Finite Group Databasebeta

← #29722540

Subgroups — #29722540

Order 15616.

910 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840029C211yesmaximal, characteristic, complemented
2#21455434C212yesmaximal, complemented
3#21454513C211yesmaximal, characteristic, complemented
4C22×#14519713C211yesmaximal, characteristic, complemented
5#21447177C212yesmaximal, complemented
6#21447112C212yesmaximal, complemented
7#21446730C211yesmaximal, characteristic, complemented
8#21446602C211yesmaximal, characteristic, complemented
9#21446283C212yesmaximal, complemented
10#21445884C211yesmaximal, characteristic, complemented
11C2×C61×#466C211yesmaximal, characteristic, complemented
12C2×#14519803C2212yes—
13C2×#14519783C2211yescharacteristic
14C22×#11758194C2212yes—
15#16840273C2211yescharacteristic
16#16840029—24no—
17#16840013—22no—
18#16840006—22no—
19#16839996—22no—
20#16839989—22no—
21#16839833—21no—
22#16839833—21no—
23C8×C22⋊D61—21no—
24C8×C22⋊D61—21no—
25C2×#14519713—24no—
26C2×#14519713C2212yes—
27C2×#14519713C2211yescharacteristic
28C2×#14519713C2211yescharacteristic
29#16839802C2212yes—
30#16839788C2211yescharacteristic
31#16839767C2212yes—
32#16839761C2212yes—
33C2×#14519694C2212yes—
34C2×#14519694C2212yes—
35#16839728C2211yescharacteristic
36#16839726C2211yescharacteristic
37#16839717C2211yescharacteristic
38#16839531C2212yes—
39#16839531C2212yes—
40C22×D8×C61C2211yescharacteristic
41C23×C8×C61C2211yescharacteristic
42C2×#121×C61C2212yes—
43C61×#466—21no—
44C61×#466—21no—
45C61×#466—21no—
46C61×#466—21no—
47C2×#107×C61C2211yescharacteristic
48C2×#102×C61C2212yes—
49C2×#11758222C2311yescharacteristic
50#14519803—24no—
51#14519803—42no—
52C2×#11758200—22no—
53C2×#11758199—22no—
54#14519783—22no—
55#14519783—22no—
56C2×#11758194—22no—
57C2×#11758194—24no—
58C2×#11758194—42no—
59C2×#11758194C2311yescharacteristic
60C2×#11758194C2311yescharacteristic
61C2×#11758194C2312yes—
62C2×#11758194C2312yes—
63C2×#11758189—21no—
64C22×#92418—22no—
65C2×C8×D61—21no—
66#14519760—21no—
67#14519756D411yescharacteristic
68#14519756D411yescharacteristic
69#14519751D411yescharacteristic
70#14519751D411yescharacteristic
71#14519746—21no—
72C4×C22⋊D61—21no—
73C4×C22⋊D61—21no—
74#14519736C2311yescharacteristic
75#14519713—22no—
76#14519713—24no—
77#14519713D414yescomplemented
78#14519710—22no—
79#14519708—22no—
80#14519706—21no—
81#14519706—21no—
82#14519700—22no—
83#14519698—22no—
84#14519698—22no—
85#14519697—22no—
86#14519694—24no—
87#14519694—42no—
88C22⋊#92418—21no—
89C22⋊#92418—21no—
90#14519689—21no—
91#14519689—21no—
92#14519687—21no—
93C8×Dic61—21no—
94C22×D4×C61—22no—
95C22×D4×C61C2312yes—
96C23×C4×C61C2311yescharacteristic
97C2×D8×C61—22no—
98C2×D8×C61—21no—
99C2×D8×C61—21no—
100C2×D8×C61—22no—
101C22×C8×C61—22no—
102C22×C8×C61—22no—
103C22×C8×C61D411yescharacteristic
104C22×C8×C61D411yescharacteristic
105C22×C8×C61D411yescharacteristic
106C22×C8×C61D411yescharacteristic
107C22×C8×C61C2311yescharacteristic
108C22×C8×C61C2311yescharacteristic
109#121×C61—22no—
110#121×C61—22no—
111#121×C61—22no—
112#121×C61—22no—
113C2×C4⋊C4×C61C2312yes—
114C2×C22⋊C4×C61—22no—
115#107×C61—21no—
116#107×C61—21no—
117#102×C61—22no—
118#102×C61—22no—
119C24×C61—22no—
120C24×C61#4111yescharacteristic
121C2×D4×C61—22no—
122C2×D4×C61—22no—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61—22no—
126C2×D4×C61—22no—
127C2×D4×C61—42no—
128C2×D4×C61—42no—
129C2×D4×C61—84no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—22no—
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61#4111yescharacteristic
140C22×C4×C61C2411yescharacteristic
141C2×C22⋊D61—21no—
142C2×C22⋊D61—21no—
143C2×C22⋊D61—21no—
144#11758222—21no—
145#11758222—21no—
146#11758222—21no—
147#11758222—21no—
148C2×D244C2×D411yescharacteristic
149C2×C4×D61—21no—
150C2×Dic122C2×D411yescharacteristic
151D8×C61—42no—
152D8×C61—42no—
153C2×C8×C61—21no—
154C2×C8×C61—21no—
155C2×C8×C61—22no—
156C2×C8×C61—22no—
157C2×C8×C61—22no—
158C2×C8×C61—22no—
159C2×C8×C61C2×D411yescharacteristic
160C2×C8×C61C2×D411yescharacteristic
161C2×C8×C61C2×D412yes—
162C2×C8×C61C2×D411yescharacteristic
163C2×C8×C61C2×D411yescharacteristic
164C2×C8×C61#4112yes—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—22no—
167C22⋊C4×C61—42no—
168C22⋊C4×C61—42no—
169C22⋊Dic61—21no—
170C22⋊Dic61—21no—
171C22⋊Dic61—21no—
172#11758200—44no—
173#11758199—44no—
174#11758198—21no—
175#11758198—21no—
176C4⋊Dic61C2×D411yescharacteristic
177C4⋊Dic61C2×D411yescharacteristic
178#11758196—21no—
179#11758196—21no—
180C4×Dic61—21no—
181#11758194—22no—
182#11758194—21no—
183#11758194—22no—
184#11758194—21no—
185#11758194—21no—
186#11758194—22no—
187#11758194—21no—
188#11758194—44no—
189#11758194C2×D414yes—
190C2×#92418—22no—
191C2×#92418—21no—
192C2×#92418—21no—
193C2×#92418—22no—
194C2×#92418—44no—
195#11758189—42no—
196C8×D61—42no—
197C23×C61—21no—
198C23×C61—21no—
199C23×C61—22no—
200C23×C61—41no—
201C23×C61—42no—
202C23×C61—42no—
203C23×C61—42no—
204C23×C61—82no—
205C23×C61—82no—
206C23×C61#12111yescharacteristic
207C23×C61#12111yescharacteristic
208C23×C61C2×#4111yescharacteristic
209C22×D61—21no—
210D4×C61—42no—
211D4×C61—42no—
212D4×C61—42no—
213D4×C61—42no—
214D4×C61—84no—
215D4×C61—84no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—42no—
227C2×C4×C61—82no—
228C2×C4×C61#12111yescharacteristic
229C2×C4×C61#12111yescharacteristic
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C2×C4×C61H(5,𝔽2)11yescharacteristic
235C2×C4×C612−1+411yescharacteristic
236C22⋊D61—42no—
237C22⋊D61—41no—
238C22⋊D61—41no—
239C22⋊D61—42no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243D244—21no—
244D244—21no—
245C4×D61—41no—
246C4×D61—41no—
247Dic122—21no—
248Dic122—21no—
249C8×C61—21no—
250C8×C61—21no—
251C8×C61#12112yes—
252C8×C61#12112yes—
253#92418—24no—
254#92418—41no—
255#92418—41no—
256#92418—42no—
257#24803—611nomaximal, Sylow
258C22×C61—21no—
259C22×C61—21no—
260C22×C61—21no—
261C22×C61—21no—
262C22×C61—21no—
263C22×C61—21no—
264C22×C61—21no—
265C22×C61—42no—
266C22×C61—42no—
267C22×C61—82no—
268C22×C61—82no—
269C22×C61—82no—
270C22×C61—162no—
271C22×C61—162no—
272C22×C61#47212yescomplemented
273C22×C61C2×#12111yescharacteristic
274C22×C61#54811yescharacteristic
275C22×C61#55311yescharacteristic
276C2×D61—41no—
277C2×D61—41no—
278C2×D61—41no—
279C4×C61—21no—
280C4×C61—21no—
281C4×C61—82no—
282C4×C61C2×#12111yescharacteristic
283C4×C61#53711yescharacteristic
284C4×C61D4211yescharacteristic
285C4×C61#54711yescharacteristic
286Dic61—41no—
287Dic61—41no—
288Dic61—41no—
289C2×#472—611no—
290#3055—612no—
291C2×#466D6111yescharacteristic, complemented
292#2929—611no—
293C22×#108—611no—
294#2063—612no—
295#2051—612no—
296#1978—611no—
297#1952—611no—
298#1900—612no—
299#1816—611no—
300C2×C61—21no—
301C2×C61—21no—
302C2×C61—22no—
303C2×C61—41no—
304C2×C61—162no—
305C2×C61#241611yescharacteristic
306C2×C61C2×#47211yescharacteristic
307C2×C61#306711yescharacteristic
308D61—81no—
309C2×AGL(1,ℤ/8)—612no—
310C22×D8C2×D6111yescharacteristic
311C2×#131—611no—
312C22×M16—612no—
313C23×C8C2×D6111yescharacteristic
314#525—611no—
315C2×#121C2×D6112yes—
316#472—1224no—
317#469—1222no—
318#468—1222no—
319#466—21no—
320#466—21no—
321#466—21no—
322#466—21no—
323#466—1222no—
324#465—1222no—
325#435—1221no—
326#435—1221no—
327C8×D4—1221no—
328C8×D4—1221no—
329C2×#108—611no—
330C2×#108—612no—
331C2×#108—611no—
332C2×#108—1224no—
333#428—612no—
334C2×#107C2×D6111yescharacteristic
335#424—611no—
336#418—612no—
337#417—612no—
338C2×#102C2×D6112yes—
339C2×#100—612no—
340C2×#100—612no—
341#408—611no—
342#407—611no—
343#405—611no—
344#343—612no—
345#343—612no—
346C61#2480311yescharacteristic, complemented, Sylow
347C2×#41—611no—
348C22×D4—22no—
349C22×D4C22×D6112yes—
350C23×C4C22×D6111yescharacteristic
351AGL(1,ℤ/8)—1224no—
352AGL(1,ℤ/8)—2442no—
353C2×SD16—1222no—
354C2×D8—22no—
355C2×D8—22no—
356C2×D8—21no—
357C2×D8—21no—
358C2×D8—1222no—
359#131—1222no—
360#131—1222no—
361C2×M16—612no—
362C2×M16—612no—
363C2×M16—611no—
364C2×M16—611no—
365C2×M16—1222no—
366C2×M16—1221no—
367C2×M16—1224no—
368C2×M16—2442no—
369C22×C8—22no—
370C22×C8—22no—
371C22×C8—1222no—
372C22×C8—1221no—
373C22×C8C22⋊D6111yescharacteristic
374C22×C8C22⋊D6111yescharacteristic
375C22×C8C22⋊D6111yescharacteristic
376C22×C8C22⋊D6111yescharacteristic
377C22×C8C22×D6111yescharacteristic
378C22×C8C22×D6111yescharacteristic
379#123—1221no—
380C22⋊Q8—611no—
381C22⋊Q8—611no—
382#121—22no—
383#121—22no—
384#121—22no—
385#121—22no—
386#121—611no—
387#121—611no—
388#120—1221no—
389C4×D4—1221no—
390C4×D4—1221no—
391#117—611no—
392C2×C4⋊C4C22×D6112yes—
393C2×C22⋊C4—22no—
394#108—614no—
395#108—1224no—
396#108—1222no—
397#107—21no—
398#107—21no—
399#107—1222no—
400#106—1222no—
401C4⋊C8—1221no—
402C4⋊C8—1221no—
403#103—1222no—
404#102—22no—
405#102—22no—
406#102—1222no—
407#102—1222no—
408#102—1222no—
409#100—1224no—
410#100—2442no—
411C22⋊C8—1221no—
412C22⋊C8—1221no—
413C22⋊C8—1221no—
414C22⋊C8—1221no—
415#97—1221no—
416C4×C8—1221no—
417C24—22no—
418C24#1175822411yescharacteristic
419#41—1221no—
420#41—1221no—
421#41—1221no—
422#41—1221no—
423C2×Q8—611no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—84no—
433C2×D4—611no—
434C2×D4—1221no—
435C2×D4—1221no—
436C2×D4—1221no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—1221no—
443C22×C4D4×D6111yescharacteristic
444C22×C4D4×D6111yescharacteristic
445C22×C4#1175822411yescharacteristic
446C22×C4C2×C22⋊D6111yescharacteristic
447C22×C4C2×C22⋊D6111yescharacteristic
448C22×C4C23×D6111yescharacteristic
449SD16—2444no—
450D8—42no—
451D8—42no—
452D8—2444no—
453M16—614no—
454M16—1222no—
455M16—1221no—
456M16—1222no—
457M16—1221no—
458M16—1221no—
459M16—1222no—
460M16—1221no—
461M16—2444no—
462M16—2442no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—1222no—
470C2×C8—1222no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8—2444no—
474C2×C8—2442no—
475C2×C8D4×D6112yes—
476C2×C8#1175822412yes—
477C2×C8C2×C22⋊D6111yescharacteristic
478C2×C8C2×C22⋊D6111yescharacteristic
479C2×C8C2×C22⋊D6111yescharacteristic
480C2×C8C2×C22⋊D6111yescharacteristic
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—611no—
484C4⋊C4—611no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—1221no—
490C22⋊C4—1221no—
491C22⋊C4—1221no—
492C22⋊C4—1221no—
493C22⋊C4—1221no—
494C42—1221no—
495C23—21no—
496C23—22no—
497C23—21no—
498C23—41no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—82no—
503C23—82no—
504C23—1221no—
505C23#1451974811yescharacteristic
506C23#1451974811yescharacteristic
507C23C2×#1175822411yescharacteristic
508Q8—1221no—
509Q8—1221no—
510D4—42no—
511D4—42no—
512D4—42no—
513D4—42no—
514D4—84no—
515D4—84no—
516D4—1221no—
517D4—1221no—
518D4—2442no—
519D4—2441no—
520D4—2442no—
521D4—2441no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—42no—
533C2×C4—82no—
534C2×C4—1221no—
535C2×C4—1221no—
536C2×C4—1221no—
537C2×C4—2441no—
538C2×C4—2441no—
539C2×C4#1451974811yescharacteristic
540C2×C4#1451974811yescharacteristic
541C2×C4C2×D4×D6111yescharacteristic
542C2×C4C2×D4×D6111yescharacteristic
543C2×C4C22×C22⋊D6111yescharacteristic
544C2×C4C2×#1175822411yescharacteristic
545C2×C4#1451982311yescharacteristic
546C2×C4#1451982411yescharacteristic
547C8—21no—
548C8—21no—
549C8—1224no—
550C8—2441no—
551C8—2441no—
552C8—2442no—
553C8#1451974912yes—
554C8#1451974912yes—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—42no—
563C22—42no—
564C22—82no—
565C22—82no—
566C22—82no—
567C22—162no—
568C22—162no—
569C22—2441no—
570C22—2441no—
571C22—2441no—
572C22#1684002912yescomplemented
573C22C2×#1451974811yescharacteristic
574C22#1684041811yescharacteristic
575C22#1684046411yescharacteristic
576C4—21no—
577C4—21no—
578C4—82no—
579C4—2441no—
580C4—2441no—
581C4—2441no—
582C4C2×#1451974911yescharacteristic
583C4#1684034011yescharacteristic
584C4#1684039611yescharacteristic
585C4#1684041211yescharacteristic
586C2—21no—
587C2—21no—
588C2—22no—
589C2—41no—
590C2—162no—
591C2—4881no—
592C2#2144926011yescharacteristic
593C2C2×#1684002911yescharacteristic
594C2#2145553511yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#466C2
socleC22×C61C2×#121
hypercenterC2×#466D61
hypocenterC61#24803
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#466

Complements

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

NH ≅ G/Ntype
1. C2×#16840029C2semidirect
2. #21455434C2semidirect
3. #21454513C2semidirect
4. C22×#14519713C2semidirect
5. #21447177C2semidirect
6. #21447112C2semidirect
7. #21446730C2semidirect
8. #21446602C2semidirect
9. #21446283C2semidirect
10. #21445884C2semidirect
11. C2×C61×#466C2semidirect
77. #14519713D4semidirect
272. C22×C61#472semidirect
291. C2×#466D61semidirect
346. C61#24803semidirect
572. C22#16840029semidirect

Maximal subgroups

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

subgroupindex
C2×#168400292
#214554342
#214545132
C22×#145197132
#214471772
#214471122
#214467302
#214466022
#214462832
#214458842
C2×C61×#4662
#2480361