Finite Group Databasebeta

← D3×#2072 #10823927

Subgroups — D3×#2072 #10823927

Order 768.

1150 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×#103C212yesmaximal
2D3×#391C211yesmaximal, characteristic
3D3×#384C211yesmaximal, characteristic
4C3×#2072C211yesmaximal, characteristic, complemented
5#70233C212yesmaximal, complemented
6#69670C211yesmaximal, characteristic, complemented
7#69589C211yesmaximal, characteristic, complemented
8#68914C212yesmaximal, complemented
9D3×#340C212yesmaximal
10D3×#336C211yesmaximal, characteristic
11#68636C211yesmaximal, characteristic, complemented
12C2×#2072—31nomaximal, Sylow
13C22×D3×Q8C2211yescharacteristic
14C22×D3×C8C2212yes—
15C2×D3×C4⋊C4—21no—
16C2×D3×C4⋊C4—21no—
17C2×D3×C4⋊C4C2211yescharacteristic
18C2×D3×C4⋊C4C2212yes—
19C2×C42×D3C2211yescharacteristic
20C2×C3×#103C2212yes—
21C3×#391C2211yescharacteristic
22C3×#384C2211yescharacteristic
23#5229C2211yescharacteristic
24C2×#835C2212yes—
25C2×#816C2212yes—
26#4975—21no—
27#4975—21no—
28#4975C2212yes—
29#4973C2211yescharacteristic
30C2×#810C2212yes—
31#4935C2211yescharacteristic
32#4929C2211yescharacteristic
33D3×#103—28no—
34D3×#95—21no—
35D3×#95—21no—
36#4646—21no—
37#4646—21no—
38C3×#340C2212yes—
39C3×#336C2211yescharacteristic
40#4551C2212yes—
41#4550C2212yes—
42#4529C2212yes—
43#4529C2212yes—
44#4524C2211yescharacteristic
45C22×#103—32no—
46C2×#391—31no—
47C2×#384—31no—
48#2072—32no—
49#2072—31no—
50#2072—31no—
51#2072—32no—
52#2072—31no—
53#2072D311yescharacteristic, direct product factor
54C2×#340—32no—
55C2×#336—31no—
56C22×C3×Q8C2311yescharacteristic
57C2×D3×Q8—22no—
58C2×D3×Q8—24no—
59C2×D3×Q8—24no—
60C2×D3×Q8D414yes—
61C22×C4×D3—21no—
62C22×C4×D3—21no—
63C22×C4×D3—22no—
64C22×C4×D3D411yescharacteristic
65C22×C4×D3D411yescharacteristic
66C22×C4×D3C2311yescharacteristic
67C22×Dic6—21no—
68C22×Dic6C2311yescharacteristic
69C22×C3×C8C2312yes—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4C2312yes—
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C42C2311yescharacteristic
75C2×C4⋊Dic3—21no—
76C2×C4⋊Dic3—21no—
77C2×C4⋊Dic3C2311yescharacteristic
78C2×C4⋊Dic3C2312yes—
79C2×Dic3⋊C4—21no—
80C2×Dic3⋊C4—22no—
81C2×Dic3⋊C4—22no—
82C2×Dic3⋊C4—22no—
83C2×C4×Dic3D411yescharacteristic
84C2×C4×Dic3D411yescharacteristic
85C2×C4×Dic3C2311yescharacteristic
86C22×C3⋊C8C2312yes—
87C2×D3×C8—24no—
88C2×D3×C8—24no—
89C2×D3×C8—24no—
90D3×C4⋊C4—24no—
91D3×C4⋊C4—24no—
92D3×C4⋊C4—24no—
93D3×C4⋊C4—44no—
94D3×C4⋊C4—44no—
95C42×D3—24no—
96C3×#103—24no—
97C3×#95—21no—
98C3×#95—21no—
99#835—24no—
100C3⋊#95—21no—
101C3⋊#95—21no—
102C3⋊#95—21no—
103C3⋊#95—21no—
104C3⋊#95—21no—
105C3⋊#95—21no—
106#816—24no—
107#810—24no—
108C23×Q8—31no—
109C23×C8—32no—
110C22×C4⋊C4—32no—
111C22×C4⋊C4—31no—
112C22×C4⋊C4—61no—
113C22×C4⋊C4—61no—
114C22×C42—31no—
115C2×#103—32no—
116C2×#103—32no—
117C2×#103—32no—
118C2×#103—68no—
119C2×#103C2×D312yes—
120#391—31no—
121#391—31no—
122#391—31no—
123#391—61no—
124#391—61no—
125#391C2×D311yescharacteristic
126#384—32no—
127#384—31no—
128#384—61no—
129#384—61no—
130#384C2×D311yescharacteristic
131C2×#95—61no—
132C2×#95—61no—
133#340—32no—
134#340—32no—
135#340—32no—
136#340C2×D312yes—
137#336—32no—
138#336—31no—
139#336C2×D311yescharacteristic
140C23×D3C2×D411yescharacteristic
141C2×C3×Q8—21no—
142C2×C3×Q8—22no—
143C2×C3×Q8—22no—
144C2×C3×Q8C2×D412yes—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—22no—
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2411yescharacteristic
151C22×Dic3—21no—
152C22×Dic3—21no—
153C22×Dic3—22no—
154C22×Dic3C2×D411yescharacteristic
155C22×Dic3C2×D411yescharacteristic
156C22×Dic3C2×D411yescharacteristic
157D3×Q8—216no—
158D3×Q8—48no—
159C2×C4×D3—22no—
160C2×C4×D3—22no—
161C2×C4×D3—22no—
162C2×C4×D3—22no—
163C2×C4×D3—24no—
164C2×C4×D3—42no—
165C2×C4×D3—42no—
166C2×C4×D3—44no—
167C2×C4×D3—42no—
168C2×C4×D3—44no—
169C2×C4×D3—44no—
170C2×C4×D3—42no—
171C2×C4×D3—42no—
172C2×C4×D3SD1614yes—
173C2×C4×D3Q1614yes—
174C2×C4×D3C2×D412yes—
175C2×C4×D3#4114yes—
176C2×C4×D3#4114yes—
177C2×C4×D3#4112yes—
178C2×Dic6—22no—
179C2×Dic6—22no—
180C2×Dic6—21no—
181C2×Dic6—24no—
182C2×Dic6—42no—
183C2×Dic6—42no—
184C2×Dic6C2×D412yes—
185C2×C3×C8—22no—
186C2×C3×C8—22no—
187C2×C3×C8—22no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—42no—
192C3×C4⋊C4—42no—
193C3×C42—22no—
194C4⋊Dic3—22no—
195C4⋊Dic3—22no—
196C4⋊Dic3—22no—
197C4⋊Dic3—42no—
198C4⋊Dic3—42no—
199Dic3⋊C4—44no—
200Dic3⋊C4—44no—
201Dic3⋊C4—42no—
202Dic3⋊C4—44no—
203C4×Dic3—22no—
204C4×Dic3—22no—
205C4×Dic3—22no—
206C2×C3⋊C8—22no—
207C2×C3⋊C8—22no—
208C2×C3⋊C8—22no—
209D3×C8—48no—
210C22×Q8—34no—
211C22×Q8—31no—
212C22×Q8—61no—
213C22×Q8—62no—
214C22×Q8—64no—
215C22×Q8—64no—
216C22×Q8C22×D311yescharacteristic
217C23×C4—31no—
218C23×C4—31no—
219C23×C4—31no—
220C23×C4—61no—
221C23×C4—62no—
222C23×C4—61no—
223C22×C8—32no—
224C22×C8—64no—
225C22×C8—64no—
226C22×C8—64no—
227C22×C8C22×D312yes—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—31no—
231C2×C4⋊C4—32no—
232C2×C4⋊C4—64no—
233C2×C4⋊C4—64no—
234C2×C4⋊C4—64no—
235C2×C4⋊C4—62no—
236C2×C4⋊C4—61no—
237C2×C4⋊C4—62no—
238C2×C4⋊C4—62no—
239C2×C4⋊C4—61no—
240C2×C4⋊C4—61no—
241C2×C4⋊C4—124no—
242C2×C4⋊C4—124no—
243C2×C4⋊C4C22×D311yescharacteristic
244C2×C4⋊C4C22×D312yes—
245C2×C42—31no—
246C2×C42—31no—
247C2×C42—31no—
248C2×C42—64no—
249C2×C42C22×D311yescharacteristic
250#103—24no—
251#103—64no—
252#103—64no—
253#103—64no—
254#95—21no—
255#95—21no—
256#95—61no—
257#95—61no—
258#95—61no—
259#95—61no—
260#95—61no—
261#95—61no—
262C23×C3C22×D411yescharacteristic
263C22×D3#12112yes—
264C22×D3#12312yes—
265C22×D3#12412yes—
266C22×D3C2×SD1612yes—
267C22×D3C2×Q1612yes—
268C22×D3AGL(1,ℤ/8)12yes—
269C22×D3#13712yes—
270C3×Q8—24no—
271C3×Q8—42no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—22no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—42no—
280C2×C3×C4—41no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4C2×SD1612yes—
286C2×C3×C4C2×Q1612yes—
287C2×C3×C4C22×D411yescharacteristic
288C2×C3×C4C2×#4112yes—
289C2×C3×C4C2×#4112yes—
290C2×C3×C4C2×#4111yescharacteristic
291C2×Dic3—21no—
292C2×Dic3—21no—
293C2×Dic3—21no—
294C2×Dic3—21no—
295C2×Dic3—22no—
296C2×Dic3—41no—
297C2×Dic3—41no—
298C2×Dic3—42no—
299C2×Dic3—41no—
300C2×Dic3—42no—
301C2×Dic3—42no—
302C2×Dic3—41no—
303C2×Dic3—41no—
304C2×Dic3#12112yes—
305C2×Dic3#12312yes—
306C2×Dic3#12412yes—
307C2×Dic3C2×SD1612yes—
308C2×Dic3C2×Q1612yes—
309C4×D3—28no—
310C4×D3—28no—
311C4×D3—48no—
312C4×D3—88no—
313C4×D3—84no—
314C4×D3#12318yes—
315C4×D3#12618yes—
316Dic6—24no—
317Dic6—42no—
318Dic6—48no—
319C3×C8—42no—
320C3⋊C8—42no—
321C24—31no—
322C2×Q8—21no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—32no—
326C2×Q8—64no—
327C2×Q8—616no—
328C2×Q8—61no—
329C2×Q8—62no—
330C2×Q8—62no—
331C2×Q8—122no—
332C2×Q8—122no—
333C2×Q8—128no—
334C2×Q8D3×D412yes—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—31no—
339C22×C4—32no—
340C22×C4—34no—
341C22×C4—32no—
342C22×C4—34no—
343C22×C4—31no—
344C22×C4—31no—
345C22×C4—34no—
346C22×C4—34no—
347C22×C4—62no—
348C22×C4—62no—
349C22×C4—62no—
350C22×C4—62no—
351C22×C4—64no—
352C22×C4—61no—
353C22×C4—61no—
354C22×C4—62no—
355C22×C4—122no—
356C22×C4—122no—
357C22×C4—122no—
358C22×C4—124no—
359C22×C4—124no—
360C22×C4—124no—
361C22×C4—122no—
362C22×C4—122no—
363C22×C4D3×D411yescharacteristic
364C22×C4D3×D411yescharacteristic
365C22×C4C23×D311yescharacteristic
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—62no—
370C2×C8—62no—
371C2×C8—62no—
372C2×C8—128no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—62no—
379C4⋊C4—62no—
380C4⋊C4—62no—
381C4⋊C4—122no—
382C4⋊C4—122no—
383C4⋊C4—124no—
384C4⋊C4—124no—
385C4⋊C4—124no—
386C4⋊C4—122no—
387C42—22no—
388C42—62no—
389C42—62no—
390C42—62no—
391C22×C3C2×#12111yescharacteristic
392C22×C3C2×#12311yescharacteristic
393C22×C3C2×#12411yescharacteristic
394C22×C3C22×SD1611yescharacteristic
395C22×C3C22×Q1611yescharacteristic
396C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
397C22×C3C2×#13711yescharacteristic
398C2×D3#39714yes—
399C2×D3#46014yes—
400C2×D3#46214yes—
401C2×D3#48314yes—
402C2×D3#48414yes—
403C2×D3#48714yes—
404C2×D3#48914yes—
405C3×C4—22no—
406C3×C4—22no—
407C3×C4—42no—
408C3×C4—81no—
409C3×C4—82no—
410C3×C4C2×#12312yes—
411C3×C4C2×#12612yes—
412Dic3—22no—
413Dic3—22no—
414Dic3—42no—
415Dic3—81no—
416Dic3—82no—
417Dic3#39714yes—
418C23—32no—
419C23—32no—
420C23—32no—
421C23—32no—
422C23—32no—
423C23—32no—
424C23—32no—
425C23C2×D3×D411yescharacteristic
426Q8—24no—
427Q8—42no—
428Q8—64no—
429Q8—128no—
430Q8—122no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—22no—
436C2×C4—32no—
437C2×C4—32no—
438C2×C4—32no—
439C2×C4—38no—
440C2×C4—38no—
441C2×C4—32no—
442C2×C4—32no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—68no—
452C2×C4—68no—
453C2×C4—61no—
454C2×C4—61no—
455C2×C4—61no—
456C2×C4—61no—
457C2×C4—62no—
458C2×C4—128no—
459C2×C4—121no—
460C2×C4—121no—
461C2×C4—121no—
462C2×C4—122no—
463C2×C4—122no—
464C2×C4—121no—
465C2×C4—121no—
466C2×C4—122no—
467C2×C4—244no—
468C2×C4—248no—
469C2×C4D3×SD1612yes—
470C2×C4D3×Q1612yes—
471C2×C4C2×D3×D411yescharacteristic
472C2×C4D3×#4112yes—
473C2×C4D3×#4112yes—
474C2×C4D3×#4111yescharacteristic
475C8—42no—
476C8—122no—
477C2×C3C2×#39711yescharacteristic
478C2×C3C2×#46011yescharacteristic
479C2×C3C2×#46211yescharacteristic
480C2×C3C2×#48311yescharacteristic
481C2×C3C2×#48411yescharacteristic
482C2×C3C2×#48711yescharacteristic
483C2×C3C2×#48911yescharacteristic
484D3#207218yesdirect product factor
485C22—34no—
486C22—34no—
487C22—34no—
488C22—34no—
489C22—34no—
490C22—34no—
491C22—34no—
492C22D3×#12111yescharacteristic
493C22D3×#12311yescharacteristic
494C22D3×#12411yescharacteristic
495C22C2×D3×SD1611yescharacteristic
496C22C2×D3×Q1611yescharacteristic
497C22D3×AGL(1,ℤ/8)11yescharacteristic
498C22D3×#13711yescharacteristic
499C4—22no—
500C4—22no—
501C4—34no—
502C4—42no—
503C4—62no—
504C4—62no—
505C4—81no—
506C4—82no—
507C4—122no—
508C4—241no—
509C4—242no—
510C4D3×#12312yes—
511C4D3×#12612yes—
512C3C2×#207211yescharacteristic, complemented, Sylow
513C2—38no—
514C2D3×#39711yescharacteristic
515C2D3×#46011yescharacteristic
516C2D3×#46211yescharacteristic
517C2D3×#48311yescharacteristic
518C2D3×#48411yescharacteristic
519C2D3×#48711yescharacteristic
520C2D3×#48911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2072C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#2072D3
hypocenterC3C2×#2072
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2072

Complements

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

NH ≅ G/Ntype
4. C3×#2072C2semidirect
5. #70233C2semidirect
6. #69670C2semidirect
7. #69589C2semidirect
8. #68914C2semidirect
11. #68636C2semidirect
53. #2072D3direct (N × H)
484. D3#2072direct (N × H)
512. C3C2×#2072semidirect

Maximal subgroups

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

subgroupindex
C2×D3×#1032
D3×#3912
D3×#3842
C3×#20722
#702332
#696702
#695892
#689142
D3×#3402
D3×#3362
#686362
C2×#20723