Finite Group Databasebeta

← Dic3×#4070 #14321820

Subgroups — Dic3×#4070 #14321820

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
1C22×C4×C5×Dic3C211yesmaximal, characteristic
2C2×C3×#4070C211yesmaximal, characteristic
3C2×Dic3×Dic10C211yesmaximal, characteristic, complemented
4#11751313C211yesmaximal, characteristic
5#11749511C212yesmaximal
6Dic3×C22⋊Dic5C212yesmaximal
7#11748857C211yesmaximal, characteristic, complemented
8Dic3×C4⋊Dic5C211yesmaximal, characteristic, complemented
9Dic3×#683C212yesmaximal, complemented
10#11748609C211yesmaximal, characteristic, complemented
11#11748590C212yesmaximal, complemented
12C4×#4070—31nomaximal, Hall
13C23×C3×C4×C5C2211yescharacteristic
14C23×C5×Dic3C2211yescharacteristic
15C23×C5×Dic3C2211yescharacteristic
16C22×C3×Dic10C2211yescharacteristic
17C2×#6719C2212yes—
18C2×C4⋊Dic15C2211yescharacteristic
19C2×C4⋊Dic15C2211yescharacteristic
20C2×C4×C5×Dic3—24no—
21C2×C4×C5×Dic3C2211yescharacteristic
22C2×C4×C5×Dic3C2211yescharacteristic
23C2×C4×C5×Dic3C2211yescharacteristic
24C2×C4×C5×Dic3C2211yescharacteristic
25C2×C3×C22⋊Dic5C2212yes—
26C2×C3×C4⋊Dic5C2211yescharacteristic
27C3×#4070C418yescomplemented
28C2×C3×#683C2212yes—
29C2×#6668C2212yes—
30C2×#6668C2212yes—
31C2×Dic3×Dic5C2212yes—
32C2×Dic3×Dic5C2212yes—
33Dic3×Dic10—24no—
34#91115C2212yes—
35#90994C2212yes—
36#90994C2212yes—
37#90994C2212yes—
38#90994C2212yes—
39Dic3×C22⋊Q8—51nomaximal, Hall
40C22×C42×C5—31no—
41C2×#4070D311yescharacteristic
42C2×C4×Dic10—31no—
43#65501—31no—
44#65499—32no—
45C4×C22⋊Dic5—32no—
46C4×C4⋊Dic5—31no—
47#65222—32no—
48C4×#683—32no—
49#65218—31no—
50#65217—31no—
51C24×C3×C5C2311yescharacteristic
52C22×C3×C4×C5—22no—
53C22×C3×C4×C5—22no—
54C22×C3×C4×C5C2×C414yes—
55C22×C3×C4×C5C2311yescharacteristic
56C22×C3×C4×C5C2311yescharacteristic
57C22×Dic15C2312yes—
58C22×Dic15C2312yes—
59C22×C5×Dic3—22no—
60C22×C5×Dic3—22no—
61C22×C5×Dic3—22no—
62C22×C5×Dic3—22no—
63C22×C5×Dic3—22no—
64C22×C5×Dic3Q814yescomplemented
65C22×C5×Dic3C2311yescharacteristic
66C22×C5×Dic3C2311yescharacteristic
67C22×C5×Dic3C2311yescharacteristic
68C22×C5×Dic3C2311yescharacteristic
69C22×C3×Dic5C2312yes—
70C22×C3×Dic5C2312yes—
71C2×C3×Dic10—22no—
72C2×C3×Dic10—22no—
73C2×C3×Dic10C2×C414yescomplemented
74#6719—24no—
75C4⋊Dic15—22no—
76C4⋊Dic15—22no—
77C4×C5×Dic3—22no—
78C4×C5×Dic3—22no—
79C4×C5×Dic3—22no—
80C4×C5×Dic3D414yes—
81C3×C22⋊Dic5C2×C418yes—
82C3×C4⋊Dic5C2×C414yescomplemented
83C3×#683C2×C418yescomplemented
84#6668—24no—
85#6668—24no—
86Dic3×Dic5—24no—
87Dic3×Dic5—24no—
88C2×C3×C22⋊Q8—51no—
89C2×Q8×Dic3—51no—
90C22×C4×Dic3D511yescharacteristic
91#5229—51no—
92#5212—51no—
93#4984—51no—
94#4980—52no—
95Dic3×C4⋊C4—52no—
96Dic3×C4⋊C4—51no—
97#4950—52no—
98Dic3×C22⋊C4—52no—
99C23×C4×C5—31no—
100C23×C4×C5—31no—
101C23×C4×C5C2×D311yescharacteristic
102C22×Dic10C2×D311yescharacteristic
103C2×C42×C5—31no—
104C2×C42×C5—31no—
105C2×C42×C5—31no—
106C2×C42×C5—31no—
107C2×C42×C5—64no—
108C2×C22⋊Dic5—32no—
109C2×C22⋊Dic5C2×D312yes—
110C2×C4⋊Dic5—31no—
111C2×C4⋊Dic5—31no—
112C2×C4⋊Dic5C2×D311yescharacteristic
113#4070Dic318yesdirect product factor
114C2×#683—32no—
115C2×#683—32no—
116C2×#683C2×D312yes—
117C2×C4×Dic5—32no—
118C2×C4×Dic5—32no—
119C4×Dic10—64no—
120#3963—32no—
121#3963—32no—
122#3963—32no—
123#3963—32no—
124#3963—32no—
125C4×C22⋊Q8—151noSylow
126C23×C3×C5—21no—
127C23×C3×C5—21no—
128C23×C3×C5—21no—
129C23×C3×C5—21no—
130C23×C3×C5C22×C412yes—
131C23×C3×C5C2×Q812yes—
132C23×C3×C5#4112yes—
133C23×C3×C5C2411yescharacteristic
134C2×C3×C4×C5—21no—
135C2×C3×C4×C5—21no—
136C2×C3×C4×C5—28no—
137C2×C3×C4×C5C22×C412yes—
138C2×C3×C4×C5C22×C412yes—
139C2×C3×C4×C5C2×D412yes—
140C2×C3×C4×C5#4112yes—
141C2×Dic15—22no—
142C2×Dic15—22no—
143C2×Dic15—22no—
144C2×Dic15—22no—
145C2×Dic15—22no—
146C2×Dic15—22no—
147C2×C5×Dic3—28no—
148C2×C5×Dic3—21no—
149C2×C5×Dic3—21no—
150C2×C5×Dic3—21no—
151C2×C5×Dic3—21no—
152C2×C5×Dic3—21no—
153C2×C5×Dic3—21no—
154C2×C5×Dic3—28no—
155C2×C5×Dic3C2×D412yes—
156C2×C5×Dic3C2×D412yes—
157C2×C5×Dic3C2×Q812yes—
158C2×C5×Dic3C2×Q812yes—
159C2×C5×Dic3#4112yes—
160C2×C5×Dic3#4112yes—
161C2×C3×Dic5—22no—
162C2×C3×Dic5—22no—
163C2×C3×Dic5—22no—
164C2×C3×Dic5—22no—
165C2×C3×Dic5C22×C414yes—
166C2×C3×Dic5C22×C414yes—
167C3×Dic10—28no—
168C22×C3×Q8—51no—
169C23×C3×C4C2×D511yescharacteristic
170C23×Dic3C2×D511yescharacteristic
171C23×Dic3C2×D511yescharacteristic
172C3×C22⋊Q8—58no—
173C2×C3×C4⋊C4—52no—
174C2×C3×C4⋊C4—51no—
175C2×C3×C22⋊C4—52no—
176C2×#217—52no—
177Q8×Dic3—104no—
178C2×C4⋊Dic3—51no—
179C2×C4⋊Dic3—51no—
180C2×C4⋊Dic3—52no—
181C2×C4⋊Dic3—52no—
182C2×C4×Dic3—24no—
183C2×C4×Dic3—52no—
184C2×C4×Dic3—52no—
185C2×C4×Dic3C2×D511yescharacteristic
186C2×C4×Dic3C2×D511yescharacteristic
187C2×C4×Dic3C2×D511yescharacteristic
188C2×C4×Dic3C2×D511yescharacteristic
189C3⋊#95—52no—
190C3⋊#95—52no—
191C3⋊#95—52no—
192C3⋊#95—52no—
193C3⋊#95—52no—
194C24×C5C22×D311yescharacteristic
195C22×C4×C5—22no—
196C22×C4×C5—22no—
197C22×C4×C5—31no—
198C22×C4×C5—34no—
199C22×C4×C5—31no—
200C22×C4×C5—31no—
201C22×C4×C5—31no—
202C22×C4×C5—62no—
203C22×C4×C5—62no—
204C22×C4×C5—62no—
205C22×C4×C5—62no—
206C22×C4×C5—62no—
207C22×C4×C5C2×Dic314yes—
208C22×C4×C5C22×D311yescharacteristic
209C22×C4×C5C22×D311yescharacteristic
210C22×Dic5—32no—
211C22×Dic5—32no—
212C22×Dic5C22×D312yes—
213C22×Dic5C22×D312yes—
214C2×Dic10—22no—
215C2×Dic10—22no—
216C2×Dic10C2×Dic314yescomplemented
217C42×C5—34no—
218C42×C5—62no—
219C42×C5—62no—
220C42×C5—62no—
221C22⋊Dic5—64no—
222C22⋊Dic5C2×Dic318yes—
223C4⋊Dic5—62no—
224C4⋊Dic5—62no—
225C4⋊Dic5C2×Dic314yescomplemented
226#683—64no—
227#683—64no—
228#683C2×Dic318yescomplemented
229C4×Dic5—64no—
230C4×Dic5—64no—
231C2×C22⋊Q8—51no—
232C2×C4×Q8—151no—
233C22×C42—31no—
234#391—151no—
235#389—151no—
236#387—152no—
237#385—152no—
238#380—151no—
239C4×C4⋊C4—152no—
240C4×C4⋊C4—151no—
241C4×C22⋊C4—152no—
242C22×C3×C5—22no—
243C22×C3×C5—22no—
244C22×C3×C5—22no—
245C22×C3×C5—22no—
246C22×C3×C5—22no—
247C22×C3×C5—22no—
248C22×C3×C5C4×Q814yescomplemented
249C22×C3×C5C23×C411yescharacteristic
250C22×C3×C5C22×D411yescharacteristic
251C22×C3×C5C22×Q811yescharacteristic
252C22×C3×C5C2×#4111yescharacteristic
253C22×C3×C5C2×#4111yescharacteristic
254C22×C3×C5C2×#4111yescharacteristic
255C22×C3×C5C2×#4111yescharacteristic
256C3×C4×C5—22no—
257C3×C4×C5C4×D414yes—
258Dic15—42no—
259Dic15—42no—
260C3×Dic5—24no—
261C3×Dic5—24no—
262C5×Dic3—22no—
263C5×Dic3—22no—
264C5×Dic3—22no—
265C5×Dic3C22⋊Q814yescomplemented
266C24×C3C22×D511yescharacteristic
267C2×C3×Q8—54no—
268C2×C3×Q8—102no—
269C2×C3×Q8—102no—
270C22×C3×C4—22no—
271C22×C3×C4—22no—
272C22×C3×C4—52no—
273C22×C3×C4—52no—
274C22×C3×C4C4×D514yes—
275C22×C3×C4C22×D511yescharacteristic
276C22×C3×C4C22×D511yescharacteristic
277C22×Dic3—22no—
278C22×Dic3—22no—
279C22×Dic3—22no—
280C22×Dic3—22no—
281C22×Dic3—22no—
282C22×Dic3—52no—
283C22×Dic3—52no—
284C22×Dic3Dic1014yescomplemented
285C22×Dic3C22×D511yescharacteristic
286C22×Dic3C22×D511yescharacteristic
287C22×Dic3C22×D511yescharacteristic
288C22×Dic3C22×D511yescharacteristic
289C3×C4⋊C4—58no—
290C3×C4⋊C4—54no—
291C3×C22⋊C4—58no—
292#217—104no—
293C4⋊Dic3—102no—
294C4⋊Dic3—102no—
295C4⋊Dic3—104no—
296C4⋊Dic3—104no—
297C4×Dic3—22no—
298C4×Dic3—22no—
299C4×Dic3—22no—
300C4×Dic3—104no—
301C4×Dic3—104no—
302C4×Dic3C22⋊D514yes—
303C23×C5—21no—
304C23×C5—21no—
305C23×C5—21no—
306C23×C5—21no—
307C23×C5D3×Q812yes—
308C23×C5#23912yes—
309C23×C5C22×Dic312yes—
310C23×C5C23×D311yescharacteristic
311C2×C4×C5—21no—
312C2×C4×C5—21no—
313C2×C4×C5—28no—
314C2×C4×C5—32no—
315C2×C4×C5—32no—
316C2×C4×C5—32no—
317C2×C4×C5—32no—
318C2×C4×C5—32no—
319C2×C4×C5—32no—
320C2×C4×C5—68no—
321C2×C4×C5—68no—
322C2×C4×C5—61no—
323C2×C4×C5—61no—
324C2×C4×C5—61no—
325C2×C4×C5—61no—
326C2×C4×C5—61no—
327C2×C4×C5—61no—
328C2×C4×C5D3×D412yes—
329C2×C4×C5#23712yes—
330C2×C4×C5C22×Dic312yes—
331C2×C4×C5C22×Dic312yes—
332C2×Dic5—22no—
333C2×Dic5—22no—
334C2×Dic5—22no—
335C2×Dic5—22no—
336C2×Dic5—62no—
337C2×Dic5—62no—
338C2×Dic5—62no—
339C2×Dic5—62no—
340C2×Dic5—62no—
341C2×Dic5—62no—
342C2×Dic5C22×Dic314yes—
343C2×Dic5C22×Dic314yes—
344Dic10—28no—
345C22×Q8—51no—
346C23×C4—31no—
347C23×C4—31no—
348C23×C4D3×D511yescharacteristic
349C22⋊Q8—58no—
350C4×Q8—304no—
351C2×C4⋊C4—52no—
352C2×C4⋊C4—51no—
353C2×C4⋊C4—152no—
354C2×C4⋊C4—152no—
355C2×C4⋊C4—151no—
356C2×C4⋊C4—151no—
357C2×C22⋊C4—52no—
358C2×C22⋊C4—152no—
359C2×C42—31no—
360C2×C42—31no—
361C2×C42—31no—
362C2×C42—31no—
363C2×C42—64no—
364C2×C42—152no—
365C2×C42—152no—
366#95—152no—
367#95—152no—
368#95—152no—
369#95—152no—
370#95—152no—
371C2×C3×C5—24no—
372C2×C3×C5C2×C4×D411yescharacteristic
373C2×C3×C5C2×C4×Q811yescharacteristic
374C2×C3×C5C4×#4111yescharacteristic
375C2×C3×C5C2×C22⋊Q811yescharacteristic
376C2×C3×C5#52511yescharacteristic
377C2×C3×C5#52911yescharacteristic
378C2×C3×C5#53311yescharacteristic
379C23×C3—21no—
380C23×C3—21no—
381C23×C3—21no—
382C23×C3—21no—
383C23×C3C2×Dic1012yes—
384C23×C3C2×C4×D512yes—
385C23×C3#70912yes—
386C23×C3C23×D511yescharacteristic
387C3×Q8—108no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—28no—
391C2×C3×C4—54no—
392C2×C3×C4—54no—
393C2×C3×C4—102no—
394C2×C3×C4—102no—
395C2×C3×C4—102no—
396C2×C3×C4—102no—
397C2×C3×C4C2×C4×D512yes—
398C2×C3×C4C2×C4×D512yes—
399C2×C3×C4#70912yes—
400C2×C3×C4C2×C22⋊D512yes—
401C2×Dic3—28no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—28no—
409C2×Dic3—102no—
410C2×Dic3—102no—
411C2×Dic3—102no—
412C2×Dic3—102no—
413C2×Dic3—102no—
414C2×Dic3—102no—
415C2×Dic3C2×Dic1012yes—
416C2×Dic3C2×Dic1012yes—
417C2×Dic3#70912yes—
418C2×Dic3#70912yes—
419C2×Dic3C2×C22⋊D512yes—
420C2×Dic3C2×C22⋊D512yes—
421C22×C5—22no—
422C22×C5—22no—
423C22×C5—22no—
424C22×C5—22no—
425C22×C5—22no—
426C22×C5—22no—
427C22×C5Q8×Dic314yescomplemented
428C22×C5C2×D3×D411yescharacteristic
429C22×C5C2×#23711yescharacteristic
430C22×C5C2×D3×Q811yescharacteristic
431C22×C5C2×#23911yescharacteristic
432C22×C5D3×#4111yescharacteristic
433C22×C5D3×#4111yescharacteristic
434C22×C5C23×Dic311yescharacteristic
435C4×C5—22no—
436C4×C5—34no—
437C4×C5—62no—
438C4×C5—62no—
439C4×C5—62no—
440C4×C5D4×Dic314yes—
441Dic5—24no—
442Dic5—24no—
443Dic5—122no—
444Dic5—122no—
445C24C2×D3×D511yescharacteristic
446C2×Q8—54no—
447C2×Q8—102no—
448C2×Q8—102no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—31no—
452C22×C4—34no—
453C22×C4—31no—
454C22×C4—31no—
455C22×C4—31no—
456C22×C4—52no—
457C22×C4—52no—
458C22×C4—62no—
459C22×C4—62no—
460C22×C4—62no—
461C22×C4—62no—
462C22×C4—62no—
463C22×C4—152no—
464C22×C4—152no—
465C22×C4D5×Dic314yes—
466C22×C4C2×D3×D511yescharacteristic
467C22×C4C2×D3×D511yescharacteristic
468C4⋊C4—58no—
469C4⋊C4—54no—
470C4⋊C4—304no—
471C4⋊C4—304no—
472C4⋊C4—302no—
473C4⋊C4—302no—
474C22⋊C4—58no—
475C22⋊C4—304no—
476C42—34no—
477C42—62no—
478C42—62no—
479C42—62no—
480C42—304no—
481C42—304no—
482C3×C5C4×C22⋊Q811yescharacteristic, complemented, Hall
483C22×C3—22no—
484C22×C3—22no—
485C22×C3—22no—
486C22×C3—22no—
487C22×C3—22no—
488C22×C3—22no—
489C22×C3C4×Dic1014yescomplemented
490C22×C3C22×Dic1011yescharacteristic
491C22×C3C22×C4×D511yescharacteristic
492C22×C3C2×#70911yescharacteristic
493C22×C3C2×#70911yescharacteristic
494C22×C3C2×#70911yescharacteristic
495C22×C3C2×#70911yescharacteristic
496C22×C3C22×C22⋊D511yescharacteristic
497C3×C4—22no—
498C3×C4—104no—
499C3×C4—104no—
500C3×C4C4×C22⋊D514yes—
501Dic3—22no—
502Dic3—22no—
503Dic3—22no—
504Dic3—202no—
505Dic3—202no—
506Dic3#407014yesdirect product factor
507C2×C5—24no—
508C2×C5#562211yescharacteristic
509C2×C5#562411yescharacteristic
510C2×C5D3×C22⋊Q811yescharacteristic
511C2×C5#562711yescharacteristic
512C2×C5C2×D4×Dic311yescharacteristic
513C2×C5C2×Q8×Dic311yescharacteristic
514C2×C5Dic3×#4111yescharacteristic
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23D3×Dic1012yes—
520C23#676912yes—
521C23C2×D5×Dic312yes—
522C23C22×D3×D511yescharacteristic
523Q8—108no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—28no—
527C2×C4—32no—
528C2×C4—32no—
529C2×C4—32no—
530C2×C4—32no—
531C2×C4—32no—
532C2×C4—32no—
533C2×C4—54no—
534C2×C4—54no—
535C2×C4—68no—
536C2×C4—68no—
537C2×C4—61no—
538C2×C4—61no—
539C2×C4—61no—
540C2×C4—61no—
541C2×C4—61no—
542C2×C4—61no—
543C2×C4—102no—
544C2×C4—102no—
545C2×C4—102no—
546C2×C4—102no—
547C2×C4—302no—
548C2×C4—302no—
549C2×C4—302no—
550C2×C4—302no—
551C2×C4—302no—
552C2×C4—302no—
553C2×C4C2×D5×Dic312yes—
554C2×C4C2×D5×Dic312yes—
555C2×C4#677912yes—
556C2×C4D3×C22⋊D512yes—
557C2×C3—24no—
558C2×C3C2×C4×Dic1011yescharacteristic
559C2×C3#6580411yescharacteristic
560C2×C3C4×#70911yescharacteristic
561C2×C3#6581311yescharacteristic
562C2×C3C2×#407011yescharacteristic
563C2×C3C2×C4×C22⋊D511yescharacteristic
564C2×C3#6612511yescharacteristic
565C5Dic3×C22⋊Q811yescharacteristic, complemented, Sylow
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22Dic3×Dic1014yescomplemented
573C22C2×D3×Dic1011yescharacteristic
574C22C2×#676911yescharacteristic
575C22D3×#70911yescharacteristic
576C22D3×#70911yescharacteristic
577C22C22×D5×Dic311yescharacteristic
578C22C2×#677911yescharacteristic
579C22C2×D3×C22⋊D511yescharacteristic
580C4—22no—
581C4—34no—
582C4—62no—
583C4—62no—
584C4—62no—
585C4—104no—
586C4—104no—
587C4—602no—
588C4—602no—
589C4Dic3×C22⋊D514yes—
590C3C4×#407011yescharacteristic, complemented, Sylow
591C2—24no—
592C2#1175417611yescharacteristic
593C2C2×Dic3×Dic1011yescharacteristic
594C2D3×#407011yescharacteristic
595C2#1175435211yescharacteristic
596C2Dic3×#70911yescharacteristic
597C2#1175444011yescharacteristic
598C2C2×Dic3×C22⋊D511yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC23×C3×C4×C5C22
socle, cosocleC23×C3×C5C24
prime radicalC24×C3×C5C23
hypercenterC23×C4D3×D5
hypocenterC3×C5C4×C22⋊Q8
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
3. C2×Dic3×Dic10C2semidirect
7. #11748857C2semidirect
8. Dic3×C4⋊Dic5C2semidirect
9. Dic3×#683C2semidirect
10. #11748609C2semidirect
11. #11748590C2semidirect
27. C3×#4070C4semidirect
64. C22×C5×Dic3Q8semidirect
73. C2×C3×Dic10C2×C4semidirect
82. C3×C4⋊Dic5C2×C4semidirect
83. C3×#683C2×C4semidirect
113. #4070Dic3direct (N × H)
216. C2×Dic10C2×Dic3semidirect
225. C4⋊Dic5C2×Dic3semidirect
228. #683C2×Dic3semidirect
248. C22×C3×C5C4×Q8semidirect
265. C5×Dic3C22⋊Q8semidirect
284. C22×Dic3Dic10semidirect
427. C22×C5Q8×Dic3semidirect
482. C3×C5C4×C22⋊Q8semidirect
489. C22×C3C4×Dic10semidirect
506. Dic3#4070direct (N × H)
565. C5Dic3×C22⋊Q8semidirect
572. C22Dic3×Dic10semidirect
590. C3C4×#4070semidirect

Maximal subgroups

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

subgroupindex
C22×C4×C5×Dic32
C2×C3×#40702
C2×Dic3×Dic102
#117513132
#117495112
Dic3×C22⋊Dic52
#117488572
Dic3×C4⋊Dic52
Dic3×#6832
#117486092
#117485902
C4×#40703
Dic3×C22⋊Q85