Finite Group Databasebeta

← #29716075

Subgroups — #29716075

Order 15616.

1378 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519776C211yesmaximal, characteristic, complemented
2C2×#16840041C212yesmaximal, complemented
3#21448325C211yesmaximal, characteristic
4#21447145C212yesmaximal, complemented
5#21446954C212yesmaximal, complemented
6#21446769C212yesmaximal, complemented
7#21446136C211yesmaximal, characteristic
8#21446134C211yesmaximal, characteristic
9#21446006C211yesmaximal, characteristic, complemented
10#21445797C211yesmaximal, characteristic, complemented
11C61×#1852C211yesmaximal, characteristic, complemented
12C23×Dic122—21no—
13C2×#14519776—28no—
14C2×#14519776—21no—
15C2×#14519776—21no—
16C2×#14519776C2212yes—
17C2×#14519776C2212yes—
18C2×#14519751C2211yescharacteristic
19C2×#14519739C2211yescharacteristic
20C2×#14519737—21no—
21C22×C4⋊Dic61C2211yescharacteristic
22C22×#11758196—21no—
23C22×C4×Dic61C2211yescharacteristic
24#16840041—28no—
25C2×#14519710C2212yes—
26C2×#14519706C2212yes—
27C2×#14519699C2212yes—
28C2×#14519697C2212yes—
29C2×C22⋊#92418C2211yescharacteristic
30#16839663C2211yescharacteristic
31#16839658C2211yescharacteristic
32#16839651—21no—
33#16839639—21no—
34C4⋊C4×Dic61C2211yescharacteristic
35#16839609—21no—
36#16839525C2212yes—
37#16839524C2212yes—
38#16839524C2212yes—
39#16839516C2212yes—
40C22×C4⋊C4×C61C2211yescharacteristic
41C2×#107×C61C2212yes—
42C2×C22⋊C8×C61C2211yescharacteristic
43C61×#380C2211yescharacteristic
44C61×#340C2212yes—
45C22×Dic122—28no—
46C22×Dic122—22no—
47C22×Dic122—22no—
48C22×Dic122—44no—
49C22×Dic122—44no—
50C22×D244C2311yescharacteristic
51C22×C22⋊D61—21no—
52C23×Dic61—21no—
53C23×Dic61D411yescharacteristic
54C23×Dic61D411yescharacteristic
55C22×C4×D61—21no—
56C22×#92418C2312yes—
57#14519776—28no—
58#14519776—28no—
59#14519776—24no—
60#14519776—24no—
61#14519776—24no—
62#14519776D418yescomplemented
63#14519751—22no—
64#14519751Q814yescomplemented
65#14519739—24no—
66#14519737—44no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—24no—
69C2×C4⋊Dic61C2312yes—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×#11758196—28no—
72C2×#11758196—22no—
73C2×#11758196—22no—
74C2×#11758196—21no—
75C2×#11758198—21no—
76C2×#11758198—21no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×C4×Dic61—24no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61D411yescharacteristic
82C2×C4×Dic61D411yescharacteristic
83C2×C4×Dic61C2311yescharacteristic
84C2×C4×Dic61C2311yescharacteristic
85#14519710—24no—
86#14519706—24no—
87#14519699—24no—
88#14519697—24no—
89C22⋊#92418—22no—
90#14519684—21no—
91#14519684—21no—
92C23×C4×C61—21no—
93C23×C4×C61C2311yescharacteristic
94C22×C8×C61C2312yes—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61—24no—
97C2×C4⋊C4×C61C2311yescharacteristic
98C2×C4⋊C4×C61C2311yescharacteristic
99C2×C4⋊C4×C61C2312yes—
100C2×C22⋊C4×C61—21no—
101#107×C61—24no—
102C22⋊C8×C61—22no—
103#95×C61—21no—
104C24×C61C2×D411yescharacteristic
105C23×D61—21no—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—24no—
109C22×C4×C61—22no—
110C22×C4×C61—21no—
111C22×C4×C61—21no—
112C22×C4×C61—42no—
113C22×C4×C61—42no—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×Q812yes—
116C22×C4×C61#4112yes—
117C22×C4×C61C2411yescharacteristic
118C2×C22⋊D61—24no—
119C2×C22⋊D61—42no—
120C2×C22⋊D61—42no—
121C22×Dic61—22no—
122C22×Dic61—22no—
123C22×Dic61—22no—
124C22×Dic61—22no—
125C22×Dic61—22no—
126C22×Dic61—24no—
127C22×Dic61—22no—
128C22×Dic61—21no—
129C22×Dic61—42no—
130C22×Dic61—42no—
131C22×Dic61D814yes—
132C22×Dic61C2×D411yescharacteristic
133C22×Dic61C2×D411yescharacteristic
134C22×Dic61C2×D411yescharacteristic
135C22×Dic61C2×D411yescharacteristic
136C2×D244—22no—
137C2×D244—22no—
138C2×D244—21no—
139C2×D244C2×Q812yescomplemented
140C2×C4×D61—42no—
141C2×C4×D61—42no—
142C2×C4×D61—42no—
143C2×Dic122—28no—
144C2×Dic122—28no—
145C2×Dic122—44no—
146C2×Dic122—432no—
147C2×Dic122—44no—
148C2×Dic122—44no—
149C2×Dic122—44no—
150C2×C8×C61—22no—
151C2×C8×C61—22no—
152C2×C8×C61—22no—
153C4⋊C4×C61—24no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61C2×D414yes—
158C22⋊C4×C61—42no—
159#11758198—24no—
160#11758198—42no—
161#11758198—42no—
162#11758198—42no—
163C4⋊Dic61—24no—
164C4⋊Dic61—22no—
165C4⋊Dic61—21no—
166C4⋊Dic61C2×D414yes—
167C4⋊Dic61C2×Q812yescomplemented
168#11758196—28no—
169#11758196—28no—
170#11758196—42no—
171C4×Dic61—22no—
172C4×Dic61—22no—
173C4×Dic61—22no—
174C4×Dic61—42no—
175C4×Dic61C2×D414yes—
176C2×#92418—22no—
177C2×#92418—22no—
178C2×#92418—22no—
179C23×C61—21no—
180C23×C61—21no—
181C23×C61—21no—
182C23×C61—21no—
183C23×C61C22⋊Q812yes—
184C23×C61C2×D812yes—
185C23×C61#13712yes—
186C23×C61C22×D411yescharacteristic
187C22×D61—22no—
188C22×D61—41no—
189C22×D61—41no—
190C22×D61—41no—
191C22×D61—41no—
192C22×D61—41no—
193C22×D61—41no—
194C2×C4×C61—21no—
195C2×C4×C61—21no—
196C2×C4×C61—24no—
197C2×C4×C61—24no—
198C2×C4×C61—24no—
199C2×C4×C61—48no—
200C2×C4×C61—42no—
201C2×C4×C61—42no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61C22⋊Q812yes—
209C2×C4×C61C22×D412yes—
210C2×C4×C61C22×Q811yescharacteristic
211C2×C4×C61C2×#4111yescharacteristic
212C2×C4×C612−1+412yes—
213C22⋊D61—48no—
214C2×Dic61—28no—
215C2×Dic61—21no—
216C2×Dic61—21no—
217C2×Dic61—21no—
218C2×Dic61—21no—
219C2×Dic61—21no—
220C2×Dic61—21no—
221C2×Dic61—28no—
222C2×Dic61—24no—
223C2×Dic61—22no—
224C2×Dic61—48no—
225C2×Dic61—41no—
226C2×Dic61—42no—
227C2×Dic61—42no—
228C2×Dic61—41no—
229C2×Dic61#12012yes—
230C2×Dic61#12012yes—
231C2×Dic61C2×D812yes—
232C2×Dic61C2×D812yes—
233C2×Dic61AGL(1,ℤ/8)12yes—
234C2×Dic61#13712yes—
235D244—24no—
236D244—42no—
237C4×D61—84no—
238Dic122—416no—
239Dic122—416no—
240C8×C61—42no—
241#92418—42no—
242#24298—611nomaximal, Sylow
243C22×C61—22no—
244C22×C61—22no—
245C22×C61—22no—
246C22×C61—22no—
247C22×C61—22no—
248C22×C61—22no—
249C22×C61#47414yescomplemented
250C22×C61C2×#12011yescharacteristic
251C22×C61C2×C22⋊Q811yescharacteristic
252C22×C61#53611yescharacteristic
253C22×C61C22×D811yescharacteristic
254C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
255C22×C61C2×#13711yescharacteristic
256C22×C61C2×#13711yescharacteristic
257C2×D61—42no—
258C2×D61—42no—
259C2×D61—42no—
260C2×D61—81no—
261C2×D61—81no—
262C2×D61—81no—
263C2×D61—81no—
264C4×C61—22no—
265C4×C61—44no—
266C4×C61—81no—
267C4×C61—81no—
268C4×C61#54812yes—
269C4×C61D4×Q812yes—
270Dic61—22no—
271Dic61—22no—
272Dic61—22no—
273Dic61—44no—
274Dic61—42no—
275Dic61#44714yes—
276C22×C4⋊Q8—611no—
277C2×#474—612no—
278C4×#121—611no—
279#2057—612no—
280#2023—612no—
281#1988—612no—
282#1874—611no—
283#1874—611no—
284#1852—611no—
285#1852D6111yescharacteristic, complemented
286#1797—611no—
287C2×C61—24no—
288C2×C61#240911yescharacteristic
289C2×C61C2×#44711yescharacteristic
290C2×C61#300411yescharacteristic
291C2×C61C2×#47411yescharacteristic
292C2×C61#307911yescharacteristic
293C2×C61#322611yescharacteristic
294C2×C61#323911yescharacteristic
295D61—82no—
296C23×Q8—1221no—
297C2×C4⋊Q8—612no—
298C2×C4⋊Q8—612no—
299C2×C4⋊Q8—1221no—
300C2×C4⋊Q8—1221no—
301C2×C4⋊Q8—1228no—
302C2×#121—611no—
303C2×C4×D4—611no—
304C2×C4×D4—1221no—
305C22×C4⋊C4—611no—
306C22×C4⋊C4—1221no—
307C22×C4⋊C4C2×D6111yescharacteristic
308C22×C42—611no—
309#474—1228no—
310C2×#107—612no—
311C2×#107C2×D6112yes—
312C2×C4⋊C8—612no—
313C2×#102—612no—
314C2×#102—612no—
315C2×C22⋊C8—611no—
316C2×C22⋊C8C2×D6111yescharacteristic
317#390—611no—
318#389—611no—
319#388—1221no—
320#386—1221no—
321#380C2×D6111yescharacteristic
322C4×C4⋊C4—611no—
323C4×C22⋊C4—1221no—
324#340—612no—
325#340—612no—
326#340—612no—
327#340C2×D6112yes—
328#336—612no—
329C61#2429811yescharacteristic, complemented, Sylow
330C22×Q8—1222no—
331C22×Q8—1222no—
332C22×Q8—1228no—
333C22×Q8—2444no—
334C22×Q8—2444no—
335C22×D4—611no—
336C22×D4—1221no—
337C23×C4—21no—
338C23×C4—611no—
339C23×C4—611no—
340C23×C4—1221no—
341C23×C4—1221no—
342C23×C4C22×D6111yescharacteristic
343C22×C8—612no—
344C22×C8C22×D6112yes—
345C4⋊Q8—618no—
346C4⋊Q8—1224no—
347C4⋊Q8—1224no—
348C4⋊Q8—1228no—
349C4⋊Q8—1228no—
350C4⋊Q8—1224no—
351#121—614no—
352#121—1222no—
353C4×D4—1224no—
354C4×D4—2444no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—612no—
358C2×C4⋊C4—611no—
359C2×C4⋊C4—1224no—
360C2×C4⋊C4—1222no—
361C2×C4⋊C4—1222no—
362C2×C4⋊C4—1228no—
363C2×C4⋊C4—1221no—
364C2×C4⋊C4—1221no—
365C2×C4⋊C4C22×D6111yescharacteristic
366C2×C4⋊C4C22×D6111yescharacteristic
367C2×C4⋊C4C22×D6112yes—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—1221no—
370C2×C22⋊C4—1221no—
371C2×C22⋊C4—1221no—
372C2×C22⋊C4—1221no—
373C2×C42—611no—
374C2×C42—611no—
375C2×C42—611no—
376C2×C42—611no—
377C2×C42—1224no—
378C2×C42—1221no—
379#107—24no—
380#107—1224no—
381C4⋊C8—1224no—
382#102—1224no—
383#102—1224no—
384C22⋊C8—22no—
385C22⋊C8—1222no—
386#95—21no—
387#95—1221no—
388#95—1221no—
389C24—1221no—
390C24D4×D6111yescharacteristic
391C2×Q8—1228no—
392C2×Q8—1228no—
393C2×Q8—2444no—
394C2×Q8—2444no—
395C2×Q8—2444no—
396C2×Q8—24432no—
397C2×Q8—2444no—
398C2×D4—612no—
399C2×D4—1224no—
400C2×D4—1222no—
401C2×D4—1221no—
402C2×D4—1222no—
403C2×D4—2442no—
404C2×D4—2442no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—611no—
414C22×C4—611no—
415C22×C4—611no—
416C22×C4—611no—
417C22×C4—614no—
418C22×C4—1222no—
419C22×C4—1222no—
420C22×C4—1222no—
421C22×C4—1222no—
422C22×C4—1222no—
423C22×C4—1224no—
424C22×C4—1221no—
425C22×C4—1222no—
426C22×C4—2442no—
427C22×C4—2442no—
428C22×C4—2442no—
429C22×C4—2442no—
430C22×C4—2442no—
431C22×C4#1175822212yes—
432C22×C4D4×D6111yescharacteristic
433C22×C4Q8×D6112yes—
434C22×C4C23×D6111yescharacteristic
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—1222no—
439C2×C8—1222no—
440C2×C8—1222no—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—614no—
446C4⋊C4—612no—
447C4⋊C4—1228no—
448C4⋊C4—1228no—
449C4⋊C4—1224no—
450C4⋊C4—1222no—
451C4⋊C4—1221no—
452C4⋊C4—2442no—
453C4⋊C4D4×D6114yes—
454C22⋊C4—42no—
455C22⋊C4—1224no—
456C22⋊C4—2442no—
457C22⋊C4—2442no—
458C22⋊C4—2442no—
459C42—614no—
460C42—1222no—
461C42—1222no—
462C42—1222no—
463C42—2442no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—1222no—
469C23—2441no—
470C23—2441no—
471C23—2441no—
472C23—2441no—
473C23—2441no—
474C23—2441no—
475C23#1451975512yes—
476C23D8×D6112yes—
477C23#1451980912yes—
478C23C2×D4×D6111yescharacteristic
479Q8—24416no—
480Q8—24416no—
481D4—1224no—
482D4—2448no—
483D4—2442no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—48no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—612no—
499C2×C4—612no—
500C2×C4—612no—
501C2×C4—612no—
502C2×C4—612no—
503C2×C4—612no—
504C2×C4—1228no—
505C2×C4—1228no—
506C2×C4—1221no—
507C2×C4—1221no—
508C2×C4—1221no—
509C2×C4—1221no—
510C2×C4—1221no—
511C2×C4—1221no—
512C2×C4—1224no—
513C2×C4—1222no—
514C2×C4—4884no—
515C2×C4—2448no—
516C2×C4—2441no—
517C2×C4—2442no—
518C2×C4—2442no—
519C2×C4—2441no—
520C2×C4#1451975512yes—
521C2×C4C2×D4×D6112yes—
522C2×C4C2×Q8×D6111yescharacteristic
523C2×C4C2×#1175822211yescharacteristic
524C2×C4#1451982412yes—
525C8—42no—
526C8—2442no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—4881no—
534C22—4881no—
535C22—4881no—
536C22—4881no—
537C22—2442no—
538C22—2442no—
539C22—2442no—
540C22#1684004114yescomplemented
541C22#120×D6111yescharacteristic
542C22C2×#1451975511yescharacteristic
543C22#1684033411yescharacteristic
544C22C2×D8×D6111yescharacteristic
545C22AGL(1,ℤ/8)×D6111yescharacteristic
546C22#137×D6111yescharacteristic
547C22C2×#1451980911yescharacteristic
548C4—22no—
549C4—44no—
550C4—81no—
551C4—81no—
552C4—614no—
553C4—1222no—
554C4—1222no—
555C4—1222no—
556C4—2444no—
557C4—2442no—
558C4#1684042112yes—
559C4#1684042512yes—
560C2—24no—
561C2—4882no—
562C2#2144918311yescharacteristic
563C2#447×D6111yescharacteristic
564C2#2145505411yescharacteristic
565C2C2×#1684004111yescharacteristic
566C2#2145563211yescharacteristic
567C2#2145693811yescharacteristic
568C2#2145705211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1852C2
socleC23×C61C22×D4
prime radicalC2×#14519751C22
hypercenter#1852D61
hypocenterC61#24298
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1852

Complements

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

NH ≅ G/Ntype
1. C22×#14519776C2semidirect
2. C2×#16840041C2semidirect
4. #21447145C2semidirect
5. #21446954C2semidirect
6. #21446769C2semidirect
9. #21446006C2semidirect
10. #21445797C2semidirect
11. C61×#1852C2semidirect
62. #14519776D4semidirect
64. #14519751Q8semidirect
139. C2×D244C2×Q8semidirect
167. C4⋊Dic61C2×Q8semidirect
249. C22×C61#474semidirect
285. #1852D61semidirect
329. C61#24298semidirect
540. C22#16840041semidirect

Maximal subgroups

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

subgroupindex
C22×#145197762
C2×#168400412
#214483252
#214471452
#214469542
#214467692
#214461362
#214461342
#214460062
#214457972
C61×#18522
#2429861