Finite Group Databasebeta

← #15874164

Subgroups — #15874164

Order 3328.

1212 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22⋊Q8×Dic13C211yesmaximal, characteristic, complemented
2C2×#11718882C211yesmaximal, characteristic
3#13115719C212yesmaximal, complemented
4#13115535C212yesmaximal, complemented
5#13115458C212yesmaximal, complemented
6#13115193C212yesmaximal, complemented
7#13115144C211yesmaximal, characteristic, complemented
8#13115026C211yesmaximal, characteristic, complemented
9#13114767C211yesmaximal, characteristic
10#13114600C211yesmaximal, characteristic, complemented
11C2×C13×#450C211yesmaximal, characteristic
12C23×D4×C13C2211yescharacteristic
13C22×C13×SD16C2212yes—
14C2×C13×C22⋊Q8C2211yescharacteristic
15C22×C22⋊Dic13—21no—
16C22×C22⋊Dic13—21no—
17C2×Q8×Dic13C2211yescharacteristic, complemented
18C22×C4⋊Dic13C2211yescharacteristic
19C22×C4×Dic13C2211yescharacteristic
20C13×#450C418yescomplemented
21C2×C13×#102C2212yes—
22C2×C13×C22⋊C8C2211yescharacteristic
23#11718891C2211yescharacteristic, complemented
24C2×#88075C2212yescomplemented
25#11718882—21no—
26#11718882—21no—
27#11718882C2212yes—
28#11718882C2212yes—
29C2×#88072C2212yes—
30C2×#88072C2212yes—
31#11718873C2211yescharacteristic
32C2×C22⋊#1055C2211yescharacteristic
33C2×#88058C2212yescomplemented
34C2×#88055C2212yescomplemented
35#11718646C2211yescharacteristic, complemented
36#11718642—21no—
37C4⋊C4×Dic13—21no—
38C4⋊C4×Dic13C2211yescharacteristic, complemented
39#11718611—21no—
40C22⋊C4×Dic13—21no—
41#11718143C2212yescomplemented
42#11718142C2212yescomplemented
43#11718121C2212yescomplemented
44C25×C13—21no—
45C22×Q8×C13C2311yescharacteristic
46C22×D4×C13—21no—
47C22×D4×C13—22no—
48C22×D4×C13—42no—
49C22×D4×C13—42no—
50C22×D4×C13C2×C414yes—
51C22×D4×C13C2312yes—
52C23×C4×C13C2311yescharacteristic
53C23×Dic13—21no—
54C23×Dic13D411yescharacteristic
55C23×Dic13D411yescharacteristic
56C2×C13×SD16—42no—
57C2×C13×SD16—42no—
58C2×C13×SD16C2×C418yescomplemented
59C22×C8×C13C2312yes—
60C13×C22⋊Q8—22no—
61C13×C22⋊Q8C2×C414yes—
62C2×C13×C4⋊C4—21no—
63C2×C13×C4⋊C4C2311yescharacteristic
64C2×C13×C22⋊C4—21no—
65C2×C22⋊Dic13—21no—
66C2×C22⋊Dic13—22no—
67C2×C22⋊Dic13—22no—
68C2×C22⋊Dic13—22no—
69C2×C22⋊Dic13—22no—
70C2×C22⋊Dic13—44no—
71C2×C22⋊Dic13—44no—
72Q8×Dic13—42no—
73C2×C4⋊Dic13—21no—
74C2×C4⋊Dic13—21no—
75C2×C4⋊Dic13—21no—
76C2×C4⋊Dic13—42no—
77C2×C4⋊Dic13C2312yes—
78C2×C4⋊Dic13C2311yescharacteristic
79C2×C4⋊Dic13C2311yescharacteristic
80C2×C4×Dic13—21no—
81C2×C4×Dic13—21no—
82C2×C4×Dic13—42no—
83C2×C4×Dic13D411yescharacteristic
84C2×C4×Dic13D411yescharacteristic
85C2×C4×Dic13C2311yescharacteristic
86C2×C4×Dic13C2311yescharacteristic
87C22×#1055C2312yes—
88C13×#102C2×C418yescomplemented
89C13×C22⋊C8C2×C414yes—
90#88075—42no—
91#88072—42no—
92#88072—42no—
93#88071—21no—
94#88071—21no—
95#88071—21no—
96#88071—21no—
97#88071—21no—
98C22⋊#1055—22no—
99#88058—42no—
100#88055—42no—
101#22049—131nomaximal, Sylow
102C24×C13—24no—
103C24×C13—22no—
104C24×C13—42no—
105C24×C13—42no—
106C24×C13—42no—
107C24×C13—42no—
108C24×C13—42no—
109C24×C13—42no—
110C24×C13C2×D411yescharacteristic
111C2×Q8×C13—21no—
112C2×Q8×C13—22no—
113C2×Q8×C13—22no—
114C2×Q8×C13C22×C412yes—
115C2×D4×C13—24no—
116C2×D4×C13—22no—
117C2×D4×C13—22no—
118C2×D4×C13—22no—
119C2×D4×C13—22no—
120C2×D4×C13—22no—
121C2×D4×C13—48no—
122C2×D4×C13—42no—
123C2×D4×C13—42no—
124C2×D4×C13—48no—
125C2×D4×C13C22×C414yes—
126C22×C4×C13—22no—
127C22×C4×C13—22no—
128C22×C4×C13—21no—
129C22×C4×C13—21no—
130C22×C4×C13C22×C412yes—
131C22×C4×C13C2×D411yescharacteristic
132C22×C4×C13#4112yes—
133C22×C4×C13C2411yescharacteristic
134C22×Dic13—22no—
135C22×Dic13—22no—
136C22×Dic13—22no—
137C22×Dic13—21no—
138C22×Dic13—21no—
139C22×Dic13—22no—
140C22×Dic13—41no—
141C22×Dic13—41no—
142C22×Dic13—41no—
143C22×Dic13—42no—
144C22×Dic13—42no—
145C22×Dic13—42no—
146C22×Dic13C2×D411yescharacteristic
147C22×Dic13C2×D411yescharacteristic
148C22×Dic13C2×D411yescharacteristic
149C22×Dic13C2×D411yescharacteristic
150C13×SD16—48no—
151C2×C8×C13—22no—
152C2×C8×C13—22no—
153C2×C8×C13C22×C414yes—
154C13×C4⋊C4—24no—
155C13×C4⋊C4—21no—
156C13×C4⋊C4C22×C412yes—
157C13×C22⋊C4—24no—
158C22⋊Dic13—44no—
159C22⋊Dic13—44no—
160C22⋊Dic13—44no—
161C22⋊Dic13—42no—
162C22⋊Dic13—44no—
163C4⋊Dic13—22no—
164C4⋊Dic13—22no—
165C4⋊Dic13—22no—
166C4⋊Dic13—22no—
167C4⋊Dic13—42no—
168C4⋊Dic13—42no—
169C4⋊Dic13—42no—
170C4×Dic13—22no—
171C4×Dic13—22no—
172C4×Dic13—41no—
173C4×Dic13—41no—
174C4×Dic13—42no—
175C4×Dic13—42no—
176C2×#1055—22no—
177C2×#1055—22no—
178C2×#1055—22no—
179C2×#450D1311yescharacteristic
180C4×C22⋊Q8—131no—
181C2×#390—131no—
182#1973—132no—
183#1936—132no—
184#1925—132no—
185#1880—132no—
186#1872—131no—
187#1852—131no—
188#1786—131no—
189#1749—131no—
190C23×C13—21no—
191C23×C13—21no—
192C23×C13—21no—
193C23×C13—21no—
194C23×C13—24no—
195C23×C13—48no—
196C23×C13—48no—
197C23×C13—42no—
198C23×C13—42no—
199C23×C13—42no—
200C23×C13—42no—
201C23×C13—42no—
202C23×C13—42no—
203C23×C13—48no—
204C23×C13—84no—
205C23×C13—84no—
206C23×C13—84no—
207C23×C13—84no—
208C23×C13C4×D412yes—
209C23×C13AGL(1,ℤ/8)12yes—
210C23×C13#13712yes—
211C23×C13C22×D411yescharacteristic
212Q8×C13—24no—
213Q8×C13—42no—
214D4×C13—24no—
215D4×C13—24no—
216D4×C13—48no—
217D4×C13—42no—
218D4×C13—42no—
219C2×C4×C13—21no—
220C2×C4×C13—21no—
221C2×C4×C13—24no—
222C2×C4×C13—24no—
223C2×C4×C13—22no—
224C2×C4×C13—22no—
225C2×C4×C13—41no—
226C2×C4×C13—41no—
227C2×C4×C13—41no—
228C2×C4×C13—41no—
229C2×C4×C13C4×D412yes—
230C2×C4×C13C23×C411yescharacteristic
231C2×C4×C13C2×#4111yescharacteristic
232C2×C4×C13H(5,𝔽2)12yes—
233C2×C4×C13H(5,𝔽2)12yes—
234C2×Dic13—21no—
235C2×Dic13—21no—
236C2×Dic13—21no—
237C2×Dic13—21no—
238C2×Dic13—21no—
239C2×Dic13—21no—
240C2×Dic13—21no—
241C2×Dic13—21no—
242C2×Dic13—21no—
243C2×Dic13—21no—
244C2×Dic13—21no—
245C2×Dic13—44no—
246C2×Dic13—44no—
247C2×Dic13—41no—
248C2×Dic13—42no—
249C2×Dic13—41no—
250C2×Dic13—41no—
251C2×Dic13—42no—
252C2×Dic13—42no—
253C2×Dic13—41no—
254C2×Dic13—41no—
255C2×Dic13—41no—
256C2×Dic13—84no—
257C2×Dic13AGL(1,ℤ/8)12yes—
258C8×C13—24no—
259#1055—42no—
260C23×D4C2×D1311yescharacteristic
261C22×SD16C2×D1312yes—
262C2×C22⋊Q8C2×D1311yescharacteristic
263C2×C4×Q8—131no—
264C22×C4⋊C4—131no—
265C22×C22⋊C4—261no—
266C22×C22⋊C4—261no—
267C22×C42—131no—
268#450Dic1318yescomplemented
269C2×#107—132no—
270C2×#103—132no—
271C2×#102—132no—
272C2×#102—132no—
273C2×#102C2×D1312yes—
274C2×C22⋊C8—131no—
275C2×C22⋊C8C2×D1311yescharacteristic
276C2×#97—132no—
277#391—131no—
278#390—132no—
279#390—132no—
280#390—261no—
281#390—261no—
282#389—131no—
283#387—261no—
284#385—261no—
285#380—131no—
286C4×C4⋊C4—131no—
287C4×C4⋊C4—261no—
288C4×C22⋊C4—261no—
289#340—132no—
290#340—132no—
291#336—132no—
292C22×C13—22no—
293C22×C13—22no—
294C22×C13—22no—
295C22×C13—22no—
296C22×C13—22no—
297C22×C13—22no—
298C22×C13—44no—
299C22×C13—44no—
300C22×C13—44no—
301C22×C13—816no—
302C22×C13—82no—
303C22×C13—82no—
304C22×C13—82no—
305C22×C13—82no—
306C22×C13#44014yes—
307C22×C13C2×C4×D411yescharacteristic
308C22×C13#53411yescharacteristic
309C22×C13#53511yescharacteristic
310C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
311C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
312C22×C13C2×#13711yescharacteristic
313C22×C13#57511yescharacteristic
314C4×C13—22no—
315C4×C13—42no—
316C4×C13—42no—
317C4×C13#51812yes—
318C4×C13#53812yes—
319Dic13—22no—
320Dic13—41no—
321Dic13—41no—
322Dic13—41no—
323Dic13—81no—
324Dic13—81no—
325Dic13—82no—
326C25—21no—
327C22×Q8C22×D1311yescharacteristic
328C22×D4—21no—
329C22×D4—22no—
330C22×D4—42no—
331C22×D4—42no—
332C22×D4C2×Dic1314yes—
333C22×D4C22×D1312yes—
334C23×C4—131no—
335C23×C4—131no—
336C23×C4—261no—
337C23×C4C22×D1311yescharacteristic
338C2×SD16—42no—
339C2×SD16—42no—
340C2×SD16C2×Dic1318yescomplemented
341C22×C8—132no—
342C22×C8C22×D1312yes—
343C22⋊Q8—22no—
344C22⋊Q8C2×Dic1314yes—
345C4×Q8—522no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—132no—
348C2×C4⋊C4—131no—
349C2×C4⋊C4—131no—
350C2×C4⋊C4—261no—
351C2×C4⋊C4—261no—
352C2×C4⋊C4—261no—
353C2×C4⋊C4—522no—
354C2×C4⋊C4C22×D1311yescharacteristic
355C2×C22⋊C4—21no—
356C2×C22⋊C4—262no—
357C2×C22⋊C4—262no—
358C2×C22⋊C4—262no—
359C2×C22⋊C4—262no—
360C2×C22⋊C4—261no—
361C2×C22⋊C4—524no—
362C2×C22⋊C4—524no—
363C2×C42—131no—
364C2×C42—131no—
365C2×C42—131no—
366C2×C42—131no—
367C2×C42—261no—
368C2×C42—261no—
369C2×C42—522no—
370#107—522no—
371#103—522no—
372#102—522no—
373#102—522no—
374#102C2×Dic1318yescomplemented
375C22⋊C8—262no—
376C22⋊C8C2×Dic1314yes—
377#97—522no—
378#95—261no—
379#95—261no—
380#95—261no—
381#95—261no—
382#95—261no—
383C2×C13—24no—
384C2×C13—84no—
385C2×C13#232111yescharacteristic
386C2×C13C2×#44011yescharacteristic
387C2×C13C4×AGL(1,ℤ/8)11yescharacteristic
388C2×C13#319311yescharacteristic
389C2×C13#319411yescharacteristic
390C2×C13#321511yescharacteristic
391C2×C13#321711yescharacteristic
392C24—24no—
393C24—22no—
394C24—42no—
395C24—42no—
396C24—42no—
397C24—42no—
398C24—42no—
399C24—42no—
400C24D4×D1311yescharacteristic
401C2×Q8—21no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8C22×Dic1312yes—
405C2×D4—24no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—48no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—48no—
415C2×D4C22×Dic1314yes—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—131no—
421C22×C4—131no—
422C22×C4—131no—
423C22×C4—131no—
424C22×C4—262no—
425C22×C4—262no—
426C22×C4—262no—
427C22×C4—261no—
428C22×C4—261no—
429C22×C4—262no—
430C22×C4—521no—
431C22×C4—521no—
432C22×C4—521no—
433C22×C4—522no—
434C22×C4—522no—
435C22×C4—522no—
436C22×C4D4×D1311yescharacteristic
437C22×C4#612912yes—
438C22×C4C22×Dic1312yes—
439C22×C4C23×D1311yescharacteristic
440SD16—48no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—262no—
444C2×C8—262no—
445C2×C8—262no—
446C2×C8C22×Dic1314yes—
447C4⋊C4—24no—
448C4⋊C4—21no—
449C4⋊C4—262no—
450C4⋊C4—262no—
451C4⋊C4—262no—
452C4⋊C4—262no—
453C4⋊C4—522no—
454C4⋊C4—522no—
455C4⋊C4—522no—
456C4⋊C4C22×Dic1312yes—
457C22⋊C4—24no—
458C22⋊C4—524no—
459C22⋊C4—524no—
460C22⋊C4—524no—
461C22⋊C4—524no—
462C22⋊C4—522no—
463C42—262no—
464C42—262no—
465C42—521no—
466C42—521no—
467C42—522no—
468C42—522no—
469C13#2204911yescharacteristic, complemented, Sylow
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—24no—
475C23—48no—
476C23—48no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—48no—
484C23—84no—
485C23—84no—
486C23—84no—
487C23—84no—
488C23#8816812yes—
489C23#8816912yes—
490C23D4×Dic1312yes—
491C23C2×D4×D1311yescharacteristic
492Q8—24no—
493Q8—42no—
494D4—24no—
495D4—24no—
496D4—48no—
497D4—42no—
498D4—42no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—132no—
510C2×C4—261no—
511C2×C4—261no—
512C2×C4—261no—
513C2×C4—261no—
514C2×C4—261no—
515C2×C4—261no—
516C2×C4—261no—
517C2×C4—261no—
518C2×C4—261no—
519C2×C4—261no—
520C2×C4—261no—
521C2×C4—524no—
522C2×C4—524no—
523C2×C4—522no—
524C2×C4—521no—
525C2×C4—521no—
526C2×C4—521no—
527C2×C4—522no—
528C2×C4—521no—
529C2×C4—521no—
530C2×C4—522no—
531C2×C4—521no—
532C2×C4—1044no—
533C2×C4D4×Dic1312yes—
534C2×C4C2×#612911yescharacteristic
535C2×C4#8825112yes—
536C2×C4#8825112yes—
537C2×C4C23×Dic1311yescharacteristic
538C8—24no—
539C8—522no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—44no—
547C22—44no—
548C22—44no—
549C22—816no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22#1171882514yes—
555C22#1171929411yescharacteristic
556C22#1171929711yescharacteristic
557C22C2×#8816811yescharacteristic
558C22C2×#8816911yescharacteristic
559C22D13×AGL(1,ℤ/8)11yescharacteristic
560C22#1171947811yescharacteristic
561C22C2×D4×Dic1311yescharacteristic
562C4—22no—
563C4—42no—
564C4—42no—
565C4—262no—
566C4—521no—
567C4—521no—
568C4—521no—
569C4—1041no—
570C4—1041no—
571C4—1042no—
572C4#1171929812yes—
573C4#1171950312yes—
574C2—24no—
575C2—84no—
576C2#1311749711yescharacteristic
577C2C2×#1171882511yescharacteristic
578C2AGL(1,ℤ/8)×Dic1311yescharacteristic
579C2#1312563911yescharacteristic
580C2#1312565311yescharacteristic
581C2#1312583611yescharacteristic
582C2#1312585711yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC2×C13×#450C2
socleC23×C13C22×D4
prime radicalC23×D4×C13C22
cosocleC22×C4×C13C24
hypercenterC2×#450D13
hypocenterC13#22049
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#450

Complements

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

NH ≅ G/Ntype
1. C22⋊Q8×Dic13C2semidirect
3. #13115719C2semidirect
4. #13115535C2semidirect
5. #13115458C2semidirect
6. #13115193C2semidirect
7. #13115144C2semidirect
8. #13115026C2semidirect
10. #13114600C2semidirect
17. C2×Q8×Dic13C22semidirect
20. C13×#450C4semidirect
23. #11718891C22semidirect
24. C2×#88075C22semidirect
33. C2×#88058C22semidirect
34. C2×#88055C22semidirect
35. #11718646C22semidirect
38. C4⋊C4×Dic13C22semidirect
41. #11718143C22semidirect
42. #11718142C22semidirect
43. #11718121C22semidirect
58. C2×C13×SD16C2×C4semidirect
88. C13×#102C2×C4semidirect
268. #450Dic13semidirect
340. C2×SD16C2×Dic13semidirect
374. #102C2×Dic13semidirect
469. C13#22049semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×Dic132
C2×#117188822
#131157192
#131155352
#131154582
#131151932
#131151442
#131150262
#131147672
#131146002
C2×C13×#4502
#2204913