Finite Group Databasebeta

← #29685208

Subgroups — #29685208

Order 15616.

1218 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
1C2×#16840291C211yesmaximal, characteristic
2C2×#16839714C211yesmaximal, characteristic, complemented
3#21448224C211yesmaximal, characteristic, complemented
4#21446788C211yesmaximal, characteristic, complemented
5#21446787C211yesmaximal, characteristic, complemented
6#21446636C211yesmaximal, characteristic, complemented
7#21446254C212yesmaximal
8#21446249C212yesmaximal, complemented
9#103×Dic61C212yesmaximal, complemented
10#102×Dic61C212yesmaximal, complemented
11C61×#1959C211yesmaximal, characteristic, complemented
12C22×#11758189C2212yes—
13#16840291—24no—
14#16840291C418yescomplemented
15C2×#14519770—21no—
16C2×#14519769C2211yescharacteristic
17C2×#14519759—21no—
18C2×#14519753C2211yescharacteristic
19C2×#14519748C2211yescharacteristic
20C2×Q8×Dic61C2211yescharacteristic
21C2×D4×Dic61C2211yescharacteristic
22C2×#14519737—21no—
23C2×#14519735—21no—
24C2×#14519733—21no—
25C2×C42×D61C2211yescharacteristic
26C2×#14519701C2212yes—
27C2×#14519698C2212yes—
28#16839714—24no—
29C2×#14519688C2211yescharacteristic
30C2×C8×Dic61C2212yes—
31#16839632C2211yescharacteristic
32#16839627—21no—
33#16839619—21no—
34#16839619C2212yes—
35C4⋊C4×Dic61—21no—
36C4⋊C4×Dic61C2212yes—
37#16839525C2212yes—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×#124×C61C2211yescharacteristic
42C2×#103×C61C2212yes—
43C2×#102×C61C2212yes—
44C2×#97×C61C2211yescharacteristic
45C61×#384C2211yescharacteristic
46C22×C22⋊D61—21no—
47C23×Dic61—21no—
48C22×C4×D61D411yescharacteristic
49C22×C4×D61D411yescharacteristic
50C22×C4×D61C2311yescharacteristic
51C2×#11758189—24no—
52C2×#11758189—24no—
53C2×#11758189—24no—
54C22×#92418C2312yes—
55#14519770—28no—
56#14519769—22no—
57#14519769C2×C414yes—
58#14519759—28no—
59#14519753—22no—
60#14519753C2×C414yes—
61#14519748—22no—
62#14519748C2×C414yes—
63Q8×Dic61—22no—
64Q8×Dic61C2×C414yes—
65D4×Dic61—22no—
66D4×Dic61C2×C414yes—
67#14519737—28no—
68#14519735—44no—
69#14519733—28no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61C2312yes—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×#11758198—21no—
82C2×#11758198—22no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—22no—
85C2×C4×Dic61D411yescharacteristic
86C2×C4×Dic61D411yescharacteristic
87C2×C4×Dic61C2311yescharacteristic
88C42×D61C2×C414yes—
89C42×D61D414yes—
90#14519701—24no—
91#14519698—24no—
92#14519688—22no—
93C8×Dic61—24no—
94#14519684—22no—
95#14519684—22no—
96C22×Q8×C61C2311yescharacteristic
97C22×D4×C61C2311yescharacteristic
98C22×C8×C61C2312yes—
99#124×C61—22no—
100#124×C61C2×C414yes—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61C2312yes—
104C2×C22⋊C4×C61—21no—
105C2×C22⋊C4×C61—21no—
106C2×C42×C61C2311yescharacteristic
107#103×C61—24no—
108#102×C61—24no—
109#97×C61—22no—
110C24×C61—21no—
111C23×D61C2×D411yescharacteristic
112C2×Q8×C61—21no—
113C2×Q8×C61—22no—
114C2×Q8×C61—22no—
115C2×Q8×C61C22×C412yes—
116C2×D4×C61—21no—
117C2×D4×C61—22no—
118C2×D4×C61—22no—
119C2×D4×C61C22×C412yes—
120C22×C4×C61—21no—
121C22×C4×C61—22no—
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2411yescharacteristic
125C2×C22⋊D61—24no—
126C2×C22⋊D61—42no—
127C2×C22⋊D61—42no—
128C22×Dic61—21no—
129C22×Dic61—21no—
130C22×Dic61—24no—
131C22×Dic61—22no—
132C22×Dic61—42no—
133C22×Dic61—42no—
134C22×Dic61C2×D411yescharacteristic
135C22×Dic61C2×D411yescharacteristic
136C22×Dic61C2×D411yescharacteristic
137C2×C4×D61—22no—
138C2×C4×D61—22no—
139C2×C4×D61—22no—
140C2×C4×D61—22no—
141C2×C4×D61—41no—
142C2×C4×D61—41no—
143C2×C4×D61C22⋊C414yes—
144C2×C4×D61C22⋊C414yes—
145C2×C4×D61C22×C412yes—
146C2×C4×D61C2×D412yes—
147C2×C8×C61—22no—
148C2×C8×C61—22no—
149C2×C8×C61#4114yes—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—42no—
153C4⋊C4×C61—42no—
154C22⋊C4×C61—24no—
155C22⋊C4×C61—24no—
156C42×C61C22×C412yes—
157C42×C61C2×D412yes—
158C22⋊Dic61—24no—
159C22⋊Dic61—24no—
160#11758198—24no—
161#11758198—44no—
162C4⋊Dic61—21no—
163C4⋊Dic61—21no—
164C4⋊Dic61—24no—
165C4⋊Dic61—22no—
166C4⋊Dic61—22no—
167C4⋊Dic61—42no—
168C4⋊Dic61—42no—
169C4⋊Dic61C22×C412yes—
170C4⋊Dic61C22×C412yes—
171#11758196—24no—
172#11758196—24no—
173#11758196—24no—
174C4×Dic61—24no—
175C4×Dic61—44no—
176C4×Dic61C22⋊C414yes—
177C4×Dic61C22⋊C414yes—
178C4×Dic61C22×C412yes—
179C4×Dic61C2×D412yes—
180C2×#92418—22no—
181C2×#92418—22no—
182C2×#92418#4114yes—
183#11758189—216no—
184C23×C61—22no—
185C23×C61—41no—
186C23×C61—41no—
187C23×C61—41no—
188C23×C61—41no—
189C23×C61—41no—
190C23×C61—41no—
191C23×C61C22×D411yescharacteristic
192C22×D61—21no—
193C22×D61—21no—
194C22×D61—21no—
195C22×D61—21no—
196C22×D61—21no—
197C22×D61—21no—
198C22×D61C2×C22⋊C412yes—
199Q8×C61—24no—
200Q8×C61—42no—
201D4×C61—24no—
202D4×C61—42no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—22no—
208C2×C4×C61—41no—
209C2×C4×C61—41no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61C2×C22⋊C412yes—
214C2×C4×C61C2×C22⋊C412yes—
215C2×C4×C61C23×C411yescharacteristic
216C2×C4×C61C22×D411yescharacteristic
217C2×C4×C61C2×#4112yes—
218C2×C4×C61C2×#4112yes—
219C22⋊D61—48no—
220C2×Dic61—21no—
221C2×Dic61—21no—
222C2×Dic61—21no—
223C2×Dic61—21no—
224C2×Dic61—22no—
225C2×Dic61—22no—
226C2×Dic61—41no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—48no—
231C2×Dic61—42no—
232C2×Dic61—42no—
233C2×Dic61—42no—
234C2×Dic61C2×C22⋊C412yes—
235C2×Dic61C2×C22⋊C412yes—
236C2×Dic61C2×C22⋊C412yes—
237C2×Dic61AGL(1,ℤ/8)12yes—
238C2×Dic61#13712yes—
239C4×D61—24no—
240C4×D61—24no—
241C4×D61—28no—
242C4×D61—44no—
243C8×C61—24no—
244#92418—24no—
245#21403—611nomaximal, Sylow
246C22×C61—42no—
247C22×C61—42no—
248C22×C61—42no—
249C22×C61—81no—
250C22×C61—81no—
251C22×C61—81no—
252C22×C61—81no—
253C22×C61C22×C22⋊C411yescharacteristic
254C22×C61#53211yescharacteristic
255C22×C61#53211yescharacteristic
256C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
257C22×C61C2×#13711yescharacteristic
258C22×C61#57511yescharacteristic
259C22×C61#57511yescharacteristic
260C2×D61—22no—
261C2×D61—22no—
262C2×D61—22no—
263C2×D61—41no—
264C2×D61—41no—
265C2×D61—41no—
266C2×D61—41no—
267C4×C61—22no—
268C4×C61—22no—
269C4×C61—42no—
270C4×C61—82no—
271C4×C61C4×#4112yes—
272C4×C61#52512yes—
273Dic61—22no—
274Dic61—22no—
275Dic61—42no—
276Dic61—42no—
277Dic61—82no—
278Dic61#41714yes—
279C2×#529—611no—
280C2×C4×M16—611no—
281#2285—611no—
282#1992—611no—
283#1992—611no—
284#1959—611no—
285#1959D6111yescharacteristic, complemented
286#1895—612no—
287#1894—612no—
288C4×#103—612no—
289C4×#102—612no—
290C2×C61—82no—
291C2×C61#229711yescharacteristic
292C2×C61C2×#41711yescharacteristic
293C2×C61#289611yescharacteristic
294C2×C61#317111yescharacteristic
295C2×C61#317111yescharacteristic
296C2×C61#317211yescharacteristic
297C2×C61#317211yescharacteristic
298D61—42no—
299C22×M16—612no—
300#529—618no—
301#529—1224no—
302C2×#126—1221no—
303C2×#125—611no—
304C2×#124C2×D6111yescharacteristic
305C2×#123—1221no—
306C2×C22⋊Q8—611no—
307C2×#121—611no—
308C2×C4×Q8—611no—
309C2×C4×D4—611no—
310C2×C4×D4—1221no—
311C2×#117—1221no—
312C2×#117—1221no—
313C22×C42—611no—
314C2×#103—612no—
315C2×#103C2×D6112yes—
316C2×#102—612no—
317C2×#102C2×D6112yes—
318C4×M16—1224no—
319C2×#97—611no—
320C2×#97C2×D6111yescharacteristic
321C2×C4×C8—612no—
322#384—611no—
323#384C2×D6111yescharacteristic
324#382—1221no—
325#380—612no—
326#380—1221no—
327C4×C4⋊C4—612no—
328C4×C4⋊C4—1221no—
329#340—612no—
330#340—612no—
331#340—612no—
332#340—612no—
333C61#2140311yescharacteristic, complemented, Sylow
334C22×Q8C22×D6111yescharacteristic
335C22×D4—1221no—
336C22×D4C22×D6111yescharacteristic
337C23×C4—611no—
338C23×C4—611no—
339C23×C4—611no—
340C23×C4—1221no—
341C2×M16—1224no—
342C2×M16—1224no—
343C2×M16—1224no—
344C22×C8—612no—
345C22×C8C22×D6112yes—
346#126—1228no—
347#125—614no—
348#125—1222no—
349#124—22no—
350#124C4×D6114yes—
351#123—1228no—
352C22⋊Q8—614no—
353C22⋊Q8—1222no—
354#121—614no—
355#121—1222no—
356C4×Q8—614no—
357C4×Q8—1222no—
358C4×D4—614no—
359C4×D4—1222no—
360C4×D4—1228no—
361#117—1228no—
362#117—2444no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—611no—
366C2×C4⋊C4—611no—
367C2×C4⋊C4—612no—
368C2×C4⋊C4—1221no—
369C2×C4⋊C4—1221no—
370C2×C4⋊C4—1221no—
371C2×C4⋊C4—1221no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4—1221no—
374C2×C4⋊C4C22×D6112yes—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—1221no—
378C2×C22⋊C4—1221no—
379C2×C22⋊C4—1221no—
380C2×C22⋊C4—1222no—
381C2×C42—611no—
382C2×C42—611no—
383C2×C42—611no—
384C2×C42—614no—
385C2×C42—614no—
386C2×C42—1221no—
387C2×C42—1222no—
388C2×C42C22×D6111yescharacteristic
389#103—24no—
390#103—1224no—
391#102—24no—
392#102—1224no—
393#97—22no—
394#97—1222no—
395C4×C8—1224no—
396#95—1222no—
397#95—1222no—
398C24—21no—
399C24—611no—
400C2×Q8—21no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8C2×C4×D6112yes—
404C2×D4—21no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—1224no—
408C2×D4—2442no—
409C2×D4—2442no—
410C2×D4C2×C4×D6112yes—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—611no—
414C22×C4—611no—
415C22×C4—611no—
416C22×C4—612no—
417C22×C4—612no—
418C22×C4—614no—
419C22×C4—614no—
420C22×C4—1222no—
421C22×C4—1222no—
422C22×C4—1224no—
423C22×C4—1222no—
424C22×C4—1222no—
425C22×C4—1221no—
426C22×C4—1221no—
427C22×C4—1222no—
428C22×C4—2441no—
429C22×C4—2441no—
430C22×C4—2442no—
431C22×C4—2442no—
432C22×C4D4×D6111yescharacteristic
433C22×C4D4×D6111yescharacteristic
434C22×C4C23×D6111yescharacteristic
435M16—12216no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—614no—
439C2×C8—1222no—
440C2×C8—1222no—
441C2×C8#1175822414yes—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—612no—
447C4⋊C4—612no—
448C4⋊C4—1224no—
449C4⋊C4—1224no—
450C4⋊C4—1221no—
451C4⋊C4—1221no—
452C4⋊C4—1224no—
453C4⋊C4—1224no—
454C4⋊C4—1222no—
455C4⋊C4—1222no—
456C4⋊C4—2442no—
457C4⋊C4—2442no—
458C22⋊C4—24no—
459C22⋊C4—24no—
460C22⋊C4—1224no—
461C22⋊C4—1224no—
462C22⋊C4—1224no—
463C22⋊C4—2444no—
464C42—612no—
465C42—612no—
466C42—614no—
467C42—614no—
468C42—1224no—
469C42—2444no—
470C42C2×C4×D6112yes—
471C42D4×D6112yes—
472C23—22no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—612no—
480C23—1221no—
481C23—1221no—
482C23—1221no—
483C23—1221no—
484C23—1221no—
485C23—1221no—
486C23C2×D4×D6111yescharacteristic
487Q8—24no—
488Q8—42no—
489D4—24no—
490D4—42no—
491D4—2448no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—612no—
503C2×C4—612no—
504C2×C4—612no—
505C2×C4—612no—
506C2×C4—612no—
507C2×C4—1228no—
508C2×C4—1221no—
509C2×C4—1221no—
510C2×C4—1221no—
511C2×C4—1221no—
512C2×C4—1224no—
513C2×C4—1224no—
514C2×C4—1222no—
515C2×C4—1222no—
516C2×C4—2444no—
517C2×C4—2448no—
518C2×C4—2442no—
519C2×C4—2442no—
520C2×C4—2442no—
521C2×C4—2441no—
522C2×C4—2441no—
523C2×C4—2441no—
524C2×C4—2441no—
525C2×C4C22⋊C4×D6112yes—
526C2×C4C22⋊C4×D6112yes—
527C2×C4C22×C4×D6111yescharacteristic
528C2×C4C2×D4×D6111yescharacteristic
529C2×C4#41×D6112yes—
530C2×C4C2×#1175822412yes—
531C8—24no—
532C8—1224no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22—1222no—
541C22—1222no—
542C22—1222no—
543C22—2441no—
544C22—2441no—
545C22—2441no—
546C22—2441no—
547C22C2×C22⋊C4×D6111yescharacteristic
548C22#1684031011yescharacteristic
549C22#1684031111yescharacteristic
550C22AGL(1,ℤ/8)×D6111yescharacteristic
551C22#137×D6111yescharacteristic
552C22#1684055811yescharacteristic
553C22#1684056111yescharacteristic
554C4—22no—
555C4—22no—
556C4—42no—
557C4—82no—
558C4—614no—
559C4—1222no—
560C4—1222no—
561C4—4882no—
562C4—2442no—
563C4—2442no—
564C4C4×#1175822412yes—
565C4#1684027212yes—
566C2—82no—
567C2—2442no—
568C2#2144830611yescharacteristic
569C2#417×D6111yescharacteristic
570C2#2145433511yescharacteristic
571C2#2145641111yescharacteristic
572C2#2145641711yescharacteristic
573C2#2145642211yescharacteristic
574C2#2145642811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1959C2
socleC23×C61C22×D4
prime radicalC2×#14519748C22
cosocleC22×C4×C61C24
hypercenter#1959D61
hypocenterC61#21403
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1959

Complements

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

NH ≅ G/Ntype
2. C2×#16839714C2semidirect
3. #21448224C2semidirect
4. #21446788C2semidirect
5. #21446787C2semidirect
6. #21446636C2semidirect
8. #21446249C2semidirect
9. #103×Dic61C2semidirect
10. #102×Dic61C2semidirect
11. C61×#1959C2semidirect
14. #16840291C4semidirect
285. #1959D61semidirect
333. C61#21403semidirect

Maximal subgroups

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

subgroupindex
C2×#168402912
C2×#168397142
#214482242
#214467882
#214467872
#214466362
#214462542
#214462492
#103×Dic612
#102×Dic612
C61×#19592
#2140361