Finite Group Databasebeta

← #29680055

Subgroups — #29680055

Order 15616.

1782 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D4×D61C211yesmaximal, characteristic
2C2×#16839817C211yesmaximal, characteristic, complemented
3#21448223C211yesmaximal, characteristic, complemented
4#21445908C212yesmaximal, complemented
5#21445818C212yesmaximal, complemented
6#21445817C212yesmaximal, complemented
7#21445748C212yesmaximal
8#21445647C211yesmaximal, characteristic, complemented
9#21445645C211yesmaximal, characteristic, complemented
10#21445644C211yesmaximal, characteristic, complemented
11C61×#1762C211yesmaximal, characteristic, complemented
12C22×D4×D61C2211yescharacteristic
13C23×C4×D61—21no—
14C2×C4×D244C2211yescharacteristic
15C2×D4×Dic61C2211yescharacteristic
16C2×#14519739C2211yescharacteristic
17C2×C4×C22⋊D61—21no—
18C2×#14519737—21no—
19C4×D4×D61—24no—
20C4×D4×D61C418yescomplemented
21C2×#14519734—21no—
22C2×C4⋊C4×D61C2211yescharacteristic
23C2×C22⋊C4×D61—21no—
24C2×C42×D61C2211yescharacteristic
25#16839817—24no—
26C2×#14519706C2212yescomplemented
27C2×#14519699C2212yes—
28C2×#14519698C2212yes—
29C2×#14519697C2212yes—
30C2×#14519689C2212yes—
31C2×#14519688C2211yescharacteristic, complemented
32#16839631C2212yescomplemented
33#16839626—21no—
34#16839621—21no—
35#16839619C2212yes—
36C4×#11758196—21no—
37C4×C4⋊Dic61C2211yescharacteristic, complemented
38#16839525C2212yescomplemented
39#16839524C2212yescomplemented
40#16839524C2212yescomplemented
41C2×C4×D4×C61C2211yescharacteristic
42C2×#102×C61C2212yes—
43C2×#97×C61C2211yescharacteristic, complemented
44C4×C4⋊C4×C61C2211yescharacteristic, complemented
45C61×#340C2212yescomplemented
46C24×D61—21no—
47C22×D244C2311yescharacteristic
48C22×C22⋊D61—21no—
49C2×D4×D61—24no—
50C2×D4×D61—24no—
51C2×D4×D61—22no—
52C2×D4×D61C2×C414yes—
53C23×Dic61—21no—
54C22×C4×D61—21no—
55C22×C4×D61—21no—
56C22×C4×D61—21no—
57C22×C4×D61—28no—
58C22×C4×D61—48no—
59C22×C4×D61D411yescharacteristic
60C22×C4×D61D411yescharacteristic
61C22×C4×D61C2311yescharacteristic
62C22×#92418C2312yes—
63C4×D244—22no—
64C4×D244C2×C414yes—
65D4×Dic61—22no—
66D4×Dic61C2×C414yes—
67#14519739—22no—
68#14519739C2×C414yes—
69C4×C22⋊D61—28no—
70#14519737—28no—
71#14519734—44no—
72C2×C4⋊Dic61C2312yes—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758196—21no—
75C2×#11758196—22no—
76C2×#11758196—22no—
77C4⋊C4×D61—22no—
78C4⋊C4×D61C2×C414yes—
79C2×C22⋊Dic61—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—22no—
83C22⋊C4×D61—28no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61D411yescharacteristic
86C2×C4×Dic61D411yescharacteristic
87C2×C4×Dic61C2311yescharacteristic
88C42×D61C2×C414yes—
89C42×D61D414yes—
90#14519706—24no—
91#14519699—24no—
92#14519698—24no—
93#14519697—24no—
94#14519689—24no—
95#14519688—22no—
96#14519684—21no—
97#14519684—22no—
98C22×D4×C61C2311yescharacteristic
99C23×C4×C61—21no—
100C22×C8×C61C2312yes—
101C4×D4×C61—22no—
102C4×D4×C61C2×C414yes—
103C2×C4⋊C4×C61C2312yes—
104C2×C4⋊C4×C61C2311yescharacteristic
105C2×C22⋊C4×C61—21no—
106C2×C42×C61—21no—
107C2×C42×C61C2311yescharacteristic
108#102×C61—24no—
109#97×C61—22no—
110#95×C61—21no—
111C24×C61—21no—
112C23×D61—28no—
113C23×D61—21no—
114C23×D61—24no—
115C23×D61—44no—
116C23×D61—42no—
117C23×D61—42no—
118C23×D61C2×D411yescharacteristic
119C2×D4×C61—21no—
120C2×D4×C61—22no—
121C2×D4×C61C22×C412yes—
122C2×D4×C61#4114yes—
123C22×C4×C61—21no—
124C22×C4×C61—24no—
125C22×C4×C61—22no—
126C22×C4×C61—44no—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2411yescharacteristic
130C2×C22⋊D61—28no—
131C2×C22⋊D61—24no—
132C22×Dic61—21no—
133C22×Dic61—24no—
134C22×Dic61—22no—
135C22×Dic61—44no—
136C22×Dic61C2×D411yescharacteristic
137C22×Dic61C2×D411yescharacteristic
138C22×Dic61C2×D411yescharacteristic
139D4×D61—216no—
140D4×D61—28no—
141D4×D61—28no—
142C2×D244—21no—
143C2×D244—22no—
144C2×D244C22×C412yes—
145C2×D244#4114yes—
146C2×C4×D61—22no—
147C2×C4×D61—22no—
148C2×C4×D61—22no—
149C2×C4×D61—22no—
150C2×C4×D61—24no—
151C2×C4×D61—24no—
152C2×C4×D61—24no—
153C2×C4×D61—24no—
154C2×C4×D61—432no—
155C2×C4×D61—44no—
156C2×C4×D61—44no—
157C2×C4×D61—44no—
158C2×C4×D61C22⋊C414yes—
159C2×C4×D61C22⋊C414yes—
160C2×C4×D61C22×C412yes—
161C2×C4×D61C2×D412yes—
162C2×C8×C61—22no—
163C2×C8×C61—22no—
164C2×C8×C61—22no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—22no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61C22×C412yes—
169C22⋊C4×C61—24no—
170C42×C61—42no—
171C42×C61C22×C412yes—
172C42×C61C2×D412yes—
173C22⋊Dic61—24no—
174#11758198—24no—
175#11758198—24no—
176#11758198—44no—
177C4⋊Dic61—21no—
178C4⋊Dic61—22no—
179C4⋊Dic61—22no—
180C4⋊Dic61C22×C412yes—
181#11758196—24no—
182#11758196—44no—
183#11758196—44no—
184C4×Dic61—42no—
185C4×Dic61C22⋊C414yes—
186C4×Dic61C22⋊C414yes—
187C4×Dic61C22×C412yes—
188C4×Dic61C2×D412yes—
189C2×#92418—22no—
190C2×#92418—22no—
191C2×#92418—22no—
192C23×C61—22no—
193C23×C61—24no—
194C23×C61—42no—
195C23×C61—41no—
196C23×C61—41no—
197C23×C61C22×D411yescharacteristic
198C22×D61—22no—
199C22×D61—22no—
200C22×D61—216no—
201C22×D61—22no—
202C22×D61—24no—
203C22×D61—48no—
204C22×D61—416no—
205C22×D61—416no—
206C22×D61—42no—
207C22×D61—41no—
208C22×D61—41no—
209C22×D61—48no—
210C22×D61C2×C22⋊C412yes—
211C22×D61AGL(1,ℤ/8)12yes—
212C22×D61#13712yes—
213D4×C61—22no—
214D4×C61—24no—
215D4×C61#11714yescomplemented
216C2×C4×C61—21no—
217C2×C4×C61—22no—
218C2×C4×C61—21no—
219C2×C4×C61—22no—
220C2×C4×C61—48no—
221C2×C4×C61—42no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61C2×C22⋊C412yes—
226C2×C4×C61C2×C22⋊C412yes—
227C2×C4×C61C23×C411yescharacteristic
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C2×#4112yes—
230C2×C4×C61C2×#4112yes—
231C22⋊D61—216no—
232C2×Dic61—22no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—22no—
236C2×Dic61—42no—
237C2×Dic61—48no—
238C2×Dic61—42no—
239C2×Dic61—42no—
240C2×Dic61—42no—
241C2×Dic61C2×C22⋊C412yes—
242C2×Dic61C2×C22⋊C412yes—
243C2×Dic61C2×C22⋊C412yes—
244C2×Dic61AGL(1,ℤ/8)14yes—
245D244—22no—
246D244—24no—
247D244#11714yescomplemented
248C4×D61—28no—
249C4×D61—22no—
250C4×D61—22no—
251C4×D61—24no—
252C4×D61—28no—
253C4×D61—48no—
254C4×D61—48no—
255C4×D61—48no—
256C8×C61—42no—
257#92418—42no—
258#20738—611nomaximal, Sylow
259C22×C61—24no—
260C22×C61—44no—
261C22×C61—44no—
262C22×C61—42no—
263C22×C61—42no—
264C22×C61C22×C22⋊C411yescharacteristic
265C22×C61#52512yes—
266C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
267C22×C61C2×AGL(1,ℤ/8)12yes—
268C22×C61C2×#13711yescharacteristic
269C2×D61—24no—
270C2×D61—24no—
271C2×D61—22no—
272C2×D61—22no—
273C2×D61—24no—
274C2×D61—432no—
275C2×D61—44no—
276C2×D61—44no—
277C2×D61—42no—
278C2×D61—42no—
279C2×D61#41714yes—
280C4×C61—22no—
281C4×C61—22no—
282C4×C61—42no—
283C4×C61—82no—
284C4×C61C2×#11711yescharacteristic
285C4×C61C2×#11711yescharacteristic
286C4×C61#52512yes—
287Dic61—22no—
288Dic61—22no—
289Dic61—42no—
290Dic61—82no—
291Dic61#41814yes—
292C22×C4×D4—611no—
293C2×#432—611no—
294#2285—611no—
295#1824—612no—
296#1802—612no—
297#1802—612no—
298#1786—612no—
299#1762—611no—
300#1762—611no—
301#1762—611no—
302#1762D6111yescharacteristic, complemented
303C2×C61—44no—
304C2×C61#228011yescharacteristic
305C2×C61C2×#41811yescharacteristic
306C2×C61C2×#41711yescharacteristic
307C2×C61#310912yes—
308C2×C61#311012yes—
309D61—24no—
310D61—44no—
311C23×D4—611no—
312C24×C4—1221no—
313C2×C4×D4—611no—
314C2×C4×D4—618no—
315C2×C4×D4—611no—
316C2×C4×D4—611no—
317C2×C4×D4—1221no—
318C2×C4×D4—1224no—
319C2×C4×D4—1221no—
320C2×C4×D4C2×D6111yescharacteristic
321C2×#117—1221no—
322C22×C4⋊C4—611no—
323C22×C22⋊C4—1221no—
324C22×C42—611no—
325#432—1224no—
326C2×C4⋊C8—612no—
327C2×#102—612no—
328C2×#102—612no—
329C2×#102—612no—
330C2×#102C2×D6112yes—
331C2×C22⋊C8—612no—
332C2×#97—611no—
333C2×#97C2×D6111yescharacteristic, complemented
334#384—612no—
335#382—1221no—
336#380—612no—
337#380—1221no—
338C4×C4⋊C4—611no—
339C4×C4⋊C4—1221no—
340C4×C4⋊C4C2×D6111yescharacteristic, complemented
341#340—612no—
342#340—612no—
343#340—612no—
344#340C2×D6112yescomplemented
345C61#2073811yescharacteristic, complemented, Sylow
346C25—1221no—
347C22×D4—614no—
348C22×D4—611no—
349C22×D4—1222no—
350C22×D4—1221no—
351C22×D4—1224no—
352C22×D4—1224no—
353C22×D4C22×D6111yescharacteristic
354C23×C4—21no—
355C23×C4—611no—
356C23×C4—611no—
357C23×C4—611no—
358C23×C4—1221no—
359C23×C4—1228no—
360C23×C4—1221no—
361C23×C4—1221no—
362C23×C4—1221no—
363C23×C4—2448no—
364C22×C8—612no—
365C22×C8C22×D6112yes—
366C4×D4—22no—
367C4×D4—614no—
368C4×D4—614no—
369C4×D4—614no—
370C4×D4—1228no—
371C4×D4—1222no—
372C4×D4—1228no—
373C4×D4—1222no—
374C4×D4—1222no—
375C4×D4C4×D6114yes—
376#117—2444no—
377C2×C4⋊C4—611no—
378C2×C4⋊C4—614no—
379C2×C4⋊C4—612no—
380C2×C4⋊C4—1222no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—1222no—
383C2×C4⋊C4—1222no—
384C2×C4⋊C4C22×D6112yes—
385C2×C4⋊C4C22×D6111yescharacteristic
386C2×C22⋊C4—21no—
387C2×C22⋊C4—1228no—
388C2×C22⋊C4—1221no—
389C2×C22⋊C4—1221no—
390C2×C22⋊C4—1221no—
391C2×C22⋊C4—1222no—
392C2×C42—21no—
393C2×C42—611no—
394C2×C42—611no—
395C2×C42—611no—
396C2×C42—614no—
397C2×C42—614no—
398C2×C42—1221no—
399C2×C42C22×D6111yescharacteristic
400C4⋊C8—1224no—
401#102—24no—
402#102—1224no—
403#102—1224no—
404#102—1224no—
405C22⋊C8—1224no—
406#97—22no—
407#97—1222no—
408#95—21no—
409#95—1222no—
410#95—1221no—
411C24—21no—
412C24—611no—
413C24—1224no—
414C24—1228no—
415C24—1221no—
416C24—2442no—
417C24—2442no—
418C24—2444no—
419C2×D4—21no—
420C2×D4—22no—
421C2×D4—612no—
422C2×D4—614no—
423C2×D4—1224no—
424C2×D4—1228no—
425C2×D4—12216no—
426C2×D4—1228no—
427C2×D4—1228no—
428C2×D4—1221no—
429C2×D4—1222no—
430C2×D4C2×C4×D6112yes—
431C2×D4#1175822214yes—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—44no—
436C22×C4—611no—
437C22×C4—611no—
438C22×C4—611no—
439C22×C4—612no—
440C22×C4—612no—
441C22×C4—614no—
442C22×C4—614no—
443C22×C4—1224no—
444C22×C4—1224no—
445C22×C4—1222no—
446C22×C4—1222no—
447C22×C4—1224no—
448C22×C4—1224no—
449C22×C4—1222no—
450C22×C4—1222no—
451C22×C4—1221no—
452C22×C4—1224no—
453C22×C4—1222no—
454C22×C4—2444no—
455C22×C4—2444no—
456C22×C4—24432no—
457C22×C4—2444no—
458C22×C4—2444no—
459C22×C4D4×D6111yescharacteristic
460C22×C4D4×D6111yescharacteristic
461C22×C4C23×D6111yescharacteristic
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—1222no—
466C2×C8—1222no—
467C2×C8—1222no—
468C4⋊C4—21no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—612no—
472C4⋊C4—1224no—
473C4⋊C4—1221no—
474C4⋊C4—1222no—
475C4⋊C4—1222no—
476C4⋊C4—2444no—
477C4⋊C4—2444no—
478C4⋊C4C2×C4×D6112yes—
479C22⋊C4—24no—
480C22⋊C4—1224no—
481C22⋊C4—1224no—
482C22⋊C4—1224no—
483C22⋊C4—2444no—
484C42—42no—
485C42—612no—
486C42—612no—
487C42—614no—
488C42—614no—
489C42—2442no—
490C42C2×C4×D6112yes—
491C42D4×D6112yes—
492C23—22no—
493C23—24no—
494C23—42no—
495C23—41no—
496C23—41no—
497C23—612no—
498C23—612no—
499C23—612no—
500C23—12216no—
501C23—1222no—
502C23—1222no—
503C23—1222no—
504C23—1224no—
505C23—2448no—
506C23—2448no—
507C23—24416no—
508C23—24416no—
509C23—2442no—
510C23—2441no—
511C23—2441no—
512C23C2×D4×D6111yescharacteristic
513D4—22no—
514D4—24no—
515D4—614no—
516D4—12216no—
517D4—1222no—
518D4—1224no—
519D4#1451973414yescomplemented
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—48no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—612no—
530C2×C4—614no—
531C2×C4—612no—
532C2×C4—612no—
533C2×C4—1228no—
534C2×C4—1228no—
535C2×C4—1222no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—1222no—
539C2×C4—1222no—
540C2×C4—1224no—
541C2×C4—1222no—
542C2×C4—2448no—
543C2×C4—2448no—
544C2×C4—2448no—
545C2×C4—2448no—
546C2×C4—2442no—
547C2×C4—2442no—
548C2×C4—2442no—
549C2×C4—2442no—
550C2×C4C22⋊C4×D6112yes—
551C2×C4C22⋊C4×D6112yes—
552C2×C4C22×C4×D6111yescharacteristic
553C2×C4C2×D4×D6111yescharacteristic
554C2×C4#41×D6112yes—
555C2×C4C2×#1175822212yes—
556C8—42no—
557C8—2442no—
558C22—24no—
559C22—44no—
560C22—44no—
561C22—42no—
562C22—42no—
563C22—614no—
564C22—1222no—
565C22—1224no—
566C22—1224no—
567C22—1222no—
568C22—1224no—
569C22—24432no—
570C22—2444no—
571C22—2444no—
572C22—2442no—
573C22—2442no—
574C22C2×C22⋊C4×D6111yescharacteristic
575C22#1684027112yes—
576C22AGL(1,ℤ/8)×D6111yescharacteristic
577C22AGL(1,ℤ/8)×D6112yes—
578C22#137×D6111yescharacteristic
579C4—22no—
580C4—22no—
581C4—42no—
582C4—82no—
583C4—614no—
584C4—1222no—
585C4—1222no—
586C4—4882no—
587C4—2442no—
588C4#117×D6111yescharacteristic
589C4C2×#1451973411yescharacteristic
590C4#1684027112yes—
591C2—44no—
592C2—1224no—
593C2—2444no—
594C2#2144820011yescharacteristic
595C2#418×D6111yescharacteristic
596C2#417×D6111yescharacteristic
597C2#2145590812yes—
598C2#2145591612yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1762C2
socleC23×C61C22×D4
prime radicalC22×D4×D61C22
cosocleC22×C4×C61C24
hypercenter#1762D61
hypocenterC61#20738
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
2. C2×#16839817C2semidirect
3. #21448223C2semidirect
4. #21445908C2semidirect
5. #21445818C2semidirect
6. #21445817C2semidirect
8. #21445647C2semidirect
9. #21445645C2semidirect
10. #21445644C2semidirect
11. C61×#1762C2semidirect
20. C4×D4×D61C4semidirect
26. C2×#14519706C22semidirect
31. C2×#14519688C22semidirect
32. #16839631C22semidirect
37. C4×C4⋊Dic61C22semidirect
38. #16839525C22semidirect
39. #16839524C22semidirect
40. #16839524C22semidirect
43. C2×#97×C61C22semidirect
44. C4×C4⋊C4×C61C22semidirect
45. C61×#340C22semidirect
215. D4×C61#117semidirect
247. D244#117semidirect
302. #1762D61semidirect
333. C2×#97C2×D61semidirect
340. C4×C4⋊C4C2×D61semidirect
344. #340C2×D61semidirect
345. C61#20738semidirect
519. D4#14519734semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D4×D612
C2×#168398172
#214482232
#214459082
#214458182
#214458172
#214457482
#214456472
#214456452
#214456442
C61×#17622
#2073861