Finite Group Databasebeta

← #29736705

Subgroups — #29736705

Order 15616.

1218 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519768C211yesmaximal, characteristic, complemented
2C2×#16840125C212yesmaximal, complemented
3#21454162C211yesmaximal, characteristic
4#21446964C212yesmaximal, complemented
5#21446963C212yesmaximal, complemented
6#21446786C212yesmaximal, complemented
7#21446137C211yesmaximal, characteristic, complemented
8#21446136C211yesmaximal, characteristic, complemented
9#21446134C211yesmaximal, characteristic, complemented
10#21445583C211yesmaximal, characteristic, complemented
11C61×#1874C211yesmaximal, characteristic, complemented
12C2×#14519773C2211yescharacteristic
13C2×#14519768—28no—
14C2×#14519768—21no—
15C2×#14519768—21no—
16C2×#14519768C2212yes—
17C2×#14519768C2212yes—
18C2×#14519751C2211yescharacteristic
19C2×#14519749—21no—
20C2×#14519749—21no—
21C22×C4⋊Dic61—21no—
22C22×C4⋊Dic61C2211yescharacteristic
23C22×#11758196—21no—
24C22×C4×Dic61C2211yescharacteristic
25#16840125—28no—
26C2×#14519699C2212yes—
27C2×#14519698C2212yes—
28C2×#14519697C2212yes—
29C2×C22⋊#92418C2211yescharacteristic
30C2×#14519687C2212yes—
31#16839665C2211yescharacteristic
32#16839663C2211yescharacteristic
33#16839662—21no—
34#16839662—21no—
35#16839631C2211yescharacteristic
36#16839525C2212yes—
37#16839524C2212yes—
38#16839524C2212yes—
39#16839516C2212yes—
40C2×#121×C61C2211yescharacteristic
41C22×C4⋊C4×C61C2211yescharacteristic
42C2×#102×C61C2212yes—
43C2×C22⋊C8×C61C2211yescharacteristic
44C61×#340C2212yes—
45C22×D244C2311yescharacteristic
46C22×C22⋊D61—21no—
47C22×C22⋊D61—21no—
48C22×C22⋊D61—21no—
49C23×Dic61—21no—
50C23×Dic61D411yescharacteristic
51C23×Dic61D411yescharacteristic
52C22×#92418C2312yes—
53#14519773—24no—
54#14519768—28no—
55#14519768—28no—
56#14519768—24no—
57#14519768—24no—
58#14519768—24no—
59#14519768D418yescomplemented
60#14519751—22no—
61#14519751—22no—
62#14519749—44no—
63#14519749—44no—
64C2×C4⋊Dic61—28no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—22no—
67C2×C4⋊Dic61—22no—
68C2×C4⋊Dic61—24no—
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2312yes—
71C2×#11758196—28no—
72C2×#11758196—22no—
73C2×#11758196—22no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61—21no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×C4×Dic61—24no—
82C2×C4×Dic61D411yescharacteristic
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61C2311yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86#14519699—24no—
87#14519698—24no—
88#14519697—24no—
89C22⋊#92418—22no—
90#14519687—24no—
91C22×D4×C61—21no—
92C22×D4×C61C2311yescharacteristic
93C23×C4×C61—21no—
94C23×C4×C61C2311yescharacteristic
95C22×C8×C61C2312yes—
96#121×C61—22no—
97#121×C61—22no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61—24no—
100C2×C4⋊C4×C61C2311yescharacteristic
101C2×C4⋊C4×C61C2312yes—
102C2×C22⋊C4×C61—21no—
103#102×C61—24no—
104C22⋊C8×C61—22no—
105C24×C61—21no—
106C24×C61C2×D411yescharacteristic
107C23×D61—21no—
108C2×D4×C61—22no—
109C2×D4×C61—21no—
110C2×D4×C61—21no—
111C2×D4×C61—22no—
112C2×D4×C61—42no—
113C2×D4×C61—42no—
114C2×D4×C61—42no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—24no—
118C22×C4×C61—22no—
119C22×C4×C61—21no—
120C22×C4×C61—42no—
121C22×C4×C61—42no—
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61#4112yes—
124C22×C4×C61#4112yes—
125C22×C4×C61C2411yescharacteristic
126C2×C22⋊D61—42no—
127C2×C22⋊D61—42no—
128C2×C22⋊D61—42no—
129C2×C22⋊D61—42no—
130C2×C22⋊D61—42no—
131C2×C22⋊D61—42no—
132C2×C22⋊D61—42no—
133C2×C22⋊D61—42no—
134C2×C22⋊D61—42no—
135C22×Dic61—24no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—22no—
141C22×Dic61—21no—
142C22×Dic61—41no—
143C22×Dic61—41no—
144C22×Dic61—42no—
145C22×Dic61—42no—
146C22×Dic61C2×D411yescharacteristic
147C22×Dic61C2×D411yescharacteristic
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
150C2×D244—22no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244—22no—
154C2×C8×C61—22no—
155C2×C8×C61—22no—
156C2×C8×C61—22no—
157C4⋊C4×C61—24no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61C2×D414yes—
162C22⋊C4×C61—42no—
163C22⋊Dic61—42no—
164C22⋊Dic61—42no—
165#11758198—42no—
166#11758198—42no—
167#11758198—42no—
168C4⋊Dic61—28no—
169C4⋊Dic61—24no—
170C4⋊Dic61—22no—
171C4⋊Dic61—28no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61C2×D414yes—
175#11758196—28no—
176#11758196—28no—
177#11758196—42no—
178#11758196—42no—
179C4×Dic61—22no—
180C4×Dic61—22no—
181C4×Dic61—22no—
182C4×Dic61C2×D414yes—
183C2×#92418—22no—
184C2×#92418—22no—
185C2×#92418—22no—
186C23×C61—21no—
187C23×C61—21no—
188C23×C61—21no—
189C23×C61—21no—
190C23×C61—41no—
191C23×C61—41no—
192C23×C61—41no—
193C23×C61—41no—
194C23×C61—41no—
195C23×C61—41no—
196C23×C61—41no—
197C23×C61#12412yes—
198C23×C61AGL(1,ℤ/8)12yes—
199C23×C61AGL(1,ℤ/8)12yes—
200C23×C61C22×D411yescharacteristic
201C22×D61—41no—
202C22×D61—41no—
203C22×D61—41no—
204C22×D61—41no—
205C22×D61—41no—
206C22×D61—41no—
207C22×D61—41no—
208D4×C61—42no—
209D4×C61—42no—
210D4×C61—84no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—24no—
214C2×C4×C61—24no—
215C2×C4×C61—24no—
216C2×C4×C61—48no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61#12412yes—
223C2×C4×C61C22×D412yes—
224C2×C4×C61C2×#4111yescharacteristic
225C2×C4×C61C2×#4111yescharacteristic
226C2×C4×C612−1+412yes—
227C22⋊D61—84no—
228C22⋊D61—84no—
229C22⋊D61—84no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—24no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—44no—
242C2×Dic61—44no—
243C2×Dic61—48no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61#12012yes—
250C2×Dic61#12012yes—
251D244—42no—
252D244—42no—
253C8×C61—42no—
254#92418—42no—
255#25913—611nomaximal, Sylow
256C22×C61—22no—
257C22×C61—22no—
258C22×C61—22no—
259C22×C61—22no—
260C22×C61—22no—
261C22×C61—22no—
262C22×C61—81no—
263C22×C61—81no—
264C22×C61—81no—
265C22×C61—81no—
266C22×C61—81no—
267C22×C61—81no—
268C22×C61—81no—
269C22×C61#49014yescomplemented
270C22×C61C2×#12011yescharacteristic
271C22×C61C2×#12411yescharacteristic
272C22×C61#53611yescharacteristic
273C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
274C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
275C22×C61#57511yescharacteristic
276C22×C61#57511yescharacteristic
277C2×D61—81no—
278C2×D61—81no—
279C2×D61—81no—
280C2×D61—81no—
281C2×D61—81no—
282C2×D61—81no—
283C2×D61—81no—
284C4×C61—22no—
285C4×C61—44no—
286C4×C61—81no—
287C4×C61#54812yes—
288C4×C61#54812yes—
289Dic61—22no—
290Dic61—22no—
291Dic61—41no—
292Dic61—41no—
293Dic61—44no—
294Dic61—81no—
295C22×#125—611no—
296C2×#490—612no—
297#2856—611no—
298#2025—612no—
299#2025—612no—
300#1992—612no—
301#1874—611no—
302#1874—611no—
303#1874—611no—
304#1874D6111yescharacteristic, complemented
305#1749—611no—
306C2×C61—24no—
307C2×C61—161no—
308C2×C61#243611yescharacteristic
309C2×C61#300611yescharacteristic
310C2×C61#300611yescharacteristic
311C2×C61C2×#49011yescharacteristic
312C2×C61#314211yescharacteristic
313C2×C61#323411yescharacteristic
314C2×C61#323411yescharacteristic
315D61—161no—
316C2×#127—611no—
317C2×#125—612no—
318C2×#125—612no—
319C2×#125—1221no—
320C2×#125—1221no—
321C2×#125—1228no—
322C2×#121—611no—
323C2×#121—1221no—
324C2×#121—1221no—
325C2×#121C2×D6111yescharacteristic
326C22×C4⋊C4—611no—
327C22×C4⋊C4—1221no—
328C22×C4⋊C4—1221no—
329C22×C4⋊C4C2×D6111yescharacteristic
330C22×C42—611no—
331#490—1228no—
332C2×#102—612no—
333C2×#102—612no—
334C2×#102—612no—
335C2×#102C2×D6112yes—
336C2×C22⋊C8—611no—
337C2×C22⋊C8C2×D6111yescharacteristic
338C2×#97—612no—
339#390—611no—
340#390—611no—
341#390—1221no—
342#390—1221no—
343#384—611no—
344#340—612no—
345#340—612no—
346#340—612no—
347#340C2×D6112yes—
348#336—612no—
349C61#2591311yescharacteristic, complemented, Sylow
350C22×D4—21no—
351C22×D4—611no—
352C22×D4—1221no—
353C22×D4—1221no—
354C22×D4—1221no—
355C22×D4C22×D6111yescharacteristic
356C23×C4—21no—
357C23×C4—611no—
358C23×C4—611no—
359C23×C4—1221no—
360C23×C4C22×D6111yescharacteristic
361C22×C8—612no—
362C22×C8C22×D6112yes—
363#127—1224no—
364#125—618no—
365#125—1224no—
366#125—1224no—
367#125—1228no—
368#125—1228no—
369#125—1224no—
370#121—22no—
371#121—22no—
372#121—1222no—
373#121—1222no—
374#121—2444no—
375#121—2444no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—24no—
378C2×C4⋊C4—612no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—1224no—
381C2×C4⋊C4—1222no—
382C2×C4⋊C4—1222no—
383C2×C4⋊C4—1222no—
384C2×C4⋊C4—1222no—
385C2×C4⋊C4—1228no—
386C2×C4⋊C4—1228no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4—1221no—
389C2×C4⋊C4—1221no—
390C2×C4⋊C4C22×D6111yescharacteristic
391C2×C4⋊C4C22×D6112yes—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—1221no—
394C2×C22⋊C4—1221no—
395C2×C22⋊C4—1221no—
396C2×C22⋊C4—1221no—
397C2×C22⋊C4—1221no—
398C2×C42—611no—
399C2×C42—611no—
400C2×C42—611no—
401C2×C42—611no—
402C2×C42—1224no—
403#102—24no—
404#102—1224no—
405#102—1224no—
406#102—1224no—
407C22⋊C8—22no—
408C22⋊C8—1222no—
409#97—1224no—
410C24—21no—
411C24—1221no—
412C24D4×D6111yescharacteristic
413C2×D4—22no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—22no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—1222no—
421C2×D4—1221no—
422C2×D4—1221no—
423C2×D4—1222no—
424C2×D4—2442no—
425C2×D4—2442no—
426C2×D4—2442no—
427C2×D4—2442no—
428C2×D4—2442no—
429C2×D4—2442no—
430C2×D4—2442no—
431C2×D4—2442no—
432C2×D4—2442no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—611no—
441C22×C4—611no—
442C22×C4—611no—
443C22×C4—611no—
444C22×C4—1222no—
445C22×C4—1222no—
446C22×C4—1224no—
447C22×C4—1222no—
448C22×C4—1222no—
449C22×C4—1222no—
450C22×C4—1221no—
451C22×C4—2441no—
452C22×C4—2441no—
453C22×C4—2442no—
454C22×C4—2442no—
455C22×C4#1175822212yes—
456C22×C4D4×D6111yescharacteristic
457C22×C4#1175822412yes—
458C22×C4C23×D6111yescharacteristic
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—1222no—
463C2×C8—1222no—
464C2×C8—1222no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—614no—
470C4⋊C4—1228no—
471C4⋊C4—1228no—
472C4⋊C4—1228no—
473C4⋊C4—1224no—
474C4⋊C4—1228no—
475C4⋊C4—1222no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—2442no—
479C4⋊C4—2442no—
480C4⋊C4D4×D6114yes—
481C22⋊C4—42no—
482C22⋊C4—2442no—
483C22⋊C4—2442no—
484C22⋊C4—2442no—
485C22⋊C4—2442no—
486C22⋊C4—2442no—
487C42—614no—
488C42—1222no—
489C42—1222no—
490C42—1222no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—2441no—
503C23—2441no—
504C23—2441no—
505C23—2441no—
506C23—2441no—
507C23—2441no—
508C23—2441no—
509C23#1451976312yes—
510C23#1451980112yes—
511C23#1451980412yes—
512C23C2×D4×D6111yescharacteristic
513D4—42no—
514D4—42no—
515D4—84no—
516D4—4884no—
517D4—4884no—
518D4—4884no—
519D4—2442no—
520D4—2442no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—48no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—612no—
533C2×C4—612no—
534C2×C4—1221no—
535C2×C4—1221no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—1224no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—2444no—
546C2×C4—2444no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2448no—
551C2×C4—2442no—
552C2×C4—2442no—
553C2×C4#1451976312yes—
554C2×C4C2×D4×D6112yes—
555C2×C4C2×#1175822411yescharacteristic
556C2×C4C2×#1175822211yescharacteristic
557C2×C4#1451982412yes—
558C8—42no—
559C8—2442no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—4881no—
574C22—4881no—
575C22—4881no—
576C22—4881no—
577C22—4881no—
578C22—4881no—
579C22—4881no—
580C22#1684012514yescomplemented
581C22#120×D6111yescharacteristic
582C22C2×#1451976311yescharacteristic
583C22#1684033411yescharacteristic
584C22C2×#1451980111yescharacteristic
585C22C2×#1451980411yescharacteristic
586C22#1684055811yescharacteristic
587C22#1684056111yescharacteristic
588C4—22no—
589C4—44no—
590C4—81no—
591C4—1222no—
592C4—1222no—
593C4—4881no—
594C4—2441no—
595C4—2441no—
596C4—2444no—
597C4#1684041812yes—
598C4#1684042112yes—
599C2—24no—
600C2—161no—
601C2—9761no—
602C2#2144947511yescharacteristic
603C2#2145506611yescharacteristic
604C2#2145507011yescharacteristic
605C2C2×#1684012511yescharacteristic
606C2#2145621711yescharacteristic
607C2#2145700611yescharacteristic
608C2#2145700811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1874C2
socleC23×C61C22×D4
prime radical#21454162C2
hypercenter#1874D61
hypocenterC61#25913
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1874

Complements

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

NH ≅ G/Ntype
1. C22×#14519768C2semidirect
2. C2×#16840125C2semidirect
4. #21446964C2semidirect
5. #21446963C2semidirect
6. #21446786C2semidirect
7. #21446137C2semidirect
8. #21446136C2semidirect
9. #21446134C2semidirect
10. #21445583C2semidirect
11. C61×#1874C2semidirect
59. #14519768D4semidirect
269. C22×C61#490semidirect
304. #1874D61semidirect
349. C61#25913semidirect
580. C22#16840125semidirect

Maximal subgroups

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

subgroupindex
C22×#145197682
C2×#168401252
#214541622
#214469642
#214469632
#214467862
#214461372
#214461362
#214461342
#214455832
C61×#18742
#2591361