Finite Group Databasebeta

← #29718988

Subgroups — #29718988

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×#16839895C211yesmaximal, characteristic
2C22×#14519697C211yesmaximal, characteristic, complemented
3C22⋊C4×D244C211yesmaximal, characteristic, complemented
4#21447223C212yesmaximal, complemented
5#21447129C212yesmaximal, complemented
6#21446248C211yesmaximal, characteristic, complemented
7#21446170C212yesmaximal, complemented
8#21445999C211yesmaximal, characteristic, complemented
9#21445905C211yesmaximal, characteristic, complemented
10#21445816C212yesmaximal, complemented
11C61×#2501C211yesmaximal, characteristic, complemented
12C23×D244C2211yescharacteristic
13C22×#11758199C2211yescharacteristic
14C22×#11758199C2211yescharacteristic
15C23×#92418C2211yescharacteristic
16C2×C4×D244C2212yes—
17C2×C22⋊C4×D61—21no—
18#16839895—24no—
19C2×C4⋊#92418C2212yescomplemented
20C2×#14519698C2212yescomplemented
21C2×#14519697—24no—
22C2×#14519697C2212yes—
23C2×#14519697C2211yescharacteristic
24C2×#14519697C2212yes—
25C2×#14519697C2211yescharacteristic
26C2×C22⋊#92418C2211yescharacteristic
27#16839664C2212yes—
28#16839640—22no—
29#16839622C2211yescharacteristic
30#16839617—21no—
31#16839524C2212yescomplemented
32#16839524C2212yescomplemented
33#16839524C2212yescomplemented
34#16839517C2212yescomplemented
35C2×#123×C61—21no—
36C2×#123×C61—21no—
37C2×#121×C61C2211yescharacteristic
38C22×C4⋊C4×C61C2211yescharacteristic
39C61×#399—22no—
40C61×#390C2212yescomplemented
41C61×#384C2212yescomplemented
42C4×C22⋊C4×C61C2211yescharacteristic
43C24×D61—21no—
44C22×D244—24no—
45C22×D244—21no—
46C22×D244—22no—
47C22×D244—42no—
48C22×D244D414yes—
49C22×D244C2311yescharacteristic
50C22×D244C2311yescharacteristic
51C22×C4×D61—22no—
52C2×#11758199—22no—
53C2×#11758199—22no—
54C2×#11758199—24no—
55C2×#11758199—42no—
56C22×#92418—22no—
57C22×#92418—42no—
58C22×#92418C2311yescharacteristic
59C22×#92418C2311yescharacteristic
60C22×#92418C2312yes—
61C4×D244—24no—
62C4×D244—42no—
63C2×C4⋊Dic61C2312yes—
64C2×C22⋊Dic61—21no—
65C2×#11758198—22no—
66C2×#11758198—22no—
67C2×#11758198—22no—
68C22⋊C4×D61—44no—
69C4⋊#92418—42no—
70#14519698—42no—
71#14519697—24no—
72#14519697—24no—
73#14519697—22no—
74#14519697—22no—
75C22⋊#92418—22no—
76#14519684—21no—
77C22×D4×C61—21no—
78C22×D4×C61C2311yescharacteristic
79C23×C4×C61—21no—
80C23×C4×C61C2311yescharacteristic
81#123×C61—44no—
82#123×C61—44no—
83#121×C61—22no—
84#121×C61—22no—
85C2×C4⋊C4×C61—22no—
86C2×C4⋊C4×C61—22no—
87C2×C4⋊C4×C61—22no—
88C2×C4⋊C4×C61—22no—
89C2×C4⋊C4×C61—21no—
90C2×C4⋊C4×C61C2311yescharacteristic
91C2×C4⋊C4×C61C2312yes—
92C2×C22⋊C4×C61—22no—
93C2×C22⋊C4×C61—22no—
94C2×C22⋊C4×C61—21no—
95C2×C22⋊C4×C61D411yescharacteristic
96C2×C22⋊C4×C61D411yescharacteristic
97C2×C42×C61C2312yes—
98#95×C61D412yes—
99C24×C61—21no—
100C24×C61C2×D411yescharacteristic
101C23×D61—21no—
102C23×D61—21no—
103C23×D61—24no—
104C23×D61—44no—
105C23×D61—42no—
106C23×D61—42no—
107C23×D61—82no—
108C2×D4×C61—21no—
109C2×D4×C61—21no—
110C2×D4×C61—22no—
111C2×D4×C61—22no—
112C2×D4×C61—42no—
113C2×D4×C61—42no—
114C2×D4×C61—42no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—21no—
119C22×C4×C61—42no—
120C22×C4×C61—42no—
121C22×C4×C61—42no—
122C22×C4×C61C2×D412yes—
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D412yes—
125C22×C4×C61C2×D412yes—
126C22×C4×C61#4112yes—
127C22×C4×C61C2411yescharacteristic
128C22×Dic61—22no—
129C2×D244—21no—
130C2×D244—22no—
131C2×D244—22no—
132C2×D244—22no—
133C2×D244—28no—
134C2×D244—24no—
135C2×D244—21no—
136C2×D244—44no—
137C2×D244—48no—
138C2×D244—44no—
139C2×D244C2×D412yes—
140C2×D244C2×D412yes—
141C2×D244#4114yes—
142C2×C4×D61—44no—
143C2×C4×D61—44no—
144C2×C4×D61—82no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—44no—
150C4⋊C4×C61—42no—
151C4⋊C4×C61—44no—
152C22⋊C4×C61—22no—
153C22⋊C4×C61—44no—
154C22⋊C4×C61—42no—
155C22⋊C4×C61—44no—
156C22⋊C4×C61D814yescomplemented
157C42×C61—22no—
158C42×C61—22no—
159C22⋊Dic61—42no—
160#11758199—44no—
161#11758199—44no—
162#11758198—44no—
163#11758198—44no—
164#11758198—82no—
165C4⋊Dic61—22no—
166C4⋊Dic61—22no—
167C2×#92418—21no—
168C2×#92418—21no—
169C2×#92418—22no—
170C2×#92418—22no—
171C2×#92418—22no—
172C2×#92418—22no—
173C2×#92418—22no—
174C2×#92418—44no—
175C23×C61—22no—
176C23×C61—22no—
177C23×C61—41no—
178C23×C61—42no—
179C23×C61—41no—
180C23×C61—41no—
181C23×C61—42no—
182C23×C61#12112yes—
183C23×C61C2×D812yes—
184C23×C61AGL(1,ℤ/8)12yes—
185C23×C61C22×D411yescharacteristic
186C22×D61—22no—
187C22×D61—24no—
188C22×D61—22no—
189C22×D61—48no—
190C22×D61—41no—
191C22×D61—41no—
192C22×D61—42no—
193C22×D61—42no—
194C22×D61—41no—
195C22×D61—41no—
196C22×D61—42no—
197C22×D61—48no—
198C22×D61—84no—
199C22×D61—88no—
200C22×D61—88no—
201D4×C61—41no—
202D4×C61—41no—
203D4×C61—42no—
204D4×C61—84no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—24no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—41no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61—41no—
218C2×C4×C61—42no—
219C2×C4×C61—41no—
220C2×C4×C61—84no—
221C2×C4×C61#12112yes—
222C2×C4×C61C2×D814yes—
223C2×C4×C61C22×D411yescharacteristic
224C2×C4×C61C2×#4112yes—
225C2×C4×C61C2×#4111yescharacteristic
226C2×C4×C61H(5,𝔽2)12yes—
227C2×Dic61—42no—
228C2×Dic61—42no—
229C2×Dic61—42no—
230D244—22no—
231D244—22no—
232D244—22no—
233D244—24no—
234D244—48no—
235D244—42no—
236D244#12314yescomplemented
237C4×D61—88no—
238#92418—41no—
239#92418—42no—
240#92418—41no—
241#24526—611nomaximal, Sylow
242C22×C61—22no—
243C22×C61—24no—
244C22×C61—22no—
245C22×C61—24no—
246C22×C61—81no—
247C22×C61—81no—
248C22×C61—82no—
249C22×C61—81no—
250C22×C61—82no—
251C22×C61#45914yes—
252C22×C61C2×#12111yescharacteristic
253C22×C61#54612yes—
254C22×C61C22×D811yescharacteristic
255C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
256C22×C61#57712yes—
257C2×D61—24no—
258C2×D61—42no—
259C2×D61—44no—
260C2×D61—42no—
261C2×D61—44no—
262C2×D61—42no—
263C2×D61—42no—
264C2×D61—42no—
265C2×D61—816no—
266C2×D61—82no—
267C2×D61—82no—
268C4×C61—22no—
269C4×C61—24no—
270C4×C61—42no—
271C4×C61—82no—
272C4×C61—81no—
273C4×C61C2×#12311yescharacteristic
274C4×C61#53411yescharacteristic
275C4×C61#55112yes—
276Dic61—82no—
277C2×#447—611no—
278C22×#102—611no—
279#2501D6111yescharacteristic, complemented
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—161no—
290C2×C61#241211yescharacteristic
291C2×C61C2×#45911yescharacteristic
292C2×C61#304211yescharacteristic
293C2×C61#329412yes—
294C2×C61#330612yes—
295D61—44no—
296D61—82no—
297C23×D4—611no—
298C22×D8—611no—
299C22×D8—611no—
300C23×C8—611no—
301C2×#123—21no—
302C2×#123—21no—
303C2×#121C2×D6111yescharacteristic
304C2×C4×D4—612no—
305C22×C4⋊C4C2×D6111yescharacteristic
306C22×C22⋊C4—1221no—
307#447—1224no—
308C2×C4⋊C8—612no—
309C2×#102—612no—
310C2×#102—611no—
311C2×#102—612no—
312C2×#102—611no—
313C2×#102—612no—
314C2×#102—1224no—
315C2×C22⋊C8—611no—
316#399—22no—
317#390—612no—
318#390C2×D6112yescomplemented
319#386—1222no—
320#384C2×D6112yescomplemented
321#380—611no—
322#379—1221no—
323C4×C22⋊C4C2×D6111yescharacteristic
324#340—612no—
325#340—612no—
326#340—612no—
327#336—612no—
328C61#2452611yescharacteristic, complemented, Sylow
329C25—1221no—
330C22×D4—21no—
331C22×D4—614no—
332C22×D4—611no—
333C22×D4—611no—
334C22×D4—1224no—
335C22×D4—1222no—
336C22×D4—1221no—
337C22×D4—2442no—
338C22×D4C22×D6111yescharacteristic
339C23×C4—21no—
340C23×C4—1222no—
341C23×C4C22×D6111yescharacteristic
342C2×D8—1222no—
343C2×D8—1222no—
344C2×D8—1224no—
345C2×D8—2442no—
346C22×C8—612no—
347C22×C8—611no—
348C22×C8—611no—
349C22×C8—1222no—
350C22×C8—2442no—
351#123—44no—
352#123—44no—
353#121—22no—
354#121—22no—
355C4×D4—1224no—
356C4×D4—2442no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—612no—
363C2×C4⋊C4C22×D6112yes—
364C2×C4⋊C4C22×D6111yescharacteristic
365C2×C22⋊C4—21no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—1222no—
369C2×C22⋊C4—1221no—
370C2×C22⋊C4—1222no—
371C2×C22⋊C4—1222no—
372C2×C22⋊C4—2444no—
373C2×C22⋊C4C22⋊D6111yescharacteristic
374C2×C22⋊C4C22⋊D6111yescharacteristic
375C2×C42C22×D6112yes—
376C4⋊C8—2442no—
377#102—1224no—
378#102—1224no—
379#102—1222no—
380#102—1222no—
381#102—2442no—
382C22⋊C8—1222no—
383#95—1221no—
384#95C22⋊D6112yes—
385C24—21no—
386C24—1221no—
387C24—1224no—
388C24—1221no—
389C24—4882no—
390C24—2444no—
391C24—2442no—
392C24—2442no—
393C24C2×C22⋊D6111yescharacteristic
394C2×D4—21no—
395C2×D4—22no—
396C2×D4—21no—
397C2×D4—22no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—612no—
402C2×D4—612no—
403C2×D4—614no—
404C2×D4—1228no—
405C2×D4—1222no—
406C2×D4—1221no—
407C2×D4—1221no—
408C2×D4—1222no—
409C2×D4—1222no—
410C2×D4—1224no—
411C2×D4—2444no—
412C2×D4—2448no—
413C2×D4—2444no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—1222no—
422C22×C4—4882no—
423C22×C4—2444no—
424C22×C4—2444no—
425C22×C4D4×D6112yes—
426C22×C4#1175822412yes—
427C22×C4C2×C22⋊D6112yes—
428C22×C4C2×C22⋊D6111yescharacteristic
429C22×C4C2×C22⋊D6112yes—
430C22×C4C23×D6111yescharacteristic
431D8—2444no—
432D8—2444no—
433C2×C8—1221no—
434C2×C8—1221no—
435C2×C8—1222no—
436C2×C8—1222no—
437C2×C8—1222no—
438C2×C8—1222no—
439C2×C8—1222no—
440C2×C8—2444no—
441C4⋊C4—22no—
442C4⋊C4—21no—
443C4⋊C4—22no—
444C4⋊C4—21no—
445C4⋊C4—44no—
446C4⋊C4—42no—
447C4⋊C4—44no—
448C4⋊C4—1222no—
449C4⋊C4—1222no—
450C22⋊C4—22no—
451C22⋊C4—44no—
452C22⋊C4—42no—
453C22⋊C4—44no—
454C22⋊C4—4882no—
455C22⋊C4—2444no—
456C22⋊C4—2442no—
457C22⋊C4—2444no—
458C22⋊C4#1175819914yescomplemented
459C42—22no—
460C42—22no—
461C23—22no—
462C23—22no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—42no—
467C23—42no—
468C23—1222no—
469C23—1224no—
470C23—1222no—
471C23—4888no—
472C23—4888no—
473C23—4884no—
474C23—2448no—
475C23—2448no—
476C23—2441no—
477C23—2441no—
478C23—2442no—
479C23—2442no—
480C23—2441no—
481C23—2441no—
482C23—2442no—
483C23#1451974812yes—
484C23C2×#1175819912yes—
485C23#1451980312yes—
486C23C22×C22⋊D6111yescharacteristic
487D4—41no—
488D4—41no—
489D4—42no—
490D4—84no—
491D4—614no—
492D4—1222no—
493D4—1222no—
494D4—1222no—
495D4—1224no—
496D4—2448no—
497D4—2442no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—84no—
514C2×C4—4888no—
515C2×C4—2442no—
516C2×C4—2442no—
517C2×C4—2442no—
518C2×C4#1451974812yes—
519C2×C4C2×#1175819914yes—
520C2×C4C2×D4×D6111yescharacteristic
521C2×C4#41×D6112yes—
522C2×C4C2×#1175822411yescharacteristic
523C2×C4#1451982312yes—
524C8—2441no—
525C8—2442no—
526C8—2441no—
527C22—24no—
528C22—24no—
529C22—22no—
530C22—22no—
531C22—82no—
532C22—81no—
533C22—82no—
534C22—81no—
535C22—81no—
536C22—1224no—
537C22—48816no—
538C22—4882no—
539C22—4882no—
540C22—2442no—
541C22—2442no—
542C22—2442no—
543C22—2444no—
544C22—2442no—
545C22—2444no—
546C22—2442no—
547C22#1683995414yes—
548C22C2×#1451974811yescharacteristic
549C22#1684040012yes—
550C22C22×#1175819911yescharacteristic
551C22C2×#1451980311yescharacteristic
552C22#1684057012yes—
553C4—24no—
554C4—22no—
555C4—42no—
556C4—82no—
557C4—81no—
558C4—4882no—
559C4#123×D6111yescharacteristic
560C4#1684032311yescharacteristic
561C4#1684043512yes—
562C2—24no—
563C2—161no—
564C2—4882no—
565C2—2444no—
566C2#2144921111yescharacteristic
567C2C2×#1683995411yescharacteristic
568C2#2145532711yescharacteristic
569C2#2145762612yes—
570C2#2145777412yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2501C2
socleC23×C61C22×D4
prime radicalC2×#16839895C2
hypercenter#2501D61
hypocenterC61#24526
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2501

Complements

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

NH ≅ G/Ntype
2. C22×#14519697C2semidirect
3. C22⋊C4×D244C2semidirect
4. #21447223C2semidirect
5. #21447129C2semidirect
6. #21446248C2semidirect
7. #21446170C2semidirect
8. #21445999C2semidirect
9. #21445905C2semidirect
10. #21445816C2semidirect
11. C61×#2501C2semidirect
19. C2×C4⋊#92418C22semidirect
20. C2×#14519698C22semidirect
31. #16839524C22semidirect
32. #16839524C22semidirect
33. #16839524C22semidirect
34. #16839517C22semidirect
40. C61×#390C22semidirect
41. C61×#384C22semidirect
156. C22⋊C4×C61D8semidirect
236. D244#123semidirect
279. #2501D61semidirect
318. #390C2×D61semidirect
320. #384C2×D61semidirect
328. C61#24526semidirect
458. C22⋊C4#11758199semidirect

Maximal subgroups

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

subgroupindex
C2×#168398952
C22×#145196972
C22⋊C4×D2442
#214472232
#214471292
#214462482
#214461702
#214459992
#214459052
#214458162
C61×#25012
#2452661