complemented
8C2×Dic3×C3⋊C8C211yesmaximal, characteristic, complemented
9C2×Dic3×D8—31nomaximal
10C2×C4×#213—31nomaximal
11C22×C32×D4C2211yescharacteristic
12C23×C3×Dic3—21no—
13C22×C3×D12C2211yescharacteristic
14C22×D3×Dic3—21no—
15C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C22×C3⋊(C3⋊C8)C2211yescharacteristic
17C2×C3×#217—21no—
18C3×D4×Dic3—22no—
19C3×D4×Dic3C2214yescomplemented
20C2×C3×#213C418yescomplemented
21C2×C3×#213C2214yes—
22C2×C3×C4⋊Dic3C2211yescharacteristic
23C2×C3×C4×Dic3C2211yescharacteristic
24C22×C3×C3⋊C8C2211yescharacteristic
25C2×(C2×D3)⋊Dic3—21no—
26Dic3×D12—22no—
27Dic3×D12C2214yescomplemented
28#63659C2214yescomplemented
29#63575C2214yescomplemented
30#63562C2214yescomplemented
31Dic3×C3⋊C8C2214yescomplemented
32C22×C3×D8—31no—
33C2×C3×C4×D4—31no—
34C2×D4×Dic3—31no—
35C2×D4×Dic3D311yescharacteristic, complemented
36C22×#213D311yescharacteristic
37C2×C4×D12—31no—
38C2×#832—31no—
39C2×#832—31no—
40C2×#832—31no—
41C2×#832—61no—
42Dic3×D8—38no—
43C2×#818—31no—
44C2×C8×Dic3—31no—
45C4×#213—38no—
46C2×#809—31no—
47C2×#807—31no—
48C2×C4×C3⋊C8—31no—
49C24×C32—21no—
50C23×C3×D3—21no—
51C2×C32×D4—21no—
52C2×C32×D4—22no—
53C2×C32×D4C2×C414yescomplemented
54C2×C32×D4C2312yes—
55C22×C32×C4C2311yescharacteristic
56C22×C3⋊Dic3—21no—
57C22×C3×Dic3—21no—
58C22×C3×Dic3—24no—
59C22×C3×Dic3—44no—
60C22×C3×Dic3D411yescharacteristic
61C22×C3×Dic3D411yescharacteristic
62C2×C3×D12—21no—
63C2×C3×D12—22no—
64C2×C3×D12C2×C414yescomplemented
65C2×C3×D12C2312yes—
66C2×D3×Dic3—24no—
67C2×D3×Dic3—44no—
68C4⋊(C3⋊Dic3)—21no—
69C4⋊(C3⋊Dic3)C2312yes—
70C2×C3⋊(C3⋊C8)—22no—
71C2×C3⋊(C3⋊C8)C2312yes—
72C3×#217—24no—
73C3×#213C2×C4116yescomplemented
74C3×C4⋊Dic3—21no—
75C3×C4⋊Dic3C2312yes—
76C3×C4×Dic3D412yes—
77C3×C4×Dic3C2312yes—
78C2×C3×C3⋊C8C2×C414yescomplemented
79C2×C3×C3⋊C8C2312yes—
80(C2×D3)⋊Dic3—24no—
81C2×C4×D8—91noSylow
82C22×C3×D4—21no—
83C22×C3×D4—31no—
84C22×C3×D4C2×D311yescharacteristic
85C22×C3×D4C2×D311yescharacteristic
86C23×C3×C4—61no—
87C23×Dic3—21no—
88C23×Dic3—61no—
89C22×D12C2×D311yescharacteristic
90C22×C4×D3—61no—
91C2×C3×D8—38no—
92C2×C3×D8—34no—
93C22×C3×C8—31no—
94C3×C4×D4—34no—
95C3×C4×D4—62no—
96C2×C3×C4⋊C4—31no—
97C2×C3×C22⋊C4—61no—
98C2×C3×C42—31no—
99C2×#217—21no—
100C2×#217—61no—
101D4×Dic3—22no—
102D4×Dic3—34no—
103D4×Dic3—62no—
104D4×Dic3C2×D314yescomplemented
105C2×#213Dic318yesdirect product factor
106C2×#213C2×D314yes—
107C2×(C2×D3)⋊C4—61no—
108C2×C4⋊Dic3—31no—
109C2×C4⋊Dic3—31no—
110C2×C4⋊Dic3—61no—
111C2×C4⋊Dic3C2×D311yescharacteristic
112C2×C4×Dic3C2×D311yescharacteristic
113C22×C3⋊C8—31no—
114C22×C3⋊C8—31no—
115C22×C3⋊C8—61no—
116C22×C3⋊C8C2×D311yescharacteristic
117C4×D12—34no—
118C4×D12—62no—
119#832—34no—
120#832—34no—
121#832—34no—
122#832—64no—
123#818—34no—
124C8×Dic3—34no—
125#809—34no—
126#807—34no—
127C4×C3⋊C8—34no—
128C23×C32—24no—
129C23×C32—22no—
130C23×C32—42no—
131C23×C32—42no—
132C23×C32C2×D411yescharacteristic
133C22×C3×D3—24no—
134C22×C3×D3—22no—
135C22×C3×D3—42no—
136C22×C3×D3—42no—
137C32×D4—24no—
138C32×D4—22no—
139C32×D4C22×C414yescomplemented
140C2×C32×C4C22×C412yes—
141C2×C32×C4C2×D411yescharacteristic
142C2×C32×C4#4112yes—
143C2×C32×C4C2411yescharacteristic
144C2×C3⋊Dic3—22no—
145C2×C3⋊Dic3—42no—
146C2×C3×Dic3—22no—
147C2×C3×Dic3—22no—
148C2×C3×Dic3—42no—
149C2×C3×Dic3—48no—
150C2×C3×Dic3D814yescomplemented
151C2×C3×Dic3C2×D412yes—
152C2×C3×Dic3C2×D412yes—
153C3×D12—24no—
154C3×D12—22no—
155C3×D12C22×C414yescomplemented
156D3×Dic3—48no—
157C3⋊(C3⋊C8)—22no—
158C3×C3⋊C8C22×C414yescomplemented
159C22×D8—31no—
160C2×C4×D4—31no—
161C2×C4×D4—91no—
162C4×D8—98no—
163C2×#107—91no—
164C2×#102—91no—
165C2×#102—91no—
166C2×C4×C8—91no—
167C24×C3—21no—
168C24×C3—21no—
169C24×C3—41no—
170C24×C3—61no—
171C23×D3—21no—
172C2×C3×D4—22no—
173C2×C3×D4—21no—
174C2×C3×D4—22no—
175C2×C3×D4—21no—
176C2×C3×D4—24no—
177C2×C3×D4—22no—
178C2×C3×D4—34no—
179C2×C3×D4—32no—
180C2×C3×D4—42no—
181C2×C3×D4—41no—
182C2×C3×D4—62no—
183C2×C3×D4—61no—
184C2×C3×D4C4×D314yescomplemented
185C2×C3×D4C2×Dic314yescomplemented
186C2×C3×D4C22×D312yes—
187C2×C3×D4C22×D312yes—
188C22×C3×C4—21no—
189C22×C3×C4—31no—
190C22×C3×C4—31no—
191C22×C3×C4—64no—
192C22×C3×C4—61no—
193C22×C3×C4—124no—
194C22×C3×C4C22×D311yescharacteristic
195C22×C3×C4C22×D311yescharacteristic
196C22×Dic3—21no—
197C22×Dic3—24no—
198C22×Dic3—44no—
199C22×Dic3—61no—
200C22×Dic3—61no—
201C22×Dic3—64no—
202C22×Dic3—121no—
203C22×Dic3—124no—
204C22×Dic3#6711yescharacteristic
205C22×Dic3#6711yescharacteristic
206C2×D12—21no—
207C2×D12—22no—
208C2×D12C2×Dic314yescomplemented
209C2×D12C22×D312yes—
210C2×C4×D3—64no—
211C2×C4×D3—124no—
212C3×D8—316no—
213C2×C3×C8—34no—
214C2×C3×C8—32no—
215C3×C4⋊C4—32no—
216C3×C4⋊C4—61no—
217C3×C22⋊C4—64no—
218C3×C42—32no—
219C3×C42—32no—
220#217—24no—
221#217—64no—
222#213C2×Dic3116yesdirect product factor
223(C2×D3)⋊C4—64no—
224C4⋊Dic3—21no—
225C4⋊Dic3—32no—
226C4⋊Dic3—32no—
227C4⋊Dic3—62no—
228C4⋊Dic3—61no—
229C4⋊Dic3—61no—
230C4⋊Dic3—121no—
231C4⋊Dic3C22×D312yes—
232C4×Dic3#6712yes—
233C4×Dic3C22×D312yes—
234C2×C3⋊C8—32no—
235C2×C3⋊C8—32no—
236C2×C3⋊C8—62no—
237C2×C3⋊C8—62no—
238C2×C3⋊C8—62no—
239C2×C3⋊C8—122no—
240C2×C3⋊C8C2×Dic314yescomplemented
241C2×C3⋊C8C22×D312yes—
242C22×C32—24no—
243C22×C32—42no—
244C22×C32—42no—
245C22×C32—48no—
246C22×C32C4×D412yes—
247C22×C32C2×D812yes—
248C22×C32#13512yes—
249C22×C32C22×D411yescharacteristic
250C2×C3×D3—24no—
251C2×C3×D3—42no—
252C2×C3×D3—42no—
253C2×C3×D3—48no—
254C32×C4C4×D412yes—
255C32×C4C23×C411yescharacteristic
256C32×C4C2×#4111yescharacteristic
257C3⋊Dic3—42no—
258C3×Dic3—22no—
259C3×Dic3—42no—
260C3×Dic3C2×D814yescomplemented
261C22×D4—31no—
262C22×D4D3211yescharacteristic
263C23×C4—61no—
264C23×C4—181no—
265C2×D8—38no—
266C2×D8—34no—
267C22×C8—31no—
268C22×C8—91no—
269C4×D4—34no—
270C4×D4—62no—
271C4×D4—94no—
272C4×D4—182no—
273C2×C4⋊C4—31no—
274C2×C4⋊C4—91no—
275C2×C22⋊C4—61no—
276C2×C22⋊C4—181no—
277C2×C42—31no—
278#107—94no—
279#102—94no—
280#102—94no—
281C4×C8—94no—
282C23×C3—24no—
283C23×C3—24no—
284C23×C3—22no—
285C23×C3—22no—
286C23×C3—21no—
287C23×C3—42no—
288C23×C3—44no—
289C23×C3—42no—
290C23×C3—42no—
291C23×C3—42no—
292C23×C3—42no—
293C23×C3—64no—
294C23×C3—62no—
295C23×C3—82no—
296C23×C3—82no—
297C23×C3—122no—
298C23×C3—122no—
299C23×C3D3×D411yescharacteristic
300C23×C3C2×#6711yescharacteristic
301C22×D3—24no—
302C22×D3—22no—
303C22×D3—42no—
304C22×D3—42no—
305C3×D4—24no—
306C3×D4—24no—
307C3×D4—22no—
308C3×D4—22no—
309C3×D4—24no—
310C3×D4—34no—
311C3×D4—42no—
312C3×D4—44no—
313C3×D4—64no—
314C3×D4—62no—
315C3×D4C2×C4×D314yescomplemented
316C3×D4C22×Dic314yescomplemented
317C2×C3×C4—21no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—21no—
321C2×C3×C4—34no—
322C2×C3×C4—32no—
323C2×C3×C4—32no—
324C2×C3×C4—62no—
325C2×C3×C4—62no—
326C2×C3×C4—128no—
327C2×C3×C4—122no—
328C2×C3×C4C2×C4×D312yes—
329C2×C3×C4#23512yes—
330C2×C3×C4D3×D411yescharacteristic
331C2×C3×C4#23712yes—
332C2×C3×C4C22×Dic312yes—
333C2×C3×C4C2×#6711yescharacteristic
334C2×C3×C4C23×D311yescharacteristic
335C2×C3×C4C23×D311yescharacteristic
336C2×Dic3—22no—
337C2×Dic3—22no—
338C2×Dic3—42no—
339C2×Dic3—48no—
340C2×Dic3—62no—
341C2×Dic3—62no—
342C2×Dic3—122no—
343C2×Dic3—122no—
344C2×Dic3—122no—
345C2×Dic3—128no—
346C2×Dic3—242no—
347C2×Dic3#21314yesdirect product factor
348C2×Dic3C2×#6712yes—
349C2×Dic3C2×#6712yes—
350D12—24no—
351D12—22no—
352D12C22×Dic314yescomplemented
353C4×D3—128no—
354C3×C8—34no—
355C3⋊C8—62no—
356C3⋊C8—62no—
357C3⋊C8—122no—
358C3⋊C8C22×Dic314yescomplemented
359C2×C32—44no—
360C2×C32C4×D814yescomplemented
361C2×C32C2×C4×D411yescharacteristic
362C2×C32C22×D811yescharacteristic
363C2×C32C2×#13511yescharacteristic
364C3×D3—44no—
365C24—21no—
366C24—61no—
367C2×D4—22no—
368C2×D4—21no—
369C2×D4—32no—
370C2×D4—34no—
371C2×D4—62no—
372C2×D4—61no—
373C2×D4D3×Dic314yescomplemented
374C2×D4C2×D3212yes—
375C22×C4—31no—
376C22×C4—31no—
377C22×C4—64no—
378C22×C4—61no—
379C22×C4—124no—
380C22×C4—184no—
381C22×C4—181no—
382C22×C4—364no—
383C22×C4C2×D3211yescharacteristic
384D8—316no—
385C2×C8—32no—
386C2×C8—34no—
387C2×C8—92no—
388C2×C8—182no—
389C4⋊C4—32no—
390C4⋊C4—61no—
391C4⋊C4—92no—
392C4⋊C4—181no—
393C22⋊C4—64no—
394C22⋊C4—184no—
395C42—32no—
396C42—32no—
397C22×C3—24no—
398C22×C3—24no—
399C22×C3—21no—
400C22×C3—22no—
401C22×C3—22no—
402C22×C3—22no—
403C22×C3—44no—
404C22×C3—48no—
405C22×C3—42no—
406C22×C3—42no—
407C22×C3—48no—
408C22×C3—42no—
409C22×C3—42no—
410C22×C3—64no—
411C22×C3—82no—
412C22×C3—82no—
413C22×C3—88no—
414C22×C3—128no—
415C22×C3—122no—
416C22×C3—122no—
417C22×C3D3×D812yes—
418C22×C3#91212yes—
419C22×C3C4×#6712yes—
420C22×C3C2×#21312yes—
421C22×C3D4×Dic312yes—
422C22×C3#95012yes—
423C22×C3C2×D3×D411yescharacteristic
424C22×C3C22×#6711yescharacteristic
425C2×D3—24no—
426C2×D3—48no—
427C2×D3—42no—
428C2×D3—42no—
429C3×C4—21no—
430C3×C4—21no—
431C3×C4—22no—
432C3×C4—34no—
433C3×C4—62no—
434C3×C4—122no—
435C3×C4C4×#6712yes—
436C3×C4D4×Dic312yes—
437C3×C4C22×C4×D311yescharacteristic
438C3×C4C2×#23511yescharacteristic
439C3×C4C2×#23711yescharacteristic
440C3×C4C23×Dic311yescharacteristic
441Dic3—22no—
442Dic3—42no—
443Dic3—122no—
444Dic3—122no—
445Dic3—242no—
446Dic3C2×#21314yesdirect product factor
447C32C2×C4×D811yescharacteristic, complemented, Sylow
448C23—22no—
449C23—24no—
450C23—42no—
451C23—42no—
452C23—62no—
453C23—64no—
454C23—122no—
455C23—122no—
456C23D3×#6711yescharacteristic
457D4—22no—
458D4—24no—
459D4—34no—
460D4—62no—
461D4—64no—
462D4C2×D3×Dic314yescomplemented
463C2×C4—32no—
464C2×C4—34no—
465C2×C4—32no—
466C2×C4—62no—
467C2×C4—62no—
468C2×C4—128no—
469C2×C4—122no—
470C2×C4—182no—
471C2×C4—368no—
472C2×C4—362no—
473C2×C4C2×D3×Dic312yes—
474C2×C4#380412yes—
475C2×C4D3×#6711yescharacteristic
476C2×C4C22×D3211yescharacteristic
477C8—34no—
478C8—182no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—24no—
483C2×C3—44no—
484C2×C3—44no—
485C2×C3—84no—
486C2×C3—124no—
487C2×C3C4×#21314yescomplemented
488C2×C3Dic3×D814yescomplemented
489C2×C3C2×D3×D811yescharacteristic
490C2×C3C2×#91211yescharacteristic
491C2×C3C2×C4×#6711yescharacteristic
492C2×C3C22×#21311yescharacteristic
493C2×C3C2×D4×Dic311yescharacteristic
494C2×C3C2×#95011yescharacteristic
495D3—44no—
496C22—24no—
497C22—42no—
498C22—42no—
499C22—48no—
500C22—64no—
501C22—122no—
502C22—122no—
503C22—128no—
504C22D3×#21312yes—
505C22#6393312yes—
506C22Dic3×#6712yes—
507C22C2×D3×#6711yescharacteristic
508C4—34no—
509C4—62no—
510C4—122no—
511C4—362no—
512C4Dic3×#6712yes—
513C4C22×D3×Dic311yescharacteristic
514C4C2×#380411yescharacteristic
515C3—21no—
516C3C2×C4×#21311yescharacteristic, complemented
517C3C2×Dic3×D811yescharacteristic, complemented
518C2—44no—
519C2—124no—
520C2Dic3×#21314yesdirect product factor
521C2C2×D3×#21311yescharacteristic
522C2C2×#6393311yescharacteristic
523C2C2×Dic3×#6711yescharacteristic
subgroupG/H
centerC23D3×#67
commutatorC32×C4C23×C4
FrattiniC2×C4C22×D32
FittingC22×C32×D4C22
socleC23×C32C2×D4
prime radicalC22×C3×#213C2
cosocleC2×C32×C4C24
hypercenterC22×D4D32
hypocenterC32C2×C4×D8
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×D4×Dic3C2semidirect
3. C2×Dic3×D12C2semidirect
4. C2×#63659C2semidirect
5. Dic3×#213C2direct (N × H)
6. C2×#63575C2semidirect
7. C2×#63562C2semidirect
8. C2×Dic3×C3⋊C8C2semidirect
19. C3×D4×Dic3C22semidirect
20. C2×C3×#213C4semidirect
27. Dic3×D12C22semidirect
28. #63659C22semidirect
29. #63575C22semidirect
30. #63562C22semidirect
31. Dic3×C3⋊C8C22semidirect
35. C2×D4×Dic3D3semidirect
53. C2×C32×D4C2×C4semidirect
64. C2×C3×D12C2×C4semidirect
73. C3×#213C2×C4semidirect
78. C2×C3×C3⋊C8C2×C4semidirect
104. D4×Dic3C2×D3semidirect
105. C2×#213Dic3direct (N × H)
139. C32×D4C22×C4semidirect
150. C2×C3×Dic3D8semidirect
155. C3×D12C22×C4semidirect
158. C3×C3⋊C8C22×C4semidirect
184. C2×C3×D4C4×D3semidirect
185. C2×C3×D4C2×Dic3semidirect
208. C2×D12C2×Dic3semidirect
222. #213C2×Dic3direct (N × H)
240. C2×C3⋊C8C2×Dic3semidirect
260. C3×Dic3C2×D8semidirect
315. C3×D4C2×C4×D3semidirect
316. C3×D4C22×Dic3semidirect
347. C2×Dic3#213direct (N × H)
352. D12C22×Dic3semidirect
358. C3⋊C8C22×Dic3semidirect
360. C2×C32C4×D8semidirect
373. C2×D4D3×Dic3semidirect
446. Dic3C2×#213direct (N × H)
447. C32C2×C4×D8semidirect
462. D4C2×D3×Dic3semidirect
487. C2×C3C4×#213semidirect
488. C2×C3Dic3×D8semidirect
516. C3C2×C4×#213semidirect
517. C3C2×Dic3×D8semidirect
520. C2Dic3×#213direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D4×Dic32
C22×C3×#2132
C2×Dic3×D122
C2×#636592
Dic3×#2132
C2×#635752
C2×#635622
C2×Dic3×C3⋊C82
C2×Dic3×D83
C2×C4×#2133