Finite Group Databasebeta

← #29679810

Subgroups — #29679810

Order 15616.

1774 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D61C211yesmaximal, characteristic
2C2×#16839725C211yesmaximal, characteristic
3#21448198C211yesmaximal, characteristic
4#21445908C214yesmaximal, complemented
5#21445904C214yesmaximal, complemented
6#21445540C211yesmaximal, characteristic, complemented
7#21445539C211yesmaximal, characteristic, complemented
8#21445538C211yesmaximal, characteristic, complemented
9C61×#1736C211yesmaximal, characteristic, complemented
10C23×C4×D61—21no—
11C23×C4×D61C2211yescharacteristic
12C22×#11758194C2211yescharacteristic
13C2×#14519736C2211yescharacteristic
14C22×C4⋊Dic61C2211yescharacteristic
15C22×#11758196—21no—
16C2×C4⋊C4×D61—24no—
17C2×C4⋊C4×D61—21no—
18C2×C4⋊C4×D61—21no—
19C2×C4⋊C4×D61C418yescomplemented
20C2×C4⋊C4×D61C2212yes—
21C2×C4⋊C4×D61C2212yes—
22#16839725—24no—
23C2×#14519689C2212yes—
24C2×#14519689C2212yes—
25#16839623—21no—
26#16839619C2212yes—
27#16839619C2212yes—
28#16839617—21no—
29C4×#11758198—21no—
30C4×#11758198—21no—
31#16839525C2214yescomplemented
32#16839524C2214yescomplemented
33#16839524C2214yescomplemented
34C22×M16×C61C2211yescharacteristic
35C2×#117×C61C2211yescharacteristic
36C22×C4⋊C4×C61C2211yescharacteristic
37C61×#340C2214yescomplemented
38C24×D61D411yescharacteristic
39C23×Dic61—21no—
40C23×Dic61D411yescharacteristic
41C22×C4×D61—22no—
42C22×C4×D61—22no—
43C22×C4×D61—22no—
44C22×C4×D61—22no—
45C22×C4×D61—28no—
46C22×C4×D61—22no—
47C22×C4×D61—22no—
48C22×C4×D61—48no—
49C22×C4×D61C2×C414yes—
50C22×C4×D61D411yescharacteristic
51C22×C4×D61D411yescharacteristic
52C22×C4×D61D414yes—
53C22×C4×D61C2311yescharacteristic
54C22×C4×D61C2311yescharacteristic
55C2×#11758194—22no—
56C2×#11758194—22no—
57C2×#11758194—22no—
58C22×#92418C2312yes—
59#14519736—22no—
60#14519736—22no—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61—22no—
63C2×C4⋊Dic61C2×C414yes—
64C2×C4⋊Dic61C2312yes—
65C2×C4⋊Dic61C2312yes—
66C2×#11758196—28no—
67C2×#11758196—22no—
68C2×#11758196—22no—
69C4⋊C4×D61—28no—
70C4⋊C4×D61—28no—
71C4⋊C4×D61—24no—
72C4⋊C4×D61—24no—
73C4⋊C4×D61C2×C418yes—
74C4⋊C4×D61C2×C418yes—
75C2×C22⋊Dic61—21no—
76C2×#11758198—22no—
77C2×#11758198—22no—
78C2×C4×Dic61—21no—
79#14519689—24no—
80#14519689—24no—
81#14519684—22no—
82#14519684—22no—
83C23×C4×C61—21no—
84C23×C4×C61C2311yescharacteristic
85C2×M16×C61—22no—
86C2×M16×C61—22no—
87C2×M16×C61—22no—
88C22×C8×C61C2312yes—
89#117×C61—22no—
90#117×C61—22no—
91C2×C4⋊C4×C61—21no—
92C2×C4⋊C4×C61—22no—
93C2×C4⋊C4×C61C2×C414yes—
94C2×C4⋊C4×C61C2312yes—
95C2×C4⋊C4×C61C2312yes—
96C2×C22⋊C4×C61—21no—
97C2×C42×C61—21no—
98C24×C61C2×D411yescharacteristic
99C23×D61—22no—
100C23×D61—22no—
101C23×D61—24no—
102C23×D61—24no—
103C23×D61C22⋊C414yes—
104C23×D61C2×D411yescharacteristic
105C23×D61C2×D411yescharacteristic
106C22×C4×C61—24no—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—44no—
110C22×C4×C61C22×C412yes—
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D412yes—
113C22×C4×C61#4112yes—
114C22×C4×C61#4112yes—
115C22×C4×C61#4112yes—
116C22×C4×C61#4112yes—
117C22×C4×C61C2411yescharacteristic
118C22×Dic61—22no—
119C22×Dic61—22no—
120C22×Dic61—24no—
121C22×Dic61—22no—
122C22×Dic61—22no—
123C22×Dic61—44no—
124C22×Dic61C22⋊C414yes—
125C22×Dic61C2×D411yescharacteristic
126C22×Dic61C2×D411yescharacteristic
127C2×C4×D61—28no—
128C2×C4×D61—28no—
129C2×C4×D61—28no—
130C2×C4×D61—28no—
131C2×C4×D61—22no—
132C2×C4×D61—22no—
133C2×C4×D61—22no—
134C2×C4×D61—22no—
135C2×C4×D61—21no—
136C2×C4×D61—21no—
137C2×C4×D61—21no—
138C2×C4×D61—21no—
139C2×C4×D61—21no—
140C2×C4×D61—21no—
141C2×C4×D61—21no—
142C2×C4×D61—21no—
143C2×C4×D61—28no—
144C2×C4×D61—28no—
145C2×C4×D61—48no—
146C2×C4×D61—432no—
147C2×C4×D61—48no—
148C2×C4×D61C22⋊C414yes—
149C2×C4×D61C22⋊C414yes—
150C2×C4×D61C22×C412yes—
151C2×C4×D61C22×C412yes—
152C2×C4×D61C2×D412yes—
153C2×C4×D61C2×D412yes—
154M16×C61—44no—
155C2×C8×C61—22no—
156C2×C8×C61—22no—
157C2×C8×C61—22no—
158C4⋊C4×C61—24no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61C22×C414yes—
163C22⋊C4×C61—42no—
164C42×C61—42no—
165C22⋊Dic61—42no—
166#11758198—44no—
167#11758198—44no—
168C4⋊Dic61—24no—
169C4⋊Dic61—22no—
170C4⋊Dic61—22no—
171C4⋊Dic61—22no—
172C4⋊Dic61C22×C414yes—
173#11758196—28no—
174#11758196—28no—
175C4×Dic61—42no—
176#11758194—44no—
177C2×#92418—22no—
178C2×#92418—22no—
179C2×#92418—22no—
180C23×C61—22no—
181C23×C61—22no—
182C23×C61C2×C22⋊C412yes—
183C23×C61#12312yes—
184C23×C61#12312yes—
185C23×C61C22×D411yescharacteristic
186C22×D61—216no—
187C22×D61—216no—
188C22×D61—21no—
189C22×D61—21no—
190C22×D61—21no—
191C22×D61—21no—
192C22×D61—22no—
193C22×D61—22no—
194C22×D61—28no—
195C22×D61—28no—
196C22×D61—28no—
197C22×D61C2×C22⋊C412yes—
198C22×D61C2×C22⋊C412yes—
199C22×D61AGL(1,ℤ/8)14yes—
200C22×D61#13714yes—
201C2×C4×C61—24no—
202C2×C4×C61—48no—
203C2×C4×C61—44no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61C2×C22⋊C412yes—
208C2×C4×C61#11714yes—
209C2×C4×C61#11714yes—
210C2×C4×C61#12314yes—
211C2×C4×C61#12314yes—
212C2×C4×C61#12312yes—
213C2×C4×C61#12312yes—
214C2×C4×C61C23×C411yescharacteristic
215C2×C4×C61C22×D411yescharacteristic
216C2×C4×C61C2×#4111yescharacteristic
217C2×C4×C61C2×#4111yescharacteristic
218C2×C4×C61C2×#4111yescharacteristic
219C2×C4×C61C2×#4111yescharacteristic
220C2×Dic61—21no—
221C2×Dic61—21no—
222C2×Dic61—21no—
223C2×Dic61—21no—
224C2×Dic61—28no—
225C2×Dic61—24no—
226C2×Dic61—48no—
227C2×Dic61—44no—
228C2×Dic61—42no—
229C2×Dic61—42no—
230C2×Dic61—42no—
231C2×Dic61C2×C22⋊C412yes—
232C2×Dic61C2×C22⋊C412yes—
233C4×D61—22no—
234C4×D61—22no—
235C4×D61—22no—
236C4×D61—22no—
237C4×D61—22no—
238C4×D61—22no—
239C4×D61—22no—
240C4×D61—22no—
241C4×D61—28no—
242C4×D61—28no—
243C4×D61—416no—
244C4×D61—416no—
245C8×C61—42no—
246#92418—42no—
247#20700—611nomaximal, Sylow
248C22×C61—24no—
249C22×C61—24no—
250C22×C61—22no—
251C22×C61—22no—
252C22×C61#38114yes—
253C22×C61C22×C22⋊C411yescharacteristic
254C22×C61C2×#12311yescharacteristic
255C22×C61C2×#12311yescharacteristic
256C22×C61C2×AGL(1,ℤ/8)12yes—
257C22×C61C2×#13712yes—
258C2×D61—232no—
259C2×D61—24no—
260C2×D61—24no—
261C2×D61—24no—
262C2×D61—24no—
263C2×D61—22no—
264C2×D61—22no—
265C2×D61—22no—
266C2×D61—22no—
267C2×D61#41814yes—
268C2×D61#41914yes—
269C4×C61—44no—
270C4×C61—82no—
271C4×C61#38114yes—
272C4×C61C2×#11711yescharacteristic
273C4×C61C2×#11711yescharacteristic
274C4×C61C2×#12311yescharacteristic
275C4×C61C2×#12311yescharacteristic
276Dic61—22no—
277Dic61—22no—
278Dic61—44no—
279Dic61—82no—
280C23×C4⋊C4—611no—
281C2×#407—611no—
282#2280—611no—
283#1824—614no—
284#1823—614no—
285#1736—611no—
286#1736—611no—
287#1736—611no—
288#1736D6111yescharacteristic, complemented
289C2×C61—24no—
290C2×C61C2×#38111yescharacteristic
291C2×C61C2×#41811yescharacteristic
292C2×C61C2×#41911yescharacteristic
293C2×C61#309212yes—
294C2×C61#309212yes—
295D61—24no—
296D61—24no—
297C24×C4—611no—
298C24×C4—1221no—
299C22×M16—611no—
300C22×M16C2×D6111yescharacteristic
301C2×#117—611no—
302C2×#117C2×D6111yescharacteristic
303C22×C4⋊C4—611no—
304C22×C4⋊C4—618no—
305C22×C4⋊C4—612no—
306C22×C4⋊C4—612no—
307C22×C4⋊C4—1221no—
308C22×C4⋊C4—1221no—
309C22×C4⋊C4—1224no—
310C22×C4⋊C4—1221no—
311C22×C4⋊C4C2×D6111yescharacteristic
312#407—1224no—
313C2×C22⋊C8—612no—
314C2×C22⋊C8—612no—
315#381—1221no—
316#380—612no—
317#380—612no—
318#379—1221no—
319C4×C22⋊C4—1221no—
320C4×C22⋊C4—1221no—
321#340—614no—
322#340—614no—
323#340—614no—
324#340C2×D6114yescomplemented
325C61#2070011yescharacteristic, complemented, Sylow
326C25—611no—
327C23×C4—21no—
328C23×C4—611no—
329C23×C4—614no—
330C23×C4—611no—
331C23×C4—611no—
332C23×C4—614no—
333C23×C4—611no—
334C23×C4—611no—
335C23×C4—1222no—
336C23×C4—1222no—
337C23×C4—1228no—
338C23×C4—1222no—
339C23×C4—1222no—
340C23×C4—1222no—
341C23×C4—1222no—
342C23×C4—1221no—
343C23×C4—2448no—
344C23×C4C22×D6111yescharacteristic
345C2×M16—22no—
346C2×M16—22no—
347C2×M16—22no—
348C2×M16—1222no—
349C2×M16—1222no—
350C2×M16—1222no—
351C22×C8—612no—
352C22×C8C22×D6112yes—
353#117—22no—
354#117—22no—
355#117—1222no—
356#117—1222no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—614no—
360C2×C4⋊C4—612no—
361C2×C4⋊C4—618no—
362C2×C4⋊C4—618no—
363C2×C4⋊C4—612no—
364C2×C4⋊C4—1224no—
365C2×C4⋊C4—1224no—
366C2×C4⋊C4—1228no—
367C2×C4⋊C4—1228no—
368C2×C4⋊C4—1222no—
369C2×C4⋊C4—1222no—
370C2×C4⋊C4—1228no—
371C2×C4⋊C4—1222no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4C4×D6114yes—
374C2×C4⋊C4C22×D6112yes—
375C2×C4⋊C4C22×D6112yes—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—1222no—
378C2×C22⋊C4—1222no—
379C2×C22⋊C4—1221no—
380C2×C42—21no—
381C2×C42—1221no—
382C22⋊C8—1224no—
383C22⋊C8—1224no—
384#95—1222no—
385#95—1222no—
386C24—611no—
387C24—611no—
388C24—614no—
389C24—1224no—
390C24—1224no—
391C24—1222no—
392C24—1222no—
393C24D4×D6111yescharacteristic
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—44no—
398C22×C4—611no—
399C22×C4—611no—
400C22×C4—614no—
401C22×C4—612no—
402C22×C4—612no—
403C22×C4—612no—
404C22×C4—612no—
405C22×C4—614no—
406C22×C4—614no—
407C22×C4—1228no—
408C22×C4—1228no—
409C22×C4—1222no—
410C22×C4—1222no—
411C22×C4—1221no—
412C22×C4—1221no—
413C22×C4—1221no—
414C22×C4—1221no—
415C22×C4—1221no—
416C22×C4—1221no—
417C22×C4—1221no—
418C22×C4—1221no—
419C22×C4—1228no—
420C22×C4—1228no—
421C22×C4—1222no—
422C22×C4—1222no—
423C22×C4—1228no—
424C22×C4—1222no—
425C22×C4—1222no—
426C22×C4—1228no—
427C22×C4—1222no—
428C22×C4—1224no—
429C22×C4—1222no—
430C22×C4—2448no—
431C22×C4—24432no—
432C22×C4—2448no—
433C22×C4—2444no—
434C22×C4C2×C4×D6112yes—
435C22×C4#1175822212yes—
436C22×C4#1175822212yes—
437C22×C4D4×D6112yes—
438C22×C4D4×D6111yescharacteristic
439C22×C4#1175822412yes—
440C22×C4#1175822612yes—
441C22×C4C23×D6111yescharacteristic
442M16—44no—
443M16—2444no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—1222no—
448C2×C8—1222no—
449C2×C8—1222no—
450C4⋊C4—22no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—614no—
455C4⋊C4—1228no—
456C4⋊C4—1228no—
457C4⋊C4—1222no—
458C4⋊C4—1224no—
459C4⋊C4—1222no—
460C4⋊C4—1222no—
461C4⋊C4C2×C4×D6114yes—
462C22⋊C4—42no—
463C22⋊C4—2444no—
464C22⋊C4—2444no—
465C22⋊C4—2442no—
466C42—42no—
467C42—2442no—
468C23—22no—
469C23—22no—
470C23—612no—
471C23—612no—
472C23—614no—
473C23—614no—
474C23—1228no—
475C23—1228no—
476C23—12216no—
477C23—12216no—
478C23—1221no—
479C23—1221no—
480C23—1221no—
481C23—1221no—
482C23—1222no—
483C23—1222no—
484C23—1228no—
485C23C22⋊C4×D6112yes—
486C23#1451975912yes—
487C23#1451976212yes—
488C23C2×D4×D6111yescharacteristic
489C2×C4—24no—
490C2×C4—48no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—612no—
496C2×C4—612no—
497C2×C4—1221no—
498C2×C4—1221no—
499C2×C4—1221no—
500C2×C4—1221no—
501C2×C4—1228no—
502C2×C4—1222no—
503C2×C4—1222no—
504C2×C4—1222no—
505C2×C4—1222no—
506C2×C4—1222no—
507C2×C4—1222no—
508C2×C4—1222no—
509C2×C4—1222no—
510C2×C4—1228no—
511C2×C4—1228no—
512C2×C4—1224no—
513C2×C4—24416no—
514C2×C4—24416no—
515C2×C4—2442no—
516C2×C4—2442no—
517C2×C4—2442no—
518C2×C4—2448no—
519C2×C4—2444no—
520C2×C4C22⋊C4×D6112yes—
521C2×C4#1451973414yes—
522C2×C4#1451973514yes—
523C2×C4#1451975914yes—
524C2×C4#1451975912yes—
525C2×C4#1451976214yes—
526C2×C4#1451976212yes—
527C2×C4C22×C4×D6111yescharacteristic
528C2×C4C2×D4×D6111yescharacteristic
529C2×C4C2×#1175822411yescharacteristic
530C2×C4C2×#1175822211yescharacteristic
531C2×C4C2×#1175822211yescharacteristic
532C2×C4C2×#1175822611yescharacteristic
533C8—42no—
534C8—2442no—
535C22—24no—
536C22—24no—
537C22—22no—
538C22—22no—
539C22—614no—
540C22—614no—
541C22—12232no—
542C22—1222no—
543C22—1222no—
544C22—1224no—
545C22—1224no—
546C22—1224no—
547C22—1224no—
548C22—1222no—
549C22—1222no—
550C22#1683962414yes—
551C22C2×C22⋊C4×D6111yescharacteristic
552C22C2×#1451975911yescharacteristic
553C22C2×#1451976211yescharacteristic
554C22AGL(1,ℤ/8)×D6112yes—
555C22#137×D6112yes—
556C4—44no—
557C4—82no—
558C4—1222no—
559C4—1222no—
560C4—4882no—
561C4—2444no—
562C4#1683962414yes—
563C4C2×#1451973411yescharacteristic
564C4C2×#1451973511yescharacteristic
565C4C2×#1451975911yescharacteristic
566C4C2×#1451976211yescharacteristic
567C2—24no—
568C2—1224no—
569C2—1224no—
570C2C2×#1683962411yescharacteristic
571C2#418×D6111yescharacteristic
572C2#419×D6111yescharacteristic
573C2#2145574012yes—
574C2#2145574512yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1736C2
socleC23×C61C22×D4
prime radicalC24×D61D4
cosocleC22×C4×C61C24
hypercenter#1736D61
hypocenterC61#20700
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
4. #21445908C2semidirect
5. #21445904C2semidirect
6. #21445540C2semidirect
7. #21445539C2semidirect
8. #21445538C2semidirect
9. C61×#1736C2semidirect
19. C2×C4⋊C4×D61C4semidirect
31. #16839525C22semidirect
32. #16839524C22semidirect
33. #16839524C22semidirect
37. C61×#340C22semidirect
288. #1736D61semidirect
324. #340C2×D61semidirect
325. C61#20700semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D612
C2×#168397252
#214481982
#214459082
#214459042
#214455402
#214455392
#214455382
C61×#17362
#2070061