Finite Group Databasebeta

← #29707922

Subgroups — #29707922

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×#16840560C211yesmaximal, characteristic
2C2×#16839810C211yesmaximal, characteristic, complemented
3C2×#16839764C211yesmaximal, characteristic, complemented
4C2×#16839763C211yesmaximal, characteristic, complemented
5#21454263C211yesmaximal, characteristic, complemented
6#21447118C214yesmaximal, complemented
7#21447111C214yesmaximal, complemented
8C2×#16839530C211yesmaximal, characteristic, complemented
9C2×C61×#413C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11#16840560—24no—
12#16840560—24no—
13C2×#14519808C2212yes—
14C2×#14519807—21no—
15C2×#14519803C2212yes—
16C2×#14519802—21no—
17C2×#14519797—22no—
18C22×#11758194C2211yescharacteristic
19C22×#11758194C2211yescharacteristic
20C22×C4⋊Dic61C2211yescharacteristic
21#16839810—24no—
22C2×#14519712C2214yescomplemented
23#16839764—22no—
24#16839764C2214yes—
25#16839763—22no—
26#16839763—22no—
27C2×#14519702—21no—
28C2×#14519701C2212yescomplemented
29C2×#14519699—21no—
30C2×#14519698C2212yescomplemented
31#16839730C2214yes—
32#16839728—22no—
33#16839725—22no—
34#16839530C2214yescomplemented
35C22×#41×C61C2211yescharacteristic
36C22×M16×C61C2211yescharacteristic
37C22×M16×C61C2211yescharacteristic
38C61×#413—24no—
39C2×#101×C61C2212yescomplemented
40C2×#100×C61C2212yescomplemented
41C2×#11758222—21no—
42C2×#11758222—21no—
43C2×#11758222—22no—
44C2×#11758222—22no—
45C2×#11758222—22no—
46C2×#11758222—22no—
47C2×#11758222D412yes—
48C2×#11758222C2312yes—
49C22×Dic122D411yescharacteristic
50C22×D244D411yescharacteristic
51C22×C22⋊D61—21no—
52C23×Dic61—21no—
53C22×C4×D61—21no—
54#14519808—24no—
55#14519808—24no—
56#14519807—28no—
57#14519803—24no—
58#14519803—24no—
59#14519802—28no—
60#14519797—216no—
61C2×#11758202—22no—
62C2×#11758200—22no—
63C2×#11758201—22no—
64C2×#11758199—22no—
65C2×#11758194—21no—
66C2×#11758194—22no—
67C2×#11758194—22no—
68C2×#11758194—22no—
69C2×#11758194D414yes—
70C2×#11758194D412yes—
71C2×#11758194D412yes—
72C2×#11758194C2312yes—
73C22×#92418—21no—
74C22×#92418C2312yes—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61C2312yes—
81#14519712—28no—
82#14519702—24no—
83#14519701—24no—
84#14519699—24no—
85#14519698—24no—
86#14519689—28no—
87C2×#41×C61—22no—
88C2×#41×C61—21no—
89C2×#41×C61—21no—
90C2×#41×C61—21no—
91C2×#41×C61—21no—
92C2×#41×C61—22no—
93C2×#41×C61—22no—
94C2×#41×C61D414yes—
95C22×Q8×C61C2311yescharacteristic
96C22×D4×C61—21no—
97C22×D4×C61C2311yescharacteristic
98C23×C4×C61—21no—
99C23×C4×C61C2311yescharacteristic
100C2×M16×C61—21no—
101C2×M16×C61—21no—
102C2×M16×C61—22no—
103C2×M16×C61—22no—
104C2×M16×C61—22no—
105C2×M16×C61—22no—
106C2×M16×C61C2312yes—
107C2×M16×C61C2312yes—
108C22×C8×C61—21no—
109C22×C8×C61—21no—
110#101×C61—24no—
111#100×C61—24no—
112C24×C61—21no—
113C24×C61#4111yescharacteristic
114C23×D61—21no—
115#41×C61—24no—
116#41×C61—24no—
117#41×C61—24no—
118#41×C61—24no—
119#41×C61—42no—
120#41×C61—42no—
121#41×C61—42no—
122#41×C61—42no—
123C2×Q8×C61—21no—
124C2×Q8×C61—21no—
125C2×Q8×C61—21no—
126C2×Q8×C61—21no—
127C2×Q8×C61—21no—
128C2×Q8×C61C2×D412yes—
129C2×D4×C61—24no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—41no—
136C2×D4×C61—41no—
137C2×D4×C61—42no—
138C2×D4×C61C2×D412yes—
139C22×C4×C61—21no—
140C22×C4×C61—24no—
141C22×C4×C61—41no—
142C22×C4×C61—41no—
143C22×C4×C61—42no—
144C22×C4×C61C2×D412yes—
145C22×C4×C61C2×D412yes—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61#4112yes—
151C22×C4×C61#4111yescharacteristic
152C22×C4×C61#4112yes—
153C22×C4×C61C2411yescharacteristic
154C2×C22⋊D61—22no—
155C2×C22⋊D61—22no—
156C2×C22⋊D61—42no—
157C2×C22⋊D61—42no—
158C22×Dic61—21no—
159C22×Dic61—22no—
160C22×Dic61—41no—
161C22×Dic61—41no—
162C22×Dic61—82no—
163#11758222—24no—
164#11758222—24no—
165#11758222—22no—
166#11758222—22no—
167#11758222—22no—
168#11758222—22no—
169#11758222—42no—
170#11758222—42no—
171#11758222—42no—
172#11758222—42no—
173C2×D244—22no—
174C2×D244—22no—
175C2×D244—21no—
176C2×D244C2×D412yes—
177C2×C4×D61—22no—
178C2×C4×D61—22no—
179C2×C4×D61—42no—
180C2×C4×D61—42no—
181C2×Dic122—22no—
182C2×Dic122—22no—
183C2×Dic122—21no—
184C2×Dic122C2×D412yes—
185M16×C61—22no—
186M16×C61—22no—
187M16×C61—22no—
188M16×C61—22no—
189M16×C61—42no—
190M16×C61—42no—
191C2×C8×C61—22no—
192C2×C8×C61—22no—
193C2×C8×C61—42no—
194C2×C8×C61—42no—
195#11758202—28no—
196#11758201—28no—
197#11758200—28no—
198#11758199—28no—
199C4⋊Dic61—22no—
200C4⋊Dic61—22no—
201C4⋊Dic61—41no—
202C4⋊Dic61—41no—
203#11758194—24no—
204#11758194—24no—
205#11758194—24no—
206#11758194—22no—
207#11758194C2×D414yes—
208C2×#92418—24no—
209C2×#92418—22no—
210C2×#92418—22no—
211C2×#92418—22no—
212C2×#92418C2×D414yes—
213C23×C61—22no—
214C23×C61—41no—
215C23×C61—41no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61—81no—
219C23×C61—81no—
220C23×C61#12111yescharacteristic
221C23×C61#12111yescharacteristic
222C23×C61#12312yes—
223C23×C61#12412yes—
224C23×C61C2×#4111yescharacteristic
225C22×D61—22no—
226C22×D61—42no—
227C22×D61—81no—
228C22×D61—81no—
229Q8×C61—22no—
230Q8×C61—22no—
231Q8×C61—41no—
232Q8×C61—41no—
233D4×C61—22no—
234D4×C61—22no—
235D4×C61—44no—
236D4×C61—44no—
237D4×C61—41no—
238D4×C61—41no—
239C2×C4×C61—22no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—44no—
243C2×C4×C61—44no—
244C2×C4×C61#12012yes—
245C2×C4×C61#12012yes—
246C2×C4×C61#12112yes—
247C2×C4×C61#12112yes—
248C2×C4×C61#12112yes—
249C2×C4×C61#12112yes—
250C2×C4×C61#12111yescharacteristic
251C2×C4×C61#12111yescharacteristic
252C2×C4×C61#12312yes—
253C2×C4×C61#12312yes—
254C2×C4×C61#12312yes—
255C2×C4×C61#12412yes—
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C22×D411yescharacteristic
260C2×C4×C61C2×#4111yescharacteristic
261C2×C4×C61C2×#4111yescharacteristic
262C22⋊D61—42no—
263C22⋊D61—42no—
264C22⋊D61—44no—
265C2×Dic61—22no—
266C2×Dic61—42no—
267C2×Dic61—42no—
268C2×Dic61—82no—
269C2×Dic61—82no—
270C2×Dic61—82no—
271D244—22no—
272D244—22no—
273D244—42no—
274C4×D61—42no—
275C4×D61—42no—
276C4×D61—44no—
277Dic122—22no—
278Dic122—22no—
279Dic122—42no—
280C8×C61—42no—
281C8×C61—42no—
282#92418—24no—
283#92418—24no—
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—82no—
295C22×C61—82no—
296C22×C61—81no—
297C22×C61—81no—
298C22×C61#39412yes—
299C22×C61#39412yes—
300C22×C61C2×#12111yescharacteristic
301C22×C61C2×#12311yescharacteristic
302C22×C61C2×#12411yescharacteristic
303C2×D61—42no—
304C2×D61—42no—
305C2×D61—42no—
306C2×D61—84no—
307C4×C61—42no—
308C4×C61#39412yes—
309C4×C61#39412yes—
310C4×C61C2×#12011yescharacteristic
311C4×C61C2×#12111yescharacteristic
312C4×C61C2×#12111yescharacteristic
313C4×C61C2×#12311yescharacteristic
314Dic61—42no—
315Dic61—82no—
316C2×#575—611no—
317C2×#430—611no—
318C2×#417—611no—
319C2×#417—611no—
320C2×#413D6111yescharacteristic, complemented
321#2879—611no—
322#2052—614no—
323#2051—614no—
324C2×#343—611no—
325C2×C61—22no—
326C2×C61—21no—
327C2×C61—21no—
328C2×C61—22no—
329C2×C61—82no—
330C2×C61C2×#39411yescharacteristic
331C2×C61#306811yescharacteristic
332C2×C61#306911yescharacteristic
333D61—82no—
334C22×#41—611no—
335C22×#41C2×D6111yescharacteristic
336#575—1224no—
337#575—1224no—
338C2×#137—612no—
339C2×#137—1221no—
340C2×AGL(1,ℤ/8)—612no—
341C2×AGL(1,ℤ/8)—1221no—
342C2×#135—1222no—
343C22×M16—611no—
344C22×M16—611no—
345C22×M16C2×D6111yescharacteristic
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—24no—
359C2×#101C2×D6112yescomplemented
360C2×#100C2×D6112yescomplemented
361#408—614no—
362#408—1222no—
363#407—1222no—
364#343—614no—
365C61#2353311yescharacteristic, complemented, Sylow
366C2×#41—22no—
367C2×#41—21no—
368C2×#41—21no—
369C2×#41—22no—
370C2×#41—21no—
371C2×#41—21no—
372C2×#41—22no—
373C2×#41—612no—
374C2×#41—612no—
375C2×#41—1221no—
376C2×#41—1221no—
377C2×#41—1222no—
378C2×#41—1222no—
379C2×#41—1222no—
380C2×#41—1222no—
381C2×#41C22⋊D6114yes—
382C22×Q8—611no—
383C22×Q8C22×D6111yescharacteristic
384C22×D4—21no—
385C22×D4—611no—
386C22×D4—1221no—
387C22×D4C22×D6111yescharacteristic
388C23×C4—21no—
389C23×C4—1221no—
390C23×C4—1221no—
391C23×C4C22×D6111yescharacteristic
392#137—1224no—
393#137—1228no—
394#137—1224no—
395AGL(1,ℤ/8)—1224no—
396AGL(1,ℤ/8)—1228no—
397AGL(1,ℤ/8)—1224no—
398#135—12216no—
399C2×Q16—1222no—
400C2×SD16—1222no—
401C2×SD16—1222no—
402C2×D8—1222no—
403C2×M16—21no—
404C2×M16—21no—
405C2×M16—22no—
406C2×M16—22no—
407C2×M16—22no—
408C2×M16—22no—
409C2×M16—612no—
410C2×M16—612no—
411C2×M16—614no—
412C2×M16—612no—
413C2×M16—1222no—
414C2×M16—1221no—
415C2×M16—1222no—
416C2×M16—1222no—
417C2×M16C22×D6112yes—
418C2×M16C22×D6112yes—
419C22×C8—21no—
420C22×C8—21no—
421C22×C8—612no—
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—24no—
435#100—24no—
436C22⋊C8—1228no—
437C24—21no—
438C24—1221no—
439C24#1175822211yescharacteristic
440#41—24no—
441#41—24no—
442#41—24no—
443#41—24no—
444#41—42no—
445#41—42no—
446#41—42no—
447#41—42no—
448#41—1222no—
449#41—1222no—
450#41—1222no—
451#41—1222no—
452#41—1224no—
453#41—1224no—
454#41—2442no—
455#41—2442no—
456#41—2442no—
457#41—2442no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—612no—
464C2×Q8—1221no—
465C2×Q8—1222no—
466C2×Q8—1222no—
467C2×Q8C2×C22⋊D6112yes—
468C2×D4—24no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—41no—
475C2×D4—41no—
476C2×D4—42no—
477C2×D4—612no—
478C2×D4—1222no—
479C2×D4—1222no—
480C2×D4—1222no—
481C2×D4—1221no—
482C2×D4—1222no—
483C2×D4—2442no—
484C2×D4—2442no—
485C2×D4C2×C22⋊D6112yes—
486C22×C4—21no—
487C22×C4—24no—
488C22×C4—41no—
489C22×C4—41no—
490C22×C4—42no—
491C22×C4—1221no—
492C22×C4—1222no—
493C22×C4—1222no—
494C22×C4—1222no—
495C22×C4—4882no—
496C22×C4—2441no—
497C22×C4—2441no—
498C22×C4—2442no—
499C22×C4—2442no—
500C22×C4#1175822212yes—
501C22×C4#1175822211yescharacteristic
502C22×C4D4×D6112yes—
503C22×C4D4×D6111yescharacteristic
504C22×C4D4×D6111yescharacteristic
505C22×C4D4×D6111yescharacteristic
506C22×C4D4×D6111yescharacteristic
507C22×C4#1175822412yes—
508C22×C4C2×C22⋊D6112yes—
509C22×C4C23×D6111yescharacteristic
510Q16—1228no—
511SD16—1228no—
512SD16—1228no—
513D8—1228no—
514M16—22no—
515M16—22no—
516M16—22no—
517M16—22no—
518M16—42no—
519M16—42no—
520M16—614no—
521M16—1222no—
522M16—1224no—
523M16—1224no—
524M16—1224no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—42no—
528C2×C8—42no—
529C2×C8—614no—
530C2×C8—1222no—
531C2×C8—1222no—
532C2×C8—1222no—
533C2×C8—1224no—
534C4⋊C4—1222no—
535C4⋊C4—1222no—
536C4⋊C4—2441no—
537C4⋊C4—2441no—
538C23—22no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—81no—
544C23—81no—
545C23—1222no—
546C23—4881no—
547C23—4881no—
548C23—2442no—
549C23#1451975011yescharacteristic
550C23#1451975011yescharacteristic
551C23#1451976012yes—
552C23#1451976312yes—
553C23C2×#1175822211yescharacteristic
554Q8—22no—
555Q8—22no—
556Q8—41no—
557Q8—41no—
558Q8—1222no—
559Q8—1222no—
560Q8—2442no—
561D4—22no—
562D4—22no—
563D4—44no—
564D4—44no—
565D4—41no—
566D4—41no—
567D4—1222no—
568D4—1222no—
569D4—2442no—
570D4—2442no—
571D4—2444no—
572D4—2442no—
573C2×C4—22no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—1222no—
579C2×C4—4882no—
580C2×C4—4882no—
581C2×C4—4882no—
582C2×C4—2442no—
583C2×C4—2442no—
584C2×C4—2442no—
585C2×C4—2442no—
586C2×C4—2444no—
587C2×C4#1451974512yes—
588C2×C4#1451974512yes—
589C2×C4#1451974912yes—
590C2×C4#1451974912yes—
591C2×C4#1451975012yes—
592C2×C4#1451975012yes—
593C2×C4#1451975011yescharacteristic
594C2×C4#1451975011yescharacteristic
595C2×C4#1451975912yes—
596C2×C4#1451975912yes—
597C2×C4#1451976012yes—
598C2×C4#1451976312yes—
599C2×C4C2×D4×D6111yescharacteristic
600C2×C4C2×D4×D6111yescharacteristic
601C2×C4C2×D4×D6111yescharacteristic
602C2×C4C22×C22⋊D6111yescharacteristic
603C2×C4C2×#1175822411yescharacteristic
604C2×C4C2×#1175822211yescharacteristic
605C8—42no—
606C8—42no—
607C8—1224no—
608C8—1224no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—82no—
619C22—82no—
620C22—81no—
621C22—81no—
622C22—4884no—
623C22—2442no—
624C22—2442no—
625C22—2442no—
626C22#1683967712yes—
627C22#1683967712yes—
628C22C2×#1451975011yescharacteristic
629C22C2×#1451976011yescharacteristic
630C22C2×#1451976311yescharacteristic
631C4—42no—
632C4—4882no—
633C4—2442no—
634C4#1683967712yes—
635C4#1683967712yes—
636C4C2×#1451974511yescharacteristic
637C4C2×#1451974911yescharacteristic
638C4C2×#1451975011yescharacteristic
639C4C2×#1451975911yescharacteristic
640C2—21no—
641C2—21no—
642C2—22no—
643C2—22no—
644C2—82no—
645C2—4882no—
646C2C2×#1683967711yescharacteristic
647C2#2145555011yescharacteristic
648C2#2145555911yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#413C2
socleC22×C61C2×#121
prime radicalC2×#16840560C2
hypercenterC2×#413D61
hypocenterC61#23533
LCSG ▷ C22×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. C2×#16839764C2semidirect
4. C2×#16839763C2semidirect
5. #21454263C2semidirect
6. #21447118C2semidirect
7. #21447111C2semidirect
8. C2×#16839530C2semidirect
9. C2×C61×#413C2semidirect
22. C2×#14519712C22semidirect
28. C2×#14519701C22semidirect
30. C2×#14519698C22semidirect
34. #16839530C22semidirect
39. C2×#101×C61C22semidirect
40. C2×#100×C61C22semidirect
320. C2×#413D61semidirect
359. C2×#101C2×D61semidirect
360. C2×#100C2×D61semidirect
365. C61#23533semidirect

Maximal subgroups

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

subgroupindex
C2×#168405602
C2×#168398102
C2×#168397642
C2×#168397632
#214542632
#214471182
#214471112
C2×#168395302
C2×C61×#4132
#2353361