Finite Group Databasebeta

← #15865954

Subgroups — #15865954

Order 3328.

1042 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13117301C211yesmaximal, characteristic, complemented
2C4×D4×Dic13C211yesmaximal, characteristic, complemented
3#13115465C211yesmaximal, characteristic, complemented
4#13115458C211yesmaximal, characteristic, complemented
5#13115423C211yesmaximal, characteristic, complemented
6#13115416C211yesmaximal, characteristic, complemented
7#13115186C211yesmaximal, characteristic
8#13115186C211yesmaximal, characteristic
9#13115099C211yesmaximal, characteristic, complemented
10#13115085C211yesmaximal, characteristic, complemented
11#13114733C211yesmaximal, characteristic, complemented
12#13114661C211yesmaximal, characteristic, complemented
13#13114661C211yesmaximal, characteristic, complemented
14#13114584C211yesmaximal, characteristic, complemented
15C2×C13×#442C211yesmaximal, characteristic
16C22×C13×D8C2211yescharacteristic
17C2×C4×D4×C13C2211yescharacteristic
18C2×C4×D4×C13C2211yescharacteristic
19C2×D4×Dic13C2211yescharacteristic
20C2×D4×Dic13C2211yescharacteristic
21C2×#88180—21no—
22C22×C4×Dic13—21no—
23C13×#442C418yescomplemented
24C2×C13×#106C2211yescharacteristic
25C2×C13×#102C2211yescharacteristic
26C2×C13×#102C2211yescharacteristic
27C2×C13×#97C2211yescharacteristic
28#11718882C2211yescharacteristic
29#11718882C2211yescharacteristic
30C2×#88072C2211yescharacteristic
31C2×#88072C2211yescharacteristic
32C2×#88072C2211yescharacteristic
33C2×#88072C2211yescharacteristic
34C4×C22⋊Dic13—21no—
35C4×C22⋊Dic13—21no—
36C2×#88058C2211yescharacteristic
37C2×#88057C2211yescharacteristic
38#11718646C2211yescharacteristic
39#11718646C2211yescharacteristic
40C4⋊C4×Dic13C2211yescharacteristic
41C4⋊C4×Dic13C2211yescharacteristic
42#11718612—21no—
43#11718612—21no—
44C22⋊C4×Dic13—21no—
45C22⋊C4×Dic13—21no—
46C4×C4⋊Dic13C2211yescharacteristic
47C4×C4⋊Dic13C2211yescharacteristic
48C42×Dic13C2211yescharacteristic
49C2×#88043C2211yescharacteristic
50#11718143C2211yescharacteristic
51#11718143C2211yescharacteristic
52#11718142C2211yescharacteristic
53#11718142C2211yescharacteristic
54#11718142C2211yescharacteristic
55#11718142C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718121C2211yescharacteristic
59#11718121C2211yescharacteristic
60C22×D4×C13C2311yescharacteristic
61C22×D4×C13C2311yescharacteristic
62C23×C4×C13—21no—
63C23×C4×C13—21no—
64C23×Dic13—21no—
65C23×Dic13—21no—
66C2×C13×D8—22no—
67C2×C13×D8—22no—
68C2×C13×D8C2×C414yes—
69C22×C8×C13C2311yescharacteristic
70C22×C8×C13C2311yescharacteristic
71C4×D4×C13—22no—
72C4×D4×C13—22no—
73C4×D4×C13C2×C414yes—
74C4×D4×C13C2×C414yescomplemented
75C2×C13×C4⋊C4C2311yescharacteristic
76C2×C13×C4⋊C4C2311yescharacteristic
77C2×C13×C22⋊C4—21no—
78C2×C13×C22⋊C4—21no—
79C2×C42×C13C2311yescharacteristic
80C2×C22⋊Dic13—21no—
81C2×C22⋊Dic13—21no—
82C2×C22⋊Dic13—21no—
83C2×C22⋊Dic13—21no—
84C2×C22⋊Dic13—21no—
85C2×C22⋊Dic13—21no—
86D4×Dic13—22no—
87D4×Dic13—22no—
88D4×Dic13—22no—
89D4×Dic13—22no—
90#88180—44no—
91C2×C4⋊Dic13—21no—
92C2×C4⋊Dic13—21no—
93C2×C4⋊Dic13C2311yescharacteristic
94C2×C4⋊Dic13C2311yescharacteristic
95C2×C4⋊Dic13C2311yescharacteristic
96C2×C4⋊Dic13C2311yescharacteristic
97C2×C4×Dic13—21no—
98C2×C4×Dic13—21no—
99C2×C4×Dic13—21no—
100C2×C4×Dic13—21no—
101C2×C4×Dic13—44no—
102C2×C4×Dic13D411yescharacteristic
103C2×C4×Dic13D411yescharacteristic
104C2×C4×Dic13D411yescharacteristic
105C2×C4×Dic13D411yescharacteristic
106C2×C4×Dic13C2311yescharacteristic
107C2×C4×Dic13C2311yescharacteristic
108C22×#1055C2311yescharacteristic
109C22×#1055C2311yescharacteristic
110C13×#106C2×C414yescomplemented
111C13×#102C2×C414yes—
112C13×#102C2×C414yescomplemented
113C13×#97C2×C414yescomplemented
114#88072—22no—
115#88072—22no—
116#88072—22no—
117#88072—22no—
118#88071—21no—
119#88071—21no—
120#88071—21no—
121#88071—21no—
122#88071—21no—
123#88071—21no—
124#88058—22no—
125#88057—22no—
126#88043—22no—
127#21122—131nomaximal, Sylow
128C24×C13—21no—
129C24×C13—21no—
130C2×D4×C13—21no—
131C2×D4×C13—21no—
132C2×D4×C13—21no—
133C2×D4×C13—21no—
134C2×D4×C13—21no—
135C2×D4×C13—21no—
136C2×D4×C13—21no—
137C2×D4×C13—21no—
138C2×D4×C13C22×C412yes—
139C2×D4×C13C22×C412yes—
140C2×D4×C13#4112yes—
141C2×D4×C13#4112yes—
142C22×C4×C13—21no—
143C22×C4×C13—21no—
144C22×C4×C13—24no—
145C22×C4×C13—24no—
146C22×C4×C13—42no—
147C22×C4×C13—42no—
148C22×C4×C13—42no—
149C22×C4×C13—42no—
150C22×C4×C13C2×D411yescharacteristic
151C22×C4×C13C2×D411yescharacteristic
152C22×C4×C13C2411yescharacteristic
153C22×Dic13—24no—
154C22×Dic13—24no—
155C22×Dic13—21no—
156C22×Dic13—21no—
157C22×Dic13—21no—
158C22×Dic13—21no—
159C22×Dic13—42no—
160C22×Dic13—42no—
161C22×Dic13—42no—
162C22×Dic13—42no—
163C22×Dic13C2×D411yescharacteristic
164C22×Dic13C2×D411yescharacteristic
165C22×Dic13C2×D411yescharacteristic
166C22×Dic13C2×D411yescharacteristic
167C13×D8—28no—
168C2×C8×C13—21no—
169C2×C8×C13—21no—
170C2×C8×C13—21no—
171C2×C8×C13—21no—
172C2×C8×C13C22×C412yes—
173C2×C8×C13C22×C412yes—
174C13×C4⋊C4—21no—
175C13×C4⋊C4—21no—
176C13×C4⋊C4C22×C412yes—
177C13×C4⋊C4C22×C412yes—
178C13×C22⋊C4—24no—
179C13×C22⋊C4—24no—
180C42×C13C22×C412yes—
181C42×C13#4112yes—
182C22⋊Dic13—24no—
183C22⋊Dic13—24no—
184C22⋊Dic13—42no—
185C22⋊Dic13—42no—
186C22⋊Dic13—42no—
187C22⋊Dic13—42no—
188C4⋊Dic13—21no—
189C4⋊Dic13—21no—
190C4⋊Dic13—21no—
191C4⋊Dic13—21no—
192C4⋊Dic13—21no—
193C4⋊Dic13—21no—
194C4⋊Dic13—42no—
195C4⋊Dic13—42no—
196C4⋊Dic13#4112yes—
197C4⋊Dic13#4112yes—
198C4×Dic13—21no—
199C4×Dic13—21no—
200C4×Dic13—21no—
201C4×Dic13—21no—
202C4×Dic13—22no—
203C4×Dic13—22no—
204C4×Dic13—41no—
205C4×Dic13—41no—
206C4×Dic13—42no—
207C4×Dic13—42no—
208C4×Dic13—42no—
209C4×Dic13—42no—
210C2×#1055—21no—
211C2×#1055—21no—
212C2×#1055—21no—
213C2×#1055—21no—
214C2×#1055—21no—
215C2×#1055—21no—
216C2×#442D1311yescharacteristic
217#2293—131no—
218C42×D4—131no—
219#1926—131no—
220#1925—131no—
221#1920—131no—
222#1919—131no—
223#1879—131no—
224#1879—131no—
225#1864—131no—
226#1862—131no—
227#1776—131no—
228#1762—131no—
229#1762—131no—
230#1745—131no—
231C23×C13—22no—
232C23×C13—22no—
233C23×C13—22no—
234C23×C13—22no—
235C23×C13—41no—
236C23×C13—41no—
237C23×C13—41no—
238C23×C13—41no—
239C23×C13—41no—
240C23×C13—41no—
241C23×C13—41no—
242C23×C13—41no—
243C23×C13—41no—
244C23×C13—41no—
245C23×C13C22×D411yescharacteristic
246D4×C13—22no—
247D4×C13—22no—
248D4×C13—22no—
249D4×C13—22no—
250D4×C13—22no—
251D4×C13—41no—
252D4×C13—41no—
253D4×C13#11714yescomplemented
254C2×C4×C13—21no—
255C2×C4×C13—21no—
256C2×C4×C13—21no—
257C2×C4×C13—21no—
258C2×C4×C13—22no—
259C2×C4×C13—22no—
260C2×C4×C13—48no—
261C2×C4×C13—48no—
262C2×C4×C13—41no—
263C2×C4×C13—41no—
264C2×C4×C13—41no—
265C2×C4×C13—41no—
266C2×C4×C13C4×D412yes—
267C2×C4×C13C4×D412yes—
268C2×C4×C13C23×C411yescharacteristic
269C2×C4×C13C2×#4111yescharacteristic
270C2×C4×C13C2×#4111yescharacteristic
271C2×C4×C13C2×#4111yescharacteristic
272C2×C4×C13C2×#4111yescharacteristic
273C2×C4×C13C2×#4111yescharacteristic
274C2×Dic13—21no—
275C2×Dic13—21no—
276C2×Dic13—21no—
277C2×Dic13—21no—
278C2×Dic13—21no—
279C2×Dic13—21no—
280C2×Dic13—22no—
281C2×Dic13—22no—
282C2×Dic13—21no—
283C2×Dic13—21no—
284C2×Dic13—21no—
285C2×Dic13—21no—
286C2×Dic13—48no—
287C2×Dic13—48no—
288C2×Dic13—41no—
289C2×Dic13—41no—
290C2×Dic13—41no—
291C2×Dic13—41no—
292C2×Dic13—41no—
293C2×Dic13—41no—
294C2×Dic13—41no—
295C2×Dic13—41no—
296C2×Dic13—41no—
297C2×Dic13—41no—
298C2×Dic13AGL(1,ℤ/8)12yes—
299C2×Dic13AGL(1,ℤ/8)12yes—
300C8×C13—22no—
301C8×C13—22no—
302#1055—41no—
303#1055—41no—
304C22×D8C2×D1311yescharacteristic
305C2×C4×D4—131no—
306C2×C4×D4—131no—
307C2×C4×D4C2×D1311yescharacteristic
308C2×C4×D4C2×D1311yescharacteristic
309C2×#117—261no—
310C22×C42—261no—
311#442Dic1318yescomplemented
312C2×#107—131no—
313C2×#106—131no—
314C2×#106C2×D1311yescharacteristic
315C2×#102—131no—
316C2×#102—131no—
317C2×#102—131no—
318C2×#102—131no—
319C2×#102C2×D1311yescharacteristic
320C2×#102C2×D1311yescharacteristic
321C2×#97—131no—
322C2×#97C2×D1311yescharacteristic
323#390—131no—
324#390—131no—
325#389—131no—
326#389—131no—
327#388—261no—
328#388—261no—
329C4×C4⋊C4—131no—
330C4×C4⋊C4—131no—
331C4×C4⋊C4—131no—
332C4×C4⋊C4—131no—
333C4×C22⋊C4—261no—
334C4×C22⋊C4—261no—
335C4×C22⋊C4—261no—
336C4×C22⋊C4—261no—
337C43—131no—
338#340—131no—
339#340—131no—
340#340—131no—
341#340—131no—
342#340—131no—
343#340—131no—
344#336—131no—
345#336—131no—
346#336—131no—
347#336—131no—
348C22×C13—24no—
349C22×C13—42no—
350C22×C13—42no—
351C22×C13—42no—
352C22×C13—42no—
353C22×C13—42no—
354C22×C13—42no—
355C22×C13—42no—
356C22×C13—42no—
357C22×C13—42no—
358C22×C13—81no—
359C22×C13—81no—
360C22×C13—81no—
361C22×C13—81no—
362C22×C13C2×C4×D411yescharacteristic
363C22×C13#52511yescharacteristic
364C22×C13#52511yescharacteristic
365C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
366C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
367C22×C13#57511yescharacteristic
368C22×C13#57511yescharacteristic
369C4×C13—22no—
370C4×C13—22no—
371C4×C13—42no—
372C4×C13—42no—
373C4×C13C2×#11711yescharacteristic
374C4×C13C4×#4111yescharacteristic
375C4×C13#52911yescharacteristic
376C4×C13#52911yescharacteristic
377Dic13—22no—
378Dic13—22no—
379Dic13—42no—
380Dic13—41no—
381Dic13—42no—
382Dic13—41no—
383Dic13—81no—
384Dic13—81no—
385C22×D4C22×D1311yescharacteristic
386C22×D4C22×D1311yescharacteristic
387C23×C4—21no—
388C23×C4—21no—
389C23×C4—261no—
390C23×C4—261no—
391C2×D8—22no—
392C2×D8—22no—
393C2×D8C2×Dic1314yes—
394C22×C8—131no—
395C22×C8—131no—
396C22×C8C22×D1311yescharacteristic
397C22×C8C22×D1311yescharacteristic
398C4×D4—22no—
399C4×D4—22no—
400C4×D4—262no—
401C4×D4—262no—
402C4×D4—262no—
403C4×D4—262no—
404C4×D4C2×Dic1314yes—
405C4×D4C2×Dic1314yescomplemented
406#117—524no—
407C2×C4⋊C4—131no—
408C2×C4⋊C4—131no—
409C2×C4⋊C4—131no—
410C2×C4⋊C4—131no—
411C2×C4⋊C4—261no—
412C2×C4⋊C4—261no—
413C2×C4⋊C4C22×D1311yescharacteristic
414C2×C4⋊C4C22×D1311yescharacteristic
415C2×C22⋊C4—21no—
416C2×C22⋊C4—21no—
417C2×C22⋊C4—261no—
418C2×C22⋊C4—261no—
419C2×C22⋊C4—261no—
420C2×C22⋊C4—261no—
421C2×C22⋊C4—261no—
422C2×C22⋊C4—261no—
423C2×C42—131no—
424C2×C42—131no—
425C2×C42—131no—
426C2×C42—131no—
427C2×C42—131no—
428C2×C42—131no—
429C2×C42—261no—
430C2×C42—261no—
431C2×C42—261no—
432C2×C42—261no—
433C2×C42—524no—
434C2×C42C22×D1311yescharacteristic
435#107—262no—
436#106—262no—
437#106C2×Dic1314yescomplemented
438#102—262no—
439#102—262no—
440#102—262no—
441#102—262no—
442#102C2×Dic1314yes—
443#102C2×Dic1314yescomplemented
444#97—262no—
445#97C2×Dic1314yescomplemented
446#95—261no—
447#95—261no—
448#95—261no—
449#95—261no—
450#95—261no—
451#95—261no—
452C2×C13—44no—
453C2×C13—82no—
454C2×C13#229011yescharacteristic
455C2×C13C4×AGL(1,ℤ/8)11yescharacteristic
456C2×C13C4×AGL(1,ℤ/8)11yescharacteristic
457C2×C13#310911yescharacteristic
458C2×C13#311111yescharacteristic
459C2×C13#311411yescharacteristic
460C2×C13#311511yescharacteristic
461C24—21no—
462C24—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4#612712yes—
472C2×D4#612712yes—
473C2×D4C22×Dic1312yes—
474C2×D4C22×Dic1312yes—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—131no—
484C22×C4—131no—
485C22×C4—131no—
486C22×C4—131no—
487C22×C4—264no—
488C22×C4—264no—
489C22×C4—261no—
490C22×C4—261no—
491C22×C4—261no—
492C22×C4—261no—
493C22×C4—522no—
494C22×C4—522no—
495C22×C4—522no—
496C22×C4—522no—
497C22×C4D4×D1311yescharacteristic
498C22×C4D4×D1311yescharacteristic
499C22×C4C23×D1311yescharacteristic
500D8—28no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—261no—
506C2×C8—261no—
507C2×C8—261no—
508C2×C8—261no—
509C2×C8—261no—
510C2×C8—261no—
511C2×C8C22×Dic1312yes—
512C2×C8C22×Dic1312yes—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—132no—
516C4⋊C4—132no—
517C4⋊C4—261no—
518C4⋊C4—261no—
519C4⋊C4—261no—
520C4⋊C4—261no—
521C4⋊C4—261no—
522C4⋊C4—261no—
523C4⋊C4—522no—
524C4⋊C4—522no—
525C4⋊C4C22×Dic1312yes—
526C4⋊C4C22×Dic1312yes—
527C22⋊C4—24no—
528C22⋊C4—24no—
529C22⋊C4—264no—
530C22⋊C4—264no—
531C22⋊C4—522no—
532C22⋊C4—522no—
533C22⋊C4—522no—
534C22⋊C4—522no—
535C42—262no—
536C42—262no—
537C42—261no—
538C42—261no—
539C42—261no—
540C42—261no—
541C42—521no—
542C42—521no—
543C42—522no—
544C42—522no—
545C42—522no—
546C42—522no—
547C42#612912yes—
548C42C22×Dic1312yes—
549C13#2112211yescharacteristic, complemented, Sylow
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23C2×D4×D1311yescharacteristic
565D4—22no—
566D4—22no—
567D4—22no—
568D4—22no—
569D4—22no—
570D4—41no—
571D4—41no—
572D4#8818014yescomplemented
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—48no—
580C2×C4—48no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—132no—
586C2×C4—132no—
587C2×C4—261no—
588C2×C4—261no—
589C2×C4—261no—
590C2×C4—261no—
591C2×C4—261no—
592C2×C4—261no—
593C2×C4—261no—
594C2×C4—261no—
595C2×C4—261no—
596C2×C4—261no—
597C2×C4—262no—
598C2×C4—262no—
599C2×C4—528no—
600C2×C4—528no—
601C2×C4—521no—
602C2×C4—521no—
603C2×C4—521no—
604C2×C4—521no—
605C2×C4—521no—
606C2×C4—521no—
607C2×C4—521no—
608C2×C4—521no—
609C2×C4—521no—
610C2×C4—521no—
611C2×C4D4×Dic1312yes—
612C2×C4D4×Dic1312yes—
613C2×C4C2×#612711yescharacteristic
614C2×C4C2×#612711yescharacteristic
615C2×C4C2×#612911yescharacteristic
616C2×C4#41×D1311yescharacteristic
617C2×C4#41×D1311yescharacteristic
618C2×C4C23×Dic1311yescharacteristic
619C8—22no—
620C8—22no—
621C8—521no—
622C8—521no—
623C22—24no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22#1171925211yescharacteristic
638C22#1171925211yescharacteristic
639C22D13×AGL(1,ℤ/8)11yescharacteristic
640C22D13×AGL(1,ℤ/8)11yescharacteristic
641C22#1171947811yescharacteristic
642C22#1171947811yescharacteristic
643C22C2×D4×Dic1311yescharacteristic
644C4—22no—
645C4—22no—
646C4—42no—
647C4—42no—
648C4—262no—
649C4—262no—
650C4—522no—
651C4—521no—
652C4—522no—
653C4—521no—
654C4—1041no—
655C4—1041no—
656C4#1171924311yescharacteristic
657C4#1171924311yescharacteristic
658C4C2×#8818011yescharacteristic
659C4#41×Dic1311yescharacteristic
660C2—44no—
661C2—82no—
662C2#1311728111yescharacteristic
663C2AGL(1,ℤ/8)×Dic1311yescharacteristic
664C2AGL(1,ℤ/8)×Dic1311yescharacteristic
665C2#1312492511yescharacteristic
666C2#1312494111yescharacteristic
667C2#1312497211yescharacteristic
668C2#1312498311yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC2×C13×#442C2
socleC23×C13C22×D4
prime radicalC22×C13×D8C22
cosocleC22×C4×C13C24
hypercenterC2×#442D13
hypocenterC13#21122
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#442

Complements

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

NH ≅ G/Ntype
1. #13117301C2semidirect
2. C4×D4×Dic13C2semidirect
3. #13115465C2semidirect
4. #13115458C2semidirect
5. #13115423C2semidirect
6. #13115416C2semidirect
9. #13115099C2semidirect
10. #13115085C2semidirect
11. #13114733C2semidirect
12. #13114661C2semidirect
13. #13114661C2semidirect
14. #13114584C2semidirect
23. C13×#442C4semidirect
74. C4×D4×C13C2×C4semidirect
110. C13×#106C2×C4semidirect
112. C13×#102C2×C4semidirect
113. C13×#97C2×C4semidirect
253. D4×C13#117semidirect
311. #442Dic13semidirect
405. C4×D4C2×Dic13semidirect
437. #106C2×Dic13semidirect
443. #102C2×Dic13semidirect
445. #97C2×Dic13semidirect
549. C13#21122semidirect
572. D4#88180semidirect

Maximal subgroups

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

subgroupindex
#131173012
C4×D4×Dic132
#131154652
#131154582
#131154232
#131154162
#131151862
#131150992
#131150852
#131147332
#131146612
#131145842
C2×C13×#4422
#2112213