Finite Group Databasebeta

← Dic6×#713 #14484692

Subgroups — Dic6×#713 #14484692

Order 1920.

2136 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11756329C211yesmaximal, characteristic, complemented
2C5×Q8×Dic6C211yesmaximal, characteristic, complemented
3C3×C4×#713C211yesmaximal, characteristic
4#11755268C212yesmaximal, complemented
5Dic3×#713C212yesmaximal
6#11753589C216yesmaximal, complemented
7#11753568C216yesmaximal, complemented
8#11752648C213yesmaximal, complemented
9Dic6×D20C213yesmaximal, complemented
10C4×D5×Dic6C213yesmaximal, complemented
11#11752506C213yesmaximal, complemented
12Q8×#713—31nomaximal, Hall
13C2×C3×#713C2211yescharacteristic
14C2×D5×Dic6C2213yes—
15C3×C4×C5×Q8C2211yescharacteristic
16Q8×Dic15C2212yes—
17#91779C2216yes—
18#91758C2213yes—
19C4×Dic30C2213yes—
20C5×Q8×Dic3C2212yes—
21C5×#888C2216yes—
22C5×#869C2213yes—
23C4×C5×Dic6C2213yes—
24C3×Q8×Dic5C2211yescharacteristic
25C3×#4039C2213yes—
26C3×#4038C2213yes—
27C3×C4×D20C2213yes—
28C3×C42×D5C2213yes—
29#91469C2216yes—
30C4×#1204C2216yes—
31#91434C2216yes—
32Dic3×D20C2216yes—
33#91422C22112yes—
34#91421C22112yes—
35D5×C4⋊Dic3C2213yes—
36D5×Dic3⋊C4C2216yes—
37C4×D5×Dic3C2216yes—
38#91388C2216yes—
39#91384C2216yes—
40#91360C2216yes—
41#91359C2216yes—
42#91358C2216yes—
43#91333C2213yes—
44Dic5×Dic6C2216yes—
45Dic5×Dic6C2211yescharacteristic
46#91323C2216yes—
47#41×Dic6—51nomaximal, Hall
48C5×Q82—31no—
49#66064—36no—
50#66064—33no—
51#66062—36no—
52#66062—33no—
53Q8×D20—33no—
54C4×#713—32no—
55C4×#713D311yescharacteristic
56C4×Q8×D5—33no—
57#65904—32no—
58#65904—31no—
59C2×C3×C5×Q8C2311yescharacteristic
60C2×Dic30C2313yes—
61C2×C5×Dic6C2316yes—
62C2×C5×Dic6C2311yescharacteristic
63C3×#713Q818yescomplemented
64C2×C3×D20C2313yes—
65C2×C3×C4×D5C2313yes—
66C2×C3×C4×D5C2313yes—
67C2×C3×C4×D5C2313yes—
68C2×#1204C2316yes—
69C2×D5×Dic3C2316yes—
70C2×D5×Dic3C2316yes—
71D5×Dic6—212no—
72C3×C5×C4⋊C4C2313yes—
73C3×C42×C5C2313yes—
74C4⋊Dic15C2311yescharacteristic
75C4⋊Dic15C2316yes—
76C4⋊Dic15C2316yes—
77#6712C2316yes—
78C4×Dic15C2316yes—
79C5×C4⋊Dic3C2313yes—
80C5×C4⋊Dic3C2316yes—
81C5×Dic3⋊C4C2316yes—
82C4×C5×Dic3C2316yes—
83C3×#685C2316yes—
84C3×C4⋊Dic5C2313yes—
85C3×C4×Dic5C2313yes—
86C3×C4×Dic5C2313yes—
87C3×C4×Dic5C2311yescharacteristic
88Dic3⋊Dic5C2312yes—
89Dic3⋊Dic5C23112yes—
90#6669C2316yes—
91#6668C2316yes—
92#6665C23112yes—
93Dic3×Dic5C2312yes—
94Dic3×Dic5C2316yes—
95C3×C4×#41—51no—
96Dic3×#41—52no—
97#5570—51no—
98#5569—52no—
99Q8×Dic6D511yescharacteristic, complemented
100#5543—53no—
101#5539—56no—
102D4×Dic6—53no—
103#5517—56no—
104#5470—53no—
105C2×C4×Dic6—53no—
106C2×#713C2×D311yescharacteristic
107C2×Q8×D5—33no—
108C5×C4⋊Q8—36no—
109C5×C4⋊Q8—33no—
110C4×C5×Q8—32no—
111C4×C5×Q8—33no—
112C4×C5×Q8C2×D311yescharacteristic
113#4092—36no—
114#4092—312no—
115Q8×Dic5—32no—
116Q8×Dic5—36no—
117Q8×Dic5—31no—
118Q8×Dic5C2×D311yescharacteristic
119#4043—36no—
120#4043—312no—
121#4039—36no—
122#4039C2×D313yes—
123#4038—36no—
124#4038C2×D313yes—
125D5×C4⋊C4—36no—
126D5×C4⋊C4—33no—
127#4036—36no—
128#4036—36no—
129#4036—36no—
130#4033—36no—
131#4033—33no—
132C4×D20—36no—
133C4×D20C2×D313yes—
134C42×D5—36no—
135C42×D5C2×D313yes—
136#4015—33no—
137#4015—36no—
138C4×Dic10—33no—
139C4×Dic10—36no—
140Q8×#41—151noSylow
141C22×C3×D5C2413yes—
142C3×C5×Q8C2×Q814yescomplemented
143C2×C3×C4×C5C2413yes—
144C2×C3×C4×C5C2413yes—
145C2×C3×C4×C5C2411yescharacteristic
146C2×Dic15C2416yes—
147C2×Dic15C2412yes—
148Dic30—26no—
149C2×C5×Dic3C2416yes—
150C2×C5×Dic3C2412yes—
151C5×Dic6—46no—
152C5×Dic6#4114yescomplemented
153C2×C3×Dic5C2413yes—
154C2×C3×Dic5C2411yescharacteristic
155C2×C3×Dic5C2411yescharacteristic
156C3×D20C2×Q8112yescomplemented
157C3×C4×D5—26no—
158C3×C4×D5—43no—
159C3×C4×D5C2×Q8112yescomplemented
160#1204—212no—
161D5×Dic3—212no—
162D5×Dic3—46no—
163C2×C3×#41—51no—
164C22×Dic6—53no—
165C3×C4×Q8—51no—
166C3×C4×Q8C2×D511yescharacteristic
167C3×C4×D4—53no—
168C3×C4×D4—53no—
169C3×#117—53no—
170C2×C3×C42—53no—
171Q8×Dic3—52no—
172Q8×Dic3C2×D512yes—
173D4×Dic3—56no—
174D4×Dic3—56no—
175#926—56no—
176C2×C4⋊Dic3—53no—
177#924—56no—
178#924—56no—
179C2×Dic3⋊C4—56no—
180C2×C4×Dic3—56no—
181#890—56no—
182#890—56no—
183#888—56no—
184#888C2×D516yes—
185#878—512no—
186#878—512no—
187#870—56no—
188#869—53no—
189#869C2×D513yes—
190C4×Dic6—51no—
191C4×Dic6—56no—
192C4×Dic6—53no—
193C4×Dic6—53no—
194C4×Dic6—56no—
195C4×Dic6—56no—
196C4×Dic6C2×D513yes—
197C2×C5×Q8—31no—
198C2×C5×Q8—36no—
199C2×C5×Q8C22×D311yescharacteristic
200#713Dic618yesdirect product factor
201Q8×D5—612no—
202C2×D20C22×D313yes—
203C2×C4×D5—36no—
204C2×C4×D5—36no—
205C2×C4×D5C22×D313yes—
206C2×C4×D5C22×D313yes—
207C2×C4×D5C22×D313yes—
208C2×Dic10—33no—
209C2×Dic10—36no—
210C5×C4⋊C4—36no—
211C5×C4⋊C4—33no—
212C5×C4⋊C4—36no—
213C5×C4⋊C4C22×D313yes—
214C42×C5—36no—
215C42×C5C22×D313yes—
216#685—312no—
217#685C22×D316yes—
218C4⋊Dic5—31no—
219C4⋊Dic5—36no—
220C4⋊Dic5—36no—
221C4⋊Dic5—32no—
222C4⋊Dic5—312no—
223C4⋊Dic5C22×D313yes—
224#683—36no—
225#683—36no—
226#683—36no—
227C4×Dic5—36no—
228C4×Dic5—32no—
229C4×Dic5—36no—
230C4×Dic5C22×D313yes—
231C4×Dic5C22×D311yescharacteristic
232C4×Dic5C22×D313yes—
233Q82—31no—
234#557—152no—
235#557—151no—
236#554—156no—
237#554—153no—
238D4×Q8—153no—
239#533—153no—
240#533—156no—
241C4×#41—51no—
242C4×#41—152no—
243C2×C4×Q8—153no—
244C22×C3×C5C2511yescharacteristic
245C2×C3×D5—23no—
246C2×C3×D5—23no—
247C2×C3×D5C22×Q816yes—
248C3×C4×C5—23no—
249C3×C4×C5C22×Q816yes—
250C3×C4×C5C2×#4112yes—
251Dic15—26no—
252Dic15—22no—
253C3×Dic5—23no—
254C3×Dic5—21no—
255C3×Dic5C22×Q812yes—
256C5×Dic3—26no—
257C5×Dic3C2×#4114yes—
258C3×#41—58no—
259C2×C3×Q8C22×D511yescharacteristic
260C2×C3×D4—53no—
261C22×C3×C4—53no—
262C22×C3×C4—53no—
263C22×C3×C4—53no—
264C22×Dic3—56no—
265C22×Dic3—56no—
266C2×Dic6—53no—
267C2×Dic6—56no—
268C2×Dic6—1012no—
269C2×Dic6C22×D516yes—
270C2×Dic6C22×D511yescharacteristic
271C3×C4⋊C4—53no—
272C3×C4⋊C4C22×D513yes—
273C3×C22⋊C4—56no—
274C3×C42—51no—
275C3×C42—53no—
276C3×C42—53no—
277C3×C42C22×D513yes—
278#217—512no—
279C4⋊Dic3—51no—
280C4⋊Dic3—56no—
281C4⋊Dic3—56no—
282C4⋊Dic3—56no—
283C4⋊Dic3C22×D513yes—
284C4⋊Dic3C22×D516yes—
285Dic3⋊C4—56no—
286Dic3⋊C4—56no—
287Dic3⋊C4—52no—
288Dic3⋊C4—512no—
289Dic3⋊C4C22×D516yes—
290C4×Dic3—56no—
291C4×Dic3—52no—
292C4×Dic3—56no—
293C4×Dic3C22×D516yes—
294C22×D5C23×D313yes—
295C5×Q8—34no—
296C5×Q8—126no—
297C5×Q8C2×Dic614yescomplemented
298C2×C4×C5—32no—
299C2×C4×C5—36no—
300C2×C4×C5C23×D313yes—
301C2×C4×C5C23×D313yes—
302C2×C4×C5C23×D311yescharacteristic
303C2×Dic5—32no—
304C2×Dic5—36no—
305C2×Dic5C23×D311yescharacteristic
306C2×Dic5C23×D313yes—
307C2×Dic5C23×D311yescharacteristic
308D20C2×Dic6112yescomplemented
309C4×D5—26no—
310C4×D5—43no—
311C4×D5—612no—
312C4×D5—126no—
313C4×D5C2×Dic6112yescomplemented
314Dic10—612no—
315Dic10—66no—
316C2×#41—51no—
317C22×Q8—153no—
318C4⋊Q8—36no—
319C4⋊Q8—33no—
320C4⋊Q8—156no—
321C4⋊Q8—153no—
322#125—156no—
323#125—156no—
324#125—156no—
325C22⋊Q8—1512no—
326C22⋊Q8—1512no—
327C22⋊Q8—156no—
328C22⋊Q8—156no—
329C4×Q8—32no—
330C4×Q8—33no—
331C4×Q8—51no—
332C4×Q8—156no—
333C4×Q8—153no—
334C4×Q8—156no—
335C4×Q8—151no—
336C4×Q8—156no—
337C4×Q8—152no—
338C4×Q8—153no—
339C4×Q8D3×D511yescharacteristic
340C4×D4—53no—
341C4×D4—53no—
342C4×D4—156no—
343C4×D4—156no—
344#117—53no—
345#117—156no—
346C2×C4⋊C4—156no—
347C2×C4⋊C4—153no—
348C2×C42—53no—
349C2×C42—156no—
350C2×C3×C5C23×Q811yescharacteristic
351C2×C3×C5C22×#4111yescharacteristic
352C2×C3×C5#58511yescharacteristic
353C3×D5—26no—
354C23×C3—53no—
355C3×Q8Q8×D514yescomplemented
356C3×D4—512no—
357C2×C3×C4—51no—
358C2×C3×C4—512no—
359C2×C3×C4—51no—
360C2×C3×C4—53no—
361C2×C3×C4—106no—
362C2×C3×C4—203no—
363C2×C3×C4C23×D513yes—
364C2×C3×C4C23×D513yes—
365C2×C3×C4C23×D511yescharacteristic
366C2×Dic3—52no—
367C2×Dic3—56no—
368C2×Dic3—1012no—
369C2×Dic3—206no—
370C2×Dic3C23×D516yes—
371C2×Dic3C23×D512yes—
372Dic6—46no—
373Dic6—106no—
374Dic6—1012no—
375Dic6#71314yesdirect product factor
376C22×C5C24×D311yescharacteristic
377C2×D5—23no—
378C2×D5—23no—
379C2×D5C22×Dic616yes—
380C4×C5—23no—
381C4×C5—34no—
382C4×C5—66no—
383C4×C5C22×Dic616yes—
384C4×C5D3×#4112yes—
385Dic5—23no—
386Dic5—21no—
387Dic5—66no—
388Dic5—62no—
389Dic5C22×Dic612yes—
390#41—58no—
391C2×Q8—31no—
392C2×Q8—36no—
393C2×Q8—156no—
394C2×Q8—153no—
395C2×Q8—3012no—
396C2×Q8C2×D3×D511yescharacteristic
397C2×D4—53no—
398C22×C4—53no—
399C22×C4—53no—
400C22×C4—53no—
401C22×C4—156no—
402C22×C4—156no—
403C4⋊C4—36no—
404C4⋊C4—33no—
405C4⋊C4—36no—
406C4⋊C4—53no—
407C4⋊C4—152no—
408C4⋊C4—156no—
409C4⋊C4—156no—
410C4⋊C4—1512no—
411C4⋊C4—151no—
412C4⋊C4—156no—
413C4⋊C4—156no—
414C4⋊C4—156no—
415C4⋊C4C2×D3×D513yes—
416C22⋊C4—56no—
417C22⋊C4—1512no—
418C42—36no—
419C42—51no—
420C42—53no—
421C42—53no—
422C42—152no—
423C42—156no—
424C42—156no—
425C42C2×D3×D513yes—
426C3×C5Q8×#4111yescharacteristic, complemented, Hall
427C22×C3—56no—
428C22×C3—103no—
429C22×C3—103no—
430C22×C3C24×D511yescharacteristic
431C3×C4—23no—
432C3×C4—52no—
433C3×C4—103no—
434C3×C4—101no—
435C3×C4C2×Q8×D516yes—
436C3×C4C2×#71312yes—
437Dic3—26no—
438Dic3—106no—
439Dic3—102no—
440Dic3C2×#71314yes—
441C2×C5C23×Dic611yescharacteristic
442C2×C5C2×D3×#4111yescharacteristic
443C2×C5#596411yescharacteristic
444D5—26no—
445C23—53no—
446Q8—34no—
447Q8—126no—
448Q8—3012no—
449Q8—306no—
450Q8D5×Dic614yescomplemented
451D4—512no—
452C2×C4—32no—
453C2×C4—36no—
454C2×C4—51no—
455C2×C4—512no—
456C2×C4—51no—
457C2×C4—53no—
458C2×C4—106no—
459C2×C4—152no—
460C2×C4—156no—
461C2×C4—203no—
462C2×C4—3012no—
463C2×C4—606no—
464C2×C4C22×D3×D513yes—
465C2×C4C22×D3×D513yes—
466C2×C4C22×D3×D511yescharacteristic
467C2×C3—106no—
468C2×C3C22×Q8×D511yescharacteristic
469C2×C3C22×#71311yescharacteristic
470C2×C3#6628511yescharacteristic
471C5#41×Dic611yescharacteristic, complemented, Sylow
472C22—56no—
473C22—103no—
474C22—103no—
475C22C23×D3×D511yescharacteristic
476C4—23no—
477C4—34no—
478C4—52no—
479C4—66no—
480C4—101no—
481C4—103no—
482C4—302no—
483C4—306no—
484C4C2×D5×Dic616yes—
485C4D3×#71312yes—
486C3Q8×#71311yescharacteristic, complemented, Sylow
487C2—106no—
488C2C22×D5×Dic611yescharacteristic
489C2C2×D3×#71311yescharacteristic
490C2#1175705911yescharacteristic
subgroupG/H
center, FrattiniC22C23×D3×D5
commutator, socle, cosocleC22×C3×C5C25
FittingC3×C4×C5×Q8C22
prime radicalC2×C3×#713C22
hypercenterC4×Q8D3×D5
hypocenterC3×C5Q8×#41
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. #11756329C2semidirect
2. C5×Q8×Dic6C2semidirect
4. #11755268C2semidirect
6. #11753589C2semidirect
7. #11753568C2semidirect
8. #11752648C2semidirect
9. Dic6×D20C2semidirect
10. C4×D5×Dic6C2semidirect
11. #11752506C2semidirect
63. C3×#713Q8semidirect
99. Q8×Dic6D5semidirect
142. C3×C5×Q8C2×Q8semidirect
152. C5×Dic6#41semidirect
156. C3×D20C2×Q8semidirect
159. C3×C4×D5C2×Q8semidirect
200. #713Dic6direct (N × H)
297. C5×Q8C2×Dic6semidirect
308. D20C2×Dic6semidirect
313. C4×D5C2×Dic6semidirect
355. C3×Q8Q8×D5semidirect
375. Dic6#713direct (N × H)
426. C3×C5Q8×#41semidirect
450. Q8D5×Dic6semidirect
471. C5#41×Dic6semidirect
486. C3Q8×#713semidirect

Maximal subgroups

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

subgroupindex
#117563292
C5×Q8×Dic62
C3×C4×#7132
#117552682
Dic3×#7132
#117535892
#117535682
#117526482
Dic6×D202
C4×D5×Dic62
#117525062
Q8×#7133
#41×Dic65