Finite Group Databasebeta

← #29695474

Subgroups — #29695474

Order 15616.

1490 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×#16840415C211yesmaximal, characteristic
2C2×#16839827C211yesmaximal, characteristic, complemented
3C2×#16839757C212yesmaximal, complemented
4#21448766C211yesmaximal, characteristic, complemented
5#21446622C211yesmaximal, characteristic, complemented
6#21446239C212yesmaximal, complemented
7#21446128C212yesmaximal, complemented
8#21445820C211yesmaximal, characteristic, complemented
9#21445758C212yesmaximal, complemented
10C4×#14519697C211yesmaximal, characteristic, complemented
11C61×#1981C211yesmaximal, characteristic, complemented
12C22×#11758226C2212yes—
13C22×C8×D61C2212yes—
14#16840415—24no—
15#16840415C418yescomplemented
16C2×#14519762—22no—
17C2×#14519753C2212yes—
18C2×#14519752—21no—
19C2×C4×D244C2211yescharacteristic
20C2×C4×D244C2211yescharacteristic
21C2×#14519735C2212yes—
22C2×C4⋊C4×D61—21no—
23#16839827—24no—
24#16839757—28no—
25C2×#14519701C2212yes—
26C2×#14519701C2212yes—
27C2×#14519699C2212yes—
28C2×#14519697C2211yescharacteristic
29C2×#14519697C2211yescharacteristic
30C2×#14519689C2212yes—
31C2×C4×#92418C2211yescharacteristic
32#16839664—21no—
33#16839654—22no—
34#16839632C2211yescharacteristic
35#16839619C2212yes—
36C4×C4⋊Dic61—21no—
37#16839525C2212yes—
38#16839524C2211yescharacteristic
39#16839524C2211yescharacteristic
40C2×C4⋊Q8×C61C2211yescharacteristic
41C2×C4⋊C8×C61C2211yescharacteristic
42C2×#103×C61C2212yes—
43C2×#103×C61C2212yes—
44C4×C4⋊C4×C61C2211yescharacteristic
45C2×#11758226—24no—
46C2×#11758226—24no—
47C2×#11758226—24no—
48C2×#11758226—42no—
49C2×#11758226—42no—
50C2×#11758226C2×C418yes—
51C22×D244—22no—
52C22×D244D411yescharacteristic
53C22×D244C2311yescharacteristic
54C22×C4×D61—21no—
55C22×C4×D61—22no—
56C22×C4×D61C2312yes—
57C22×#92418C2311yescharacteristic
58C22×#92418C2311yescharacteristic
59C2×C8×D61—24no—
60C2×C8×D61—24no—
61C2×C8×D61—42no—
62#14519762—216no—
63#14519753—24no—
64#14519753C2×C418yes—
65#14519752—28no—
66C4×D244—22no—
67C4×D244C2×C414yes—
68C4×D244C2×C414yes—
69C4×D244D414yes—
70#14519735—24no—
71#14519735—42no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61D411yescharacteristic
75C2×C4⋊Dic61C2312yes—
76C2×C4⋊Dic61C2311yescharacteristic
77C2×#11758196—22no—
78C4⋊C4×D61—28no—
79C2×#11758198—22no—
80C2×#11758198—21no—
81C2×#11758198—22no—
82C2×#11758198—22no—
83C2×#11758198D412yes—
84C2×C4×Dic61—22no—
85#14519701—24no—
86#14519701—42no—
87#14519699—24no—
88#14519697—22no—
89#14519697D414yescomplemented
90#14519689—24no—
91C4×#92418—22no—
92#14519684—22no—
93C22×Q8×C61C2312yes—
94C22×C8×C61C2312yes—
95C4⋊Q8×C61—22no—
96C4⋊Q8×C61C2×C414yes—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61C2311yescharacteristic
100C2×C4⋊C4×C61C2311yescharacteristic
101C2×C42×C61—21no—
102C2×C42×C61C2311yescharacteristic
103C4⋊C8×C61—22no—
104#103×C61—24no—
105#103×C61—42no—
106#95×C61—21no—
107C23×D61—21no—
108C23×D61C2×D412yes—
109C2×Q8×C61—22no—
110C2×Q8×C61—22no—
111C2×Q8×C61—22no—
112C2×Q8×C61—22no—
113C2×Q8×C61—22no—
114C2×Q8×C61C22×C414yes—
115C22×C4×C61—22no—
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2411yescharacteristic
121C22×Dic61—21no—
122C22×Dic61—22no—
123C22×Dic61C2×D412yes—
124#11758226—216no—
125#11758226—48no—
126#11758226—48no—
127#11758226—48no—
128C2×D244—22no—
129C2×D244—22no—
130C2×D244—21no—
131C2×D244—21no—
132C2×D244—21no—
133C2×D244—28no—
134C2×D244—44no—
135C2×D244—44no—
136C2×D244C22⋊C414yes—
137C2×D244C22×C412yes—
138C2×D244C2×D412yes—
139C2×D244#4112yes—
140C2×C4×D61—22no—
141C2×C4×D61—22no—
142C2×C4×D61—22no—
143C2×C4×D61—22no—
144C2×C4×D61—24no—
145C2×C4×D61—28no—
146C2×C4×D61—44no—
147C2×C4×D61—44no—
148C2×C4×D61—42no—
149C2×C4×D61—42no—
150C2×C4×D61C22×C414yes—
151C2×C4×D61C2×D414yes—
152C2×C8×C61—22no—
153C2×C8×C61—22no—
154C2×C8×C61—22no—
155C4⋊C4×C61—24no—
156C4⋊C4×C61—24no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61C2×D412yes—
161C42×C61—42no—
162C42×C61C22×C412yes—
163C42×C61C2×D412yes—
164#11758198—28no—
165#11758198—24no—
166#11758198—44no—
167#11758198—82no—
168#11758198C22⋊C418yes—
169C4⋊Dic61—21no—
170C4⋊Dic61—22no—
171C4⋊Dic61—22no—
172C4⋊Dic61—42no—
173C4⋊Dic61—81no—
174C4⋊Dic61C22⋊C414yes—
175C4⋊Dic61C22×C412yes—
176#11758196—28no—
177C4×Dic61—44no—
178C2×#92418—21no—
179C2×#92418—21no—
180C2×#92418—21no—
181C2×#92418—21no—
182C2×#92418—21no—
183C2×#92418C2×D412yes—
184C8×D61—48no—
185C23×C61C22×D411yescharacteristic
186C22×D61—22no—
187C22×D61—22no—
188C22×D61—22no—
189C22×D61—22no—
190C22×D61—22no—
191C22×D61—22no—
192C22×D61—22no—
193C22×D61—22no—
194C22×D61—22no—
195C22×D61—41no—
196C22×D61—41no—
197C22×D61—41no—
198C22×D61—41no—
199C22×D61C2×C22⋊C414yes—
200Q8×C61—24no—
201Q8×C61—24no—
202Q8×C61—42no—
203Q8×C61—42no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—24no—
207C2×C4×C61—22no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61C2×C22⋊C412yes—
216C2×C4×C61C2×C22⋊C412yes—
217C2×C4×C61#13512yes—
218C2×C4×C61#13512yes—
219C2×C4×C61C23×C411yescharacteristic
220C2×C4×C61C22×D411yescharacteristic
221C2×C4×C61C22×D411yescharacteristic
222C2×C4×C61C2×#4111yescharacteristic
223C2×C4×C61H(5,𝔽2)11yescharacteristic
224C2×C4×C61H(5,𝔽2)11yescharacteristic
225C2×Dic61—22no—
226C2×Dic61—22no—
227C2×Dic61—22no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—42no—
231C2×Dic61—42no—
232C2×Dic61—42no—
233C2×Dic61C2×C22⋊C414yes—
234D244—22no—
235D244—22no—
236D244—22no—
237D244—28no—
238D244—416no—
239D244C4×D414yes—
240C4×D61—24no—
241C4×D61—24no—
242C4×D61—24no—
243C4×D61—24no—
244C4×D61—416no—
245C4×D61—48no—
246C8×C61—42no—
247#92418—22no—
248#92418—41no—
249#22417—611nomaximal, Sylow
250C22×C61C22×C22⋊C411yescharacteristic
251C22×C61D4211yescharacteristic
252C22×C61#54611yescharacteristic
253C22×C61C2×#13511yescharacteristic
254C22×C61C2×#13511yescharacteristic
255C22×C61#57711yescharacteristic
256C22×C61#57811yescharacteristic
257C2×D61—24no—
258C2×D61—24no—
259C2×D61—24no—
260C2×D61—24no—
261C2×D61—42no—
262C2×D61—42no—
263C2×D61—42no—
264C2×D61—42no—
265C2×D61—42no—
266C2×D61—42no—
267C2×D61—42no—
268C2×D61—42no—
269C2×D61—42no—
270C2×D61—42no—
271C4×C61—22no—
272C4×C61—44no—
273C4×C61—42no—
274C4×C61—81no—
275C4×C61#41614yescomplemented
276C4×C61C2×C4×D411yescharacteristic
277C4×C61#51811yescharacteristic
278C4×C61D4211yescharacteristic
279C4×C61#54611yescharacteristic
280Dic61—24no—
281Dic61—42no—
282Dic61—82no—
283C2×#547—611no—
284C2×C8×D4—611no—
285C2×#416—612no—
286#2354—611no—
287#1981D6111yescharacteristic, complemented
288#1956—611no—
289#1892—612no—
290#1873—612no—
291#1802—611no—
292#1788—612no—
293C4×#102—611no—
294C2×C61D4×C22⋊C411yescharacteristic
295C2×C61C2×#41611yescharacteristic
296C2×C61#289811yescharacteristic
297C2×C61#328311yescharacteristic
298C2×C61#328411yescharacteristic
299C2×C61#330211yescharacteristic
300C2×C61#330311yescharacteristic
301D61—44no—
302D61—44no—
303C22×#41—612no—
304C23×C8—612no—
305#547—618no—
306#547—1224no—
307C2×C4⋊Q8C2×D6111yescharacteristic
308C2×#123—1222no—
309C2×C22⋊Q8—612no—
310C2×#121—1221no—
311C2×C4×D4—611no—
312C2×C4×D4—611no—
313C2×#117—612no—
314C22×C4⋊C4—1221no—
315C8×D4—1224no—
316C2×C4⋊C8C2×D6111yescharacteristic
317#416—1228no—
318C2×#103—612no—
319C2×#103—612no—
320C2×#103C2×D6112yes—
321C2×#103C2×D6112yes—
322C2×#102—611no—
323C2×#102—612no—
324C2×#102—611no—
325C2×C22⋊C8—612no—
326C2×C4×C8—611no—
327#390—1221no—
328#388—1222no—
329#384—611no—
330#380—612no—
331C4×C4⋊C4—1221no—
332C4×C4⋊C4C2×D6111yescharacteristic
333#340—612no—
334#340—611no—
335#340—611no—
336C61#2241711yescharacteristic, complemented, Sylow
337C2×#41—618no—
338C2×#41—1224no—
339C2×#41—1224no—
340C2×#41—1224no—
341C2×#41—2442no—
342C2×#41—2442no—
343C22×Q8C22×D6112yes—
344C22×D4—611no—
345C22×D4—611no—
346C22×D4—1222no—
347C23×C4—612no—
348C23×C4—1222no—
349C23×C4—1221no—
350C22×C8—611no—
351C22×C8—611no—
352C22×C8—1224no—
353C22×C8—1224no—
354C22×C8—2442no—
355C22×C8C22×D6112yes—
356C4⋊Q8—22no—
357C4⋊Q8C4×D6114yes—
358#123—12216no—
359C22⋊Q8—618no—
360C22⋊Q8—1224no—
361#121—1228no—
362C4×D4—614no—
363C4×D4—614no—
364C4×D4—614no—
365C4×D4—1222no—
366#117—1224no—
367#117—2442no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—611no—
371C2×C4⋊C4—611no—
372C2×C4⋊C4—612no—
373C2×C4⋊C4—1228no—
374C2×C4⋊C4—1222no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4C22×D6111yescharacteristic
378C2×C4⋊C4C22×D6111yescharacteristic
379C2×C22⋊C4—612no—
380C2×C22⋊C4—1222no—
381C2×C22⋊C4—1221no—
382C2×C22⋊C4—1222no—
383C2×C22⋊C4—1222no—
384C2×C42—21no—
385C2×C42—1222no—
386C2×C42C22×D6111yescharacteristic
387C4⋊C8—22no—
388#103—24no—
389#103—42no—
390#103—1224no—
391#103—2442no—
392#102—614no—
393#102—1222no—
394#102—1224no—
395C22⋊C8—1224no—
396C4×C8—1222no—
397#95—21no—
398#95—1222no—
399C24—612no—
400C24—1221no—
401#41—12216no—
402#41—2448no—
403#41—2448no—
404#41—2448no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8C2×C4×D6114yes—
411C2×D4—612no—
412C2×D4—612no—
413C2×D4—612no—
414C2×D4—614no—
415C2×D4—1228no—
416C2×D4—1221no—
417C2×D4—1221no—
418C2×D4—1221no—
419C2×D4—1222no—
420C2×D4—1222no—
421C2×D4—2444no—
422C2×D4—2444no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—612no—
427C22×C4—614no—
428C22×C4—614no—
429C22×C4—1228no—
430C22×C4—1222no—
431C22×C4—1222no—
432C22×C4—1222no—
433C22×C4—1222no—
434C22×C4—1221no—
435C22×C4—1224no—
436C22×C4—1222no—
437C22×C4—2444no—
438C22×C4—2444no—
439C22×C4—2442no—
440C22×C4—2442no—
441C22×C4D4×D6111yescharacteristic
442C22×C4D4×D6111yescharacteristic
443C22×C4C23×D6111yescharacteristic
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—612no—
448C2×C8—1221no—
449C2×C8—1221no—
450C2×C8—1221no—
451C2×C8—1221no—
452C2×C8—1221no—
453C2×C8—2448no—
454C4⋊C4—21no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—614no—
460C4⋊C4—612no—
461C4⋊C4—1228no—
462C4⋊C4—1221no—
463C4⋊C4—1222no—
464C4⋊C4—1222no—
465C4⋊C4—4881no—
466C4⋊C4—2442no—
467C4⋊C4D4×D6112yes—
468C22⋊C4—618no—
469C22⋊C4—1228no—
470C22⋊C4—1224no—
471C22⋊C4—4882no—
472C22⋊C4—2444no—
473C42—42no—
474C42—2444no—
475C42C2×C4×D6112yes—
476C42D4×D6112yes—
477C23—614no—
478C23—1222no—
479C23—1222no—
480C23—1222no—
481C23—1222no—
482C23—1222no—
483C23—1222no—
484C23—1222no—
485C23—1222no—
486C23—1222no—
487C23—2441no—
488C23—2441no—
489C23—2441no—
490C23—2441no—
491C23C2×D4×D6111yescharacteristic
492Q8—24no—
493Q8—24no—
494Q8—42no—
495Q8—42no—
496D4—614no—
497D4—1222no—
498D4—1228no—
499D4—1222no—
500D4—1222no—
501D4—24416no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—614no—
514C2×C4—1222no—
515C2×C4—1222no—
516C2×C4—1224no—
517C2×C4—1224no—
518C2×C4—1224no—
519C2×C4—1224no—
520C2×C4—1222no—
521C2×C4—24416no—
522C2×C4—2442no—
523C2×C4—2442no—
524C2×C4—2442no—
525C2×C4—2441no—
526C2×C4—2441no—
527C2×C4—2448no—
528C2×C4C22⋊C4×D6112yes—
529C2×C4C22⋊C4×D6112yes—
530C2×C4#1451979812yes—
531C2×C4#1451980012yes—
532C2×C4C22×C4×D6111yescharacteristic
533C2×C4C2×D4×D6111yescharacteristic
534C2×C4C2×D4×D6111yescharacteristic
535C2×C4#41×D6111yescharacteristic
536C2×C4#1451982211yescharacteristic
537C2×C4#1451982311yescharacteristic
538C8—42no—
539C8—1222no—
540C8—2441no—
541C22—1224no—
542C22—1224no—
543C22—1224no—
544C22—1224no—
545C22—2442no—
546C22—2442no—
547C22—2442no—
548C22—2442no—
549C22—2442no—
550C22—2442no—
551C22—2442no—
552C22—2442no—
553C22—2442no—
554C22—2442no—
555C22C2×C22⋊C4×D6111yescharacteristic
556C22#1684039611yescharacteristic
557C22#1684040911yescharacteristic
558C22C2×#1451979811yescharacteristic
559C22C2×#1451980011yescharacteristic
560C22#1684057111yescharacteristic
561C22#1684057811yescharacteristic
562C4—22no—
563C4—42no—
564C4—44no—
565C4—81no—
566C4—1224no—
567C4—4882no—
568C4—2442no—
569C4#1683975714yescomplemented
570C4C4×D4×D6111yescharacteristic
571C4#1684023411yescharacteristic
572C4#1684039711yescharacteristic
573C4#1684040511yescharacteristic
574C2—2444no—
575C2—2444no—
576C2#2144864011yescharacteristic
577C2C2×#1683975711yescharacteristic
578C2#2145435111yescharacteristic
579C2#2145753611yescharacteristic
580C2#2145754411yescharacteristic
581C2#2145773911yescharacteristic
582C2#2145775011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1981C2
socleC23×C61C22×D4
prime radicalC2×#16840415C2
cosocleC22×C4×C61C24
hypercenter#1981D61
hypocenterC61#22417
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1981

Complements

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

NH ≅ G/Ntype
2. C2×#16839827C2semidirect
3. C2×#16839757C2semidirect
4. #21448766C2semidirect
5. #21446622C2semidirect
6. #21446239C2semidirect
7. #21446128C2semidirect
8. #21445820C2semidirect
9. #21445758C2semidirect
10. C4×#14519697C2semidirect
11. C61×#1981C2semidirect
15. #16840415C4semidirect
89. #14519697D4semidirect
275. C4×C61#416semidirect
287. #1981D61semidirect
336. C61#22417semidirect
569. C4#16839757semidirect

Maximal subgroups

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

subgroupindex
C2×#168404152
C2×#168398272
C2×#168397572
#214487662
#214466222
#214462392
#214461282
#214458202
#214457582
C4×#145196972
C61×#19812
#2241761