Finite Group Databasebeta

← #29691254

Subgroups — #29691254

Order 15616.

1212 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839915C211yesmaximal, characteristic
2#21448347C211yesmaximal, characteristic, complemented
3C2×#16839663C211yesmaximal, characteristic
4#21446698C212yesmaximal, complemented
5#21446514C212yesmaximal, complemented
6#21446438C212yesmaximal, complemented
7#21446175C212yesmaximal, complemented
8#21446126C211yesmaximal, characteristic, complemented
9#21445749C211yesmaximal, characteristic
10#21445583C211yesmaximal, characteristic, complemented
11C61×#1852C211yesmaximal, characteristic, complemented
12C23×D244C2211yescharacteristic
13C22×#11758190C2212yes—
14C2×#14519756C2211yescharacteristic
15C2×#14519742C2211yescharacteristic, complemented
16C22×#11758198—21no—
17C22×#11758198—21no—
18C22×C4×Dic61C2211yescharacteristic
19#16839915C418yescomplemented
20C2×#14519700C2212yescomplemented
21C2×#14519699C2212yes—
22C2×#14519697C2212yes—
23C2×#14519697C2212yes—
24C2×C22⋊#92418C2211yescharacteristic
25C2×#14519687C2212yescomplemented
26#16839667C2211yescharacteristic, complemented
27#16839663—21no—
28#16839663—21no—
29#16839663C2212yes—
30#16839663C2212yes—
31#16839658C2211yescharacteristic, complemented
32#16839644—21no—
33#16839634—21no—
34C4⋊C4×Dic61C2211yescharacteristic, complemented
35#16839613—21no—
36C22⋊C4×Dic61—21no—
37#16839524C2212yescomplemented
38#16839516C2212yescomplemented
39C22×C4⋊C4×C61C2211yescharacteristic
40C2×#107×C61C2212yescomplemented
41C2×C22⋊C8×C61C2211yescharacteristic
42C61×#380C2211yescharacteristic
43C61×#340C2212yescomplemented
44C24×D61—21no—
45C22×Dic122C2311yescharacteristic
46C22×D244—21no—
47C22×D244—22no—
48C22×D244—42no—
49C22×D244—42no—
50C22×D244C2×C414yes—
51C22×D244C2312yes—
52C23×Dic61D411yescharacteristic
53C23×Dic61D411yescharacteristic
54C2×#11758190—42no—
55C2×#11758190—42no—
56C2×#11758190C2×C418yescomplemented
57C22×#92418C2312yes—
58#14519756—22no—
59#14519756C2×C414yes—
60#14519742—42no—
61C2×C4⋊Dic61C2311yescharacteristic
62C2×#11758196—21no—
63C2×#11758196—21no—
64C2×#11758196—21no—
65C2×C22⋊Dic61—21no—
66C2×#11758198—22no—
67C2×#11758198—22no—
68C2×#11758198—22no—
69C2×#11758198—22no—
70C2×#11758198—44no—
71C2×#11758198—44no—
72C2×C4×Dic61—21no—
73C2×C4×Dic61—21no—
74C2×C4×Dic61—42no—
75C2×C4×Dic61D411yescharacteristic
76C2×C4×Dic61D411yescharacteristic
77C2×C4×Dic61C2311yescharacteristic
78C2×C4×Dic61C2311yescharacteristic
79#14519700—42no—
80#14519699C2×C418yescomplemented
81#14519697—42no—
82#14519697—42no—
83C22⋊#92418—22no—
84#14519687—42no—
85#14519684—21no—
86#14519684—21no—
87#14519684—21no—
88#14519684—21no—
89C23×C4×C61—21no—
90C23×C4×C61C2311yescharacteristic
91C22×C8×C61C2312yes—
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61—42no—
94C2×C4⋊C4×C61C2311yescharacteristic
95C2×C4⋊C4×C61C2312yes—
96C2×C4⋊C4×C61C2311yescharacteristic
97C2×C22⋊C4×C61—21no—
98#107×C61—42no—
99C22⋊C8×C61C2×C414yes—
100#95×C61—21no—
101C24×C61C2×D411yescharacteristic
102C23×D61—24no—
103C23×D61—22no—
104C23×D61—42no—
105C23×D61—42no—
106C23×D61—42no—
107C23×D61—42no—
108C23×D61—42no—
109C23×D61—42no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—21no—
113C22×C4×C61—21no—
114C22×C4×C61—22no—
115C22×C4×C61—42no—
116C22×C4×C61—42no—
117C22×C4×C61—42no—
118C22×C4×C61C22×C412yes—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61#4112yes—
121C22×C4×C61C2411yescharacteristic
122C22×Dic61—22no—
123C22×Dic61—22no—
124C22×Dic61—21no—
125C22×Dic61—21no—
126C22×Dic61—22no—
127C22×Dic61—41no—
128C22×Dic61—41no—
129C22×Dic61—41no—
130C22×Dic61C2×D411yescharacteristic
131C22×Dic61C2×D411yescharacteristic
132C22×Dic61C2×D411yescharacteristic
133C22×Dic61C2×D411yescharacteristic
134C2×D244—22no—
135C2×D244—22no—
136C2×D244—22no—
137C2×D244—22no—
138C2×D244—24no—
139C2×D244—22no—
140C2×D244—48no—
141C2×D244—42no—
142C2×D244—42no—
143C2×D244—48no—
144C2×D244C22×C414yes—
145C2×Dic122—22no—
146C2×Dic122—22no—
147C2×Dic122—21no—
148C2×Dic122C22×C412yes—
149C2×C8×C61—22no—
150C2×C8×C61—22no—
151C2×C8×C61C22×C414yes—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—42no—
157C22⋊C4×C61—42no—
158C22⋊Dic61—24no—
159#11758198—44no—
160#11758198—44no—
161#11758198—44no—
162#11758198—44no—
163C4⋊Dic61—21no—
164C4⋊Dic61C22×C412yes—
165#11758196—24no—
166#11758196—42no—
167#11758196—42no—
168C4×Dic61—22no—
169C4×Dic61—22no—
170C4×Dic61—41no—
171C4×Dic61—41no—
172C4×Dic61—42no—
173C4×Dic61—42no—
174C2×#92418—22no—
175C2×#92418—22no—
176C2×#92418—22no—
177#11758190—48no—
178C23×C61—21no—
179C23×C61—21no—
180C23×C61—21no—
181C23×C61—21no—
182C23×C61C4×D412yes—
183C23×C61AGL(1,ℤ/8)12yes—
184C23×C61#13712yes—
185C23×C61C22×D411yescharacteristic
186C22×D61—24no—
187C22×D61—48no—
188C22×D61—48no—
189C22×D61—42no—
190C22×D61—42no—
191C22×D61—42no—
192C22×D61—42no—
193C22×D61—42no—
194C22×D61—42no—
195C22×D61—48no—
196C22×D61—84no—
197C22×D61—84no—
198C22×D61—84no—
199C22×D61—84no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—24no—
203C2×C4×C61—24no—
204C2×C4×C61—41no—
205C2×C4×C61—42no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—84no—
214C2×C4×C61C4×D412yes—
215C2×C4×C61C23×C411yescharacteristic
216C2×C4×C61C2×#4111yescharacteristic
217C2×C4×C61H(5,𝔽2)12yes—
218C2×C4×C61H(5,𝔽2)12yes—
219C2×Dic61—21no—
220C2×Dic61—21no—
221C2×Dic61—22no—
222C2×Dic61—22no—
223C2×Dic61—21no—
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—21no—
229C2×Dic61—21no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—44no—
233C2×Dic61—44no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61AGL(1,ℤ/8)12yes—
239D244—24no—
240D244—24no—
241D244—48no—
242D244—42no—
243D244—42no—
244Dic122—24no—
245Dic122—42no—
246C8×C61—24no—
247#92418—42no—
248#22049—611nomaximal, Sylow
249C22×C61—22no—
250C22×C61—22no—
251C22×C61—22no—
252C22×C61—22no—
253C22×C61—22no—
254C22×C61—22no—
255C22×C61#44014yes—
256C22×C61C2×C4×D411yescharacteristic
257C22×C61#53411yescharacteristic
258C22×C61#53511yescharacteristic
259C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
260C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
261C22×C61C2×#13711yescharacteristic
262C22×C61#57511yescharacteristic
263C2×D61—44no—
264C2×D61—44no—
265C2×D61—44no—
266C2×D61—816no—
267C2×D61—82no—
268C2×D61—82no—
269C2×D61—82no—
270C2×D61—82no—
271C4×C61—22no—
272C4×C61—81no—
273C4×C61—81no—
274C4×C61—82no—
275C4×C61#51812yes—
276C4×C61#53812yes—
277Dic61—22no—
278Dic61—42no—
279Dic61—41no—
280Dic61—41no—
281Dic61—41no—
282Dic61—42no—
283C2×#450—611no—
284C4×C22⋊Q8—611no—
285C2×#390—611no—
286#1973—612no—
287#1936—612no—
288#1925—612no—
289#1880—612no—
290#1872—611no—
291#1852D6111yescharacteristic, complemented
292#1786—611no—
293#1749—611no—
294C2×C61—24no—
295C2×C61#232111yescharacteristic
296C2×C61C2×#44011yescharacteristic
297C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
298C2×C61#319311yescharacteristic
299C2×C61#319411yescharacteristic
300C2×C61#321511yescharacteristic
301C2×C61#321711yescharacteristic
302D61—84no—
303C23×D4—611no—
304C22×SD16—612no—
305C2×C22⋊Q8—611no—
306C2×C4×Q8—611no—
307C22×C4⋊C4C2×D6111yescharacteristic
308C22×C22⋊C4—1221no—
309C22×C22⋊C4—1221no—
310C22×C42—611no—
311#450—618no—
312C2×#107C2×D6112yescomplemented
313C2×#103—612no—
314C2×#102—612no—
315C2×#102—612no—
316C2×#102—612no—
317C2×C22⋊C8—611no—
318C2×C22⋊C8C2×D6111yescharacteristic
319C2×#97—612no—
320#391—611no—
321#390—612no—
322#390—612no—
323#390—1221no—
324#390—1221no—
325#389—611no—
326#387—1221no—
327#385—1221no—
328#380C2×D6111yescharacteristic
329C4×C4⋊C4—611no—
330C4×C4⋊C4—1221no—
331C4×C22⋊C4—1221no—
332#340—612no—
333#340C2×D6112yescomplemented
334#336—612no—
335C61#2204911yescharacteristic, complemented, Sylow
336C25—1221no—
337C22×Q8—611no—
338C22×D4—614no—
339C22×D4—612no—
340C22×D4—1221no—
341C22×D4—1222no—
342C22×D4—2442no—
343C22×D4—2442no—
344C23×C4—21no—
345C23×C4—611no—
346C23×C4—611no—
347C23×C4C22×D6111yescharacteristic
348C2×SD16—618no—
349C2×SD16—2442no—
350C2×SD16—2442no—
351C22×C8—612no—
352C22×C8C22×D6112yes—
353C22⋊Q8—614no—
354C22⋊Q8—1222no—
355C4×Q8—2442no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—42no—
358C2×C4⋊C4—611no—
359C2×C4⋊C4—1221no—
360C2×C4⋊C4—1221no—
361C2×C4⋊C4—1221no—
362C2×C4⋊C4C22×D6112yes—
363C2×C4⋊C4C22×D6111yescharacteristic
364C2×C4⋊C4C22×D6111yescharacteristic
365C2×C22⋊C4—21no—
366C2×C22⋊C4—1221no—
367C2×C22⋊C4—1222no—
368C2×C22⋊C4—1222no—
369C2×C22⋊C4—1222no—
370C2×C22⋊C4—1222no—
371C2×C22⋊C4—2444no—
372C2×C22⋊C4—2444no—
373C2×C42—611no—
374C2×C42—611no—
375C2×C42—611no—
376C2×C42—611no—
377C2×C42—1221no—
378C2×C42—1221no—
379C2×C42—2442no—
380#107—42no—
381#103—2442no—
382#102—618no—
383#102—2442no—
384#102—2442no—
385C22⋊C8—1222no—
386C22⋊C8C4×D6114yes—
387#97—2442no—
388#95—21no—
389#95—1221no—
390#95—1221no—
391#95—1221no—
392#95—1221no—
393C24—1222no—
394C24—1224no—
395C24—2442no—
396C24—2442no—
397C24—2442no—
398C24—2442no—
399C24—2442no—
400C24—2442no—
401C24D4×D6111yescharacteristic
402C2×Q8—612no—
403C2×Q8—1221no—
404C2×Q8—1222no—
405C2×Q8—1222no—
406C2×D4—614no—
407C2×D4—1222no—
408C2×D4—1222no—
409C2×D4—1222no—
410C2×D4—1222no—
411C2×D4—1222no—
412C2×D4—1224no—
413C2×D4—2448no—
414C2×D4—2442no—
415C2×D4—2442no—
416C2×D4—2448no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—611no—
426C22×C4—611no—
427C22×C4—611no—
428C22×C4—611no—
429C22×C4—1222no—
430C22×C4—1222no—
431C22×C4—1221no—
432C22×C4—1221no—
433C22×C4—1222no—
434C22×C4—2441no—
435C22×C4—2441no—
436C22×C4—2441no—
437C22×C4C2×C4×D6112yes—
438C22×C4D4×D6111yescharacteristic
439C22×C4#1175822612yes—
440C22×C4C23×D6111yescharacteristic
441SD16—2448no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—1222no—
445C2×C8—1222no—
446C2×C8—1222no—
447C2×C8C2×C4×D6114yes—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—42no—
453C4⋊C4—612no—
454C4⋊C4—1221no—
455C4⋊C4—1224no—
456C4⋊C4—2442no—
457C4⋊C4—2442no—
458C22⋊C4—42no—
459C22⋊C4—1224no—
460C22⋊C4—2444no—
461C22⋊C4—2444no—
462C22⋊C4—2444no—
463C22⋊C4—2444no—
464C42—1222no—
465C42—1222no—
466C42—2441no—
467C42—2441no—
468C42—2442no—
469C42—2442no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—1224no—
475C23—4884no—
476C23—4884no—
477C23—4884no—
478C23—4884no—
479C23—2448no—
480C23—2448no—
481C23—2442no—
482C23—2442no—
483C23—2442no—
484C23—2442no—
485C23—2442no—
486C23—2442no—
487C23—2448no—
488C23#1451973912yes—
489C23#1451980112yes—
490C23#1451980912yes—
491C23C2×D4×D6111yescharacteristic
492Q8—1224no—
493Q8—2442no—
494D4—1224no—
495D4—1224no—
496D4—2448no—
497D4—2442no—
498D4—2442no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—84no—
513C2×C4—612no—
514C2×C4—1221no—
515C2×C4—1221no—
516C2×C4—1222no—
517C2×C4—1222no—
518C2×C4—1221no—
519C2×C4—1221no—
520C2×C4—1221no—
521C2×C4—1221no—
522C2×C4—1221no—
523C2×C4—1221no—
524C2×C4—1221no—
525C2×C4—1221no—
526C2×C4—1221no—
527C2×C4—2444no—
528C2×C4—2444no—
529C2×C4—2441no—
530C2×C4—2441no—
531C2×C4—2441no—
532C2×C4—2441no—
533C2×C4#1451973912yes—
534C2×C4C22×C4×D6111yescharacteristic
535C2×C4C2×#1175822611yescharacteristic
536C2×C4#1451982212yes—
537C2×C4#1451982312yes—
538C8—24no—
539C8—2442no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—48816no—
547C22—4882no—
548C22—4882no—
549C22—4882no—
550C22—4882no—
551C22—2444no—
552C22—2444no—
553C22—2444no—
554C22#1683986114yes—
555C22C2×#1451973911yescharacteristic
556C22#1684032311yescharacteristic
557C22#1684032711yescharacteristic
558C22AGL(1,ℤ/8)×D6111yescharacteristic
559C22C2×#1451980111yescharacteristic
560C22C2×#1451980911yescharacteristic
561C22#1684056111yescharacteristic
562C4—22no—
563C4—81no—
564C4—82no—
565C4—81no—
566C4—1222no—
567C4—2442no—
568C4—2442no—
569C4—2441no—
570C4—2441no—
571C4—2441no—
572C4#1684023412yes—
573C4#1684035212yes—
574C2—24no—
575C2—4884no—
576C2#2144847611yescharacteristic
577C2C2×#1683986111yescharacteristic
578C2#2145462611yescharacteristic
579C2#2145662411yescharacteristic
580C2#2145663211yescharacteristic
581C2#2145682111yescharacteristic
582C2#2145683511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1852C2
socleC23×C61C22×D4
prime radicalC23×D244C22
cosocleC22×C4×C61C24
hypercenter#1852D61
hypocenterC61#22049
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1852

Complements

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

NH ≅ G/Ntype
2. #21448347C2semidirect
4. #21446698C2semidirect
5. #21446514C2semidirect
6. #21446438C2semidirect
7. #21446175C2semidirect
8. #21446126C2semidirect
10. #21445583C2semidirect
11. C61×#1852C2semidirect
15. C2×#14519742C22semidirect
19. #16839915C4semidirect
20. C2×#14519700C22semidirect
25. C2×#14519687C22semidirect
26. #16839667C22semidirect
31. #16839658C22semidirect
34. C4⋊C4×Dic61C22semidirect
37. #16839524C22semidirect
38. #16839516C22semidirect
40. C2×#107×C61C22semidirect
43. C61×#340C22semidirect
56. C2×#11758190C2×C4semidirect
80. #14519699C2×C4semidirect
291. #1852D61semidirect
312. C2×#107C2×D61semidirect
333. #340C2×D61semidirect
335. C61#22049semidirect

Maximal subgroups

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

subgroupindex
C2×#168399152
#214483472
C2×#168396632
#214466982
#214465142
#214464382
#214461752
#214461262
#214457492
#214455832
C61×#18522
#2204961