Finite Group Databasebeta

← #29721567

Subgroups — #29721567

Order 15616.

1118 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
1C2×#16839960C212yesmaximal, complemented
2C2×C4×#11758201C211yesmaximal, characteristic, complemented
3#21450766C211yesmaximal, characteristic
4Q8×C4⋊Dic61C211yesmaximal, characteristic, complemented
5#21447189C212yesmaximal, complemented
6#21447052C212yesmaximal, complemented
7#21446626C211yesmaximal, characteristic, complemented
8#21446535C211yesmaximal, characteristic, complemented
9#21446176C212yesmaximal, complemented
10#21445820C211yesmaximal, characteristic, complemented
11C2×C61×#460C211yesmaximal, characteristic, complemented
12C22×#11758201C2211yescharacteristic
13C2×#14519773C2212yes—
14C2×#14519748—21no—
15C2×#14519748—21no—
16C2×Q8×Dic61C2212yes—
17C2×C4×D244C2211yescharacteristic
18#16839960—28no—
19C4×#11758201—24no—
20C2×#14519708C2211yescharacteristic
21C2×#14519706C2212yes—
22C2×#14519701C2211yescharacteristic
23C2×#14519699C2212yes—
24C2×#14519698C2212yes—
25C2×#14519698C2212yes—
26C2×#14519697C2211yescharacteristic
27C2×C4×#92418C2211yescharacteristic
28#16839670—22no—
29#16839665C2212yes—
30#16839661—22no—
31#16839661C2212yes—
32#16839632—21no—
33#16839632C2211yescharacteristic
34C4×C4⋊Dic61—21no—
35C4×C4⋊Dic61C2211yescharacteristic
36#16839525C2212yes—
37#16839516C2212yes—
38C22×SD16×C61C2212yes—
39C2×#127×C61C2211yescharacteristic
40C2×C4×Q8×C61C2211yescharacteristic
41C61×#460—24no—
42C2×C4⋊C8×C61C2211yescharacteristic
43C2×#102×C61C2212yes—
44C22×D244C2311yescharacteristic
45C22×C22⋊D61—22no—
46C22×C22⋊D61—22no—
47C22×C4×D61—21no—
48C2×#11758201—22no—
49C2×#11758201—22no—
50C2×#11758201D414yescomplemented
51C22×#92418C2311yescharacteristic
52C22×#92418C2311yescharacteristic
53#14519773—24no—
54#14519773—42no—
55#14519748—44no—
56#14519748—44no—
57Q8×Dic61—24no—
58Q8×Dic61—42no—
59C4×D244—22no—
60C4×D244—22no—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61—22no—
63C2×C4⋊Dic61—21no—
64C2×C4⋊Dic61—22no—
65C2×C4⋊Dic61D411yescharacteristic
66C2×C4⋊Dic61D411yescharacteristic
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2312yes—
69C2×C22⋊Dic61—22no—
70C2×C22⋊Dic61—22no—
71C2×#11758198—21no—
72C2×C4×Dic61—22no—
73C2×C4×Dic61C2312yes—
74#14519708—22no—
75#14519706—24no—
76#14519701—22no—
77#14519699—24no—
78#14519698—24no—
79#14519698—42no—
80#14519697—22no—
81C4×#92418—22no—
82#14519684—22no—
83#14519684D412yes—
84C22×Q8×C61C2311yescharacteristic
85C22×D4×C61—21no—
86C22×D4×C61—21no—
87C22×D4×C61C2312yes—
88C2×SD16×C61—24no—
89C2×SD16×C61—24no—
90C2×SD16×C61—24no—
91C22×C8×C61C2312yes—
92#127×C61—22no—
93#127×C61—22no—
94C4×Q8×C61—22no—
95C4×Q8×C61D414yescomplemented
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61C2311yescharacteristic
98C2×C42×C61—21no—
99C2×C42×C61C2311yescharacteristic
100C4⋊C8×C61—22no—
101#102×C61—24no—
102C24×C61—22no—
103C23×D61—21no—
104C2×Q8×C61—21no—
105C2×Q8×C61—21no—
106C2×Q8×C61—21no—
107C2×Q8×C61C2×D412yes—
108C2×Q8×C61C2×D412yes—
109C2×Q8×C61#4112yes—
110C2×D4×C61—22no—
111C2×D4×C61—22no—
112C2×D4×C61—22no—
113C2×D4×C61—22no—
114C2×D4×C61—22no—
115C2×D4×C61—22no—
116C2×D4×C61—42no—
117C2×D4×C61—42no—
118C2×D4×C61—42no—
119C2×D4×C61—42no—
120C2×D4×C61—81no—
121C2×D4×C61—81no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2411yescharacteristic
127C2×C22⋊D61—44no—
128C2×C22⋊D61—44no—
129C2×C22⋊D61—44no—
130C2×C22⋊D61—44no—
131C2×C22⋊D61—82no—
132C2×C22⋊D61—82no—
133C22×Dic61—21no—
134C22×Dic61—22no—
135C22×Dic61C2×D412yes—
136C22×Dic61C2×D412yes—
137C2×D244—21no—
138C2×D244—21no—
139C2×D244—21no—
140C2×D244—21no—
141C2×D244—21no—
142C2×D244C2×D412yes—
143C2×C4×D61—42no—
144C2×C4×D61—42no—
145C2×C4×D61—42no—
146SD16×C61—48no—
147C2×C8×C61—22no—
148C2×C8×C61—22no—
149C2×C8×C61—22no—
150C4⋊C4×C61—24no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61C2×D412yes—
153C42×C61—24no—
154C42×C61C2×D412yes—
155C42×C61#4112yes—
156C22⋊Dic61—44no—
157C22⋊Dic61—44no—
158#11758201—28no—
159#11758198—42no—
160C4⋊Dic61—22no—
161C4⋊Dic61—22no—
162C4⋊Dic61—21no—
163C4⋊Dic61—21no—
164C4⋊Dic61—22no—
165C4⋊Dic61—44no—
166C4⋊Dic61—42no—
167C4⋊Dic61—82no—
168C4⋊Dic61—81no—
169C4⋊Dic61SD1614yescomplemented
170C4×Dic61—22no—
171C4×Dic61—22no—
172C4×Dic61—44no—
173C2×#92418—21no—
174C2×#92418—21no—
175C2×#92418—21no—
176C2×#92418—21no—
177C2×#92418—21no—
178C2×#92418C2×D412yes—
179C23×C61—42no—
180C23×C61—42no—
181C23×C61—42no—
182C23×C61—42no—
183C23×C61—42no—
184C23×C61—42no—
185C23×C61—42no—
186C23×C61C22×D411yescharacteristic
187C22×D61—22no—
188C22×D61—41no—
189C22×D61—41no—
190C22×D61—41no—
191C22×D61—41no—
192C22×D61—41no—
193C22×D61—41no—
194Q8×C61—22no—
195Q8×C61—22no—
196Q8×C61—22no—
197Q8×C61#12114yescomplemented
198D4×C61—42no—
199D4×C61—42no—
200D4×C61—42no—
201D4×C61—42no—
202D4×C61—84no—
203D4×C61—84no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—22no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61#12112yes—
213C2×C4×C61#12112yes—
214C2×C4×C61C2×SD1612yes—
215C2×C4×C61AGL(1,ℤ/8)12yes—
216C2×C4×C61C22×D411yescharacteristic
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C2×#4111yescharacteristic
219C2×C4×C61C2×#4111yescharacteristic
220C2×C4×C61H(5,𝔽2)11yescharacteristic
221C2×C4×C612−1+411yescharacteristic
222C22⋊D61—88no—
223C22⋊D61—88no—
224C2×Dic61—22no—
225C2×Dic61—22no—
226C2×Dic61—22no—
227C2×Dic61—22no—
228C2×Dic61—22no—
229C2×Dic61—41no—
230C2×Dic61—42no—
231C2×Dic61—41no—
232C2×Dic61—42no—
233C2×Dic61—41no—
234C2×Dic61—42no—
235C2×Dic61C2×SD1614yes—
236D244—22no—
237D244—22no—
238D244—41no—
239D244—41no—
240C4×D61—84no—
241C8×C61—42no—
242#92418—22no—
243#92418—41no—
244#24728—611nomaximal, Sylow
245C22×C61—82no—
246C22×C61—82no—
247C22×C61—82no—
248C22×C61—82no—
249C22×C61—82no—
250C22×C61—82no—
251C22×C61—82no—
252C22×C61C2×#12111yescharacteristic
253C22×C61D4211yescharacteristic
254C22×C61#54811yescharacteristic
255C22×C61C22×SD1611yescharacteristic
256C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
257C22×C61#57611yescharacteristic
258C22×C61#57711yescharacteristic
259C2×D61—42no—
260C2×D61—42no—
261C2×D61—42no—
262C2×D61—81no—
263C2×D61—81no—
264C2×D61—81no—
265C2×D61—81no—
266C4×C61—24no—
267C4×C61—22no—
268C4×C61—42no—
269C4×C61#46014yescomplemented
270C4×C61C2×#12111yescharacteristic
271C4×C61#53711yescharacteristic
272C4×C61#54711yescharacteristic
273C4×C61#55311yescharacteristic
274Dic61—24no—
275Dic61—42no—
276Dic61—82no—
277Dic61—81no—
278C2×#460—612no—
279C2×#460D6111yescharacteristic, complemented
280C2×C4×SD16—611no—
281#2541—611no—
282Q8×C4⋊C4—611no—
283#2065—612no—
284#2041—612no—
285#1957—611no—
286#1939—611no—
287#1880—612no—
288#1802—611no—
289C2×C61—162no—
290C2×C61#241611yescharacteristic
291C2×C61C2×#46011yescharacteristic
292C2×C61#304511yescharacteristic
293C2×C61#327411yescharacteristic
294C2×C61D4×SD1611yescharacteristic
295C2×C61#336211yescharacteristic
296C2×C61#336811yescharacteristic
297D61—82no—
298C22×SD16—611no—
299C22×SD16C2×D6112yes—
300C2×#127—612no—
301C2×#127C2×D6111yescharacteristic
302C2×#121—1221no—
303C2×#121—1221no—
304C2×C4×Q8—612no—
305C2×C4×Q8C2×D6111yescharacteristic
306C2×C4×D4—611no—
307#460—24no—
308#460—1228no—
309C4×SD16—1224no—
310C2×#106—611no—
311C2×C4⋊C8—612no—
312C2×C4⋊C8C2×D6111yescharacteristic
313C2×#103—611no—
314C2×#102—612no—
315C2×#102—612no—
316C2×#102—611no—
317C2×#102—612no—
318C2×#102C2×D6112yes—
319C2×C4×C8—611no—
320#392—1222no—
321#390—612no—
322#389—612no—
323#389—1222no—
324#384—611no—
325#384—1221no—
326C4×C4⋊C4—611no—
327C4×C4⋊C4—1221no—
328#340—612no—
329#336—612no—
330C61#2472811yescharacteristic, complemented, Sylow
331C22×Q8C22×D6111yescharacteristic
332C22×D4—21no—
333C22×D4—21no—
334C22×D4—611no—
335C22×D4—1222no—
336C22×D4—1222no—
337C22×D4C22×D6112yes—
338C23×C4—1221no—
339C2×SD16—24no—
340C2×SD16—24no—
341C2×SD16—24no—
342C2×SD16—614no—
343C2×SD16—1222no—
344C2×SD16—1222no—
345C22×C8—611no—
346C22×C8—611no—
347C22×C8C22×D6112yes—
348#127—22no—
349#127—22no—
350#127—1224no—
351#127—2442no—
352#121—2444no—
353#121—2444no—
354C4×Q8—22no—
355C4×Q8—1224no—
356C4×Q8—2442no—
357C4×Q8C22⋊D6114yescomplemented
358C4×D4—1222no—
359C4×D4—1222no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—611no—
362C2×C4⋊C4—611no—
363C2×C4⋊C4—611no—
364C2×C4⋊C4—612no—
365C2×C4⋊C4—1222no—
366C2×C4⋊C4—1221no—
367C2×C4⋊C4—1222no—
368C2×C4⋊C4—1221no—
369C2×C4⋊C4C22×D6111yescharacteristic
370C2×C22⋊C4—1222no—
371C2×C22⋊C4—1222no—
372C2×C22⋊C4—1221no—
373C2×C42—21no—
374C2×C42—612no—
375C2×C42—1222no—
376C2×C42C22×D6111yescharacteristic
377#106—1222no—
378C4⋊C8—22no—
379C4⋊C8—1224no—
380#103—1222no—
381#102—24no—
382#102—1224no—
383#102—1224no—
384#102—1222no—
385#102—2442no—
386C4×C8—1222no—
387#95—612no—
388#95—1222no—
389C24—22no—
390C24—1221no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8D4×D6112yes—
395C2×Q8#1175822412yes—
396C2×Q8C2×C22⋊D6112yes—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—81no—
408C2×D4—81no—
409C2×D4—612no—
410C2×D4—1221no—
411C2×D4—1221no—
412C2×D4—1221no—
413C2×D4—1221no—
414C2×D4—1221no—
415C2×D4—4882no—
416C2×D4—4882no—
417C2×D4—2444no—
418C2×D4—2444no—
419C2×D4—2444no—
420C2×D4—2444no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—612no—
424C22×C4—612no—
425C22×C4—1222no—
426C22×C4—1221no—
427C22×C4—2442no—
428C22×C4—2442no—
429C22×C4—2442no—
430C22×C4D4×D6111yescharacteristic
431C22×C4D4×D6111yescharacteristic
432C22×C4C23×D6111yescharacteristic
433SD16—48no—
434SD16—1228no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—612no—
439C2×C8—1221no—
440C2×C8—1221no—
441C2×C8—1221no—
442C2×C8—1221no—
443C2×C8—1221no—
444C4⋊C4—21no—
445C4⋊C4—24no—
446C4⋊C4—614no—
447C4⋊C4—1222no—
448C4⋊C4—1222no—
449C4⋊C4—1221no—
450C4⋊C4—1221no—
451C4⋊C4—1222no—
452C4⋊C4—4881no—
453C4⋊C4—4882no—
454C4⋊C4—2444no—
455C4⋊C4—2442no—
456C4⋊C4C2×C22⋊D6112yes—
457C22⋊C4—2444no—
458C22⋊C4—2444no—
459C22⋊C4—2442no—
460C42—24no—
461C42—1222no—
462C42—1222no—
463C42—2444no—
464C42#1175822412yes—
465C42C2×C22⋊D6112yes—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—1222no—
474C23—2441no—
475C23—2441no—
476C23—2441no—
477C23—2441no—
478C23—2441no—
479C23—2441no—
480C23C2×D4×D6111yescharacteristic
481Q8—22no—
482Q8—22no—
483Q8—22no—
484Q8#1451974814yescomplemented
485D4—42no—
486D4—42no—
487D4—42no—
488D4—42no—
489D4—84no—
490D4—84no—
491D4—1222no—
492D4—1222no—
493D4—4888no—
494D4—4888no—
495D4—2441no—
496D4—2441no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—614no—
506C2×C4—1222no—
507C2×C4—1222no—
508C2×C4—1222no—
509C2×C4—1222no—
510C2×C4—1222no—
511C2×C4—4884no—
512C2×C4—2441no—
513C2×C4—2441no—
514C2×C4—2441no—
515C2×C4—2442no—
516C2×C4—2442no—
517C2×C4—2442no—
518C2×C4#1451974912yes—
519C2×C4#1451974912yes—
520C2×C4SD16×D6112yes—
521C2×C4#1451980412yes—
522C2×C4C2×D4×D6111yescharacteristic
523C2×C4C22×C22⋊D6111yescharacteristic
524C2×C4C2×#1175822411yescharacteristic
525C2×C4C2×#1175822411yescharacteristic
526C2×C4#1451982311yescharacteristic
527C2×C4#1451982411yescharacteristic
528C8—42no—
529C8—1222no—
530C8—2441no—
531C22—82no—
532C22—82no—
533C22—82no—
534C22—82no—
535C22—82no—
536C22—82no—
537C22—82no—
538C22—4881no—
539C22—4881no—
540C22—4881no—
541C22—4881no—
542C22—2442no—
543C22—2442no—
544C22—2442no—
545C22C2×#1451974911yescharacteristic
546C22#1684039611yescharacteristic
547C22#1684041811yescharacteristic
548C22C2×SD16×D6111yescharacteristic
549C22C2×#1451980411yescharacteristic
550C22#1684056611yescharacteristic
551C22#1684057211yescharacteristic
552C4—22no—
553C4—24no—
554C4—42no—
555C4—1224no—
556C4—4882no—
557C4—4881no—
558C4—2442no—
559C4#1683996014yescomplemented
560C4C2×#1451974811yescharacteristic
561C4#1684034011yescharacteristic
562C4#1684041211yescharacteristic
563C4#1684046411yescharacteristic
564C2—162no—
565C2—4882no—
566C2#2144926011yescharacteristic
567C2C2×#1683996011yescharacteristic
568C2#2145536211yescharacteristic
569C2#2145743411yescharacteristic
570C2#2145751011yescharacteristic
571C2#2145848511yescharacteristic
572C2#2145855211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#460C2
socleC23×C61C22×D4
prime radical#21450766C2
hypercenterC2×#460D61
hypocenterC61#24728
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#460

Complements

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

NH ≅ G/Ntype
1. C2×#16839960C2semidirect
2. C2×C4×#11758201C2semidirect
4. Q8×C4⋊Dic61C2semidirect
5. #21447189C2semidirect
6. #21447052C2semidirect
7. #21446626C2semidirect
8. #21446535C2semidirect
9. #21446176C2semidirect
10. #21445820C2semidirect
11. C2×C61×#460C2semidirect
50. C2×#11758201D4semidirect
95. C4×Q8×C61D4semidirect
169. C4⋊Dic61SD16semidirect
197. Q8×C61#121semidirect
269. C4×C61#460semidirect
279. C2×#460D61semidirect
330. C61#24728semidirect
357. C4×Q8C22⋊D61semidirect
484. Q8#14519748semidirect
559. C4#16839960semidirect

Maximal subgroups

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

subgroupindex
C2×#168399602
C2×C4×#117582012
#214507662
Q8×C4⋊Dic612
#214471892
#214470522
#214466262
#214465352
#214461762
#214458202
C2×C61×#4602
#2472861