Finite Group Databasebeta

← #29721566

Subgroups — #29721566

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×#16839962C211yesmaximal, characteristic, complemented
2C2×#16839960C212yesmaximal, complemented
3#21450765C211yesmaximal, characteristic
4Q8×C4⋊Dic61C211yesmaximal, characteristic, complemented
5#21447188C212yesmaximal, complemented
6#21447052C212yesmaximal, complemented
7#21446625C211yesmaximal, characteristic, complemented
8#21446537C211yesmaximal, characteristic, complemented
9#21446172C212yesmaximal, complemented
10#21445819C211yesmaximal, characteristic, complemented
11C2×C4×SD16×C61C211yesmaximal, characteristic, complemented
12C22×#11758201C2212yes—
13C2×#14519774C2211yescharacteristic
14C2×#14519773C2212yes—
15C2×#14519751—21no—
16C2×#14519751—21no—
17C2×Q8×Dic61C2212yes—
18#16839962—24no—
19#16839960—28no—
20C2×C4⋊#92418C2211yescharacteristic
21C2×#14519706C2212yes—
22C2×#14519699C2212yes—
23C2×#14519699C2212yes—
24C2×#14519698C2212yes—
25C2×#14519697C2212yes—
26#16839671—22no—
27#16839663C2212yes—
28#16839661—22no—
29#16839661C2212yes—
30#16839632—21no—
31#16839632C2211yescharacteristic
32C4×C4⋊Dic61—21no—
33C4×C4⋊Dic61C2211yescharacteristic
34#16839524C2212yes—
35#16839516C2212yes—
36C22×SD16×C61C2211yescharacteristic
37C2×C4×Q8×C61C2211yescharacteristic
38C2×C4×D4×C61C2211yescharacteristic
39C4×SD16×C61—24no—
40C2×#106×C61C2211yescharacteristic
41C2×#103×C61C2211yescharacteristic
42C2×#102×C61C2211yescharacteristic
43C2×C4×C8×C61C2211yescharacteristic
44C22×D244—21no—
45C22×D244—21no—
46C22×D244C2312yes—
47C22×C22⋊D61—22no—
48C22×C22⋊D61—22no—
49C2×#11758201—24no—
50C2×#11758201—24no—
51C2×#11758201—24no—
52C22×#92418C2312yes—
53#14519774—22no—
54#14519774—22no—
55#14519773—24no—
56#14519773—42no—
57#14519751—44no—
58#14519751—44no—
59Q8×Dic61—24no—
60Q8×Dic61—42no—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61—22no—
63C2×C4⋊Dic61—21no—
64C2×C4⋊Dic61—22no—
65C2×C4⋊Dic61D411yescharacteristic
66C2×C4⋊Dic61D411yescharacteristic
67C2×C4⋊Dic61C2312yes—
68C2×#11758198—22no—
69C2×#11758198—22no—
70C2×C4×Dic61—22no—
71C2×C4×Dic61C2312yes—
72C4⋊#92418—22no—
73#14519706—24no—
74#14519699—24no—
75#14519699—42no—
76#14519698—24no—
77#14519697—24no—
78#14519684—22no—
79#14519684D412yes—
80C22×Q8×C61C2311yescharacteristic
81C22×D4×C61C2311yescharacteristic
82C23×C4×C61—21no—
83C2×SD16×C61—22no—
84C2×SD16×C61—22no—
85C2×SD16×C61D414yescomplemented
86C22×C8×C61C2311yescharacteristic
87C22×C8×C61C2311yescharacteristic
88C4×Q8×C61—22no—
89C4×Q8×C61D414yescomplemented
90C4×D4×C61—22no—
91C4×D4×C61—22no—
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61C2311yescharacteristic
94C2×C4⋊C4×C61C2311yescharacteristic
95C2×C22⋊C4×C61—21no—
96C2×C42×C61—21no—
97C2×C42×C61C2311yescharacteristic
98#106×C61—22no—
99#103×C61—22no—
100#102×C61—22no—
101C4×C8×C61—22no—
102C24×C61—21no—
103C23×D61—22no—
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—21no—
111C2×D4×C61—21no—
112C2×D4×C61—21no—
113C2×D4×C61—21no—
114C2×D4×C61—21no—
115C2×D4×C61C2×D412yes—
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61—42no—
120C22×C4×C61—42no—
121C22×C4×C61—42no—
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2411yescharacteristic
125C2×C22⋊D61—44no—
126C2×C22⋊D61—44no—
127C2×C22⋊D61—44no—
128C2×C22⋊D61—44no—
129C2×C22⋊D61—82no—
130C2×C22⋊D61—82no—
131C22×Dic61—22no—
132C22×Dic61C2×D412yes—
133C22×Dic61C2×D412yes—
134C2×D244—22no—
135C2×D244—22no—
136C2×D244—22no—
137C2×D244—22no—
138C2×D244—22no—
139C2×D244—22no—
140C2×D244—42no—
141C2×D244—42no—
142C2×D244—42no—
143C2×D244—42no—
144C2×D244—81no—
145C2×D244—81no—
146SD16×C61—28no—
147C2×C8×C61—21no—
148C2×C8×C61—21no—
149C2×C8×C61—21no—
150C2×C8×C61—21no—
151C2×C8×C61—21no—
152C2×C8×C61C2×D412yes—
153C4⋊C4×C61—24no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61C2×D412yes—
158C22⋊C4×C61—42no—
159C42×C61—24no—
160C42×C61C2×D412yes—
161C42×C61#4112yes—
162#11758201—48no—
163#11758198—44no—
164#11758198—44no—
165C4⋊Dic61—22no—
166C4⋊Dic61—22no—
167C4⋊Dic61—22no—
168C4⋊Dic61—44no—
169C4⋊Dic61—42no—
170C4⋊Dic61—82no—
171C4⋊Dic61—81no—
172C4⋊Dic61SD1614yescomplemented
173C4×Dic61—22no—
174C4×Dic61—22no—
175C4×Dic61—44no—
176C2×#92418—22no—
177C2×#92418—22no—
178C2×#92418—22no—
179C23×C61—22no—
180C23×C61—41no—
181C23×C61—41no—
182C23×C61—41no—
183C23×C61—41no—
184C23×C61—41no—
185C23×C61—41no—
186C23×C61C22×D411yescharacteristic
187C22×D61—42no—
188C22×D61—42no—
189C22×D61—42no—
190C22×D61—42no—
191C22×D61—42no—
192C22×D61—42no—
193C22×D61—42no—
194Q8×C61—22no—
195Q8×C61—22no—
196Q8×C61—22no—
197Q8×C61#12114yescomplemented
198D4×C61—22no—
199D4×C61—22no—
200D4×C61—41no—
201D4×C61—41no—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—22no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—22no—
208C2×C4×C61—41no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—84no—
214C2×C4×C61#12112yes—
215C2×C4×C61#12112yes—
216C2×C4×C61C2×SD1612yes—
217C2×C4×C61AGL(1,ℤ/8)12yes—
218C2×C4×C61C22×D411yescharacteristic
219C2×C4×C61C22×D411yescharacteristic
220C2×C4×C61C2×#4111yescharacteristic
221C2×C4×C61C2×#4111yescharacteristic
222C2×C4×C61H(5,𝔽2)11yescharacteristic
223C2×C4×C612−1+411yescharacteristic
224C22⋊D61—88no—
225C22⋊D61—88no—
226C2×Dic61—22no—
227C2×Dic61—22no—
228C2×Dic61—22no—
229C2×Dic61—22no—
230C2×Dic61—22no—
231C2×Dic61—42no—
232C2×Dic61—42no—
233C2×Dic61—42no—
234C2×Dic61C2×SD1614yes—
235D244—42no—
236D244—42no—
237D244—42no—
238D244—42no—
239D244—84no—
240D244—84no—
241C8×C61—22no—
242C8×C61—41no—
243#92418—42no—
244#24728—611nomaximal, Sylow
245C22×C61—42no—
246C22×C61—42no—
247C22×C61—42no—
248C22×C61—81no—
249C22×C61—81no—
250C22×C61—81no—
251C22×C61—81no—
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—82no—
260C2×D61—82no—
261C2×D61—82no—
262C2×D61—82no—
263C2×D61—82no—
264C2×D61—82no—
265C2×D61—82no—
266C4×C61—24no—
267C4×C61—22no—
268C4×C61—42no—
269C4×C61—81no—
270C4×C61#46014yescomplemented
271C4×C61C2×#12111yescharacteristic
272C4×C61#53711yescharacteristic
273C4×C61#54711yescharacteristic
274C4×C61#55311yescharacteristic
275Dic61—24no—
276Dic61—42no—
277Dic61—82no—
278C2×#460—612no—
279C2×#460—611no—
280C2×C4×SD16D6111yescharacteristic, complemented
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—82no—
290C2×C61#241611yescharacteristic
291C2×C61C2×#46011yescharacteristic
292C2×C61#304511yescharacteristic
293C2×C61#327411yescharacteristic
294C2×C61D4×SD1611yescharacteristic
295C2×C61#336211yescharacteristic
296C2×C61#336811yescharacteristic
297D61—162no—
298C22×SD16—612no—
299C22×SD16C2×D6111yescharacteristic
300C2×#127—612no—
301C2×#127—611no—
302C2×#121—1221no—
303C2×#121—1221no—
304C2×C4×Q8—612no—
305C2×C4×Q8C2×D6111yescharacteristic
306C2×C4×D4C2×D6111yescharacteristic
307#460—1228no—
308#460—1224no—
309C4×SD16—24no—
310C2×#106C2×D6111yescharacteristic
311C2×C4⋊C8—612no—
312C2×C4⋊C8—611no—
313C2×#103C2×D6111yescharacteristic
314C2×#102—612no—
315C2×#102—612no—
316C2×#102—612no—
317C2×#102—612no—
318C2×#102C2×D6111yescharacteristic
319C2×C4×C8C2×D6111yescharacteristic
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—612no—
333C22×D4—1222no—
334C22×D4—1222no—
335C22×D4—1221no—
336C22×D4—1221no—
337C22×D4C22×D6111yescharacteristic
338C23×C4—21no—
339C2×SD16—22no—
340C2×SD16—22no—
341C2×SD16—1224no—
342C2×SD16—1224no—
343C2×SD16—1224no—
344C2×SD16D24414yescomplemented
345C22×C8—612no—
346C22×C8C22×D6111yescharacteristic
347C22×C8C22×D6111yescharacteristic
348#127—1224no—
349#127—1222no—
350#127—1222no—
351#127—2442no—
352#121—2444no—
353#121—2444no—
354C4×Q8—22no—
355C4×Q8—1224no—
356C4×Q8—2442no—
357C4×Q8C22⋊D6114yescomplemented
358C4×D4—22no—
359C4×D4—22no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—611no—
362C2×C4⋊C4—611no—
363C2×C4⋊C4—612no—
364C2×C4⋊C4—1222no—
365C2×C4⋊C4—1221no—
366C2×C4⋊C4—1222no—
367C2×C4⋊C4—1221no—
368C2×C4⋊C4C22×D6111yescharacteristic
369C2×C4⋊C4C22×D6111yescharacteristic
370C2×C22⋊C4—21no—
371C2×C22⋊C4—1222no—
372C2×C22⋊C4—1222no—
373C2×C42—21no—
374C2×C42—612no—
375C2×C42—1222no—
376C2×C42C22×D6111yescharacteristic
377#106—22no—
378C4⋊C8—1224no—
379C4⋊C8—1222no—
380#103—22no—
381#102—22no—
382#102—1224no—
383#102—1224no—
384#102—1224no—
385#102—2442no—
386C4×C8—22no—
387#95—612no—
388#95—1222no—
389C24—21no—
390C24—1222no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8C2×D24412yes—
395C2×Q8#1175822212yes—
396C2×Q8C2×C22⋊D6112yes—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—1222no—
403C2×D4—1222no—
404C2×D4—1222no—
405C2×D4—1222no—
406C2×D4—1222no—
407C2×D4—1222no—
408C2×D4—4881no—
409C2×D4—4881no—
410C2×D4—4882no—
411C2×D4—4882no—
412C2×D4—2444no—
413C2×D4—2444no—
414C2×D4—2444no—
415C2×D4—2444no—
416C2×D4—2442no—
417C2×D4—2442no—
418C2×D4—2442no—
419C2×D4—2442no—
420C2×D4C2×D24412yes—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—612no—
428C22×C4—612no—
429C22×C4—1222no—
430C22×C4D4×D6111yescharacteristic
431C22×C4D4×D6111yescharacteristic
432C22×C4C23×D6111yescharacteristic
433SD16—28no—
434SD16—2448no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—1222no—
441C2×C8—1222no—
442C2×C8—1222no—
443C2×C8C2×D24412yes—
444C4⋊C4—21no—
445C4⋊C4—24no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—614no—
449C4⋊C4—1222no—
450C4⋊C4—1222no—
451C4⋊C4—1222no—
452C4⋊C4—4881no—
453C4⋊C4—4882no—
454C4⋊C4—2444no—
455C4⋊C4—2442no—
456C4⋊C4C2×C22⋊D6112yes—
457C22⋊C4—42no—
458C22⋊C4—2444no—
459C22⋊C4—2444no—
460C42—24no—
461C42—1222no—
462C42—1222no—
463C42—2444no—
464C42#1175822412yes—
465C42C2×C22⋊D6112yes—
466C23—22no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—2442no—
474C23—2442no—
475C23—2442no—
476C23—2442no—
477C23—2442no—
478C23—2442no—
479C23—2442no—
480C23C2×D4×D6111yescharacteristic
481Q8—22no—
482Q8—22no—
483Q8—22no—
484Q8#1451975114yescomplemented
485D4—22no—
486D4—22no—
487D4—41no—
488D4—41no—
489D4—4884no—
490D4—4888no—
491D4—4888no—
492D4—4884no—
493D4—2442no—
494D4—2442no—
495D4—2442no—
496D4—2442no—
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—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—84no—
509C2×C4—614no—
510C2×C4—1222no—
511C2×C4—1222no—
512C2×C4—1222no—
513C2×C4—1222no—
514C2×C4—1222no—
515C2×C4—2442no—
516C2×C4—2442no—
517C2×C4—2442no—
518C2×C4#1451974912yes—
519C2×C4#1451974912yes—
520C2×C4SD16×D6112yes—
521C2×C4#1451980412yes—
522C2×C4C22×C22⋊D6111yescharacteristic
523C2×C4C22×D24411yescharacteristic
524C2×C4C2×#1175822411yescharacteristic
525C2×C4C2×#1175822211yescharacteristic
526C2×C4#1451982311yescharacteristic
527C2×C4#1451982411yescharacteristic
528C8—22no—
529C8—41no—
530C8—2442no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22—4882no—
539C22—4882no—
540C22—4882no—
541C22—4882no—
542C22—4882no—
543C22—4882no—
544C22—4882no—
545C22C2×#1451974911yescharacteristic
546C22D4×D24411yescharacteristic
547C22#1684042111yescharacteristic
548C22C2×SD16×D6111yescharacteristic
549C22C2×#1451980411yescharacteristic
550C22#1684056411yescharacteristic
551C22#1684056911yescharacteristic
552C4—24no—
553C4—22no—
554C4—42no—
555C4—81no—
556C4—1224no—
557C4—4882no—
558C4—2442no—
559C4#1683996014yescomplemented
560C4C2×#1451975111yescharacteristic
561C4#1684034011yescharacteristic
562C4#1684041611yescharacteristic
563C4#1684045811yescharacteristic
564C2—82no—
565C2—9762no—
566C2#2144925911yescharacteristic
567C2C2×#1683996011yescharacteristic
568C2#2145535611yescharacteristic
569C2#2145744011yescharacteristic
570C2SD16×D24411yescharacteristic
571C2#2145848811yescharacteristic
572C2#2145854611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C4×SD16×C61C2
socleC23×C61C22×D4
prime radical#21450765C2
hypercenterC2×C4×SD16D61
hypocenterC61#24728
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#16839962C2semidirect
2. C2×#16839960C2semidirect
4. Q8×C4⋊Dic61C2semidirect
5. #21447188C2semidirect
6. #21447052C2semidirect
7. #21446625C2semidirect
8. #21446537C2semidirect
9. #21446172C2semidirect
10. #21445819C2semidirect
11. C2×C4×SD16×C61C2semidirect
85. C2×SD16×C61D4semidirect
89. C4×Q8×C61D4semidirect
172. C4⋊Dic61SD16semidirect
197. Q8×C61#121semidirect
270. C4×C61#460semidirect
280. C2×C4×SD16D61semidirect
330. C61#24728semidirect
344. C2×SD16D244semidirect
357. C4×Q8C22⋊D61semidirect
484. Q8#14519751semidirect
559. C4#16839960semidirect

Maximal subgroups

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

subgroupindex
C2×#168399622
C2×#168399602
#214507652
Q8×C4⋊Dic612
#214471882
#214470522
#214466252
#214465372
#214461722
#214458192
C2×C4×SD16×C612
#2472861