Finite Group Databasebeta

← Dic3×#4008 #14295619

Subgroups — Dic3×#4008 #14295619

Order 1920.

720 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Dic3×#711C211yesmaximal, characteristic
2C2×#91237C211yesmaximal, characteristic
3C2×C3×#4008C211yesmaximal, characteristic
4C4×Dic3×AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
5Dic3×#700C211yesmaximal, characteristic, complemented
6#11746526C211yesmaximal, characteristic, complemented
7#11746524C211yesmaximal, characteristic, complemented
8C4×#4008—31nomaximal, Hall
9C2×C3×#711C2211yescharacteristic
10C2×C4×ΓL(1,𝔽16)C2211yescharacteristic
11C2×#6758C2211yescharacteristic
12C2×C3×C4×AGL(1,𝔽5)C2211yescharacteristic
13C2×C3×#700C2211yescharacteristic
14C2×Dic3×AGL(1,𝔽5)—21no—
15C2×Dic3×AGL(1,𝔽5)—21no—
16D4×Dic15C411yescharacteristic, complemented
17C5×D4×Dic3C411yescharacteristic, complemented
18#91559—21no—
19Dic3×C22⋊D5—21no—
20#91528—21no—
21C2×Dic3×Dic5—21no—
22C4×D5×Dic3C2211yescharacteristic
23#91358C2211yescharacteristic
24#91331C411yescharacteristic, complemented
25Dic3×Dic10C411yescharacteristic, complemented
26#91237C414yescomplemented
27C3×#4008C414yescomplemented
28#91169—21no—
29Dic3×#170—21no—
30Dic3×#104—51nomaximal, Hall
31C4×#711—31no—
32C2×#4008—31no—
33C2×#4008D311yescharacteristic
34C42×AGL(1,𝔽5)—31no—
35C4×#700—31no—
36#64875—31no—
37#64875—31no—
38C22×ΓL(1,𝔽16)—21no—
39C22×C3×AGL(1,𝔽5)—21no—
40C2×C3×C5×D4C2×C411yescharacteristic
41C22×Dic15—21no—
42C22×C5×Dic3—21no—
43C2×C3×C22⋊D5—21no—
44C22×C3×Dic5—21no—
45C3×#711—21no—
46C3×#711—21no—
47C3×#711C2×C412yes—
48C3×#711C2×C412yes—
49C2×C3×C4×D5C2311yescharacteristic
50C2×C3×Dic10C2×C411yescharacteristic
51C2×D5×Dic3D411yescharacteristic
52C2×D5×Dic3D411yescharacteristic
53C2×C3⋊#170—21no—
54C4×ΓL(1,𝔽16)—21no—
55C4×ΓL(1,𝔽16)C2×C412yescomplemented
56#6758—21no—
57#6758C2×C412yescomplemented
58C2×C3×#170—21no—
59C3×C4×AGL(1,𝔽5)—21no—
60C3×C4×AGL(1,𝔽5)C2×C412yescomplemented
61C3×#700—21no—
62C3×#700C2×C412yescomplemented
63Dic3×AGL(1,𝔽5)—24no—
64Dic3×AGL(1,𝔽5)—24no—
65#6719—21no—
66C4⋊Dic15C2×C411yescharacteristic
67C4×Dic15C2×C411yescharacteristic
68C5×#217—21no—
69C5×C4⋊Dic3C2×C411yescharacteristic
70C4×C5×Dic3C2×C411yescharacteristic
71#6668—21no—
72#6668—21no—
73#6665—21no—
74Dic3×Dic5—21no—
75Dic3×Dic5—21no—
76Dic3×Dic5—21no—
77Dic3×Dic5D411yescharacteristic
78Dic3×Dic5D411yescharacteristic
79Dic3×#41—51no—
80C2×C3×#104—51no—
81C2×#837—51no—
82Dic3×M16—51no—
83C42×Dic3—51no—
84#4555—51no—
85#4523—51no—
86C2×#711C2×D311yescharacteristic
87C2×C4×AGL(1,𝔽5)—31no—
88C2×C4×AGL(1,𝔽5)—61no—
89C2×C4×AGL(1,𝔽5)—61no—
90C2×C4×AGL(1,𝔽5)C2×D311yescharacteristic
91C2×#700—31no—
92C2×#700C2×D311yescharacteristic
93C4×C5×D4—31no—
94D4×Dic5—31no—
95C4×C22⋊D5—61no—
96C2×C4×Dic5—61no—
97#4038—31no—
98#4033—31no—
99#4027—61no—
100#4023—61no—
101C42×D5—31no—
102C4×Dic10—31no—
103#4008—34no—
104#4008Dic314yesdirect product factor
105#4002—61no—
106C4×#170—61no—
107C4×#104—151noSylow
108C23×C3×C5—21no—
109C22×C3×D5C2×D411yescharacteristic
110C2×ΓL(1,𝔽16)—22no—
111C2×ΓL(1,𝔽16)—22no—
112C2×ΓL(1,𝔽16)—22no—
113C2×C3×AGL(1,𝔽5)—22no—
114C2×C3×AGL(1,𝔽5)—22no—
115C2×C3×AGL(1,𝔽5)—22no—
116C3×C5×D4—21no—
117C3×C5×D4C4212yescomplemented
118C2×C3×C4×C5C22×C411yescharacteristic
119C2×Dic15—21no—
120C2×Dic15—42no—
121C2×Dic15C22⋊C411yescharacteristic
122C2×Dic15C22⋊C411yescharacteristic
123C2×C5×Dic3—21no—
124C2×C5×Dic3—42no—
125C2×C5×Dic3C22⋊C411yescharacteristic
126C2×C5×Dic3C22⋊C411yescharacteristic
127C3×C22⋊D5—22no—
128C3×C22⋊D5—22no—
129C2×C3×Dic5—21no—
130C2×C3×Dic5—22no—
131C2×C3×Dic5—22no—
132C2×C3×Dic5C2×D411yescharacteristic
133C3×C4×D5C22×C411yescharacteristic
134C3×C4×D5C22×C411yescharacteristic
135C3×C4×D5#4111yescharacteristic
136C3×C4×D5#4111yescharacteristic
137C3×Dic10—21no—
138C3×Dic10C4212yescomplemented
139D5×Dic3—22no—
140D5×Dic3—22no—
141C3⋊#170—22no—
142C3×#170—22no—
143C2×C3×#41—51no—
144C2×C3×M16—51no—
145C2×C3×C42—51no—
146Q8×Dic3—51no—
147Q8×Dic3—51no—
148D4×Dic3—51no—
149D4×Dic3—101no—
150D4×Dic3—101no—
151D4×Dic3AGL(1,𝔽5)11yescharacteristic, complemented
152#926—51no—
153#926—101no—
154C2×C4×Dic3—51no—
155C2×C4×Dic3—51no—
156C2×C4×Dic3—101no—
157C2×C4×Dic3—101no—
158C2×C4×Dic3—101no—
159C2×#208—51no—
160C3×#104—54no—
161#837—54no—
162#815—101no—
163C8×Dic3—101no—
164C22×AGL(1,𝔽5)—21no—
165C22×AGL(1,𝔽5)—61no—
166C2×C5×D4C4×D311yescharacteristic
167C22×C4×C5—61no—
168C2×C22⋊D5—21no—
169C22×Dic5—21no—
170C22×Dic5—61no—
171#711—21no—
172#711—21no—
173#711C4×D312yes—
174#711C2×Dic312yes—
175C2×C4×D5—31no—
176C2×C4×D5—31no—
177C2×C4×D5C22×D311yescharacteristic
178C2×Dic10C4×D311yescharacteristic
179C2×#170—21no—
180C2×#170—61no—
181C4×AGL(1,𝔽5)—21no—
182C4×AGL(1,𝔽5)—32no—
183C4×AGL(1,𝔽5)—64no—
184C4×AGL(1,𝔽5)—64no—
185C4×AGL(1,𝔽5)—61no—
186C4×AGL(1,𝔽5)C2×Dic312yescomplemented
187#700—21no—
188#700—32no—
189#700—61no—
190#700C2×Dic312yescomplemented
191C5×C4⋊C4—31no—
192C5×C22⋊C4—61no—
193C42×C5—31no—
194C22⋊Dic5—61no—
195#685—61no—
196C4⋊Dic5—31no—
197#683—61no—
198#683—61no—
199C4×Dic5—31no—
200C4×Dic5—31no—
201C4×Dic5—31no—
202C4×Dic5—61no—
203C4×Dic5—61no—
204C4×Dic5—61no—
205C4×#41—151no—
206C2×#104—51no—
207C2×#104—151no—
208C4×M16—151no—
209C43—151no—
210#339—151no—
211#339—151no—
212C22×C3×C5—22no—
213C22×C3×C5—41no—
214C22×C3×C5—41no—
215C22×C3×C5C2×C22⋊C411yescharacteristic
216C2×C3×D5—21no—
217C2×C3×D5—21no—
218C2×C3×D5C4×D411yescharacteristic
219C2×C3×D5C4×D411yescharacteristic
220ΓL(1,𝔽16)—24no—
221C3×AGL(1,𝔽5)—24no—
222C3×C4×C5C2×C4211yescharacteristic
223C3×C4×C5#11711yescharacteristic
224Dic15—21no—
225Dic15—41no—
226Dic15#10412yescomplemented
227C3×Dic5—22no—
228C3×Dic5C4×D411yescharacteristic
229C3×Dic5C4×D411yescharacteristic
230C5×Dic3—21no—
231C5×Dic3—41no—
232C5×Dic3#10412yescomplemented
233C3×#41—52no—
234C3×#41—52no—
235C3×#41—101no—
236C3×#41—101no—
237C2×C3×Q8—51no—
238C2×C3×D4—101no—
239C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
240C22×C3×C4—51no—
241C22×C3×C4—101no—
242C22×C3×C4—101no—
243C22×Dic3—21no—
244C22×Dic3—51no—
245C22×Dic3—51no—
246C22×Dic3—101no—
247C22×Dic3—101no—
248C3×M16—52no—
249C3×M16—101no—
250C2×C3×C8—101no—
251C3×C42—52no—
252C3×C42—101no—
253#217—21no—
254#217—101no—
255#217—101no—
256C4⋊Dic3—51no—
257C4⋊Dic3—101no—
258C4⋊Dic3—101no—
259C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
260C4×Dic3—51no—
261C4×Dic3—51no—
262C4×Dic3—51no—
263C4×Dic3—52no—
264C4×Dic3—101no—
265C4×Dic3—101no—
266C4×Dic3—101no—
267C4×Dic3—104no—
268C4×Dic3—104no—
269C4×Dic3—101no—
270C4×Dic3C2×AGL(1,𝔽5)11yescharacteristic
271#208—52no—
272#208—101no—
273C2×C3⋊C8—101no—
274C23×C5—21no—
275C22×D5D3×D411yescharacteristic
276C2×AGL(1,𝔽5)—22no—
277C2×AGL(1,𝔽5)—22no—
278C2×AGL(1,𝔽5)—22no—
279C2×AGL(1,𝔽5)—62no—
280C2×AGL(1,𝔽5)—62no—
281C2×AGL(1,𝔽5)—62no—
282C5×D4—21no—
283C5×D4C4×Dic312yescomplemented
284C2×C4×C5—31no—
285C2×C4×C5—31no—
286C2×C4×C5—61no—
287C2×C4×C5—122no—
288C2×C4×C5C2×C4×D311yescharacteristic
289C22⋊D5—22no—
290C22⋊D5—22no—
291C2×Dic5—21no—
292C2×Dic5—22no—
293C2×Dic5—22no—
294C2×Dic5—31no—
295C2×Dic5—31no—
296C2×Dic5—61no—
297C2×Dic5—122no—
298C2×Dic5D3×D411yescharacteristic
299C4×D5—62no—
300C4×D5—62no—
301C4×D5C2×C4×D311yescharacteristic
302C4×D5#23711yescharacteristic
303C4×D5#23711yescharacteristic
304C4×D5C22×Dic311yescharacteristic
305Dic10—21no—
306Dic10C4×Dic312yescomplemented
307#170—22no—
308#170—62no—
309C2×#41—51no—
310C2×M16—51no—
311C2×M16—151no—
312C4×Q8—151no—
313C4×Q8—151no—
314C4×D4—31no—
315C4×D4—151no—
316C4×D4—301no—
317C4×D4—301no—
318#117—151no—
319#117—301no—
320C2×C42—51no—
321C2×C42—151no—
322C2×C42—151no—
323C2×C42—301no—
324C2×C42—301no—
325C2×C42—301no—
326#104—54no—
327#104—154no—
328#97—301no—
329C4×C8—301no—
330C2×C3×C5—42no—
331C2×C3×C5C4×C22⋊C411yescharacteristic
332C2×C3×C5C2×#10411yescharacteristic
333C2×C3×C5C2×#10411yescharacteristic
334C3×D5—21no—
335C3×D5—21no—
336C23×C3—21no—
337C23×C3—51no—
338C3×Q8—52no—
339C3×Q8—101no—
340C3×D4—21no—
341C3×D4—102no—
342C3×D4—102no—
343C3×D4C4×AGL(1,𝔽5)12yescomplemented
344C2×C3×C4—51no—
345C2×C3×C4—51no—
346C2×C3×C4—51no—
347C2×C3×C4—51no—
348C2×C3×C4—51no—
349C2×C3×C4—102no—
350C2×C3×C4—102no—
351C2×C3×C4—102no—
352C2×C3×C4—102no—
353C2×C3×C4—102no—
354C2×C3×C4—101no—
355C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
356C2×Dic3—21no—
357C2×Dic3—42no—
358C2×Dic3—51no—
359C2×Dic3—51no—
360C2×Dic3—102no—
361C2×Dic3—102no—
362C2×Dic3—102no—
363C2×Dic3—102no—
364C2×Dic3—102no—
365C2×Dic3—101no—
366C2×Dic3—202no—
367C2×Dic3C22⋊AGL(1,𝔽5)11yescharacteristic
368C2×Dic3C22⋊AGL(1,𝔽5)11yescharacteristic
369C3×C8—102no—
370C3⋊C8—102no—
371C22×C5—22no—
372C22×C5—41no—
373C22×C5—41no—
374C22×C5D3×C22⋊C411yescharacteristic
375C2×D5—21no—
376C2×D5—21no—
377C2×D5#67⋊C411yescharacteristic
378C2×D5D4×Dic311yescharacteristic
379AGL(1,𝔽5)—24no—
380AGL(1,𝔽5)—64no—
381C4×C5—32no—
382C4×C5—61no—
383C4×C5—121no—
384C4×C5#87711yescharacteristic
385C4×C5C2×C4×Dic311yescharacteristic
386Dic5—22no—
387Dic5—32no—
388Dic5—61no—
389Dic5—121no—
390Dic5#67⋊C411yescharacteristic
391Dic5D4×Dic311yescharacteristic
392#41—52no—
393#41—52no—
394#41—101no—
395#41—101no—
396C2×Q8—51no—
397C2×D4—101no—
398C2×D4D3×AGL(1,𝔽5)11yescharacteristic
399C22×C4—51no—
400C22×C4—61no—
401C22×C4—101no—
402C22×C4—101no—
403C22×C4—151no—
404C22×C4—151no—
405C22×C4—301no—
406C22×C4—301no—
407M16—52no—
408M16—101no—
409M16—152no—
410M16—301no—
411C2×C8—101no—
412C2×C8—301no—
413C4⋊C4—31no—
414C4⋊C4—151no—
415C4⋊C4—301no—
416C4⋊C4—301no—
417C22⋊C4—61no—
418C22⋊C4—301no—
419C22⋊C4—301no—
420C42—31no—
421C42—52no—
422C42—101no—
423C42—151no—
424C42—151no—
425C42—152no—
426C42—151no—
427C42—301no—
428C42—301no—
429C42—301no—
430C42—301no—
431C42—304no—
432C42—304no—
433C3×C5C4×#10411yescharacteristic, complemented, Hall
434C22×C3—22no—
435C22×C3—41no—
436C22×C3—41no—
437C22×C3—51no—
438C22×C3—51no—
439C22×C3—101no—
440C22×C3—101no—
441C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
442C3×C4—51no—
443C3×C4—51no—
444C3×C4—104no—
445C3×C4—102no—
446C3×C4C2×C4×AGL(1,𝔽5)11yescharacteristic
447C3×C4#413011yescharacteristic
448Dic3—21no—
449Dic3—41no—
450Dic3—52no—
451Dic3—104no—
452Dic3—101no—
453Dic3—201no—
454Dic3#400812yesdirect product factor
455C2×C5—42no—
456C2×C5D3×#10411yescharacteristic
457C2×C5D3×#10411yescharacteristic
458C2×C5Dic3×C22⋊C411yescharacteristic
459D5—21no—
460D5—21no—
461C23—21no—
462C23—51no—
463Q8—52no—
464Q8—101no—
465D4—21no—
466D4—102no—
467D4—102no—
468D4Dic3×AGL(1,𝔽5)12yescomplemented
469C2×C4—31no—
470C2×C4—31no—
471C2×C4—51no—
472C2×C4—51no—
473C2×C4—51no—
474C2×C4—51no—
475C2×C4—51no—
476C2×C4—61no—
477C2×C4—102no—
478C2×C4—102no—
479C2×C4—102no—
480C2×C4—102no—
481C2×C4—102no—
482C2×C4—101no—
483C2×C4—122no—
484C2×C4—151no—
485C2×C4—151no—
486C2×C4—302no—
487C2×C4—302no—
488C2×C4—302no—
489C2×C4—302no—
490C2×C4—302no—
491C2×C4—301no—
492C2×C4—602no—
493C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
494C8—102no—
495C8—302no—
496C2×C3—42no—
497C2×C3—101no—
498C2×C3—101no—
499C2×C3C4×C22⋊AGL(1,𝔽5)11yescharacteristic
500C2×C3C2×#400811yescharacteristic
501C2×C3D5⋊#10411yescharacteristic
502C5Dic3×#10411yescharacteristic, complemented, Sylow
503C22—22no—
504C22—41no—
505C22—41no—
506C22—51no—
507C22—51no—
508C22—101no—
509C22—101no—
510C22D3×C22⋊AGL(1,𝔽5)11yescharacteristic
511C4—32no—
512C4—51no—
513C4—51no—
514C4—61no—
515C4—104no—
516C4—102no—
517C4—121no—
518C4—152no—
519C4—304no—
520C4—301no—
521C4—601no—
522C4C2×Dic3×AGL(1,𝔽5)11yescharacteristic
523C4#9192311yescharacteristic
524C3C4×#400811yescharacteristic, complemented, Sylow
525C2—42no—
526C2—101no—
527C2—101no—
528C2#1175193511yescharacteristic
529C2D3×#400811yescharacteristic
530C2Dic3×C22⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
centerC22D3×C22⋊AGL(1,𝔽5)
commutatorC3×C4×C5C2×C42
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC2×C3×C5×D4C2×C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC2×C3×#711C22
cosocleC2×C3×C4×D5C23
hypercenterC2×D4D3×AGL(1,𝔽5)
hypocenterC3×C5C4×#104
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
4. C4×Dic3×AGL(1,𝔽5)C2semidirect
5. Dic3×#700C2semidirect
6. #11746526C2semidirect
7. #11746524C2semidirect
16. D4×Dic15C4semidirect
17. C5×D4×Dic3C4semidirect
24. #91331C4semidirect
25. Dic3×Dic10C4semidirect
26. #91237C4semidirect
27. C3×#4008C4semidirect
55. C4×ΓL(1,𝔽16)C2×C4semidirect
57. #6758C2×C4semidirect
60. C3×C4×AGL(1,𝔽5)C2×C4semidirect
62. C3×#700C2×C4semidirect
104. #4008Dic3direct (N × H)
117. C3×C5×D4C42semidirect
138. C3×Dic10C42semidirect
151. D4×Dic3AGL(1,𝔽5)semidirect
186. C4×AGL(1,𝔽5)C2×Dic3semidirect
190. #700C2×Dic3semidirect
226. Dic15#104semidirect
232. C5×Dic3#104semidirect
283. C5×D4C4×Dic3semidirect
306. Dic10C4×Dic3semidirect
343. C3×D4C4×AGL(1,𝔽5)semidirect
433. C3×C5C4×#104semidirect
454. Dic3#4008direct (N × H)
468. D4Dic3×AGL(1,𝔽5)semidirect
502. C5Dic3×#104semidirect
524. C3C4×#4008semidirect

Maximal subgroups

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

subgroupindex
Dic3×#7112
C2×#912372
C2×C3×#40082
C4×Dic3×AGL(1,𝔽5)2
Dic3×#7002
#117465262
#117465242
C4×#40083
Dic3×#1045