Finite Group Databasebeta

← #15872598

Subgroups — #15872598

Order 3328.

1222 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11719327C211yesmaximal, characteristic
2C2×#11718379C211yesmaximal, characteristic
3C2×#11718637C211yesmaximal, characteristic
4#13115795C212yesmaximal, complemented
5#13115636C212yesmaximal, complemented
6#13115144C211yesmaximal, characteristic
7#13115143C211yesmaximal, characteristic
8#13115141C211yesmaximal, characteristic
9#13114840C212yesmaximal, complemented
10#13114839C212yesmaximal, complemented
11C13×#1872C211yesmaximal, characteristic
12C2×C13×C22⋊Q8C2211yescharacteristic
13C22×C13×C4⋊C4C2211yescharacteristic
14C2×Q8×Dic13C2211yescharacteristic
15C2×#88198C2211yescharacteristic
16C22×C4⋊Dic13C2211yescharacteristic
17C2×#88178C2211yescharacteristic
18C22×#6101—21no—
19C22×#6101—21no—
20C22×C4×Dic13C2211yescharacteristic
21#11719327—24no—
22#11719327C418yescomplemented
23C2×#88143—21no—
24C2×#88142C2211yescharacteristic
25C2×#88141—21no—
26C2×#88141C2211yescharacteristic
27C2×#88132—21no—
28C2×#88131—21no—
29C2×C13×#103C2212yes—
30C2×C13×C22⋊C8C2211yescharacteristic
31C2×#88075C2212yes—
32C2×C22⋊#1055C2211yescharacteristic
33C2×#88056C2212yes—
34C2×#88054C2212yes—
35C2×C8×Dic13C2212yes—
36#11718637—21no—
37#11718637—21no—
38#11718637C2212yes—
39#11718637C2212yes—
40C2×#88050C2212yes—
41#11718379—24no—
42C13×#340C2212yes—
43#11718143C2212yes—
44#11718121C2212yes—
45#11718121C2212yes—
46C22×Q8×C13C2311yescharacteristic
47C23×C4×C13—21no—
48C23×C4×C13C2311yescharacteristic
49C23×Dic13—21no—
50C23×Dic13D411yescharacteristic
51C23×Dic13D411yescharacteristic
52C22×Dic26—21no—
53C22×Dic26C2311yescharacteristic
54C22×C8×C13C2312yes—
55C13×C22⋊Q8—22no—
56C13×C22⋊Q8C2×C414yes—
57C2×C13×C4⋊C4—21no—
58C2×C13×C4⋊C4—21no—
59C2×C13×C4⋊C4—42no—
60C2×C13×C4⋊C4C2312yes—
61C2×C13×C4⋊C4C2311yescharacteristic
62C2×C13×C22⋊C4—21no—
63C2×C22⋊Dic13—21no—
64Q8×Dic13—22no—
65Q8×Dic13C2×C414yescomplemented
66#88198—22no—
67#88198C2×C414yescomplemented
68C2×C4⋊Dic13—21no—
69C2×C4⋊Dic13—21no—
70C2×C4⋊Dic13—42no—
71C2×C4⋊Dic13C2312yes—
72C2×C4⋊Dic13C2311yescharacteristic
73#88178—22no—
74#88178C2×C414yes—
75C2×#6101—21no—
76C2×#6101—21no—
77C2×#6101—21no—
78C2×#6101—21no—
79C2×#6101—21no—
80C2×#6101—22no—
81C2×#6101—22no—
82C2×#6101—22no—
83C2×#6101—22no—
84C2×#6101—44no—
85C2×#6101—44no—
86C2×C4×Dic13—21no—
87C2×C4×Dic13—21no—
88C2×C4×Dic13C2×C414yes—
89C2×C4×Dic13D411yescharacteristic
90C2×C4×Dic13D411yescharacteristic
91C2×C4×Dic13D414yes—
92C2×C4×Dic13C2311yescharacteristic
93C2×C4×Dic13C2311yescharacteristic
94C22×#1055C2312yes—
95#88143—28no—
96#88142—22no—
97#88142C2×C414yescomplemented
98#88141—28no—
99#88141—22no—
100#88141C2×C414yescomplemented
101#88132—28no—
102#88131—28no—
103C13×#103—42no—
104C13×C22⋊C8—22no—
105#88075—42no—
106C22⋊#1055—22no—
107#88056—42no—
108#88054—24no—
109C8×Dic13—24no—
110#88050—42no—
111#21860—131nomaximal, Sylow
112C24×C13C2×D411yescharacteristic
113C2×Q8×C13—21no—
114C2×Q8×C13—22no—
115C2×Q8×C13—22no—
116C2×Q8×C13C22×C412yes—
117C22×C4×C13—22no—
118C22×C4×C13—22no—
119C22×C4×C13—21no—
120C22×C4×C13—21no—
121C22×C4×C13—22no—
122C22×C4×C13—42no—
123C22×C4×C13—42no—
124C22×C4×C13—42no—
125C22×C4×C13C22×C412yes—
126C22×C4×C13C2×D411yescharacteristic
127C22×C4×C13C2×D412yes—
128C22×C4×C13C2411yescharacteristic
129C22×Dic13—22no—
130C22×Dic13—22no—
131C22×Dic13—21no—
132C22×Dic13—21no—
133C22×Dic13—22no—
134C22×Dic13—42no—
135C22×Dic13—42no—
136C22×Dic13—42no—
137C22×Dic13C22⋊C414yes—
138C22×Dic13C22⋊C414yes—
139C22×Dic13SD1614yes—
140C22×Dic13Q1614yes—
141C22×Dic13C2×D411yescharacteristic
142C22×Dic13C2×D411yescharacteristic
143C22×Dic13C2×D411yescharacteristic
144C22×Dic13C2×D411yescharacteristic
145C2×Dic26—22no—
146C2×Dic26—22no—
147C2×Dic26—21no—
148C2×Dic26—24no—
149C2×Dic26—42no—
150C2×Dic26—42no—
151C2×Dic26C22×C412yes—
152C2×C8×C13—22no—
153C2×C8×C13—22no—
154C2×C8×C13—22no—
155C13×C4⋊C4—24no—
156C13×C4⋊C4—21no—
157C13×C4⋊C4—22no—
158C13×C4⋊C4—22no—
159C13×C4⋊C4—42no—
160C13×C4⋊C4C22×C412yes—
161C13×C22⋊C4—24no—
162C22⋊Dic13—24no—
163C4⋊Dic13—21no—
164C4⋊Dic13—24no—
165C4⋊Dic13—22no—
166C4⋊Dic13—22no—
167C4⋊Dic13—42no—
168C4⋊Dic13C22×C412yes—
169#6101—24no—
170#6101—24no—
171#6101—24no—
172#6101—24no—
173#6101—24no—
174#6101—44no—
175#6101—44no—
176#6101—44no—
177#6101—44no—
178C4×Dic13—24no—
179C4×Dic13—24no—
180C4×Dic13C22⋊C414yes—
181C4×Dic13C22⋊C414yes—
182C4×Dic13C22×C412yes—
183C4×Dic13C22×C412yes—
184C4×Dic13C2×D412yes—
185C4×Dic13C2×D412yes—
186C2×#1055—22no—
187C2×#1055—22no—
188C2×#1055—22no—
189C2×#533—131no—
190C2×#431—131no—
191C2×#384—131no—
192#1990—132no—
193#1956—132no—
194#1872—131no—
195#1872—131no—
196#1872—131no—
197#1872D1311yescharacteristic
198#1803—132no—
199#1803—132no—
200C23×C13—21no—
201C23×C13—21no—
202C23×C13—21no—
203C23×C13—21no—
204C23×C13C2×C22⋊C412yes—
205C23×C13C2×SD1612yes—
206C23×C13C2×Q1612yes—
207C23×C13C22×D411yescharacteristic
208Q8×C13—24no—
209Q8×C13—42no—
210C2×C4×C13—21no—
211C2×C4×C13—21no—
212C2×C4×C13—24no—
213C2×C4×C13—24no—
214C2×C4×C13—22no—
215C2×C4×C13—22no—
216C2×C4×C13—41no—
217C2×C4×C13—41no—
218C2×C4×C13—41no—
219C2×C4×C13—41no—
220C2×C4×C13—42no—
221C2×C4×C13—42no—
222C2×C4×C13—42no—
223C2×C4×C13—84no—
224C2×C4×C13C2×C22⋊C412yes—
225C2×C4×C13C23×C411yescharacteristic
226C2×C4×C13C22×D411yescharacteristic
227C2×C4×C13H(5,𝔽2)12yes—
228C2×C4×C132−1+412yes—
229C2×Dic13—21no—
230C2×Dic13—21no—
231C2×Dic13—28no—
232C2×Dic13—22no—
233C2×Dic13—22no—
234C2×Dic13—41no—
235C2×Dic13—41no—
236C2×Dic13—41no—
237C2×Dic13—41no—
238C2×Dic13—42no—
239C2×Dic13—42no—
240C2×Dic13—42no—
241C2×Dic13—84no—
242C2×Dic13#103116yescomplemented
243C2×Dic13C2×C22⋊C412yes—
244C2×Dic13C2×C22⋊C412yes—
245C2×Dic13C2×C22⋊C412yes—
246C2×Dic13C2×C22⋊C412yes—
247C2×Dic13C2×SD1612yes—
248C2×Dic13C2×SD1612yes—
249C2×Dic13C2×Q1612yes—
250C2×Dic13C2×Q1612yes—
251C2×Dic13AGL(1,ℤ/8)12yes—
252C2×Dic13#13712yes—
253Dic26—24no—
254Dic26—42no—
255Dic26—48no—
256C8×C13—42no—
257#1055—42no—
258#533—138no—
259#533—264no—
260C2×C4⋊Q8—131no—
261C2×C4⋊Q8—131no—
262C2×#125—261no—
263C2×C22⋊Q8—131no—
264C2×C22⋊Q8—261no—
265C2×C22⋊Q8C2×D1311yescharacteristic
266C2×C4×Q8—131no—
267C2×C4×Q8—131no—
268C2×C4×Q8—261no—
269C2×#117—261no—
270C22×C4⋊C4—131no—
271C22×C4⋊C4—261no—
272C22×C4⋊C4—261no—
273C22×C4⋊C4C2×D1311yescharacteristic
274C22×C42—131no—
275#431—264no—
276C2×C4⋊C8—132no—
277C2×#103—132no—
278C2×#103—132no—
279C2×#103—132no—
280C2×#103C2×D1312yes—
281C2×C22⋊C8—131no—
282C2×C22⋊C8C2×D1311yescharacteristic
283C2×C4×C8—132no—
284#384—132no—
285#384—132no—
286#384—261no—
287#384—261no—
288#340—132no—
289#340—132no—
290#340—132no—
291#340C2×D1312yes—
292C22×C13—22no—
293C22×C13—22no—
294C22×C13—22no—
295C22×C13—22no—
296C22×C13—22no—
297C22×C13—22no—
298C22×C13C2×#10314yes—
299C22×C13C22×C22⋊C411yescharacteristic
300C22×C13#53511yescharacteristic
301C22×C13#53611yescharacteristic
302C22×C13C22×SD1611yescharacteristic
303C22×C13C22×Q1611yescharacteristic
304C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
305C22×C13C2×#13711yescharacteristic
306C4×C13—22no—
307C4×C13—42no—
308C4×C13—42no—
309C4×C13—82no—
310C4×C13#52012yes—
311C4×C13#53712yes—
312Dic13—22no—
313Dic13—42no—
314Dic13—42no—
315Dic13—82no—
316Dic13C2×#10314yes—
317Dic13C2×#10314yes—
318Dic13#41714yes—
319C22×Q8—131no—
320C22×Q8—261no—
321C22×Q8C22×D1311yescharacteristic
322C23×C4—21no—
323C23×C4—131no—
324C23×C4—131no—
325C23×C4—261no—
326C23×C4C22×D1311yescharacteristic
327C22×C8—132no—
328C22×C8C22×D1312yes—
329C4⋊Q8—134no—
330C4⋊Q8—134no—
331C4⋊Q8—262no—
332C4⋊Q8—262no—
333#125—268no—
334C22⋊Q8—22no—
335C22⋊Q8—134no—
336C22⋊Q8—268no—
337C22⋊Q8—262no—
338C22⋊Q8C4×D1314yes—
339C4×Q8—134no—
340C4×Q8—134no—
341C4×Q8—262no—
342C4×Q8—268no—
343C4×Q8—262no—
344#117—268no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—42no—
348C2×C4⋊C4—131no—
349C2×C4⋊C4—132no—
350C2×C4⋊C4—261no—
351C2×C4⋊C4—261no—
352C2×C4⋊C4—261no—
353C2×C4⋊C4—261no—
354C2×C4⋊C4—261no—
355C2×C4⋊C4—261no—
356C2×C4⋊C4—262no—
357C2×C4⋊C4—262no—
358C2×C4⋊C4—262no—
359C2×C4⋊C4—262no—
360C2×C4⋊C4—261no—
361C2×C4⋊C4—522no—
362C2×C4⋊C4—524no—
363C2×C4⋊C4—524no—
364C2×C4⋊C4C22×D1312yes—
365C2×C4⋊C4C22×D1311yescharacteristic
366C2×C22⋊C4—21no—
367C2×C22⋊C4—261no—
368C2×C42—134no—
369C2×C42—131no—
370C2×C42—131no—
371C2×C42—134no—
372C2×C42—131no—
373C2×C42—131no—
374C2×C42—261no—
375C2×C42—261no—
376C4⋊C8—264no—
377#103—42no—
378#103—522no—
379#103—522no—
380#103—522no—
381C22⋊C8—22no—
382C22⋊C8—262no—
383C4×C8—264no—
384C2×C13—24no—
385C2×C13#231111yescharacteristic
386C2×C13C22×#10311yescharacteristic
387C2×C13C2×#41711yescharacteristic
388C2×C13#320811yescharacteristic
389C2×C13#320911yescharacteristic
390C2×C13#322911yescharacteristic
391C2×C13#323011yescharacteristic
392C24D4×D1311yescharacteristic
393C2×Q8—21no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—132no—
397C2×Q8—264no—
398C2×Q8—261no—
399C2×Q8—262no—
400C2×Q8—262no—
401C2×Q8—522no—
402C2×Q8—522no—
403C2×Q8C2×C4×D1312yes—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—134no—
413C22×C4—134no—
414C22×C4—134no—
415C22×C4—131no—
416C22×C4—131no—
417C22×C4—131no—
418C22×C4—134no—
419C22×C4—131no—
420C22×C4—262no—
421C22×C4—262no—
422C22×C4—261no—
423C22×C4—261no—
424C22×C4—262no—
425C22×C4—522no—
426C22×C4—522no—
427C22×C4—522no—
428C22×C4C2×C4×D1312yes—
429C22×C4D4×D1311yescharacteristic
430C22×C4D4×D1312yes—
431C22×C4C23×D1311yescharacteristic
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—262no—
436C2×C8—262no—
437C2×C8—262no—
438C4⋊C4—24no—
439C4⋊C4—21no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—42no—
443C4⋊C4—132no—
444C4⋊C4—264no—
445C4⋊C4—264no—
446C4⋊C4—264no—
447C4⋊C4—264no—
448C4⋊C4—264no—
449C4⋊C4—264no—
450C4⋊C4—261no—
451C4⋊C4—262no—
452C4⋊C4—262no—
453C4⋊C4—522no—
454C4⋊C4—524no—
455C4⋊C4—524no—
456C4⋊C4—524no—
457C4⋊C4—524no—
458C4⋊C4C2×C4×D1312yes—
459C22⋊C4—24no—
460C22⋊C4—264no—
461C42—134no—
462C42—134no—
463C42—132no—
464C42—132no—
465C42—132no—
466C42—132no—
467C42—264no—
468C42—264no—
469C13#2186011yescharacteristic, complemented, Sylow
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23C22⋊C4×D1312yes—
475C23SD16×D1312yes—
476C23Q16×D1312yes—
477C23C2×D4×D1311yescharacteristic
478Q8—24no—
479Q8—42no—
480Q8—264no—
481Q8—528no—
482Q8—522no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—84no—
497C2×C4—132no—
498C2×C4—132no—
499C2×C4—132no—
500C2×C4—132no—
501C2×C4—132no—
502C2×C4—132no—
503C2×C4—132no—
504C2×C4—132no—
505C2×C4—132no—
506C2×C4—1316no—
507C2×C4—132no—
508C2×C4—268no—
509C2×C4—261no—
510C2×C4—261no—
511C2×C4—262no—
512C2×C4—262no—
513C2×C4—521no—
514C2×C4—521no—
515C2×C4—521no—
516C2×C4—521no—
517C2×C4—522no—
518C2×C4—522no—
519C2×C4—522no—
520C2×C4—1044no—
521C2×C4C22⋊C4×D1312yes—
522C2×C4C22×C4×D1311yescharacteristic
523C2×C4C2×D4×D1311yescharacteristic
524C2×C4#8825112yes—
525C2×C4#8825712yes—
526C8—42no—
527C8—522no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22D13×#10314yes—
535C22C2×C22⋊C4×D1311yescharacteristic
536C22#1171929711yescharacteristic
537C22#1171935311yescharacteristic
538C22C2×SD16×D1311yescharacteristic
539C22C2×Q16×D1311yescharacteristic
540C22D13×AGL(1,ℤ/8)11yescharacteristic
541C22D13×#13711yescharacteristic
542C4—22no—
543C4—42no—
544C4—42no—
545C4—82no—
546C4—134no—
547C4—134no—
548C4—134no—
549C4—262no—
550C4—522no—
551C4—522no—
552C4—1042no—
553C4#1171925112yes—
554C4#1171930212yes—
555C2—24no—
556C2#1311742311yescharacteristic
557C2C2×D13×#10311yescharacteristic
558C2D13×#41711yescharacteristic
559C2#1312577511yescharacteristic
560C2#1312578211yescharacteristic
561C2#1312597811yescharacteristic
562C2#1312598511yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#1872C2
socleC23×C13C22×D4
prime radicalC24×C13C2×D4
cosocleC22×C4×C13C24
hypercenter#1872D13
hypocenterC13#21860
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
4. #13115795C2semidirect
5. #13115636C2semidirect
9. #13114840C2semidirect
10. #13114839C2semidirect
22. #11719327C4semidirect
65. Q8×Dic13C2×C4semidirect
67. #88198C2×C4semidirect
97. #88142C2×C4semidirect
100. #88141C2×C4semidirect
242. C2×Dic13#103semidirect
469. C13#21860semidirect

Maximal subgroups

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

subgroupindex
C2×#117193272
C2×#117183792
C2×#117186372
#131157952
#131156362
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