Finite Group Databasebeta

← #15874157

Subgroups — #15874157

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
1C2×#11718707C211yesmaximal, characteristic
2C2×#11718600C211yesmaximal, characteristic
3#13115720C212yesmaximal, complemented
4#13115532C212yesmaximal, complemented
5#13115456C212yesmaximal, complemented
6#13115191C212yesmaximal, complemented
7#13115144C211yesmaximal, characteristic, complemented
8#13115023C211yesmaximal, characteristic, complemented
9#13114766C211yesmaximal, characteristic
10#13114598C211yesmaximal, characteristic, complemented
11C4×C13×C22⋊Q8C211yesmaximal, characteristic, complemented
12C23×D52C2211yescharacteristic
13C2×C13×C22⋊Q8C2211yescharacteristic
14C2×C4×Q8×C13C2211yescharacteristic, complemented
15C22×C42×C13C2211yescharacteristic
16C22×#6106C2212yes—
17C22×#6103—21no—
18C22×#6103—21no—
19C22×C4⋊Dic13C2211yescharacteristic
20C13×#391C2211yescharacteristic, complemented
21C13×#389C2211yescharacteristic, complemented
22C13×#387—21no—
23C13×#385—21no—
24C13×#380C2211yescharacteristic
25C4×C13×C4⋊C4—21no—
26C4×C13×C4⋊C4C2211yescharacteristic, complemented
27C4×C13×C22⋊C4—21no—
28C2×#88075C2212yescomplemented
29C2×C22⋊#1055C2211yescharacteristic
30C2×C22⋊#1055C2211yescharacteristic
31#11718707C418yescomplemented
32C2×#88049C2212yes—
33C2×#88049C2212yes—
34C2×#88049C2212yes—
35C2×#88047C2212yescomplemented
36#11718600—21no—
37#11718600—21no—
38#11718600C2212yes—
39#11718600C2212yes—
40C2×#88043C2212yescomplemented
41#11718121C2212yescomplemented
42#11718121C2212yescomplemented
43#11718116C2212yescomplemented
44C24×D13—21no—
45C22×Q8×C13C2311yescharacteristic
46C23×C4×C13D411yescharacteristic
47C23×C4×C13D411yescharacteristic
48C23×C4×C13C2311yescharacteristic
49C23×Dic13—21no—
50C22×D52—21no—
51C22×D52—22no—
52C22×D52—42no—
53C22×D52—42no—
54C22×D52C2×C414yes—
55C22×D52C2312yes—
56C13×C22⋊Q8—22no—
57C13×C22⋊Q8C2×C414yes—
58C4×Q8×C13—42no—
59C2×C13×C4⋊C4—21no—
60C2×C13×C4⋊C4—21no—
61C2×C13×C4⋊C4—21no—
62C2×C13×C4⋊C4C2311yescharacteristic
63C2×C13×C4⋊C4C2311yescharacteristic
64C2×C13×C4⋊C4C2311yescharacteristic
65C2×C13×C22⋊C4—21no—
66C2×C13×C22⋊C4—21no—
67C2×C42×C13—21no—
68C2×C42×C13—21no—
69C2×C42×C13—42no—
70C2×C42×C13D411yescharacteristic
71C2×C42×C13D411yescharacteristic
72C2×C42×C13C2311yescharacteristic
73C2×C42×C13C2311yescharacteristic
74C2×#6106—42no—
75C2×#6106—42no—
76C2×#6106C2×C418yescomplemented
77C2×#6103—22no—
78C2×#6103—22no—
79C2×#6103—22no—
80C2×#6103—22no—
81C2×#6103—44no—
82C2×#6103—44no—
83C2×C4⋊Dic13—21no—
84C2×C4⋊Dic13—42no—
85C2×C4⋊Dic13C2312yes—
86C22×#1055C2312yes—
87C22×#1055C2312yes—
88C13×#95—21no—
89C13×#95—21no—
90C13×#95—21no—
91C13×#95—21no—
92C13×#95—21no—
93#88075—42no—
94C22⋊#1055—22no—
95C22⋊#1055C2×C414yes—
96#88049—42no—
97#88049—42no—
98#88049C2×C418yescomplemented
99#88047—42no—
100#88043—42no—
101#22049—131nomaximal, Sylow
102C24×C13C2×D411yescharacteristic
103C23×D13—24no—
104C23×D13—22no—
105C23×D13—42no—
106C23×D13—42no—
107C23×D13—42no—
108C23×D13—42no—
109C23×D13—42no—
110C23×D13—42no—
111C2×Q8×C13—21no—
112C2×Q8×C13—22no—
113C2×Q8×C13—22no—
114C2×Q8×C13C22×C412yes—
115C22×C4×C13—22no—
116C22×C4×C13—22no—
117C22×C4×C13—22no—
118C22×C4×C13—22no—
119C22×C4×C13—21no—
120C22×C4×C13—21no—
121C22×C4×C13—22no—
122C22×C4×C13—21no—
123C22×C4×C13—21no—
124C22×C4×C13—41no—
125C22×C4×C13—41no—
126C22×C4×C13—41no—
127C22×C4×C13C22×C412yes—
128C22×C4×C13C2×D411yescharacteristic
129C22×C4×C13C2×D411yescharacteristic
130C22×C4×C13C2×D411yescharacteristic
131C22×C4×C13C2×D411yescharacteristic
132C22×C4×C13C2×D411yescharacteristic
133C22×C4×C13#4112yes—
134C22×C4×C13C2411yescharacteristic
135C22×Dic13—22no—
136C22×Dic13—42no—
137C22×Dic13—42no—
138C22×Dic13—42no—
139C2×D52—22no—
140C2×D52—22no—
141C2×D52—22no—
142C2×D52—22no—
143C2×D52—24no—
144C2×D52—22no—
145C2×D52—48no—
146C2×D52—42no—
147C2×D52—42no—
148C2×D52—48no—
149C2×D52C22×C414yes—
150C13×C4⋊C4—24no—
151C13×C4⋊C4—21no—
152C13×C4⋊C4—22no—
153C13×C4⋊C4—22no—
154C13×C4⋊C4—42no—
155C13×C4⋊C4—42no—
156C13×C4⋊C4C22×C412yes—
157C13×C22⋊C4—24no—
158C13×C22⋊C4—42no—
159C42×C13—22no—
160C42×C13—22no—
161C42×C13—41no—
162C42×C13—41no—
163C42×C13—42no—
164C42×C13—42no—
165#6106—48no—
166#6103—44no—
167#6103—44no—
168#6103—44no—
169#6103—44no—
170C4⋊Dic13—22no—
171C4⋊Dic13—22no—
172C4⋊Dic13—42no—
173C2×#1055—22no—
174C2×#1055—22no—
175C2×#1055—22no—
176C2×#1055—22no—
177C2×#1055—22no—
178C2×#1055C22×C414yes—
179C2×#450—131no—
180C4×C22⋊Q8D1311yescharacteristic, complemented
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×C13C4×D412yes—
195C23×C13AGL(1,ℤ/8)12yes—
196C23×C13#13712yes—
197C23×C13C22×D411yescharacteristic
198C22×D13—24no—
199C22×D13—48no—
200C22×D13—48no—
201C22×D13—42no—
202C22×D13—42no—
203C22×D13—42no—
204C22×D13—42no—
205C22×D13—42no—
206C22×D13—42no—
207C22×D13—48no—
208C22×D13—84no—
209C22×D13—84no—
210C22×D13—84no—
211C22×D13—84no—
212Q8×C13—24no—
213Q8×C13—42no—
214C2×C4×C13—21no—
215C2×C4×C13—21no—
216C2×C4×C13—21no—
217C2×C4×C13—21no—
218C2×C4×C13—24no—
219C2×C4×C13—24no—
220C2×C4×C13—22no—
221C2×C4×C13—22no—
222C2×C4×C13—21no—
223C2×C4×C13—21no—
224C2×C4×C13—21no—
225C2×C4×C13—21no—
226C2×C4×C13—21no—
227C2×C4×C13—21no—
228C2×C4×C13—21no—
229C2×C4×C13—21no—
230C2×C4×C13—21no—
231C2×C4×C13—44no—
232C2×C4×C13—44no—
233C2×C4×C13—41no—
234C2×C4×C13—41no—
235C2×C4×C13—41no—
236C2×C4×C13—41no—
237C2×C4×C13—41no—
238C2×C4×C13—41no—
239C2×C4×C13—41no—
240C2×C4×C13—41no—
241C2×C4×C13—41no—
242C2×C4×C13—41no—
243C2×C4×C13C4×D412yes—
244C2×C4×C13AGL(1,ℤ/8)12yes—
245C2×C4×C13C23×C411yescharacteristic
246C2×C4×C13C2×#4111yescharacteristic
247C2×C4×C13H(5,𝔽2)12yes—
248C2×C4×C13H(5,𝔽2)12yes—
249C2×Dic13—42no—
250C2×Dic13—42no—
251C2×Dic13—42no—
252C2×Dic13—84no—
253D52—24no—
254D52—24no—
255D52—48no—
256D52—42no—
257D52—42no—
258#1055—24no—
259#1055—42no—
260C23×D4—131no—
261C22×SD16—132no—
262C2×C22⋊Q8C2×D1311yescharacteristic
263C2×C4×Q8C2×D1311yescharacteristic, complemented
264C22×C4⋊C4—131no—
265C22×C22⋊C4—261no—
266C22×C22⋊C4—261no—
267C22×C42C2×D1311yescharacteristic
268#450—138no—
269C2×#107—132no—
270C2×#103—132no—
271C2×#102—132no—
272C2×#102—132no—
273C2×#102—132no—
274C2×C22⋊C8—131no—
275C2×C22⋊C8—131no—
276C2×#97—132no—
277#391C2×D1311yescharacteristic, complemented
278#390—132no—
279#390—132no—
280#390—261no—
281#390—261no—
282#389C2×D1311yescharacteristic, complemented
283#387—21no—
284#385—21no—
285#380C2×D1311yescharacteristic
286C4×C4⋊C4—21no—
287C4×C4⋊C4C2×D1311yescharacteristic, complemented
288C4×C22⋊C4—21no—
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#44014yes—
299C22×C13C2×C4×D411yescharacteristic
300C22×C13#53411yescharacteristic
301C22×C13#53511yescharacteristic
302C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
303C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
304C22×C13C2×#13711yescharacteristic
305C22×C13#57511yescharacteristic
306C2×D13—44no—
307C2×D13—44no—
308C2×D13—44no—
309C2×D13—816no—
310C2×D13—82no—
311C2×D13—82no—
312C2×D13—82no—
313C2×D13—82no—
314C4×C13—22no—
315C4×C13—22no—
316C4×C13—42no—
317C4×C13—41no—
318C4×C13—41no—
319C4×C13—41no—
320C4×C13—42no—
321C4×C13—81no—
322C4×C13—81no—
323C4×C13#51812yes—
324C4×C13#53812yes—
325Dic13—82no—
326C25—261no—
327C22×Q8C22×D1311yescharacteristic
328C22×D4—134no—
329C22×D4—132no—
330C22×D4—262no—
331C22×D4—261no—
332C22×D4—522no—
333C22×D4—522no—
334C23×C4—261no—
335C23×C4C22⋊D1311yescharacteristic
336C23×C4C22⋊D1311yescharacteristic
337C23×C4C22×D1311yescharacteristic
338C2×SD16—138no—
339C2×SD16—522no—
340C2×SD16—522no—
341C22×C8—132no—
342C22×C8—132no—
343C22⋊Q8—22no—
344C22⋊Q8C4×D1314yes—
345C4×Q8—42no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—132no—
350C2×C4⋊C4—261no—
351C2×C4⋊C4—522no—
352C2×C4⋊C4C22×D1311yescharacteristic
353C2×C4⋊C4C22×D1311yescharacteristic
354C2×C4⋊C4C22×D1311yescharacteristic
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—262no—
358C2×C22⋊C4—262no—
359C2×C22⋊C4—262no—
360C2×C22⋊C4—262no—
361C2×C22⋊C4—524no—
362C2×C22⋊C4—524no—
363C2×C42—21no—
364C2×C42—21no—
365C2×C42—42no—
366C2×C42C22⋊D1311yescharacteristic
367C2×C42C22⋊D1311yescharacteristic
368C2×C42C22×D1311yescharacteristic
369C2×C42C22×D1311yescharacteristic
370#107—522no—
371#103—522no—
372#102—138no—
373#102—522no—
374#102—522no—
375C22⋊C8—134no—
376C22⋊C8—262no—
377#97—522no—
378#95—21no—
379#95—21no—
380#95—21no—
381#95—21no—
382#95—21no—
383C2×C13—24no—
384C2×C13#232111yescharacteristic
385C2×C13C2×#44011yescharacteristic
386C2×C13C4×AGL(1,ℤ/8)11yescharacteristic
387C2×C13#319311yescharacteristic
388C2×C13#319411yescharacteristic
389C2×C13#321511yescharacteristic
390C2×C13#321711yescharacteristic
391D13—84no—
392C24—264no—
393C24—262no—
394C24—522no—
395C24—522no—
396C24—522no—
397C24—522no—
398C24—522no—
399C24—522no—
400C24C2×C22⋊D1311yescharacteristic
401C2×Q8—21no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8C2×C4×D1312yes—
405C2×D4—134no—
406C2×D4—264no—
407C2×D4—262no—
408C2×D4—262no—
409C2×D4—262no—
410C2×D4—262no—
411C2×D4—262no—
412C2×D4—528no—
413C2×D4—522no—
414C2×D4—522no—
415C2×D4—528no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—262no—
429C22×C4—522no—
430C22×C4—522no—
431C22×C4—522no—
432C22×C4C2×C4×D1312yes—
433C22×C4#612712yes—
434C22×C4C2×C22⋊D1311yescharacteristic
435C22×C4C2×C22⋊D1311yescharacteristic
436C22×C4C2×C22⋊D1311yescharacteristic
437C22×C4C2×C22⋊D1311yescharacteristic
438C22×C4C2×C22⋊D1311yescharacteristic
439C22×C4C23×D1311yescharacteristic
440SD16—528no—
441C2×C8—134no—
442C2×C8—262no—
443C2×C8—262no—
444C2×C8—262no—
445C2×C8—262no—
446C2×C8—262no—
447C4⋊C4—24no—
448C4⋊C4—21no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—262no—
454C4⋊C4—262no—
455C4⋊C4—522no—
456C4⋊C4C2×C4×D1312yes—
457C22⋊C4—24no—
458C22⋊C4—42no—
459C22⋊C4—524no—
460C22⋊C4—524no—
461C22⋊C4—524no—
462C22⋊C4—524no—
463C42—22no—
464C42—22no—
465C42—41no—
466C42—41no—
467C42—42no—
468C42—42no—
469C13#2204911yescharacteristic, complemented, Sylow
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—264no—
475C23—528no—
476C23—528no—
477C23—528no—
478C23—522no—
479C23—522no—
480C23—522no—
481C23—522no—
482C23—522no—
483C23—522no—
484C23—1044no—
485C23—1044no—
486C23—1044no—
487C23—1044no—
488C23C4×C22⋊D1312yes—
489C23#8819612yes—
490C23#8820312yes—
491C23C22×C22⋊D1311yescharacteristic
492Q8—24no—
493Q8—42no—
494D4—264no—
495D4—264no—
496D4—528no—
497D4—522no—
498D4—522no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—522no—
529C2×C4—522no—
530C2×C4—522no—
531C2×C4—1044no—
532C2×C4C4×C22⋊D1312yes—
533C2×C4#8820312yes—
534C2×C4C22×C4×D1311yescharacteristic
535C2×C4C2×#612711yescharacteristic
536C2×C4#8825112yes—
537C2×C4#8825612yes—
538C8—264no—
539C8—522no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—524no—
547C22—524no—
548C22—524no—
549C22—10416no—
550C22—1042no—
551C22—1042no—
552C22—1042no—
553C22—1042no—
554C22#1171868714yes—
555C22C2×C4×C22⋊D1311yescharacteristic
556C22#1171950911yescharacteristic
557C22C2×#8819611yescharacteristic
558C22C2×#8820311yescharacteristic
559C22C2×#8820311yescharacteristic
560C22#1171952711yescharacteristic
561C22#1171953511yescharacteristic
562C4—22no—
563C4—22no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—42no—
568C4—42no—
569C4—81no—
570C4—81no—
571C4—1042no—
572C4#1171925012yes—
573C4#1171926512yes—
574C2—24no—
575C2—1044no—
576C2#1311749011yescharacteristic
577C2C2×#1171868711yescharacteristic
578C2C4×#8820311yescharacteristic
579C2#1312563811yescharacteristic
580C2#1312564611yescharacteristic
581C2#1312583511yescharacteristic
582C2#1312585111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC4×C13×C22⋊Q8C2
socleC23×C13C22×D4
prime radicalC23×D52C22
cosocleC22×C4×C13C24
hypercenterC4×C22⋊Q8D13
hypocenterC13#22049
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #13115720C2semidirect
4. #13115532C2semidirect
5. #13115456C2semidirect
6. #13115191C2semidirect
7. #13115144C2semidirect
8. #13115023C2semidirect
10. #13114598C2semidirect
11. C4×C13×C22⋊Q8C2semidirect
14. C2×C4×Q8×C13C22semidirect
20. C13×#391C22semidirect
21. C13×#389C22semidirect
26. C4×C13×C4⋊C4C22semidirect
28. C2×#88075C22semidirect
31. #11718707C4semidirect
35. C2×#88047C22semidirect
40. C2×#88043C22semidirect
41. #11718121C22semidirect
42. #11718121C22semidirect
43. #11718116C22semidirect
76. C2×#6106C2×C4semidirect
98. #88049C2×C4semidirect
180. C4×C22⋊Q8D13semidirect
263. C2×C4×Q8C2×D13semidirect
277. #391C2×D13semidirect
282. #389C2×D13semidirect
287. C4×C4⋊C4C2×D13semidirect
469. C13#22049semidirect

Maximal subgroups

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

subgroupindex
C2×#117187072
C2×#117186002
#131157202
#131155322
#131154562
#131151912
#131151442
#131150232
#131147662
#131145982
C4×C13×C22⋊Q82
#2204913