Finite Group Databasebeta

← #29722539

Subgroups — #29722539

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×#16840032C211yesmaximal, characteristic, complemented
2#21455440C212yesmaximal, complemented
3C2×#16839998C211yesmaximal, characteristic, complemented
4C22×#14519713C211yesmaximal, characteristic, complemented
5#21447178C212yesmaximal, complemented
6#21447110C212yesmaximal, complemented
7#21446729C211yesmaximal, characteristic, complemented
8#21446604C211yesmaximal, characteristic, complemented
9#21446280C212yesmaximal, complemented
10#21445883C211yesmaximal, characteristic, complemented
11C61×#2929C211yesmaximal, characteristic, complemented
12C2×#14519802C2212yes—
13C22×D488C2211yescharacteristic
14C22×#11758194C2212yes—
15C2×#14519751C2212yes—
16#16840032—24no—
17#16840013—22no—
18#16840010—22no—
19#16839998—21no—
20#16839998—21no—
21#16839998—21no—
22#16839998—21no—
23#16839996—22no—
24#16839993—22no—
25C2×#14519713—24no—
26C2×#14519713C2212yes—
27C2×#14519713C2211yescharacteristic
28C2×#14519713C2211yescharacteristic
29#16839805C2212yes—
30C2×#14519711C2211yescharacteristic
31#16839766C2212yes—
32#16839763C2212yes—
33C2×#14519699C2212yes—
34C2×#14519693C2212yes—
35C2×#14519693C2212yes—
36#16839530C2212yes—
37#16839530C2212yes—
38C2×#131×C61C2211yescharacteristic
39C23×C8×C61C2211yescharacteristic
40C61×#525C2211yescharacteristic
41C61×#435—21no—
42C61×#435—21no—
43C8×D4×C61—21no—
44C8×D4×C61—21no—
45C61×#424C2211yescharacteristic
46C61×#408C2211yescharacteristic
47C61×#407C2211yescharacteristic
48C61×#405C2211yescharacteristic
49C22×D244C2312yes—
50C22×C22⋊D61—22no—
51#14519802—24no—
52#14519802—42no—
53C2×#11758201—22no—
54C2×D488—21no—
55C2×D488—21no—
56C2×D488—22no—
57C2×D488—22no—
58C2×#11758199—22no—
59C2×#11758194—22no—
60C2×#11758194—24no—
61C2×#11758194—42no—
62C2×#11758194C2312yes—
63C2×#11758194C2312yes—
64C22×#92418—22no—
65#14519751—22no—
66#14519751—22no—
67#14519751—22no—
68#14519751—22no—
69C2×C4⋊Dic61C2312yes—
70C2×#11758198—22no—
71#14519713—22no—
72#14519713—24no—
73#14519713D414yescomplemented
74#14519711—21no—
75#14519711—21no—
76#14519710—22no—
77#14519708—22no—
78#14519701—22no—
79#14519699—22no—
80#14519699—22no—
81#14519698—22no—
82#14519697—22no—
83#14519697—22no—
84#14519693—24no—
85#14519693—42no—
86C2×#41×C61C2311yescharacteristic
87C23×C4×C61C2311yescharacteristic
88#131×C61—22no—
89#131×C61—22no—
90C2×M16×C61—21no—
91C2×M16×C61C2311yescharacteristic
92C2×M16×C61C2311yescharacteristic
93C22×C8×C61—22no—
94C22×C8×C61—22no—
95C22×C8×C61—21no—
96C22×C8×C61D411yescharacteristic
97C22×C8×C61D411yescharacteristic
98C22×C8×C61D411yescharacteristic
99C22×C8×C61D411yescharacteristic
100C22×C8×C61C2311yescharacteristic
101C22×C8×C61C2311yescharacteristic
102#123×C61—21no—
103C22⋊Q8×C61D411yescharacteristic
104C22⋊Q8×C61D411yescharacteristic
105#121×C61D411yescharacteristic
106#121×C61D411yescharacteristic
107#120×C61—21no—
108C4×D4×C61—21no—
109C4×D4×C61—21no—
110#117×C61C2311yescharacteristic
111C4⋊C8×C61—21no—
112C4⋊C8×C61—21no—
113C22⋊C8×C61—21no—
114C22⋊C8×C61—21no—
115C22⋊C8×C61—21no—
116C22⋊C8×C61—21no—
117#97×C61—21no—
118C4×C8×C61—21no—
119C24×C61#4111yescharacteristic
120C23×D61—22no—
121#41×C61—21no—
122#41×C61—21no—
123#41×C61—21no—
124#41×C61—21no—
125C2×Q8×C61C2×D411yescharacteristic
126C2×D4×C61—21no—
127C2×D4×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61C2×D411yescharacteristic
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
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—42no—
142C2×C22⋊D61—42no—
143C2×C22⋊D61—84no—
144C22×Dic61—22no—
145C2×D244—22no—
146C2×D244—22no—
147C2×D244—22no—
148C2×D244—22no—
149C2×D244—22no—
150C2×D244—22no—
151M16×C61—21no—
152M16×C61—21no—
153M16×C61—21no—
154M16×C61—21no—
155M16×C61—42no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—22no—
159C2×C8×C61—22no—
160C2×C8×C61—22no—
161C2×C8×C61—22no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61—42no—
165C2×C8×C61C2×D411yescharacteristic
166C2×C8×C61C2×D411yescharacteristic
167C2×C8×C61C2×D412yes—
168C2×C8×C61C2×D411yescharacteristic
169C2×C8×C61C2×D411yescharacteristic
170C2×C8×C61#4112yes—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61C2×D411yescharacteristic
174C4⋊C4×C61C2×D411yescharacteristic
175C22⋊C4×C61—21no—
176C22⋊C4×C61—21no—
177C22⋊C4×C61—21no—
178C22⋊C4×C61—21no—
179C22⋊C4×C61—21no—
180C42×C61—21no—
181#11758201—44no—
182#11758199—44no—
183#11758198—42no—
184#11758198—42no—
185C4⋊Dic61—22no—
186C4⋊Dic61—22no—
187#11758194—22no—
188#11758194—22no—
189#11758194—22no—
190#11758194—44no—
191#11758194C2×D414yes—
192C2×#92418—22no—
193C2×#92418—22no—
194C2×#92418—44no—
195D488—42no—
196D488—42no—
197C23×C61—21no—
198C23×C61—21no—
199C23×C61—22no—
200C23×C61—21no—
201C23×C61—41no—
202C23×C61#12111yescharacteristic
203C23×C61#12111yescharacteristic
204C23×C61C2×#4111yescharacteristic
205C22×D61—42no—
206C22×D61—42no—
207C22×D61—42no—
208C22×D61—82no—
209C22×D61—82no—
210Q8×C61—21no—
211Q8×C61—21no—
212D4×C61—21no—
213D4×C61—21no—
214D4×C61—42no—
215D4×C61—41no—
216D4×C61—41no—
217D4×C61—42no—
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—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61#12111yescharacteristic
234C2×C4×C61#12111yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C2×C4×C61H(5,𝔽2)11yescharacteristic
240C2×C4×C612−1+411yescharacteristic
241C22⋊D61—84no—
242C22⋊D61—84no—
243C2×Dic61—42no—
244C2×Dic61—82no—
245D244—42no—
246D244—42no—
247D244—42no—
248D244—42no—
249C8×C61—21no—
250C8×C61—21no—
251C8×C61—41no—
252C8×C61—41no—
253C8×C61#12112yes—
254C8×C61#12112yes—
255#92418—24no—
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—41no—
268C22×C61—41no—
269C22×C61—41no—
270C22×C61#47212yescomplemented
271C22×C61C2×#12111yescharacteristic
272C22×C61#54811yescharacteristic
273C22×C61#55311yescharacteristic
274C2×D61—82no—
275C2×D61—82no—
276C2×D61—82no—
277C2×D61—162no—
278C2×D61—162no—
279C4×C61—21no—
280C4×C61—21no—
281C4×C61—41no—
282C4×C61—41no—
283C4×C61—41no—
284C4×C61C2×#12111yescharacteristic
285C4×C61#53711yescharacteristic
286C4×C61D4211yescharacteristic
287C4×C61#54711yescharacteristic
288Dic61—82no—
289C2×#472—611no—
290#3055—612no—
291C2×#466—611no—
292#2929D6111yescharacteristic, complemented
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—81no—
305C2×C61#241611yescharacteristic
306C2×C61C2×#47211yescharacteristic
307C2×C61#306711yescharacteristic
308D61—162no—
309C2×AGL(1,ℤ/8)—612no—
310C22×D8—611no—
311C2×#131C2×D6111yescharacteristic
312C22×M16—612no—
313C23×C8C2×D6111yescharacteristic
314#525C2×D6111yescharacteristic
315C2×#121—612no—
316#472—1224no—
317#469—1222no—
318#468—1222no—
319#466—1221no—
320#466—1221no—
321#466—1221no—
322#466—1221no—
323#466—1222no—
324#465—1222no—
325#435—21no—
326#435—21no—
327C8×D4—21no—
328C8×D4—21no—
329C2×#108—611no—
330C2×#108—612no—
331C2×#108—611no—
332C2×#108—1224no—
333#428—612no—
334C2×#107—611no—
335#424C2×D6111yescharacteristic
336#418—612no—
337#417—612no—
338C2×#102—612no—
339C2×#100—612no—
340C2×#100—612no—
341#408C2×D6111yescharacteristic
342#407C2×D6111yescharacteristic
343#405C2×D6111yescharacteristic
344#343—612no—
345#343—612no—
346C61#2480311yescharacteristic, complemented, Sylow
347C2×#41C22×D6111yescharacteristic
348C22×D4—612no—
349C22×D4—1222no—
350C23×C4C22×D6111yescharacteristic
351AGL(1,ℤ/8)—1224no—
352AGL(1,ℤ/8)—2442no—
353C2×SD16—1222no—
354C2×D8—1222no—
355C2×D8—1222no—
356C2×D8—1222no—
357C2×D8—1221no—
358C2×D8—1221no—
359#131—22no—
360#131—22no—
361C2×M16—21no—
362C2×M16—612no—
363C2×M16—612no—
364C2×M16—1222no—
365C2×M16—1224no—
366C2×M16—2442no—
367C2×M16C22×D6111yescharacteristic
368C2×M16C22×D6111yescharacteristic
369C22×C8—22no—
370C22×C8—22no—
371C22×C8—21no—
372C22×C8—1222no—
373C22×C8C22⋊D6111yescharacteristic
374C22×C8C22⋊D6111yescharacteristic
375C22×C8C22⋊D6111yescharacteristic
376C22×C8C22⋊D6111yescharacteristic
377C22×C8C22×D6111yescharacteristic
378C22×C8C22×D6111yescharacteristic
379#123—21no—
380C22⋊Q8D24411yescharacteristic
381C22⋊Q8D24411yescharacteristic
382#121—1222no—
383#121—1222no—
384#121—1222no—
385#121—1222no—
386#121D24411yescharacteristic
387#121D24411yescharacteristic
388#120—21no—
389C4×D4—21no—
390C4×D4—21no—
391#117C22×D6111yescharacteristic
392C2×C4⋊C4—612no—
393C2×C22⋊C4—1222no—
394#108—614no—
395#108—1224no—
396#108—1222no—
397#107—1221no—
398#107—1221no—
399#107—1222no—
400#106—1222no—
401C4⋊C8—21no—
402C4⋊C8—21no—
403#103—1222no—
404#102—1222no—
405#102—1222no—
406#102—1222no—
407#102—1222no—
408#102—1222no—
409#100—1224no—
410#100—2442no—
411C22⋊C8—21no—
412C22⋊C8—21no—
413C22⋊C8—21no—
414C22⋊C8—21no—
415#97—21no—
416C4×C8—21no—
417C24—1222no—
418C24#1175822211yescharacteristic
419#41—21no—
420#41—21no—
421#41—21no—
422#41—21no—
423C2×Q8C2×D24411yescharacteristic
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—1222no—
428C2×D4—1222no—
429C2×D4—1222no—
430C2×D4—1222no—
431C2×D4—1222no—
432C2×D4—1222no—
433C2×D4—4884no—
434C2×D4—2442no—
435C2×D4—2442no—
436C2×D4C2×D24411yescharacteristic
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—1222no—
443C22×C4C2×D24411yescharacteristic
444C22×C4C2×D24411yescharacteristic
445C22×C4#1175822211yescharacteristic
446C22×C4C2×C22⋊D6111yescharacteristic
447C22×C4C2×C22⋊D6111yescharacteristic
448C22×C4C23×D6111yescharacteristic
449SD16—2444no—
450D8—2444no—
451D8—2442no—
452D8—2442no—
453M16—21no—
454M16—21no—
455M16—21no—
456M16—21no—
457M16—42no—
458M16—614no—
459M16—1222no—
460M16—1222no—
461M16—1222no—
462M16—2444no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—42no—
472C2×C8—1222no—
473C2×C8—1222no—
474C2×C8—2444no—
475C2×C8D4×D6112yes—
476C2×C8#1175822412yes—
477C2×C8C2×C22⋊D6111yescharacteristic
478C2×C8C2×C22⋊D6111yescharacteristic
479C2×C8C2×C22⋊D6111yescharacteristic
480C2×C8C2×C22⋊D6111yescharacteristic
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—1222no—
484C4⋊C4—1222no—
485C4⋊C4C2×D24411yescharacteristic
486C4⋊C4C2×D24411yescharacteristic
487C22⋊C4—21no—
488C22⋊C4—21no—
489C22⋊C4—21no—
490C22⋊C4—21no—
491C22⋊C4—21no—
492C22⋊C4—2442no—
493C22⋊C4—2442no—
494C42—21no—
495C23—21no—
496C23—22no—
497C23—21no—
498C23—21no—
499C23—41no—
500C23—4882no—
501C23—4882no—
502C23—2442no—
503C23—2442no—
504C23—2442no—
505C23#1451975111yescharacteristic
506C23#1451975111yescharacteristic
507C23C2×#1175822211yescharacteristic
508Q8—21no—
509Q8—21no—
510D4—21no—
511D4—21no—
512D4—42no—
513D4—41no—
514D4—42no—
515D4—41no—
516D4—4884no—
517D4—4884no—
518D4—2442no—
519D4—2442no—
520D4—2442no—
521D4—2442no—
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—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—4882no—
538C2×C4—2442no—
539C2×C4#1451975111yescharacteristic
540C2×C4#1451975111yescharacteristic
541C2×C4C2×D4×D6111yescharacteristic
542C2×C4C22×C22⋊D6111yescharacteristic
543C2×C4C22×D24411yescharacteristic
544C2×C4C2×#1175822411yescharacteristic
545C2×C4#1451982311yescharacteristic
546C2×C4#1451982411yescharacteristic
547C8—21no—
548C8—21no—
549C8—41no—
550C8—41no—
551C8—1224no—
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—41no—
565C22—41no—
566C22—41no—
567C22—9762no—
568C22—9762no—
569C22—4882no—
570C22—4882no—
571C22—4882no—
572C22#1684003212yescomplemented
573C22C2×#1451975111yescharacteristic
574C22#1684042111yescharacteristic
575C22#1684045811yescharacteristic
576C4—21no—
577C4—21no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—4882no—
582C4C2×#1451974911yescharacteristic
583C4#1684034011yescharacteristic
584C4D4×D24411yescharacteristic
585C4#1684041611yescharacteristic
586C2—21no—
587C2—21no—
588C2—22no—
589C2—41no—
590C2—81no—
591C2—9762no—
592C2#2144925911yescharacteristic
593C2C2×#1684003211yescharacteristic
594C2#2145553911yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2929C2
socleC22×C61C2×#121
hypercenter#2929D61
hypocenterC61#24803
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2929

Complements

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

NH ≅ G/Ntype
1. C2×#16840032C2semidirect
2. #21455440C2semidirect
3. C2×#16839998C2semidirect
4. C22×#14519713C2semidirect
5. #21447178C2semidirect
6. #21447110C2semidirect
7. #21446729C2semidirect
8. #21446604C2semidirect
9. #21446280C2semidirect
10. #21445883C2semidirect
11. C61×#2929C2semidirect
73. #14519713D4semidirect
270. C22×C61#472semidirect
292. #2929D61semidirect
346. C61#24803semidirect
572. C22#16840032semidirect

Maximal subgroups

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

subgroupindex
C2×#168400322
#214554402
C2×#168399982
C22×#145197132
#214471782
#214471102
#214467292
#214466042
#214462802
#214458832
C61×#29292
#2480361