Finite Group Databasebeta

← #29685222

Subgroups — #29685222

Order 15616.

1422 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840307C211yesmaximal, characteristic
2C2×M16×Dic61C211yesmaximal, characteristic, complemented
3#21448214C211yesmaximal, characteristic, complemented
4#21446790C212yesmaximal, complemented
5#21446641C212yesmaximal, complemented
6#21446258C211yesmaximal, characteristic, complemented
7#21446257C211yesmaximal, characteristic, complemented
8#21446255C211yesmaximal, characteristic, complemented
9#102×Dic61C212yesmaximal, complemented
10#21445633C212yesmaximal, complemented
11C61×#1895C211yesmaximal, characteristic, complemented
12C22×#11758224—21no—
13C22×#11758194C2211yescharacteristic
14#16840307—24no—
15#16840307C418yescomplemented
16C2×#14519776C2211yescharacteristic
17C2×#14519773C2211yescharacteristic
18C2×#14519763—21no—
19C2×#14519751C2211yescharacteristic
20C2×#14519749—21no—
21C2×D4×Dic61C2211yescharacteristic
22C2×#14519739C2211yescharacteristic
23C2×#14519737—21no—
24C22×C4×Dic61C2211yescharacteristic
25C2×#14519699C2212yes—
26C2×#14519698C2212yes—
27C2×#14519697C2212yes—
28M16×Dic61—24no—
29C2×#14519687C2212yes—
30C2×C8×Dic61C2212yes—
31#16839629—21no—
32#16839627C2212yes—
33#16839623—21no—
34#16839622C2211yescharacteristic
35C4⋊C4×Dic61C2212yes—
36C22⋊C4×Dic61—21no—
37#16839607—21no—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C22×M16×C61C2211yescharacteristic
42C2×#121×C61C2211yescharacteristic
43C2×#102×C61C2212yes—
44C61×#380C2211yescharacteristic
45C61×#340C2212yes—
46C2×#11758224—28no—
47C2×#11758224—48no—
48C22×Dic122—21no—
49C22×D244C2311yescharacteristic
50C22×C22⋊D61—21no—
51C22×C22⋊D61—21no—
52C22×C22⋊D61—21no—
53C23×Dic61—21no—
54C23×Dic61D411yescharacteristic
55C23×Dic61D411yescharacteristic
56C22×C4×D61—21no—
57C2×#11758194—22no—
58C2×#11758194—24no—
59C22×#92418C2312yes—
60#14519776—22no—
61#14519776C2×C414yes—
62#14519773—22no—
63#14519773C2×C414yes—
64#14519763—28no—
65#14519751—22no—
66#14519751C2×C414yes—
67#14519749—28no—
68D4×Dic61—22no—
69D4×Dic61C2×C414yes—
70#14519739—22no—
71#14519739C2×C414yes—
72#14519737—28no—
73C2×C4⋊Dic61C2312yes—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758196—21no—
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61—21no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×C4×Dic61—22no—
81C2×C4×Dic61C2×C414yes—
82C2×C4×Dic61D411yescharacteristic
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61D414yes—
85C2×C4×Dic61C2311yescharacteristic
86C2×C4×Dic61C2311yescharacteristic
87#14519699—24no—
88#14519698—24no—
89#14519697—24no—
90#14519687—24no—
91C8×Dic61—24no—
92#14519684—21no—
93#14519684—22no—
94#14519684—22no—
95#14519684—22no—
96C22×D4×C61—21no—
97C22×D4×C61C2311yescharacteristic
98C23×C4×C61C2311yescharacteristic
99C2×M16×C61—22no—
100C2×M16×C61—24no—
101C22×C8×C61C2312yes—
102#121×C61—22no—
103#121×C61C2×C414yes—
104C2×C4⋊C4×C61C2312yes—
105C2×C4⋊C4×C61C2311yescharacteristic
106C2×C22⋊C4×C61—21no—
107C2×C22⋊C4×C61—21no—
108#102×C61—24no—
109#95×C61—21no—
110C24×C61—21no—
111C24×C61C2×D411yescharacteristic
112C23×D61—21no—
113C2×D4×C61—24no—
114C2×D4×C61—21no—
115C2×D4×C61—22no—
116C2×D4×C61—22no—
117C2×D4×C61—44no—
118C2×D4×C61C22×C412yes—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—21no—
122C22×C4×C61—22no—
123C22×C4×C61C22×C412yes—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D412yes—
126C22×C4×C61C2411yescharacteristic
127C2×C22⋊D61—24no—
128C2×C22⋊D61—24no—
129C2×C22⋊D61—24no—
130C2×C22⋊D61—42no—
131C2×C22⋊D61—42no—
132C2×C22⋊D61—44no—
133C2×C22⋊D61—44no—
134C22×Dic61—24no—
135C22×Dic61—21no—
136C22×Dic61—24no—
137C22×Dic61—22no—
138C22×Dic61—41no—
139C22×Dic61—41no—
140C22×Dic61—44no—
141C22×Dic61C22⋊C414yes—
142C22×Dic61C22⋊C414yes—
143C22×Dic61C2×D411yescharacteristic
144C22×Dic61C2×D411yescharacteristic
145C22×Dic61C2×D411yescharacteristic
146C22×Dic61C2×D411yescharacteristic
147#11758224—432no—
148C2×D244—22no—
149C2×D244—22no—
150C2×D244—21no—
151C2×D244C22×C412yes—
152C2×C4×D61—24no—
153C2×C4×D61—44no—
154C2×Dic122—24no—
155C2×Dic122—44no—
156M16×C61—28no—
157C2×C8×C61—22no—
158C2×C8×C61—22no—
159C2×C8×C61#4114yescomplemented
160C4⋊C4×C61—21no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61C22×C412yes—
164C22⋊C4×C61—24no—
165C22⋊C4×C61—42no—
166C22⋊Dic61—24no—
167C22⋊Dic61—42no—
168#11758198—24no—
169#11758198—24no—
170C4⋊Dic61—21no—
171C4⋊Dic61—22no—
172C4⋊Dic61—22no—
173C4⋊Dic61C22×C412yes—
174#11758196—24no—
175C4×Dic61—44no—
176C4×Dic61C22⋊C414yes—
177C4×Dic61C22⋊C414yes—
178C4×Dic61C22×C412yes—
179C4×Dic61C22×C412yes—
180C4×Dic61C2×D412yes—
181C4×Dic61C2×D412yes—
182#11758194—28no—
183C2×#92418—22no—
184C2×#92418—22no—
185C2×#92418#4114yescomplemented
186C23×C61—22no—
187C23×C61—22no—
188C23×C61—21no—
189C23×C61—21no—
190C23×C61—22no—
191C23×C61—42no—
192C23×C61—42no—
193C23×C61—41no—
194C23×C61—41no—
195C23×C61C2×C22⋊C412yes—
196C23×C61C22×D411yescharacteristic
197C22×D61—22no—
198C22×D61—42no—
199C22×D61—42no—
200C22×D61—41no—
201C22×D61—41no—
202D4×C61—24no—
203D4×C61—48no—
204D4×C61—41no—
205D4×C61—41no—
206C2×C4×C61—22no—
207C2×C4×C61—24no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61C2×C22⋊C412yes—
216C2×C4×C61C23×C411yescharacteristic
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C2×#4112yes—
219C2×C4×C61C2×#4112yes—
220C22⋊D61—48no—
221C22⋊D61—48no—
222C22⋊D61—48no—
223C2×Dic61—22no—
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—28no—
229C2×Dic61—22no—
230C2×Dic61—44no—
231C2×Dic61—42no—
232C2×Dic61—48no—
233C2×Dic61—42no—
234C2×Dic61—42no—
235C2×Dic61—42no—
236C2×Dic61C2×C22⋊C412yes—
237C2×Dic61C2×C22⋊C412yes—
238C2×Dic61C2×C22⋊C412yes—
239C2×Dic61C2×C22⋊C412yes—
240C2×Dic61AGL(1,ℤ/8)14yes—
241D244—24no—
242D244—41no—
243D244—41no—
244C4×D61—48no—
245Dic122—48no—
246C8×C61—24no—
247#92418—24no—
248#21405—611nomaximal, Sylow
249C22×C61—22no—
250C22×C61—22no—
251C22×C61—22no—
252C22×C61—42no—
253C22×C61—42no—
254C22×C61—42no—
255C22×C61—42no—
256C22×C61—42no—
257C22×C61—82no—
258C22×C61—82no—
259C22×C61C22×C22⋊C411yescharacteristic
260C22×C61#53212yes—
261C22×C61C2×AGL(1,ℤ/8)12yes—
262C22×C61#57511yescharacteristic
263C22×C61#57511yescharacteristic
264C2×D61—42no—
265C2×D61—42no—
266C2×D61—42no—
267C2×D61—82no—
268C2×D61—82no—
269C4×C61—22no—
270C4×C61—42no—
271C4×C61—82no—
272C4×C61C4×#4112yes—
273C4×C61#52511yescharacteristic
274C4×C61#52511yescharacteristic
275Dic61—22no—
276Dic61—22no—
277Dic61—22no—
278Dic61—42no—
279Dic61—82no—
280Dic61#41814yes—
281C2×#532—611no—
282C2×C4×M16—611no—
283#2283—611no—
284#1993—612no—
285#1961—612no—
286#1895—611no—
287#1895—611no—
288#1895—611no—
289#1895D6111yescharacteristic, complemented
290C4×#102—612no—
291C4×#102—612no—
292C2×C61—42no—
293C2×C61—82no—
294C2×C61#229711yescharacteristic
295C2×C61C2×#41811yescharacteristic
296C2×C61#289611yescharacteristic
297C2×C61#317112yes—
298C2×C61#317112yes—
299D61—82no—
300C22×#41—1221no—
301C22×M16—611no—
302C22×M16C2×D6111yescharacteristic
303#532—618no—
304#532—1224no—
305C2×C4⋊Q8—611no—
306C2×#127—611no—
307C2×#124—1221no—
308C2×#121—611no—
309C2×#121—1221no—
310C2×#121C2×D6111yescharacteristic
311C2×C4×D4—611no—
312C2×C4×D4—611no—
313C2×C4×D4—1221no—
314C22×C42—611no—
315C2×#102—612no—
316C2×#102—612no—
317C2×#102—612no—
318C2×#102C2×D6112yes—
319C4×M16—1224no—
320C2×#97—612no—
321C2×C4×C8—612no—
322#383—1221no—
323#382—612no—
324#381—1221no—
325#380—611no—
326#380C2×D6111yescharacteristic
327C4×C4⋊C4—612no—
328C4×C22⋊C4—1221no—
329#376—1221no—
330#340—612no—
331#340—612no—
332#340—612no—
333#340C2×D6112yes—
334C61#2140511yescharacteristic, complemented, Sylow
335C2×#41—1228no—
336C2×#41—2448no—
337C22×Q8—1221no—
338C22×D4—21no—
339C22×D4—611no—
340C22×D4—1221no—
341C22×D4—1221no—
342C22×D4—1221no—
343C22×D4C22×D6111yescharacteristic
344C23×C4—611no—
345C23×C4—611no—
346C23×C4—1221no—
347C23×C4—1221no—
348C23×C4C22×D6111yescharacteristic
349C2×M16—22no—
350C2×M16—24no—
351C2×M16—1224no—
352C2×M16—1222no—
353C22×C8—612no—
354C22×C8C22×D6112yes—
355C4⋊Q8—614no—
356C4⋊Q8—1222no—
357#127—614no—
358#127—1222no—
359#124—1228no—
360#121—22no—
361#121—614no—
362#121—1228no—
363#121—1222no—
364#121C4×D6114yes—
365C4×D4—614no—
366C4×D4—614no—
367C4×D4—1228no—
368C4×D4—1222no—
369C4×D4—1222no—
370C2×C4⋊C4—611no—
371C2×C4⋊C4—612no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4C22×D6112yes—
374C2×C4⋊C4C22×D6111yescharacteristic
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—1221no—
378C2×C22⋊C4—1221no—
379C2×C22⋊C4—1221no—
380C2×C22⋊C4—1221no—
381C2×C42—614no—
382C2×C42—611no—
383C2×C42—611no—
384C2×C42—614no—
385C2×C42—611no—
386C2×C42—611no—
387C2×C42—1222no—
388#102—24no—
389#102—1224no—
390#102—1224no—
391#102—1224no—
392#97—1224no—
393C4×C8—1224no—
394#95—21no—
395#95—1222no—
396#95—1222no—
397#95—1222no—
398#95—1221no—
399C24—21no—
400C24—1221no—
401C24D4×D6111yescharacteristic
402#41—24432no—
403C2×Q8—1224no—
404C2×Q8—2444no—
405C2×D4—24no—
406C2×D4—21no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—44no—
410C2×D4—612no—
411C2×D4—1224no—
412C2×D4—1224no—
413C2×D4—1224no—
414C2×D4—1221no—
415C2×D4—1222no—
416C2×D4—1222no—
417C2×D4—2444no—
418C2×D4—2442no—
419C2×D4—2442no—
420C2×D4—2444no—
421C2×D4C2×C4×D6112yes—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—611no—
427C22×C4—611no—
428C22×C4—611no—
429C22×C4—611no—
430C22×C4—614no—
431C22×C4—614no—
432C22×C4—1224no—
433C22×C4—1224no—
434C22×C4—1224no—
435C22×C4—1221no—
436C22×C4—1222no—
437C22×C4—2444no—
438C22×C4—2444no—
439C22×C4—2441no—
440C22×C4—2441no—
441C22×C4C2×C4×D6112yes—
442C22×C4D4×D6112yes—
443C22×C4D4×D6111yescharacteristic
444C22×C4C23×D6111yescharacteristic
445M16—28no—
446M16—1228no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—614no—
450C2×C8—1222no—
451C2×C8—1222no—
452C2×C8#1175822414yescomplemented
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—612no—
457C4⋊C4—1224no—
458C4⋊C4—1221no—
459C4⋊C4—1222no—
460C4⋊C4—1222no—
461C4⋊C4C2×C4×D6112yes—
462C22⋊C4—24no—
463C22⋊C4—42no—
464C22⋊C4—1224no—
465C22⋊C4—1224no—
466C22⋊C4—1224no—
467C22⋊C4—2442no—
468C42—612no—
469C42—612no—
470C42—612no—
471C42—612no—
472C42—614no—
473C42—614no—
474C42—2444no—
475C23—22no—
476C23—21no—
477C23—21no—
478C23—22no—
479C23—22no—
480C23—42no—
481C23—41no—
482C23—41no—
483C23—42no—
484C23—1222no—
485C23—2442no—
486C23—2441no—
487C23—2441no—
488C23—2442no—
489C23C22⋊C4×D6112yes—
490C23C2×D4×D6111yescharacteristic
491Q8—2448no—
492D4—24no—
493D4—48no—
494D4—41no—
495D4—41no—
496D4—1224no—
497D4—2448no—
498D4—2448no—
499D4—2448no—
500D4—2441no—
501D4—2441no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—612no—
512C2×C4—614no—
513C2×C4—612no—
514C2×C4—612no—
515C2×C4—612no—
516C2×C4—1228no—
517C2×C4—1222no—
518C2×C4—1221no—
519C2×C4—1221no—
520C2×C4—1221no—
521C2×C4—1221no—
522C2×C4—1222no—
523C2×C4—2448no—
524C2×C4—2448no—
525C2×C4—2444no—
526C2×C4—2442no—
527C2×C4—2442no—
528C2×C4—2442no—
529C2×C4—2442no—
530C2×C4C22⋊C4×D6112yes—
531C2×C4C22×C4×D6111yescharacteristic
532C2×C4C2×D4×D6111yescharacteristic
533C2×C4#41×D6112yes—
534C2×C4C2×#1175822412yes—
535C8—24no—
536C8—1224no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—82no—
546C22—82no—
547C22—4882no—
548C22—4882no—
549C22—2442no—
550C22—2442no—
551C22—2442no—
552C22C2×C22⋊C4×D6111yescharacteristic
553C22#1684030712yes—
554C22AGL(1,ℤ/8)×D6112yes—
555C22#1684055811yescharacteristic
556C22#1684056111yescharacteristic
557C4—22no—
558C4—42no—
559C4—82no—
560C4—614no—
561C4—1222no—
562C4—1222no—
563C4—1222no—
564C4—4882no—
565C4—2442no—
566C4C4×#1175822412yes—
567C4#1684026811yescharacteristic
568C4#1684027011yescharacteristic
569C2—42no—
570C2—82no—
571C2—4882no—
572C2#2144830411yescharacteristic
573C2#418×D6111yescharacteristic
574C2#2145433511yescharacteristic
575C2#2145640712yes—
576C2#2145640912yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1895C2
socleC23×C61C22×D4
prime radicalC2×#16840307C2
cosocleC22×C4×C61C24
hypercenter#1895D61
hypocenterC61#21405
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1895

Complements

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

NH ≅ G/Ntype
2. C2×M16×Dic61C2semidirect
3. #21448214C2semidirect
4. #21446790C2semidirect
5. #21446641C2semidirect
6. #21446258C2semidirect
7. #21446257C2semidirect
8. #21446255C2semidirect
9. #102×Dic61C2semidirect
10. #21445633C2semidirect
11. C61×#1895C2semidirect
15. #16840307C4semidirect
159. C2×C8×C61#41semidirect
185. C2×#92418#41semidirect
289. #1895D61semidirect
334. C61#21405semidirect
452. C2×C8#11758224semidirect

Maximal subgroups

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

subgroupindex
C2×#168403072
C2×M16×Dic612
#214482142
#214467902
#214466412
#214462582
#214462572
#214462552
#102×Dic612
#214456332
C61×#18952
#2140561