Finite Group Databasebeta

← #29718995

Subgroups — #29718995

Order 15616.

1280 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839895C211yesmaximal, characteristic
2C22×#14519699C211yesmaximal, characteristic, complemented
3#21450291C211yesmaximal, characteristic, complemented
4#21448639C211yesmaximal, characteristic, complemented
5#21447226C212yesmaximal, complemented
6#21447128C212yesmaximal, complemented
7#21446164C212yesmaximal, complemented
8#21445997C211yesmaximal, characteristic, complemented
9#21445903C211yesmaximal, characteristic, complemented
10#21445818C212yesmaximal, complemented
11C61×#1893C211yesmaximal, characteristic, complemented
12C23×D244C2211yescharacteristic
13C22×#11758199C2211yescharacteristic
14C22×#11758199C2211yescharacteristic
15C2×#14519758—21no—
16C2×#14519758—21no—
17C2×#14519739C2212yes—
18C22×C4⋊Dic61C2211yescharacteristic
19C2×C22⋊C4×D61—21no—
20#16839895—24no—
21C2×#14519706C2212yescomplemented
22C2×#14519699—24no—
23C2×#14519699C2212yes—
24C2×#14519699C2212yes—
25C2×#14519697C2211yescharacteristic
26C2×#14519697C2211yescharacteristic
27C2×C22⋊#92418C2211yescharacteristic
28#16839698—22no—
29#16839665C2212yescomplemented
30#16839663C2212yes—
31#16839640—22no—
32#16839631C2212yescomplemented
33#16839617—21no—
34C4×C22⋊Dic61C2211yescharacteristic
35#16839524C2212yescomplemented
36#16839524C2212yescomplemented
37#16839516C2212yescomplemented
38C23×C8×C61C2211yescharacteristic
39C2×#121×C61C2211yescharacteristic
40C2×#102×C61C2212yescomplemented
41C61×#380C2211yescharacteristic
42C61×#340C2212yescomplemented
43C24×D61—21no—
44C22×D244—24no—
45C22×D244—21no—
46C22×D244—22no—
47C22×D244—42no—
48C22×D244D414yes—
49C22×D244C2311yescharacteristic
50C22×D244C2311yescharacteristic
51C23×Dic61—21no—
52C22×C4×D61—22no—
53C2×#11758199—22no—
54C2×#11758199—22no—
55C2×#11758199—24no—
56C2×#11758199—42no—
57C22×#92418C2311yescharacteristic
58C22×#92418C2311yescharacteristic
59#14519758—44no—
60#14519758—44no—
61#14519739—24no—
62#14519739—42no—
63C2×C4⋊Dic61—22no—
64C2×C4⋊Dic61—22no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61C2312yes—
67C2×#11758196—22no—
68C2×#11758196—22no—
69C2×C22⋊Dic61—22no—
70C2×C22⋊Dic61—22no—
71C2×C22⋊Dic61D411yescharacteristic
72C2×C22⋊Dic61D411yescharacteristic
73C2×#11758198—22no—
74C2×#11758198—22no—
75C2×#11758198—22no—
76C22⋊C4×D61—44no—
77C2×C4×Dic61C2312yes—
78#14519706—42no—
79#14519699—24no—
80#14519699—24no—
81#14519697—22no—
82#14519697—22no—
83C22⋊#92418—22no—
84#14519684D412yes—
85C22×D4×C61—21no—
86C22×D4×C61C2311yescharacteristic
87C23×C4×C61C2311yescharacteristic
88C22×C8×C61—22no—
89C22×C8×C61—42no—
90C22×C8×C61C2312yes—
91#121×C61—22no—
92#121×C61—22no—
93C2×C4⋊C4×C61C2312yes—
94C2×C4⋊C4×C61C2311yescharacteristic
95C2×C22⋊C4×C61—21no—
96C2×C22⋊C4×C61—21no—
97#102×C61—42no—
98#95×C61—21no—
99C24×C61—21no—
100C24×C61C2×D411yescharacteristic
101C23×D61—21no—
102C23×D61—21no—
103C23×D61—24no—
104C23×D61—44no—
105C23×D61—42no—
106C23×D61—42no—
107C23×D61—82no—
108C2×D4×C61—21no—
109C2×D4×C61—21no—
110C2×D4×C61—22no—
111C2×D4×C61—22no—
112C2×D4×C61—42no—
113C2×D4×C61—42no—
114C2×D4×C61—42no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—21no—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D412yes—
121C22×C4×C61#4112yes—
122C22×C4×C61C2411yescharacteristic
123C22×Dic61—22no—
124C22×Dic61—42no—
125C22×Dic61—42no—
126C22×Dic61—42no—
127C22×Dic61C2×D412yes—
128C22×Dic61C2×D412yes—
129C2×D244—21no—
130C2×D244—22no—
131C2×D244—22no—
132C2×D244—22no—
133C2×D244—28no—
134C2×D244—24no—
135C2×D244—21no—
136C2×D244—44no—
137C2×D244—48no—
138C2×D244—44no—
139C2×D244C2×D412yes—
140C2×D244C2×D412yes—
141C2×D244#4114yes—
142C2×C4×D61—44no—
143C2×C4×D61—44no—
144C2×C4×D61—82no—
145C2×C8×C61—22no—
146C2×C8×C61—22no—
147C2×C8×C61—22no—
148C2×C8×C61—44no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—22no—
153C22⋊C4×C61—42no—
154C22⋊C4×C61—42no—
155C22⋊Dic61—22no—
156C22⋊Dic61—44no—
157C22⋊Dic61—44no—
158C22⋊Dic61D814yescomplemented
159#11758199—44no—
160#11758199—44no—
161#11758198—44no—
162#11758198—44no—
163#11758198—82no—
164C4⋊Dic61—22no—
165C4⋊Dic61—22no—
166C4⋊Dic61—42no—
167#11758196—44no—
168#11758196—44no—
169C4×Dic61—22no—
170C4×Dic61—22no—
171C2×#92418—21no—
172C2×#92418—21no—
173C2×#92418—22no—
174C2×#92418—22no—
175C23×C61—22no—
176C23×C61—22no—
177C23×C61—41no—
178C23×C61—42no—
179C23×C61—41no—
180C23×C61—41no—
181C23×C61—42no—
182C23×C61#12112yes—
183C23×C61C2×D812yes—
184C23×C61AGL(1,ℤ/8)12yes—
185C23×C61C22×D411yescharacteristic
186C22×D61—22no—
187C22×D61—24no—
188C22×D61—22no—
189C22×D61—48no—
190C22×D61—41no—
191C22×D61—41no—
192C22×D61—42no—
193C22×D61—42no—
194C22×D61—41no—
195C22×D61—41no—
196C22×D61—42no—
197C22×D61—48no—
198C22×D61—84no—
199C22×D61—88no—
200C22×D61—88no—
201D4×C61—41no—
202D4×C61—41no—
203D4×C61—42no—
204D4×C61—84no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—22no—
208C2×C4×C61—24no—
209C2×C4×C61—41no—
210C2×C4×C61—42no—
211C2×C4×C61—41no—
212C2×C4×C61—42no—
213C2×C4×C61—41no—
214C2×C4×C61—42no—
215C2×C4×C61#12112yes—
216C2×C4×C61C22×D411yescharacteristic
217C2×C4×C61C2×#4112yes—
218C2×C4×C61C2×#4111yescharacteristic
219C2×C4×C61H(5,𝔽2)12yes—
220C2×Dic61—22no—
221C2×Dic61—22no—
222C2×Dic61—22no—
223C2×Dic61—22no—
224C2×Dic61—22no—
225C2×Dic61—42no—
226C2×Dic61—42no—
227C2×Dic61—42no—
228C2×Dic61—84no—
229C2×Dic61C2×D814yes—
230D244—22no—
231D244—22no—
232D244—22no—
233D244—24no—
234D244—48no—
235D244—42no—
236D244#12314yescomplemented
237C4×D61—88no—
238C8×C61—42no—
239#92418—41no—
240#92418—41no—
241#24526—611nomaximal, Sylow
242C22×C61—22no—
243C22×C61—24no—
244C22×C61—22no—
245C22×C61—24no—
246C22×C61—81no—
247C22×C61—81no—
248C22×C61—82no—
249C22×C61—81no—
250C22×C61—82no—
251C22×C61#45914yes—
252C22×C61C2×#12111yescharacteristic
253C22×C61#54612yes—
254C22×C61C22×D811yescharacteristic
255C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
256C22×C61#57712yes—
257C2×D61—24no—
258C2×D61—42no—
259C2×D61—44no—
260C2×D61—42no—
261C2×D61—44no—
262C2×D61—42no—
263C2×D61—42no—
264C2×D61—42no—
265C2×D61—816no—
266C2×D61—82no—
267C2×D61—82no—
268C4×C61—22no—
269C4×C61—81no—
270C4×C61—82no—
271C4×C61C2×#12311yescharacteristic
272C4×C61#53411yescharacteristic
273C4×C61#55112yes—
274Dic61—24no—
275Dic61—42no—
276Dic61—82no—
277C2×#447—611no—
278C22×#102—611no—
279#2501—611no—
280D4×C22⋊C4—611no—
281#2071—612no—
282#2054—612no—
283#1893D6111yescharacteristic, complemented
284#1879—612no—
285#1850—611no—
286#1823—611no—
287#1802—612no—
288C2×C61—24no—
289C2×C61—161no—
290C2×C61#241211yescharacteristic
291C2×C61C2×#45911yescharacteristic
292C2×C61#304211yescharacteristic
293C2×C61#329412yes—
294C2×C61#330612yes—
295D61—44no—
296D61—82no—
297C23×D4—611no—
298C22×D8—611no—
299C22×D8—611no—
300C23×C8C2×D6111yescharacteristic
301C2×#123—1221no—
302C2×#123—1221no—
303C2×#121C2×D6111yescharacteristic
304C2×C4×D4—612no—
305C22×C4⋊C4—611no—
306C22×C22⋊C4—1221no—
307#447—1224no—
308C2×C4⋊C8—612no—
309C2×#102—612no—
310C2×#102—612no—
311C2×#102—611no—
312C2×#102—611no—
313C2×#102—1224no—
314C2×#102C2×D6112yescomplemented
315C2×C22⋊C8—611no—
316#399—1222no—
317#390—612no—
318#390—612no—
319#386—1222no—
320#384—612no—
321#380C2×D6111yescharacteristic
322#379—1221no—
323C4×C22⋊C4—611no—
324#340—612no—
325#340—612no—
326#340C2×D6112yescomplemented
327#336—612no—
328C61#2452611yescharacteristic, complemented, Sylow
329C25—1221no—
330C22×D4—21no—
331C22×D4—614no—
332C22×D4—611no—
333C22×D4—611no—
334C22×D4—1224no—
335C22×D4—1222no—
336C22×D4—1221no—
337C22×D4—2442no—
338C22×D4C22×D6111yescharacteristic
339C23×C4—1222no—
340C23×C4—1221no—
341C23×C4C22×D6111yescharacteristic
342C2×D8—1222no—
343C2×D8—1222no—
344C2×D8—1224no—
345C2×D8—2442no—
346C22×C8—22no—
347C22×C8—42no—
348C22×C8—611no—
349C22×C8—611no—
350C22×C8C22×D6112yes—
351#123—2444no—
352#123—2444no—
353#121—22no—
354#121—22no—
355C4×D4—1224no—
356C4×D4—2442no—
357C2×C4⋊C4—612no—
358C2×C4⋊C4—1222no—
359C2×C4⋊C4—1222no—
360C2×C4⋊C4—1222no—
361C2×C4⋊C4—1222no—
362C2×C4⋊C4—1221no—
363C2×C4⋊C4C22×D6112yes—
364C2×C4⋊C4C22×D6111yescharacteristic
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—611no—
368C2×C22⋊C4—611no—
369C2×C22⋊C4—1222no—
370C2×C22⋊C4—1222no—
371C2×C22⋊C4—1222no—
372C2×C22⋊C4—1222no—
373C2×C22⋊C4—1222no—
374C2×C22⋊C4—2444no—
375C2×C42—612no—
376C4⋊C8—2442no—
377#102—42no—
378#102—1222no—
379#102—1222no—
380#102—1224no—
381#102—1224no—
382C22⋊C8—1222no—
383#95—21no—
384#95—612no—
385C24—21no—
386C24—1221no—
387C24—1224no—
388C24—1221no—
389C24—4882no—
390C24—2444no—
391C24—2442no—
392C24—2442no—
393C24D4×D6111yescharacteristic
394C2×D4—21no—
395C2×D4—22no—
396C2×D4—21no—
397C2×D4—22no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—612no—
402C2×D4—612no—
403C2×D4—614no—
404C2×D4—1228no—
405C2×D4—1222no—
406C2×D4—1221no—
407C2×D4—1221no—
408C2×D4—1222no—
409C2×D4—1222no—
410C2×D4—1224no—
411C2×D4—2444no—
412C2×D4—2448no—
413C2×D4—2444no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—612no—
419C22×C4—612no—
420C22×C4—1222no—
421C22×C4—4882no—
422C22×C4—2444no—
423C22×C4—2444no—
424C22×C4—2442no—
425C22×C4—2442no—
426C22×C4—2442no—
427C22×C4#1175822212yes—
428C22×C4D4×D6112yes—
429C22×C4D4×D6111yescharacteristic
430C22×C4C23×D6111yescharacteristic
431D8—2444no—
432D8—2444no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—44no—
437C2×C8—1221no—
438C2×C8—1221no—
439C2×C8—1222no—
440C2×C8—1222no—
441C4⋊C4—21no—
442C4⋊C4—22no—
443C4⋊C4—21no—
444C4⋊C4—22no—
445C4⋊C4—1222no—
446C4⋊C4—1222no—
447C4⋊C4—2444no—
448C4⋊C4—2444no—
449C4⋊C4—2442no—
450C22⋊C4—42no—
451C22⋊C4—42no—
452C22⋊C4—614no—
453C22⋊C4—1222no—
454C22⋊C4—4882no—
455C22⋊C4—2444no—
456C22⋊C4—2444no—
457C22⋊C4—2444no—
458C22⋊C4—2444no—
459C42—1222no—
460C42—1222no—
461C23—22no—
462C23—22no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—42no—
467C23—42no—
468C23—1222no—
469C23—1224no—
470C23—1222no—
471C23—4888no—
472C23—4888no—
473C23—4884no—
474C23—2448no—
475C23—2448no—
476C23—2441no—
477C23—2441no—
478C23—2442no—
479C23—2442no—
480C23—2441no—
481C23—2441no—
482C23—2442no—
483C23#1451975012yes—
484C23D8×D6112yes—
485C23#1451980412yes—
486C23C2×D4×D6111yescharacteristic
487D4—41no—
488D4—41no—
489D4—42no—
490D4—84no—
491D4—614no—
492D4—1222no—
493D4—1222no—
494D4—1222no—
495D4—1224no—
496D4—2448no—
497D4—2442no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—614no—
509C2×C4—1222no—
510C2×C4—1222no—
511C2×C4—1222no—
512C2×C4—1222no—
513C2×C4—1222no—
514C2×C4—4888no—
515C2×C4—4884no—
516C2×C4—2442no—
517C2×C4—2442no—
518C2×C4—2442no—
519C2×C4#1451975012yes—
520C2×C4C2×D4×D6111yescharacteristic
521C2×C4#41×D6112yes—
522C2×C4C2×#1175822211yescharacteristic
523C2×C4#1451982212yes—
524C8—42no—
525C8—2441no—
526C8—2441no—
527C22—22no—
528C22—24no—
529C22—24no—
530C22—22no—
531C22—81no—
532C22—82no—
533C22—81no—
534C22—82no—
535C22—81no—
536C22—1224no—
537C22—48816no—
538C22—4882no—
539C22—4882no—
540C22—2442no—
541C22—2442no—
542C22—2442no—
543C22—2444no—
544C22—2442no—
545C22—2444no—
546C22—2442no—
547C22#1683995614yes—
548C22C2×#1451975011yescharacteristic
549C22#1684040112yes—
550C22C2×D8×D6111yescharacteristic
551C22C2×#1451980411yescharacteristic
552C22#1684056912yes—
553C4—22no—
554C4—81no—
555C4—82no—
556C4—1224no—
557C4—4882no—
558C4—2442no—
559C4#123×D6111yescharacteristic
560C4#1684032211yescharacteristic
561C4#1684044112yes—
562C2—24no—
563C2—161no—
564C2—4882no—
565C2—2444no—
566C2#2144921811yescharacteristic
567C2C2×#1683995611yescharacteristic
568C2#2145533411yescharacteristic
569C2#2145762712yes—
570C2#2145778612yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1893C2
socleC23×C61C22×D4
prime radicalC2×#16839895C2
hypercenter#1893D61
hypocenterC61#24526
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1893

Complements

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

NH ≅ G/Ntype
2. C22×#14519699C2semidirect
3. #21450291C2semidirect
4. #21448639C2semidirect
5. #21447226C2semidirect
6. #21447128C2semidirect
7. #21446164C2semidirect
8. #21445997C2semidirect
9. #21445903C2semidirect
10. #21445818C2semidirect
11. C61×#1893C2semidirect
21. C2×#14519706C22semidirect
29. #16839665C22semidirect
32. #16839631C22semidirect
35. #16839524C22semidirect
36. #16839524C22semidirect
37. #16839516C22semidirect
40. C2×#102×C61C22semidirect
42. C61×#340C22semidirect
158. C22⋊Dic61D8semidirect
236. D244#123semidirect
283. #1893D61semidirect
314. C2×#102C2×D61semidirect
326. #340C2×D61semidirect
328. C61#24526semidirect

Maximal subgroups

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

subgroupindex
C2×#168398952
C22×#145196992
#214502912
#214486392
#214472262
#214471282
#214461642
#214459972
#214459032
#214458182
C61×#18932
#2452661