Finite Group Databasebeta

← #14417008

Subgroups — #14417008

Order 1920.

1012 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×AGL(1,ℤ/8)C211yesmaximal, characteristic
2#11756586C211yesmaximal, characteristic, complemented
3#11756557C211yesmaximal, characteristic, complemented
4#11751334C211yesmaximal, characteristic, complemented
5C2×#91116C211yesmaximal, characteristic, complemented
6#11751265C212yesmaximal, complemented
7SD16×Dic15C212yesmaximal, complemented
8#11751253C212yesmaximal, complemented
9D8×Dic15C212yesmaximal, complemented
10#11751223C211yesmaximal, characteristic, complemented
11#11751222C211yesmaximal, characteristic, complemented
12#10615661—31nomaximal, Hall
13C2×C3×C5×#41C2211yescharacteristic
14C22×C3×C5×D4C2211yescharacteristic
15C3×C5×AGL(1,ℤ/8)C418yescomplemented
16C2×C3×C5×SD16C2212yes—
17C2×C3×C5×D8C2212yes—
18C2×C3×C5×M16C2211yescharacteristic
19C2×#6719—22no—
20Q8×Dic15C2212yescomplemented
21D4×Dic15—22no—
22D4×Dic15—21no—
23D4×Dic15—21no—
24D4×Dic15C2212yes—
25D4×Dic15C2212yes—
26D4×Dic15C2212yescomplemented
27#91815—21no—
28#91815C2211yescharacteristic
29#91815C2211yescharacteristic
30C2×C4⋊Dic15—21no—
31C2×C4⋊Dic15C2211yescharacteristic
32C22×C3⋊#168C2211yescharacteristic
33#91119C2211yescharacteristic, complemented
34#91119C2211yescharacteristic, complemented
35#91116C2212yes—
36#91116C2212yes—
37#91116C2211yescharacteristic, complemented
38#91116C2211yescharacteristic, complemented
39#91102C2212yescomplemented
40#91101C2212yescomplemented
41#91099C2212yescomplemented
42C8×Dic15C2212yescomplemented
43#83930—51nomaximal, Hall
44C2×C5×AGL(1,ℤ/8)D311yescharacteristic
45#66161—31no—
46#66132—31no—
47#65522—31no—
48C2×#3964—31no—
49#65453—32no—
50SD16×Dic5—32no—
51#65441—32no—
52D8×Dic5—32no—
53#65411—31no—
54#65410—31no—
55C24×C3×C5—21no—
56C3×C5×#41—22no—
57C3×C5×#41C2×C414yescomplemented
58C2×C3×C5×Q8C2311yescharacteristic
59C2×C3×C5×D4—21no—
60C2×C3×C5×D4—21no—
61C2×C3×C5×D4—22no—
62C2×C3×C5×D4C2×C414yescomplemented
63C2×C3×C5×D4C2312yes—
64C2×C3×C5×D4C2311yescharacteristic
65C22×C3×C4×C5—21no—
66C22×C3×C4×C5C2311yescharacteristic
67C22×Dic15—21no—
68C22×Dic15—22no—
69C22×Dic15—22no—
70C22×Dic15—22no—
71C3×C5×SD16C2×C418yescomplemented
72C3×C5×D8C2×C418yescomplemented
73C3×C5×M16C2×C414yescomplemented
74C2×C3×C5×C8C2312yes—
75#6719—21no—
76#6719—22no—
77#6719—22no—
78#6719—22no—
79#6719D412yes—
80C4⋊Dic15—21no—
81C4⋊Dic15—22no—
82C4⋊Dic15D411yescharacteristic
83C4⋊Dic15D411yescharacteristic
84C4⋊Dic15C2312yes—
85C4⋊Dic15C2311yescharacteristic
86C4⋊Dic15C2311yescharacteristic
87C4×Dic15—21no—
88C4×Dic15—22no—
89C4×Dic15C2312yes—
90C2×C3⋊#168—22no—
91C2×C3⋊#168C2311yescharacteristic
92C2×C3⋊#168C2311yescharacteristic
93C2×C3×AGL(1,ℤ/8)D511yescharacteristic
94#5827—51no—
95#5798—51no—
96#5233—51no—
97C2×#832—51no—
98#5164—52no—
99Dic3×SD16—52no—
100#5152—52no—
101Dic3×D8—52no—
102#5122—51no—
103#5121—51no—
104C2×C5×#41C2×D311yescharacteristic
105C22×C5×D4C2×D311yescharacteristic
106C5×AGL(1,ℤ/8)Dic318yescomplemented
107C2×C5×SD16C2×D312yes—
108C2×C5×D8C2×D312yes—
109C2×C5×M16C2×D311yescharacteristic
110C2×C22⋊Dic5—62no—
111Q8×Dic5—32no—
112D4×Dic5—32no—
113D4×Dic5—32no—
114D4×Dic5—32no—
115D4×Dic5—61no—
116D4×Dic5—61no—
117D4×Dic5—62no—
118#4072—31no—
119#4072—31no—
120#4072—61no—
121C2×C4⋊Dic5—31no—
122C2×C4⋊Dic5—61no—
123C22×#168—31no—
124#3967—31no—
125#3967—31no—
126#3964—31no—
127#3964—32no—
128#3964—31no—
129#3964—32no—
130#3950—32no—
131#3949—32no—
132#3947—32no—
133C8×Dic5—32no—
134#2946—151noSylow
135C23×C3×C5—21no—
136C23×C3×C5—22no—
137C23×C3×C5—24no—
138C23×C3×C5—81no—
139C23×C3×C5—81no—
140C23×C3×C5C2×D411yescharacteristic
141C3×C5×Q8—21no—
142C3×C5×Q8C22×C412yes—
143C3×C5×D4—24no—
144C3×C5×D4—24no—
145C3×C5×D4—22no—
146C3×C5×D4—21no—
147C3×C5×D4C22×C412yes—
148C3×C5×D4C22×C414yes—
149C2×C3×C4×C5—21no—
150C2×C3×C4×C5—24no—
151C2×C3×C4×C5C22×C412yes—
152C2×C3×C4×C5C2×D411yescharacteristic
153C2×C3×C4×C5#4112yes—
154C2×C3×C4×C5C2411yescharacteristic
155C2×Dic15—22no—
156C2×Dic15—21no—
157C2×Dic15—21no—
158C2×Dic15—82no—
159C2×Dic15—82no—
160C2×Dic15—82no—
161C2×Dic15—81no—
162C2×Dic15C2×D412yes—
163C2×Dic15C2×D412yes—
164C3×C5×C8C22×C414yes—
165C3⋊#168—21no—
166C3⋊#168—21no—
167C2×C3×#41C2×D511yescharacteristic
168C22×C3×D4C2×D511yescharacteristic
169C3×AGL(1,ℤ/8)Dic518yescomplemented
170C2×C3×SD16C2×D512yes—
171C2×C3×D8C2×D512yes—
172C2×C3×M16C2×D511yescharacteristic
173C2×#217—102no—
174Q8×Dic3—52no—
175D4×Dic3—52no—
176D4×Dic3—52no—
177D4×Dic3—52no—
178D4×Dic3—101no—
179D4×Dic3—101no—
180D4×Dic3—102no—
181#926—51no—
182#926—51no—
183#926—101no—
184C2×C4⋊Dic3—51no—
185C2×C4⋊Dic3—101no—
186C22×C3⋊C8—51no—
187#835—51no—
188#835—51no—
189#832—51no—
190#832—52no—
191#832—51no—
192#832—52no—
193#818—52no—
194#817—52no—
195#815—52no—
196C8×Dic3—52no—
197C24×C5—21no—
198C5×#41—22no—
199C5×#41C2×Dic314yescomplemented
200C2×C5×Q8C22×D311yescharacteristic
201C2×C5×D4—21no—
202C2×C5×D4—21no—
203C2×C5×D4—22no—
204C2×C5×D4C2×Dic314yescomplemented
205C2×C5×D4C22×D311yescharacteristic
206C2×C5×D4C22×D312yes—
207C22×C4×C5—21no—
208C22×C4×C5C22×D311yescharacteristic
209C22×Dic5—62no—
210C22×Dic5—62no—
211C22×Dic5—62no—
212C22×Dic5—61no—
213C5×SD16C2×Dic318yescomplemented
214C5×D8C2×Dic318yescomplemented
215C5×M16C2×Dic314yescomplemented
216C2×C5×C8C22×D312yes—
217C22⋊Dic5—32no—
218C22⋊Dic5—62no—
219C22⋊Dic5—62no—
220C22⋊Dic5—62no—
221C22⋊Dic5—61no—
222C4⋊Dic5—31no—
223C4⋊Dic5—31no—
224C4⋊Dic5—31no—
225C4⋊Dic5—32no—
226C4⋊Dic5—31no—
227C4⋊Dic5—62no—
228C4⋊Dic5—61no—
229C4×Dic5—32no—
230C4×Dic5—62no—
231C4×Dic5—61no—
232C2×#168—31no—
233C2×#168—31no—
234C2×#168—62no—
235C2×AGL(1,ℤ/8)D1511yescharacteristic
236#520—151no—
237#518—151no—
238#442—152no—
239#440—152no—
240C4×SD16—152no—
241C4×D8—152no—
242#428—151no—
243#416—151no—
244C2×#102—151no—
245#405—151no—
246C22×C3×C5—21no—
247C22×C3×C5—21no—
248C22×C3×C5—22no—
249C22×C3×C5—24no—
250C22×C3×C5—41no—
251C22×C3×C5—42no—
252C22×C3×C5—42no—
253C22×C3×C5—41no—
254C22×C3×C5—84no—
255C22×C3×C5C4×D412yes—
256C22×C3×C5C22×D411yescharacteristic
257C3×C4×C5—22no—
258C3×C4×C5C4×D412yes—
259C3×C4×C5C23×C411yescharacteristic
260C3×C4×C5C2×#4111yescharacteristic
261Dic15—22no—
262Dic15—22no—
263Dic15—42no—
264Dic15—41no—
265Dic15—41no—
266C24×C3—21no—
267C3×#41—22no—
268C3×#41C2×Dic514yescomplemented
269C2×C3×Q8C22×D511yescharacteristic
270C2×C3×D4—21no—
271C2×C3×D4—21no—
272C2×C3×D4—22no—
273C2×C3×D4C2×Dic514yescomplemented
274C2×C3×D4C22×D511yescharacteristic
275C2×C3×D4C22×D512yes—
276C22×C3×C4—21no—
277C22×C3×C4C22×D511yescharacteristic
278C22×Dic3—102no—
279C22×Dic3—102no—
280C22×Dic3—102no—
281C22×Dic3—101no—
282C3×SD16C2×Dic518yescomplemented
283C3×D8C2×Dic518yescomplemented
284C3×M16C2×Dic514yescomplemented
285C2×C3×C8C22×D512yes—
286#217—52no—
287#217—102no—
288#217—102no—
289#217—102no—
290#217—101no—
291C4⋊Dic3—51no—
292C4⋊Dic3—51no—
293C4⋊Dic3—51no—
294C4⋊Dic3—52no—
295C4⋊Dic3—51no—
296C4⋊Dic3—102no—
297C4⋊Dic3—101no—
298C4×Dic3—52no—
299C4×Dic3—102no—
300C4×Dic3—101no—
301C2×C3⋊C8—51no—
302C2×C3⋊C8—51no—
303C2×C3⋊C8—102no—
304C23×C5—21no—
305C23×C5—22no—
306C23×C5—24no—
307C23×C5—81no—
308C23×C5—81no—
309C23×C5D3×D411yescharacteristic
310C5×Q8—21no—
311C5×Q8C22×Dic312yes—
312C5×D4—24no—
313C5×D4—24no—
314C5×D4—22no—
315C5×D4—21no—
316C5×D4C22×Dic312yes—
317C5×D4C22×Dic314yes—
318C2×C4×C5—21no—
319C2×C4×C5—24no—
320C2×C4×C5D3×D411yescharacteristic
321C2×C4×C5#23712yes—
322C2×C4×C5C22×Dic312yes—
323C2×C4×C5C23×D311yescharacteristic
324C2×Dic5—32no—
325C2×Dic5—32no—
326C2×Dic5—62no—
327C2×Dic5—61no—
328C2×Dic5—61no—
329C2×Dic5—242no—
330C2×Dic5—242no—
331C2×Dic5—242no—
332C2×Dic5—241no—
333C5×C8C22×Dic314yes—
334#168—61no—
335#168—61no—
336C2×#41C2×D1511yescharacteristic
337C22×D4C2×D1511yescharacteristic
338AGL(1,ℤ/8)Dic1518yescomplemented
339C2×SD16C2×D1512yes—
340C2×D8C2×D1512yes—
341C2×M16C2×D1511yescharacteristic
342C22×C8—151no—
343C4×Q8—152no—
344C4×D4—152no—
345C4×D4—152no—
346C4×D4—152no—
347C4×D4—302no—
348C4×D4—301no—
349C4×D4—301no—
350#117—151no—
351#117—151no—
352#117—301no—
353C2×C4⋊C4—151no—
354C2×C4⋊C4—301no—
355C2×C22⋊C4—302no—
356#107—152no—
357#106—152no—
358#103—151no—
359#103—151no—
360#102—152no—
361#102—152no—
362#102—151no—
363#102—151no—
364#97—152no—
365C4×C8—152no—
366C2×C3×C5—22no—
367C2×C3×C5—42no—
368C2×C3×C5—44no—
369C2×C3×C5C2×C4×D411yescharacteristic
370C2×C3×C5#57611yescharacteristic
371C2×C3×C5#57711yescharacteristic
372C23×C3—21no—
373C23×C3—22no—
374C23×C3—24no—
375C23×C3—81no—
376C23×C3—81no—
377C23×C3D4×D511yescharacteristic
378C3×Q8—21no—
379C3×Q8C22×Dic512yes—
380C3×D4—24no—
381C3×D4—22no—
382C3×D4—21no—
383C3×D4—24no—
384C3×D4C22×Dic512yes—
385C3×D4C22×Dic514yes—
386C2×C3×C4—21no—
387C2×C3×C4—24no—
388C2×C3×C4D4×D511yescharacteristic
389C2×C3×C4#71112yes—
390C2×C3×C4C22×Dic512yes—
391C2×C3×C4C23×D511yescharacteristic
392C2×Dic3—52no—
393C2×Dic3—52no—
394C2×Dic3—102no—
395C2×Dic3—101no—
396C2×Dic3—101no—
397C2×Dic3—402no—
398C2×Dic3—402no—
399C2×Dic3—402no—
400C2×Dic3—401no—
401C3×C8C22×Dic514yes—
402C3⋊C8—101no—
403C3⋊C8—101no—
404C22×C5—21no—
405C22×C5—21no—
406C22×C5—22no—
407C22×C5—24no—
408C22×C5—41no—
409C22×C5—42no—
410C22×C5—42no—
411C22×C5—41no—
412C22×C5—84no—
413C22×C5D4×Dic312yes—
414C22×C5C2×D3×D411yescharacteristic
415C4×C5—22no—
416C4×C5D4×Dic312yes—
417C4×C5C2×#23711yescharacteristic
418C4×C5C23×Dic311yescharacteristic
419Dic5—62no—
420Dic5—62no—
421Dic5—122no—
422Dic5—121no—
423Dic5—121no—
424C24—21no—
425#41—22no—
426#41C2×Dic1514yescomplemented
427C2×Q8C22×D1511yescharacteristic
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—22no—
431C2×D4C2×Dic1514yescomplemented
432C2×D4C22×D1511yescharacteristic
433C2×D4C22×D1512yes—
434C22×C4—21no—
435C22×C4—302no—
436C22×C4—302no—
437C22×C4—302no—
438C22×C4—301no—
439C22×C4C22×D1511yescharacteristic
440SD16C2×Dic1518yescomplemented
441D8C2×Dic1518yescomplemented
442M16C2×Dic1514yescomplemented
443C2×C8—151no—
444C2×C8—151no—
445C2×C8—302no—
446C2×C8C22×D1512yes—
447C4⋊C4—151no—
448C4⋊C4—151no—
449C4⋊C4—152no—
450C4⋊C4—151no—
451C4⋊C4—151no—
452C4⋊C4—302no—
453C4⋊C4—301no—
454C22⋊C4—152no—
455C22⋊C4—302no—
456C22⋊C4—302no—
457C22⋊C4—302no—
458C22⋊C4—301no—
459C42—152no—
460C42—302no—
461C42—301no—
462C3×C5#294611yescharacteristic, complemented, Hall
463C22×C3—21no—
464C22×C3—21no—
465C22×C3—22no—
466C22×C3—24no—
467C22×C3—41no—
468C22×C3—42no—
469C22×C3—42no—
470C22×C3—41no—
471C22×C3—84no—
472C22×C3D4×Dic512yes—
473C22×C3C2×D4×D511yescharacteristic
474C3×C4—22no—
475C3×C4D4×Dic512yes—
476C3×C4C2×#71111yescharacteristic
477C3×C4C23×Dic511yescharacteristic
478Dic3—102no—
479Dic3—102no—
480Dic3—202no—
481Dic3—201no—
482Dic3—201no—
483C2×C5—22no—
484C2×C5—44no—
485C2×C5—42no—
486C2×C5#577411yescharacteristic
487C2×C5#577511yescharacteristic
488C2×C5C2×D4×Dic311yescharacteristic
489C23—21no—
490C23—22no—
491C23—24no—
492C23—81no—
493C23—81no—
494C23D4×D1511yescharacteristic
495Q8—21no—
496Q8C22×Dic1512yes—
497D4—24no—
498D4—24no—
499D4—22no—
500D4—21no—
501D4C22×Dic1512yes—
502D4C22×Dic1514yes—
503C2×C4—21no—
504C2×C4—24no—
505C2×C4—152no—
506C2×C4—152no—
507C2×C4—301no—
508C2×C4—301no—
509C2×C4—302no—
510C2×C4—1202no—
511C2×C4—1202no—
512C2×C4—1202no—
513C2×C4—1201no—
514C2×C4D4×D1511yescharacteristic
515C2×C4#681912yes—
516C2×C4C22×Dic1512yes—
517C2×C4C23×D1511yescharacteristic
518C8—301no—
519C8—301no—
520C8C22×Dic1514yes—
521C2×C3—22no—
522C2×C3—44no—
523C2×C3—42no—
524C2×C3#6610811yescharacteristic
525C2×C3#6610911yescharacteristic
526C2×C3C2×D4×Dic511yescharacteristic
527C5#8393011yescharacteristic, complemented, Sylow
528C22—21no—
529C22—21no—
530C22—22no—
531C22—24no—
532C22—41no—
533C22—42no—
534C22—42no—
535C22—41no—
536C22—84no—
537C22D4×Dic1512yes—
538C22C2×D4×D1511yescharacteristic
539C4—22no—
540C4—302no—
541C4—302no—
542C4—601no—
543C4—602no—
544C4—601no—
545C4D4×Dic1512yes—
546C4C2×#681911yescharacteristic
547C4C23×Dic1511yescharacteristic
548C3#1061566111yescharacteristic, complemented, Sylow
549C2—22no—
550C2—44no—
551C2—42no—
552C2#1175653311yescharacteristic
553C2#1175653411yescharacteristic
554C2C2×D4×Dic1511yescharacteristic
subgroupG/H
centerC22C2×D4×D15
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C23×D15
Fitting, prime radicalC2×C3×C5×AGL(1,ℤ/8)C2
socleC22×C3×C5C22×D4
cosocleC2×C3×C4×C5C24
hypercenterC2×AGL(1,ℤ/8)D15
hypocenterC3×C5#2946
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
2. #11756586C2semidirect
3. #11756557C2semidirect
4. #11751334C2semidirect
5. C2×#91116C2semidirect
6. #11751265C2semidirect
7. SD16×Dic15C2semidirect
8. #11751253C2semidirect
9. D8×Dic15C2semidirect
10. #11751223C2semidirect
11. #11751222C2semidirect
15. C3×C5×AGL(1,ℤ/8)C4semidirect
20. Q8×Dic15C22semidirect
26. D4×Dic15C22semidirect
33. #91119C22semidirect
34. #91119C22semidirect
37. #91116C22semidirect
38. #91116C22semidirect
39. #91102C22semidirect
40. #91101C22semidirect
41. #91099C22semidirect
42. C8×Dic15C22semidirect
57. C3×C5×#41C2×C4semidirect
62. C2×C3×C5×D4C2×C4semidirect
71. C3×C5×SD16C2×C4semidirect
72. C3×C5×D8C2×C4semidirect
73. C3×C5×M16C2×C4semidirect
106. C5×AGL(1,ℤ/8)Dic3semidirect
169. C3×AGL(1,ℤ/8)Dic5semidirect
199. C5×#41C2×Dic3semidirect
204. C2×C5×D4C2×Dic3semidirect
213. C5×SD16C2×Dic3semidirect
214. C5×D8C2×Dic3semidirect
215. C5×M16C2×Dic3semidirect
268. C3×#41C2×Dic5semidirect
273. C2×C3×D4C2×Dic5semidirect
282. C3×SD16C2×Dic5semidirect
283. C3×D8C2×Dic5semidirect
284. C3×M16C2×Dic5semidirect
338. AGL(1,ℤ/8)Dic15semidirect
426. #41C2×Dic15semidirect
431. C2×D4C2×Dic15semidirect
440. SD16C2×Dic15semidirect
441. D8C2×Dic15semidirect
442. M16C2×Dic15semidirect
462. C3×C5#2946semidirect
527. C5#83930semidirect
548. C3#10615661semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C5×AGL(1,ℤ/8)2
#117565862
#117565572
#117513342
C2×#911162
#117512652
SD16×Dic152
#117512532
D8×Dic152
#117512232
#117512222
#106156613
#839305