Finite Group Databasebeta

← #29721600

Subgroups — #29721600

Order 15616.

1086 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×#16839975C212yesmaximal, complemented
2C2×C4×#11758200C211yesmaximal, characteristic, complemented
3#21450822C211yesmaximal, characteristic, complemented
4D4×C4⋊Dic61C211yesmaximal, characteristic, complemented
5#21447189C212yesmaximal, complemented
6#21447059C212yesmaximal, complemented
7#21446626C211yesmaximal, characteristic, complemented
8#21446535C211yesmaximal, characteristic, complemented
9#21446169C212yesmaximal, complemented
10#21445825C211yesmaximal, characteristic, complemented
11C2×C61×#459C211yesmaximal, characteristic
12C22×#11758200C2211yescharacteristic
13C2×#14519764C2212yes—
14C2×C4×Dic122C2211yescharacteristic
15C2×D4×Dic61C2212yes—
16C22×C4⋊Dic61—21no—
17#16839975—28no—
18C4×#11758200—24no—
19C2×#14519708C2211yescharacteristic
20C2×#14519706C2212yes—
21C2×#14519702C2212yes—
22C2×#14519700C2211yescharacteristic
23C2×#14519698C2212yes—
24C2×#14519698C2211yescharacteristic
25C2×#14519698C2212yes—
26C2×C4×#92418C2211yescharacteristic
27#16839676—22no—
28#16839665—21no—
29#16839665—21no—
30#16839665C2212yes—
31#16839661C2212yes—
32#16839637—22no—
33#16839632C2211yescharacteristic
34#16839622—21no—
35C4×C4⋊Dic61C2211yescharacteristic
36#16839525C2212yes—
37#16839516C2212yes—
38C22×D8×C61C2212yes—
39C2×#127×C61C2211yescharacteristic
40C2×C4×D4×C61C2211yescharacteristic
41C61×#459—24no—
42C2×C4⋊C8×C61C2211yescharacteristic
43C2×#102×C61C2212yes—
44C22×Dic122C2311yescharacteristic
45C23×Dic61—22no—
46C2×#11758200—22no—
47C2×#11758200—22no—
48C2×#11758200D414yescomplemented
49C22×#92418C2311yescharacteristic
50C22×#92418C2311yescharacteristic
51#14519764—24no—
52#14519764—42no—
53C4×Dic122—22no—
54C4×Dic122—22no—
55D4×Dic61—24no—
56D4×Dic61—42no—
57C2×C4⋊Dic61—21no—
58C2×C4⋊Dic61—22no—
59C2×C4⋊Dic61—21no—
60C2×C4⋊Dic61—44no—
61C2×C4⋊Dic61D411yescharacteristic
62C2×C4⋊Dic61D411yescharacteristic
63C2×C4⋊Dic61C2312yes—
64C2×C4⋊Dic61C2311yescharacteristic
65C2×#11758196—21no—
66C2×C22⋊Dic61—22no—
67C2×C22⋊Dic61—22no—
68C2×C22⋊Dic61—22no—
69C2×C22⋊Dic61—22no—
70C2×C22⋊Dic61—22no—
71C2×C4×Dic61—21no—
72C2×C4×Dic61C2312yes—
73#14519708—22no—
74#14519706—24no—
75#14519702—24no—
76#14519700—22no—
77#14519698—22no—
78#14519698—24no—
79#14519698—42no—
80C4×#92418—22no—
81#14519684—22no—
82#14519684D412yes—
83C22×D4×C61—21no—
84C22×D4×C61—21no—
85C22×D4×C61C2312yes—
86C22×D4×C61C2311yescharacteristic
87C23×C4×C61—21no—
88C2×D8×C61—24no—
89C2×D8×C61—24no—
90C2×D8×C61—24no—
91C22×C8×C61C2312yes—
92#127×C61—22no—
93#127×C61—22no—
94C4×D4×C61—22no—
95C4×D4×C61D414yes—
96C2×C4⋊C4×C61C2311yescharacteristic
97C2×C22⋊C4×C61—21no—
98C2×C42×C61C2311yescharacteristic
99C4⋊C8×C61—22no—
100#102×C61—24no—
101C24×C61—21no—
102C24×C61—22no—
103C2×D4×C61—22no—
104C2×D4×C61—22no—
105C2×D4×C61—21no—
106C2×D4×C61—22no—
107C2×D4×C61—22no—
108C2×D4×C61—21no—
109C2×D4×C61—22no—
110C2×D4×C61—21no—
111C2×D4×C61—22no—
112C2×D4×C61—42no—
113C2×D4×C61—42no—
114C2×D4×C61—42no—
115C2×D4×C61—42no—
116C2×D4×C61—81no—
117C2×D4×C61—81no—
118C2×D4×C61C2×D412yes—
119C2×D4×C61C2×D412yes—
120C2×D4×C61#4112yes—
121C22×C4×C61—21no—
122C22×C4×C61—24no—
123C22×C4×C61—42no—
124C22×C4×C61—42no—
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2411yescharacteristic
128C22×Dic61—21no—
129C22×Dic61—22no—
130C22×Dic61—21no—
131C22×Dic61—44no—
132C22×Dic61—44no—
133C22×Dic61—82no—
134C22×Dic61C2×D412yes—
135C22×Dic61C2×D412yes—
136C2×Dic122—21no—
137C2×Dic122—21no—
138C2×Dic122—21no—
139C2×Dic122—21no—
140C2×Dic122—21no—
141C2×Dic122C2×D412yes—
142D8×C61—48no—
143C2×C8×C61—22no—
144C2×C8×C61—22no—
145C2×C8×C61—22no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61C2×D412yes—
148C22⋊C4×C61—24no—
149C42×C61C2×D412yes—
150C42×C61#4112yes—
151C22⋊Dic61—44no—
152C22⋊Dic61—44no—
153C22⋊Dic61—44no—
154C22⋊Dic61—82no—
155C22⋊Dic61—82no—
156#11758200—28no—
157C4⋊Dic61—22no—
158C4⋊Dic61—22no—
159C4⋊Dic61—22no—
160C4⋊Dic61—21no—
161C4⋊Dic61—21no—
162C4⋊Dic61—22no—
163C4⋊Dic61—44no—
164C4⋊Dic61—42no—
165C4⋊Dic61—81no—
166#11758196—42no—
167C4×Dic61—22no—
168C4×Dic61—22no—
169C4×Dic61—42no—
170C2×#92418—21no—
171C2×#92418—21no—
172C2×#92418—21no—
173C2×#92418—21no—
174C2×#92418—21no—
175C2×#92418C2×D412yes—
176C23×C61—22no—
177C23×C61—22no—
178C23×C61—22no—
179C23×C61—41no—
180C23×C61—41no—
181C23×C61—41no—
182C23×C61—41no—
183C23×C61—42no—
184C23×C61—42no—
185C23×C61—42no—
186C23×C61—42no—
187C23×C61—42no—
188C23×C61—42no—
189C23×C61—42no—
190C23×C61C22×D411yescharacteristic
191D4×C61—22no—
192D4×C61—22no—
193D4×C61—22no—
194D4×C61—42no—
195D4×C61—42no—
196D4×C61—42no—
197D4×C61—42no—
198D4×C61—84no—
199D4×C61—84no—
200D4×C61#12114yes—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—22no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—48no—
207C2×C4×C61#12112yes—
208C2×C4×C61#12112yes—
209C2×C4×C61#13512yes—
210C2×C4×C61AGL(1,ℤ/8)12yes—
211C2×C4×C61C22×D411yescharacteristic
212C2×C4×C61C22×D411yescharacteristic
213C2×C4×C61C2×#4111yescharacteristic
214C2×C4×C61C2×#4111yescharacteristic
215C2×C4×C61H(5,𝔽2)11yescharacteristic
216C2×C4×C612−1+411yescharacteristic
217C2×Dic61—22no—
218C2×Dic61—22no—
219C2×Dic61—22no—
220C2×Dic61—22no—
221C2×Dic61—22no—
222C2×Dic61—22no—
223C2×Dic61—22no—
224C2×Dic61—41no—
225C2×Dic61—41no—
226C2×Dic61—41no—
227C2×Dic61—42no—
228C2×Dic61—42no—
229C2×Dic61—42no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61—88no—
233Dic122—22no—
234Dic122—22no—
235Dic122—41no—
236Dic122—41no—
237C8×C61—42no—
238#92418—22no—
239#92418—41no—
240#24731—611nomaximal, Sylow
241C22×C61—24no—
242C22×C61—42no—
243C22×C61—42no—
244C22×C61—42no—
245C22×C61—42no—
246C22×C61—42no—
247C22×C61—42no—
248C22×C61—82no—
249C22×C61—82no—
250C22×C61—82no—
251C22×C61—82no—
252C22×C61—82no—
253C22×C61—82no—
254C22×C61—82no—
255C22×C61C2×#12111yescharacteristic
256C22×C61D4211yescharacteristic
257C22×C61#54811yescharacteristic
258C22×C61C2×#13511yescharacteristic
259C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
260C22×C61#57611yescharacteristic
261C22×C61#57711yescharacteristic
262C4×C61—22no—
263C4×C61—42no—
264C4×C61#46314yescomplemented
265C4×C61C2×#12111yescharacteristic
266C4×C61#53711yescharacteristic
267C4×C61#54711yescharacteristic
268C4×C61#55311yescharacteristic
269Dic61—42no—
270Dic61—42no—
271Dic61—42no—
272Dic61—82no—
273Dic61—81no—
274C2×#463—612no—
275C2×#459D6111yescharacteristic
276C2×C4×SD16—611no—
277#2548—611no—
278D4×C4⋊C4—611no—
279#2065—612no—
280#2042—612no—
281#1957—611no—
282#1939—611no—
283#1879—612no—
284#1803—611no—
285C2×C61—44no—
286C2×C61—162no—
287C2×C61#241611yescharacteristic
288C2×C61C2×#46311yescharacteristic
289C2×C61#304511yescharacteristic
290C2×C61#327411yescharacteristic
291C2×C61#328411yescharacteristic
292C2×C61#336111yescharacteristic
293C2×C61#337311yescharacteristic
294C22×SD16—611no—
295C22×D8C2×D6112yes—
296C2×#127C2×D6111yescharacteristic
297C2×#124—612no—
298C2×C4×Q8—611no—
299C2×C4×D4—612no—
300C2×C4×D4C2×D6111yescharacteristic
301C22×C4⋊C4—1221no—
302#463—1228no—
303#459—24no—
304C4×SD16—1224no—
305C2×#106—611no—
306C2×C4⋊C8—612no—
307C2×C4⋊C8C2×D6111yescharacteristic
308C2×#103—612no—
309C2×#103—611no—
310C2×#102—612no—
311C2×#102—611no—
312C2×#102—612no—
313C2×#102C2×D6112yes—
314C2×C4×C8—611no—
315#394—1222no—
316#390—612no—
317#390—1221no—
318#390—1221no—
319#389—612no—
320#385—1222no—
321#384—611no—
322#380—1221no—
323C4×C4⋊C4—611no—
324#340—612no—
325#336—612no—
326C61#2473111yescharacteristic, complemented, Sylow
327C22×Q8—611no—
328C22×D4—21no—
329C22×D4—21no—
330C22×D4C22×D6112yes—
331C22×D4C22×D6111yescharacteristic
332C23×C4—21no—
333C23×C4—1222no—
334C2×SD16—614no—
335C2×SD16—1222no—
336C2×SD16—1222no—
337C2×D8—24no—
338C2×D8—24no—
339C2×D8—24no—
340C22×C8—611no—
341C22×C8—611no—
342C22×C8C22×D6112yes—
343#127—22no—
344#127—22no—
345#124—1224no—
346#124—2442no—
347C4×Q8—1222no—
348C4×Q8—1222no—
349C4×D4—22no—
350C4×D4—1224no—
351C4×D4—2442no—
352C4×D4C22⋊D6114yes—
353C2×C4⋊C4—611no—
354C2×C4⋊C4—611no—
355C2×C4⋊C4—612no—
356C2×C4⋊C4—611no—
357C2×C4⋊C4—1221no—
358C2×C4⋊C4—1222no—
359C2×C4⋊C4—1221no—
360C2×C4⋊C4—1221no—
361C2×C4⋊C4—2444no—
362C2×C4⋊C4C22×D6111yescharacteristic
363C2×C22⋊C4—21no—
364C2×C22⋊C4—1222no—
365C2×C22⋊C4—1222no—
366C2×C22⋊C4—1222no—
367C2×C22⋊C4—1222no—
368C2×C22⋊C4—1222no—
369C2×C42—612no—
370C2×C42—1221no—
371C2×C42C22×D6111yescharacteristic
372#106—1222no—
373C4⋊C8—22no—
374C4⋊C8—1224no—
375#103—1224no—
376#103—1222no—
377#102—24no—
378#102—1224no—
379#102—1222no—
380#102—2442no—
381C4×C8—1222no—
382#95—612no—
383#95—1222no—
384C24—21no—
385C24—22no—
386C2×Q8—612no—
387C2×Q8—1221no—
388C2×Q8—1221no—
389C2×Q8—1221no—
390C2×Q8—1221no—
391C2×Q8—1221no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—21no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—21no—
398C2×D4—22no—
399C2×D4—21no—
400C2×D4—22no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—81no—
406C2×D4—81no—
407C2×D4D4×D6112yes—
408C2×D4#1175822412yes—
409C2×D4C2×C22⋊D6112yes—
410C22×C4—21no—
411C22×C4—24no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—612no—
415C22×C4—612no—
416C22×C4—1221no—
417C22×C4—1222no—
418C22×C4—1221no—
419C22×C4—4882no—
420C22×C4—2444no—
421C22×C4—2444no—
422C22×C4D4×D6111yescharacteristic
423C22×C4D4×D6111yescharacteristic
424C22×C4C23×D6111yescharacteristic
425SD16—1228no—
426D8—48no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—612no—
431C2×C8—1221no—
432C2×C8—1221no—
433C2×C8—1221no—
434C2×C8—1221no—
435C2×C8—1221no—
436C4⋊C4—21no—
437C4⋊C4—1222no—
438C4⋊C4—1222no—
439C4⋊C4—1222no—
440C4⋊C4—1221no—
441C4⋊C4—1221no—
442C4⋊C4—1222no—
443C4⋊C4—4881no—
444C4⋊C4—2444no—
445C4⋊C4—2442no—
446C4⋊C4—2442no—
447C4⋊C4C2×C22⋊D6112yes—
448C22⋊C4—24no—
449C22⋊C4—4882no—
450C22⋊C4—4882no—
451C22⋊C4—2444no—
452C22⋊C4—2444no—
453C22⋊C4—2444no—
454C42—1222no—
455C42—1222no—
456C42—2442no—
457C42#1175822412yes—
458C42C2×C22⋊D6112yes—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23C2×D4×D6111yescharacteristic
474Q8—1222no—
475Q8—1222no—
476Q8—2441no—
477Q8—2441no—
478D4—22no—
479D4—22no—
480D4—22no—
481D4—42no—
482D4—42no—
483D4—42no—
484D4—42no—
485D4—84no—
486D4—84no—
487D4#1451974814yes—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—48no—
494C2×C4—1222no—
495C2×C4—1222no—
496C2×C4—1222no—
497C2×C4—1222no—
498C2×C4—1222no—
499C2×C4—1222no—
500C2×C4—1222no—
501C2×C4—4888no—
502C2×C4—2441no—
503C2×C4—2442no—
504C2×C4—2441no—
505C2×C4—2442no—
506C2×C4—2441no—
507C2×C4—2442no—
508C2×C4—2441no—
509C2×C4—2441no—
510C2×C4#1451974912yes—
511C2×C4#1451974912yes—
512C2×C4#1451979612yes—
513C2×C4#1451980112yes—
514C2×C4C2×D4×D6111yescharacteristic
515C2×C4C22×C22⋊D6111yescharacteristic
516C2×C4C2×#1175822411yescharacteristic
517C2×C4C2×#1175822411yescharacteristic
518C2×C4#1451982311yescharacteristic
519C2×C4#1451982411yescharacteristic
520C8—42no—
521C8—1222no—
522C8—2441no—
523C22—24no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—82no—
531C22—82no—
532C22—82no—
533C22—82no—
534C22—82no—
535C22—82no—
536C22—82no—
537C22C2×#1451974911yescharacteristic
538C22#1684039611yescharacteristic
539C22#1684041811yescharacteristic
540C22C2×#1451979611yescharacteristic
541C22C2×#1451980111yescharacteristic
542C22#1684056611yescharacteristic
543C22#1684057211yescharacteristic
544C4—22no—
545C4—42no—
546C4—4881no—
547C4—4882no—
548C4—2442no—
549C4—2442no—
550C4—2442no—
551C4#1683997514yescomplemented
552C4C2×#1451974811yescharacteristic
553C4#1684034011yescharacteristic
554C4#1684041211yescharacteristic
555C4#1684046411yescharacteristic
556C2—44no—
557C2—162no—
558C2#2144926011yescharacteristic
559C2C2×#1683997511yescharacteristic
560C2#2145536211yescharacteristic
561C2#2145744211yescharacteristic
562C2#2145755211yescharacteristic
563C2#2145848311yescharacteristic
564C2#2145861511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC2×C61×#459C2
socleC23×C61C22×D4
hypercenterC2×#459D61
hypocenterC61#24731
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#459

Complements

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

NH ≅ G/Ntype
1. C2×#16839975C2semidirect
2. C2×C4×#11758200C2semidirect
3. #21450822C2semidirect
4. D4×C4⋊Dic61C2semidirect
5. #21447189C2semidirect
6. #21447059C2semidirect
7. #21446626C2semidirect
8. #21446535C2semidirect
9. #21446169C2semidirect
10. #21445825C2semidirect
48. C2×#11758200D4semidirect
264. C4×C61#463semidirect
326. C61#24731semidirect
551. C4#16839975semidirect

Maximal subgroups

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

subgroupindex
C2×#168399752
C2×C4×#117582002
#214508222
D4×C4⋊Dic612
#214471892
#214470592
#214466262
#214465352
#214461692
#214458252
C2×C61×#4592
#2473161