Finite Group Databasebeta

← #29828807

Subgroups — #29828807

Order 15616.

2008 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839957C212yesmaximal, complemented
2C2×#16839896C212yesmaximal, complemented
3#21454164C211yesmaximal, characteristic, complemented
4#21446167C214yesmaximal, complemented
5#21446136C212yesmaximal, complemented
6#21445819C212yesmaximal, complemented
7#21445799C211yesmaximal, characteristic, complemented
8C22×C4×D4×C61C211yesmaximal, characteristic, complemented
9C22×#11758199C2214yes—
10C2×#14519774C2211yescharacteristic, complemented
11C2×#14519751—21no—
12C2×#14519751—21no—
13C2×#14519751C2212yes—
14#16839957—28no—
15#16839896—28no—
16C2×C4⋊#92418C2212yescomplemented
17C2×#14519698C2214yes—
18C2×#14519697C2214yescomplemented
19C2×C22⋊#92418C2212yes—
20#16839664—21no—
21#16839664—21no—
22#16839664C2212yescomplemented
23#16839632C2211yescharacteristic, complemented
24#16839524C2214yescomplemented
25#16839517C2212yescomplemented
26C23×D4×C61C2211yescharacteristic
27C24×C4×C61—21no—
28C2×C4×D4×C61—28no—
29C2×C4×D4×C61—21no—
30C2×C4×D4×C61—21no—
31C2×C4×D4×C61C2212yes—
32C2×C4×D4×C61C2212yes—
33C22×C4⋊C4×C61C2211yescharacteristic
34C22×C22⋊C4×C61—21no—
35C22×C42×C61C2211yescharacteristic
36C22×D244—21no—
37C22×D244—21no—
38C22×D244C2312yes—
39C22×C22⋊D61—22no—
40C2×#11758199—28no—
41C2×#11758199—28no—
42C2×#11758199—28no—
43C22×#92418C2314yes—
44#14519774—24no—
45#14519751—24no—
46#14519751—24no—
47#14519751—44no—
48#14519751—44no—
49C2×C4⋊Dic61—21no—
50C2×C4⋊Dic61—21no—
51C2×C4⋊Dic61C2312yes—
52C2×#11758198—22no—
53C2×#11758198—22no—
54C2×#11758198—22no—
55C4⋊#92418—24no—
56#14519698—28no—
57#14519697—28no—
58C22⋊#92418—24no—
59C25×C61—21no—
60C22×D4×C61—24no—
61C22×D4×C61—24no—
62C22×D4×C61—21no—
63C22×D4×C61—22no—
64C22×D4×C61D414yescomplemented
65C22×D4×C61C2312yes—
66C23×C4×C61—28no—
67C23×C4×C61—21no—
68C23×C4×C61—22no—
69C23×C4×C61—22no—
70C23×C4×C61—44no—
71C23×C4×C61—44no—
72C23×C4×C61D411yescharacteristic
73C23×C4×C61D411yescharacteristic
74C23×C4×C61C2311yescharacteristic
75C4×D4×C61—28no—
76C4×D4×C61—28no—
77C4×D4×C61—24no—
78C4×D4×C61—24no—
79C4×D4×C61—24no—
80C4×D4×C61D418yescomplemented
81C2×C4⋊C4×C61—21no—
82C2×C4⋊C4×C61—24no—
83C2×C4⋊C4×C61C2312yes—
84C2×C22⋊C4×C61—28no—
85C2×C22⋊C4×C61—22no—
86C2×C22⋊C4×C61—22no—
87C2×C42×C61—24no—
88C2×C42×C61D411yescharacteristic
89C2×C42×C61D411yescharacteristic
90C2×C42×C61C2311yescharacteristic
91C2×C42×C61C2311yescharacteristic
92C24×C61—24no—
93C24×C61—24no—
94C24×C61—24no—
95C24×C61—22no—
96C24×C61—42no—
97C24×C61—42no—
98C24×C61—42no—
99C24×C61—42no—
100C24×C61C2×D411yescharacteristic
101C23×D61—22no—
102C2×D4×C61—24no—
103C2×D4×C61—24no—
104C2×D4×C61—216no—
105C2×D4×C61—216no—
106C2×D4×C61—24no—
107C2×D4×C61—22no—
108C2×D4×C61—22no—
109C2×D4×C61—22no—
110C2×D4×C61C2×D414yes—
111C2×D4×C61C2×D414yes—
112C2×D4×C61#4114yes—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—24no—
118C22×C4×C61—28no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—28no—
123C22×C4×C61—41no—
124C22×C4×C61—44no—
125C22×C4×C61—432no—
126C22×C4×C61—42no—
127C22×C4×C61—42no—
128C22×C4×C61—44no—
129C22×C4×C61—44no—
130C22×C4×C61—44no—
131C22×C4×C61D814yescomplemented
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D412yes—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61#4112yes—
139C22×C4×C61C2411yescharacteristic
140C2×C22⋊D61—44no—
141C2×C22⋊D61—44no—
142C2×C22⋊D61—44no—
143C22×Dic61—22no—
144C2×D244—24no—
145C2×D244—22no—
146C2×D244—22no—
147C2×D244—24no—
148C2×D244—42no—
149C2×D244—42no—
150C2×D244—42no—
151C2×D244—42no—
152C2×D244—42no—
153C2×D244—42no—
154C4⋊C4×C61—24no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61C2×D414yes—
157C22⋊C4×C61—28no—
158C22⋊C4×C61—28no—
159C42×C61—22no—
160C42×C61—22no—
161C42×C61—22no—
162C42×C61C2×D414yes—
163#11758199—416no—
164#11758198—44no—
165#11758198—44no—
166#11758198—44no—
167C4⋊Dic61—22no—
168C4⋊Dic61—22no—
169C4⋊Dic61—42no—
170C4⋊Dic61—42no—
171C2×#92418—24no—
172C2×#92418—24no—
173C2×#92418—24no—
174C23×C61—21no—
175C23×C61—21no—
176C23×C61—21no—
177C23×C61—21no—
178C23×C61—216no—
179C23×C61—24no—
180C23×C61—24no—
181C23×C61—24no—
182C23×C61—48no—
183C23×C61—48no—
184C23×C61—48no—
185C23×C61—48no—
186C23×C61—48no—
187C23×C61—42no—
188C23×C61—42no—
189C23×C61—42no—
190C23×C61—42no—
191C23×C61—48no—
192C23×C61#12112yes—
193C23×C61C2×D812yes—
194C23×C61AGL(1,ℤ/8)12yes—
195C23×C61C22×D411yescharacteristic
196C22×D61—42no—
197C22×D61—42no—
198C22×D61—42no—
199C22×D61—42no—
200C22×D61—42no—
201C22×D61—42no—
202C22×D61—42no—
203D4×C61—216no—
204D4×C61—24no—
205D4×C61—24no—
206D4×C61—24no—
207D4×C61#12118yescomplemented
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—24no—
215C2×C4×C61—24no—
216C2×C4×C61—28no—
217C2×C4×C61—21no—
218C2×C4×C61—24no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—44no—
222C2×C4×C61—416no—
223C2×C4×C61—48no—
224C2×C4×C61—42no—
225C2×C4×C61—42no—
226C2×C4×C61—416no—
227C2×C4×C61#12012yes—
228C2×C4×C61#12012yes—
229C2×C4×C61#12112yes—
230C2×C4×C61C2×D812yes—
231C2×C4×C61C2×D812yes—
232C2×C4×C61AGL(1,ℤ/8)12yes—
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C22×D412yes—
235C2×C4×C61C2×#4112yes—
236C2×C4×C61C2×#4111yescharacteristic
237C22⋊D61—88no—
238C2×Dic61—42no—
239C2×Dic61—42no—
240C2×Dic61—42no—
241D244—44no—
242D244—44no—
243D244—84no—
244D244—84no—
245#92418—44no—
246#32902—611nomaximal, Sylow
247C22×C61—22no—
248C22×C61—22no—
249C22×C61—22no—
250C22×C61—22no—
251C22×C61—22no—
252C22×C61—22no—
253C22×C61—28no—
254C22×C61—432no—
255C22×C61—44no—
256C22×C61—44no—
257C22×C61—44no—
258C22×C61—44no—
259C22×C61—44no—
260C22×C61—44no—
261C22×C61#45914yescomplemented
262C22×C61C2×#12011yescharacteristic
263C22×C61C2×#12111yescharacteristic
264C22×C61#52511yescharacteristic
265C22×C61C22×D811yescharacteristic
266C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
267C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
268C22×C61#57511yescharacteristic
269C2×D61—82no—
270C2×D61—82no—
271C2×D61—82no—
272C2×D61—82no—
273C2×D61—82no—
274C2×D61—82no—
275C2×D61—82no—
276C4×C61—22no—
277C4×C61—22no—
278C4×C61—22no—
279C4×C61—44no—
280C4×C61—41no—
281C4×C61#44714yescomplemented
282C4×C61C2×#12112yes—
283C4×C61#52512yes—
284Dic61—82no—
285C22×C4×D4D6111yescharacteristic, complemented
286C2×#459—612no—
287C2×#447—612no—
288#2856—611no—
289#1879—614no—
290#1874—612no—
291#1802—612no—
292#1797—611no—
293C2×C61—24no—
294C2×C61—48no—
295C2×C61#284711yescharacteristic
296C2×C61C2×#44711yescharacteristic
297C2×C61#300611yescharacteristic
298C2×C61C2×#45911yescharacteristic
299C2×C61#303611yescharacteristic
300C2×C61#310011yescharacteristic
301C2×C61#310911yescharacteristic
302D61—162no—
303C23×D4C2×D6111yescharacteristic
304C24×C4—21no—
305C22×D8—614no—
306C2×#127—611no—
307C2×#121—612no—
308C2×#121—1221no—
309C2×#121—1221no—
310C2×C4×D4—28no—
311C2×C4×D4—21no—
312C2×C4×D4—21no—
313C2×C4×D4C2×D6112yes—
314C2×C4×D4C2×D6112yes—
315C22×C4⋊C4C2×D6111yescharacteristic
316C22×C22⋊C4—21no—
317C22×C42C2×D6111yescharacteristic
318#459—1228no—
319#447—1228no—
320C2×C4⋊C8—612no—
321C2×#102—614no—
322C2×#102—614no—
323C2×C22⋊C8—612no—
324#390—612no—
325#390—1221no—
326#390—1221no—
327#384—611no—
328#340—614no—
329#336—612no—
330C61#3290211yescharacteristic, complemented, Sylow
331C25—21no—
332C22×D4—24no—
333C22×D4—24no—
334C22×D4—22no—
335C22×D4—21no—
336C22×D4—612no—
337C22×D4—1222no—
338C22×D4—1221no—
339C22×D4—1221no—
340C22×D4D24414yescomplemented
341C22×D4C22×D6112yes—
342C23×C4—28no—
343C23×C4—21no—
344C23×C4—22no—
345C23×C4—22no—
346C23×C4—44no—
347C23×C4—44no—
348C23×C4C22⋊D6111yescharacteristic
349C23×C4C22⋊D6111yescharacteristic
350C23×C4C22×D6111yescharacteristic
351C2×D8—1228no—
352C2×D8—1228no—
353C2×D8—1228no—
354C22×C8—614no—
355#127—1224no—
356#121—1224no—
357#121—1224no—
358#121—2444no—
359#121—2444no—
360C4×D4—28no—
361C4×D4—28no—
362C4×D4—24no—
363C4×D4—24no—
364C4×D4—24no—
365C4×D4C22⋊D6118yescomplemented
366C2×C4⋊C4—21no—
367C2×C4⋊C4—24no—
368C2×C4⋊C4—612no—
369C2×C4⋊C4—1221no—
370C2×C4⋊C4—1221no—
371C2×C4⋊C4C22×D6112yes—
372C2×C22⋊C4—28no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—1222no—
376C2×C22⋊C4—1222no—
377C2×C22⋊C4—1222no—
378C2×C42—24no—
379C2×C42C22⋊D6111yescharacteristic
380C2×C42C22⋊D6111yescharacteristic
381C2×C42C22×D6111yescharacteristic
382C2×C42C22×D6111yescharacteristic
383C4⋊C8—1224no—
384#102—1228no—
385#102—1228no—
386C22⋊C8—1224no—
387C24—24no—
388C24—24no—
389C24—22no—
390C24—24no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24—1222no—
396C24C2×C22⋊D6111yescharacteristic
397C2×D4—24no—
398C2×D4—24no—
399C2×D4—216no—
400C2×D4—216no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—24no—
405C2×D4—1224no—
406C2×D4—1222no—
407C2×D4—1222no—
408C2×D4—1224no—
409C2×D4—2442no—
410C2×D4—2442no—
411C2×D4—2444no—
412C2×D4—2444no—
413C2×D4—2444no—
414C2×D4—2442no—
415C2×D4—2442no—
416C2×D4—2442no—
417C2×D4—2442no—
418C2×D4C2×D24414yes—
419C2×D4#1175822214yes—
420C2×D4C2×C22⋊D6114yes—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—28no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—28no—
431C22×C4—41no—
432C22×C4—44no—
433C22×C4—432no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—1222no—
440C22×C4#1175819914yescomplemented
441C22×C4C2×D24412yes—
442C22×C4#1175822212yes—
443C22×C4C2×C22⋊D6111yescharacteristic
444C22×C4C2×C22⋊D6111yescharacteristic
445C22×C4C2×C22⋊D6111yescharacteristic
446C22×C4C2×C22⋊D6111yescharacteristic
447C22×C4C2×C22⋊D6111yescharacteristic
448C22×C4C23×D6111yescharacteristic
449D8—24416no—
450C2×C8—1224no—
451C2×C8—1224no—
452C2×C8—1224no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—1222no—
456C4⋊C4—1222no—
457C4⋊C4—2442no—
458C4⋊C4—2442no—
459C4⋊C4C2×C22⋊D6114yes—
460C22⋊C4—28no—
461C22⋊C4—28no—
462C22⋊C4—2444no—
463C22⋊C4—2444no—
464C22⋊C4—2444no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42C2×C22⋊D6114yes—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—216no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—48no—
478C23—48no—
479C23—48no—
480C23—48no—
481C23—48no—
482C23—48no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—2442no—
488C23—2442no—
489C23—2442no—
490C23—2442no—
491C23—2442no—
492C23—2442no—
493C23—2442no—
494C23#1451975112yes—
495C23C2×#1175819912yes—
496C23#1451980212yes—
497C23C22×C22⋊D6111yescharacteristic
498D4—216no—
499D4—24no—
500D4—24no—
501D4—24no—
502D4—4884no—
503D4—4888no—
504D4—4884no—
505D4—2444no—
506D4—2444no—
507D4#1451975118yescomplemented
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—28no—
517C2×C4—21no—
518C2×C4—24no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—44no—
522C2×C4—416no—
523C2×C4—48no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—416no—
527C2×C4—2442no—
528C2×C4—2442no—
529C2×C4—2442no—
530C2×C4#1451974612yes—
531C2×C4#1451974612yes—
532C2×C4#1451975112yes—
533C2×C4C2×#1175819912yes—
534C2×C4C2×#1175819912yes—
535C2×C4#1451980312yes—
536C2×C4C22×C22⋊D6112yes—
537C2×C4C22×D24411yescharacteristic
538C2×C4C2×#1175822212yes—
539C2×C4C2×#1175822211yescharacteristic
540C8—2444no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—28no—
548C22—432no—
549C22—44no—
550C22—44no—
551C22—44no—
552C22—44no—
553C22—44no—
554C22—44no—
555C22—4882no—
556C22—4882no—
557C22—4882no—
558C22—4882no—
559C22—4882no—
560C22—4882no—
561C22—4882no—
562C22#1683995714yescomplemented
563C22C2×#1451974611yescharacteristic
564C22C2×#1451975111yescharacteristic
565C22#1684027311yescharacteristic
566C22C22×#1175819911yescharacteristic
567C22C2×#1451980211yescharacteristic
568C22C2×#1451980311yescharacteristic
569C22#1684056011yescharacteristic
570C4—22no—
571C4—22no—
572C4—22no—
573C4—44no—
574C4—41no—
575C4—4882no—
576C4#1683989614yescomplemented
577C4C2×#1451975112yes—
578C4#1684027312yes—
579C2—24no—
580C2—48no—
581C2—9762no—
582C2#2145410111yescharacteristic
583C2C2×#1683989611yescharacteristic
584C2#2145507111yescharacteristic
585C2C2×#1683995711yescharacteristic
586C2#2145528311yescharacteristic
587C2#2145581911yescharacteristic
588C2#2145591011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC22×C4×D4×C61C2
socleC23×C61C22×D4
hypercenterC22×C4×D4D61
hypocenterC61#32902
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16839957C2semidirect
2. C2×#16839896C2semidirect
3. #21454164C2semidirect
4. #21446167C2semidirect
5. #21446136C2semidirect
6. #21445819C2semidirect
7. #21445799C2semidirect
8. C22×C4×D4×C61C2semidirect
10. C2×#14519774C22semidirect
16. C2×C4⋊#92418C22semidirect
18. C2×#14519697C22semidirect
22. #16839664C22semidirect
23. #16839632C22semidirect
24. #16839524C22semidirect
25. #16839517C22semidirect
64. C22×D4×C61D4semidirect
80. C4×D4×C61D4semidirect
131. C22×C4×C61D8semidirect
207. D4×C61#121semidirect
261. C22×C61#459semidirect
281. C4×C61#447semidirect
285. C22×C4×D4D61semidirect
330. C61#32902semidirect
340. C22×D4D244semidirect
365. C4×D4C22⋊D61semidirect
440. C22×C4#11758199semidirect
507. D4#14519751semidirect
562. C22#16839957semidirect
576. C4#16839896semidirect

Maximal subgroups

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

subgroupindex
C2×#168399572
C2×#168398962
#214541642
#214461672
#214461362
#214458192
#214457992
C22×C4×D4×C612
#3290261