Finite Group Databasebeta

← #15863898

Subgroups — #15863898

Order 3328.

1550 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×Dic13C211yesmaximal, characteristic
2C2×#11718782C211yesmaximal, characteristic, complemented
3C2×#11718893C211yesmaximal, characteristic
4C2×#11718646C211yesmaximal, characteristic, complemented
5#13115106C212yesmaximal, complemented
6#13115099C212yesmaximal, complemented
7#13115092C212yesmaximal, complemented
8#13115085C212yesmaximal, complemented
9#13114555C211yesmaximal, characteristic, complemented
10#13114529C211yesmaximal, characteristic, complemented
11C2×C13×#417C211yesmaximal, characteristic
12C22×C13×#41C2211yescharacteristic
13C22×C13×M16C2211yescharacteristic
14C22×C13×C4⋊C4C2211yescharacteristic
15#41×Dic13—24no—
16#41×Dic13—24no—
17C2×Q8×Dic13C2211yescharacteristic
18C2×Q8×Dic13C2211yescharacteristic
19C2×D4×Dic13—21no—
20C2×D4×Dic13—21no—
21C2×D4×Dic13C2211yescharacteristic
22C2×D4×Dic13C2211yescharacteristic
23C2×#88180—21no—
24C2×#88180C2211yescharacteristic
25C22×C4⋊Dic13—21no—
26C22×C4⋊Dic13C2211yescharacteristic
27C22×C4×Dic13—21no—
28C22×C4×Dic13C2211yescharacteristic
29C22×#6099C2211yescharacteristic
30C13×#417C418yescomplemented
31C2×C13×#103C2212yes—
32C2×C13×#102C2212yes—
33#11718893—24no—
34C2×#88075C2212yes—
35C2×#88072C2212yes—
36C2×#88071—21no—
37#11718782—24no—
38C2×#88058C2212yes—
39C2×#88057C2212yes—
40#11718646—21no—
41#11718646—21no—
42#11718646C2212yes—
43#11718646C2212yes—
44#11718142C2212yes—
45#11718142C2212yes—
46#11718121C2212yes—
47#11718121C2212yes—
48C2×C13×#41—22no—
49C2×C13×#41—22no—
50C2×C13×#41—22no—
51C2×C13×#41C2×C414yes—
52C2×C13×#41D414yes—
53C2×C13×#41Q814yes—
54C22×Q8×C13C2311yescharacteristic
55C22×D4×C13—21no—
56C22×D4×C13C2311yescharacteristic
57C23×C4×C13—21no—
58C23×C4×C13—21no—
59C23×C4×C13C2311yescharacteristic
60C23×Dic13—21no—
61C23×Dic13—21no—
62C23×Dic13—21no—
63C23×Dic13D411yescharacteristic
64C23×Dic13D411yescharacteristic
65C2×C13×M16—22no—
66C2×C13×M16—22no—
67C2×C13×M16C2×C414yescomplemented
68C22×C8×C13C2312yes—
69C2×C13×C4⋊C4—21no—
70C2×C13×C4⋊C4—22no—
71C2×C13×C4⋊C4C2×C414yescomplemented
72C2×C13×C4⋊C4C2312yes—
73C2×C22⋊Dic13—21no—
74C2×C22⋊Dic13—21no—
75C2×C22⋊Dic13—21no—
76Q8×Dic13—22no—
77Q8×Dic13—22no—
78Q8×Dic13—22no—
79Q8×Dic13—22no—
80D4×Dic13—28no—
81D4×Dic13—28no—
82D4×Dic13—22no—
83D4×Dic13—22no—
84D4×Dic13—22no—
85D4×Dic13—22no—
86#88180—28no—
87#88180—22no—
88#88180D414yes—
89C2×C4⋊Dic13—21no—
90C2×C4⋊Dic13—21no—
91C2×C4⋊Dic13—22no—
92C2×C4⋊Dic13—22no—
93C2×C4⋊Dic13—21no—
94C2×C4⋊Dic13—22no—
95C2×C4⋊Dic13—22no—
96C2×C4⋊Dic13—44no—
97C2×C4⋊Dic13C2312yes—
98C2×C4⋊Dic13C2311yescharacteristic
99C2×C4⋊Dic13C2311yescharacteristic
100C2×C4×Dic13—28no—
101C2×C4×Dic13—21no—
102C2×C4×Dic13—22no—
103C2×C4×Dic13—22no—
104C2×C4×Dic13—21no—
105C2×C4×Dic13—21no—
106C2×C4×Dic13D411yescharacteristic
107C2×C4×Dic13D411yescharacteristic
108C2×C4×Dic13C2311yescharacteristic
109C2×C4×Dic13C2311yescharacteristic
110C2×#6099—22no—
111C2×#6099—22no—
112C2×#6099—22no—
113C22×#1055C2312yes—
114C13×#103C2×C418yescomplemented
115C13×#102C2×C418yescomplemented
116#88075—24no—
117#88072—24no—
118#88071—22no—
119#88071—22no—
120#88058—24no—
121#88057—24no—
122#20875—131nomaximal, Sylow
123C24×C13—21no—
124C24×C13C2×D411yescharacteristic
125C13×#41—28no—
126C13×#41—28no—
127C13×#41—28no—
128C13×#41C4⋊C4116yescomplemented
129C2×Q8×C13—21no—
130C2×Q8×C13—21no—
131C2×Q8×C13—21no—
132C2×Q8×C13C22×C412yes—
133C2×Q8×C13C2×D412yes—
134C2×Q8×C13C2×Q812yes—
135C2×D4×C13—24no—
136C2×D4×C13—24no—
137C2×D4×C13—24no—
138C2×D4×C13—21no—
139C2×D4×C13—21no—
140C2×D4×C13—21no—
141C2×D4×C13C22×C412yes—
142C2×D4×C13C2×D412yes—
143C2×D4×C13C2×Q812yes—
144C22×C4×C13—21no—
145C22×C4×C13—24no—
146C22×C4×C13—24no—
147C22×C4×C13—24no—
148C22×C4×C13—24no—
149C22×C4×C13—22no—
150C22×C4×C13—42no—
151C22×C4×C13—42no—
152C22×C4×C13C22×C412yes—
153C22×C4×C13C2×D412yes—
154C22×C4×C13C2×D411yescharacteristic
155C22×C4×C13C2×D412yes—
156C22×C4×C13C2×Q812yes—
157C22×C4×C13#4112yes—
158C22×C4×C13#4112yes—
159C22×C4×C13C2411yescharacteristic
160C22×Dic13—24no—
161C22×Dic13—24no—
162C22×Dic13—22no—
163C22×Dic13—22no—
164C22×Dic13—21no—
165C22×Dic13—21no—
166C22×Dic13—22no—
167C22×Dic13—22no—
168C22×Dic13—22no—
169C22×Dic13—41no—
170C22×Dic13—41no—
171C22×Dic13—42no—
172C22×Dic13—42no—
173C22×Dic13—42no—
174C22×Dic13—42no—
175C22×Dic13—42no—
176C22×Dic13—42no—
177C22×Dic13—42no—
178C22×Dic13C2×D411yescharacteristic
179C22×Dic13C2×D411yescharacteristic
180C22×Dic13C2×D411yescharacteristic
181C22×Dic13C2×D411yescharacteristic
182C13×M16—28no—
183C2×C8×C13—22no—
184C2×C8×C13—22no—
185C2×C8×C13C22×C414yes—
186C13×C4⋊C4—24no—
187C13×C4⋊C4—22no—
188C13×C4⋊C4C22×C414yes—
189C22⋊Dic13—24no—
190C22⋊Dic13—24no—
191C22⋊Dic13—24no—
192C4⋊Dic13—24no—
193C4⋊Dic13—24no—
194C4⋊Dic13—21no—
195C4⋊Dic13—21no—
196C4⋊Dic13—22no—
197C4⋊Dic13—22no—
198C4⋊Dic13—44no—
199C4⋊Dic13—42no—
200C4⋊Dic13—44no—
201C4⋊Dic13C2×D412yes—
202C4⋊Dic13C2×D412yes—
203C4×Dic13—24no—
204C4×Dic13—24no—
205C4×Dic13—22no—
206C4×Dic13—24no—
207C4×Dic13—22no—
208C4×Dic13—21no—
209C4×Dic13—21no—
210C4×Dic13—21no—
211C4×Dic13—21no—
212#6099—44no—
213C2×#1055—22no—
214C2×#1055—22no—
215C2×#1055—22no—
216C2×C4×#41—131no—
217C2×#428—131no—
218C2×#417—131no—
219C2×#417D1311yescharacteristic
220C2×#389—131no—
221#1865—132no—
222#1864—132no—
223#1863—132no—
224#1862—132no—
225#1736—131no—
226#1728—131no—
227C23×C13—21no—
228C23×C13—21no—
229C23×C13—21no—
230C23×C13—21no—
231C23×C13—22no—
232C23×C13—22no—
233C23×C13—22no—
234C23×C13—41no—
235C23×C13—41no—
236C23×C13—41no—
237C23×C13—41no—
238C23×C13C4×D412yes—
239C23×C13#12012yes—
240C23×C13C22⋊Q812yes—
241C23×C13C22×D411yescharacteristic
242Q8×C13—22no—
243Q8×C13—22no—
244Q8×C13—22no—
245Q8×C13C2×C4⋊C414yescomplemented
246D4×C13—216no—
247D4×C13—22no—
248D4×C13—22no—
249D4×C13—22no—
250D4×C13C2×C4⋊C414yescomplemented
251C2×C4×C13—22no—
252C2×C4×C13—22no—
253C2×C4×C13—22no—
254C2×C4×C13—216no—
255C2×C4×C13—24no—
256C2×C4×C13—48no—
257C2×C4×C13—42no—
258C2×C4×C13—42no—
259C2×C4×C13C2×C4⋊C414yes—
260C2×C4×C13C4×D414yes—
261C2×C4×C13C4×D412yes—
262C2×C4×C13#12012yes—
263C2×C4×C13C22⋊Q814yes—
264C2×C4×C13C22⋊Q812yes—
265C2×C4×C13#12314yes—
266C2×C4×C13C23×C411yescharacteristic
267C2×C4×C13C22×D411yescharacteristic
268C2×C4×C13C22×D411yescharacteristic
269C2×C4×C13C22×Q811yescharacteristic
270C2×C4×C13C2×#4111yescharacteristic
271C2×C4×C13C2×#4111yescharacteristic
272C2×Dic13—21no—
273C2×Dic13—21no—
274C2×Dic13—21no—
275C2×Dic13—21no—
276C2×Dic13—21no—
277C2×Dic13—21no—
278C2×Dic13—22no—
279C2×Dic13—22no—
280C2×Dic13—21no—
281C2×Dic13—21no—
282C2×Dic13—21no—
283C2×Dic13—21no—
284C2×Dic13—44no—
285C2×Dic13—44no—
286C2×Dic13—48no—
287C2×Dic13—48no—
288C2×Dic13—41no—
289C2×Dic13—41no—
290C2×Dic13—41no—
291C2×Dic13—41no—
292C2×Dic13—42no—
293C2×Dic13—42no—
294C2×Dic13—42no—
295C2×Dic13—84no—
296C2×Dic13AGL(1,ℤ/8)12yes—
297C2×Dic13#13712yes—
298C8×C13—24no—
299#1055—42no—
300C22×#41C2×D1311yescharacteristic
301C22×M16—131no—
302C22×M16C2×D1311yescharacteristic
303C4×#41—264no—
304C4×#41—264no—
305C2×C4×Q8—131no—
306C2×C4×Q8—131no—
307C2×C4×D4—131no—
308C2×C4×D4—131no—
309C2×C4×D4—261no—
310C2×C4×D4—261no—
311C2×#117—131no—
312C2×#117—261no—
313C22×C4⋊C4—131no—
314C22×C4⋊C4—261no—
315C22×C4⋊C4C2×D1311yescharacteristic
316C22×C42—131no—
317C22×C42—261no—
318#428—264no—
319C2×#107—132no—
320C2×#106—132no—
321#417—264no—
322#417Dic1318yescomplemented
323C2×#103—132no—
324C2×#103C2×D1312yes—
325C2×#102—132no—
326C2×#102C2×D1312yes—
327#389—132no—
328#389—132no—
329#389—261no—
330#389—261no—
331C2×#95—261no—
332#340—132no—
333#340—132no—
334#336—132no—
335#336—132no—
336C22×C13—22no—
337C22×C13—22no—
338C22×C13—22no—
339C22×C13—22no—
340C22×C13—22no—
341C22×C13—22no—
342C22×C13—24no—
343C22×C13—42no—
344C22×C13—42no—
345C22×C13—42no—
346C22×C13—42no—
347C22×C13—42no—
348C22×C13—42no—
349C22×C13#38514yes—
350C22×C13C2×C4×D411yescharacteristic
351C22×C13C2×#12011yescharacteristic
352C22×C13C2×C22⋊Q811yescharacteristic
353C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
354C22×C13C2×#13711yescharacteristic
355C22×C13#57511yescharacteristic
356C22×C13#57511yescharacteristic
357C4×C13—24no—
358C4×C13—44no—
359C4×C13#38514yes—
360C4×C13C22×C4⋊C411yescharacteristic
361C4×C13C2×C4×D411yescharacteristic
362C4×C13C2×C22⋊Q811yescharacteristic
363C4×C13C2×#12311yescharacteristic
364Dic13—22no—
365Dic13—22no—
366Dic13—41no—
367Dic13—42no—
368Dic13—41no—
369Dic13—42no—
370Dic13—82no—
371C2×#41—22no—
372C2×#41—22no—
373C2×#41—22no—
374C2×#41Dic2614yes—
375C2×#41D5214yes—
376C2×#41C2×Dic1314yes—
377C22×Q8C22×D1311yescharacteristic
378C22×D4—21no—
379C22×D4C22×D1311yescharacteristic
380C23×C4—21no—
381C23×C4—21no—
382C23×C4—131no—
383C23×C4—131no—
384C23×C4—261no—
385C23×C4—261no—
386C23×C4—261no—
387C23×C4C22×D1311yescharacteristic
388C2×M16—22no—
389C2×M16—22no—
390C2×M16—262no—
391C2×M16—262no—
392C2×M16—262no—
393C2×M16C2×Dic1314yescomplemented
394C22×C8—132no—
395C22×C8C22×D1312yes—
396C4×Q8—262no—
397C4×Q8—262no—
398C4×Q8—262no—
399C4×Q8—262no—
400C4×D4—262no—
401C4×D4—262no—
402C4×D4—268no—
403C4×D4—268no—
404C4×D4—262no—
405C4×D4—262no—
406#117—134no—
407#117—268no—
408#117—262no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—22no—
411C2×C4⋊C4—131no—
412C2×C4⋊C4—131no—
413C2×C4⋊C4—132no—
414C2×C4⋊C4—261no—
415C2×C4⋊C4—261no—
416C2×C4⋊C4—262no—
417C2×C4⋊C4—262no—
418C2×C4⋊C4—262no—
419C2×C4⋊C4—262no—
420C2×C4⋊C4—261no—
421C2×C4⋊C4—524no—
422C2×C4⋊C4C2×Dic1314yescomplemented
423C2×C4⋊C4C22×D1312yes—
424C2×C22⋊C4—261no—
425C2×C22⋊C4—261no—
426C2×C22⋊C4—261no—
427C2×C42—131no—
428C2×C42—131no—
429C2×C42—131no—
430C2×C42—131no—
431C2×C42—261no—
432C2×C42—261no—
433C2×C42—268no—
434C2×C42—262no—
435C2×C42—261no—
436C2×C42—262no—
437#107—264no—
438#106—264no—
439#103—264no—
440#103C2×Dic1318yescomplemented
441#102—264no—
442#102C2×Dic1318yescomplemented
443#95—262no—
444#95—262no—
445C2×C13—24no—
446C2×C13—44no—
447C2×C13C2×#38511yescharacteristic
448C2×C13C4×AGL(1,ℤ/8)11yescharacteristic
449C2×C13C4×#13711yescharacteristic
450C2×C13#300611yescharacteristic
451C2×C13#300711yescharacteristic
452C2×C13#307911yescharacteristic
453C2×C13#308011yescharacteristic
454C24—21no—
455C24D4×D1311yescharacteristic
456#41—28no—
457#41—28no—
458#41—28no—
459#41C4⋊Dic13116yescomplemented
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8C2×Dic2612yes—
464C2×Q8C2×D5212yes—
465C2×Q8C22×Dic1312yes—
466C2×D4—24no—
467C2×D4—24no—
468C2×D4—24no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4C2×Dic2612yes—
473C2×D4C2×D5212yes—
474C2×D4C22×Dic1312yes—
475C22×C4—21no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—22no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—131no—
484C22×C4—131no—
485C22×C4—131no—
486C22×C4—131no—
487C22×C4—262no—
488C22×C4—262no—
489C22×C4—264no—
490C22×C4—264no—
491C22×C4—262no—
492C22×C4—262no—
493C22×C4—261no—
494C22×C4—261no—
495C22×C4—262no—
496C22×C4—522no—
497C22×C4—522no—
498C22×C4—521no—
499C22×C4—521no—
500C22×C4—522no—
501C22×C4—522no—
502C22×C4—522no—
503C22×C4—522no—
504C22×C4—522no—
505C22×C4C2×Dic2612yes—
506C22×C4C2×D5212yes—
507C22×C4D4×D1311yescharacteristic
508C22×C4D4×D1312yes—
509C22×C4#612912yes—
510C22×C4#612912yes—
511C22×C4C22×Dic1312yes—
512C22×C4C23×D1311yescharacteristic
513M16—28no—
514M16—524no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—262no—
518C2×C8—262no—
519C2×C8—262no—
520C2×C8C22×Dic1314yes—
521C4⋊C4—24no—
522C4⋊C4—22no—
523C4⋊C4—132no—
524C4⋊C4—132no—
525C4⋊C4—264no—
526C4⋊C4—264no—
527C4⋊C4—261no—
528C4⋊C4—261no—
529C4⋊C4—262no—
530C4⋊C4—262no—
531C4⋊C4—524no—
532C4⋊C4—524no—
533C4⋊C4—522no—
534C4⋊C4C22×Dic1314yes—
535C22⋊C4—264no—
536C22⋊C4—264no—
537C22⋊C4—264no—
538C42—264no—
539C42—264no—
540C42—262no—
541C42—262no—
542C42—264no—
543C42—261no—
544C42—261no—
545C42—261no—
546C42—261no—
547C13#2087511yescharacteristic, complemented, Sylow
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23#8813212yes—
560C23#8813612yes—
561C23D4×Dic1312yes—
562C23C2×D4×D1311yescharacteristic
563Q8—22no—
564Q8—22no—
565Q8—22no—
566Q8C2×C4⋊Dic1314yescomplemented
567D4—216no—
568D4—22no—
569D4—22no—
570D4—22no—
571D4C2×C4⋊Dic1314yescomplemented
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—216no—
576C2×C4—24no—
577C2×C4—48no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—132no—
581C2×C4—132no—
582C2×C4—261no—
583C2×C4—261no—
584C2×C4—261no—
585C2×C4—261no—
586C2×C4—262no—
587C2×C4—262no—
588C2×C4—261no—
589C2×C4—261no—
590C2×C4—261no—
591C2×C4—261no—
592C2×C4—261no—
593C2×C4—261no—
594C2×C4—524no—
595C2×C4—524no—
596C2×C4—528no—
597C2×C4—528no—
598C2×C4—521no—
599C2×C4—521no—
600C2×C4—521no—
601C2×C4—521no—
602C2×C4—522no—
603C2×C4—522no—
604C2×C4—522no—
605C2×C4—1044no—
606C2×C4#8813214yes—
607C2×C4#8813212yes—
608C2×C4#8813612yes—
609C2×C4#8814014yes—
610C2×C4C2×C4⋊Dic1314yes—
611C2×C4D4×Dic1314yes—
612C2×C4D4×Dic1312yes—
613C2×C4C22×Dic2611yescharacteristic
614C2×C4C22×D5211yescharacteristic
615C2×C4C2×D4×D1311yescharacteristic
616C2×C4C2×#612911yescharacteristic
617C2×C4C2×#612911yescharacteristic
618C2×C4C23×Dic1311yescharacteristic
619C8—24no—
620C8—522no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—24no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22#1171861114yes—
635C22C2×#8813211yescharacteristic
636C22C2×#8813611yescharacteristic
637C22D13×AGL(1,ℤ/8)11yescharacteristic
638C22#1171947811yescharacteristic
639C22D13×#13711yescharacteristic
640C22#1171948211yescharacteristic
641C22C2×D4×Dic1311yescharacteristic
642C4—24no—
643C4—44no—
644C4—262no—
645C4—262no—
646C4—521no—
647C4—522no—
648C4—521no—
649C4—522no—
650C4—1042no—
651C4#1171861114yes—
652C4C2×#8813211yescharacteristic
653C4C2×#8814011yescharacteristic
654C4C22×C4⋊Dic1311yescharacteristic
655C4C2×D4×Dic1311yescharacteristic
656C2—24no—
657C2—44no—
658C2C2×#1171861111yescharacteristic
659C2AGL(1,ℤ/8)×Dic1311yescharacteristic
660C2#137×Dic1311yescharacteristic
661C2#1312408911yescharacteristic
662C2#1312409711yescharacteristic
663C2#1312464711yescharacteristic
664C2#1312465611yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC2×C13×#417C2
socleC23×C13C22×D4
prime radicalC22×C13×#41C22
cosocleC22×C4×C13C24
hypercenterC2×#417D13
hypocenterC13#20875
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
2. C2×#11718782C2semidirect
4. C2×#11718646C2semidirect
5. #13115106C2semidirect
6. #13115099C2semidirect
7. #13115092C2semidirect
8. #13115085C2semidirect
9. #13114555C2semidirect
10. #13114529C2semidirect
30. C13×#417C4semidirect
67. C2×C13×M16C2×C4semidirect
71. C2×C13×C4⋊C4C2×C4semidirect
114. C13×#103C2×C4semidirect
115. C13×#102C2×C4semidirect
128. C13×#41C4⋊C4semidirect
245. Q8×C13C2×C4⋊C4semidirect
250. D4×C13C2×C4⋊C4semidirect
322. #417Dic13semidirect
393. C2×M16C2×Dic13semidirect
422. C2×C4⋊C4C2×Dic13semidirect
440. #103C2×Dic13semidirect
442. #102C2×Dic13semidirect
459. #41C4⋊Dic13semidirect
547. C13#20875semidirect
566. Q8C2×C4⋊Dic13semidirect
571. D4C2×C4⋊Dic13semidirect

Maximal subgroups

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

subgroupindex
C2×#41×Dic132
C2×#117187822
C2×#117188932
C2×#117186462
#131151062
#131150992
#131150922
#131150852
#131145552
#131145292
C2×C13×#4172
#2087513