Finite Group Databasebeta

← #29709148

Subgroups — #29709148

Order 15616.

1092 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D8×Dic61C212yesmaximal, complemented
2#127×Dic61C211yesmaximal, characteristic, complemented
3#21448281C211yesmaximal, characteristic, complemented
4#21447292C211yesmaximal, characteristic, complemented
5#21447222C211yesmaximal, characteristic, complemented
6#21447205C212yesmaximal, complemented
7#21447127C212yesmaximal, complemented
8#21447050C211yesmaximal, characteristic, complemented
9#21445667C211yesmaximal, characteristic, complemented
10#102×Dic61C212yesmaximal, complemented
11C61×#2011C211yesmaximal, characteristic
12C2×D4×Dic61—21no—
13C2×D4×Dic61—21no—
14C2×D4×Dic61C2212yes—
15C2×D4×Dic61C2211yescharacteristic
16C2×#14519733—21no—
17D8×Dic61—28no—
18C2×#14519711C2212yes—
19C2×#14519706C2212yes—
20C2×#14519698C2212yes—
21C2×#14519698C2212yes—
22C2×#14519698C2212yes—
23C2×C8×Dic61C2212yes—
24#16839665—21no—
25#16839665—21no—
26#16839665C2212yes—
27#16839665C2211yescharacteristic
28#16839660C2211yescharacteristic
29#16839658C2211yescharacteristic
30#16839656—21no—
31#16839650—21no—
32#16839632C2211yescharacteristic
33C4⋊C4×Dic61C2211yescharacteristic
34C22⋊C4×Dic61—21no—
35C42×Dic61C2211yescharacteristic
36#16839525C2212yes—
37#16839524C2212yes—
38#16839517C2211yescharacteristic
39C22×D8×C61C2212yes—
40C2×#127×C61C2211yescharacteristic
41C2×#102×C61C2212yes—
42C61×#390C2211yescharacteristic
43C61×#336C2211yescharacteristic
44C23×Dic61—21no—
45C23×Dic61—22no—
46C22×#92418C2312yes—
47D4×Dic61—24no—
48D4×Dic61—22no—
49D4×Dic61—24no—
50D4×Dic61—44no—
51D4×Dic61—44no—
52D4×Dic61D414yescomplemented
53#14519733—44no—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61C2312yes—
57C2×C4⋊Dic61C2311yescharacteristic
58C2×C4⋊Dic61C2311yescharacteristic
59C2×#11758196—21no—
60C2×#11758196—21no—
61C2×C22⋊Dic61—21no—
62C2×C22⋊Dic61—21no—
63C2×C22⋊Dic61—22no—
64C2×C22⋊Dic61—22no—
65C2×C22⋊Dic61—22no—
66C2×C22⋊Dic61—21no—
67C2×C4×Dic61—21no—
68C2×C4×Dic61D411yescharacteristic
69C2×C4×Dic61D411yescharacteristic
70C2×C4×Dic61D411yescharacteristic
71C2×C4×Dic61D411yescharacteristic
72C2×C4×Dic61C2311yescharacteristic
73C2×C4×Dic61C2311yescharacteristic
74#14519711—24no—
75#14519706—24no—
76#14519698—24no—
77#14519698—24no—
78#14519698—24no—
79C8×Dic61—24no—
80#14519684—21no—
81#14519684—21no—
82#14519684—21no—
83C22×D4×C61—21no—
84C22×D4×C61—21no—
85C22×D4×C61C2311yescharacteristic
86C22×D4×C61C2312yes—
87C2×D8×C61—24no—
88C2×D8×C61—24no—
89C2×D8×C61—24no—
90C22×C8×C61C2312yes—
91#127×C61—24no—
92C2×C4⋊C4×C61C2311yescharacteristic
93C2×C22⋊C4×C61—21no—
94C2×C22⋊C4×C61—21no—
95C2×C42×C61C2311yescharacteristic
96#102×C61—24no—
97C24×C61—21no—
98C24×C61—22no—
99C2×D4×C61—24no—
100C2×D4×C61—24no—
101C2×D4×C61—22no—
102C2×D4×C61—22no—
103C2×D4×C61—22no—
104C2×D4×C61—22no—
105C2×D4×C61—22no—
106C2×D4×C61—22no—
107C2×D4×C61—22no—
108C2×D4×C61—21no—
109C2×D4×C61—42no—
110C2×D4×C61—42no—
111C2×D4×C61—42no—
112C2×D4×C61—42no—
113C2×D4×C61C2×D412yes—
114C2×D4×C61#4114yes—
115C22×C4×C61—21no—
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2411yescharacteristic
119C22×Dic61—21no—
120C22×Dic61—24no—
121C22×Dic61—22no—
122C22×Dic61—21no—
123C22×Dic61—42no—
124C22×Dic61—42no—
125C22×Dic61—44no—
126C22×Dic61—44no—
127C22×Dic61—44no—
128C22×Dic61C2×D411yescharacteristic
129C22×Dic61C2×D411yescharacteristic
130C22×Dic61C2×D411yescharacteristic
131C22×Dic61C2×D411yescharacteristic
132D8×C61—216no—
133C2×C8×C61—22no—
134C2×C8×C61—22no—
135C2×C8×C61#4114yes—
136C4⋊C4×C61—22no—
137C22⋊C4×C61—42no—
138C22⋊C4×C61—42no—
139C42×C61—22no—
140C22⋊Dic61—24no—
141C22⋊Dic61—42no—
142C22⋊Dic61—44no—
143C22⋊Dic61—44no—
144C22⋊Dic61—42no—
145C22⋊Dic61—44no—
146C4⋊Dic61—21no—
147C4⋊Dic61—22no—
148C4⋊Dic61—22no—
149C4⋊Dic61—22no—
150C4⋊Dic61—42no—
151C4⋊Dic61—42no—
152C4⋊Dic61C2×D412yes—
153#11758196—42no—
154#11758196—42no—
155C4×Dic61—22no—
156C4×Dic61—22no—
157C4×Dic61—22no—
158C4×Dic61—41no—
159C4×Dic61—42no—
160C4×Dic61D814yescomplemented
161C4×Dic61C2×D412yes—
162C4×Dic61#4112yes—
163C2×#92418—22no—
164C2×#92418—22no—
165C2×#92418—22no—
166C23×C61—22no—
167C23×C61—24no—
168C23×C61—42no—
169C23×C61—41no—
170C23×C61—42no—
171C23×C61—41no—
172C23×C61—41no—
173C23×C61—42no—
174C23×C61—41no—
175C23×C61—42no—
176C23×C61—42no—
177C23×C61—41no—
178C23×C61—41no—
179C23×C61—42no—
180C23×C61C22×D411yescharacteristic
181D4×C61—24no—
182D4×C61—24no—
183D4×C61—24no—
184D4×C61—48no—
185D4×C61—48no—
186D4×C61—42no—
187D4×C61—42no—
188D4×C61—42no—
189C2×C4×C61—21no—
190C2×C4×C61—21no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—41no—
194C2×C4×C61—41no—
195C2×C4×C61—41no—
196C2×C4×C61C2×D812yes—
197C2×C4×C61#13512yes—
198C2×C4×C61C22×D411yescharacteristic
199C2×C4×C61C2×#4112yes—
200C2×C4×C61C2×#4112yes—
201C2×C4×C61C2×#4111yescharacteristic
202C2×Dic61—21no—
203C2×Dic61—21no—
204C2×Dic61—21no—
205C2×Dic61—21no—
206C2×Dic61—22no—
207C2×Dic61—21no—
208C2×Dic61—21no—
209C2×Dic61—21no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61—48no—
213C2×Dic61—42no—
214C2×Dic61—41no—
215C2×Dic61—41no—
216C2×Dic61—42no—
217C2×Dic61—42no—
218C2×Dic61—41no—
219C2×Dic61—88no—
220C2×Dic61#12112yes—
221C2×Dic61#12112yes—
222C2×Dic61C2×D812yes—
223C2×Dic61C2×D812yes—
224C2×Dic61AGL(1,ℤ/8)12yes—
225C8×C61—24no—
226#92418—42no—
227#23676—611nomaximal, Sylow
228C22×C61—44no—
229C22×C61—44no—
230C22×C61—42no—
231C22×C61—44no—
232C22×C61—42no—
233C22×C61—42no—
234C22×C61—81no—
235C22×C61—81no—
236C22×C61—81no—
237C22×C61—82no—
238C22×C61—82no—
239C22×C61—81no—
240C22×C61—82no—
241C22×C61—82no—
242C22×C61C2×#12111yescharacteristic
243C22×C61#52511yescharacteristic
244C22×C61#53211yescharacteristic
245C22×C61C22×D811yescharacteristic
246C22×C61C2×#13511yescharacteristic
247C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
248C22×C61#57511yescharacteristic
249C4×C61—22no—
250C4×C61—22no—
251C4×C61—81no—
252C4×C61#52512yes—
253C4×C61#52912yes—
254Dic61—22no—
255Dic61—22no—
256Dic61—41no—
257Dic61—42no—
258Dic61—81no—
259Dic61—82no—
260Dic61#45914yes—
261C2×C4×D8—612no—
262C4×#127—611no—
263#2293—611no—
264#2083—611no—
265#2071—611no—
266#2067—612no—
267#2054—612no—
268#2041—611no—
269#2011D6111yescharacteristic
270C4×C4⋊C8—611no—
271C4×#102—612no—
272C2×C61—84no—
273C2×C61—82no—
274C2×C61#239211yescharacteristic
275C2×C61C2×#45911yescharacteristic
276C2×C61#303511yescharacteristic
277C2×C61#310011yescharacteristic
278C2×C61#310511yescharacteristic
279C2×C61#316911yescharacteristic
280C2×C61#317111yescharacteristic
281C22×D8C2×D6112yes—
282C2×#127C2×D6111yescharacteristic
283C2×C4×D4—611no—
284C2×C4×D4—612no—
285C2×C4×D4—1221no—
286C2×C4×D4—1221no—
287C2×#117—1221no—
288C4×D8—1228no—
289C2×#107—612no—
290C2×C4⋊C8—612no—
291C2×#102—612no—
292C2×#102—612no—
293C2×#102—612no—
294C2×#102C2×D6112yes—
295C2×C4×C8—612no—
296#390—612no—
297#390—611no—
298#390—1221no—
299#390—1221no—
300#390C2×D6111yescharacteristic
301#389—611no—
302#389—611no—
303#388—1221no—
304#388—1221no—
305#384—611no—
306C4×C4⋊C4—611no—
307C4×C22⋊C4—1221no—
308C43—611no—
309#340—612no—
310#340—612no—
311#336—611no—
312#336C2×D6111yescharacteristic
313C61#2367611yescharacteristic, complemented, Sylow
314C22×D4—21no—
315C22×D4—21no—
316C22×D4C22×D6111yescharacteristic
317C22×D4C22×D6112yes—
318C23×C4—1221no—
319C23×C4—1222no—
320C2×D8—24no—
321C2×D8—24no—
322C2×D8—24no—
323C22×C8—612no—
324C22×C8C22×D6112yes—
325#127—24no—
326C4×D4—614no—
327C4×D4—1222no—
328C4×D4—1224no—
329C4×D4—1224no—
330C4×D4—2444no—
331C4×D4—2444no—
332#117—2444no—
333C2×C4⋊C4—611no—
334C2×C4⋊C4—612no—
335C2×C4⋊C4—611no—
336C2×C4⋊C4—1221no—
337C2×C4⋊C4—1221no—
338C2×C4⋊C4—1221no—
339C2×C4⋊C4—1221no—
340C2×C4⋊C4C22×D6111yescharacteristic
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—1221no—
344C2×C22⋊C4—1221no—
345C2×C22⋊C4—1222no—
346C2×C22⋊C4—1222no—
347C2×C22⋊C4—1222no—
348C2×C22⋊C4—1221no—
349C2×C42—611no—
350C2×C42—611no—
351C2×C42—611no—
352C2×C42—611no—
353C2×C42—611no—
354C2×C42—611no—
355C2×C42—1221no—
356C2×C42C22×D6111yescharacteristic
357#107—1224no—
358C4⋊C8—1224no—
359#102—24no—
360#102—1224no—
361#102—1224no—
362#102—1224no—
363C4×C8—1224no—
364#95—1221no—
365#95—1221no—
366#95—1221no—
367C24—21no—
368C24—22no—
369C2×D4—24no—
370C2×D4—24no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—21no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—42no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4D4×D6112yes—
384C2×D4#1175822414yes—
385C22×C4—21no—
386C22×C4—611no—
387C22×C4—611no—
388C22×C4—611no—
389C22×C4—611no—
390C22×C4—1224no—
391C22×C4—1221no—
392C22×C4—1221no—
393C22×C4—1222no—
394C22×C4—2444no—
395C22×C4—2444no—
396C22×C4—2444no—
397C22×C4—2442no—
398C22×C4—2442no—
399C22×C4D4×D6111yescharacteristic
400C22×C4D4×D6111yescharacteristic
401C22×C4C23×D6111yescharacteristic
402D8—216no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—1222no—
406C2×C8—1222no—
407C2×C8—1222no—
408C2×C8#1175822414yes—
409C4⋊C4—22no—
410C4⋊C4—612no—
411C4⋊C4—1221no—
412C4⋊C4—1222no—
413C4⋊C4—1222no—
414C4⋊C4—1222no—
415C4⋊C4—2442no—
416C4⋊C4—2442no—
417C4⋊C4—2442no—
418C4⋊C4—2442no—
419C22⋊C4—42no—
420C22⋊C4—42no—
421C22⋊C4—1224no—
422C22⋊C4—2442no—
423C22⋊C4—2444no—
424C22⋊C4—2444no—
425C22⋊C4—2444no—
426C22⋊C4—2442no—
427C42—22no—
428C42—612no—
429C42—612no—
430C42—614no—
431C42—1222no—
432C42—1222no—
433C42—1222no—
434C42—2441no—
435C42—2442no—
436C23—22no—
437C23—24no—
438C23—42no—
439C23—41no—
440C23—41no—
441C23—42no—
442C23—41no—
443C23—42no—
444C23—42no—
445C23—41no—
446C23—41no—
447C23—42no—
448C23—42no—
449C23—41no—
450C23C2×D4×D6111yescharacteristic
451D4—24no—
452D4—24no—
453D4—24no—
454D4—48no—
455D4—48no—
456D4—42no—
457D4—42no—
458D4—42no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—612no—
467C2×C4—612no—
468C2×C4—612no—
469C2×C4—612no—
470C2×C4—612no—
471C2×C4—1221no—
472C2×C4—1221no—
473C2×C4—1221no—
474C2×C4—1221no—
475C2×C4—1222no—
476C2×C4—1221no—
477C2×C4—1221no—
478C2×C4—1221no—
479C2×C4—4888no—
480C2×C4—2448no—
481C2×C4—2442no—
482C2×C4—2441no—
483C2×C4—2441no—
484C2×C4—2442no—
485C2×C4—2442no—
486C2×C4—2441no—
487C2×C4—2441no—
488C2×C4—2441no—
489C2×C4D8×D6112yes—
490C2×C4#1451979612yes—
491C2×C4C2×D4×D6111yescharacteristic
492C2×C4#41×D6111yescharacteristic
493C2×C4C2×#1175822412yes—
494C2×C4C2×#1175822412yes—
495C8—24no—
496C8—2442no—
497C22—42no—
498C22—44no—
499C22—42no—
500C22—42no—
501C22—44no—
502C22—44no—
503C22—81no—
504C22—82no—
505C22—82no—
506C22—81no—
507C22—82no—
508C22—82no—
509C22—81no—
510C22—81no—
511C22#121×D6111yescharacteristic
512C22#1684026811yescharacteristic
513C22#1684031011yescharacteristic
514C22C2×D8×D6111yescharacteristic
515C22C2×#1451979611yescharacteristic
516C22AGL(1,ℤ/8)×D6111yescharacteristic
517C22#1684055811yescharacteristic
518C4—22no—
519C4—22no—
520C4—81no—
521C4—614no—
522C4—1222no—
523C4—1222no—
524C4—4882no—
525C4—4881no—
526C4—2441no—
527C4—2442no—
528C4#1684026812yes—
529C4#1684029312yes—
530C2—82no—
531C2—84no—
532C2#2144905411yescharacteristic
533C2#459×D6111yescharacteristic
534C2#2145526711yescharacteristic
535C2#2145581411yescharacteristic
536C2#2145586111yescharacteristic
537C2#2145639611yescharacteristic
538C2#2145641111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#2011C2
socleC23×C61C22×D4
hypercenter#2011D61
hypocenterC61#23676
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2011

Complements

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

NH ≅ G/Ntype
1. C2×D8×Dic61C2semidirect
2. #127×Dic61C2semidirect
3. #21448281C2semidirect
4. #21447292C2semidirect
5. #21447222C2semidirect
6. #21447205C2semidirect
7. #21447127C2semidirect
8. #21447050C2semidirect
9. #21445667C2semidirect
10. #102×Dic61C2semidirect
52. D4×Dic61D4semidirect
160. C4×Dic61D8semidirect
313. C61#23676semidirect

Maximal subgroups

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

subgroupindex
C2×D8×Dic612
#127×Dic612
#214482812
#214472922
#214472222
#214472052
#214471272
#214470502
#214456672
#102×Dic612
C61×#20112
#2367661