Finite Group Databasebeta

← #15864281

Subgroups — #15864281

Order 3328.

2234 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×Dic13C211yesmaximal, characteristic, complemented
2C2×#11718893C211yesmaximal, characteristic, complemented
3C2×#11718876C211yesmaximal, characteristic, complemented
4#13117327C211yesmaximal, characteristic, complemented
5#13115186C214yesmaximal, complemented
6#13115172C214yesmaximal, complemented
7#13114556C211yesmaximal, characteristic, complemented
8#13114529C211yesmaximal, characteristic, complemented
9C22×C13×AGL(1,ℤ/8)C211yesmaximal, characteristic
10C22×C13×#41C2211yescharacteristic
11C23×D4×C13C2211yescharacteristic
12C24×Dic13—21no—
13C2×C13×AGL(1,ℤ/8)—24no—
14C2×C13×AGL(1,ℤ/8)—24no—
15C2×C13×AGL(1,ℤ/8)C418yescomplemented
16C22×C13×SD16C2212yes—
17C22×C13×D8C2212yes—
18C22×C13×M16C2211yescharacteristic
19C22×C22⋊Dic13—21no—
20C2×D4×Dic13—24no—
21C2×D4×Dic13—24no—
22C2×D4×Dic13—22no—
23C2×D4×Dic13C2214yes—
24C2×#88180C2211yescharacteristic
25C22×C4⋊Dic13C2211yescharacteristic
26C22×C4×Dic13C2211yescharacteristic
27C22×#6099C2211yescharacteristic
28#11718893—24no—
29#11718891C2212yescomplemented
30C2×#88075C2212yescomplemented
31#11718882C2212yescomplemented
32#11718881—21no—
33#11718881—21no—
34#11718876—24no—
35C2×#88072C2214yes—
36C2×#88072C2212yescomplemented
37C4×C22⋊Dic13—22no—
38#11718143C2214yescomplemented
39#11718142C2214yescomplemented
40C25×C13—21no—
41C2×C13×#41—22no—
42C2×C13×#41—22no—
43C2×C13×#41—22no—
44C2×C13×#41—22no—
45C2×C13×#41—22no—
46C2×C13×#41C2×C414yescomplemented
47C22×Q8×C13C2311yescharacteristic
48C22×D4×C13—21no—
49C22×D4×C13—21no—
50C22×D4×C13—22no—
51C22×D4×C13—22no—
52C22×D4×C13—22no—
53C22×D4×C13C2×C414yes—
54C22×D4×C13D414yes—
55C22×D4×C13D414yes—
56C22×D4×C13C2311yescharacteristic
57C22×D4×C13C2312yes—
58C23×C4×C13—21no—
59C23×C4×C13C2311yescharacteristic
60C23×Dic13—28no—
61C23×Dic13—21no—
62C23×Dic13—24no—
63C23×Dic13—48no—
64C23×Dic13D411yescharacteristic
65C23×Dic13D411yescharacteristic
66C13×AGL(1,ℤ/8)—232no—
67C2×C13×SD16—24no—
68C2×C13×SD16—24no—
69C2×C13×SD16C2×C418yescomplemented
70C2×C13×D8—24no—
71C2×C13×D8—24no—
72C2×C13×D8C2×C418yescomplemented
73C2×C13×M16—22no—
74C2×C13×M16—22no—
75C2×C13×M16C2×C414yescomplemented
76C22×C8×C13C2312yes—
77C2×C22⋊Dic13—28no—
78C2×C22⋊Dic13—24no—
79C2×C22⋊Dic13—21no—
80C2×C22⋊Dic13—22no—
81C2×C22⋊Dic13—22no—
82D4×Dic13—216no—
83D4×Dic13—28no—
84D4×Dic13—28no—
85#88180—22no—
86#88180—22no—
87C2×C4⋊Dic13—21no—
88C2×C4⋊Dic13—22no—
89C2×C4⋊Dic13D414yes—
90C2×C4⋊Dic13C2312yes—
91C2×C4⋊Dic13C2312yes—
92C2×C4×Dic13—22no—
93C2×C4×Dic13—22no—
94C2×C4×Dic13—21no—
95C2×C4×Dic13D412yes—
96C2×C4×Dic13C2312yes—
97C2×#6099—22no—
98C2×#6099—22no—
99C2×#6099—22no—
100C22×#1055C2312yes—
101#88075—24no—
102#88072—28no—
103#88072—24no—
104#88071—22no—
105#88071—22no—
106#20925—131nomaximal, Sylow
107C24×C13—24no—
108C24×C13—24no—
109C24×C13—24no—
110C24×C13—22no—
111C24×C13—21no—
112C24×C13—44no—
113C24×C13—44no—
114C24×C13C2×D411yescharacteristic
115C13×#41—28no—
116C13×#41—28no—
117C13×#41—44no—
118C13×#41—44no—
119C2×Q8×C13—21no—
120C2×Q8×C13—21no—
121C2×Q8×C13—21no—
122C2×Q8×C13—21no—
123C2×Q8×C13—21no—
124C2×Q8×C13C22×C412yescomplemented
125C2×D4×C13—24no—
126C2×D4×C13—28no—
127C2×D4×C13—24no—
128C2×D4×C13—24no—
129C2×D4×C13—28no—
130C2×D4×C13—24no—
131C2×D4×C13—28no—
132C2×D4×C13—22no—
133C2×D4×C13—21no—
134C2×D4×C13—21no—
135C2×D4×C13—21no—
136C2×D4×C13—22no—
137C2×D4×C13—22no—
138C2×D4×C13—21no—
139C2×D4×C13—21no—
140C2×D4×C13—42no—
141C2×D4×C13—42no—
142C2×D4×C13C22⋊C4116yescomplemented
143C2×D4×C13C22×C414yes—
144C2×D4×C13C22×C412yescomplemented
145C2×D4×C13C2×D414yes—
146C2×D4×C13C2×D414yes—
147C22×C4×C13—21no—
148C22×C4×C13—24no—
149C22×C4×C13—42no—
150C22×C4×C13—42no—
151C22×C4×C13C22×C412yes—
152C22×C4×C13C2×D411yescharacteristic
153C22×C4×C13C2×D412yes—
154C22×C4×C13C2×D412yes—
155C22×C4×C13C2×D412yes—
156C22×C4×C13#4112yes—
157C22×C4×C13#4112yes—
158C22×C4×C13C2411yescharacteristic
159C22×Dic13—22no—
160C22×Dic13—24no—
161C22×Dic13—216no—
162C22×Dic13—22no—
163C22×Dic13—24no—
164C22×Dic13—22no—
165C22×Dic13—22no—
166C22×Dic13—42no—
167C22×Dic13—432no—
168C22×Dic13—48no—
169C22×Dic13—44no—
170C22×Dic13—48no—
171C22×Dic13C2×D412yes—
172C22×Dic13C2×D412yes—
173C13×SD16—216no—
174C13×D8—216no—
175C13×M16—28no—
176C2×C8×C13—22no—
177C2×C8×C13—22no—
178C2×C8×C13C22×C414yescomplemented
179C22⋊Dic13—216no—
180C22⋊Dic13—44no—
181C22⋊Dic13—42no—
182C22⋊Dic13—44no—
183C4⋊Dic13—24no—
184C4⋊Dic13—22no—
185C4⋊Dic13—22no—
186C4⋊Dic13—22no—
187C4⋊Dic13C2×D414yes—
188C4×Dic13—24no—
189C4×Dic13—22no—
190C4×Dic13—22no—
191C4×Dic13—42no—
192#6099—44no—
193C2×#1055—22no—
194C2×#1055—22no—
195C2×#1055—22no—
196C22×AGL(1,ℤ/8)D1311yescharacteristic
197C22×C4×D4—131no—
198C2×#417—131no—
199C2×#418—131no—
200#2297—131no—
201#1879—134no—
202#1877—134no—
203#1736—131no—
204#1728—131no—
205C23×C13—22no—
206C23×C13—22no—
207C23×C13—24no—
208C23×C13—24no—
209C23×C13—24no—
210C23×C13—216no—
211C23×C13—22no—
212C23×C13—48no—
213C23×C13—416no—
214C23×C13—48no—
215C23×C13—416no—
216C23×C13—44no—
217C23×C13—44no—
218C23×C13—41no—
219C23×C13—41no—
220C23×C13—42no—
221C23×C13—42no—
222C23×C13C4×D412yes—
223C23×C13#12012yes—
224C23×C13#12112yes—
225C23×C13C22×D411yescharacteristic
226Q8×C13—22no—
227Q8×C13—22no—
228Q8×C13—41no—
229Q8×C13—41no—
230D4×C13—22no—
231D4×C13—24no—
232D4×C13—22no—
233D4×C13—24no—
234D4×C13—24no—
235D4×C13—216no—
236D4×C13—48no—
237D4×C13—41no—
238D4×C13—41no—
239D4×C13C2×C22⋊C418yes—
240C2×C4×C13—22no—
241C2×C4×C13—41no—
242C2×C4×C13—41no—
243C2×C4×C13—48no—
244C2×C4×C13C2×C22⋊C414yes—
245C2×C4×C13C4×D414yes—
246C2×C4×C13C4×D412yes—
247C2×C4×C13#12012yes—
248C2×C4×C13#12114yes—
249C2×C4×C13#12112yes—
250C2×C4×C13#12314yes—
251C2×C4×C13C23×C411yescharacteristic
252C2×C4×C13C22×D411yescharacteristic
253C2×C4×C13C22×D411yescharacteristic
254C2×C4×C13C22×D411yescharacteristic
255C2×C4×C13C2×#4111yescharacteristic
256C2×C4×C13C2×#4111yescharacteristic
257C2×Dic13—28no—
258C2×Dic13—22no—
259C2×Dic13—22no—
260C2×Dic13—22no—
261C2×Dic13—24no—
262C2×Dic13—22no—
263C2×Dic13—22no—
264C2×Dic13—432no—
265C2×Dic13—44no—
266C2×Dic13—48no—
267C2×Dic13—44no—
268C2×Dic13—42no—
269C2×Dic13—42no—
270C2×Dic13—42no—
271C2×Dic13AGL(1,ℤ/8)14yescomplemented
272C8×C13—24no—
273#1055—42no—
274C22×#41C2×D1311yescharacteristic
275C23×D4C2×D1311yescharacteristic
276C24×C4—261no—
277C2×AGL(1,ℤ/8)—24no—
278C2×AGL(1,ℤ/8)—24no—
279C2×AGL(1,ℤ/8)Dic1318yescomplemented
280C22×SD16C2×D1312yes—
281C22×D8C2×D1312yes—
282C22×M16—131no—
283C22×M16C2×D1311yescharacteristic
284C2×C4×D4—134no—
285C2×C4×D4—262no—
286C2×C4×D4—264no—
287C2×C4×D4—264no—
288C2×#117—131no—
289C22×C4⋊C4—131no—
290C22×C22⋊C4—261no—
291C22×C42—131no—
292#418—264no—
293#417—264no—
294C2×#103—132no—
295C2×#102—134no—
296C2×#102—132no—
297#391—132no—
298#390—132no—
299#386—261no—
300#386—261no—
301C4×C22⋊C4—262no—
302#340—134no—
303#336—134no—
304C22×C13—24no—
305C22×C13—22no—
306C22×C13—22no—
307C22×C13—24no—
308C22×C13—28no—
309C22×C13—432no—
310C22×C13—48no—
311C22×C13—44no—
312C22×C13—44no—
313C22×C13—48no—
314C22×C13—44no—
315C22×C13—42no—
316C22×C13—81no—
317C22×C13—81no—
318C22×C13—82no—
319C22×C13#38614yes—
320C22×C13C2×C4×D411yescharacteristic
321C22×C13C2×#12011yescharacteristic
322C22×C13C2×#12111yescharacteristic
323C22×C13C2×AGL(1,ℤ/8)12yes—
324C22×C13#57512yes—
325C4×C13—42no—
326C4×C13#38614yes—
327C4×C13C22×C22⋊C411yescharacteristic
328C4×C13C2×C4×D411yescharacteristic
329C4×C13C2×#12111yescharacteristic
330C4×C13C2×#12311yescharacteristic
331Dic13—24no—
332Dic13—44no—
333Dic13—42no—
334Dic13—82no—
335C25—21no—
336C2×#41—22no—
337C2×#41—22no—
338C2×#41—22no—
339C2×#41—22no—
340C2×#41—22no—
341C2×#41C2×Dic1314yescomplemented
342C22×Q8C22×D1311yescharacteristic
343C22×D4—21no—
344C22×D4—21no—
345C22×D4—22no—
346C22×D4—22no—
347C22×D4—22no—
348C22×D4C2×Dic1314yes—
349C22×D4C22⋊D1314yes—
350C22×D4C22⋊D1314yes—
351C22×D4C22×D1311yescharacteristic
352C22×D4C22×D1312yes—
353C23×C4—21no—
354C23×C4—131no—
355C23×C4—131no—
356C23×C4—264no—
357C23×C4—268no—
358C23×C4—261no—
359C23×C4—528no—
360C23×C4C22×D1311yescharacteristic
361AGL(1,ℤ/8)—232no—
362C2×SD16—24no—
363C2×SD16—24no—
364C2×SD16C2×Dic1318yescomplemented
365C2×D8—24no—
366C2×D8—24no—
367C2×D8C2×Dic1318yescomplemented
368C2×M16—22no—
369C2×M16—22no—
370C2×M16—262no—
371C2×M16—262no—
372C2×M16—262no—
373C2×M16C2×Dic1314yescomplemented
374C22×C8—132no—
375C22×C8C22×D1312yes—
376C4×D4—268no—
377C4×D4—2616no—
378C4×D4—268no—
379#117—262no—
380#117—262no—
381C2×C4⋊C4—134no—
382C2×C4⋊C4—132no—
383C2×C4⋊C4—132no—
384C2×C4⋊C4—261no—
385C2×C4⋊C4—262no—
386C2×C22⋊C4—264no—
387C2×C22⋊C4—268no—
388C2×C22⋊C4—262no—
389C2×C22⋊C4—262no—
390C2×C22⋊C4—261no—
391C2×C42—132no—
392C2×C42—132no—
393C2×C42—262no—
394C2×C42—262no—
395C2×C42—261no—
396#103—264no—
397#102—264no—
398#102—268no—
399#95—262no—
400#95—262no—
401C2×C13—24no—
402C2×C13—48no—
403C2×C13—82no—
404C2×C13C2×#38611yescharacteristic
405C2×C13C4×AGL(1,ℤ/8)12yes—
406C2×C13#300311yescharacteristic
407C2×C13#300511yescharacteristic
408C2×C13#303612yes—
409C24—24no—
410C24—24no—
411C24—24no—
412C24—22no—
413C24—21no—
414C24—44no—
415C24—44no—
416C24D4×D1311yescharacteristic
417#41—28no—
418#41—28no—
419#41—44no—
420#41—44no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8C22×Dic1312yescomplemented
427C2×D4—24no—
428C2×D4—28no—
429C2×D4—24no—
430C2×D4—28no—
431C2×D4—24no—
432C2×D4—28no—
433C2×D4—24no—
434C2×D4—22no—
435C2×D4—21no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4C22⋊Dic13116yescomplemented
445C2×D4C22×Dic1314yes—
446C2×D4C22×Dic1312yescomplemented
447C2×D4C2×C22⋊D1314yes—
448C2×D4C2×C22⋊D1314yes—
449C22×C4—21no—
450C22×C4—24no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—132no—
454C22×C4—132no—
455C22×C4—264no—
456C22×C4—262no—
457C22×C4—2616no—
458C22×C4—262no—
459C22×C4—264no—
460C22×C4—262no—
461C22×C4—262no—
462C22×C4—528no—
463C22×C4—522no—
464C22×C4—5232no—
465C22×C4—528no—
466C22×C4—524no—
467C22×C4D4×D1311yescharacteristic
468C22×C4D4×D1312yes—
469C22×C4#612912yes—
470C22×C4#612912yes—
471C22×C4C22×Dic1312yes—
472C22×C4C2×C22⋊D1312yes—
473C22×C4C2×C22⋊D1312yes—
474C22×C4C23×D1311yescharacteristic
475SD16—216no—
476D8—216no—
477M16—28no—
478M16—524no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—262no—
482C2×C8—262no—
483C2×C8—262no—
484C2×C8C22×Dic1314yescomplemented
485C4⋊C4—134no—
486C4⋊C4—264no—
487C4⋊C4—262no—
488C4⋊C4—262no—
489C4⋊C4—262no—
490C22⋊C4—2616no—
491C22⋊C4—524no—
492C22⋊C4—524no—
493C22⋊C4—522no—
494C42—264no—
495C42—262no—
496C42—262no—
497C42—522no—
498C13#2092511yescharacteristic, complemented, Sylow
499C23—22no—
500C23—22no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—216no—
505C23—22no—
506C23—48no—
507C23—416no—
508C23—416no—
509C23—48no—
510C23—44no—
511C23—44no—
512C23—42no—
513C23—42no—
514C23—41no—
515C23—41no—
516C23D4×Dic1312yes—
517C23#8819112yes—
518C23#8819312yes—
519C23C2×D4×D1311yescharacteristic
520Q8—22no—
521Q8—22no—
522Q8—41no—
523Q8—41no—
524D4—24no—
525D4—22no—
526D4—24no—
527D4—24no—
528D4—22no—
529D4—216no—
530D4—48no—
531D4—41no—
532D4—41no—
533D4C2×C22⋊Dic1318yes—
534C2×C4—22no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—48no—
538C2×C4—134no—
539C2×C4—268no—
540C2×C4—262no—
541C2×C4—262no—
542C2×C4—262no—
543C2×C4—262no—
544C2×C4—262no—
545C2×C4—264no—
546C2×C4—5232no—
547C2×C4—524no—
548C2×C4—528no—
549C2×C4—524no—
550C2×C4—522no—
551C2×C4—522no—
552C2×C4—522no—
553C2×C4D4×Dic1314yes—
554C2×C4D4×Dic1312yes—
555C2×C4#8818914yes—
556C2×C4#8819112yes—
557C2×C4#8819314yes—
558C2×C4#8819312yes—
559C2×C4C2×C22⋊Dic1314yes—
560C2×C4C2×D4×D1311yescharacteristic
561C2×C4C2×#612911yescharacteristic
562C2×C4C2×#612911yescharacteristic
563C2×C4C23×Dic1311yescharacteristic
564C2×C4C22×C22⋊D1311yescharacteristic
565C2×C4C22×C22⋊D1311yescharacteristic
566C8—24no—
567C8—522no—
568C22—24no—
569C22—24no—
570C22—22no—
571C22—22no—
572C22—28no—
573C22—432no—
574C22—48no—
575C22—48no—
576C22—44no—
577C22—44no—
578C22—44no—
579C22—42no—
580C22—82no—
581C22—81no—
582C22—81no—
583C22#1171888114yes—
584C22D13×AGL(1,ℤ/8)12yes—
585C22#1171947812yes—
586C22C2×D4×Dic1311yescharacteristic
587C22C2×#8819111yescharacteristic
588C22C2×#8819311yescharacteristic
589C4—42no—
590C4—264no—
591C4—524no—
592C4—522no—
593C4—1042no—
594C4#1171888114yes—
595C4C2×D4×Dic1311yescharacteristic
596C4C2×#8818911yescharacteristic
597C4C2×#8819311yescharacteristic
598C4C22×C22⋊Dic1311yescharacteristic
599C2—24no—
600C2—48no—
601C2—82no—
602C2C2×#1171888111yescharacteristic
603C2AGL(1,ℤ/8)×Dic1312yes—
604C2#1312406111yescharacteristic
605C2#1312407511yescharacteristic
606C2#1312429712yes—
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
Fitting, prime radicalC22×C13×AGL(1,ℤ/8)C2
socleC23×C13C22×D4
cosocleC22×C4×C13C24
hypercenterC22×AGL(1,ℤ/8)D13
hypocenterC13#20925
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C22×D4×Dic13C2semidirect
2. C2×#11718893C2semidirect
3. C2×#11718876C2semidirect
4. #13117327C2semidirect
5. #13115186C2semidirect
6. #13115172C2semidirect
7. #13114556C2semidirect
8. #13114529C2semidirect
15. C2×C13×AGL(1,ℤ/8)C4semidirect
29. #11718891C22semidirect
30. C2×#88075C22semidirect
31. #11718882C22semidirect
36. C2×#88072C22semidirect
38. #11718143C22semidirect
39. #11718142C22semidirect
46. C2×C13×#41C2×C4semidirect
69. C2×C13×SD16C2×C4semidirect
72. C2×C13×D8C2×C4semidirect
75. C2×C13×M16C2×C4semidirect
124. C2×Q8×C13C22×C4semidirect
142. C2×D4×C13C22⋊C4semidirect
144. C2×D4×C13C22×C4semidirect
178. C2×C8×C13C22×C4semidirect
271. C2×Dic13AGL(1,ℤ/8)semidirect
279. C2×AGL(1,ℤ/8)Dic13semidirect
341. C2×#41C2×Dic13semidirect
364. C2×SD16C2×Dic13semidirect
367. C2×D8C2×Dic13semidirect
373. C2×M16C2×Dic13semidirect
426. C2×Q8C22×Dic13semidirect
444. C2×D4C22⋊Dic13semidirect
446. C2×D4C22×Dic13semidirect
484. C2×C8C22×Dic13semidirect
498. C13#20925semidirect

Maximal subgroups

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

subgroupindex
C22×D4×Dic132
C2×#117188932
C2×#117188762
#131173272
#131151862
#131151722
#131145562
#131145292
C22×C13×AGL(1,ℤ/8)2
#2092513