Finite Group Databasebeta

← C2×D3×#501 #11664793

Subgroups — C2×D3×#501 #11664793

Order 768.

2142 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×C4⋊Q8C211yesmaximal, characteristic
2C2×D3×#125C211yesmaximal, characteristic
3C2×C3×#501C211yesmaximal, characteristic, complemented
4C2×#5081C211yesmaximal, characteristic, complemented
5C2×#5064C211yesmaximal, characteristic, complemented
6D3×#501C2116yesmaximal, direct product factor
7C2×D3×#107C212yesmaximal
8C2×#4876C212yesmaximal, complemented
9C2×D3×#106C212yesmaximal
10C2×#4856C212yesmaximal, complemented
11C2×D3×#97C211yesmaximal, characteristic
12C2×#4702C211yesmaximal, characteristic, complemented
13C22×#501—31nomaximal, Sylow
14C22×D3×Q8—21no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#125C2211yescharacteristic
17C2×#944—21no—
18C22×D3×C8C2212yes—
19D3×C4⋊Q8—24no—
20D3×C4⋊Q8C2218yes—
21D3×#125—24no—
22D3×#125C2218yes—
23C2×D3×C4⋊C4—21no—
24C2×D3×C4⋊C4—21no—
25C2×D3×C4⋊C4—21no—
26C2×D3×C4⋊C4C2211yescharacteristic
27C2×D3×C4⋊C4C2212yes—
28C2×D3×C4⋊C4C2211yescharacteristic
29C2×#890C2211yescharacteristic
30C2×#890C2211yescharacteristic
31C2×#889—21no—
32C2×#889—21no—
33C2×#888—21no—
34C2×#888C2212yes—
35C2×C42×D3C2211yescharacteristic
36C2×#870C2211yescharacteristic
37C2×#869C2211yescharacteristic
38C3×#501C2218yescomplemented
39C2×C3×#107C2212yes—
40C2×C3×#106C2212yes—
41C2×C3×#97C2211yescharacteristic
42C2×#818C2212yes—
43C2×#817C2212yes—
44C2×#815C2212yes—
45#5081C2218yescomplemented
46#5064C2218yescomplemented
47C2×#808C2212yes—
48C2×#808C2212yes—
49C2×#807C2212yes—
50C2×#807C2212yes—
51C2×#803C2211yescharacteristic
52D3×#107C22116yes—
53#4876C22116yescomplemented
54D3×#106C22116yes—
55#4856C22116yescomplemented
56D3×#97C2218yes—
57#4702C2218yescomplemented
58C22×C4⋊Q8—31no—
59C22×#125—31no—
60C2×#501—316no—
61C2×#501—32no—
62C2×#501—32no—
63C2×#501—31no—
64C2×#501—31no—
65C2×#501—31no—
66C2×#501D311yescharacteristic, direct product factor
67C22×#107—32no—
68C22×#106—32no—
69C22×#97—31no—
70C22×C3×Q8—21no—
71C2×D3×Q8—28no—
72C2×D3×Q8—48no—
73C22×C4×D3—22no—
74C22×C4×D3—21no—
75C22×C4×D3—21no—
76C22×C4×D3D411yescharacteristic
77C22×C4×D3D411yescharacteristic
78C22×C4×D3C2311yescharacteristic
79C22×Dic6—21no—
80C22×Dic6—22no—
81C22×C3×C8C2312yes—
82C3×C4⋊Q8—22no—
83C3×C4⋊Q8C2314yes—
84C3×#125—22no—
85C3×#125C2314yes—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2312yes—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C42C2311yescharacteristic
93#944—28no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3C2312yes—
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—22no—
103C2×Dic3⋊C4—22no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×C4×Dic3D411yescharacteristic
109C2×C4×Dic3D411yescharacteristic
110C2×C4×Dic3C2311yescharacteristic
111C22×C3⋊C8C2312yes—
112C2×D3×C8Q8116yescomplemented
113C2×D3×C8C2318yes—
114D3×C4⋊C4—22no—
115D3×C4⋊C4—22no—
116D3×C4⋊C4—28no—
117D3×C4⋊C4—28no—
118D3×C4⋊C4—24no—
119D3×C4⋊C4—28no—
120D3×C4⋊C4C2314yes—
121D3×C4⋊C4C2318yes—
122D3×C4⋊C4C2314yes—
123#890—22no—
124#890—22no—
125#890C2314yes—
126#890C2314yes—
127#889—28no—
128#889—28no—
129#888—24no—
130#888—28no—
131#888C2318yes—
132C42×D3D414yes—
133C42×D3C2314yes—
134#870—22no—
135#870C2314yes—
136#869—22no—
137#869C2314yes—
138C3×#107C2318yes—
139C3×#106C2318yes—
140C3×#97C2314yes—
141#818C2318yes—
142#817C2318yes—
143#815C2318yes—
144#808C2318yes—
145#808C2318yes—
146#807C2318yes—
147#807C2318yes—
148#803C2314yes—
149C23×Q8—61no—
150C23×C8—32no—
151C2×C4⋊Q8—38no—
152C2×C4⋊Q8—32no—
153C2×C4⋊Q8—31no—
154C2×C4⋊Q8—61no—
155C2×C4⋊Q8—64no—
156C2×C4⋊Q8—61no—
157C2×C4⋊Q8C2×D311yescharacteristic
158C2×#125—38no—
159C2×#125—31no—
160C2×#125—31no—
161C2×#125—31no—
162C2×#125—61no—
163C2×#125—61no—
164C2×#125—64no—
165C2×#125C2×D311yescharacteristic
166C22×C4⋊C4—31no—
167C22×C4⋊C4—32no—
168C22×C4⋊C4—31no—
169C22×C4⋊C4—61no—
170C22×C4⋊C4—61no—
171C22×C4⋊C4—61no—
172C22×C42—31no—
173#501—38no—
174#501—38no—
175#501—316no—
176#501—316no—
177#501—38no—
178#501C2×D318yesdirect product factor
179C2×#107—32no—
180C2×#107—32no—
181C2×#107—32no—
182C2×#107—316no—
183C2×#107C2×D312yes—
184C2×#106—32no—
185C2×#106—32no—
186C2×#106—32no—
187C2×#106—316no—
188C2×#106C2×D312yes—
189C2×#97—32no—
190C2×#97—31no—
191C2×#97—38no—
192C2×#97C2×D311yescharacteristic
193C23×D3C2×D411yescharacteristic
194C2×C3×Q8—24no—
195C2×C3×Q8—44no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—22no—
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2411yescharacteristic
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—22no—
205C22×Dic3C2×D411yescharacteristic
206C22×Dic3C2×D411yescharacteristic
207C22×Dic3C2×D411yescharacteristic
208D3×Q8—432no—
209C2×C4×D3—24no—
210C2×C4×D3—24no—
211C2×C4×D3—24no—
212C2×C4×D3—24no—
213C2×C4×D3—28no—
214C2×C4×D3—44no—
215C2×C4×D3—48no—
216C2×C4×D3—44no—
217C2×C4×D3C2×D414yes—
218C2×C4×D3C2×D414yes—
219C2×C4×D3C2×D412yes—
220C2×C4×D3C2×Q818yes—
221C2×C4×D3C2412yes—
222C2×Dic6—24no—
223C2×Dic6—28no—
224C2×Dic6—48no—
225C2×Dic6—44no—
226C2×C3×C8C2×Q818yescomplemented
227C2×C3×C8C2414yes—
228C3×C4⋊C4—24no—
229C3×C4⋊C4—22no—
230C3×C4⋊C4—24no—
231C3×C4⋊C4—24no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4C2412yes—
235C3×C4⋊C4C2414yes—
236C3×C4⋊C4C2412yes—
237C3×C42C2×D412yes—
238C3×C42C2412yes—
239C4⋊Dic3—22no—
240C4⋊Dic3—24no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—24no—
244C4⋊Dic3—24no—
245C4⋊Dic3C2412yes—
246C4⋊Dic3C2414yes—
247C4⋊Dic3C2412yes—
248Dic3⋊C4—24no—
249Dic3⋊C4—24no—
250Dic3⋊C4—28no—
251Dic3⋊C4—28no—
252Dic3⋊C4—24no—
253Dic3⋊C4—24no—
254Dic3⋊C4—24no—
255Dic3⋊C4—24no—
256C4×Dic3C2×D412yes—
257C4×Dic3C2×D414yes—
258C4×Dic3C2×D414yes—
259C4×Dic3C2412yes—
260C2×C3⋊C8C2×Q818yescomplemented
261C2×C3⋊C8C2414yes—
262D3×C8C2×Q8132yescomplemented
263C22×Q8—21no—
264C22×Q8—62no—
265C22×Q8—68no—
266C22×Q8—61no—
267C22×Q8—128no—
268C23×C4—31no—
269C23×C4—31no—
270C23×C4—31no—
271C23×C4—61no—
272C23×C4—61no—
273C23×C4—62no—
274C22×C8—32no—
275C22×C8—316no—
276C22×C8—38no—
277C22×C8C22×D312yes—
278C4⋊Q8—22no—
279C4⋊Q8—38no—
280C4⋊Q8—34no—
281C4⋊Q8—68no—
282C4⋊Q8—64no—
283C4⋊Q8—68no—
284C4⋊Q8—62no—
285C4⋊Q8C22×D314yes—
286#125—22no—
287#125—34no—
288#125—34no—
289#125—34no—
290#125—68no—
291#125—68no—
292#125—62no—
293#125—62no—
294#125—62no—
295#125C22×D314yes—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—32no—
302C2×C4⋊C4—34no—
303C2×C4⋊C4—34no—
304C2×C4⋊C4—38no—
305C2×C4⋊C4—62no—
306C2×C4⋊C4—62no—
307C2×C4⋊C4—64no—
308C2×C4⋊C4—68no—
309C2×C4⋊C4—68no—
310C2×C4⋊C4—68no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—62no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4—61no—
317C2×C4⋊C4—61no—
318C2×C4⋊C4—62no—
319C2×C4⋊C4—61no—
320C2×C4⋊C4—61no—
321C2×C4⋊C4—61no—
322C2×C4⋊C4C22×D311yescharacteristic
323C2×C4⋊C4C22×D311yescharacteristic
324C2×C4⋊C4C22×D312yes—
325C2×C42—31no—
326C2×C42—31no—
327C2×C42—31no—
328C2×C42—34no—
329C2×C42—34no—
330C2×C42C22×D311yescharacteristic
331#107—38no—
332#107—38no—
333#107—38no—
334#107C22×D318yes—
335#106—38no—
336#106—38no—
337#106—38no—
338#106C22×D318yes—
339#97—38no—
340#97—34no—
341#97C22×D314yes—
342C23×C3C22×D411yescharacteristic
343C22×D3C4⋊Q814yes—
344C22×D3AGL(1,ℤ/8)14yes—
345C22×D3#13714yes—
346C22×D3C22×D412yes—
347C3×Q8—48no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—24no—
353C2×C3×C4—44no—
354C2×C3×C4—42no—
355C2×C3×C4—42no—
356C2×C3×C4C22×D412yes—
357C2×C3×C4C22×D412yes—
358C2×C3×C4C22×D411yescharacteristic
359C2×C3×C4C22×Q814yes—
360C2×C3×C4C2511yescharacteristic
361C2×Dic3—22no—
362C2×Dic3—22no—
363C2×Dic3—22no—
364C2×Dic3—22no—
365C2×Dic3—24no—
366C2×Dic3—44no—
367C2×Dic3—42no—
368C2×Dic3—42no—
369C2×Dic3C4⋊Q814yes—
370C2×Dic3C22×D412yes—
371C2×Dic3C22×D412yes—
372C2×Dic3C22×D412yes—
373C4×D3—28no—
374C4×D3—28no—
375C4×D3—48no—
376C4×D3—416no—
377C4×D3—48no—
378C4×D3C4⋊Q818yes—
379C4×D3C22×Q818yes—
380Dic6—416no—
381Dic6—48no—
382C3×C8C22×Q818yescomplemented
383C3⋊C8C22×Q818yescomplemented
384C24—31no—
385C2×Q8—24no—
386C2×Q8—44no—
387C2×Q8—68no—
388C2×Q8—64no—
389C2×Q8—124no—
390C2×Q8—128no—
391C2×Q8—1232no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—31no—
396C22×C4—32no—
397C22×C4—38no—
398C22×C4—32no—
399C22×C4—31no—
400C22×C4—31no—
401C22×C4—34no—
402C22×C4—34no—
403C22×C4—64no—
404C22×C4—64no—
405C22×C4—64no—
406C22×C4—64no—
407C22×C4—68no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4—62no—
411C22×C4—128no—
412C22×C4—124no—
413C22×C4—124no—
414C22×C4D3×D411yescharacteristic
415C22×C4D3×D411yescharacteristic
416C22×C4C23×D311yescharacteristic
417C2×C8—38no—
418C2×C8—34no—
419C2×C8—332no—
420C2×C8D3×Q818yescomplemented
421C2×C8C23×D314yes—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—22no—
428C4⋊C4—32no—
429C4⋊C4—34no—
430C4⋊C4—32no—
431C4⋊C4—64no—
432C4⋊C4—64no—
433C4⋊C4—68no—
434C4⋊C4—64no—
435C4⋊C4—64no—
436C4⋊C4—64no—
437C4⋊C4—64no—
438C4⋊C4—68no—
439C4⋊C4—61no—
440C4⋊C4—61no—
441C4⋊C4—62no—
442C4⋊C4—64no—
443C4⋊C4—64no—
444C4⋊C4—64no—
445C4⋊C4C23×D312yes—
446C4⋊C4C23×D314yes—
447C4⋊C4C23×D312yes—
448C42—34no—
449C42—34no—
450C42—32no—
451C42—32no—
452C42D3×D412yes—
453C42C23×D312yes—
454C22×C3C2×C4⋊Q812yes—
455C22×C3C2×AGL(1,ℤ/8)12yes—
456C22×C3C2×#13712yes—
457C22×C3C23×D411yescharacteristic
458C2×D3#501116yesdirect product factor
459C2×D3C2×C4⋊Q814yes—
460C2×D3C2×AGL(1,ℤ/8)14yes—
461C2×D3C2×#13714yes—
462C3×C4—22no—
463C3×C4—22no—
464C3×C4—44no—
465C3×C4—42no—
466C3×C4—42no—
467C3×C4C2×C4⋊Q812yes—
468C3×C4C23×Q812yes—
469Dic3—22no—
470Dic3—22no—
471Dic3—44no—
472Dic3—42no—
473Dic3—42no—
474Dic3C2×C4⋊Q814yes—
475C23—32no—
476C23—34no—
477C23—34no—
478C23—34no—
479C23C2×D3×D411yescharacteristic
480Q8—48no—
481Q8—128no—
482Q8—1216no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—32no—
489C2×C4—34no—
490C2×C4—38no—
491C2×C4—38no—
492C2×C4—32no—
493C2×C4—32no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—68no—
498C2×C4—68no—
499C2×C4—62no—
500C2×C4—62no—
501C2×C4—62no—
502C2×C4—62no—
503C2×C4—64no—
504C2×C4—1216no—
505C2×C4—128no—
506C2×C4—128no—
507C2×C4—124no—
508C2×C4—122no—
509C2×C4—122no—
510C2×C4C2×D3×D412yes—
511C2×C4C2×D3×D412yes—
512C2×C4C2×D3×D411yescharacteristic
513C2×C4C2×D3×Q814yes—
514C2×C4C24×D311yescharacteristic
515C8—38no—
516C8C2×D3×Q818yescomplemented
517C2×C3C2×#50114yescomplemented
518C2×C3C22×C4⋊Q811yescharacteristic
519C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
520C2×C3C22×#13711yescharacteristic
521D3C2×#50118yesdirect product factor
522C22—316no—
523C22—34no—
524C22—34no—
525C22—34no—
526C22D3×C4⋊Q812yes—
527C22D3×AGL(1,ℤ/8)12yes—
528C22D3×#13712yes—
529C22C22×D3×D411yescharacteristic
530C4—22no—
531C4—22no—
532C4—34no—
533C4—44no—
534C4—42no—
535C4—42no—
536C4—62no—
537C4—62no—
538C4—124no—
539C4—122no—
540C4—122no—
541C4D3×C4⋊Q812yes—
542C4C22×D3×Q812yes—
543C3C22×#50111yescharacteristic, complemented, Sylow
544C2—38no—
545C2D3×#50114yesdirect product factor
546C2C2×D3×C4⋊Q811yescharacteristic
547C2C2×D3×AGL(1,ℤ/8)11yescharacteristic
548C2C2×D3×#13711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#501C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenterC2×#501D3
hypocenterC3C22×#501
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
3. C2×C3×#501C2semidirect
4. C2×#5081C2semidirect
5. C2×#5064C2semidirect
6. D3×#501C2direct (N × H)
8. C2×#4876C2semidirect
10. C2×#4856C2semidirect
12. C2×#4702C2semidirect
38. C3×#501C22semidirect
45. #5081C22semidirect
46. #5064C22semidirect
53. #4876C22semidirect
55. #4856C22semidirect
57. #4702C22semidirect
66. C2×#501D3direct (N × H)
112. C2×D3×C8Q8semidirect
178. #501C2×D3direct (N × H)
226. C2×C3×C8C2×Q8semidirect
260. C2×C3⋊C8C2×Q8semidirect
262. D3×C8C2×Q8semidirect
382. C3×C8C22×Q8semidirect
383. C3⋊C8C22×Q8semidirect
420. C2×C8D3×Q8semidirect
458. C2×D3#501direct (N × H)
516. C8C2×D3×Q8semidirect
517. C2×C3C2×#501semidirect
521. D3C2×#501direct (N × H)
543. C3C22×#501semidirect
545. C2D3×#501direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D3×C4⋊Q82
C2×D3×#1252
C2×C3×#5012
C2×#50812
C2×#50642
D3×#5012
C2×D3×#1072
C2×#48762
C2×D3×#1062
C2×#48562
C2×D3×#972
C2×#47022
C22×#5013