Finite Group Databasebeta

← #29706871

Subgroups — #29706871

Order 15616.

2206 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519803C211yesmaximal, characteristic, complemented
2C2×#16840340C211yesmaximal, characteristic, complemented
3C2×#16839756C211yesmaximal, characteristic, complemented
4C22×#14519698C211yesmaximal, characteristic, complemented
5C2×#16839729C211yesmaximal, characteristic, complemented
6#21446822C214yesmaximal, complemented
7#21446815C214yesmaximal, complemented
8#21445540C211yesmaximal, characteristic, complemented
9C2×C61×#535C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C2×#14519803—24no—
12C2×#14519803—24no—
13C2×#14519803—24no—
14C22×#11758200C2212yes—
15C22×#11758199C2212yes—
16C22×#11758194C2211yescharacteristic
17C23×#92418C2211yescharacteristic
18#16840340—24no—
19#16840340—24no—
20C2×#14519756C2212yescomplemented
21C2×#14519753—21no—
22C2×#14519751C2212yescomplemented
23C2×#14519749—22no—
24C2×#14519748—21no—
25C22×C4⋊Dic61C2211yescharacteristic
26#16839756—24no—
27C2×#14519700C2212yescomplemented
28C2×#14519698—24no—
29C2×#14519698C2214yes—
30C2×#14519697C2212yescomplemented
31#16839729—24no—
32C2×C22⋊#92418C2214yescomplemented
33#16839524C2214yescomplemented
34C22×#41×C61C2211yescharacteristic
35C23×D4×C61C2211yescharacteristic
36C61×#535—24no—
37C61×#535—24no—
38C2×#127×C61—21no—
39C2×#124×C61—21no—
40C2×#121×C61C2214yes—
41C2×#120×C61—22no—
42C2×#117×C61C2211yescharacteristic
43C2×#11758222—22no—
44C2×#11758222—22no—
45C2×#11758222—22no—
46C2×#11758222—22no—
47C2×#11758222—22no—
48C2×#11758222D414yescomplemented
49C22×Dic122C2311yescharacteristic
50C22×D244C2311yescharacteristic
51C22×C22⋊D61—21no—
52C22×C22⋊D61—22no—
53C23×Dic61—21no—
54C22×C4×D61—21no—
55#14519803—232no—
56C2×#11758200—24no—
57C2×#11758200—24no—
58C2×#11758200—24no—
59C2×#11758199—24no—
60C2×#11758199—24no—
61C2×#11758199—24no—
62C2×#11758194—22no—
63C2×#11758194—22no—
64C2×#11758194D414yescomplemented
65C22×#92418—22no—
66C22×#92418—22no—
67C22×#92418—22no—
68C22×#92418C2312yes—
69C22×#92418C2312yes—
70#14519756—24no—
71#14519756—24no—
72#14519753—28no—
73#14519751—24no—
74#14519751—24no—
75#14519749—216no—
76#14519748—28no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61—22no—
79C2×C4⋊Dic61D414yescomplemented
80C2×C4⋊Dic61C2312yes—
81C2×#11758196—22no—
82C2×C22⋊Dic61—22no—
83C2×#11758198—22no—
84#14519700—24no—
85#14519698—28no—
86#14519697—24no—
87C22⋊#92418—28no—
88C25×C61—21no—
89C2×#41×C61—22no—
90C2×#41×C61—22no—
91C2×#41×C61—22no—
92C2×#41×C61—22no—
93C2×#41×C61—22no—
94C2×#41×C61—22no—
95C22×Q8×C61D411yescharacteristic
96C22×D4×C61—24no—
97C22×D4×C61—22no—
98C22×D4×C61—22no—
99C22×D4×C61—22no—
100C22×D4×C61—21no—
101C22×D4×C61—22no—
102C22×D4×C61—22no—
103C22×D4×C61D411yescharacteristic
104C22×D4×C61D412yes—
105C22×D4×C61D414yescomplemented
106C22×D4×C61C2312yes—
107C23×C4×C61C2312yes—
108C23×C4×C61C2311yescharacteristic
109#127×C61—28no—
110#124×C61—28no—
111#121×C61—28no—
112#121×C61—28no—
113#120×C61—216no—
114#117×C61—22no—
115#117×C61D414yescomplemented
116C2×C4⋊C4×C61C2312yes—
117C2×C22⋊C4×C61—21no—
118C2×C22⋊C4×C61—24no—
119C2×C42×C61—21no—
120C24×C61—22no—
121C24×C61—24no—
122C24×C61—42no—
123C24×C61—42no—
124C24×C61—82no—
125C24×C61—82no—
126C24×C61C2×D412yes—
127C24×C61C2×D411yescharacteristic
128C23×D61—21no—
129#41×C61—44no—
130#41×C61—44no—
131#41×C61—44no—
132#41×C61—44no—
133C2×Q8×C61—22no—
134C2×Q8×C61—22no—
135C2×Q8×C61—22no—
136C2×D4×C61—24no—
137C2×D4×C61—24no—
138C2×D4×C61—24no—
139C2×D4×C61—216no—
140C2×D4×C61—28no—
141C2×D4×C61—24no—
142C2×D4×C61—22no—
143C2×D4×C61—22no—
144C2×D4×C61—28no—
145C2×D4×C61—22no—
146C2×D4×C61—22no—
147C2×D4×C61—22no—
148C2×D4×C61—22no—
149C2×D4×C61—22no—
150C2×D4×C61—22no—
151C2×D4×C61—42no—
152C2×D4×C61—44no—
153C2×D4×C61—42no—
154C2×D4×C61—44no—
155C2×D4×C61—84no—
156C2×D4×C61—84no—
157C2×D4×C61C2×D414yes—
158C22×C4×C61—22no—
159C22×C4×C61—22no—
160C22×C4×C61—22no—
161C22×C4×C61—22no—
162C22×C4×C61—22no—
163C22×C4×C61—22no—
164C22×C4×C61—22no—
165C22×C4×C61C2×D412yes—
166C22×C4×C61C2×D411yescharacteristic
167C22×C4×C61C2×D412yes—
168C22×C4×C61C2×D412yes—
169C22×C4×C61C2×D412yes—
170C22×C4×C61C2×D412yes—
171C22×C4×C61C2×D412yes—
172C22×C4×C61#4112yes—
173C22×C4×C61C2411yescharacteristic
174C2×C22⋊D61—24no—
175C2×C22⋊D61—28no—
176C2×C22⋊D61—42no—
177C2×C22⋊D61—42no—
178C2×C22⋊D61—84no—
179C22×Dic61—21no—
180C22×Dic61—22no—
181C22×Dic61—24no—
182C22×Dic61—82no—
183#11758222—28no—
184#11758222—28no—
185#11758222—44no—
186#11758222—44no—
187C2×D244—21no—
188C2×D244—21no—
189C2×D244—21no—
190C2×D244—21no—
191C2×D244—21no—
192C2×D244C2×D412yes—
193C2×C4×D61—24no—
194C2×C4×D61—42no—
195C2×C4×D61—42no—
196C2×Dic122—21no—
197C2×Dic122—21no—
198C2×Dic122—21no—
199C2×Dic122—21no—
200C2×Dic122—21no—
201C2×Dic122C2×D412yes—
202C4⋊C4×C61—22no—
203C4⋊C4×C61C2×D414yes—
204C22⋊C4×C61—216no—
205C22⋊C4×C61—24no—
206C42×C61—24no—
207C22⋊Dic61—28no—
208#11758200—216no—
209#11758199—216no—
210#11758198—28no—
211C4⋊Dic61—24no—
212C4⋊Dic61—22no—
213C4⋊Dic61C2×D414yes—
214#11758196—28no—
215#11758194—28no—
216C2×#92418—22no—
217C2×#92418—22no—
218C2×#92418—22no—
219C2×#92418—22no—
220C2×#92418—22no—
221C2×#92418—44no—
222C2×#92418C2×D414yes—
223C23×C61—22no—
224C23×C61—22no—
225C23×C61—22no—
226C23×C61—24no—
227C23×C61—22no—
228C23×C61—22no—
229C23×C61—24no—
230C23×C61—24no—
231C23×C61—42no—
232C23×C61—42no—
233C23×C61—44no—
234C23×C61—44no—
235C23×C61—48no—
236C23×C61—88no—
237C23×C61—84no—
238C23×C61—84no—
239C23×C61—88no—
240C23×C61#12012yes—
241C23×C61#12012yes—
242C23×C61#12112yes—
243C23×C61C22×D411yescharacteristic
244C22×D61—22no—
245C22×D61—41no—
246C22×D61—41no—
247C22×D61—42no—
248C22×D61—42no—
249Q8×C61—44no—
250D4×C61—24no—
251D4×C61—24no—
252D4×C61—48no—
253D4×C61—42no—
254D4×C61—42no—
255D4×C61—48no—
256D4×C61—44no—
257D4×C61—816no—
258C2×C4×C61—22no—
259C2×C4×C61—24no—
260C2×C4×C61—22no—
261C2×C4×C61—22no—
262C2×C4×C61—42no—
263C2×C4×C61—42no—
264C2×C4×C61—42no—
265C2×C4×C61—42no—
266C2×C4×C61—42no—
267C2×C4×C61—42no—
268C2×C4×C61#12014yes—
269C2×C4×C61#12012yes—
270C2×C4×C61#12012yes—
271C2×C4×C61#12114yes—
272C2×C4×C61#12114yes—
273C2×C4×C61#12112yes—
274C2×C4×C61#12714yes—
275C2×C4×C61C22×D411yescharacteristic
276C2×C4×C61C22×D411yescharacteristic
277C2×C4×C61C22×D411yescharacteristic
278C2×C4×C61C22×D411yescharacteristic
279C2×C4×C61C22×D411yescharacteristic
280C2×C4×C61C2×#4111yescharacteristic
281C22⋊D61—48no—
282C22⋊D61—88no—
283C2×Dic61—22no—
284C2×Dic61—24no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—44no—
288C2×Dic61—84no—
289D244—22no—
290D244—22no—
291D244—41no—
292D244—41no—
293C4×D61—48no—
294Dic122—22no—
295Dic122—22no—
296Dic122—41no—
297Dic122—41no—
298#92418—24no—
299#92418—42no—
300#23393—611nomaximal, Sylow
301C22×C61—22no—
302C22×C61—22no—
303C22×C61—24no—
304C22×C61—24no—
305C22×C61—42no—
306C22×C61—44no—
307C22×C61—42no—
308C22×C61—42no—
309C22×C61—44no—
310C22×C61—48no—
311C22×C61—44no—
312C22×C61—816no—
313C22×C61—82no—
314C22×C61—82no—
315C22×C61—84no—
316C22×C61#39214yes—
317C22×C61C2×#12011yescharacteristic
318C22×C61C2×#12011yescharacteristic
319C22×C61C2×#12111yescharacteristic
320C22×C61#57612yes—
321C22×C61#57712yes—
322C2×D61—42no—
323C2×D61—42no—
324C2×D61—42no—
325C2×D61—82no—
326C2×D61—82no—
327C4×C61—42no—
328C4×C61—44no—
329C4×C61#39214yes—
330C4×C61C2×#12011yescharacteristic
331C4×C61C2×#12111yescharacteristic
332C4×C61C2×#12111yescharacteristic
333C4×C61C2×#12711yescharacteristic
334Dic61—42no—
335Dic61—44no—
336C22×AGL(1,ℤ/8)—611no—
337C2×#537—611no—
338C2×#535D6111yescharacteristic, complemented
339C2×#416—611no—
340C22×#102—611no—
341C2×#408—611no—
342#2000—614no—
343#1999—614no—
344#1736—611no—
345C2×C61—24no—
346C2×C61—82no—
347C2×C61—84no—
348C2×C61C2×#39211yescharacteristic
349C2×C61#300811yescharacteristic
350C2×C61#301011yescharacteristic
351C2×C61#301212yes—
352C2×C61#306212yes—
353D61—82no—
354C22×#41—611no—
355C22×#41C2×D6111yescharacteristic
356C23×D4C2×D6111yescharacteristic
357C2×AGL(1,ℤ/8)—1224no—
358C2×AGL(1,ℤ/8)—1224no—
359C2×AGL(1,ℤ/8)—1224no—
360C22×SD16—612no—
361C22×D8—612no—
362C22×M16—611no—
363C23×C8—611no—
364#537—1224no—
365#537—1224no—
366#535—24no—
367#535—24no—
368C2×#127—21no—
369C2×#124—21no—
370C2×C22⋊Q8—612no—
371C2×C22⋊Q8—1221no—
372C2×#121—612no—
373C2×#121—1221no—
374C2×#121—1222no—
375C2×#121C2×D6114yes—
376C2×#120—22no—
377C2×#117C2×D6111yescharacteristic
378C22×C4⋊C4—611no—
379#416—1224no—
380C2×#103—612no—
381C2×#102—614no—
382C2×#102—612no—
383C2×#102—1224no—
384#408—1224no—
385C2×C22⋊C8—614no—
386#340—614no—
387C61#2339311yescharacteristic, complemented, Sylow
388C25—21no—
389C2×#41—22no—
390C2×#41—22no—
391C2×#41—22no—
392C2×#41—22no—
393C2×#41—22no—
394C2×#41—22no—
395C2×#41—614no—
396C2×#41—1222no—
397C2×#41—1222no—
398C2×#41—1222no—
399C2×#41—1222no—
400C2×#41—1222no—
401C22×Q8—611no—
402C22×Q8C22⋊D6111yescharacteristic
403C22×D4—24no—
404C22×D4—22no—
405C22×D4—22no—
406C22×D4—22no—
407C22×D4—22no—
408C22×D4—21no—
409C22×D4—22no—
410C22×D4—611no—
411C22×D4—1222no—
412C22×D4—1221no—
413C22×D4C22⋊D6112yes—
414C22×D4C22⋊D6111yescharacteristic
415C22×D4C22⋊D6114yescomplemented
416C22×D4C22×D6112yes—
417C23×C4—1221no—
418C23×C4—1221no—
419C23×C4C22×D6112yes—
420C23×C4C22×D6111yescharacteristic
421AGL(1,ℤ/8)—12232no—
422C2×SD16—1224no—
423C2×SD16—1224no—
424C2×SD16—1224no—
425C2×D8—1224no—
426C2×D8—1224no—
427C2×D8—1224no—
428C2×M16—614no—
429C2×M16—1222no—
430C2×M16—1222no—
431C22×C8—612no—
432C22×C8—612no—
433C22×C8—1222no—
434C22×C8—1222no—
435C22×C8—1222no—
436#127—28no—
437#124—28no—
438C22⋊Q8—1224no—
439C22⋊Q8—1224no—
440C22⋊Q8—1228no—
441#121—28no—
442#121—28no—
443#121—1224no—
444#121—1224no—
445#121—1228no—
446#121—12216no—
447#120—216no—
448#117—22no—
449#117C22⋊D6114yescomplemented
450C2×C4⋊C4—614no—
451C2×C4⋊C4—612no—
452C2×C4⋊C4—1222no—
453C2×C4⋊C4—1221no—
454C2×C4⋊C4—1222no—
455C2×C4⋊C4C22×D6112yes—
456C2×C22⋊C4—21no—
457C2×C22⋊C4—24no—
458C2×C22⋊C4—1222no—
459C2×C22⋊C4—1222no—
460C2×C42—21no—
461#103—1224no—
462#102—1228no—
463#102—1224no—
464C22⋊C8—1228no—
465C24—22no—
466C24—24no—
467C24—42no—
468C24—42no—
469C24—82no—
470C24—82no—
471C24—1221no—
472C24C2×C22⋊D6112yes—
473C24C2×C22⋊D6111yescharacteristic
474#41—44no—
475#41—44no—
476#41—44no—
477#41—44no—
478#41—1228no—
479#41—1228no—
480#41—2444no—
481#41—2444no—
482C2×Q8—22no—
483C2×Q8—22no—
484C2×Q8—22no—
485C2×Q8—612no—
486C2×Q8—1221no—
487C2×Q8—1221no—
488C2×Q8—1221no—
489C2×Q8—1221no—
490C2×Q8—1221no—
491C2×D4—24no—
492C2×D4—24no—
493C2×D4—24no—
494C2×D4—216no—
495C2×D4—22no—
496C2×D4—22no—
497C2×D4—22no—
498C2×D4—22no—
499C2×D4—22no—
500C2×D4—28no—
501C2×D4—22no—
502C2×D4—24no—
503C2×D4—22no—
504C2×D4—28no—
505C2×D4—22no—
506C2×D4—42no—
507C2×D4—44no—
508C2×D4—42no—
509C2×D4—44no—
510C2×D4—84no—
511C2×D4—84no—
512C2×D4—612no—
513C2×D4—1228no—
514C2×D4—1224no—
515C2×D4—1221no—
516C2×D4—1221no—
517C2×D4—1221no—
518C2×D4—1221no—
519C2×D4—1221no—
520C2×D4—4884no—
521C2×D4—2442no—
522C2×D4—2442no—
523C2×D4C2×C22⋊D6114yes—
524C22×C4—22no—
525C22×C4—22no—
526C22×C4—22no—
527C22×C4—22no—
528C22×C4—22no—
529C22×C4—22no—
530C22×C4—22no—
531C22×C4—1224no—
532C22×C4—1222no—
533C22×C4—1221no—
534C22×C4—1224no—
535C22×C4—4882no—
536C22×C4—2442no—
537C22×C4—2442no—
538C22×C4D4×D6112yes—
539C22×C4D4×D6112yes—
540C22×C4D4×D6112yes—
541C22×C4#1175822412yes—
542C22×C4C2×C22⋊D6112yes—
543C22×C4C2×C22⋊D6112yes—
544C22×C4C2×C22⋊D6112yes—
545C22×C4C2×C22⋊D6111yescharacteristic
546C22×C4C23×D6111yescharacteristic
547SD16—12216no—
548D8—12216no—
549M16—1228no—
550C2×C8—614no—
551C2×C8—1222no—
552C2×C8—1222no—
553C2×C8—1222no—
554C2×C8—1222no—
555C2×C8—1222no—
556C2×C8—2444no—
557C4⋊C4—22no—
558C4⋊C4—614no—
559C4⋊C4—1228no—
560C4⋊C4—1222no—
561C4⋊C4—1224no—
562C4⋊C4C2×C22⋊D6114yes—
563C22⋊C4—216no—
564C22⋊C4—24no—
565C22⋊C4—1228no—
566C22⋊C4—1228no—
567C42—24no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—24no—
572C23—22no—
573C23—22no—
574C23—24no—
575C23—24no—
576C23—42no—
577C23—42no—
578C23—44no—
579C23—44no—
580C23—48no—
581C23—88no—
582C23—88no—
583C23—84no—
584C23—84no—
585C23—1222no—
586C23—2441no—
587C23—2441no—
588C23—2442no—
589C23—2442no—
590C23#1451974512yes—
591C23#1451974612yes—
592C23#1451974912yes—
593C23C22×C22⋊D6111yescharacteristic
594Q8—44no—
595Q8—1222no—
596Q8—1222no—
597Q8—2441no—
598Q8—2441no—
599D4—24no—
600D4—24no—
601D4—48no—
602D4—42no—
603D4—42no—
604D4—48no—
605D4—44no—
606D4—816no—
607D4—1222no—
608D4—1222no—
609D4—4888no—
610D4—2448no—
611D4—2441no—
612D4—2441no—
613C2×C4—22no—
614C2×C4—24no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—1224no—
624C2×C4—1222no—
625C2×C4—4884no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4—2444no—
629C2×C4—2448no—
630C2×C4#1451974514yes—
631C2×C4#1451974512yes—
632C2×C4#1451974612yes—
633C2×C4#1451974814yes—
634C2×C4#1451974914yes—
635C2×C4#1451974912yes—
636C2×C4#1451977314yes—
637C2×C4C2×D4×D6111yescharacteristic
638C2×C4C2×D4×D6111yescharacteristic
639C2×C4C2×D4×D6111yescharacteristic
640C2×C4C22×C22⋊D6111yescharacteristic
641C2×C4C22×C22⋊D6111yescharacteristic
642C2×C4C2×#1175822411yescharacteristic
643C8—1224no—
644C8—2442no—
645C22—24no—
646C22—24no—
647C22—22no—
648C22—22no—
649C22—44no—
650C22—48no—
651C22—42no—
652C22—44no—
653C22—44no—
654C22—42no—
655C22—42no—
656C22—816no—
657C22—84no—
658C22—82no—
659C22—82no—
660C22—4882no—
661C22—4882no—
662C22—2442no—
663C22—2442no—
664C22—2442no—
665C22#1683967014yes—
666C22C2×#1451974511yescharacteristic
667C22C2×#1451974611yescharacteristic
668C22C2×#1451974911yescharacteristic
669C22#1684056512yes—
670C22#1684057312yes—
671C4—44no—
672C4—42no—
673C4—2442no—
674C4—2444no—
675C4#1683967014yes—
676C4C2×#1451974511yescharacteristic
677C4C2×#1451974811yescharacteristic
678C4C2×#1451974911yescharacteristic
679C4C2×#1451977311yescharacteristic
680C2—24no—
681C2—82no—
682C2—84no—
683C2—4882no—
684C2C2×#1683967011yescharacteristic
685C2#2145508511yescharacteristic
686C2#2145510311yescharacteristic
687C2#2145512312yes—
688C2#2145549512yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#535C2
socleC23×C61C22×D4
hypercenterC2×#535D61
hypocenterC61#23393
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#535

Complements

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

NH ≅ G/Ntype
1. C22×#14519803C2semidirect
2. C2×#16840340C2semidirect
3. C2×#16839756C2semidirect
4. C22×#14519698C2semidirect
5. C2×#16839729C2semidirect
6. #21446822C2semidirect
7. #21446815C2semidirect
8. #21445540C2semidirect
9. C2×C61×#535C2semidirect
20. C2×#14519756C22semidirect
22. C2×#14519751C22semidirect
27. C2×#14519700C22semidirect
30. C2×#14519697C22semidirect
32. C2×C22⋊#92418C22semidirect
33. #16839524C22semidirect
48. C2×#11758222D4semidirect
64. C2×#11758194D4semidirect
79. C2×C4⋊Dic61D4semidirect
105. C22×D4×C61D4semidirect
115. #117×C61D4semidirect
338. C2×#535D61semidirect
387. C61#23393semidirect
415. C22×D4C22⋊D61semidirect
449. #117C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C22×#145198032
C2×#168403402
C2×#168397562
C22×#145196982
C2×#168397292
#214468222
#214468152
#214455402
C2×C61×#5352
#2339361