Finite Group Databasebeta

← #29707893

Subgroups — #29707893

Order 15616.

1350 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519712C211yesmaximal, characteristic, complemented
2#21454338C211yesmaximal, characteristic, complemented
3C2×#16839758C211yesmaximal, characteristic, complemented
4C2×#16839746C211yesmaximal, characteristic, complemented
5C2×#16839729C211yesmaximal, characteristic, complemented
6#21447115C214yesmaximal, complemented
7#21447108C214yesmaximal, complemented
8#21445557C211yesmaximal, characteristic, complemented
9C2×C61×#575C211yesmaximal, characteristic
10C22×#11758194C2211yescharacteristic
11C22×#11758194C2211yescharacteristic
12C23×#92418C2211yescharacteristic
13C2×#14519736C2211yescharacteristic
14C2×#14519712—24no—
15C2×#14519712C2214yescomplemented
16#16839771—21no—
17#16839767—21no—
18#16839763C2214yes—
19#16839758—22no—
20#16839758—22no—
21#16839758—22no—
22C2×#14519701C2212yescomplemented
23C2×#14519698C2212yescomplemented
24#16839746—24no—
25C2×#14519696C2212yescomplemented
26C2×#14519694C2212yescomplemented
27#16839729—22no—
28#16839729C2214yes—
29C2×C22⋊#92418—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
41C2×#11758194—21no—
42C2×#11758194—21no—
43C2×#11758194—22no—
44C2×#11758194—22no—
45C2×#11758194—22no—
46C2×#11758194—22no—
47C2×#11758194C2312yes—
48C2×#11758194C2312yes—
49C22×#92418—22no—
50C22×#92418—22no—
51C22×#92418—21no—
52C22×#92418—21no—
53C22×#92418D412yes—
54C22×#92418D412yes—
55C22×#92418C2312yes—
56#14519736—21no—
57#14519736—21no—
58#14519736—21no—
59#14519736—21no—
60C2×C4⋊Dic61C2312yes—
61C2×C22⋊Dic61—21no—
62C2×C4×Dic61—21no—
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—
182Q16×C61—28no—
183SD16×C61—28no—
184SD16×C61—28no—
185D8×C61—28no—
186M16×C61—24no—
187M16×C61—22no—
188M16×C61C2×D414yes—
189C2×C8×C61—24no—
190C2×C8×C61—22no—
191C22⋊Dic61—41no—
192C22⋊Dic61—41no—
193C4⋊Dic61—22no—
194C4⋊Dic61—22no—
195C4×Dic61—41no—
196C4×Dic61—41no—
197#11758194—22no—
198#11758194—22no—
199#11758194—22no—
200#11758194—22no—
201#11758194—42no—
202#11758194—42no—
203C2×#92418—24no—
204C2×#92418—24no—
205C2×#92418—22no—
206C2×#92418—22no—
207C2×#92418—22no—
208C2×#92418—22no—
209C2×#92418—42no—
210C2×#92418—42no—
211C2×#92418C2×D414yes—
212C23×C61—22no—
213C23×C61—22no—
214C23×C61—42no—
215C23×C61—44no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61—42no—
219C23×C61—82no—
220C23×C61#12111yescharacteristic
221C23×C61#12111yescharacteristic
222C23×C61#12312yes—
223C23×C61#12412yes—
224C23×C61C2×#4111yescharacteristic
225Q8×C61—22no—
226Q8×C61—22no—
227Q8×C61—22no—
228Q8×C61—22no—
229Q8×C61—41no—
230Q8×C61—41no—
231Q8×C61—42no—
232D4×C61—22no—
233D4×C61—22no—
234D4×C61—22no—
235D4×C61—22no—
236D4×C61—42no—
237D4×C61—42no—
238D4×C61—44no—
239D4×C61—44no—
240D4×C61—44no—
241D4×C61—41no—
242D4×C61—41no—
243D4×C61—42no—
244C2×C4×C61—22no—
245C2×C4×C61—22no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—42no—
249C2×C4×C61—42no—
250C2×C4×C61—42no—
251C2×C4×C61—44no—
252C2×C4×C61—44no—
253C2×C4×C61—44no—
254C2×C4×C61#12012yes—
255C2×C4×C61#12012yes—
256C2×C4×C61#12112yes—
257C2×C4×C61#12112yes—
258C2×C4×C61#12112yes—
259C2×C4×C61#12112yes—
260C2×C4×C61#12111yescharacteristic
261C2×C4×C61#12111yescharacteristic
262C2×C4×C61#12312yes—
263C2×C4×C61#12312yes—
264C2×C4×C61#12312yes—
265C2×C4×C61#12412yes—
266C2×C4×C61C22×D411yescharacteristic
267C2×C4×C61C22×D411yescharacteristic
268C2×C4×C61C22×D411yescharacteristic
269C2×C4×C61C22×D411yescharacteristic
270C2×C4×C61C2×#4111yescharacteristic
271C2×C4×C61C2×#4111yescharacteristic
272C2×Dic61—42no—
273C2×Dic61—82no—
274C8×C61—24no—
275#92418—24no—
276#92418—42no—
277#92418—42no—
278#23530—611nomaximal, Sylow
279C22×C61—22no—
280C22×C61—22no—
281C22×C61—22no—
282C22×C61—42no—
283C22×C61—42no—
284C22×C61—42no—
285C22×C61—42no—
286C22×C61—44no—
287C22×C61—82no—
288C22×C61—84no—
289C22×C61—82no—
290C22×C61—82no—
291C22×C61#39412yes—
292C22×C61#39412yes—
293C22×C61C2×#12111yescharacteristic
294C22×C61C2×#12311yescharacteristic
295C22×C61C2×#12411yescharacteristic
296C4×C61—42no—
297C4×C61—42no—
298C4×C61#39412yes—
299C4×C61#39412yes—
300C4×C61C2×#12011yescharacteristic
301C4×C61C2×#12111yescharacteristic
302C4×C61C2×#12111yescharacteristic
303C4×C61C2×#12311yescharacteristic
304Dic61—82no—
305C2×#575D6111yescharacteristic
306C22×#108—611no—
307#2896—611no—
308C2×#416—611no—
309C2×#413—611no—
310C2×#408—611no—
311#2052—614no—
312#2051—614no—
313#1742—611no—
314C2×C61—21no—
315C2×C61—21no—
316C2×C61—22no—
317C2×C61—22no—
318C2×C61—82no—
319C2×C61—82no—
320C2×C61C2×#39411yescharacteristic
321C2×C61#306712yes—
322C22×#41C2×D6111yescharacteristic
323C22×#41C2×D6111yescharacteristic
324#575—24no—
325#575—24no—
326C2×#137—21no—
327C2×#137C2×D6112yes—
328C2×AGL(1,ℤ/8)—21no—
329C2×AGL(1,ℤ/8)C2×D6112yes—
330C2×#135—22no—
331C22×M16—611no—
332C22×M16—611no—
333C22×M16C2×D6111yescharacteristic
334C23×C8—611no—
335C2×#117—611no—
336C2×#108—614no—
337C2×#108—1224no—
338#419—1221no—
339#418—1221no—
340#417—614no—
341#416—1222no—
342#416—1222no—
343#416—1222no—
344C2×#103—612no—
345C2×#102—612no—
346#413—1224no—
347C2×#101—612no—
348C2×#100—612no—
349#408—614no—
350#408—1222no—
351C2×C22⋊C8—1222no—
352#343—614no—
353C61#2353011yescharacteristic, complemented, Sylow
354C2×#41—21no—
355C2×#41—21no—
356C2×#41—22no—
357C2×#41—21no—
358C2×#41—21no—
359C2×#41—22no—
360C2×#41—22no—
361C2×#41—22no—
362C2×#41—21no—
363C2×#41—21no—
364C2×#41—22no—
365C2×#41—22no—
366C2×#41—22no—
367C2×#41C22⋊D6112yes—
368C2×#41C22⋊D6114yes—
369C2×#41C22×D6112yes—
370C22×Q8C22⋊D6111yescharacteristic
371C22×Q8C22×D6111yescharacteristic
372C22×D4—21no—
373C22×D4—21no—
374C22×D4C22⋊D6111yescharacteristic
375C22×D4C22×D6111yescharacteristic
376C23×C4—21no—
377C23×C4—21no—
378C23×C4C22×D6111yescharacteristic
379#137—28no—
380#137—24no—
381#137—24no—
382AGL(1,ℤ/8)—28no—
383AGL(1,ℤ/8)—24no—
384AGL(1,ℤ/8)—24no—
385#135—216no—
386C2×Q16—22no—
387C2×SD16—22no—
388C2×SD16—22no—
389C2×D8—22no—
390C2×M16—21no—
391C2×M16—22no—
392C2×M16—612no—
393C2×M16—612no—
394C2×M16—1221no—
395C2×M16—1221no—
396C2×M16—1222no—
397C2×M16—1222no—
398C2×M16—1222no—
399C2×M16—1222no—
400C2×M16C22⋊D6114yes—
401C2×M16C22×D6112yes—
402C22×C8—21no—
403C22×C8—612no—
404C22×C8—612no—
405C22×C8—612no—
406C22×C8—1222no—
407C22×C8—1222no—
408C22×C8—1221no—
409C22×C8—1221no—
410#117—1221no—
411#117—1221no—
412#117—1221no—
413#117—1221no—
414C2×C4⋊C4—612no—
415C2×C22⋊C4—1221no—
416C2×C42—1221no—
417#108—1228no—
418#103—1224no—
419#103—1224no—
420#102—1224no—
421#102—1224no—
422#101—1224no—
423#100—1224no—
424C22⋊C8—1228no—
425C24—21no—
426C24—21no—
427C24#1175822411yescharacteristic
428#41—22no—
429#41—22no—
430#41—22no—
431#41—22no—
432#41—24no—
433#41—24no—
434#41—24no—
435#41—24no—
436#41—24no—
437#41—24no—
438#41—42no—
439#41—42no—
440#41—42no—
441#41—42no—
442#41—42no—
443#41—42no—
444#41—42no—
445#41—42no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—22no—
453C2×Q8—22no—
454C2×Q8C2×C22⋊D6112yes—
455C2×Q8C2×C22⋊D6112yes—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—24no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—22no—
465C2×D4—21no—
466C2×D4—22no—
467C2×D4—41no—
468C2×D4—41no—
469C2×D4—42no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4C2×C22⋊D6112yes—
473C2×D4C2×C22⋊D6112yes—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—24no—
479C22×C4—42no—
480C22×C4—41no—
481C22×C4—42no—
482C22×C4—41no—
483C22×C4—42no—
484C22×C4—1222no—
485C22×C4D4×D6111yescharacteristic
486C22×C4D4×D6111yescharacteristic
487C22×C4#1175822412yes—
488C22×C4#1175822412yes—
489C22×C4#1175822411yescharacteristic
490C22×C4C2×C22⋊D6112yes—
491C22×C4C2×C22⋊D6112yes—
492C22×C4C2×C22⋊D6111yescharacteristic
493C22×C4C2×C22⋊D6111yescharacteristic
494C22×C4C23×D6111yescharacteristic
495Q16—28no—
496SD16—28no—
497SD16—28no—
498D8—28no—
499M16—22no—
500M16—24no—
501M16—1222no—
502M16—1222no—
503M16—1222no—
504M16—1222no—
505M16—2442no—
506M16—2442no—
507M16C2×C22⋊D6114yes—
508C2×C8—22no—
509C2×C8—24no—
510C2×C8—614no—
511C2×C8—1222no—
512C2×C8—1222no—
513C2×C8—1224no—
514C2×C8—1224no—
515C2×C8—1222no—
516C2×C8—1222no—
517C2×C8—2442no—
518C2×C8—2442no—
519C4⋊C4—1222no—
520C4⋊C4—1222no—
521C22⋊C4—2441no—
522C22⋊C4—2441no—
523C42—2441no—
524C42—2441no—
525C23—22no—
526C23—22no—
527C23—42no—
528C23—44no—
529C23—41no—
530C23—41no—
531C23—42no—
532C23—82no—
533C23#1451974911yescharacteristic
534C23#1451974911yescharacteristic
535C23#1451975812yes—
536C23#1451976412yes—
537C23C2×#1175822411yescharacteristic
538Q8—22no—
539Q8—22no—
540Q8—22no—
541Q8—22no—
542Q8—41no—
543Q8—41no—
544Q8—42no—
545D4—22no—
546D4—22no—
547D4—22no—
548D4—22no—
549D4—42no—
550D4—42no—
551D4—44no—
552D4—44no—
553D4—44no—
554D4—41no—
555D4—41no—
556D4—42no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—4882no—
568C2×C4—2442no—
569C2×C4#1451974612yes—
570C2×C4#1451974612yes—
571C2×C4#1451974812yes—
572C2×C4#1451974812yes—
573C2×C4#1451974912yes—
574C2×C4#1451974912yes—
575C2×C4#1451974911yescharacteristic
576C2×C4#1451974911yescharacteristic
577C2×C4#1451975812yes—
578C2×C4#1451975812yes—
579C2×C4#1451975812yes—
580C2×C4#1451976412yes—
581C2×C4C2×D4×D6111yescharacteristic
582C2×C4C22×C22⋊D6111yescharacteristic
583C2×C4C22×C22⋊D6111yescharacteristic
584C2×C4C22×C22⋊D6111yescharacteristic
585C2×C4C2×#1175822411yescharacteristic
586C2×C4C2×#1175822411yescharacteristic
587C8—24no—
588C8—1224no—
589C8—2442no—
590C8—2442no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—44no—
599C22—82no—
600C22—84no—
601C22—82no—
602C22—82no—
603C22#1683967612yes—
604C22#1683967612yes—
605C22C2×#1451974911yescharacteristic
606C22C2×#1451975811yescharacteristic
607C22C2×#1451976411yescharacteristic
608C4—42no—
609C4—42no—
610C4—4882no—
611C4#1683967612yes—
612C4#1683967612yes—
613C4C2×#1451974611yescharacteristic
614C4C2×#1451974811yescharacteristic
615C4C2×#1451974911yescharacteristic
616C4C2×#1451975811yescharacteristic
617C2—21no—
618C2—21no—
619C2—22no—
620C2—22no—
621C2—82no—
622C2—82no—
623C2C2×#1683967611yescharacteristic
624C2#2145553612yes—
subgroupG/H
centerC2×C4#14519749
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC2×C61×#575C2
socleC22×C61C2×#121
hypercenterC2×#575D61
hypocenterC61#23530
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#575

Complements

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

NH ≅ G/Ntype
1. C22×#14519712C2semidirect
2. #21454338C2semidirect
3. C2×#16839758C2semidirect
4. C2×#16839746C2semidirect
5. C2×#16839729C2semidirect
6. #21447115C2semidirect
7. #21447108C2semidirect
8. #21445557C2semidirect
15. C2×#14519712C22semidirect
22. C2×#14519701C22semidirect
23. C2×#14519698C22semidirect
25. C2×#14519696C22semidirect
26. C2×#14519694C22semidirect
30. #16839530C22semidirect
353. C61#23530semidirect

Maximal subgroups

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

subgroupindex
C22×#145197122
#214543382
C2×#168397582
C2×#168397462
C2×#168397292
#214471152
#214471082
#214455572
C2×C61×#5752
#2353061