Finite Group Databasebeta

← D3×#5074 #11874347

Subgroups — D3×#5074 #11874347

Order 1152.

1408 conjugacy classes of subgroups.

The subgroup lattice has 607 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×D3×#127C211yesmaximal, characteristic, complemented
2C2×D3×#214C212yesmaximal, complemented
3C2×D3×#213C212yesmaximal, complemented
4D3×#875C211yesmaximal, characteristic, complemented
5#10592773C211yesmaximal, characteristic, complemented
6C3×#5074C211yesmaximal, characteristic, complemented
7#10591472C212yesmaximal, complemented
8#10591469C212yesmaximal, complemented
9#10590371C211yesmaximal, characteristic, complemented
10#10590354C211yesmaximal, characteristic, complemented
11D3×#803C211yesmaximal, characteristic, complemented
12C2×#5074—31nomaximal
13D3×#496—31nomaximal
14C2×C3×D3×D4—22no—
15C2×C3×D3×D4C2212yes—
16C2×D3×D12C2211yescharacteristic
17C2×D3×Dic6C2211yescharacteristic
18C32×#127C2211yescharacteristic, complemented
19C2×#3754C2212yescomplemented
20C2×#3753C2212yescomplemented
21#64084C2211yescharacteristic, complemented
22C3×#940—22no—
23C3×#940C2212yescomplemented
24C2×C3×#214C2212yescomplemented
25C2×C3×#213C2212yescomplemented
26C3×#875C2211yescharacteristic, complemented
27C3×#874C2211yescharacteristic, complemented
28C3×C42×D3C2211yescharacteristic
29D3×#214—28no—
30D3×#213—28no—
31D3×(C2×D3)⋊C4—22no—
32#63864C2211yescharacteristic, complemented
33#63860C2211yescharacteristic, complemented
34#63852—22no—
35C2×#3716C2212yescomplemented
36C2×#3715C2212yescomplemented
37C2×#3714C2212yescomplemented
38C2×#3713C2212yescomplemented
39C2×D3×C3⋊C8C2212yes—
40#63630C2211yescharacteristic, complemented
41C3×#803C2211yescharacteristic, complemented
42#63554C2212yescomplemented
43C2×C3×#127—31no—
44C22×#214—32no—
45C22×#213—32no—
46C2×D3×SD16—32no—
47C2×D3×D8—32no—
48D3×#127D311yescharacteristic, complemented
49D3×#124—31no—
50C2×#875—31no—
51C3×#496—31no—
52#5176—32no—
53#5154—32no—
54#5074—31no—
55#5074—31no—
56#5074—32no—
57#5074—32no—
58#5074—31no—
59#5074—31no—
60#5074—61no—
61#5074D311yescharacteristic, direct product factor
62#5058—31no—
63C2×#803—31no—
64#4712—31no—
65D3×#97—31no—
66C23×C3×D3—22no—
67C22×D32—21no—
68C2×C32×D4—22no—
69C2×C32×D4C2312yes—
70C2×C4⋊(C3⋊D3)C2311yescharacteristic
71C2×#634C2311yescharacteristic
72C2×C3×#67—22no—
73C2×C3×#67—24no—
74C3×D3×D4—24no—
75C3×D3×D4—24no—
76C3×D3×D4—48no—
77C2×C3×D12—22no—
78C2×C3×D12C2311yescharacteristic
79C2×C3×D12C2312yes—
80C2×C3×C4×D3D412yes—
81C2×C3×C4×D3C2311yescharacteristic
82C2×C3×Dic6C2311yescharacteristic
83C2×#627—21no—
84C2×#626—21no—
85C2×D3×Dic3—21no—
86D3×D12—24no—
87D3×Dic6—24no—
88C32×C42C2311yescharacteristic
89#3754—24no—
90#3753—24no—
91#3752—22no—
92C2×C3⋊(C3⋊C8)C2312yes—
93C3×#214—24no—
94C3×#213—24no—
95C3×(C2×D3)⋊C4—22no—
96C3×C4×Dic3D412yes—
97C3×C4×Dic3C2311yescharacteristic
98C2×C3×C3⋊C8C2312yes—
99#3722—22no—
100(C2×D3)⋊Dic3—22no—
101#3716—24no—
102#3715—24no—
103#3714—24no—
104#3713—24no—
105D3×C3⋊C8D418yescomplemented
106C2×#496—91noSylow
107C22×C3×D4—32no—
108C22×C3×D4—62no—
109C2×D3×Q8—31no—
110C2×D3×D4—22no—
111C2×D3×D4—31no—
112C2×D3×D4C2×D312yes—
113C22×D12—31no—
114C22×Dic6—31no—
115C2×C3×SD16—32no—
116C2×C3×D8—32no—
117C3×#127—21no—
118C3×#127—32no—
119C3×#127—31no—
120C3×#127—62no—
121C3×#127C2×D311yescharacteristic, complemented
122C3×#127C2×D311yescharacteristic, complemented
123C3×#124—31no—
124C2×C3×C42—31no—
125#947—31no—
126C2×#215—32no—
127#940—22no—
128#940C2×D312yescomplemented
129#936—31no—
130C2×#214—32no—
131C2×#214—32no—
132C2×#214—32no—
133C2×#214—32no—
134C2×#214—62no—
135C2×#214—68no—
136C2×#214C2×D312yescomplemented
137C2×#213—32no—
138C2×#213—32no—
139C2×#213—32no—
140C2×#213—32no—
141C2×#213—32no—
142C2×#213—62no—
143C2×#213—68no—
144C2×#213C2×D312yescomplemented
145C2×(C2×D3)⋊C4—62no—
146C22×C3⋊C8—32no—
147D3×SD16—68no—
148D3×D8—68no—
149C2×D24—32no—
150C2×#204—32no—
151C2×D3×C8—32no—
152#885—62no—
153D3×C22⋊C4—62no—
154#875—31no—
155#875—31no—
156#875—31no—
157#875—31no—
158#875—61no—
159#875—62no—
160#875C2×D311yescharacteristic, complemented
161#874C2×D311yescharacteristic, complemented
162C42×D3C2×D311yescharacteristic
163C3×#97—31no—
164#815—32no—
165#803—31no—
166#803—31no—
167#803—32no—
168#803—61no—
169#803C2×D311yescharacteristic, complemented
170C23×C32—22no—
171C22×C3⋊D3—21no—
172C22×C3×D3—21no—
173C22×C3×D3—22no—
174C22×C3×D3—44no—
175C22×C3×D3—48no—
176C22×C3×D3C2×D411yescharacteristic
177C2×D32—42no—
178C2×D32—44no—
179C32×D4—22no—
180C32×D4—22no—
181C32×D4—44no—
182C2×C32×C4C2×D412yes—
183C2×C32×C4C2411yescharacteristic
184C2×C3⋊Dic3—21no—
185C4⋊(C3⋊D3)—22no—
186#634—22no—
187C3×#67—48no—
188C3×#67—44no—
189C2×C3×Dic3—21no—
190C2×C3×Dic3C2×D412yes—
191C2×C3×Dic3C2×D411yescharacteristic
192C3×D12—22no—
193C3×D12—22no—
194C3×D12—22no—
195C3×D12—44no—
196C3×C4×D3—24no—
197C3×C4×D3C2×D414yes—
198C3×Dic6—22no—
199#627—42no—
200#626—42no—
201D3×Dic3—42no—
202C3⋊(C3⋊C8)C2×D414yescomplemented
203C3×C3⋊C8C2×D414yescomplemented
204C22×SD16—92no—
205C22×D8—92no—
206C2×#127—31no—
207C2×#124—91no—
208#496—31no—
209#496—91no—
210#496—91no—
211#496—92no—
212#496—92no—
213#496—91no—
214C2×#97—91no—
215C24×C3—62no—
216C23×D3—22no—
217C23×D3—61no—
218C23×D3—61no—
219C2×C3×Q8—31no—
220C2×C3×D4—22no—
221C2×C3×D4—22no—
222C2×C3×D4—22no—
223C2×C3×D4—32no—
224C2×C3×D4—31no—
225C2×C3×D4—42no—
226C2×C3×D4—64no—
227C2×C3×D4—64no—
228C2×C3×D4—64no—
229C2×C3×D4—62no—
230C2×C3×D4—62no—
231C2×C3×D4—128no—
232C2×C3×D4C22×D312yes—
233C2×C3×D4C22×D312yes—
234C22×C3×C4—31no—
235C22×C3×C4—32no—
236C2×#67—22no—
237C2×#67—24no—
238C2×#67—61no—
239C22×Dic3—61no—
240D3×Q8—64no—
241D3×D4—24no—
242D3×D4—24no—
243D3×D4—48no—
244D3×D4—64no—
245C2×D12—22no—
246C2×D12—31no—
247C2×D12—31no—
248C2×D12—61no—
249C2×D12—64no—
250C2×D12—61no—
251C2×D12C22×D312yes—
252C2×D12C22×D311yescharacteristic
253C2×C4×D3—61no—
254C2×C4×D3#6712yes—
255C2×C4×D3C22×D311yescharacteristic
256C2×Dic6—31no—
257C2×Dic6—31no—
258C2×Dic6—61no—
259C2×Dic6—64no—
260C2×Dic6—61no—
261C2×Dic6—61no—
262C2×Dic6C22×D311yescharacteristic
263C3×SD16—64no—
264C3×D8—64no—
265C2×C3×C8—32no—
266C3×C22⋊C4—62no—
267C3×C42—21no—
268C3×C42—32no—
269C3×C42—31no—
270C3×C42C22×D311yescharacteristic
271C3×C42C22×D311yescharacteristic
272#217—62no—
273#215—64no—
274#214—24no—
275#214—64no—
276#214—64no—
277#214—64no—
278#214—64no—
279#214—124no—
280#213—24no—
281#213—64no—
282#213—64no—
283#213—64no—
284#213—64no—
285#213—64no—
286#213—124no—
287(C2×D3)⋊C4—22no—
288(C2×D3)⋊C4—62no—
289(C2×D3)⋊C4—62no—
290(C2×D3)⋊C4—62no—
291(C2×D3)⋊C4—62no—
292(C2×D3)⋊C4—62no—
293(C2×D3)⋊C4—122no—
294C4×Dic3#6712yes—
295C4×Dic3C22×D311yescharacteristic
296C2×C3⋊C8—32no—
297C2×C3⋊C8—32no—
298C2×C3⋊C8—38no—
299C2×C3⋊C8—62no—
300C2×C3⋊C8C22×D312yes—
301D24—64no—
302#204—64no—
303D3×C8—38no—
304C22×C32—42no—
305C22×C32—44no—
306C22×C32C22×D411yescharacteristic
307C2×C3⋊D3—41no—
308C2×C3⋊D3—42no—
309C2×C3×D3—41no—
310C2×C3×D3—42no—
311C2×C3×D3—44no—
312C2×C3×D3—42no—
313C2×C3×D3—88no—
314C2×C3×D3#12712yes—
315C2×C3×D3AGL(1,ℤ/8)14yes—
316D32—84no—
317C32×C4—22no—
318C32×C4C22×D412yes—
319C3⋊Dic3—41no—
320C3×Dic3—22no—
321C3×Dic3—41no—
322C3×Dic3#12712yes—
323C22×Q8—91no—
324C22×D4—32no—
325C22×D4—62no—
326C22×D4—91no—
327C2×SD16—32no—
328C2×SD16—92no—
329C2×SD16—92no—
330C2×SD16—92no—
331C2×SD16—188no—
332C2×D8—32no—
333C2×D8—92no—
334C2×D8—92no—
335C2×D8—92no—
336C2×D8—188no—
337C22×C8—92no—
338#127—32no—
339#127—31no—
340#127—62no—
341#127D3211yescharacteristic, complemented
342#124—31no—
343#124—91no—
344#124—91no—
345#124—91no—
346#124—182no—
347C2×C22⋊C4—182no—
348C2×C42—31no—
349#97—31no—
350#97—92no—
351#97—91no—
352C23×C3—22no—
353C23×C3—22no—
354C23×C3—31no—
355C23×C3—42no—
356C23×C3—62no—
357C23×C3—61no—
358C23×C3—124no—
359C23×C3—128no—
360C22×D3—21no—
361C22×D3—22no—
362C22×D3—48no—
363C22×D3—44no—
364C22×D3—61no—
365C22×D3—61no—
366C22×D3—121no—
367C22×D3—122no—
368C22×D3—124no—
369C22×D3—124no—
370C22×D3—122no—
371C22×D3C2×#6711yescharacteristic
372C3×Q8—62no—
373C3×D4—22no—
374C3×D4—22no—
375C3×D4—22no—
376C3×D4—22no—
377C3×D4—42no—
378C3×D4—42no—
379C3×D4—44no—
380C3×D4—44no—
381C3×D4—62no—
382C3×D4—62no—
383C3×D4—62no—
384C3×D4—84no—
385C3×D4—124no—
386C3×D4—124no—
387C3×D4—128no—
388C2×C3×C4—21no—
389C2×C3×C4—22no—
390C2×C3×C4—31no—
391C2×C3×C4—34no—
392C2×C3×C4—32no—
393C2×C3×C4—64no—
394C2×C3×C4—61no—
395C2×C3×C4D3×D412yes—
396C2×C3×C4C2×#6712yes—
397C2×C3×C4C23×D311yescharacteristic
398C2×C3×C4C23×D311yescharacteristic
399#67—48no—
400#67—44no—
401#67—122no—
402C2×Dic3—21no—
403C2×Dic3—61no—
404C2×Dic3—61no—
405C2×Dic3—121no—
406C2×Dic3—122no—
407C2×Dic3C2×#6712yes—
408C2×Dic3C2×#6711yescharacteristic
409D12—22no—
410D12—22no—
411D12—22no—
412D12—44no—
413D12—62no—
414D12—62no—
415D12—122no—
416D12—122no—
417C4×D3—24no—
418C4×D3—122no—
419C4×D3D3×D414yes—
420Dic6—22no—
421Dic6—62no—
422Dic6—62no—
423Dic6—122no—
424Dic6—122no—
425Dic6—122no—
426C3×C8—34no—
427C3⋊C8—34no—
428C3⋊C8—34no—
429C3⋊C8—64no—
430C3⋊C8D3×D414yescomplemented
431C2×C32—82no—
432C2×C32C2×#12711yescharacteristic
433C2×C32C2×AGL(1,ℤ/8)12yes—
434C3⋊D3—81no—
435C3×D3—81no—
436C3×D3—82no—
437C3×D3#49614yescomplemented
438C24—62no—
439C24—181no—
440C2×Q8—31no—
441C2×Q8—91no—
442C2×Q8—181no—
443C2×Q8—184no—
444C2×D4—22no—
445C2×D4—31no—
446C2×D4—32no—
447C2×D4—64no—
448C2×D4—64no—
449C2×D4—64no—
450C2×D4—62no—
451C2×D4—62no—
452C2×D4—91no—
453C2×D4—128no—
454C2×D4—184no—
455C2×D4—181no—
456C2×D4C2×D3212yes—
457C22×C4—31no—
458C22×C4—32no—
459C22×C4—181no—
460SD16—64no—
461SD16—184no—
462SD16—184no—
463SD16—184no—
464D8—64no—
465D8—184no—
466D8—184no—
467D8—184no—
468C2×C8—32no—
469C2×C8—98no—
470C2×C8—92no—
471C22⋊C4—62no—
472C22⋊C4—182no—
473C22⋊C4—182no—
474C22⋊C4—182no—
475C42—31no—
476C42—32no—
477C42C2×D3211yescharacteristic
478C22×C3—21no—
479C22×C3—32no—
480C22×C3—34no—
481C22×C3—42no—
482C22×C3—44no—
483C22×C3—44no—
484C22×C3—42no—
485C22×C3—82no—
486C22×C3—84no—
487C22×C3—122no—
488C22×C3—124no—
489C22×C3—122no—
490C22×C3—121no—
491C22×C3—248no—
492C22×C3C2×D3×D411yescharacteristic
493C22×C3C22×#6711yescharacteristic
494C2×D3—41no—
495C2×D3—42no—
496C2×D3—44no—
497C2×D3—42no—
498C2×D3—88no—
499C2×D3—121no—
500C2×D3—122no—
501C2×D3—122no—
502C2×D3—121no—
503C2×D3—241no—
504C2×D3—242no—
505C2×D3—244no—
506C2×D3—244no—
507C2×D3#93214yes—
508C2×D3#94012yes—
509C3×C4—22no—
510C3×C4—22no—
511C3×C4—22no—
512C3×C4—32no—
513C3×C4—42no—
514C3×C4—62no—
515C3×C4—121no—
516C3×C4C2×D3×D412yes—
517C3×C4C2×D3×D412yes—
518Dic3—22no—
519Dic3—41no—
520Dic3—121no—
521Dic3—121no—
522Dic3—241no—
523Dic3#94012yes—
524C32C2×#49611yescharacteristic, complemented, Sylow
525C23—22no—
526C23—31no—
527C23—61no—
528C23—62no—
529C23—124no—
530C23—128no—
531C23—181no—
532C23—362no—
533C23—364no—
534Q8—62no—
535Q8—182no—
536Q8—362no—
537D4—22no—
538D4—22no—
539D4—44no—
540D4—62no—
541D4—62no—
542D4—62no—
543D4—124no—
544D4—124no—
545D4—128no—
546D4—182no—
547D4—362no—
548C2×C4—31no—
549C2×C4—34no—
550C2×C4—32no—
551C2×C4—64no—
552C2×C4—61no—
553C2×C4—181no—
554C2×C4—362no—
555C2×C4D3×#6712yes—
556C2×C4C22×D3211yescharacteristic
557C8—34no—
558C8—94no—
559C2×C3—22no—
560C2×C3—21no—
561C2×C3—34no—
562C2×C3—82no—
563C2×C3—82no—
564C2×C3—162no—
565C2×C3—242no—
566C2×C3—241no—
567C2×C3D3×#12711yescharacteristic
568C2×C3D3×AGL(1,ℤ/8)12yes—
569C2×C3C2×#93212yes—
570C2×C3C2×#94011yescharacteristic
571D3—81no—
572D3—82no—
573D3—241no—
574D3—241no—
575D3—481no—
576D3#507414yesdirect product factor
577C22—34no—
578C22—32no—
579C22—44no—
580C22—42no—
581C22—124no—
582C22—122no—
583C22—122no—
584C22—121no—
585C22—248no—
586C22—362no—
587C22—361no—
588C22—724no—
589C22C2×D3×#6711yescharacteristic
590C4—22no—
591C4—32no—
592C4—62no—
593C4—121no—
594C4—361no—
595C4D32×D412yes—
596C3—21no—
597C3D3×#49611yescharacteristic, complemented
598C3C2×#507411yescharacteristic, complemented
599C2—34no—
600C2—82no—
601C2—242no—
602C2—241no—
603C2—721no—
604C2D3×#93212yes—
605C2D3×#94011yescharacteristic
subgroupG/H
centerC22C2×D3×#67
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×#127C22
socleC22×C32C22×D4
hypercenter#127D32
hypocenterC32C2×#496
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #127

Complements

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

NH ≅ G/Ntype
1. C3×D3×#127C2semidirect
2. C2×D3×#214C2semidirect
3. C2×D3×#213C2semidirect
4. D3×#875C2semidirect
5. #10592773C2semidirect
6. C3×#5074C2semidirect
7. #10591472C2semidirect
8. #10591469C2semidirect
9. #10590371C2semidirect
10. #10590354C2semidirect
11. D3×#803C2semidirect
18. C32×#127C22semidirect
19. C2×#3754C22semidirect
20. C2×#3753C22semidirect
21. #64084C22semidirect
23. C3×#940C22semidirect
24. C2×C3×#214C22semidirect
25. C2×C3×#213C22semidirect
26. C3×#875C22semidirect
27. C3×#874C22semidirect
32. #63864C22semidirect
33. #63860C22semidirect
35. C2×#3716C22semidirect
36. C2×#3715C22semidirect
37. C2×#3714C22semidirect
38. C2×#3713C22semidirect
40. #63630C22semidirect
41. C3×#803C22semidirect
42. #63554C22semidirect
48. D3×#127D3semidirect
61. #5074D3direct (N × H)
105. D3×C3⋊C8D4semidirect
121. C3×#127C2×D3semidirect
122. C3×#127C2×D3semidirect
128. #940C2×D3semidirect
136. C2×#214C2×D3semidirect
144. C2×#213C2×D3semidirect
160. #875C2×D3semidirect
161. #874C2×D3semidirect
169. #803C2×D3semidirect
202. C3⋊(C3⋊C8)C2×D4semidirect
203. C3×C3⋊C8C2×D4semidirect
341. #127D32semidirect
430. C3⋊C8D3×D4semidirect
437. C3×D3#496semidirect
524. C32C2×#496semidirect
576. D3#5074direct (N × H)
597. C3D3×#496semidirect
598. C3C2×#5074semidirect

Maximal subgroups

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

subgroupindex
C3×D3×#1272
C2×D3×#2142
C2×D3×#2132
D3×#8752
#105927732
C3×#50742
#105914722
#105914692
#105903712
#105903542
D3×#8032
C2×#50743
D3×#4963