Finite Group Databasebeta

← #29707303

Subgroups — #29707303

Order 15616.

1294 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840317C211yesmaximal, characteristic, complemented
2C2×#16839755C212yesmaximal
3#21448306C211yesmaximal, characteristic
4#21446969C212yesmaximal, complemented
5#21446963C212yesmaximal, complemented
6#21446961C211yesmaximal, characteristic, complemented
7#21446957C211yesmaximal, characteristic, complemented
8#21446955C211yesmaximal, characteristic, complemented
9#21445534C212yesmaximal, complemented
10#336×D61C211yesmaximal, characteristic, complemented
11C61×#2023C211yesmaximal, characteristic, complemented
12C22×#11758224C2211yescharacteristic
13C22×C8×D61C2212yes—
14#16840317—28no—
15C2×#14519777C2211yescharacteristic
16C2×#14519768C2212yes—
17C2×#14519757—21no—
18C2×#14519753—21no—
19C2×Q8×Dic61—21no—
20C2×#14519742—21no—
21C2×#14519735C2211yescharacteristic
22C2×#14519735C2212yes—
23C2×C42×D61C2211yescharacteristic
24#16839755—28no—
25C2×#14519702C2212yes—
26C2×#14519700C2212yes—
27C2×#14519698C2212yes—
28#16839668C2211yescharacteristic
29#16839667C2211yescharacteristic
30#16839665C2211yescharacteristic
31#16839638—21no—
32#16839638—21no—
33C22⋊C4×Dic61—21no—
34C22⋊C4×Dic61—21no—
35#16839525C2212yes—
36#16839524C2212yes—
37#16839524C2212yes—
38#16839517C2211yescharacteristic
39#16839516C2212yes—
40C2×C4⋊Q8×C61C2211yescharacteristic
41C2×#102×C61C2212yes—
42C61×#390C2211yescharacteristic
43C61×#340C2212yes—
44C61×#336C2211yescharacteristic
45C2×#11758224—24no—
46C2×#11758224—24no—
47C2×#11758224—22no—
48C2×#11758224Q814yescomplemented
49C22×Dic122—21no—
50C22×Dic122C2311yescharacteristic
51C22×C22⋊D61—21no—
52C23×Dic61—21no—
53C22×C4×D61D411yescharacteristic
54C22×C4×D61D411yescharacteristic
55C22×C4×D61C2311yescharacteristic
56C22×#92418C2312yes—
57C2×C8×D61—24no—
58C2×C8×D61—24no—
59C2×C8×D61—24no—
60#14519777—24no—
61#14519768—24no—
62#14519768—24no—
63#14519757—28no—
64#14519753—28no—
65Q8×Dic61—28no—
66#14519742—28no—
67#14519735—24no—
68#14519735—24no—
69#14519735D418yes—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—22no—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2312yes—
74C2×#11758196—22no—
75C2×C22⋊Dic61—21no—
76C2×C22⋊Dic61—21no—
77C2×#11758198—22no—
78C2×#11758198—21no—
79C2×C4×Dic61—22no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61D411yescharacteristic
82C2×C4×Dic61D411yescharacteristic
83C2×C4×Dic61C2311yescharacteristic
84C42×D61—24no—
85#14519702—24no—
86#14519700—24no—
87#14519698—24no—
88#14519684—21no—
89#14519684—21no—
90#14519684—21no—
91#14519684—21no—
92C22×Q8×C61—21no—
93C22×D4×C61C2311yescharacteristic
94C22×C8×C61C2312yes—
95C4⋊Q8×C61—24no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61C2311yescharacteristic
98C2×C4⋊C4×C61C2312yes—
99C2×C22⋊C4×C61—21no—
100C2×C22⋊C4×C61—21no—
101C2×C42×C61C2311yescharacteristic
102#102×C61—24no—
103C24×C61—21no—
104C23×D61C2×D411yescharacteristic
105C2×Q8×C61—24no—
106C2×Q8×C61—42no—
107C2×Q8×C61—42no—
108C2×D4×C61—22no—
109C2×D4×C61—22no—
110C2×D4×C61—21no—
111C2×D4×C61C2×Q812yes—
112C22×C4×C61—22no—
113C22×C4×C61—21no—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2411yescharacteristic
117C2×C22⋊D61—24no—
118C2×C22⋊D61—42no—
119C2×C22⋊D61—42no—
120C22×Dic61—21no—
121C22×Dic61—22no—
122C22×Dic61—24no—
123C22×Dic61—21no—
124C22×Dic61—42no—
125C22×Dic61—42no—
126C22×Dic61C2×D411yescharacteristic
127C22×Dic61C2×D411yescharacteristic
128C22×Dic61C2×D411yescharacteristic
129#11758224—216no—
130#11758224—48no—
131C2×C4×D61—22no—
132C2×C4×D61—22no—
133C2×C4×D61—22no—
134C2×C4×D61—22no—
135C2×C4×D61—41no—
136C2×C4×D61—41no—
137C2×C4×D61C2×D414yes—
138C2×C4×D61C2×Q812yes—
139C2×C4×D61#4114yes—
140C2×C4×D61#4112yes—
141C2×Dic122—22no—
142C2×Dic122—22no—
143C2×Dic122—24no—
144C2×Dic122—21no—
145C2×Dic122—42no—
146C2×Dic122—42no—
147C2×Dic122C2×Q812yes—
148C2×C8×C61—22no—
149C2×C8×C61—22no—
150C2×C8×C61—22no—
151C4⋊C4×C61—24no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61C2×D414yes—
155C22⋊C4×C61—42no—
156C22⋊C4×C61—42no—
157C42×C61—22no—
158C22⋊Dic61—42no—
159C22⋊Dic61—42no—
160#11758198—28no—
161#11758198—42no—
162C4⋊Dic61—24no—
163C4⋊Dic61—28no—
164C4⋊Dic61—22no—
165C4⋊Dic61—22no—
166C4⋊Dic61C2×D414yes—
167#11758196—28no—
168C4×Dic61—22no—
169C4×Dic61—22no—
170C4×Dic61—22no—
171C4×Dic61—28no—
172C4×Dic61—42no—
173C2×#92418—22no—
174C2×#92418—22no—
175C2×#92418—22no—
176C8×D61—48no—
177C23×C61—22no—
178C23×C61—41no—
179C23×C61—41no—
180C23×C61—41no—
181C23×C61—41no—
182C23×C61—41no—
183C23×C61—41no—
184C23×C61C22×D411yescharacteristic
185C22×D61—21no—
186C22×D61—21no—
187C22×D61—21no—
188C22×D61—21no—
189C22×D61#12012yes—
190C22×D61C22⋊Q812yes—
191C22×D61#12412yes—
192Q8×C61—48no—
193D4×C61—24no—
194D4×C61—42no—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—21no—
198C2×C4×C61—21no—
199C2×C4×C61—24no—
200C2×C4×C61—42no—
201C2×C4×C61—42no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61—41no—
205C2×C4×C61#13512yes—
206C2×C4×C61#13512yes—
207C2×C4×C61C22×D412yes—
208C2×C4×C61C22×Q811yescharacteristic
209C2×C4×C61C2×#4112yes—
210C2×C4×C61C2×#4111yescharacteristic
211C22⋊D61—48no—
212C2×Dic61—21no—
213C2×Dic61—21no—
214C2×Dic61—24no—
215C2×Dic61—22no—
216C2×Dic61—21no—
217C2×Dic61—21no—
218C2×Dic61—21no—
219C2×Dic61—21no—
220C2×Dic61—41no—
221C2×Dic61—41no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224C2×Dic61—48no—
225C2×Dic61—41no—
226C2×Dic61—41no—
227C2×Dic61—41no—
228C2×Dic61#12012yes—
229C2×Dic61C22⋊Q812yes—
230C2×Dic61#12412yes—
231C4×D61—44no—
232C4×D61—44no—
233C4×D61C22⋊Q818yes—
234C4×D61#12418yes—
235Dic122—24no—
236Dic122—48no—
237Dic122—42no—
238C8×C61—42no—
239#92418—42no—
240#23451—611nomaximal, Sylow
241C22×C61—42no—
242C22×C61—42no—
243C22×C61—42no—
244C22×C61—81no—
245C22×C61—81no—
246C22×C61—81no—
247C22×C61—81no—
248C22×C61C2×#12011yescharacteristic
249C22×C61C2×C22⋊Q811yescharacteristic
250C22×C61C2×#12411yescharacteristic
251C22×C61C2×#13511yescharacteristic
252C22×C61C2×#13511yescharacteristic
253C22×C61#57511yescharacteristic
254C22×C61#57511yescharacteristic
255C2×D61—22no—
256C2×D61—22no—
257C2×D61—22no—
258C2×D61—22no—
259C2×D61—22no—
260C2×D61—22no—
261C2×D61#39314yes—
262C4×C61—22no—
263C4×C61—22no—
264C4×C61—44no—
265C4×C61—81no—
266C4×C61C2×C22⋊Q812yes—
267C4×C61C2×#12412yes—
268Dic61—42no—
269Dic61—44no—
270Dic61—41no—
271Dic61—41no—
272Dic61—81no—
273Dic61#39314yes—
274C2×#533—611no—
275C2×#416—612no—
276#2297—611no—
277#2026—612no—
278#2025—612no—
279#2024—611no—
280#2024—611no—
281#2023—611no—
282#2023D6111yescharacteristic, complemented
283#1735—612no—
284C2×#336—611no—
285C2×C61—82no—
286C2×C61C2×#39311yescharacteristic
287C2×C61#300211yescharacteristic
288C2×C61#300211yescharacteristic
289C2×C61#307611yescharacteristic
290C2×C61#307611yescharacteristic
291C2×C61#313111yescharacteristic
292C2×C61#313511yescharacteristic
293D61—24no—
294C22×#41—611no—
295C23×C8—612no—
296#533—1228no—
297C2×C4⋊Q8—611no—
298C2×C4⋊Q8C2×D6111yescharacteristic
299C2×#125—612no—
300C2×C22⋊Q8—1221no—
301C2×C22⋊Q8—1221no—
302C2×C4×Q8—1221no—
303C2×C4×Q8—1221no—
304C2×#117—612no—
305C2×#117—611no—
306C22×C42—611no—
307#416—1228no—
308C2×#103—612no—
309C2×#103—612no—
310C2×#102—612no—
311C2×#102C2×D6112yes—
312#391—611no—
313#391—611no—
314#390—611no—
315#390C2×D6111yescharacteristic
316#386—1221no—
317#386—1221no—
318C4×C22⋊C4—1221no—
319C4×C22⋊C4—1221no—
320#340—612no—
321#340—612no—
322#340—612no—
323#340C2×D6112yes—
324#336—612no—
325#336—611no—
326#336C2×D6111yescharacteristic
327C61#2345111yescharacteristic, complemented, Sylow
328C2×#41—614no—
329C2×#41—1222no—
330C2×#41—1224no—
331C2×#41—1224no—
332C22×Q8—21no—
333C22×Q8—611no—
334C22×Q8—1221no—
335C22×D4—1221no—
336C22×D4C22×D6111yescharacteristic
337C23×C4—611no—
338C23×C4—611no—
339C23×C4—611no—
340C23×C4—1221no—
341C22×C8—612no—
342C22×C8—1224no—
343C22×C8—1224no—
344C22×C8—1224no—
345C22×C8C22×D6112yes—
346C4⋊Q8—24no—
347C4⋊Q8—1224no—
348#125—1224no—
349#125—1224no—
350C22⋊Q8—1228no—
351C22⋊Q8—1228no—
352C4×Q8—1228no—
353C4×Q8—1228no—
354#117—618no—
355#117—1224no—
356#117—1224no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—612no—
359C2×C4⋊C4—611no—
360C2×C4⋊C4—1222no—
361C2×C4⋊C4—1222no—
362C2×C4⋊C4—1221no—
363C2×C4⋊C4C22×D6111yescharacteristic
364C2×C4⋊C4C22×D6112yes—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—1222no—
368C2×C22⋊C4—1221no—
369C2×C22⋊C4—1221no—
370C2×C22⋊C4—1221no—
371C2×C42—611no—
372C2×C42—611no—
373C2×C42—611no—
374C2×C42—1224no—
375C2×C42—1222no—
376C2×C42—1221no—
377C2×C42C22×D6111yescharacteristic
378#103—1224no—
379#103—1224no—
380#102—24no—
381#102—1224no—
382#95—1221no—
383#95—1221no—
384#95—1221no—
385#95—1221no—
386C24—21no—
387C24—611no—
388#41—12216no—
389#41—2448no—
390C2×Q8—24no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×Q8—612no—
394C2×Q8—1224no—
395C2×Q8—1221no—
396C2×Q8—1222no—
397C2×Q8—1222no—
398C2×Q8—2442no—
399C2×Q8—2442no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—21no—
403C2×D4—1224no—
404C2×D4—2442no—
405C2×D4—2442no—
406C2×D4Q8×D6112yes—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—611no—
410C22×C4—614no—
411C22×C4—612no—
412C22×C4—614no—
413C22×C4—612no—
414C22×C4—611no—
415C22×C4—611no—
416C22×C4—1222no—
417C22×C4—1222no—
418C22×C4—1224no—
419C22×C4—1221no—
420C22×C4—1222no—
421C22×C4—1222no—
422C22×C4—1222no—
423C22×C4—1221no—
424C22×C4—2441no—
425C22×C4—2441no—
426C22×C4—2442no—
427C22×C4—2442no—
428C22×C4D4×D6111yescharacteristic
429C22×C4D4×D6111yescharacteristic
430C22×C4C23×D6111yescharacteristic
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—1222no—
435C2×C8—1222no—
436C2×C8—1222no—
437C2×C8—2448no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—614no—
442C4⋊C4—1228no—
443C4⋊C4—1228no—
444C4⋊C4—1224no—
445C4⋊C4—1222no—
446C4⋊C4—1222no—
447C4⋊C4D4×D6114yes—
448C22⋊C4—42no—
449C22⋊C4—42no—
450C22⋊C4—1228no—
451C22⋊C4—2442no—
452C22⋊C4—2442no—
453C22⋊C4—2442no—
454C42—22no—
455C42—1222no—
456C42—1222no—
457C42—1228no—
458C42—1222no—
459C42—2442no—
460C23—22no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—612no—
468C23—612no—
469C23—612no—
470C23—1221no—
471C23—1221no—
472C23—1221no—
473C23—1221no—
474C23C2×D4×D6111yescharacteristic
475Q8—48no—
476Q8—1224no—
477Q8—2448no—
478Q8—2442no—
479D4—24no—
480D4—42no—
481D4—2448no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—24no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—612no—
493C2×C4—612no—
494C2×C4—612no—
495C2×C4—618no—
496C2×C4—618no—
497C2×C4—1221no—
498C2×C4—1221no—
499C2×C4—1224no—
500C2×C4—1222no—
501C2×C4—1221no—
502C2×C4—1221no—
503C2×C4—1221no—
504C2×C4—1221no—
505C2×C4—2444no—
506C2×C4—2444no—
507C2×C4—2448no—
508C2×C4—2441no—
509C2×C4—2441no—
510C2×C4—2441no—
511C2×C4—2441no—
512C2×C4—2441no—
513C2×C4—2442no—
514C2×C4—2442no—
515C2×C4#1451979612yes—
516C2×C4#1451979812yes—
517C2×C4C2×D4×D6112yes—
518C2×C4C2×Q8×D6111yescharacteristic
519C2×C4#41×D6112yes—
520C2×C4#41×D6111yescharacteristic
521C8—42no—
522C8—2442no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—81no—
527C22—81no—
528C22—81no—
529C22—81no—
530C22—614no—
531C22—1222no—
532C22—1222no—
533C22—1222no—
534C22—1222no—
535C22—1222no—
536C22—1222no—
537C22#120×D6111yescharacteristic
538C22C22⋊Q8×D6111yescharacteristic
539C22#124×D6111yescharacteristic
540C22C2×#1451979611yescharacteristic
541C22C2×#1451979811yescharacteristic
542C22#1684055811yescharacteristic
543C22#1684056111yescharacteristic
544C4—22no—
545C4—22no—
546C4—44no—
547C4—81no—
548C4—614no—
549C4—4881no—
550C4—2442no—
551C4—2441no—
552C4—2441no—
553C4—2444no—
554C4C22⋊Q8×D6112yes—
555C4#124×D6112yes—
556C2—82no—
557C2—1224no—
558C2#393×D6111yescharacteristic
559C2#2145503211yescharacteristic
560C2#2145503611yescharacteristic
561C2#2145560011yescharacteristic
562C2#2145560811yescharacteristic
563C2#2145613811yescharacteristic
564C2#2145616111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2023C2
socleC23×C61C22×D4
prime radicalC22×#11758224C22
hypercenter#2023D61
hypocenterC61#23451
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2023

Complements

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

NH ≅ G/Ntype
1. C2×#16840317C2semidirect
4. #21446969C2semidirect
5. #21446963C2semidirect
6. #21446961C2semidirect
7. #21446957C2semidirect
8. #21446955C2semidirect
9. #21445534C2semidirect
10. #336×D61C2semidirect
11. C61×#2023C2semidirect
48. C2×#11758224Q8semidirect
282. #2023D61semidirect
327. C61#23451semidirect

Maximal subgroups

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

subgroupindex
C2×#168403172
C2×#168397552
#214483062
#214469692
#214469632
#214469612
#214469572
#214469552
#214455342
#336×D612
C61×#20232
#2345161