Finite Group Databasebeta

← D3×#3166 #11664838

Subgroups — D3×#3166 #11664838

Order 768.

1902 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×#533C211yesmaximal, characteristic
2C2×D3×#125C211yesmaximal, characteristic
3C3×#3166C211yesmaximal, characteristic, complemented
4#83342C211yesmaximal, characteristic, complemented
5#83015C211yesmaximal, characteristic, complemented
6#81977C214yesmaximal, complemented
7D3×#501C214yesmaximal, complemented
8#81919C214yesmaximal, complemented
9D3×#499C214yesmaximal, complemented
10D3×#428C214yesmaximal
11#79204C214yesmaximal, complemented
12C4×D3×M16C211yesmaximal, characteristic
13#78507C211yesmaximal, characteristic, complemented
14C2×#3166—31nomaximal, Sylow
15C3×#533C2211yescharacteristic
16C2×C3×#125C2211yescharacteristic
17C2×D3×M16C2212yes—
18#5728C2212yes—
19D3×C4⋊Q8C2212yes—
20#5688C2212yes—
21D3×#125—22no—
22D3×#125—22no—
23D3×#125C2212yes—
24D3×#125C2212yes—
25D3×#125C2212yes—
26D3×#125C2212yes—
27D3×#125C2212yes—
28D3×C22⋊Q8—22no—
29#5622—22no—
30#5578—22no—
31C4×D3×Q8—22no—
32D3×#117C2212yes—
33#5517C2212yes—
34C2×D3×C4⋊C4—22no—
35C2×D3×C4⋊C4C2212yes—
36C2×#890C2212yes—
37C2×#889—22no—
38C2×C42×D3C2211yescharacteristic
39#5470C2211yescharacteristic
40C2×#870C2211yescharacteristic
41C3×#501C2214yescomplemented
42C3×#499C2214yescomplemented
43C3×#428C2214yes—
44C3×C4×M16C2211yescharacteristic
45#5121C2214yes—
46Dic3×M16C2212yes—
47#5081C2214yescomplemented
48#5065C2214yescomplemented
49#5065C2214yescomplemented
50#5064C2214yescomplemented
51#4995C2214yes—
52#4964C2214yes—
53C4×#208C2211yescharacteristic
54#4881C2218yes—
55D3×#107C2218yes—
56#4878C2218yescomplemented
57#4876C2218yescomplemented
58#4861C2218yes—
59D3×#106C2218yes—
60#4858C2218yescomplemented
61#4856C2218yescomplemented
62#4707C2212yes—
63D3×#97C2212yes—
64#4702C2214yescomplemented
65C4×ΓL(1,𝔽25)C2212yes—
66C4×D3×C8C2212yes—
67#4683C2214yescomplemented
68C2×#533—31no—
69C22×#125—31no—
70#3166—34no—
71#3166—34no—
72#3166—34no—
73#3166—31no—
74#3166—31no—
75#3166—31no—
76#3166D311yescharacteristic, direct product factor
77C2×#501—34no—
78C2×#499—34no—
79C2×#428—34no—
80C2×C4×M16—31no—
81C2×D3×Q8—22no—
82C22×C4×D3—22no—
83C22×C4×D3D412yes—
84C22×C4×D3C2311yescharacteristic
85C2×C3×M16C2312yes—
86C3×C4⋊Q8C2312yes—
87C3×#125—22no—
88C3×#125C2312yes—
89C3×#125C2312yes—
90C3×#125C2312yes—
91C3×C22⋊Q8—22no—
92C3×C4×Q8—22no—
93C3×#117C2312yes—
94C2×C3×C4⋊C4—22no—
95C2×C3×C4⋊C4C2312yes—
96C2×C3×C42C2311yescharacteristic
97#946—22no—
98Q8×Dic3—22no—
99#944—22no—
100#926C2312yes—
101C2×C4⋊Dic3—22no—
102C2×C4⋊Dic3C2312yes—
103#924—22no—
104C2×Dic3⋊C4—22no—
105C2×Dic3⋊C4—22no—
106C2×Dic3⋊C4—22no—
107C2×C4×Dic3D412yes—
108C2×C4×Dic3C2311yescharacteristic
109C2×#208C2312yes—
110D3×M16Q8116yescomplemented
111C2×ΓL(1,𝔽25)C2314yes—
112C2×D3×C8C2314yes—
113#897—22no—
114#896—24no—
115#892C2314yes—
116D3×C4⋊C4—22no—
117D3×C4⋊C4—22no—
118D3×C4⋊C4—24no—
119D3×C4⋊C4—24no—
120D3×C4⋊C4—24no—
121D3×C4⋊C4—24no—
122D3×C4⋊C4—24no—
123D3×C4⋊C4—22no—
124D3×C4⋊C4C2314yes—
125D3×C4⋊C4C2314yes—
126D3×C4⋊C4C2314yes—
127#890—22no—
128#890—22no—
129#890C2314yes—
130#890C2314yes—
131#890C2314yes—
132#889—24no—
133#889—24no—
134#889—24no—
135#889—24no—
136#889—24no—
137#888—22no—
138#888C2314yes—
139#887—24no—
140#887—22no—
141D3×C22⋊C4—22no—
142#878—24no—
143#877—22no—
144#872—22no—
145C42×D3—22no—
146C42×D3D412yes—
147C42×D3D412yes—
148C42×D3C2311yescharacteristic
149C42×D3C2311yescharacteristic
150C42×D3C2311yescharacteristic
151C42×D3C2311yescharacteristic
152#870—22no—
153#870C2312yes—
154#870C2312yes—
155#870C2312yes—
156#869C2312yes—
157C4×Dic6—22no—
158C4×Dic6—24no—
159C3×#107C2318yes—
160C3×#106C2318yes—
161C3×#97C2312yes—
162C3×C4×C8C2312yes—
163#818C2318yes—
164#817C2318yes—
165#815C2314yes—
166C8×Dic3C2314yes—
167#808C2318yes—
168#808C2318yes—
169#807C2318yes—
170#807C2318yes—
171#803C2312yes—
172C4×C3⋊C8C2312yes—
173C22×M16—32no—
174#533—32no—
175#533—32no—
176#533—32no—
177#533—31no—
178#533—62no—
179#533—62no—
180#533C2×D311yescharacteristic
181C2×C4⋊Q8—32no—
182C2×#125—32no—
183C2×#125—32no—
184C2×#125—32no—
185C2×#125—32no—
186C2×#125—32no—
187C2×#125—32no—
188C2×#125—31no—
189C2×#125—62no—
190C2×#125—62no—
191C2×#125—62no—
192C2×#125C2×D311yescharacteristic
193C2×C22⋊Q8—62no—
194C2×C4×Q8—62no—
195C2×#117—32no—
196C22×C4⋊C4—32no—
197C22×C4⋊C4—62no—
198C22×C42—31no—
199#501—34no—
200#501—34no—
201#501—38no—
202#501—38no—
203#501—34no—
204#501C2×D314yescomplemented
205#499—34no—
206#499—34no—
207#499—38no—
208#499—38no—
209#499—34no—
210#499C2×D314yescomplemented
211#428—34no—
212#428—34no—
213#428—34no—
214#428—38no—
215#428—38no—
216#428C2×D314yes—
217C2×#107—38no—
218C2×#106—38no—
219C4×M16—32no—
220C4×M16—31no—
221C4×M16—32no—
222C4×M16—32no—
223C4×M16C2×D311yescharacteristic
224C2×#97—32no—
225C2×C4×C8—32no—
226C23×D3C2×D411yescharacteristic
227C2×C3×Q8—22no—
228C22×C3×C4—22no—
229C22×C3×C4C2×D412yes—
230C22×C3×C4C2411yescharacteristic
231C22×Dic3—22no—
232C22×Dic3C2×D412yes—
233C22×Dic3C2×D411yescharacteristic
234D3×Q8—48no—
235C2×C4×D3—24no—
236C2×C4×D3—24no—
237C2×C4×D3—24no—
238C2×C4×D3—44no—
239C2×C4×D3—48no—
240C2×C4×D3C2×D412yes—
241C2×C4×D3C2×D412yes—
242C2×C4×D3C2×D412yes—
243C2×C4×D3C2×D412yes—
244C2×C4×D3C2×D411yescharacteristic
245C2×C4×D3C2×D411yescharacteristic
246C2×C4×D3C2×Q818yes—
247C2×C4×D3C2411yescharacteristic
248C2×C4×D3C2411yescharacteristic
249C2×Dic6—22no—
250C2×Dic6—24no—
251C3×M16C2×Q818yescomplemented
252C2×C3×C8C2414yes—
253C3×C4⋊C4—22no—
254C3×C4⋊C4—24no—
255C3×C4⋊C4—24no—
256C3×C4⋊C4—24no—
257C3×C4⋊C4—22no—
258C3×C4⋊C4C2414yes—
259C3×C4⋊C4C2414yes—
260C3×C22⋊C4—22no—
261C3×C42—22no—
262C3×C42C2×D412yes—
263C3×C42C2411yescharacteristic
264C3×C42C2411yescharacteristic
265#217—22no—
266(C2×D3)⋊C4—24no—
267C4⋊Dic3—22no—
268C4⋊Dic3—22no—
269C4⋊Dic3—24no—
270C4⋊Dic3—24no—
271C4⋊Dic3—24no—
272C4⋊Dic3C2414yes—
273C4⋊Dic3C2414yes—
274Dic3⋊C4—24no—
275Dic3⋊C4—24no—
276Dic3⋊C4—24no—
277Dic3⋊C4—24no—
278Dic3⋊C4—24no—
279Dic3⋊C4—24no—
280Dic3⋊C4—24no—
281Dic3⋊C4—24no—
282Dic3⋊C4—24no—
283Dic3⋊C4—24no—
284C4×Dic3—22no—
285C4×Dic3—24no—
286C4×Dic3C2×D412yes—
287C4×Dic3C2×D412yes—
288C4×Dic3C2×D412yes—
289C4×Dic3C2×D412yes—
290C4×Dic3C2×D412yes—
291C4×Dic3C2411yescharacteristic
292C4×Dic3C2411yescharacteristic
293#208C2×Q818yescomplemented
294C2×C3⋊C8C2414yes—
295ΓL(1,𝔽25)C2×Q8116yescomplemented
296D3×C8C2×Q8116yes—
297C22×Q8—62no—
298C23×C4—31no—
299C23×C4—32no—
300C23×C4—62no—
301C2×M16—32no—
302C2×M16—316no—
303C2×M16—34no—
304C2×M16C22×D312yes—
305C22×C8—34no—
306C4⋊Q8—34no—
307C4⋊Q8—32no—
308C4⋊Q8—62no—
309C4⋊Q8—62no—
310C4⋊Q8C22×D312yes—
311#125—22no—
312#125—34no—
313#125—34no—
314#125—34no—
315#125—32no—
316#125—32no—
317#125—32no—
318#125—64no—
319#125—64no—
320#125—64no—
321#125—64no—
322#125—62no—
323#125—62no—
324#125—64no—
325#125—62no—
326#125C22×D312yes—
327#125C22×D312yes—
328#125C22×D312yes—
329C22⋊Q8—22no—
330C22⋊Q8—64no—
331C22⋊Q8—64no—
332C22⋊Q8—62no—
333C22⋊Q8—62no—
334C22⋊Q8—62no—
335C4×Q8—22no—
336C4×Q8—64no—
337C4×Q8—64no—
338C4×Q8—62no—
339C4×Q8—62no—
340C4×Q8—62no—
341#117—32no—
342#117—34no—
343#117—62no—
344#117—62no—
345#117C22×D312yes—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—32no—
348C2×C4⋊C4—34no—
349C2×C4⋊C4—34no—
350C2×C4⋊C4—34no—
351C2×C4⋊C4—62no—
352C2×C4⋊C4—62no—
353C2×C4⋊C4—64no—
354C2×C4⋊C4—64no—
355C2×C4⋊C4—64no—
356C2×C4⋊C4—64no—
357C2×C4⋊C4—64no—
358C2×C4⋊C4—62no—
359C2×C4⋊C4—62no—
360C2×C4⋊C4—62no—
361C2×C4⋊C4—62no—
362C2×C4⋊C4—62no—
363C2×C4⋊C4C22×D312yes—
364C2×C22⋊C4—62no—
365C2×C42—32no—
366C2×C42—31no—
367C2×C42—32no—
368C2×C42—32no—
369C2×C42—31no—
370C2×C42—31no—
371C2×C42—31no—
372C2×C42—31no—
373C2×C42—62no—
374C2×C42C22×D311yescharacteristic
375#107—38no—
376#107—38no—
377#107—38no—
378#107C22×D318yes—
379#106—38no—
380#106—38no—
381#106—38no—
382#106C22×D318yes—
383#97—34no—
384#97—32no—
385#97C22×D312yes—
386C4×C8—34no—
387C4×C8—32no—
388C4×C8C22×D312yes—
389C23×C3C22×D411yescharacteristic
390C22×D3—24no—
391C22×D3C4⋊Q814yes—
392C22×D3C22×D411yescharacteristic
393C22×D3C22×D411yescharacteristic
394C3×Q8—44no—
395C2×C3×C4—24no—
396C2×C3×C4—24no—
397C2×C3×C4—42no—
398C2×C3×C4—44no—
399C2×C3×C4C22×D412yes—
400C2×C3×C4C22×D412yes—
401C2×C3×C4C22×D411yescharacteristic
402C2×C3×C4C22×Q814yes—
403C2×C3×C4C2511yescharacteristic
404C2×Dic3—24no—
405C2×Dic3—24no—
406C2×Dic3—42no—
407C2×Dic3—44no—
408C2×Dic3C4⋊Q814yes—
409C2×Dic3C22×D412yes—
410C2×Dic3C22×D412yes—
411C2×Dic3C22×D411yescharacteristic
412C2×Dic3C22×D411yescharacteristic
413C4×D3—24no—
414C4×D3—24no—
415C4×D3—42no—
416C4×D3—42no—
417C4×D3—48no—
418C4×D3—48no—
419C4×D3—48no—
420C4×D3C4⋊Q814yes—
421C4×D3C4⋊Q814yes—
422C4×D3C22×Q814yes—
423C4×D3C22×Q814yes—
424Dic6—48no—
425Dic6—44no—
426C3×C8C22×Q818yes—
427C3⋊C8C22×Q818yes—
428C24—31no—
429C2×Q8—22no—
430C2×Q8—64no—
431C2×Q8—62no—
432C2×Q8—128no—
433C22×C4—22no—
434C22×C4—31no—
435C22×C4—31no—
436C22×C4—31no—
437C22×C4—38no—
438C22×C4—31no—
439C22×C4—31no—
440C22×C4—32no—
441C22×C4—32no—
442C22×C4—32no—
443C22×C4—32no—
444C22×C4—32no—
445C22×C4—64no—
446C22×C4—64no—
447C22×C4—64no—
448C22×C4—62no—
449C22×C4—124no—
450C22×C4—128no—
451C22×C4D3×D412yes—
452C22×C4C23×D311yescharacteristic
453M16—38no—
454M16—316no—
455M16D3×Q818yescomplemented
456C2×C8—34no—
457C2×C8—316no—
458C2×C8C23×D314yes—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—34no—
465C4⋊C4—34no—
466C4⋊C4—64no—
467C4⋊C4—64no—
468C4⋊C4—64no—
469C4⋊C4—64no—
470C4⋊C4—64no—
471C4⋊C4—64no—
472C4⋊C4—64no—
473C4⋊C4—64no—
474C4⋊C4—64no—
475C4⋊C4—64no—
476C4⋊C4—62no—
477C4⋊C4—64no—
478C4⋊C4—64no—
479C4⋊C4—64no—
480C4⋊C4—62no—
481C4⋊C4C23×D314yes—
482C4⋊C4C23×D314yes—
483C22⋊C4—22no—
484C22⋊C4—64no—
485C22⋊C4—62no—
486C42—22no—
487C42—32no—
488C42—32no—
489C42—32no—
490C42—32no—
491C42—32no—
492C42—31no—
493C42—31no—
494C42—64no—
495C42—62no—
496C42D3×D412yes—
497C42C23×D311yescharacteristic
498C42C23×D311yescharacteristic
499C22×C3—22no—
500C22×C3C2×C4⋊Q812yes—
501C22×C3C23×D411yescharacteristic
502C2×D3—22no—
503C2×D3—28no—
504C2×D3C2×C4⋊Q812yes—
505C2×D3C2×C4⋊Q812yes—
506C2×D3#57514yes—
507C3×C4—22no—
508C3×C4—22no—
509C3×C4—44no—
510C3×C4—44no—
511C3×C4C2×C4⋊Q812yes—
512C3×C4C23×Q812yes—
513Dic3—22no—
514Dic3—22no—
515Dic3—44no—
516Dic3—44no—
517Dic3C2×C4⋊Q812yes—
518Dic3C2×C4⋊Q812yes—
519C23—31no—
520C23—31no—
521C23—34no—
522C23—64no—
523C23C2×D3×D411yescharacteristic
524Q8—44no—
525Q8—124no—
526Q8—128no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—31no—
530C2×C4—31no—
531C2×C4—34no—
532C2×C4—34no—
533C2×C4—34no—
534C2×C4—34no—
535C2×C4—34no—
536C2×C4—32no—
537C2×C4—32no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—64no—
541C2×C4—64no—
542C2×C4—64no—
543C2×C4—64no—
544C2×C4—122no—
545C2×C4—122no—
546C2×C4—128no—
547C2×C4—128no—
548C2×C4—128no—
549C2×C4—122no—
550C2×C4—124no—
551C2×C4C2×D3×D412yes—
552C2×C4C2×D3×D412yes—
553C2×C4C2×D3×D411yescharacteristic
554C2×C4C2×D3×Q814yes—
555C2×C4C24×D311yescharacteristic
556C8—38no—
557C8C2×D3×Q818yes—
558C2×C3—22no—
559C2×C3C22×C4⋊Q811yescharacteristic
560C2×C3C2×#57512yes—
561D3—22no—
562D3#316614yesdirect product factor
563C22—22no—
564C22—34no—
565C22—32no—
566C22—32no—
567C22—68no—
568C22—62no—
569C22D3×C4⋊Q812yes—
570C22C22×D3×D411yescharacteristic
571C4—22no—
572C4—22no—
573C4—32no—
574C4—32no—
575C4—44no—
576C4—44no—
577C4—62no—
578C4—62no—
579C4—124no—
580C4—124no—
581C4D3×C4⋊Q812yes—
582C4C22×D3×Q812yes—
583C3C2×#316611yescharacteristic, complemented, Sylow
584C2—22no—
585C2—34no—
586C2—62no—
587C2C2×D3×C4⋊Q811yescharacteristic
588C2D3×#57512yes—
subgroupG/H
centerC22C22×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×#3166C2
socleC22×C3C23×D4
prime radicalC23×D3C2×D4
hypercenter#3166D3
hypocenterC3C2×#3166
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3166

Complements

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

NH ≅ G/Ntype
3. C3×#3166C2semidirect
4. #83342C2semidirect
5. #83015C2semidirect
6. #81977C2semidirect
7. D3×#501C2semidirect
8. #81919C2semidirect
9. D3×#499C2semidirect
11. #79204C2semidirect
13. #78507C2semidirect
41. C3×#501C22semidirect
42. C3×#499C22semidirect
47. #5081C22semidirect
48. #5065C22semidirect
49. #5065C22semidirect
50. #5064C22semidirect
56. #4878C22semidirect
57. #4876C22semidirect
60. #4858C22semidirect
61. #4856C22semidirect
64. #4702C22semidirect
67. #4683C22semidirect
76. #3166D3direct (N × H)
110. D3×M16Q8semidirect
204. #501C2×D3semidirect
210. #499C2×D3semidirect
251. C3×M16C2×Q8semidirect
293. #208C2×Q8semidirect
295. ΓL(1,𝔽25)C2×Q8semidirect
455. M16D3×Q8semidirect
562. D3#3166direct (N × H)
583. C3C2×#3166semidirect

Maximal subgroups

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

subgroupindex
D3×#5332
C2×D3×#1252
C3×#31662
#833422
#830152
#819772
D3×#5012
#819192
D3×#4992
D3×#4282
#792042
C4×D3×M162
#785072
C2×#31663