Finite Group Databasebeta

← #29722533

Subgroups — #29722533

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×#16840033C211yesmaximal, characteristic, complemented
2#21455438C212yesmaximal, complemented
3C2×#16839998C211yesmaximal, characteristic, complemented
4#21454513C211yesmaximal, characteristic, complemented
5#21447174C212yesmaximal, complemented
6#21447113C212yesmaximal, complemented
7#21446727C211yesmaximal, characteristic, complemented
8#21446605C211yesmaximal, characteristic, complemented
9#21446282C212yesmaximal, complemented
10#21445880C211yesmaximal, characteristic, complemented
11C22×#108×C61C211yesmaximal, characteristic, complemented
12C2×#14519805C2212yes—
13C22×D488C2211yescharacteristic
14C2×#14519783C2211yescharacteristic
15#16840273C2211yescharacteristic
16C2×#14519751C2212yes—
17#16840033—24no—
18#16840015—22no—
19#16840008—22no—
20#16839998—22no—
21#16839998—21no—
22#16839998—21no—
23#16839998—21no—
24#16839998—21no—
25#16839991—22no—
26#16839833—21no—
27#16839833—21no—
28C8×C22⋊D61—21no—
29C8×C22⋊D61—21no—
30#16839804C2212yes—
31C2×#14519711C2211yescharacteristic
32#16839788C2211yescharacteristic
33#16839768C2212yes—
34#16839764C2212yes—
35C2×#14519699C2212yes—
36C2×#14519693C2212yes—
37C2×#14519693C2212yes—
38#16839728C2211yescharacteristic
39#16839726C2211yescharacteristic
40#16839717C2211yescharacteristic
41#16839530C2212yes—
42#16839530C2212yes—
43C22×M16×C61C2212yes—
44C23×C8×C61C2211yescharacteristic
45C2×#108×C61—24no—
46C2×#108×C61C2212yes—
47C2×#108×C61C2211yescharacteristic
48C2×#108×C61C2211yescharacteristic
49C2×#11758222C2311yescharacteristic
50C22×D244C2312yes—
51C22×C22⋊D61—22no—
52#14519805—24no—
53#14519805—42no—
54C2×#11758190—22no—
55C2×D488—22no—
56C2×D488—21no—
57C2×D488—21no—
58C2×D488—22no—
59C2×D488—22no—
60#14519783—22no—
61#14519783—22no—
62C2×#11758194C2311yescharacteristic
63C2×#11758194C2311yescharacteristic
64C2×#11758189—21no—
65C2×C8×D61—21no—
66#14519760—21no—
67#14519756D411yescharacteristic
68#14519756D411yescharacteristic
69#14519751—22no—
70#14519751—22no—
71#14519751—22no—
72#14519751—22no—
73#14519751D411yescharacteristic
74#14519751D411yescharacteristic
75#14519746—21no—
76C4×C22⋊D61—21no—
77C4×C22⋊D61—21no—
78#14519736C2311yescharacteristic
79C2×C4⋊Dic61C2312yes—
80C2×#11758198—22no—
81#14519711—21no—
82#14519711—21no—
83#14519711—22no—
84#14519709—22no—
85#14519706—21no—
86#14519706—21no—
87#14519702—22no—
88#14519699—22no—
89#14519699—22no—
90#14519699—22no—
91#14519699—22no—
92#14519699—22no—
93#14519693—24no—
94#14519693—42no—
95C22⋊#92418—21no—
96C22⋊#92418—21no—
97#14519689—21no—
98#14519689—21no—
99#14519687—21no—
100C8×Dic61—21no—
101C23×C4×C61C2311yescharacteristic
102C2×M16×C61—22no—
103C2×M16×C61—24no—
104C2×M16×C61—42no—
105C2×M16×C61C2312yes—
106C2×M16×C61C2312yes—
107C22×C8×C61—22no—
108C22×C8×C61—22no—
109C22×C8×C61—22no—
110C22×C8×C61D411yescharacteristic
111C22×C8×C61D411yescharacteristic
112C22×C8×C61D411yescharacteristic
113C22×C8×C61D411yescharacteristic
114C22×C8×C61C2311yescharacteristic
115C22×C8×C61C2311yescharacteristic
116#108×C61—24no—
117#108×C61—22no—
118#108×C61D414yescomplemented
119C24×C61#4111yescharacteristic
120C23×D61—22no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61#4111yescharacteristic
130C22×C4×C61C2411yescharacteristic
131C2×C22⋊D61—21no—
132C2×C22⋊D61—21no—
133C2×C22⋊D61—21no—
134C2×C22⋊D61—42no—
135C2×C22⋊D61—42no—
136C2×C22⋊D61—84no—
137C22×Dic61—22no—
138#11758222—21no—
139#11758222—21no—
140#11758222—21no—
141#11758222—21no—
142C2×D244—22no—
143C2×D244—22no—
144C2×D244—22no—
145C2×D244—22no—
146C2×D244—22no—
147C2×D244—22no—
148C2×D244C2×D411yescharacteristic
149C2×C4×D61—21no—
150C2×Dic122C2×D411yescharacteristic
151M16×C61—22no—
152M16×C61—22no—
153M16×C61—22no—
154M16×C61—44no—
155M16×C61C2×D414yes—
156C2×C8×C61—21no—
157C2×C8×C61—22no—
158C2×C8×C61—21no—
159C2×C8×C61—22no—
160C2×C8×C61—22no—
161C2×C8×C61—22no—
162C2×C8×C61—22no—
163C2×C8×C61—22no—
164C2×C8×C61—44no—
165C2×C8×C61C2×D411yescharacteristic
166C2×C8×C61C2×D411yescharacteristic
167C2×C8×C61C2×D411yescharacteristic
168C2×C8×C61C2×D411yescharacteristic
169C2×C8×C61C2×D412yes—
170C2×C8×C61#4112yes—
171C22⋊Dic61—21no—
172C22⋊Dic61—21no—
173C22⋊Dic61—21no—
174#11758198—21no—
175#11758198—21no—
176#11758198—42no—
177#11758198—42no—
178C4⋊Dic61—22no—
179C4⋊Dic61—22no—
180C4⋊Dic61C2×D411yescharacteristic
181C4⋊Dic61C2×D411yescharacteristic
182#11758196—21no—
183#11758196—21no—
184C4×Dic61—21no—
185#11758194—21no—
186#11758194—21no—
187#11758194—21no—
188#11758194—21no—
189C2×#92418—21no—
190C2×#92418—21no—
191D488—44no—
192D488—42no—
193D488—42no—
194#11758190—44no—
195#11758189—42no—
196C8×D61—42no—
197C23×C61—21no—
198C23×C61—21no—
199C23×C61—22no—
200C23×C61—41no—
201C23×C61#12111yescharacteristic
202C23×C61#12111yescharacteristic
203C23×C61C2×#4111yescharacteristic
204C22×D61—21no—
205C22×D61—42no—
206C22×D61—42no—
207C22×D61—42no—
208C22×D61—82no—
209C22×D61—82no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61#12111yescharacteristic
221C2×C4×C61#12111yescharacteristic
222C2×C4×C61C22×D411yescharacteristic
223C2×C4×C61C22×D411yescharacteristic
224C2×C4×C61C22×D411yescharacteristic
225C2×C4×C61C2×#4111yescharacteristic
226C2×C4×C61H(5,𝔽2)11yescharacteristic
227C2×C4×C612−1+411yescharacteristic
228C22⋊D61—42no—
229C22⋊D61—41no—
230C22⋊D61—41no—
231C22⋊D61—42no—
232C22⋊D61—84no—
233C22⋊D61—84no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—42no—
238C2×Dic61—82no—
239D244—21no—
240D244—21no—
241D244—42no—
242D244—42no—
243D244—42no—
244D244—42no—
245C4×D61—41no—
246C4×D61—41no—
247Dic122—21no—
248Dic122—21no—
249C8×C61—21no—
250C8×C61—21no—
251C8×C61—24no—
252C8×C61—42no—
253C8×C61#12112yes—
254C8×C61#12112yes—
255#92418—41no—
256#92418—41no—
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#47212yescomplemented
268C22×C61C2×#12111yescharacteristic
269C22×C61#54811yescharacteristic
270C22×C61#55311yescharacteristic
271C2×D61—41no—
272C2×D61—41no—
273C2×D61—41no—
274C2×D61—82no—
275C2×D61—82no—
276C2×D61—82no—
277C2×D61—162no—
278C2×D61—162no—
279C4×C61—21no—
280C4×C61—21no—
281C4×C61C2×#12111yescharacteristic
282C4×C61#53711yescharacteristic
283C4×C61D4211yescharacteristic
284C4×C61#54711yescharacteristic
285Dic61—41no—
286Dic61—41no—
287Dic61—41no—
288Dic61—82no—
289C2×#472—611no—
290#3055—612no—
291C2×#466—611no—
292#2929—611no—
293C22×#108D6111yescharacteristic, complemented
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#241611yescharacteristic
305C2×C61C2×#47211yescharacteristic
306C2×C61#306711yescharacteristic
307D61—81no—
308D61—162no—
309C2×AGL(1,ℤ/8)—612no—
310C22×D8—611no—
311C2×#131—611no—
312C22×M16C2×D6112yes—
313C23×C8C2×D6111yescharacteristic
314#525—611no—
315C2×#121—612no—
316#472—1224no—
317#469—1222no—
318#468—1222no—
319#466—1222no—
320#466—1221no—
321#466—1221no—
322#466—1221no—
323#466—1221no—
324#465—1222no—
325#435—1221no—
326#435—1221no—
327C8×D4—1221no—
328C8×D4—1221no—
329C2×#108—24no—
330C2×#108C2×D6112yes—
331C2×#108C2×D6111yescharacteristic
332C2×#108C2×D6111yescharacteristic
333#428—612no—
334C2×#107—611no—
335#424—611no—
336#418—612no—
337#417—612no—
338C2×#102—612no—
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—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—1221no—
356C2×D8—1221no—
357C2×D8—1222no—
358C2×D8—1222no—
359#131—1222no—
360#131—1222no—
361C2×M16—22no—
362C2×M16—24no—
363C2×M16—42no—
364C2×M16—611no—
365C2×M16—611no—
366C2×M16—1221no—
367C2×M16C22×D6112yes—
368C2×M16C22×D6112yes—
369C22×C8—22no—
370C22×C8—22no—
371C22×C8—22no—
372C22×C8—1221no—
373C22×C8D24411yescharacteristic
374C22×C8D24411yescharacteristic
375C22×C8D24411yescharacteristic
376C22×C8D24411yescharacteristic
377C22×C8C22×D6111yescharacteristic
378C22×C8C22×D6111yescharacteristic
379#123—1221no—
380C22⋊Q8—611no—
381C22⋊Q8—611no—
382#121—611no—
383#121—611no—
384#121—1222no—
385#121—1222no—
386#121—1222no—
387#121—1222no—
388#120—1221no—
389C4×D4—1221no—
390C4×D4—1221no—
391#117—611no—
392C2×C4⋊C4—612no—
393C2×C22⋊C4—1222no—
394#108—24no—
395#108—22no—
396#108C22⋊D6114yescomplemented
397#107—1222no—
398#107—1221no—
399#107—1221no—
400#106—1222no—
401C4⋊C8—1221no—
402C4⋊C8—1221no—
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—1221no—
412C22⋊C8—1221no—
413C22⋊C8—1221no—
414C22⋊C8—1221no—
415#97—1221no—
416C4×C8—1221no—
417C24—1222no—
418C24#1175822611yescharacteristic
419#41—1221no—
420#41—1221no—
421#41—1221no—
422#41—1221no—
423C2×Q8—611no—
424C2×D4—611no—
425C2×D4—1221no—
426C2×D4—1221no—
427C2×D4—1221no—
428C2×D4—1222no—
429C2×D4—1222no—
430C2×D4—1222no—
431C2×D4—1222no—
432C2×D4—1222no—
433C2×D4—1222no—
434C2×D4—4884no—
435C2×D4—2442no—
436C2×D4—2442no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—1221no—
442C22×C4—1222no—
443C22×C4C2×D24411yescharacteristic
444C22×C4C2×D24411yescharacteristic
445C22×C4D4×D6111yescharacteristic
446C22×C4D4×D6111yescharacteristic
447C22×C4#1175822611yescharacteristic
448C22×C4C23×D6111yescharacteristic
449SD16—2444no—
450D8—2444no—
451D8—2442no—
452D8—2442no—
453M16—22no—
454M16—22no—
455M16—22no—
456M16—44no—
457M16—1221no—
458M16—1221no—
459M16—1221no—
460M16—1221no—
461M16—2442no—
462M16C2×C22⋊D6114yes—
463C2×C8—22no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—44no—
472C2×C8—1221no—
473C2×C8—1221no—
474C2×C8—2442no—
475C2×C8C2×D24411yescharacteristic
476C2×C8C2×D24411yescharacteristic
477C2×C8C2×D24411yescharacteristic
478C2×C8C2×D24411yescharacteristic
479C2×C8#1175822212yes—
480C2×C8C2×C22⋊D6112yes—
481C4⋊C4—611no—
482C4⋊C4—611no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1222no—
486C4⋊C4—1222no—
487C22⋊C4—1221no—
488C22⋊C4—1221no—
489C22⋊C4—1221no—
490C22⋊C4—1221no—
491C22⋊C4—1221no—
492C22⋊C4—2442no—
493C22⋊C4—2442no—
494C42—1221no—
495C23—21no—
496C23—22no—
497C23—21no—
498C23—41no—
499C23—1221no—
500C23—4882no—
501C23—4882no—
502C23—2442no—
503C23—2442no—
504C23—2442no—
505C23#1451975211yescharacteristic
506C23#1451975211yescharacteristic
507C23C2×#1175822611yescharacteristic
508Q8—1221no—
509Q8—1221no—
510D4—1221no—
511D4—1221no—
512D4—4884no—
513D4—4884no—
514D4—2442no—
515D4—2441no—
516D4—2442no—
517D4—2441no—
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—1221no—
533C2×C4—1221no—
534C2×C4—1221no—
535C2×C4—4882no—
536C2×C4—2442no—
537C2×C4—2441no—
538C2×C4—2441no—
539C2×C4#1451975211yescharacteristic
540C2×C4#1451975211yescharacteristic
541C2×C4C2×D4×D6111yescharacteristic
542C2×C4C22×C22⋊D6111yescharacteristic
543C2×C4C22×D24411yescharacteristic
544C2×C4C2×#1175822211yescharacteristic
545C2×C4#1451982211yescharacteristic
546C2×C4#1451982511yescharacteristic
547C8—21no—
548C8—21no—
549C8—24no—
550C8—42no—
551C8—2441no—
552C8—2441no—
553C8#1451975112yes—
554C8#1451975112yes—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—42no—
563C22—42no—
564C22—9762no—
565C22—9762no—
566C22—4882no—
567C22—4882no—
568C22—4882no—
569C22—2441no—
570C22—2441no—
571C22—2441no—
572C22#1684003312yescomplemented
573C22C2×#1451975211yescharacteristic
574C22#1684042311yescharacteristic
575C22#1684046011yescharacteristic
576C4—21no—
577C4—21no—
578C4—4882no—
579C4—2441no—
580C4—2441no—
581C4—2441no—
582C4C2×#1451975111yescharacteristic
583C4#1684034311yescharacteristic
584C4D4×C22⋊D6111yescharacteristic
585C4#1684041311yescharacteristic
586C2—21no—
587C2—21no—
588C2—22no—
589C2—41no—
590C2—9762no—
591C2—4881no—
592C2#2144925311yescharacteristic
593C2C2×#1684003311yescharacteristic
594C2#2145554011yescharacteristic
subgroupG/H
centerC22C2×#14519752
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC22×#108×C61C2
socleC22×C61C2×#121
hypercenterC22×#108D61
hypocenterC61#24803
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#108

Complements

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

NH ≅ G/Ntype
1. C2×#16840033C2semidirect
2. #21455438C2semidirect
3. C2×#16839998C2semidirect
4. #21454513C2semidirect
5. #21447174C2semidirect
6. #21447113C2semidirect
7. #21446727C2semidirect
8. #21446605C2semidirect
9. #21446282C2semidirect
10. #21445880C2semidirect
11. C22×#108×C61C2semidirect
118. #108×C61D4semidirect
267. C22×C61#472semidirect
293. C22×#108D61semidirect
346. C61#24803semidirect
396. #108C22⋊D61semidirect
572. C22#16840033semidirect

Maximal subgroups

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

subgroupindex
C2×#168400332
#214554382
C2×#168399982
#214545132
#214471742
#214471132
#214467272
#214466052
#214462822
#214458802
C22×#108×C612
#2480361