Finite Group Databasebeta

← #29707907

Subgroups — #29707907

Order 15616.

1454 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519805C211yesmaximal, characteristic
2#21454339C211yesmaximal, characteristic, complemented
3C2×#16839768C211yesmaximal, characteristic, complemented
4C22×#14519693C211yesmaximal, characteristic, complemented
5#21454263C211yesmaximal, characteristic, complemented
6#21447113C218yesmaximal, complemented
7#21445557C211yesmaximal, characteristic, complemented
8C2×C61×#430C211yesmaximal, characteristic, complemented
9C22×#11758222C2211yescharacteristic
10C23×D244C2211yescharacteristic
11C2×#14519805—24no—
12C2×#14519805—22no—
13C2×#14519805—24no—
14C2×#14519805C2214yes—
15C22×#11758190—21no—
16C22×D488—21no—
17C22×#11758194C2211yescharacteristic
18C2×#14519736C2211yescharacteristic
19#16839772—21no—
20#16839768—22no—
21#16839768—22no—
22#16839768—21no—
23#16839764C2214yescomplemented
24#16839759—22no—
25C2×#14519699C2214yescomplemented
26C2×#14519693—24no—
27C2×#14519693C2214yescomplemented
28#16839730—22no—
29#16839728C2214yescomplemented
30#16839725—22no—
31#16839530C2214yescomplemented
32C22×M16×C61C2211yescharacteristic
33C22×M16×C61C2211yescharacteristic
34C22×M16×C61C2211yescharacteristic
35C61×#430—24no—
36C2×#108×C61C2214yescomplemented
37C24×D61—21no—
38C2×#11758222—21no—
39C2×#11758222—21no—
40C2×#11758222—22no—
41C2×#11758222—22no—
42C2×#11758222—22no—
43C2×#11758222—22no—
44C2×#11758222D412yes—
45C2×#11758222C2312yes—
46C22×Dic122D411yescharacteristic
47C22×D244—21no—
48C22×D244—21no—
49C22×D244—21no—
50C22×D244—21no—
51C22×D244—22no—
52C22×D244—22no—
53C22×D244—22no—
54C22×D244—21no—
55C22×D244D414yes—
56C22×D244D411yescharacteristic
57C22×D244C2312yes—
58C22×C22⋊D61—21no—
59C22×C4×D61—21no—
60#14519805—216no—
61#14519805—28no—
62#14519805—28no—
63C2×#11758190—28no—
64C2×#11758190—24no—
65C2×D488—28no—
66C2×D488—24no—
67C2×#11758194—21no—
68C2×#11758194—22no—
69C2×#11758194—22no—
70C2×#11758194C2312yes—
71C22×#92418—21no—
72#14519736—21no—
73#14519736—21no—
74#14519736—21no—
75#14519736—21no—
76C2×C4⋊Dic61C2312yes—
77C2×C22⋊Dic61—21no—
78C2×C4×Dic61—21no—
79#14519702—24no—
80#14519699—24no—
81#14519699—28no—
82#14519693—28no—
83#14519689—28no—
84C23×C4×C61C2311yescharacteristic
85C2×M16×C61—22no—
86C2×M16×C61—21no—
87C2×M16×C61—22no—
88C2×M16×C61—22no—
89C2×M16×C61—21no—
90C2×M16×C61—22no—
91C2×M16×C61—22no—
92C2×M16×C61D414yes—
93C2×M16×C61D412yes—
94C2×M16×C61D412yes—
95C2×M16×C61C2312yes—
96C2×M16×C61C2312yes—
97C22×C8×C61—21no—
98C22×C8×C61—21no—
99C22×C8×C61C2312yes—
100#108×C61—28no—
101C24×C61#4111yescharacteristic
102C23×D61—21no—
103C23×D61—24no—
104C23×D61—22no—
105C23×D61—41no—
106C23×D61—41no—
107C23×D61—42no—
108C23×D61—84no—
109C22×C4×C61C2×D412yes—
110C22×C4×C61C2×D412yes—
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61#4112yes—
116C22×C4×C61#4111yescharacteristic
117C22×C4×C61#4112yes—
118C22×C4×C61C2411yescharacteristic
119C2×C22⋊D61—22no—
120C2×C22⋊D61—22no—
121C2×C22⋊D61—42no—
122C2×C22⋊D61—42no—
123C22×Dic61—21no—
124C22×Dic61—22no—
125#11758222—24no—
126#11758222—24no—
127#11758222—22no—
128#11758222—22no—
129#11758222—22no—
130#11758222—22no—
131#11758222—42no—
132#11758222—42no—
133#11758222—42no—
134#11758222—42no—
135C2×D244—24no—
136C2×D244—24no—
137C2×D244—22no—
138C2×D244—24no—
139C2×D244—24no—
140C2×D244—22no—
141C2×D244—22no—
142C2×D244—22no—
143C2×D244—22no—
144C2×D244—22no—
145C2×D244—21no—
146C2×D244—24no—
147C2×D244—22no—
148C2×D244—42no—
149C2×D244—42no—
150C2×D244—41no—
151C2×D244—41no—
152C2×D244—42no—
153C2×D244—42no—
154C2×D244—42no—
155C2×D244C2×D414yes—
156C2×D244C2×D412yes—
157C2×C4×D61—22no—
158C2×C4×D61—22no—
159C2×C4×D61—42no—
160C2×C4×D61—42no—
161C2×Dic122—22no—
162C2×Dic122—22no—
163C2×Dic122—21no—
164C2×Dic122C2×D412yes—
165M16×C61—24no—
166M16×C61—24no—
167M16×C61—24no—
168M16×C61—22no—
169M16×C61—22no—
170M16×C61—22no—
171M16×C61—42no—
172M16×C61C2×D414yes—
173C2×C8×C61—24no—
174C2×C8×C61—22no—
175C2×C8×C61—22no—
176C2×C8×C61—22no—
177C2×C8×C61—22no—
178C2×C8×C61—42no—
179C2×C8×C61C2×D414yes—
180C22⋊Dic61—41no—
181C22⋊Dic61—41no—
182C4⋊Dic61—22no—
183C4⋊Dic61—22no—
184C4×Dic61—41no—
185C4×Dic61—41no—
186#11758194—22no—
187#11758194—22no—
188#11758194—42no—
189C2×#92418—22no—
190C2×#92418—42no—
191D488—216no—
192#11758190—216no—
193C23×C61—42no—
194C23×C61#12111yescharacteristic
195C23×C61#12111yescharacteristic
196C23×C61#12312yes—
197C23×C61#12412yes—
198C23×C61C2×#4111yescharacteristic
199C22×D61—22no—
200C22×D61—24no—
201C22×D61—42no—
202C22×D61—42no—
203C22×D61—42no—
204C22×D61—44no—
205C22×D61—44no—
206C22×D61—82no—
207C22×D61—84no—
208C22×D61—84no—
209C22×D61—88no—
210C22×D61—84no—
211C22×D61—88no—
212C2×C4×C61#12012yes—
213C2×C4×C61#12012yes—
214C2×C4×C61#12112yes—
215C2×C4×C61#12112yes—
216C2×C4×C61#12112yes—
217C2×C4×C61#12112yes—
218C2×C4×C61#12111yescharacteristic
219C2×C4×C61#12111yescharacteristic
220C2×C4×C61#12312yes—
221C2×C4×C61#12312yes—
222C2×C4×C61#12312yes—
223C2×C4×C61#12412yes—
224C2×C4×C61C22×D411yescharacteristic
225C2×C4×C61C22×D411yescharacteristic
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C2×#4111yescharacteristic
229C2×C4×C61C2×#4111yescharacteristic
230C22⋊D61—42no—
231C22⋊D61—42no—
232C22⋊D61—44no—
233C2×Dic61—22no—
234C2×Dic61—42no—
235C2×Dic61—42no—
236C2×Dic61—82no—
237D244—24no—
238D244—22no—
239D244—24no—
240D244—22no—
241D244—42no—
242D244—42no—
243D244—44no—
244D244—44no—
245D244—42no—
246C4×D61—42no—
247C4×D61—42no—
248C4×D61—44no—
249Dic122—22no—
250Dic122—22no—
251Dic122—42no—
252C8×C61—24no—
253C8×C61—24no—
254C8×C61—42no—
255#92418—42no—
256#23531—611nomaximal, Sylow
257C22×C61—22no—
258C22×C61—22no—
259C22×C61—22no—
260C22×C61—42no—
261C22×C61—42no—
262C22×C61#39412yes—
263C22×C61#39412yes—
264C22×C61C2×#12111yescharacteristic
265C22×C61C2×#12311yescharacteristic
266C22×C61C2×#12411yescharacteristic
267C2×D61—42no—
268C2×D61—44no—
269C2×D61—44no—
270C2×D61—84no—
271C2×D61—88no—
272C2×D61—88no—
273C2×D61—84no—
274C2×D61—84no—
275C2×D61—84no—
276C4×C61#39412yes—
277C4×C61#39412yes—
278C4×C61C2×#12011yescharacteristic
279C4×C61C2×#12111yescharacteristic
280C4×C61C2×#12111yescharacteristic
281C4×C61C2×#12311yescharacteristic
282Dic61—42no—
283Dic61—82no—
284C22×AGL(1,ℤ/8)—611no—
285C2×#430D6111yescharacteristic, complemented
286#2896—611no—
287C2×#418—611no—
288C22×#100—611no—
289#2879—611no—
290#2051—618no—
291#1742—611no—
292C2×C61—22no—
293C2×C61—21no—
294C2×C61—21no—
295C2×C61—22no—
296C2×C61C2×#39411yescharacteristic
297C2×C61#306812yes—
298D61—82no—
299D61—84no—
300C22×#41—611no—
301C23×D4—611no—
302C2×AGL(1,ℤ/8)—614no—
303C2×AGL(1,ℤ/8)—1224no—
304C2×AGL(1,ℤ/8)—1222no—
305C2×AGL(1,ℤ/8)—1224no—
306C22×SD16—1221no—
307C22×D8—1221no—
308C22×M16—611no—
309C22×M16C2×D6111yescharacteristic
310C22×M16C2×D6111yescharacteristic
311C22×M16C2×D6111yescharacteristic
312C2×#117—611no—
313#430—24no—
314C2×#108C2×D6114yescomplemented
315#419—1221no—
316#418—1221no—
317#418—1222no—
318#418—1222no—
319#417—614no—
320#416—1222no—
321C2×#102—614no—
322C2×#100—614no—
323C2×#100—1224no—
324#408—614no—
325#408—1222no—
326#407—1222no—
327#343—614no—
328C61#2353111yescharacteristic, complemented, Sylow
329C25—1221no—
330C2×#41—612no—
331C2×#41—612no—
332C2×#41—1221no—
333C2×#41—1221no—
334C2×#41—1222no—
335C2×#41—1222no—
336C2×#41—1222no—
337C2×#41—1222no—
338C22×Q8—611no—
339C22×D4—611no—
340C22×D4—614no—
341C22×D4—612no—
342C22×D4—1221no—
343C22×D4—1221no—
344C22×D4—1221no—
345C22×D4—1222no—
346C22×D4—1221no—
347C22×D4—1221no—
348C22×D4—1222no—
349C22×D4—1221no—
350C22×D4—1222no—
351C23×C4—1221no—
352C23×C4C22×D6111yescharacteristic
353AGL(1,ℤ/8)—1228no—
354AGL(1,ℤ/8)—12216no—
355AGL(1,ℤ/8)—1228no—
356C2×SD16—1224no—
357C2×SD16—1228no—
358C2×D8—1224no—
359C2×D8—1228no—
360C2×M16—22no—
361C2×M16—21no—
362C2×M16—22no—
363C2×M16—22no—
364C2×M16—21no—
365C2×M16—22no—
366C2×M16—22no—
367C2×M16—612no—
368C2×M16—1221no—
369C2×M16—1222no—
370C2×M16—1222no—
371C2×M16D24412yes—
372C2×M16D24412yes—
373C2×M16C22⋊D6114yes—
374C2×M16C22×D6112yes—
375C2×M16C22×D6112yes—
376C22×C8—21no—
377C22×C8—21no—
378C22×C8—1221no—
379C22×C8C22×D6112yes—
380#117—1221no—
381#117—1221no—
382#117—1221no—
383#117—1221no—
384C2×C4⋊C4—612no—
385C2×C22⋊C4—1221no—
386C2×C42—1221no—
387#108—28no—
388#103—1224no—
389#102—1224no—
390#102—1228no—
391#100—1228no—
392C22⋊C8—1228no—
393C24—1222no—
394C24—1221no—
395C24—1224no—
396C24—4884no—
397C24—2441no—
398C24—2441no—
399C24—2442no—
400C24#1175822611yescharacteristic
401#41—1222no—
402#41—1222no—
403#41—1222no—
404#41—1222no—
405#41—1224no—
406#41—1224no—
407#41—2442no—
408#41—2442no—
409#41—2442no—
410#41—2442no—
411C2×Q8—612no—
412C2×Q8—1221no—
413C2×Q8—1222no—
414C2×Q8—1222no—
415C2×D4—614no—
416C2×D4—612no—
417C2×D4—1222no—
418C2×D4—1222no—
419C2×D4—1222no—
420C2×D4—1222no—
421C2×D4—1222no—
422C2×D4—1222no—
423C2×D4—1222no—
424C2×D4—1222no—
425C2×D4—1224no—
426C2×D4—1224no—
427C2×D4—1224no—
428C2×D4—1224no—
429C2×D4—1221no—
430C2×D4—1222no—
431C2×D4—1224no—
432C2×D4—2442no—
433C2×D4—2442no—
434C2×D4—2442no—
435C2×D4—2442no—
436C2×D4—2441no—
437C2×D4—2441no—
438C2×D4—2442no—
439C2×D4—2442no—
440C2×D4—2442no—
441C22×C4—1221no—
442C22×C4—1222no—
443C22×C4—1222no—
444C22×C4—1222no—
445C22×C4—2442no—
446C22×C4—2442no—
447C22×C4C2×D24411yescharacteristic
448C22×C4C2×D24411yescharacteristic
449C22×C4#1175822212yes—
450C22×C4D4×D6112yes—
451C22×C4D4×D6111yescharacteristic
452C22×C4D4×D6111yescharacteristic
453C22×C4#1175822612yes—
454C22×C4#1175822611yescharacteristic
455C22×C4C2×C22⋊D6112yes—
456C22×C4C23×D6111yescharacteristic
457SD16—12216no—
458D8—12216no—
459M16—22no—
460M16—24no—
461M16—24no—
462M16—24no—
463M16—22no—
464M16—22no—
465M16—42no—
466M16—1222no—
467M16—1222no—
468M16—2442no—
469M16C2×C22⋊D6114yes—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—24no—
474C2×C8—22no—
475C2×C8—42no—
476C2×C8—1222no—
477C2×C8—2442no—
478C2×C8C2×D24414yes—
479C4⋊C4—1222no—
480C4⋊C4—1222no—
481C22⋊C4—2441no—
482C22⋊C4—2441no—
483C42—2441no—
484C42—2441no—
485C23—42no—
486C23—1224no—
487C23—1222no—
488C23—4884no—
489C23—4884no—
490C23—4884no—
491C23—4882no—
492C23—4888no—
493C23—4888no—
494C23—2442no—
495C23—2442no—
496C23—2442no—
497C23—2444no—
498C23—2444no—
499C23#1451975211yescharacteristic
500C23#1451975211yescharacteristic
501C23#1451976212yes—
502C23#1451976612yes—
503C23C2×#1175822611yescharacteristic
504Q8—1222no—
505Q8—1222no—
506Q8—2442no—
507D4—1222no—
508D4—1224no—
509D4—1224no—
510D4—1222no—
511D4—2442no—
512D4—2442no—
513D4—2444no—
514D4—2442no—
515D4—2442no—
516D4—2444no—
517D4—2444no—
518D4—2442no—
519C2×C4—1222no—
520C2×C4—4882no—
521C2×C4—2442no—
522C2×C4—2442no—
523C2×C4—2442no—
524C2×C4—2442no—
525C2×C4—2444no—
526C2×C4#1451974512yes—
527C2×C4#1451974512yes—
528C2×C4#1451975112yes—
529C2×C4#1451975112yes—
530C2×C4#1451975212yes—
531C2×C4#1451975212yes—
532C2×C4#1451975211yescharacteristic
533C2×C4#1451975211yescharacteristic
534C2×C4#1451976212yes—
535C2×C4#1451976212yes—
536C2×C4#1451976212yes—
537C2×C4#1451976612yes—
538C2×C4C2×D4×D6111yescharacteristic
539C2×C4C2×D4×D6111yescharacteristic
540C2×C4C22×C22⋊D6111yescharacteristic
541C2×C4C22×D24411yescharacteristic
542C2×C4C2×#1175822211yescharacteristic
543C2×C4C2×#1175822611yescharacteristic
544C8—24no—
545C8—24no—
546C8—42no—
547C8—2442no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—42no—
552C22—42no—
553C22—4884no—
554C22—4888no—
555C22—4884no—
556C22—4888no—
557C22—4884no—
558C22—4884no—
559C22—2442no—
560C22—2444no—
561C22—2444no—
562C22#1683967912yes—
563C22#1683967912yes—
564C22C2×#1451975211yescharacteristic
565C22C2×#1451976211yescharacteristic
566C22C2×#1451976611yescharacteristic
567C4—4882no—
568C4—2442no—
569C4#1683967912yes—
570C4#1683967912yes—
571C4C2×#1451974511yescharacteristic
572C4C2×#1451975111yescharacteristic
573C4C2×#1451975211yescharacteristic
574C4C2×#1451976211yescharacteristic
575C2—22no—
576C2—21no—
577C2—21no—
578C2—22no—
579C2—4884no—
580C2—4882no—
581C2C2×#1683967911yescharacteristic
582C2#2145554812yes—
subgroupG/H
centerC22C2×#14519752
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#430C2
socleC22×C61C2×#121
prime radicalC22×#14519805C2
hypercenterC2×#430D61
hypocenterC61#23531
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#430

Complements

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

NH ≅ G/Ntype
2. #21454339C2semidirect
3. C2×#16839768C2semidirect
4. C22×#14519693C2semidirect
5. #21454263C2semidirect
6. #21447113C2semidirect
7. #21445557C2semidirect
8. C2×C61×#430C2semidirect
23. #16839764C22semidirect
25. C2×#14519699C22semidirect
27. C2×#14519693C22semidirect
29. #16839728C22semidirect
31. #16839530C22semidirect
36. C2×#108×C61C22semidirect
285. C2×#430D61semidirect
314. C2×#108C2×D61semidirect
328. C61#23531semidirect

Maximal subgroups

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

subgroupindex
C22×#145198052
#214543392
C2×#168397682
C22×#145196932
#214542632
#214471132
#214455572
C2×C61×#4302
#2353161