Finite Group Databasebeta

← Dic3×#4043 #14321807

Subgroups — Dic3×#4043 #14321807

Order 1920.

1312 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4043C211yesmaximal, characteristic
2C2×C4×D5×Dic3C211yesmaximal, characteristic
3C2×Dic3×Dic10C211yesmaximal, characteristic, complemented
4#11751085C211yesmaximal, characteristic, complemented
5C5×Dic3×C4⋊C4C211yesmaximal, characteristic, complemented
6#11748900C212yesmaximal
7Dic3×#685C212yesmaximal
8#11748759C211yesmaximal, characteristic
9#11748697C212yesmaximal, complemented
10Dic3×C4⋊Dic5C212yesmaximal, complemented
11#11748610C211yesmaximal, characteristic, complemented
12C4×#4043—31nomaximal, Hall
13C22×C3×C4×D5C2211yescharacteristic
14C22×C3×Dic10C2211yescharacteristic
15C22×D5×Dic3C2211yescharacteristic
16C22×D5×Dic3C2211yescharacteristic
17C2×C3×C5×C4⋊C4C2211yescharacteristic
18C2×C4⋊Dic15C2212yes—
19C2×C4⋊Dic15C2211yescharacteristic
20C2×C4×Dic15C2211yescharacteristic
21C2×C5×C4⋊Dic3C2211yescharacteristic
22C2×C4×C5×Dic3C2212yes—
23C2×C4×C5×Dic3C2211yescharacteristic
24C2×C3×#685C2212yes—
25C2×C3×C4⋊Dic5C2212yes—
26C3×#4043C418yescomplemented
27C2×#6668C2212yes—
28C2×#6665C2212yes—
29C2×Dic3×Dic5C2212yes—
30C2×Dic3×Dic5C2211yescharacteristic
31C2×Dic3×Dic5C2211yescharacteristic
32C4×D5×Dic3—24no—
33Dic3×Dic10—24no—
34#91115C2212yes—
35C5×C3⋊#95C2212yes—
36#90994C2212yes—
37#90994C2212yes—
38#90994C2212yes—
39Dic3×C22⋊Q8—51nomaximal, Hall
40C2×#4043D311yescharacteristic
41C2×C42×D5—31no—
42C2×C4×Dic10—31no—
43C4×C5×C4⋊C4—31no—
44#65276—31no—
45#65274—31no—
46#65263—31no—
47C4×#685—32no—
48C4×C4⋊Dic5—32no—
49#64957—32no—
50#64942—32no—
51C23×C3×D5C2311yescharacteristic
52C22×C3×C4×C5C2312yes—
53C22×C3×C4×C5C2311yescharacteristic
54C22×Dic15C2312yes—
55C22×Dic15C2311yescharacteristic
56C22×Dic15C2311yescharacteristic
57C22×C5×Dic3C2312yes—
58C22×C5×Dic3C2311yescharacteristic
59C22×C5×Dic3C2311yescharacteristic
60C22×C3×Dic5C2311yescharacteristic
61C22×C3×Dic5C2312yes—
62C2×C3×C4×D5—22no—
63C2×C3×C4×D5—22no—
64C2×C3×C4×D5C2×C414yes—
65C2×C3×Dic10—22no—
66C2×C3×Dic10—22no—
67C2×C3×Dic10C2×C414yescomplemented
68C2×D5×Dic3—22no—
69C2×D5×Dic3—22no—
70C2×D5×Dic3—22no—
71C2×D5×Dic3—22no—
72C2×D5×Dic3—22no—
73C2×D5×Dic3Q814yescomplemented
74C3×C5×C4⋊C4C2×C414yescomplemented
75C4⋊Dic15—22no—
76C4⋊Dic15—24no—
77C4×Dic15—22no—
78C5×C4⋊Dic3—22no—
79C4×C5×Dic3—24no—
80C4×C5×Dic3D414yes—
81C3×#685C2×C418yes—
82C3×C4⋊Dic5C2×C418yescomplemented
83#6668—24no—
84#6665—24no—
85Dic3×Dic5—24no—
86Dic3×Dic5—22no—
87Dic3×Dic5—22no—
88C2×C3×C22⋊Q8—51no—
89C2×Q8×Dic3—51no—
90C22×C4×Dic3—51no—
91#5229—51no—
92#5212—51no—
93#4984—51no—
94#4980—52no—
95Dic3×C4⋊C4—52no—
96Dic3×C4⋊C4D511yescharacteristic, complemented
97#4950—52no—
98Dic3×C22⋊C4—52no—
99C22×C4×D5—31no—
100C22×C4×D5—31no—
101C22×C4×D5C2×D311yescharacteristic
102C22×Dic10C2×D311yescharacteristic
103C2×C5×C4⋊C4—31no—
104C2×C5×C4⋊C4C2×D311yescharacteristic
105C2×C42×C5—31no—
106C2×C42×C5—32no—
107C2×#685—32no—
108C2×#685C2×D312yes—
109C2×C4⋊Dic5—31no—
110C2×C4⋊Dic5—32no—
111C2×C4⋊Dic5C2×D312yes—
112C2×#683—32no—
113C2×C4×Dic5—31no—
114C2×C4×Dic5—32no—
115C2×C4×Dic5—31no—
116C2×C4×Dic5—31no—
117#4043Dic318yesdirect product factor
118C42×D5—64no—
119C4×Dic10—64no—
120C5×#95—32no—
121#3963—32no—
122#3963—32no—
123#3963—32no—
124#3963—32no—
125C4×C22⋊Q8—151noSylow
126C23×C3×C5C2411yescharacteristic
127C22×C3×D5—21no—
128C22×C3×D5—21no—
129C22×C3×D5—21no—
130C22×C3×D5—21no—
131C22×C3×D5C22×C412yes—
132C22×C3×D5C2×Q812yes—
133C22×C3×D5#4112yes—
134C2×C3×C4×C5—22no—
135C2×C3×C4×C5—22no—
136C2×C3×C4×C5C22×C414yes—
137C2×C3×C4×C5C22×C412yes—
138C2×C3×C4×C5C2×D412yes—
139C2×C3×C4×C5#4112yes—
140C2×Dic15—21no—
141C2×Dic15—21no—
142C2×Dic15—21no—
143C2×Dic15—21no—
144C2×Dic15—22no—
145C2×Dic15—22no—
146C2×Dic15—22no—
147C2×Dic15C2×Q812yes—
148C2×Dic15#4112yes—
149C2×C5×Dic3—21no—
150C2×C5×Dic3—21no—
151C2×C5×Dic3—22no—
152C2×C5×Dic3—22no—
153C2×C5×Dic3—22no—
154C2×C5×Dic3C2×D412yes—
155C2×C5×Dic3C2×D412yes—
156C2×C5×Dic3C2×Q812yes—
157C2×C5×Dic3#4112yes—
158C2×C3×Dic5—21no—
159C2×C3×Dic5—22no—
160C2×C3×Dic5—22no—
161C2×C3×Dic5—21no—
162C2×C3×Dic5C22×C412yes—
163C2×C3×Dic5C22×C414yes—
164C3×C4×D5—28no—
165C3×Dic10—28no—
166D5×Dic3—28no—
167D5×Dic3—28no—
168C22×C3×Q8—51no—
169C23×C3×C4—51no—
170C23×Dic3—51no—
171C23×Dic3—51no—
172C3×C22⋊Q8—58no—
173C2×C3×C4⋊C4—52no—
174C2×C3×C4⋊C4C2×D511yescharacteristic
175C2×C3×C22⋊C4—52no—
176C2×#217—52no—
177Q8×Dic3—104no—
178C2×C4⋊Dic3—52no—
179C2×C4⋊Dic3—51no—
180C2×C4⋊Dic3—52no—
181C2×C4⋊Dic3C2×D511yescharacteristic
182C2×C4×Dic3—51no—
183C2×C4×Dic3—52no—
184C2×C4×Dic3—51no—
185C2×C4×Dic3—51no—
186C2×C4×Dic3—104no—
187C2×C4×Dic3C2×D512yes—
188C2×C4×Dic3C2×D511yescharacteristic
189C3⋊#95—52no—
190C3⋊#95—52no—
191C3⋊#95—52no—
192C3⋊#95—52no—
193C3⋊#95C2×D512yes—
194C23×D5C22×D311yescharacteristic
195C22×C4×C5—31no—
196C22×C4×C5—31no—
197C22×C4×C5—32no—
198C22×C4×C5C22×D312yes—
199C22×C4×C5C22×D311yescharacteristic
200C22×Dic5—31no—
201C22×Dic5—31no—
202C22×Dic5—32no—
203C22×Dic5C22×D312yes—
204C22×Dic5C22×D311yescharacteristic
205C2×C4×D5—22no—
206C2×C4×D5—22no—
207C2×C4×D5—34no—
208C2×C4×D5—62no—
209C2×C4×D5—62no—
210C2×C4×D5—62no—
211C2×C4×D5—62no—
212C2×C4×D5—62no—
213C2×C4×D5C2×Dic314yes—
214C2×Dic10—22no—
215C2×Dic10—22no—
216C2×Dic10C2×Dic314yescomplemented
217C5×C4⋊C4—62no—
218C5×C4⋊C4C2×Dic314yescomplemented
219C42×C5—34no—
220C42×C5—64no—
221#685—64no—
222#685C2×Dic318yes—
223C4⋊Dic5—64no—
224C4⋊Dic5—62no—
225C4⋊Dic5C2×Dic318yescomplemented
226#683—64no—
227C4×Dic5—64no—
228C4×Dic5—62no—
229C4×Dic5—62no—
230C4×Dic5—62no—
231C2×C22⋊Q8—51no—
232C2×C4×Q8—151no—
233C22×C42—151no—
234#391—151no—
235#389—151no—
236#387—152no—
237#385—152no—
238#380—151no—
239C4×C4⋊C4—31no—
240C4×C4⋊C4—152no—
241C4×C22⋊C4—152no—
242C22×C3×C5C23×C411yescharacteristic
243C22×C3×C5C22×D411yescharacteristic
244C22×C3×C5C22×Q811yescharacteristic
245C22×C3×C5C2×#4111yescharacteristic
246C22×C3×C5C2×#4111yescharacteristic
247C22×C3×C5C2×#4111yescharacteristic
248C22×C3×C5C2×#4111yescharacteristic
249C2×C3×D5—22no—
250C2×C3×D5—22no—
251C2×C3×D5—22no—
252C2×C3×D5—22no—
253C2×C3×D5—22no—
254C2×C3×D5—22no—
255C2×C3×D5C4×Q814yescomplemented
256C3×C4×C5—24no—
257C3×C4×C5C4×D414yes—
258Dic15—22no—
259Dic15—22no—
260Dic15—42no—
261C3×Dic5—24no—
262C3×Dic5—22no—
263C5×Dic3—22no—
264C5×Dic3—42no—
265C5×Dic3C22⋊Q814yescomplemented
266C24×C3—51no—
267C2×C3×Q8—54no—
268C2×C3×Q8—102no—
269C2×C3×Q8—102no—
270C22×C3×C4—51no—
271C22×C3×C4—52no—
272C22×C3×C4—54no—
273C22×C3×C4—102no—
274C22×C3×C4—102no—
275C22×C3×C4C22×D512yes—
276C22×C3×C4C22×D511yescharacteristic
277C22×Dic3—51no—
278C22×Dic3—51no—
279C22×Dic3—54no—
280C22×Dic3—52no—
281C22×Dic3—102no—
282C22×Dic3—102no—
283C22×Dic3—102no—
284C22×Dic3—102no—
285C22×Dic3—102no—
286C22×Dic3C22×D512yes—
287C22×Dic3C22×D511yescharacteristic
288C22×Dic3C22×D511yescharacteristic
289C3×C4⋊C4—58no—
290C3×C4⋊C4C4×D514yescomplemented
291C3×C22⋊C4—58no—
292#217—104no—
293C4⋊Dic3—22no—
294C4⋊Dic3—104no—
295C4⋊Dic3—102no—
296C4⋊Dic3—104no—
297C4×Dic3—24no—
298C4×Dic3—102no—
299C4×Dic3—102no—
300C4×Dic3—102no—
301C4×Dic3—104no—
302C4×Dic3D2014yes—
303C23×C5C23×D311yescharacteristic
304C22×D5—21no—
305C22×D5—21no—
306C22×D5—21no—
307C22×D5—21no—
308C22×D5D3×Q812yes—
309C22×D5#23912yes—
310C22×D5C22×Dic312yes—
311C2×C4×C5—22no—
312C2×C4×C5—22no—
313C2×C4×C5—32no—
314C2×C4×C5—32no—
315C2×C4×C5—32no—
316C2×C4×C5—32no—
317C2×C4×C5—61no—
318C2×C4×C5—61no—
319C2×C4×C5—62no—
320C2×C4×C5—62no—
321C2×C4×C5—62no—
322C2×C4×C5D3×D412yes—
323C2×C4×C5#23712yes—
324C2×C4×C5C22×Dic314yes—
325C2×C4×C5C22×Dic312yes—
326C2×Dic5—21no—
327C2×Dic5—22no—
328C2×Dic5—22no—
329C2×Dic5—21no—
330C2×Dic5—32no—
331C2×Dic5—32no—
332C2×Dic5—61no—
333C2×Dic5—61no—
334C2×Dic5—61no—
335C2×Dic5—61no—
336C2×Dic5—62no—
337C2×Dic5—62no—
338C2×Dic5—62no—
339C2×Dic5C22×Dic312yes—
340C2×Dic5C22×Dic314yes—
341C4×D5—28no—
342C4×D5—68no—
343C4×D5—68no—
344Dic10—28no—
345C22×Q8—51no—
346C23×C4—51no—
347C23×C4—151no—
348C23×C4—151no—
349C22⋊Q8—58no—
350C4×Q8—304no—
351C2×C4⋊C4—31no—
352C2×C4⋊C4—52no—
353C2×C4⋊C4—152no—
354C2×C4⋊C4—151no—
355C2×C4⋊C4—152no—
356C2×C4⋊C4D3×D511yescharacteristic
357C2×C22⋊C4—52no—
358C2×C22⋊C4—152no—
359C2×C42—31no—
360C2×C42—32no—
361C2×C42—152no—
362C2×C42—151no—
363C2×C42—151no—
364C2×C42—151no—
365C2×C42—304no—
366#95—32no—
367#95—152no—
368#95—152no—
369#95—152no—
370#95—152no—
371C2×C3×C5C2×C4×D411yescharacteristic
372C2×C3×C5C2×C4×Q811yescharacteristic
373C2×C3×C5C4×#4111yescharacteristic
374C2×C3×C5C2×C22⋊Q811yescharacteristic
375C2×C3×C5#52511yescharacteristic
376C2×C3×C5#52911yescharacteristic
377C2×C3×C5#53311yescharacteristic
378C3×D5—24no—
379C23×C3—52no—
380C23×C3—52no—
381C23×C3—52no—
382C23×C3—101no—
383C23×C3—101no—
384C23×C3—101no—
385C23×C3—101no—
386C23×C3C23×D511yescharacteristic
387C3×Q8—108no—
388C2×C3×C4—22no—
389C2×C3×C4—22no—
390C2×C3×C4—54no—
391C2×C3×C4—52no—
392C2×C3×C4—101no—
393C2×C3×C4—102no—
394C2×C3×C4—101no—
395C2×C3×C4—102no—
396C2×C3×C4—108no—
397C2×C3×C4C2×C4×D514yes—
398C2×C3×C4C2×C4×D512yes—
399C2×C3×C4C2×D2012yes—
400C2×C3×C4#70912yes—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—22no—
404C2×Dic3—22no—
405C2×Dic3—22no—
406C2×Dic3—52no—
407C2×Dic3—52no—
408C2×Dic3—108no—
409C2×Dic3—101no—
410C2×Dic3—101no—
411C2×Dic3—101no—
412C2×Dic3—101no—
413C2×Dic3—108no—
414C2×Dic3—102no—
415C2×Dic3—102no—
416C2×Dic3—102no—
417C2×Dic3C2×D2012yes—
418C2×Dic3C2×D2012yes—
419C2×Dic3#71112yes—
420C2×Dic3Q8×D512yes—
421C22×C5C2×D3×D411yescharacteristic
422C22×C5C2×#23711yescharacteristic
423C22×C5C2×D3×Q811yescharacteristic
424C22×C5C2×#23911yescharacteristic
425C22×C5D3×#4111yescharacteristic
426C22×C5D3×#4111yescharacteristic
427C22×C5C23×Dic311yescharacteristic
428C2×D5—22no—
429C2×D5—22no—
430C2×D5—22no—
431C2×D5—22no—
432C2×D5—22no—
433C2×D5—22no—
434C2×D5Q8×Dic314yescomplemented
435C4×C5—24no—
436C4×C5—34no—
437C4×C5—62no—
438C4×C5—122no—
439C4×C5D4×Dic314yes—
440Dic5—24no—
441Dic5—22no—
442Dic5—62no—
443Dic5—62no—
444Dic5—122no—
445C24—51no—
446C2×Q8—54no—
447C2×Q8—102no—
448C2×Q8—102no—
449C22×C4—31no—
450C22×C4—31no—
451C22×C4—32no—
452C22×C4—51no—
453C22×C4—52no—
454C22×C4—54no—
455C22×C4—102no—
456C22×C4—102no—
457C22×C4—154no—
458C22×C4—151no—
459C22×C4—151no—
460C22×C4—152no—
461C22×C4—302no—
462C22×C4—302no—
463C22×C4—302no—
464C22×C4—302no—
465C22×C4—302no—
466C22×C4C2×D3×D512yes—
467C22×C4C2×D3×D511yescharacteristic
468C4⋊C4—58no—
469C4⋊C4—62no—
470C4⋊C4—304no—
471C4⋊C4—304no—
472C4⋊C4—302no—
473C4⋊C4D5×Dic314yescomplemented
474C22⋊C4—58no—
475C22⋊C4—304no—
476C42—34no—
477C42—64no—
478C42—302no—
479C42—302no—
480C42—304no—
481C42—302no—
482C3×C5C4×C22⋊Q811yescharacteristic, complemented, Hall
483C22×C3—54no—
484C22×C3—102no—
485C22×C3—102no—
486C22×C3—102no—
487C22×C3—102no—
488C22×C3—102no—
489C22×C3—102no—
490C22×C3C22×C4×D511yescharacteristic
491C22×C3C22×D2011yescharacteristic
492C22×C3C2×#70911yescharacteristic
493C22×C3C2×#71111yescharacteristic
494C22×C3C2×Q8×D511yescharacteristic
495C22×C3D5×#4111yescharacteristic
496C22×C3D5×#4111yescharacteristic
497C3×C4—24no—
498C3×C4—102no—
499C3×C4—104no—
500C3×C4C4×D2014yes—
501Dic3—22no—
502Dic3—42no—
503Dic3—102no—
504Dic3—102no—
505Dic3—202no—
506Dic3#404314yesdirect product factor
507C2×C5#562211yescharacteristic
508C2×C5#562411yescharacteristic
509C2×C5D3×C22⋊Q811yescharacteristic
510C2×C5#562711yescharacteristic
511C2×C5C2×D4×Dic311yescharacteristic
512C2×C5C2×Q8×Dic311yescharacteristic
513C2×C5Dic3×#4111yescharacteristic
514D5—24no—
515C23—52no—
516C23—52no—
517C23—52no—
518C23—101no—
519C23—101no—
520C23—101no—
521C23—101no—
522C23C22×D3×D511yescharacteristic
523Q8—108no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—32no—
527C2×C4—32no—
528C2×C4—32no—
529C2×C4—32no—
530C2×C4—54no—
531C2×C4—52no—
532C2×C4—61no—
533C2×C4—61no—
534C2×C4—62no—
535C2×C4—62no—
536C2×C4—62no—
537C2×C4—101no—
538C2×C4—102no—
539C2×C4—101no—
540C2×C4—102no—
541C2×C4—108no—
542C2×C4—152no—
543C2×C4—152no—
544C2×C4—308no—
545C2×C4—308no—
546C2×C4—301no—
547C2×C4—301no—
548C2×C4—301no—
549C2×C4—301no—
550C2×C4—302no—
551C2×C4—302no—
552C2×C4—302no—
553C2×C4#676512yes—
554C2×C4D3×D2012yes—
555C2×C4C2×D5×Dic314yes—
556C2×C4C2×D5×Dic312yes—
557C2×C3—104no—
558C2×C3C2×C4×D2011yescharacteristic
559C2×C3C2×#404311yescharacteristic
560C2×C3#6585811yescharacteristic
561C2×C3C4×#71111yescharacteristic
562C2×C3#6589111yescharacteristic
563C2×C3C4×Q8×D511yescharacteristic
564C2×C3#6591211yescharacteristic
565C5Dic3×C22⋊Q811yescharacteristic, complemented, Sylow
566C22—54no—
567C22—102no—
568C22—102no—
569C22—102no—
570C22—102no—
571C22—102no—
572C22—102no—
573C22C2×#676511yescharacteristic
574C22C2×D3×D2011yescharacteristic
575C22D3×#71111yescharacteristic
576C22#9202411yescharacteristic
577C22D3×Q8×D511yescharacteristic
578C22D5×#23911yescharacteristic
579C22C22×D5×Dic311yescharacteristic
580C4—24no—
581C4—34no—
582C4—62no—
583C4—102no—
584C4—104no—
585C4—122no—
586C4—302no—
587C4—302no—
588C4—602no—
589C4Dic3×D2014yes—
590C3C4×#404311yescharacteristic, complemented, Sylow
591C2—104no—
592C2#1175344311yescharacteristic
593C2#1175352811yescharacteristic
594C2D3×#404311yescharacteristic
595C2#1175388011yescharacteristic
596C2C2×Dic3×D2011yescharacteristic
597C2Dic3×#71111yescharacteristic
598C2Q8×D5×Dic311yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C5×C4⋊C4C22
socle, cosocleC23×C3×C5C24
prime radicalC23×C3×D5C23
hypercenterC2×C4⋊C4D3×D5
hypocenterC3×C5C4×C22⋊Q8
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×Dic3×Dic10C2semidirect
4. #11751085C2semidirect
5. C5×Dic3×C4⋊C4C2semidirect
9. #11748697C2semidirect
10. Dic3×C4⋊Dic5C2semidirect
11. #11748610C2semidirect
26. C3×#4043C4semidirect
67. C2×C3×Dic10C2×C4semidirect
73. C2×D5×Dic3Q8semidirect
74. C3×C5×C4⋊C4C2×C4semidirect
82. C3×C4⋊Dic5C2×C4semidirect
96. Dic3×C4⋊C4D5semidirect
117. #4043Dic3direct (N × H)
216. C2×Dic10C2×Dic3semidirect
218. C5×C4⋊C4C2×Dic3semidirect
225. C4⋊Dic5C2×Dic3semidirect
255. C2×C3×D5C4×Q8semidirect
265. C5×Dic3C22⋊Q8semidirect
290. C3×C4⋊C4C4×D5semidirect
434. C2×D5Q8×Dic3semidirect
473. C4⋊C4D5×Dic3semidirect
482. C3×C5C4×C22⋊Q8semidirect
506. Dic3#4043direct (N × H)
565. C5Dic3×C22⋊Q8semidirect
590. C3C4×#4043semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#40432
C2×C4×D5×Dic32
C2×Dic3×Dic102
#117510852
C5×Dic3×C4⋊C42
#117489002
Dic3×#6852
#117487592
#117486972
Dic3×C4⋊Dic52
#117486102
C4×#40433
Dic3×C22⋊Q85