Finite Group Databasebeta

← #29718992

Subgroups — #29718992

Order 15616.

1280 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839896C211yesmaximal, characteristic
2#21450286C211yesmaximal, characteristic, complemented
3D4×C22⋊Dic61C211yesmaximal, characteristic, complemented
4#21447225C212yesmaximal, complemented
5#21447128C212yesmaximal, complemented
6#21446247C211yesmaximal, characteristic, complemented
7#21446165C212yesmaximal, complemented
8#21445997C211yesmaximal, characteristic, complemented
9#21445905C211yesmaximal, characteristic, complemented
10#21445817C212yesmaximal, complemented
11C22×#102×C61C211yesmaximal, characteristic, complemented
12C22×#11758199C2211yescharacteristic
13C22×#11758199C2211yescharacteristic
14C2×#14519760—21no—
15C2×#14519760—21no—
16C2×#14519751C2211yescharacteristic
17C2×D4×Dic61C2212yes—
18C22×C22⋊Dic61—21no—
19#16839896—24no—
20C2×#14519706C2212yescomplemented
21C2×#14519699C2212yescomplemented
22C2×#14519698C2211yescharacteristic
23C2×#14519698C2211yescharacteristic
24C2×C22⋊#92418C2211yescharacteristic
25#16839697—22no—
26#16839665C2212yes—
27#16839663C2212yescomplemented
28#16839642—22no—
29#16839631C2212yescomplemented
30#16839622C2211yescharacteristic
31#16839618—21no—
32C4×C22⋊Dic61C2211yescharacteristic
33#16839525C2212yescomplemented
34#16839524C2212yescomplemented
35#16839524C2212yescomplemented
36#16839516C2212yescomplemented
37C23×D4×C61C2211yescharacteristic
38C23×C8×C61C2211yescharacteristic
39C22×C4⋊C4×C61C2211yescharacteristic
40C2×#102×C61—24no—
41C2×#102×C61C2212yes—
42C2×#102×C61C2212yes—
43C22×D244C2311yescharacteristic
44C22×C22⋊D61—21no—
45C23×Dic61—22no—
46C2×#11758199—22no—
47C2×#11758199—22no—
48C2×#11758199—24no—
49C2×#11758199—42no—
50C22×#92418C2311yescharacteristic
51C22×#92418C2311yescharacteristic
52#14519760—44no—
53#14519760—44no—
54#14519751—22no—
55#14519751—22no—
56D4×Dic61—24no—
57D4×Dic61—42no—
58C2×C4⋊Dic61C2312yes—
59C2×C4⋊Dic61C2311yescharacteristic
60C2×#11758196—22no—
61C2×#11758196—22no—
62C2×C22⋊Dic61—21no—
63C2×C22⋊Dic61—22no—
64C2×C22⋊Dic61—22no—
65C2×C22⋊Dic61—22no—
66C2×C22⋊Dic61—44no—
67C2×C22⋊Dic61D411yescharacteristic
68C2×C22⋊Dic61D411yescharacteristic
69C2×#11758198—21no—
70C2×#11758198—22no—
71C2×#11758198—22no—
72C2×C4×Dic61C2312yes—
73#14519706—42no—
74#14519699—42no—
75#14519698—22no—
76#14519698—22no—
77C22⋊#92418—22no—
78#14519684—21no—
79#14519684D412yes—
80C25×C61—21no—
81C22×D4×C61—24no—
82C22×D4×C61—21no—
83C22×D4×C61—22no—
84C22×D4×C61—42no—
85C22×D4×C61D414yes—
86C22×D4×C61C2311yescharacteristic
87C22×D4×C61C2311yescharacteristic
88C23×C4×C61—21no—
89C23×C4×C61C2311yescharacteristic
90C22×C8×C61—22no—
91C22×C8×C61—42no—
92C22×C8×C61C2312yes—
93C2×C4⋊C4×C61—22no—
94C2×C4⋊C4×C61—22no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61C2312yes—
97#102×C61—24no—
98#102×C61—24no—
99C24×C61—21no—
100C24×C61—21no—
101C24×C61—24no—
102C24×C61—44no—
103C24×C61—42no—
104C24×C61—42no—
105C24×C61—82no—
106C24×C61C2×D411yescharacteristic
107C23×D61—21no—
108C2×D4×C61—28no—
109C2×D4×C61—24no—
110C2×D4×C61—21no—
111C2×D4×C61—22no—
112C2×D4×C61—21no—
113C2×D4×C61—22no—
114C2×D4×C61—22no—
115C2×D4×C61—44no—
116C2×D4×C61—48no—
117C2×D4×C61—44no—
118C2×D4×C61C2×D412yes—
119C2×D4×C61C2×D412yes—
120C2×D4×C61#4114yes—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—42no—
125C22×C4×C61—42no—
126C22×C4×C61—42no—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D412yes—
129C22×C4×C61#4112yes—
130C22×C4×C61C2411yescharacteristic
131C2×C22⋊D61—42no—
132C2×C22⋊D61—42no—
133C2×C22⋊D61—42no—
134C22×Dic61—22no—
135C22×Dic61—21no—
136C22×Dic61—44no—
137C22×Dic61—44no—
138C22×Dic61—82no—
139C22×Dic61C2×D412yes—
140C22×Dic61C2×D412yes—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244—22no—
144C2×D244—22no—
145C2×C8×C61—22no—
146C2×C8×C61—22no—
147C2×C8×C61—22no—
148C2×C8×C61—44no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—42no—
152C22⋊Dic61—22no—
153C22⋊Dic61—44no—
154C22⋊Dic61—42no—
155C22⋊Dic61—44no—
156C22⋊Dic61—82no—
157C22⋊Dic61D814yescomplemented
158#11758199—44no—
159#11758199—44no—
160#11758198—44no—
161#11758198—42no—
162#11758198—44no—
163C4⋊Dic61—21no—
164C4⋊Dic61—22no—
165C4⋊Dic61—21no—
166C4⋊Dic61—22no—
167#11758196—44no—
168#11758196—44no—
169C4×Dic61—22no—
170C4×Dic61—22no—
171C2×#92418—21no—
172C2×#92418—21no—
173C2×#92418—22no—
174C2×#92418—22no—
175C23×C61—22no—
176C23×C61—22no—
177C23×C61—22no—
178C23×C61—24no—
179C23×C61—22no—
180C23×C61—48no—
181C23×C61—41no—
182C23×C61—41no—
183C23×C61—42no—
184C23×C61—42no—
185C23×C61—41no—
186C23×C61—41no—
187C23×C61—42no—
188C23×C61—48no—
189C23×C61—84no—
190C23×C61—88no—
191C23×C61—88no—
192C23×C61#12112yes—
193C23×C61C2×D812yes—
194C23×C61AGL(1,ℤ/8)12yes—
195C23×C61C22×D411yescharacteristic
196C22×D61—41no—
197C22×D61—42no—
198C22×D61—41no—
199C22×D61—41no—
200C22×D61—42no—
201D4×C61—22no—
202D4×C61—22no—
203D4×C61—22no—
204D4×C61—24no—
205D4×C61—48no—
206D4×C61—42no—
207D4×C61#12314yescomplemented
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—24no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—84no—
216C2×C4×C61#12112yes—
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C2×#4112yes—
219C2×C4×C61C2×#4111yescharacteristic
220C2×C4×C61H(5,𝔽2)12yes—
221C22⋊D61—84no—
222C2×Dic61—22no—
223C2×Dic61—22no—
224C2×Dic61—22no—
225C2×Dic61—22no—
226C2×Dic61—22no—
227C2×Dic61—41no—
228C2×Dic61—42no—
229C2×Dic61—41no—
230C2×Dic61—42no—
231C2×Dic61—41no—
232C2×Dic61—42no—
233C2×Dic61—88no—
234C2×Dic61C2×D814yes—
235D244—41no—
236D244—41no—
237D244—42no—
238C8×C61—42no—
239#92418—41no—
240#92418—41no—
241#24526—611nomaximal, Sylow
242C22×C61—22no—
243C22×C61—24no—
244C22×C61—22no—
245C22×C61—24no—
246C22×C61—24no—
247C22×C61—42no—
248C22×C61—44no—
249C22×C61—42no—
250C22×C61—44no—
251C22×C61—42no—
252C22×C61—42no—
253C22×C61—42no—
254C22×C61—816no—
255C22×C61—82no—
256C22×C61—82no—
257C22×C61#45914yes—
258C22×C61C2×#12111yescharacteristic
259C22×C61#54612yes—
260C22×C61C22×D811yescharacteristic
261C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
262C22×C61#57712yes—
263C2×D61—81no—
264C2×D61—81no—
265C2×D61—82no—
266C2×D61—81no—
267C2×D61—82no—
268C4×C61—22no—
269C4×C61—82no—
270C4×C61C2×#12311yescharacteristic
271C4×C61#53411yescharacteristic
272C4×C61#55112yes—
273Dic61—24no—
274Dic61—42no—
275Dic61—81no—
276Dic61—82no—
277C2×#447—611no—
278C22×#102D6111yescharacteristic, complemented
279#2501—611no—
280D4×C22⋊C4—611no—
281#2071—612no—
282#2054—612no—
283#1893—611no—
284#1879—612no—
285#1850—611no—
286#1823—611no—
287#1802—612no—
288C2×C61—24no—
289C2×C61—44no—
290C2×C61—82no—
291C2×C61#241211yescharacteristic
292C2×C61C2×#45911yescharacteristic
293C2×C61#304211yescharacteristic
294C2×C61#329412yes—
295C2×C61#330612yes—
296D61—161no—
297C23×D4C2×D6111yescharacteristic
298C22×D8—611no—
299C22×D8—611no—
300C23×C8C2×D6111yescharacteristic
301C2×#123—1221no—
302C2×#123—1221no—
303C2×#121—611no—
304C2×C4×D4—612no—
305C22×C4⋊C4C2×D6111yescharacteristic
306C22×C22⋊C4—1221no—
307#447—1224no—
308C2×C4⋊C8—612no—
309C2×#102—24no—
310C2×#102—611no—
311C2×#102—611no—
312C2×#102—612no—
313C2×#102C2×D6112yes—
314C2×#102C2×D6112yes—
315C2×C22⋊C8—611no—
316#399—1222no—
317#390—612no—
318#390—612no—
319#386—1222no—
320#384—612no—
321#380—611no—
322#379—1221no—
323C4×C22⋊C4—611no—
324#340—612no—
325#340—612no—
326#340—612no—
327#336—612no—
328C61#2452611yescharacteristic, complemented, Sylow
329C25—21no—
330C22×D4—24no—
331C22×D4—21no—
332C22×D4—22no—
333C22×D4—42no—
334C22×D4—611no—
335C22×D4—1221no—
336C22×D4C22⋊D6114yes—
337C22×D4C22×D6111yescharacteristic
338C22×D4C22×D6111yescharacteristic
339C23×C4—21no—
340C23×C4—1222no—
341C23×C4C22×D6111yescharacteristic
342C2×D8—1224no—
343C2×D8—1222no—
344C2×D8—1222no—
345C2×D8—2442no—
346C22×C8—22no—
347C22×C8—42no—
348C22×C8—611no—
349C22×C8—611no—
350C22×C8C22×D6112yes—
351#123—2444no—
352#123—2444no—
353#121—1222no—
354#121—1222no—
355C4×D4—1224no—
356C4×D4—2442no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—612no—
361C2×C4⋊C4—611no—
362C2×C4⋊C4—1222no—
363C2×C4⋊C4—1222no—
364C2×C4⋊C4C22×D6112yes—
365C2×C22⋊C4—611no—
366C2×C22⋊C4—611no—
367C2×C22⋊C4—1222no—
368C2×C22⋊C4—1222no—
369C2×C22⋊C4—1222no—
370C2×C22⋊C4—1222no—
371C2×C22⋊C4—1222no—
372C2×C22⋊C4—1221no—
373C2×C22⋊C4—1221no—
374C2×C22⋊C4—2444no—
375C2×C42—612no—
376C4⋊C8—2442no—
377#102—24no—
378#102—24no—
379#102—1222no—
380#102—1222no—
381#102—2442no—
382C22⋊C8—1222no—
383#95—612no—
384#95—1221no—
385C24—21no—
386C24—21no—
387C24—24no—
388C24—44no—
389C24—42no—
390C24—42no—
391C24—82no—
392C24—1221no—
393C24D4×D6111yescharacteristic
394C2×D4—28no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—24no—
401C2×D4—44no—
402C2×D4—48no—
403C2×D4—44no—
404C2×D4—1221no—
405C2×D4—1221no—
406C2×D4—1222no—
407C2×D4—1222no—
408C2×D4—2442no—
409C2×D4—2442no—
410C2×D4—2442no—
411C2×D4#1175822214yes—
412C2×D4C2×C22⋊D6112yes—
413C2×D4C2×C22⋊D6112yes—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—612no—
421C22×C4—612no—
422C22×C4—1222no—
423C22×C4—1221no—
424C22×C4—4882no—
425C22×C4—2444no—
426C22×C4—2444no—
427C22×C4D4×D6111yescharacteristic
428C22×C4#1175822412yes—
429C22×C4C2×C22⋊D6112yes—
430C22×C4C23×D6111yescharacteristic
431D8—2444no—
432D8—2444no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—44no—
437C2×C8—1221no—
438C2×C8—1221no—
439C2×C8—1222no—
440C2×C8—1222no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—42no—
444C4⋊C4—1221no—
445C4⋊C4—1221no—
446C4⋊C4—1222no—
447C4⋊C4—1222no—
448C4⋊C4—2444no—
449C4⋊C4—2444no—
450C22⋊C4—614no—
451C22⋊C4—1222no—
452C22⋊C4—4882no—
453C22⋊C4—2444no—
454C22⋊C4—2444no—
455C22⋊C4—2444no—
456C22⋊C4—2444no—
457C22⋊C4—2442no—
458C22⋊C4—2442no—
459C42—1222no—
460C42—1222no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—24no—
465C23—22no—
466C23—48no—
467C23—41no—
468C23—41no—
469C23—42no—
470C23—48no—
471C23—41no—
472C23—41no—
473C23—42no—
474C23—42no—
475C23—84no—
476C23—88no—
477C23—88no—
478C23—2441no—
479C23—2441no—
480C23—2441no—
481C23—2442no—
482C23—2442no—
483C23#1451974912yes—
484C23D8×D6112yes—
485C23#1451980112yes—
486C23C2×D4×D6111yescharacteristic
487D4—22no—
488D4—22no—
489D4—22no—
490D4—24no—
491D4—48no—
492D4—42no—
493D4—4884no—
494D4—2441no—
495D4—2441no—
496D4—2442no—
497D4#1451976014yescomplemented
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—84no—
506C2×C4—614no—
507C2×C4—1222no—
508C2×C4—1222no—
509C2×C4—1222no—
510C2×C4—1222no—
511C2×C4—1222no—
512C2×C4—4888no—
513C2×C4—2441no—
514C2×C4—2442no—
515C2×C4—2441no—
516C2×C4—2441no—
517C2×C4—2442no—
518C2×C4—2442no—
519C2×C4#1451974912yes—
520C2×C4C22×C22⋊D6111yescharacteristic
521C2×C4C2×#1175822411yescharacteristic
522C2×C4C2×#1175822212yes—
523C2×C4#1451982212yes—
524C8—42no—
525C8—2441no—
526C8—2441no—
527C22—24no—
528C22—24no—
529C22—22no—
530C22—22no—
531C22—24no—
532C22—44no—
533C22—42no—
534C22—44no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—816no—
540C22—82no—
541C22—82no—
542C22—4882no—
543C22—4881no—
544C22—4882no—
545C22—4881no—
546C22—4881no—
547C22#1683995514yes—
548C22C2×#1451974911yescharacteristic
549C22#1684040212yes—
550C22C2×D8×D6111yescharacteristic
551C22C2×#1451980111yescharacteristic
552C22#1684056812yes—
553C4—22no—
554C4—82no—
555C4—1224no—
556C4—4881no—
557C4—4882no—
558C4—2442no—
559C4C2×#1451976011yescharacteristic
560C4#1684032111yescharacteristic
561C4#1684043812yes—
562C2—24no—
563C2—44no—
564C2—82no—
565C2—9761no—
566C2#2144921511yescharacteristic
567C2C2×#1683995511yescharacteristic
568C2#2145532811yescharacteristic
569C2#2145762812yes—
570C2#2145777612yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC22×#102×C61C2
socleC23×C61C22×D4
prime radicalC2×#16839896C2
hypercenterC22×#102D61
hypocenterC61#24526
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#102

Complements

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

NH ≅ G/Ntype
2. #21450286C2semidirect
3. D4×C22⋊Dic61C2semidirect
4. #21447225C2semidirect
5. #21447128C2semidirect
6. #21446247C2semidirect
7. #21446165C2semidirect
8. #21445997C2semidirect
9. #21445905C2semidirect
10. #21445817C2semidirect
11. C22×#102×C61C2semidirect
20. C2×#14519706C22semidirect
21. C2×#14519699C22semidirect
27. #16839663C22semidirect
29. #16839631C22semidirect
33. #16839525C22semidirect
34. #16839524C22semidirect
35. #16839524C22semidirect
36. #16839516C22semidirect
157. C22⋊Dic61D8semidirect
207. D4×C61#123semidirect
278. C22×#102D61semidirect
328. C61#24526semidirect
497. D4#14519760semidirect

Maximal subgroups

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

subgroupindex
C2×#168398962
#214502862
D4×C22⋊Dic612
#214472252
#214471282
#214462472
#214461652
#214459972
#214459052
#214458172
C22×#102×C612
#2452661