Finite Group Databasebeta

← #29611358

Subgroups — #29611358

Order 15616.

1662 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519767C211yesmaximal, characteristic
2C2×#16839626C211yesmaximal, characteristic
3C2×C42×Dic61C211yesmaximal, characteristic
4#21444054C211yesmaximal, characteristic
5#21444053C212yesmaximal
6#21444052C211yesmaximal, characteristic
7#21444051C211yesmaximal, characteristic
8#21444050C212yesmaximal
9#21444047C212yesmaximal
10#21444047C212yesmaximal
11C61×#1436C211yesmaximal, characteristic
12C2×#14519767—24no—
13C2×#14519767—24no—
14C2×#14519767C418yescomplemented
15C22×C4⋊Dic61C2211yescharacteristic
16C22×#11758196C2212yes—
17C22×#11758196C2211yescharacteristic
18C22×#11758196C2211yescharacteristic
19C22×#11758196C2211yescharacteristic
20C22×#11758196C2211yescharacteristic
21C22×C4×Dic61C2212yes—
22C22×C4×Dic61C2211yescharacteristic
23C22×C4×Dic61C2212yes—
24C22×C4×Dic61C2211yescharacteristic
25#16839626—24no—
26C2×#14519684C2212yes—
27C2×#14519684C2211yescharacteristic
28C2×#14519684C2212yes—
29C2×#14519684C2212yes—
30C2×#14519684C2212yes—
31C2×#14519684C2212yes—
32C2×#14519684C2212yes—
33C2×#14519684C2212yes—
34C42×Dic61—24no—
35C22×C4⋊C4×C61C2211yescharacteristic
36C22×C42×C61C2211yescharacteristic
37C2×#95×C61C2211yescharacteristic
38C2×#95×C61C2212yes—
39C2×#95×C61C2212yes—
40C23×Dic61C2312yes—
41C23×Dic61C2311yescharacteristic
42C23×Dic61C2311yescharacteristic
43C23×Dic61C2311yescharacteristic
44C23×Dic61C2312yes—
45C23×Dic61C2311yescharacteristic
46#14519767—232no—
47C2×C4⋊Dic61—22no—
48C2×C4⋊Dic61—22no—
49C2×C4⋊Dic61C2×C414yes—
50C2×#11758196—22no—
51C2×#11758196—22no—
52C2×#11758196—22no—
53C2×#11758196—22no—
54C2×#11758196—22no—
55C2×#11758196—22no—
56C2×#11758196—22no—
57C2×#11758196—22no—
58C2×#11758196—24no—
59C2×#11758196—24no—
60C2×#11758196—24no—
61C2×#11758196C2×C414yes—
62C2×#11758196C2×C414yes—
63C2×#11758196C2×C414yes—
64C2×#11758196C2×C414yes—
65C2×C4×Dic61—22no—
66C2×C4×Dic61—24no—
67C2×C4×Dic61—24no—
68C2×C4×Dic61—24no—
69C2×C4×Dic61—24no—
70C2×C4×Dic61—24no—
71C2×C4×Dic61—24no—
72C2×C4×Dic61C2×C414yes—
73C2×C4×Dic61D414yes—
74C2×C4×Dic61D414yes—
75C2×C4×Dic61Q818yes—
76#14519684—22no—
77#14519684—24no—
78#14519684—24no—
79#14519684—24no—
80#14519684—24no—
81#14519684—24no—
82#14519684—24no—
83#14519684—24no—
84C23×C4×C61C2312yes—
85C23×C4×C61C2311yescharacteristic
86C23×C4×C61C2311yescharacteristic
87C23×C4×C61C2311yescharacteristic
88C23×C4×C61C2312yes—
89C2×C4⋊C4×C61—22no—
90C2×C4⋊C4×C61—22no—
91C2×C4⋊C4×C61C2×C414yes—
92C2×C42×C61—22no—
93C2×C42×C61—24no—
94#95×C61—22no—
95#95×C61—24no—
96#95×C61—24no—
97C24×C61C2411yescharacteristic
98C22×C4×C61—22no—
99C22×C4×C61—22no—
100C22×C4×C61—22no—
101C22×C4×C61—22no—
102C22×C4×C61—22no—
103C22×C4×C61—22no—
104C22×C4×C61—21no—
105C22×C4×C61—21no—
106C22×C4×C61—21no—
107C22×C4×C61—21no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61C22×C412yes—
120C22×C4×C61C22×C412yes—
121C22×C4×C61C22×C412yes—
122C22×C4×C61C2×D412yes—
123C22×C4×C61C2×D412yes—
124C22×C4×C61C2×Q814yes—
125C22×C4×C61#4114yes—
126C22×C4×C61#4114yes—
127C22×Dic61—22no—
128C22×Dic61—22no—
129C22×Dic61—22no—
130C22×Dic61—22no—
131C22×Dic61—22no—
132C22×Dic61—22no—
133C22×Dic61—21no—
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61—21no—
137C22×Dic61—22no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—22no—
141C22×Dic61—22no—
142C22×Dic61—22no—
143C22×Dic61—22no—
144C22×Dic61—22no—
145C22×Dic61—22no—
146C22×Dic61—22no—
147C22×Dic61—22no—
148C22×Dic61C22×C412yes—
149C22×Dic61C22×C412yes—
150C22×Dic61C22×C412yes—
151C22×Dic61C22×C412yes—
152C22×Dic61C2×D412yes—
153C22×Dic61C2×D412yes—
154C22×Dic61C2×D412yes—
155C22×Dic61C2×D412yes—
156C22×Dic61C2×Q814yes—
157C22×Dic61C2×Q814yes—
158C22×Dic61#4114yes—
159C22×Dic61#4114yes—
160C22×Dic61#4114yes—
161C4⋊C4×C61—28no—
162C42×C61—28no—
163C4⋊Dic61—28no—
164#11758196—28no—
165#11758196—28no—
166#11758196—28no—
167#11758196—28no—
168#11758196—48no—
169C4×Dic61—216no—
170C4×Dic61—28no—
171C4×Dic61—216no—
172C4×Dic61C22⋊C4116yes—
173C23×C61C23×C411yescharacteristic
174C23×C61C22×D411yescharacteristic
175C23×C61C22×D411yescharacteristic
176C23×C61C22×Q812yes—
177C23×C61C2×#4112yes—
178C23×C61C2×#4112yes—
179C23×C61C2×#4112yes—
180C23×C61C2×#4112yes—
181C23×C61C2×#4112yes—
182C2×C4×C61—24no—
183C2×C4×C61—24no—
184C2×C4×C61—22no—
185C2×C4×C61—24no—
186C2×C4×C61—22no—
187C2×C4×C61—24no—
188C2×C4×C61—24no—
189C2×C4×C61—24no—
190C2×C4×C61—24no—
191C2×C4×C61—24no—
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61—24no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—41no—
198C2×C4×C61—41no—
199C2×C4×C61—42no—
200C2×C4×C61—42no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61C2×C22⋊C414yes—
210C2×C4×C61#12518yescomplemented
211C2×Dic61—22no—
212C2×Dic61—22no—
213C2×Dic61—24no—
214C2×Dic61—24no—
215C2×Dic61—24no—
216C2×Dic61—24no—
217C2×Dic61—24no—
218C2×Dic61—24no—
219C2×Dic61—24no—
220C2×Dic61—24no—
221C2×Dic61—22no—
222C2×Dic61—22no—
223C2×Dic61—24no—
224C2×Dic61—22no—
225C2×Dic61—22no—
226C2×Dic61—41no—
227C2×Dic61—41no—
228C2×Dic61—42no—
229C2×Dic61—42no—
230C2×Dic61—42no—
231C2×Dic61—42no—
232C2×Dic61—42no—
233C2×Dic61—42no—
234C2×Dic61—42no—
235C2×Dic61—42no—
236C2×Dic61—42no—
237C2×Dic61—42no—
238C2×Dic61C2×C22⋊C414yes—
239C2×Dic61C2×C22⋊C414yes—
240C2×Dic61C4×Q818yes—
241C2×Dic61C22⋊Q818yes—
242C2×Dic61#12414yes—
243C2×Dic61#12518yescomplemented
244C2×Dic61C4⋊Q814yes—
245#14709—611nomaximal, Sylow
246C22×C61C22×C22⋊C411yescharacteristic
247C22×C61C2×C4×Q812yes—
248C22×C61C4×#4112yes—
249C22×C61C4×#4111yescharacteristic
250C22×C61C4×#4111yescharacteristic
251C22×C61C2×C22⋊Q812yes—
252C22×C61#52511yescharacteristic
253C22×C61#52511yescharacteristic
254C22×C61#52512yes—
255C22×C61C2×#12411yescharacteristic
256C22×C61C2×#12512yes—
257C22×C61C2×#12512yes—
258C22×C61#52912yes—
259C22×C61#52912yes—
260C22×C61#52912yes—
261C22×C61#52912yes—
262C22×C61C2×C4⋊Q811yescharacteristic
263C22×C61#53212yes—
264C22×C61#53212yes—
265C22×C61#53312yes—
266C22×C61#53312yes—
267C4×C61—28no—
268C4×C61—24no—
269C4×C61—42no—
270C4×C61—42no—
271C4×C61—82no—
272Dic61—28no—
273Dic61—24no—
274Dic61—42no—
275Dic61—42no—
276Dic61—82no—
277Dic61#39118yes—
278C22×#125—611no—
279C2×#382—611no—
280C2×C43—611no—
281#1436—611no—
282#1436—612no—
283#1436—612no—
284#1436—611no—
285#1436—611no—
286#1436D6111yescharacteristic
287C4⋊#95—612no—
288C4⋊#95—612no—
289C2×C61C2×#39111yescharacteristic
290C2×C61#229711yescharacteristic
291C2×C61#229711yescharacteristic
292C2×C61C4×#12512yes—
293C2×C61C4×#12512yes—
294C2×C61#285112yes—
295C2×C61#285112yes—
296C2×C61#285912yes—
297C2×C61#286112yes—
298C2×#125—618no—
299C2×#125—1224no—
300C2×#125—1224no—
301C22×C4⋊C4—611no—
302C22×C4⋊C4—611no—
303C22×C4⋊C4—611no—
304C22×C4⋊C4—611no—
305C22×C4⋊C4—611no—
306C22×C4⋊C4—612no—
307C22×C4⋊C4C2×D6111yescharacteristic
308C22×C42—611no—
309C22×C42—612no—
310C22×C42—612no—
311C22×C42—611no—
312C22×C42C2×D6111yescharacteristic
313#382—1224no—
314C2×#95—612no—
315C2×#95—612no—
316C2×#95—612no—
317C2×#95—612no—
318C2×#95—612no—
319C2×#95—612no—
320C2×#95—612no—
321C2×#95—611no—
322C2×#95C2×D6112yes—
323C2×#95C2×D6112yes—
324C2×#95C2×D6111yescharacteristic
325C43—1224no—
326C61#1470911yescharacteristic, complemented, Sylow
327C23×C4—611no—
328C23×C4—611no—
329C23×C4—612no—
330C23×C4—611no—
331C23×C4—611no—
332C23×C4—612no—
333C23×C4C22×D6112yes—
334C23×C4C22×D6111yescharacteristic
335C23×C4C22×D6111yescharacteristic
336C23×C4C22×D6111yescharacteristic
337C23×C4C22×D6112yes—
338#125—12232no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—614no—
342C2×C4⋊C4—614no—
343C2×C4⋊C4—614no—
344C2×C4⋊C4—614no—
345C2×C4⋊C4—614no—
346C2×C4⋊C4—1222no—
347C2×C4⋊C4—1222no—
348C2×C4⋊C4—1222no—
349C2×C4⋊C4—1222no—
350C2×C4⋊C4—1222no—
351C2×C4⋊C4—1222no—
352C2×C4⋊C4—1222no—
353C2×C4⋊C4—1222no—
354C2×C4⋊C4—1222no—
355C2×C4⋊C4—1222no—
356C2×C4⋊C4—1224no—
357C2×C4⋊C4—1224no—
358C2×C4⋊C4—1224no—
359C2×C4⋊C4C4×D6114yes—
360C2×C42—22no—
361C2×C42—24no—
362C2×C42—614no—
363C2×C42—614no—
364C2×C42—614no—
365C2×C42—618no—
366C2×C42—1224no—
367C2×C42—1224no—
368C2×C42—1224no—
369C2×C42—1224no—
370C2×C42—1224no—
371C2×C42—1224no—
372C2×C42—1222no—
373#95—24no—
374#95—22no—
375#95—24no—
376#95—1224no—
377#95—1224no—
378#95—1224no—
379#95—1224no—
380#95—1224no—
381#95—1224no—
382#95—1222no—
383#95—1224no—
384C24C23×D6111yescharacteristic
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—612no—
407C22×C4—612no—
408C22×C4—612no—
409C22×C4—614no—
410C22×C4—614no—
411C22×C4—612no—
412C22×C4—612no—
413C22×C4—612no—
414C22×C4—614no—
415C22×C4—614no—
416C22×C4—614no—
417C22×C4—612no—
418C22×C4—612no—
419C22×C4—1222no—
420C22×C4—1222no—
421C22×C4—1222no—
422C22×C4—1222no—
423C22×C4—1222no—
424C22×C4—1222no—
425C22×C4—1221no—
426C22×C4—1221no—
427C22×C4—1221no—
428C22×C4—1221no—
429C22×C4—1222no—
430C22×C4—1222no—
431C22×C4—1222no—
432C22×C4—1222no—
433C22×C4—1222no—
434C22×C4—1222no—
435C22×C4—1222no—
436C22×C4—1222no—
437C22×C4—1222no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4C2×C4×D6112yes—
441C22×C4C2×C4×D6112yes—
442C22×C4C2×C4×D6112yes—
443C22×C4#1175822214yes—
444C22×C4D4×D6112yes—
445C22×C4D4×D6112yes—
446C22×C4#1175822414yes—
447C22×C4Q8×D6114yes—
448C4⋊C4—28no—
449C4⋊C4—1228no—
450C4⋊C4—1228no—
451C4⋊C4—1228no—
452C4⋊C4—1228no—
453C4⋊C4—1228no—
454C4⋊C4—2448no—
455C42—28no—
456C42—6116no—
457C42—12216no—
458C42—1228no—
459C42—12216no—
460C23C22×C4×D6111yescharacteristic
461C23C2×D4×D6111yescharacteristic
462C23C2×D4×D6111yescharacteristic
463C23C2×Q8×D6112yes—
464C23#41×D6112yes—
465C23#41×D6112yes—
466C23#41×D6112yes—
467C23C2×#1175822412yes—
468C23C2×#1175822212yes—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—614no—
497C2×C4—618no—
498C2×C4—618no—
499C2×C4—614no—
500C2×C4—614no—
501C2×C4—614no—
502C2×C4—618no—
503C2×C4—1222no—
504C2×C4—1222no—
505C2×C4—1224no—
506C2×C4—1224no—
507C2×C4—1224no—
508C2×C4—1224no—
509C2×C4—1224no—
510C2×C4—1224no—
511C2×C4—1224no—
512C2×C4—1224no—
513C2×C4—1222no—
514C2×C4—1222no—
515C2×C4—1224no—
516C2×C4—1222no—
517C2×C4—1222no—
518C2×C4—2442no—
519C2×C4—2442no—
520C2×C4—2442no—
521C2×C4—2442no—
522C2×C4—2442no—
523C2×C4—2442no—
524C2×C4—2442no—
525C2×C4—2441no—
526C2×C4—2441no—
527C2×C4—2442no—
528C2×C4—2442no—
529C2×C4—2442no—
530C2×C4C22⋊C4×D6114yes—
531C2×C4#1451976718yescomplemented
532C22C2×C22⋊C4×D6111yescharacteristic
533C22C4×Q8×D6112yes—
534C22C4×#1175822412yes—
535C22#1684023011yescharacteristic
536C22C4×#1175822211yescharacteristic
537C22C22⋊Q8×D6112yes—
538C22#1684026811yescharacteristic
539C22#1684027011yescharacteristic
540C22#1684027112yes—
541C22#124×D6111yescharacteristic
542C22#125×D6112yes—
543C22C2×#1451976712yes—
544C22#1684028912yes—
545C22#1684029012yes—
546C22#1684029212yes—
547C22#1684029512yes—
548C22C4⋊Q8×D6111yescharacteristic
549C22#1684030712yes—
550C22#1684030912yes—
551C22#1684031412yes—
552C22#1684031612yes—
553C4—28no—
554C4—24no—
555C4—42no—
556C4—42no—
557C4—82no—
558C4—618no—
559C4—1228no—
560C4—1224no—
561C4—4882no—
562C4—2442no—
563C4—2442no—
564C2#391×D6111yescharacteristic
565C2#2144830411yescharacteristic
566C2#2144830511yescharacteristic
567C2C4×#1451976712yes—
568C2#2144837612yes—
569C2#2145412512yes—
570C2#2145412612yes—
571C2#2145417812yes—
572C2#2145418812yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC61×#1436C2
socle, prime radical, cosocleC24×C61C24
hypercenter#1436D61
hypocenterC61#14709
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ #1436

Complements

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

NH ≅ G/Ntype
14. C2×#14519767C4semidirect
210. C2×C4×C61#125semidirect
243. C2×Dic61#125semidirect
326. C61#14709semidirect
531. C2×C4#14519767semidirect

Maximal subgroups

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

subgroupindex
C22×#145197672
C2×#168396262
C2×C42×Dic612
#214440542
#214440532
#214440522
#214440512
#214440502
#214440472
C61×#14362
#1470961