Finite Group Databasebeta

← #29695468

Subgroups — #29695468

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×#16840416C211yesmaximal, characteristic
2C2×#16839827C211yesmaximal, characteristic, complemented
3C2×#16839755C212yesmaximal, complemented
4#21448770C211yesmaximal, characteristic, complemented
5#21446741C211yesmaximal, characteristic, complemented
6#21446620C211yesmaximal, characteristic, complemented
7#21446241C212yesmaximal, complemented
8#21446130C212yesmaximal, complemented
9#21445756C212yesmaximal, complemented
10C4×#14519698C211yesmaximal, characteristic, complemented
11C61×#1802C211yesmaximal, characteristic, complemented
12C22×#11758224C2212yes—
13C22×C8×D61C2212yes—
14#16840416—24no—
15#16840416C418yescomplemented
16C2×#14519777C2211yescharacteristic
17C2×#14519761—22no—
18C2×#14519757C2212yes—
19C2×#14519751—21no—
20C2×C4×D244C2211yescharacteristic
21C2×#14519735C2212yes—
22C22×C4⋊Dic61—21no—
23#16839827—24no—
24#16839755—28no—
25C2×#14519702C2212yes—
26C2×#14519702C2212yes—
27C2×#14519700C2212yes—
28C2×#14519700C2212yes—
29C2×#14519698C2211yescharacteristic
30C2×#14519698C2211yescharacteristic
31C2×#14519689C2212yes—
32C2×C4×#92418C2211yescharacteristic
33#16839663—21no—
34#16839652—22no—
35#16839619C2212yes—
36C4⋊C4×Dic61—21no—
37C4×C4⋊Dic61C2211yescharacteristic
38#16839524C2211yescharacteristic
39#16839524C2211yescharacteristic
40C2×C4×D4×C61C2211yescharacteristic
41C2×C4⋊C8×C61C2211yescharacteristic
42C2×#102×C61C2212yes—
43C61×#384C2211yescharacteristic
44C61×#340C2212yes—
45C2×#11758224—24no—
46C2×#11758224—24no—
47C2×#11758224—24no—
48C2×#11758224—42no—
49C2×#11758224—42no—
50C2×#11758224C2×C418yes—
51C22×Dic122C2312yes—
52C22×D244D411yescharacteristic
53C22×C22⋊D61—22no—
54C23×Dic61—22no—
55C22×C4×D61C2312yes—
56C22×#92418C2311yescharacteristic
57C22×#92418C2311yescharacteristic
58C2×C8×D61—24no—
59C2×C8×D61—24no—
60C2×C8×D61—42no—
61#14519777—22no—
62#14519777C2×C414yes—
63#14519761—216no—
64#14519757—24no—
65#14519757C2×C418yes—
66#14519751—28no—
67C4×D244C2×C414yes—
68C4×D244D414yes—
69#14519735—24no—
70#14519735—42no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—22no—
74C2×C4⋊Dic61—28no—
75C2×C4⋊Dic61D411yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×#11758198—22no—
79C2×#11758198—22no—
80C2×#11758198—22no—
81C2×#11758198D412yes—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—22no—
84#14519702—24no—
85#14519702—42no—
86#14519700—24no—
87#14519700—42no—
88#14519698—22no—
89#14519698D414yescomplemented
90#14519689—24no—
91C4×#92418—22no—
92#14519684—21no—
93#14519684—22no—
94C22×D4×C61C2311yescharacteristic
95C23×C4×C61—21no—
96C22×C8×C61C2312yes—
97C4×D4×C61—22no—
98C4×D4×C61C2×C414yes—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61C2312yes—
102C2×C4⋊C4×C61C2311yescharacteristic
103C2×C22⋊C4×C61—21no—
104C2×C42×C61C2311yescharacteristic
105C4⋊C8×C61—22no—
106#102×C61—24no—
107C24×C61—21no—
108C23×D61C2×D412yes—
109C2×D4×C61—21no—
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61C22×C412yes—
113C2×D4×C61C2×D412yes—
114C2×D4×C61#4112yes—
115C22×C4×C61—21no—
116C22×C4×C61—24no—
117C22×C4×C61—22no—
118C22×C4×C61—42no—
119C22×C4×C61—42no—
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2411yescharacteristic
123C2×C22⋊D61—28no—
124C2×C22⋊D61—44no—
125C2×C22⋊D61—44no—
126C22×Dic61—22no—
127C22×Dic61—21no—
128C22×Dic61—28no—
129C22×Dic61—21no—
130C22×Dic61—44no—
131C22×Dic61—44no—
132C22×Dic61C2×D412yes—
133#11758224—216no—
134#11758224—48no—
135#11758224—48no—
136#11758224—48no—
137C2×D244—22no—
138C2×D244—22no—
139C2×D244C22⋊C414yes—
140C2×C4×D61—22no—
141C2×C4×D61—22no—
142C2×C4×D61—22no—
143C2×C4×D61—22no—
144C2×C4×D61C22×C414yes—
145C2×C4×D61C2×D414yes—
146C2×Dic122—22no—
147C2×Dic122—22no—
148C2×Dic122—22no—
149C2×Dic122—22no—
150C2×Dic122—22no—
151C2×Dic122C22×C414yes—
152C2×C8×C61—22no—
153C2×C8×C61—22no—
154C2×C8×C61—22no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—42no—
159C4⋊C4×C61—81no—
160C4⋊C4×C61C22×C412yes—
161C22⋊C4×C61—24no—
162C42×C61C22×C412yes—
163C42×C61C2×D412yes—
164#11758198—28no—
165#11758198—44no—
166#11758198—82no—
167#11758198C22⋊C418yes—
168C4⋊Dic61—24no—
169C4⋊Dic61—24no—
170C4⋊Dic61—21no—
171C4⋊Dic61—28no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61C22⋊C414yes—
175C4⋊Dic61C2×D412yes—
176C4×Dic61—44no—
177C4×Dic61—42no—
178C2×#92418—21no—
179C2×#92418—21no—
180C2×#92418—21no—
181C2×#92418—21no—
182C2×#92418—21no—
183C2×#92418C2×D412yes—
184C8×D61—48no—
185C23×C61—22no—
186C23×C61—22no—
187C23×C61—22no—
188C23×C61—41no—
189C23×C61—41no—
190C23×C61—41no—
191C23×C61—41no—
192C23×C61C22×D411yescharacteristic
193C22×D61—22no—
194C22×D61—22no—
195C22×D61—22no—
196C22×D61—22no—
197C22×D61—22no—
198C22×D61—22no—
199C22×D61C2×C22⋊C414yes—
200D4×C61—22no—
201D4×C61—22no—
202D4×C61—22no—
203D4×C61C4×D414yes—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—22no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61—48no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61C2×C22⋊C412yes—
214C2×C4×C61C2×C22⋊C412yes—
215C2×C4×C61#13512yes—
216C2×C4×C61#13512yes—
217C2×C4×C61C23×C411yescharacteristic
218C2×C4×C61C22×D411yescharacteristic
219C2×C4×C61C22×D411yescharacteristic
220C2×C4×C61C2×#4111yescharacteristic
221C2×C4×C61H(5,𝔽2)11yescharacteristic
222C2×C4×C61H(5,𝔽2)11yescharacteristic
223C22⋊D61—416no—
224C2×Dic61—22no—
225C2×Dic61—22no—
226C2×Dic61—24no—
227C2×Dic61—22no—
228C2×Dic61—416no—
229C2×Dic61—42no—
230C2×Dic61—42no—
231C2×Dic61—41no—
232C2×Dic61—41no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61C2×C22⋊C414yes—
237D244—28no—
238C4×D61—24no—
239C4×D61—24no—
240C4×D61—24no—
241C4×D61—24no—
242Dic122—24no—
243Dic122—24no—
244Dic122—42no—
245Dic122—42no—
246C8×C61—42no—
247#92418—22no—
248#92418—41no—
249#22417—611nomaximal, Sylow
250C22×C61—24no—
251C22×C61—42no—
252C22×C61—42no—
253C22×C61—42no—
254C22×C61—42no—
255C22×C61—42no—
256C22×C61—42no—
257C22×C61C22×C22⋊C411yescharacteristic
258C22×C61D4211yescharacteristic
259C22×C61#54611yescharacteristic
260C22×C61C2×#13511yescharacteristic
261C22×C61C2×#13511yescharacteristic
262C22×C61#57711yescharacteristic
263C22×C61#57811yescharacteristic
264C2×D61—24no—
265C2×D61—24no—
266C2×D61—24no—
267C2×D61—42no—
268C2×D61—42no—
269C2×D61—42no—
270C2×D61—42no—
271C4×C61—22no—
272C4×C61—42no—
273C4×C61—82no—
274C4×C61#41614yescomplemented
275C4×C61C2×C4×D411yescharacteristic
276C4×C61#51811yescharacteristic
277C4×C61D4211yescharacteristic
278C4×C61#54611yescharacteristic
279Dic61—24no—
280Dic61—44no—
281Dic61—42no—
282Dic61—81no—
283C2×#547—611no—
284C2×C8×D4—611no—
285C2×#416—612no—
286#2354—611no—
287#1981—611no—
288#1956—611no—
289#1892—612no—
290#1873—612no—
291#1802D6111yescharacteristic, complemented
292#1788—612no—
293C4×#102—611no—
294C2×C61—44no—
295C2×C61D4×C22⋊C411yescharacteristic
296C2×C61C2×#41611yescharacteristic
297C2×C61#289811yescharacteristic
298C2×C61#328311yescharacteristic
299C2×C61#328411yescharacteristic
300C2×C61#330211yescharacteristic
301C2×C61#330311yescharacteristic
302D61—44no—
303C22×#41—612no—
304C23×C8—612no—
305#547—618no—
306#547—1224no—
307C2×C4⋊Q8—611no—
308C2×#123—1222no—
309C2×C22⋊Q8—612no—
310C2×#121—1221no—
311C2×C4×D4—611no—
312C2×C4×D4C2×D6111yescharacteristic
313C2×#117—612no—
314C22×C4⋊C4—1221no—
315C8×D4—1224no—
316C2×C4⋊C8C2×D6111yescharacteristic
317#416—1228no—
318C2×#103—612no—
319C2×#103—612no—
320C2×#103—612no—
321C2×#103—612no—
322C2×#102—611no—
323C2×#102—611no—
324C2×#102C2×D6112yes—
325C2×C22⋊C8—612no—
326C2×C4×C8—611no—
327#390—1221no—
328#388—1222no—
329#384C2×D6111yescharacteristic
330#380—612no—
331C4×C4⋊C4—611no—
332C4×C4⋊C4—1221no—
333#340—611no—
334#340—611no—
335#340C2×D6112yes—
336C61#2241711yescharacteristic, complemented, Sylow
337C2×#41—618no—
338C2×#41—1224no—
339C2×#41—1224no—
340C2×#41—1224no—
341C2×#41—2442no—
342C2×#41—2442no—
343C22×Q8—612no—
344C22×D4—611no—
345C22×D4—1222no—
346C22×D4C22×D6111yescharacteristic
347C23×C4—21no—
348C23×C4—612no—
349C23×C4—1222no—
350C22×C8—611no—
351C22×C8—611no—
352C22×C8—1224no—
353C22×C8—1224no—
354C22×C8—2442no—
355C22×C8C22×D6112yes—
356C4⋊Q8—614no—
357C4⋊Q8—1222no—
358#123—12216no—
359C22⋊Q8—618no—
360C22⋊Q8—1224no—
361#121—1228no—
362C4×D4—22no—
363C4×D4—614no—
364C4×D4—614no—
365C4×D4C4×D6114yes—
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—611no—
373C2×C4⋊C4—1228no—
374C2×C4⋊C4—1222no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4C22×D6112yes—
378C2×C4⋊C4C22×D6111yescharacteristic
379C2×C22⋊C4—21no—
380C2×C22⋊C4—612no—
381C2×C22⋊C4—1222no—
382C2×C22⋊C4—1222no—
383C2×C22⋊C4—1222no—
384C2×C42—1222no—
385C2×C42—1221no—
386C2×C42C22×D6111yescharacteristic
387C4⋊C8—22no—
388#103—1224no—
389#103—1224no—
390#103—2442no—
391#103—2442no—
392#102—24no—
393#102—614no—
394#102—1222no—
395C22⋊C8—1224no—
396C4×C8—1222no—
397#95—1222no—
398#95—1221no—
399C24—21no—
400C24—612no—
401#41—12216no—
402#41—2448no—
403#41—2448no—
404#41—2448no—
405C2×Q8—614no—
406C2×Q8—1222no—
407C2×Q8—1222no—
408C2×Q8—1222no—
409C2×Q8—1222no—
410C2×Q8—1222no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—614no—
415C2×D4—1228no—
416C2×D4—1222no—
417C2×D4—1222no—
418C2×D4—2444no—
419C2×D4—2444no—
420C2×D4C2×C4×D6112yes—
421C2×D4D4×D6112yes—
422C2×D4#1175822612yes—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—612no—
429C22×C4—614no—
430C22×C4—614no—
431C22×C4—1228no—
432C22×C4—1222no—
433C22×C4—1222no—
434C22×C4—1222no—
435C22×C4—1222no—
436C22×C4—1221no—
437C22×C4—1222no—
438C22×C4—1221no—
439C22×C4—2444no—
440C22×C4—2444no—
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—22no—
456C4⋊C4—22no—
457C4⋊C4—42no—
458C4⋊C4—81no—
459C4⋊C4—614no—
460C4⋊C4—612no—
461C4⋊C4—1228no—
462C4⋊C4—1221no—
463C4⋊C4—1224no—
464C4⋊C4—1224no—
465C4⋊C4—1221no—
466C4⋊C4—1221no—
467C4⋊C4C2×C4×D6112yes—
468C22⋊C4—24no—
469C22⋊C4—618no—
470C22⋊C4—1228no—
471C22⋊C4—4882no—
472C22⋊C4—2444no—
473C42—2444no—
474C42—2442no—
475C42C2×C4×D6112yes—
476C42D4×D6112yes—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—614no—
485C23—1222no—
486C23—1222no—
487C23—1222no—
488C23—1222no—
489C23—1222no—
490C23—1222no—
491C23C2×D4×D6111yescharacteristic
492Q8—1224no—
493Q8—1224no—
494Q8—2442no—
495Q8—2442no—
496D4—22no—
497D4—22no—
498D4—22no—
499D4—1228no—
500D4—24416no—
501D4#1451973914yes—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—48no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—614no—
512C2×C4—1222no—
513C2×C4—1222no—
514C2×C4—1224no—
515C2×C4—1224no—
516C2×C4—1224no—
517C2×C4—1224no—
518C2×C4—1224no—
519C2×C4—1222no—
520C2×C4—24416no—
521C2×C4—2441no—
522C2×C4—2441no—
523C2×C4—2441no—
524C2×C4—2441no—
525C2×C4—2441no—
526C2×C4—2442no—
527C2×C4—2442no—
528C2×C4C22⋊C4×D6112yes—
529C2×C4C22⋊C4×D6112yes—
530C2×C4#1451979612yes—
531C2×C4#1451979812yes—
532C2×C4C22×C4×D6111yescharacteristic
533C2×C4C2×D4×D6111yescharacteristic
534C2×C4C2×D4×D6111yescharacteristic
535C2×C4C2×#1175822611yescharacteristic
536C2×C4#1451982311yescharacteristic
537C2×C4#1451982311yescharacteristic
538C8—42no—
539C8—1222no—
540C8—2441no—
541C22—24no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—1224no—
549C22—1224no—
550C22—1224no—
551C22—2442no—
552C22—2442no—
553C22—2442no—
554C22—2442no—
555C22C2×C22⋊C4×D6111yescharacteristic
556C22#1684039611yescharacteristic
557C22#1684040311yescharacteristic
558C22C2×#1451979611yescharacteristic
559C22C2×#1451979811yescharacteristic
560C22#1684057211yescharacteristic
561C22#1684057811yescharacteristic
562C4—22no—
563C4—42no—
564C4—82no—
565C4—1224no—
566C4—4881no—
567C4—2442no—
568C4—2444no—
569C4#1683975514yescomplemented
570C4C2×#1451973911yescharacteristic
571C4#1684023611yescharacteristic
572C4#1684039611yescharacteristic
573C4#1684040311yescharacteristic
574C2—44no—
575C2—2444no—
576C2#2144863911yescharacteristic
577C2C2×#1683975511yescharacteristic
578C2#2145435011yescharacteristic
579C2#2145753611yescharacteristic
580C2#2145755211yescharacteristic
581C2#2145773311yescharacteristic
582C2#2145774411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1802C2
socleC23×C61C22×D4
prime radicalC2×#16840416C2
cosocleC22×C4×C61C24
hypercenter#1802D61
hypocenterC61#22417
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
2. C2×#16839827C2semidirect
3. C2×#16839755C2semidirect
4. #21448770C2semidirect
5. #21446741C2semidirect
6. #21446620C2semidirect
7. #21446241C2semidirect
8. #21446130C2semidirect
9. #21445756C2semidirect
10. C4×#14519698C2semidirect
11. C61×#1802C2semidirect
15. #16840416C4semidirect
89. #14519698D4semidirect
274. C4×C61#416semidirect
291. #1802D61semidirect
336. C61#22417semidirect
569. C4#16839755semidirect

Maximal subgroups

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

subgroupindex
C2×#168404162
C2×#168398272
C2×#168397552
#214487702
#214467412
#214466202
#214462412
#214461302
#214457562
C4×#145196982
C61×#18022
#2241761