Finite Group Databasebeta

← #29684029

Subgroups — #29684029

Order 15616.

1562 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519802C211yesmaximal, characteristic
2C2×#16839810C211yesmaximal, characteristic, complemented
3#21454338C211yesmaximal, characteristic, complemented
4C2×#16839768C211yesmaximal, characteristic, complemented
5C22×#14519693C211yesmaximal, characteristic, complemented
6#21454237C211yesmaximal, characteristic, complemented
7#21446726C214yesmaximal, complemented
8#21446723C214yesmaximal, complemented
9C2×C61×#413C211yesmaximal, characteristic, complemented
10C23×D244C2211yescharacteristic
11C2×#14519802—22no—
12C2×#14519802—22no—
13C2×#14519802—24no—
14C2×#14519802C418yescomplemented
15C22×#11758201C2212yes—
16C22×#11758199C2212yes—
17C22×#11758194C2211yescharacteristic
18C22×#11758194C2211yescharacteristic
19C2×#14519736C2211yescharacteristic
20#16839810—24no—
21C2×#14519712C2214yescomplemented
22#16839771C2212yescomplemented
23#16839768—22no—
24#16839768C2214yes—
25#16839767C2212yescomplemented
26#16839763—22no—
27#16839758—22no—
28C2×#14519699—22no—
29C2×#14519693—24no—
30C2×#14519693C2214yes—
31#16839720C2214yescomplemented
32#16839717—22no—
33M16×Dic61—22no—
34C22×#41×C61C2211yescharacteristic
35C22×M16×C61C2211yescharacteristic
36C22×M16×C61C2211yescharacteristic
37C61×#413—24no—
38C2×#101×C61C2212yescomplemented
39C2×#100×C61C2212yescomplemented
40C24×D61—21no—
41C22×D244—21no—
42C22×D244—21no—
43C22×D244—21no—
44C22×D244—21no—
45C22×D244—22no—
46C22×D244—22no—
47C22×D244—22no—
48C22×D244—21no—
49C22×D244C2×C414yes—
50C22×D244C2312yes—
51#14519802—216no—
52#14519802—216no—
53C2×#11758201—24no—
54C2×#11758201—24no—
55C2×#11758201C2×C418yescomplemented
56C2×#11758199—24no—
57C2×#11758199—24no—
58C2×#11758199C2×C418yescomplemented
59C2×#11758194—21no—
60C2×#11758194—22no—
61C2×#11758194—22no—
62C2×#11758194C2×C414yes—
63C2×#11758194D412yes—
64C2×#11758194D412yes—
65C2×#11758194D414yes—
66C2×#11758194C2312yes—
67C22×#92418—21no—
68C22×#92418C2312yes—
69#14519736—21no—
70#14519736—21no—
71#14519736D412yes—
72#14519736C2312yes—
73C2×C4⋊Dic61D412yes—
74C2×C22⋊Dic61—21no—
75C2×C4×Dic61—21no—
76#14519712—28no—
77#14519701—24no—
78#14519699—28no—
79#14519698—24no—
80#14519693—28no—
81#14519687—24no—
82C8×Dic61—24no—
83C2×#41×C61—22no—
84C2×#41×C61—21no—
85C2×#41×C61—21no—
86C2×#41×C61—21no—
87C2×#41×C61—21no—
88C2×#41×C61—22no—
89C2×#41×C61—22no—
90C2×#41×C61C2×C414yes—
91C22×Q8×C61C2311yescharacteristic
92C22×D4×C61—21no—
93C22×D4×C61C2311yescharacteristic
94C23×C4×C61—21no—
95C23×C4×C61C2311yescharacteristic
96C2×M16×C61—21no—
97C2×M16×C61—22no—
98C2×M16×C61—21no—
99C2×M16×C61—22no—
100C2×M16×C61—22no—
101C2×M16×C61D414yes—
102C2×M16×C61C2312yes—
103C2×M16×C61C2312yes—
104C22×C8×C61—21no—
105C22×C8×C61—21no—
106#101×C61—24no—
107#100×C61—24no—
108C24×C61—21no—
109C24×C61#4111yescharacteristic
110C23×D61—24no—
111C23×D61—22no—
112C23×D61—41no—
113C23×D61—41no—
114C23×D61—42no—
115C23×D61—84no—
116#41×C61—24no—
117#41×C61—24no—
118#41×C61—24no—
119#41×C61—24no—
120#41×C61—42no—
121#41×C61—42no—
122#41×C61—42no—
123#41×C61—42no—
124C2×Q8×C61—21no—
125C2×Q8×C61—21no—
126C2×Q8×C61—21no—
127C2×Q8×C61—21no—
128C2×Q8×C61—21no—
129C2×Q8×C61C22×C412yes—
130C2×D4×C61—24no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—41no—
137C2×D4×C61—41no—
138C2×D4×C61—42no—
139C2×D4×C61C22×C412yes—
140C22×C4×C61—21no—
141C22×C4×C61—24no—
142C22×C4×C61—41no—
143C22×C4×C61—41no—
144C22×C4×C61—42no—
145C22×C4×C61C22×C412yes—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D412yes—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D412yes—
152C22×C4×C61#4111yescharacteristic
153C22×C4×C61#4112yes—
154C22×C4×C61C2411yescharacteristic
155C22×Dic61—22no—
156C2×D244—24no—
157C2×D244—24no—
158C2×D244—24no—
159C2×D244—24no—
160C2×D244—22no—
161C2×D244—22no—
162C2×D244—22no—
163C2×D244—22no—
164C2×D244—24no—
165C2×D244—22no—
166C2×D244—42no—
167C2×D244—42no—
168C2×D244—41no—
169C2×D244—41no—
170C2×D244—42no—
171C2×D244—42no—
172C2×D244—42no—
173C2×D244C22×C414yes—
174M16×C61—24no—
175M16×C61—22no—
176M16×C61—22no—
177M16×C61—22no—
178M16×C61—42no—
179M16×C61C2×D414yes—
180C2×C8×C61—24no—
181C2×C8×C61—22no—
182C2×C8×C61—22no—
183C2×C8×C61—42no—
184C22⋊Dic61—22no—
185C22⋊Dic61—22no—
186#11758201—216no—
187#11758199—216no—
188C4⋊Dic61—22no—
189C4⋊Dic61C2×D414yes—
190C4×Dic61—22no—
191C4×Dic61—22no—
192#11758194—22no—
193#11758194—22no—
194#11758194—42no—
195#11758194C22⋊C418yes—
196#11758194C22⋊C418yes—
197C2×#92418—22no—
198C2×#92418—42no—
199C2×#92418C22×C414yes—
200C2×#92418C2×D414yes—
201C2×#92418C2×D414yes—
202C23×C61—22no—
203C23×C61—42no—
204C23×C61—41no—
205C23×C61—41no—
206C23×C61—82no—
207C23×C61C4×D412yes—
208C23×C61#12111yescharacteristic
209C23×C61#12111yescharacteristic
210C23×C61#12412yes—
211C23×C61C2×#4111yescharacteristic
212C22×D61—24no—
213C22×D61—42no—
214C22×D61—42no—
215C22×D61—44no—
216C22×D61—44no—
217C22×D61—84no—
218C22×D61—84no—
219C22×D61—88no—
220C22×D61—84no—
221C22×D61—88no—
222Q8×C61—22no—
223Q8×C61—22no—
224Q8×C61—41no—
225Q8×C61—41no—
226D4×C61—22no—
227D4×C61—22no—
228D4×C61—44no—
229D4×C61—44no—
230D4×C61—41no—
231D4×C61—41no—
232C2×C4×C61—22no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—44no—
236C2×C4×C61—44no—
237C2×C4×C61C2×C22⋊C412yes—
238C2×C4×C61C2×C22⋊C412yes—
239C2×C4×C61C4×D412yes—
240C2×C4×C61C4×D412yes—
241C2×C4×C61C4×D412yes—
242C2×C4×C61#12112yes—
243C2×C4×C61#12112yes—
244C2×C4×C61#12111yescharacteristic
245C2×C4×C61#12111yescharacteristic
246C2×C4×C61#12412yes—
247C2×C4×C61#12712yes—
248C2×C4×C61#12712yes—
249C2×C4×C61C23×C411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×Dic61—24no—
256C2×Dic61—82no—
257D244—24no—
258D244—24no—
259D244—42no—
260D244—42no—
261D244—44no—
262D244—44no—
263C8×C61—24no—
264C8×C61—42no—
265#92418—42no—
266#92418C2×C22⋊C418yes—
267#21249—611nomaximal, Sylow
268C22×C61—22no—
269C22×C61—22no—
270C22×C61—22no—
271C22×C61—42no—
272C22×C61—42no—
273C22×C61—42no—
274C22×C61—82no—
275C22×C61—82no—
276C22×C61—82no—
277C22×C61#39012yes—
278C22×C61#39012yes—
279C22×C61C2×C4×D411yescharacteristic
280C22×C61C2×#12111yescharacteristic
281C22×C61C2×#12411yescharacteristic
282C2×D61—44no—
283C2×D61—88no—
284C2×D61—84no—
285C2×D61—88no—
286C2×D61—84no—
287C2×D61—84no—
288C4×C61—42no—
289C4×C61#39012yes—
290C4×C61#39012yes—
291C4×C61C22×C22⋊C411yescharacteristic
292C4×C61C2×C4×D411yescharacteristic
293C4×C61C2×#12111yescharacteristic
294C4×C61C2×#12711yescharacteristic
295Dic61—44no—
296C22×AGL(1,ℤ/8)—611no—
297C2×#430—611no—
298#2896—611no—
299C2×#418—611no—
300C2×#413D6111yescharacteristic, complemented
301C22×#100—611no—
302#2873—611no—
303#1978—614no—
304#1977—614no—
305C2×C61—21no—
306C2×C61—21no—
307C2×C61—22no—
308C2×C61—22no—
309C2×C61—82no—
310C2×C61C2×#39011yescharacteristic
311C2×C61#306812yes—
312D61—84no—
313C22×#41C2×D6111yescharacteristic
314C23×D4—611no—
315C2×AGL(1,ℤ/8)—618no—
316C2×AGL(1,ℤ/8)—1222no—
317C2×AGL(1,ℤ/8)—1222no—
318C2×AGL(1,ℤ/8)—1224no—
319C22×SD16—612no—
320C22×D8—612no—
321C22×M16—611no—
322C22×M16—611no—
323C22×M16C2×D6111yescharacteristic
324C22×M16C2×D6111yescharacteristic
325C2×#117—611no—
326#430—1224no—
327C2×#108—614no—
328#419—612no—
329#418—612no—
330#418—614no—
331#418—1222no—
332#417—1222no—
333#416—1222no—
334C2×#102—1222no—
335#413—24no—
336C2×#101C2×D6112yescomplemented
337C2×#100—614no—
338C2×#100—1224no—
339C2×#100C2×D6112yescomplemented
340#405—614no—
341#405—1222no—
342C4×M16—1222no—
343C61#2124911yescharacteristic, complemented, Sylow
344C25—1221no—
345C2×#41—22no—
346C2×#41—21no—
347C2×#41—21no—
348C2×#41—22no—
349C2×#41—21no—
350C2×#41—21no—
351C2×#41—22no—
352C2×#41C4×D6114yes—
353C22×Q8C22×D6111yescharacteristic
354C22×D4—21no—
355C22×D4—614no—
356C22×D4—612no—
357C22×D4—1221no—
358C22×D4—1221no—
359C22×D4—1221no—
360C22×D4—1222no—
361C22×D4—1221no—
362C22×D4—1221no—
363C22×D4—1222no—
364C22×D4—1222no—
365C22×D4C22×D6111yescharacteristic
366C23×C4—21no—
367C23×C4C22×D6111yescharacteristic
368AGL(1,ℤ/8)—12216no—
369AGL(1,ℤ/8)—12216no—
370C2×SD16—618no—
371C2×SD16—1224no—
372C2×SD16—1224no—
373C2×D8—618no—
374C2×D8—1224no—
375C2×D8—1224no—
376C2×M16—21no—
377C2×M16—22no—
378C2×M16—21no—
379C2×M16—22no—
380C2×M16—22no—
381C2×M16—614no—
382C2×M16—612no—
383C2×M16—612no—
384C2×M16—614no—
385C2×M16—612no—
386C2×M16—1221no—
387C2×M16—1222no—
388C2×M16—1222no—
389C2×M16C22⋊D6114yes—
390C2×M16C22×D6112yes—
391C2×M16C22×D6112yes—
392C22×C8—21no—
393C22×C8—21no—
394C22×C8—612no—
395C22×C8—1221no—
396#117—612no—
397#117—612no—
398#117—1221no—
399#117—1221no—
400C2×C4⋊C4—612no—
401C2×C22⋊C4—1221no—
402C2×C42—1221no—
403#108—1228no—
404#103—1224no—
405#102—1228no—
406#102—1224no—
407#101—24no—
408#100—24no—
409#100—1228no—
410#97—1224no—
411C4×C8—1224no—
412C24—21no—
413C24—1222no—
414C24—1224no—
415C24—4884no—
416C24—2441no—
417C24—2441no—
418C24—2442no—
419C24#1175822211yescharacteristic
420#41—24no—
421#41—24no—
422#41—24no—
423#41—24no—
424#41—42no—
425#41—42no—
426#41—42no—
427#41—42no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8C2×C4×D6112yes—
434C2×D4—24no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—41no—
441C2×D4—41no—
442C2×D4—42no—
443C2×D4—614no—
444C2×D4—1222no—
445C2×D4—1222no—
446C2×D4—1222no—
447C2×D4—1222no—
448C2×D4—1222no—
449C2×D4—1224no—
450C2×D4—1224no—
451C2×D4—1224no—
452C2×D4—1224no—
453C2×D4—1224no—
454C2×D4—2442no—
455C2×D4—2442no—
456C2×D4—2441no—
457C2×D4—2441no—
458C2×D4—2442no—
459C2×D4—2442no—
460C2×D4—2442no—
461C2×D4C2×C4×D6112yes—
462C22×C4—21no—
463C22×C4—24no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—42no—
467C22×C4—1222no—
468C22×C4C2×C4×D6112yes—
469C22×C4#1175822211yescharacteristic
470C22×C4D4×D6111yescharacteristic
471C22×C4D4×D6111yescharacteristic
472C22×C4D4×D6112yes—
473C22×C4D4×D6111yescharacteristic
474C22×C4D4×D6111yescharacteristic
475C22×C4#1175822412yes—
476C22×C4C2×C22⋊D6112yes—
477C22×C4C23×D6111yescharacteristic
478SD16—12216no—
479D8—12216no—
480M16—22no—
481M16—24no—
482M16—22no—
483M16—22no—
484M16—42no—
485M16—618no—
486M16—618no—
487M16—1222no—
488M16—1222no—
489M16—2442no—
490M16C2×C22⋊D6114yes—
491C2×C8—22no—
492C2×C8—24no—
493C2×C8—22no—
494C2×C8—42no—
495C2×C8—614no—
496C2×C8—614no—
497C2×C8—614no—
498C2×C8—1222no—
499C2×C8—2442no—
500C4⋊C4—614no—
501C4⋊C4—1222no—
502C22⋊C4—1222no—
503C22⋊C4—1222no—
504C42—1222no—
505C42—1222no—
506C23—22no—
507C23—42no—
508C23—41no—
509C23—41no—
510C23—82no—
511C23—1224no—
512C23—4884no—
513C23—4884no—
514C23—4884no—
515C23—4888no—
516C23—4888no—
517C23—2442no—
518C23—2442no—
519C23—2444no—
520C23—2444no—
521C23C4×C22⋊D6112yes—
522C23#1451975011yescharacteristic
523C23#1451975011yescharacteristic
524C23#1451976312yes—
525C23C2×#1175822211yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—41no—
529Q8—41no—
530D4—22no—
531D4—22no—
532D4—44no—
533D4—44no—
534D4—41no—
535D4—41no—
536D4—1224no—
537D4—1224no—
538D4—2442no—
539D4—2442no—
540D4—2444no—
541D4—2444no—
542C2×C4—22no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—1224no—
548C2×C4—4882no—
549C2×C4C22⋊C4×D6112yes—
550C2×C4C22⋊C4×D6112yes—
551C2×C4#1451973712yes—
552C2×C4#1451973712yes—
553C2×C4C4×C22⋊D6112yes—
554C2×C4#1451974912yes—
555C2×C4#1451974912yes—
556C2×C4#1451975011yescharacteristic
557C2×C4#1451975011yescharacteristic
558C2×C4#1451976312yes—
559C2×C4#1451977312yes—
560C2×C4#1451977312yes—
561C2×C4C22×C4×D6111yescharacteristic
562C2×C4C2×D4×D6111yescharacteristic
563C2×C4C2×D4×D6111yescharacteristic
564C2×C4C2×D4×D6111yescharacteristic
565C2×C4C22×C22⋊D6111yescharacteristic
566C2×C4C2×#1175822411yescharacteristic
567C8—24no—
568C8—42no—
569C8—618no—
570C8—2442no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—82no—
578C22—82no—
579C22—82no—
580C22—4884no—
581C22—4884no—
582C22—4888no—
583C22—4888no—
584C22—4884no—
585C22—2444no—
586C22#1683966212yes—
587C22#1683966212yes—
588C22C2×C4×C22⋊D6111yescharacteristic
589C22C2×#1451975011yescharacteristic
590C22C2×#1451976311yescharacteristic
591C4—42no—
592C4—2444no—
593C4#1683966212yes—
594C4#1683966212yes—
595C4C2×C22⋊C4×D6111yescharacteristic
596C4C2×#1451973711yescharacteristic
597C4C2×#1451974911yescharacteristic
598C4C2×#1451977311yescharacteristic
599C2—21no—
600C2—21no—
601C2—22no—
602C2—22no—
603C2—82no—
604C2—4884no—
605C2C2×#1683966211yescharacteristic
606C2#2145555012yes—
subgroupG/H
centerC22C2×#14519750
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#413C2
socleC22×C61C2×#121
prime radicalC22×#14519802C2
cosocleC22×C4×C61C24
hypercenterC2×#413D61
hypocenterC61#21249
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#413

Complements

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

NH ≅ G/Ntype
2. C2×#16839810C2semidirect
3. #21454338C2semidirect
4. C2×#16839768C2semidirect
5. C22×#14519693C2semidirect
6. #21454237C2semidirect
7. #21446726C2semidirect
8. #21446723C2semidirect
9. C2×C61×#413C2semidirect
14. C2×#14519802C4semidirect
21. C2×#14519712C22semidirect
22. #16839771C22semidirect
25. #16839767C22semidirect
31. #16839720C22semidirect
38. C2×#101×C61C22semidirect
39. C2×#100×C61C22semidirect
55. C2×#11758201C2×C4semidirect
58. C2×#11758199C2×C4semidirect
300. C2×#413D61semidirect
336. C2×#101C2×D61semidirect
339. C2×#100C2×D61semidirect
343. C61#21249semidirect

Maximal subgroups

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

subgroupindex
C22×#145198022
C2×#168398102
#214543382
C2×#168397682
C22×#145196932
#214542372
#214467262
#214467232
C2×C61×#4132
#2124961