Finite Group Databasebeta

← #29707918

Subgroups — #29707918

Order 15616.

1358 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839810C211yesmaximal, characteristic, complemented
2C2×#16839763C211yesmaximal, characteristic, complemented
3C2×#16839763C211yesmaximal, characteristic, complemented
4C2×#16839746C211yesmaximal, characteristic, complemented
5#21454264C211yesmaximal, characteristic, complemented
6#21447115C214yesmaximal, complemented
7#21447108C214yesmaximal, complemented
8C2×#16839530C211yesmaximal, characteristic, complemented
9C2×C61×#575C211yesmaximal, characteristic
10C22×#11758194C2211yescharacteristic
11C22×#11758194C2211yescharacteristic
12C22×#11758194C2211yescharacteristic
13C22×C4⋊Dic61C2211yescharacteristic
14#16839810—24no—
15C2×#14519712C2214yescomplemented
16#16839763—22no—
17#16839763—22no—
18#16839763—22no—
19#16839763C2214yes—
20C2×#14519701—21no—
21C2×#14519701C2212yescomplemented
22C2×#14519698—21no—
23C2×#14519698C2212yescomplemented
24#16839746—24no—
25C2×#14519696C2212yescomplemented
26C2×#14519694C2212yescomplemented
27#16839729—22no—
28#16839729C2214yes—
29#16839726—22no—
30#16839530C2214yescomplemented
31C22×#41×C61C2211yescharacteristic
32C22×#41×C61C2211yescharacteristic
33C61×#575—24no—
34C61×#575—24no—
35C2×#137×C61—21no—
36C2×#137×C61C2212yes—
37C2×AGL(1,ℤ/8)×C61—21no—
38C2×AGL(1,ℤ/8)×C61C2212yes—
39C2×#135×C61—22no—
40C22×M16×C61C2211yescharacteristic
41C23×Dic61—21no—
42C2×#11758194—22no—
43C2×#11758194—22no—
44C2×#11758194—21no—
45C2×#11758194—21no—
46C2×#11758194—22no—
47C2×#11758194—22no—
48C2×#11758194—22no—
49C2×#11758194—22no—
50C2×#11758194D412yes—
51C2×#11758194D412yes—
52C2×#11758194C2312yes—
53C2×#11758194C2312yes—
54C22×#92418—21no—
55C22×#92418—21no—
56C22×#92418C2312yes—
57C2×C4⋊Dic61—21no—
58C2×C4⋊Dic61—21no—
59C2×C4⋊Dic61—21no—
60C2×C4⋊Dic61—21no—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61C2312yes—
63#14519712—28no—
64#14519701—24no—
65#14519701—24no—
66#14519698—24no—
67#14519698—24no—
68#14519696—24no—
69#14519694—24no—
70C22⋊#92418—28no—
71C2×#41×C61—21no—
72C2×#41×C61—21no—
73C2×#41×C61—22no—
74C2×#41×C61—21no—
75C2×#41×C61—21no—
76C2×#41×C61—22no—
77C2×#41×C61—22no—
78C2×#41×C61—22no—
79C2×#41×C61—21no—
80C2×#41×C61—21no—
81C2×#41×C61—22no—
82C2×#41×C61—22no—
83C2×#41×C61—22no—
84C2×#41×C61D412yes—
85C2×#41×C61D414yes—
86C2×#41×C61C2312yes—
87C22×Q8×C61D411yescharacteristic
88C22×Q8×C61C2311yescharacteristic
89C22×D4×C61—21no—
90C22×D4×C61—21no—
91C22×D4×C61D411yescharacteristic
92C22×D4×C61C2311yescharacteristic
93C23×C4×C61—21no—
94C23×C4×C61—21no—
95C23×C4×C61C2311yescharacteristic
96#137×C61—28no—
97#137×C61—24no—
98#137×C61—24no—
99AGL(1,ℤ/8)×C61—28no—
100AGL(1,ℤ/8)×C61—24no—
101AGL(1,ℤ/8)×C61—24no—
102#135×C61—216no—
103C2×Q16×C61—22no—
104C2×SD16×C61—22no—
105C2×SD16×C61—22no—
106C2×D8×C61—22no—
107C2×M16×C61—21no—
108C2×M16×C61—22no—
109C2×M16×C61D414yes—
110C2×M16×C61C2312yes—
111C22×C8×C61—21no—
112C24×C61—21no—
113C24×C61—21no—
114C24×C61#4111yescharacteristic
115#41×C61—24no—
116#41×C61—24no—
117#41×C61—24no—
118#41×C61—24no—
119#41×C61—24no—
120#41×C61—24no—
121#41×C61—22no—
122#41×C61—22no—
123#41×C61—22no—
124#41×C61—22no—
125#41×C61—42no—
126#41×C61—42no—
127#41×C61—42no—
128#41×C61—42no—
129#41×C61—42no—
130#41×C61—42no—
131#41×C61—42no—
132#41×C61—42no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—22no—
136C2×Q8×C61—22no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61C2×D412yes—
142C2×Q8×C61C2×D412yes—
143C2×D4×C61—22no—
144C2×D4×C61—24no—
145C2×D4×C61—22no—
146C2×D4×C61—22no—
147C2×D4×C61—21no—
148C2×D4×C61—22no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61—41no—
155C2×D4×C61—41no—
156C2×D4×C61—42no—
157C2×D4×C61—42no—
158C2×D4×C61—42no—
159C2×D4×C61C2×D412yes—
160C2×D4×C61C2×D412yes—
161C22×C4×C61—21no—
162C22×C4×C61—21no—
163C22×C4×C61—22no—
164C22×C4×C61—24no—
165C22×C4×C61—22no—
166C22×C4×C61—41no—
167C22×C4×C61—41no—
168C22×C4×C61—42no—
169C22×C4×C61—42no—
170C22×C4×C61—42no—
171C22×C4×C61C2×D412yes—
172C22×C4×C61C2×D412yes—
173C22×C4×C61C2×D411yescharacteristic
174C22×C4×C61C2×D411yescharacteristic
175C22×C4×C61C2×D411yescharacteristic
176C22×C4×C61C2×D411yescharacteristic
177C22×C4×C61#4112yes—
178C22×C4×C61#4111yescharacteristic
179C22×C4×C61#4112yes—
180C22×C4×C61C2411yescharacteristic
181C22×Dic61—22no—
182C22×Dic61—41no—
183C22×Dic61—41no—
184C22×Dic61—82no—
185Q16×C61—28no—
186SD16×C61—28no—
187SD16×C61—28no—
188D8×C61—28no—
189M16×C61—24no—
190M16×C61—22no—
191M16×C61C2×D414yes—
192C2×C8×C61—24no—
193C2×C8×C61—22no—
194C4⋊Dic61—22no—
195C4⋊Dic61—22no—
196C4⋊Dic61—41no—
197C4⋊Dic61—41no—
198#11758194—24no—
199#11758194—24no—
200#11758194—22no—
201#11758194—22no—
202#11758194—22no—
203#11758194—22no—
204#11758194—42no—
205#11758194—42no—
206C2×#92418—22no—
207C2×#92418—22no—
208C2×#92418—22no—
209C2×#92418—22no—
210C2×#92418—42no—
211C2×#92418—42no—
212C2×#92418C2×D414yes—
213C23×C61—22no—
214C23×C61—22no—
215C23×C61—41no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61—42no—
220C23×C61—81no—
221C23×C61—81no—
222C23×C61—81no—
223C23×C61—81no—
224C23×C61#12111yescharacteristic
225C23×C61#12111yescharacteristic
226C23×C61#12312yes—
227C23×C61#12412yes—
228C23×C61C2×#4111yescharacteristic
229Q8×C61—22no—
230Q8×C61—22no—
231Q8×C61—22no—
232Q8×C61—22no—
233Q8×C61—41no—
234Q8×C61—41no—
235Q8×C61—42no—
236D4×C61—22no—
237D4×C61—22no—
238D4×C61—22no—
239D4×C61—22no—
240D4×C61—42no—
241D4×C61—42no—
242D4×C61—44no—
243D4×C61—44no—
244D4×C61—44no—
245D4×C61—41no—
246D4×C61—41no—
247D4×C61—42no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—42no—
253C2×C4×C61—42no—
254C2×C4×C61—42no—
255C2×C4×C61—44no—
256C2×C4×C61—44no—
257C2×C4×C61—44no—
258C2×C4×C61#12012yes—
259C2×C4×C61#12012yes—
260C2×C4×C61#12112yes—
261C2×C4×C61#12112yes—
262C2×C4×C61#12112yes—
263C2×C4×C61#12112yes—
264C2×C4×C61#12111yescharacteristic
265C2×C4×C61#12111yescharacteristic
266C2×C4×C61#12312yes—
267C2×C4×C61#12312yes—
268C2×C4×C61#12312yes—
269C2×C4×C61#12412yes—
270C2×C4×C61C22×D411yescharacteristic
271C2×C4×C61C22×D411yescharacteristic
272C2×C4×C61C22×D411yescharacteristic
273C2×C4×C61C22×D411yescharacteristic
274C2×C4×C61C2×#4111yescharacteristic
275C2×C4×C61C2×#4111yescharacteristic
276C2×Dic61—42no—
277C2×Dic61—82no—
278C2×Dic61—82no—
279C2×Dic61—82no—
280C8×C61—24no—
281#92418—24no—
282#92418—42no—
283#92418—42no—
284#23533—611nomaximal, Sylow
285C22×C61—21no—
286C22×C61—21no—
287C22×C61—21no—
288C22×C61—21no—
289C22×C61—21no—
290C22×C61—21no—
291C22×C61—42no—
292C22×C61—42no—
293C22×C61—42no—
294C22×C61—42no—
295C22×C61—42no—
296C22×C61—42no—
297C22×C61—82no—
298C22×C61—82no—
299C22×C61—84no—
300C22×C61—81no—
301C22×C61—81no—
302C22×C61#39412yes—
303C22×C61#39412yes—
304C22×C61C2×#12111yescharacteristic
305C22×C61C2×#12311yescharacteristic
306C22×C61C2×#12411yescharacteristic
307C4×C61—42no—
308C4×C61—42no—
309C4×C61#39412yes—
310C4×C61#39412yes—
311C4×C61C2×#12011yescharacteristic
312C4×C61C2×#12111yescharacteristic
313C4×C61C2×#12111yescharacteristic
314C4×C61C2×#12311yescharacteristic
315Dic61—82no—
316C2×#575D6111yescharacteristic
317C2×#430—611no—
318C2×#417—611no—
319C2×#417—611no—
320C2×#413—611no—
321#2879—611no—
322#2052—614no—
323#2051—614no—
324C2×#343—611no—
325C2×C61—21no—
326C2×C61—21no—
327C2×C61—22no—
328C2×C61—22no—
329C2×C61—82no—
330C2×C61—82no—
331C2×C61C2×#39411yescharacteristic
332C2×C61#306811yescharacteristic
333C2×C61#306911yescharacteristic
334C22×#41C2×D6111yescharacteristic
335C22×#41C2×D6111yescharacteristic
336#575—24no—
337#575—24no—
338C2×#137—21no—
339C2×#137C2×D6112yes—
340C2×AGL(1,ℤ/8)—21no—
341C2×AGL(1,ℤ/8)C2×D6112yes—
342C2×#135—22no—
343C22×M16—611no—
344C22×M16—611no—
345C22×M16—611no—
346C22×M16C2×D6111yescharacteristic
347C22×C4⋊C4—611no—
348#430—1224no—
349C2×#108—614no—
350#417—614no—
351#417—1222no—
352#417—1222no—
353#417—1222no—
354C2×#103—612no—
355C2×#103—1221no—
356C2×#102—612no—
357C2×#102—1221no—
358#413—1224no—
359C2×#101—612no—
360C2×#100—612no—
361#408—614no—
362#408—1222no—
363#407—1222no—
364#343—614no—
365C61#2353311yescharacteristic, complemented, Sylow
366C2×#41—21no—
367C2×#41—21no—
368C2×#41—22no—
369C2×#41—21no—
370C2×#41—21no—
371C2×#41—22no—
372C2×#41—22no—
373C2×#41—22no—
374C2×#41—21no—
375C2×#41—21no—
376C2×#41—22no—
377C2×#41—22no—
378C2×#41—22no—
379C2×#41C22⋊D6112yes—
380C2×#41C22⋊D6114yes—
381C2×#41C22×D6112yes—
382C22×Q8C22⋊D6111yescharacteristic
383C22×Q8C22×D6111yescharacteristic
384C22×D4—21no—
385C22×D4—21no—
386C22×D4C22⋊D6111yescharacteristic
387C22×D4C22×D6111yescharacteristic
388C23×C4—21no—
389C23×C4—21no—
390C23×C4—1221no—
391C23×C4C22×D6111yescharacteristic
392#137—28no—
393#137—24no—
394#137—24no—
395AGL(1,ℤ/8)—28no—
396AGL(1,ℤ/8)—24no—
397AGL(1,ℤ/8)—24no—
398#135—216no—
399C2×Q16—22no—
400C2×SD16—22no—
401C2×SD16—22no—
402C2×D8—22no—
403C2×M16—21no—
404C2×M16—22no—
405C2×M16—612no—
406C2×M16—612no—
407C2×M16—612no—
408C2×M16—612no—
409C2×M16—1222no—
410C2×M16—1222no—
411C2×M16—1221no—
412C2×M16—1221no—
413C2×M16—1222no—
414C2×M16—1222no—
415C2×M16—1222no—
416C2×M16—1222no—
417C2×M16C22⋊D6114yes—
418C2×M16C22×D6112yes—
419C22×C8—21no—
420C22×C8—612no—
421C22×C8—1221no—
422C22×C8—1221no—
423C2×C4⋊C4—612no—
424C2×C4⋊C4—1221no—
425C2×C4⋊C4—1221no—
426C2×C4⋊C4—1221no—
427C2×C4⋊C4—1221no—
428C2×C4⋊C4—1221no—
429#108—1228no—
430#103—1224no—
431#103—1224no—
432#102—1224no—
433#102—1224no—
434#101—1224no—
435#100—1224no—
436C22⋊C8—1228no—
437C24—21no—
438C24—21no—
439C24#1175822411yescharacteristic
440#41—22no—
441#41—22no—
442#41—22no—
443#41—22no—
444#41—24no—
445#41—24no—
446#41—24no—
447#41—24no—
448#41—24no—
449#41—24no—
450#41—42no—
451#41—42no—
452#41—42no—
453#41—42no—
454#41—42no—
455#41—42no—
456#41—42no—
457#41—42no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—22no—
465C2×Q8—22no—
466C2×Q8C2×C22⋊D6112yes—
467C2×Q8C2×C22⋊D6112yes—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—24no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—22no—
477C2×D4—21no—
478C2×D4—22no—
479C2×D4—41no—
480C2×D4—41no—
481C2×D4—42no—
482C2×D4—42no—
483C2×D4—42no—
484C2×D4C2×C22⋊D6112yes—
485C2×D4C2×C22⋊D6112yes—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—24no—
491C22×C4—42no—
492C22×C4—41no—
493C22×C4—41no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—1222no—
497C22×C4—4882no—
498C22×C4—2441no—
499C22×C4—2441no—
500C22×C4D4×D6111yescharacteristic
501C22×C4D4×D6111yescharacteristic
502C22×C4#1175822412yes—
503C22×C4#1175822412yes—
504C22×C4#1175822411yescharacteristic
505C22×C4C2×C22⋊D6112yes—
506C22×C4C2×C22⋊D6112yes—
507C22×C4C2×C22⋊D6111yescharacteristic
508C22×C4C2×C22⋊D6111yescharacteristic
509C22×C4C23×D6111yescharacteristic
510Q16—28no—
511SD16—28no—
512SD16—28no—
513D8—28no—
514M16—22no—
515M16—24no—
516M16—1224no—
517M16—1224no—
518M16—1222no—
519M16—1222no—
520M16—1222no—
521M16—1222no—
522M16—2442no—
523M16—2442no—
524M16C2×C22⋊D6114yes—
525C2×C8—22no—
526C2×C8—24no—
527C2×C8—614no—
528C2×C8—1222no—
529C2×C8—1222no—
530C2×C8—1222no—
531C2×C8—1222no—
532C2×C8—2442no—
533C2×C8—2442no—
534C4⋊C4—1222no—
535C4⋊C4—1222no—
536C4⋊C4—2441no—
537C4⋊C4—2441no—
538C23—22no—
539C23—22no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—42no—
545C23—81no—
546C23—81no—
547C23—81no—
548C23—81no—
549C23#1451974911yescharacteristic
550C23#1451974911yescharacteristic
551C23#1451975812yes—
552C23#1451976412yes—
553C23C2×#1175822411yescharacteristic
554Q8—22no—
555Q8—22no—
556Q8—22no—
557Q8—22no—
558Q8—41no—
559Q8—41no—
560Q8—42no—
561D4—22no—
562D4—22no—
563D4—22no—
564D4—22no—
565D4—42no—
566D4—42no—
567D4—44no—
568D4—44no—
569D4—44no—
570D4—41no—
571D4—41no—
572D4—42no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—4882no—
584C2×C4—4882no—
585C2×C4—4882no—
586C2×C4—2442no—
587C2×C4#1451974612yes—
588C2×C4#1451974612yes—
589C2×C4#1451974812yes—
590C2×C4#1451974812yes—
591C2×C4#1451974912yes—
592C2×C4#1451974912yes—
593C2×C4#1451974911yescharacteristic
594C2×C4#1451974911yescharacteristic
595C2×C4#1451975812yes—
596C2×C4#1451975812yes—
597C2×C4#1451975812yes—
598C2×C4#1451976412yes—
599C2×C4C2×D4×D6111yescharacteristic
600C2×C4C22×C22⋊D6111yescharacteristic
601C2×C4C22×C22⋊D6111yescharacteristic
602C2×C4C22×C22⋊D6111yescharacteristic
603C2×C4C2×#1175822411yescharacteristic
604C2×C4C2×#1175822411yescharacteristic
605C8—24no—
606C8—1224no—
607C8—2442no—
608C8—2442no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—82no—
622C22—82no—
623C22—84no—
624C22—81no—
625C22—81no—
626C22#1683967612yes—
627C22#1683967612yes—
628C22C2×#1451974911yescharacteristic
629C22C2×#1451975811yescharacteristic
630C22C2×#1451976411yescharacteristic
631C4—42no—
632C4—42no—
633C4—4882no—
634C4#1683967612yes—
635C4#1683967612yes—
636C4C2×#1451974611yescharacteristic
637C4C2×#1451974811yescharacteristic
638C4C2×#1451974911yescharacteristic
639C4C2×#1451975811yescharacteristic
640C2—21no—
641C2—21no—
642C2—22no—
643C2—22no—
644C2—82no—
645C2—82no—
646C2C2×#1683967611yescharacteristic
647C2#2145554911yescharacteristic
648C2#2145555811yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC2×C61×#575C2
socleC22×C61C2×#121
hypercenterC2×#575D61
hypocenterC61#23533
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#575

Complements

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

NH ≅ G/Ntype
1. C2×#16839810C2semidirect
2. C2×#16839763C2semidirect
3. C2×#16839763C2semidirect
4. C2×#16839746C2semidirect
5. #21454264C2semidirect
6. #21447115C2semidirect
7. #21447108C2semidirect
8. C2×#16839530C2semidirect
15. C2×#14519712C22semidirect
21. C2×#14519701C22semidirect
23. C2×#14519698C22semidirect
25. C2×#14519696C22semidirect
26. C2×#14519694C22semidirect
30. #16839530C22semidirect
365. C61#23533semidirect

Maximal subgroups

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

subgroupindex
C2×#168398102
C2×#168397632
C2×#168397462
#214542642
#214471152
#214471082
C2×#168395302
C2×C61×#5752
#2353361