Finite Group Databasebeta

← #29718986

Subgroups — #29718986

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
2C22×#14519698C211yesmaximal, characteristic, complemented
3#21450290C211yesmaximal, characteristic, complemented
4#21447222C212yesmaximal, complemented
5#21447129C212yesmaximal, complemented
6#21446245C211yesmaximal, characteristic, complemented
7#21446167C212yesmaximal, complemented
8#21445999C211yesmaximal, characteristic, complemented
9#21445903C211yesmaximal, characteristic, complemented
10#21445819C212yesmaximal, complemented
11D4×C22⋊C4×C61C211yesmaximal, characteristic, complemented
12C22×#11758199C2211yescharacteristic
13C22×#11758199C2211yescharacteristic
14C23×#92418C2211yescharacteristic
15C2×#14519761—21no—
16C2×#14519761—21no—
17C2×#14519751C2211yescharacteristic
18C22×C4⋊Dic61C2211yescharacteristic
19#16839896—24no—
20C2×C4⋊#92418C2212yescomplemented
21C2×#14519698—24no—
22C2×#14519698C2212yes—
23C2×#14519698C2211yescharacteristic
24C2×#14519698C2212yes—
25C2×#14519698C2211yescharacteristic
26C2×#14519697C2212yescomplemented
27C2×C22⋊#92418C2211yescharacteristic
28#16839699—22no—
29#16839664C2212yescomplemented
30#16839632C2212yescomplemented
31#16839524C2212yescomplemented
32#16839524C2212yescomplemented
33#16839524C2212yescomplemented
34#16839517C2212yescomplemented
35C23×D4×C61C2211yescharacteristic
36C2×C4×D4×C61C2212yes—
37C22×C22⋊C4×C61—21no—
38C61×#390C2212yes—
39C61×#386—22no—
40C61×#380C2211yescharacteristic
41C61×#379—21no—
42C4×C22⋊C4×C61C2211yescharacteristic
43C22×D244C2311yescharacteristic
44C22×C22⋊D61—21no—
45C23×Dic61—21no—
46C2×#11758199—22no—
47C2×#11758199—22no—
48C2×#11758199—24no—
49C2×#11758199—42no—
50C22×#92418—22no—
51C22×#92418—42no—
52C22×#92418C2311yescharacteristic
53C22×#92418C2311yescharacteristic
54C22×#92418C2312yes—
55#14519761—44no—
56#14519761—44no—
57#14519751—22no—
58#14519751—22no—
59C2×C4⋊Dic61—22no—
60C2×C4⋊Dic61—22no—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61—22no—
63C2×C4⋊Dic61—22no—
64C2×C4⋊Dic61C2312yes—
65C2×C4⋊Dic61C2311yescharacteristic
66C2×#11758198—21no—
67C2×#11758198—22no—
68C2×#11758198—22no—
69C4⋊#92418—42no—
70#14519698—24no—
71#14519698—24no—
72#14519698—22no—
73#14519698—22no—
74#14519697—42no—
75C22⋊#92418—22no—
76C25×C61—21no—
77C22×D4×C61—24no—
78C22×D4×C61—21no—
79C22×D4×C61—22no—
80C22×D4×C61—42no—
81C22×D4×C61D414yes—
82C22×D4×C61C2311yescharacteristic
83C22×D4×C61C2311yescharacteristic
84C23×C4×C61—22no—
85C23×C4×C61C2311yescharacteristic
86C4×D4×C61—24no—
87C4×D4×C61—42no—
88C2×C4⋊C4×C61C2312yes—
89C2×C22⋊C4×C61—21no—
90C2×C22⋊C4×C61—22no—
91C2×C22⋊C4×C61—22no—
92C2×C22⋊C4×C61—22no—
93C2×C22⋊C4×C61—44no—
94C2×C22⋊C4×C61D411yescharacteristic
95C2×C22⋊C4×C61D411yescharacteristic
96C2×C42×C61C2312yes—
97#95×C61—21no—
98#95×C61D412yes—
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—44no—
125C22×C4×C61—44no—
126C22×C4×C61—82no—
127C22×C4×C61C2×D412yes—
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D412yes—
130C22×C4×C61C2×D412yes—
131C22×C4×C61#4112yes—
132C22×C4×C61C2411yescharacteristic
133C2×C22⋊D61—42no—
134C2×C22⋊D61—42no—
135C2×C22⋊D61—42no—
136C22×Dic61—22no—
137C22×Dic61—21no—
138C22×Dic61—42no—
139C22×Dic61—42no—
140C22×Dic61—42no—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244—22no—
144C2×D244—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—22no—
147C22⋊C4×C61—22no—
148C22⋊C4×C61—44no—
149C22⋊C4×C61—42no—
150C22⋊C4×C61—44no—
151C22⋊C4×C61—82no—
152C22⋊C4×C61D814yescomplemented
153C42×C61—22no—
154C42×C61—22no—
155#11758199—44no—
156#11758199—44no—
157#11758198—44no—
158#11758198—42no—
159#11758198—44no—
160C4⋊Dic61—22no—
161C4⋊Dic61—21no—
162C4⋊Dic61—22no—
163C4⋊Dic61—21no—
164C4⋊Dic61—44no—
165C4⋊Dic61—42no—
166C4⋊Dic61—44no—
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—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—22no—
212C2×C4×C61—22no—
213C2×C4×C61—24no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61—88no—
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—
227C22⋊D61—84no—
228C2×Dic61—42no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—42no—
232C2×Dic61—41no—
233C2×Dic61—42no—
234C2×Dic61—84no—
235D244—41no—
236D244—41no—
237D244—42no—
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—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—24no—
270C4×C61—42no—
271C4×C61—82no—
272C4×C61C2×#12311yescharacteristic
273C4×C61#53411yescharacteristic
274C4×C61#55112yes—
275Dic61—82no—
276Dic61—81no—
277C2×#447—611no—
278C22×#102—611no—
279#2501—611no—
280D4×C22⋊C4D6111yescharacteristic, complemented
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×C8—611no—
301C2×#123—1221no—
302C2×#123—1221no—
303C2×#121—611no—
304C2×C4×D4C2×D6112yes—
305C22×C4⋊C4—611no—
306C22×C22⋊C4—21no—
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—1222no—
317#390—612no—
318#390C2×D6112yes—
319#386—22no—
320#384—612no—
321#380C2×D6111yescharacteristic
322#379—21no—
323C4×C22⋊C4C2×D6111yescharacteristic
324#340—612no—
325#340—612no—
326#340—612no—
327#336—612no—
328C61#2452611yescharacteristic, complemented, Sylow
329C25—21no—
330C22×D4—24no—
331C22×D4—22no—
332C22×D4—21no—
333C22×D4—42no—
334C22×D4—611no—
335C22×D4—1221no—
336C22×D4D24414yes—
337C22×D4C22×D6111yescharacteristic
338C22×D4C22×D6111yescharacteristic
339C23×C4—22no—
340C23×C4—1221no—
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—2444no—
352#123—2444no—
353#121—1222no—
354#121—1222no—
355C4×D4—24no—
356C4×D4—42no—
357C2×C4⋊C4—612no—
358C2×C4⋊C4—611no—
359C2×C4⋊C4—1222no—
360C2×C4⋊C4—1222no—
361C2×C4⋊C4—1221no—
362C2×C4⋊C4—1222no—
363C2×C4⋊C4—1222no—
364C2×C4⋊C4C22×D6112yes—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—44no—
370C2×C22⋊C4—1221no—
371C2×C22⋊C4—1222no—
372C2×C22⋊C4—1222no—
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—21no—
384#95C22⋊D6112yes—
385C24—21no—
386C24—21no—
387C24—24no—
388C24—44no—
389C24—42no—
390C24—42no—
391C24—82no—
392C24—1221no—
393C24C2×C22⋊D6111yescharacteristic
394C2×D4—28no—
395C2×D4—22no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—24no—
401C2×D4—44no—
402C2×D4—48no—
403C2×D4—44no—
404C2×D4—1221no—
405C2×D4—1222no—
406C2×D4—1221no—
407C2×D4—1222no—
408C2×D4—2442no—
409C2×D4—2442no—
410C2×D4—2442no—
411C2×D4C2×D24412yes—
412C2×D4C2×D24412yes—
413C2×D4#1175822414yes—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—82no—
420C22×C4—1222no—
421C22×C4—1221no—
422C22×C4—2442no—
423C22×C4—2442no—
424C22×C4—2442no—
425C22×C4C2×D24412yes—
426C22×C4#1175822212yes—
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—22no—
443C4⋊C4—1222no—
444C4⋊C4—1221no—
445C4⋊C4—1222no—
446C4⋊C4—1221no—
447C4⋊C4—2444no—
448C4⋊C4—2442no—
449C4⋊C4—2444no—
450C22⋊C4—22no—
451C22⋊C4—44no—
452C22⋊C4—42no—
453C22⋊C4—44no—
454C22⋊C4—82no—
455C22⋊C4—2444no—
456C22⋊C4—2442no—
457C22⋊C4—2444no—
458C22⋊C4#1175819914yescomplemented
459C42—22no—
460C42—22no—
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#1451975112yes—
484C23C2×#1175819912yes—
485C23#1451980212yes—
486C23C22×C22⋊D6111yescharacteristic
487D4—22no—
488D4—22no—
489D4—22no—
490D4—24no—
491D4—48no—
492D4—42no—
493D4—4884no—
494D4—2441no—
495D4—2441no—
496D4—2442no—
497D4#1451976114yescomplemented
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—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—88no—
511C2×C4—4884no—
512C2×C4—2441no—
513C2×C4—2442no—
514C2×C4—2442no—
515C2×C4—2441no—
516C2×C4—2442no—
517C2×C4—2441no—
518C2×C4#1451975112yes—
519C2×C4C2×#1175819914yes—
520C2×C4C22×D24411yescharacteristic
521C2×C4C2×#1175822412yes—
522C2×C4C2×#1175822211yescharacteristic
523C2×C4#1451982212yes—
524C8—2441no—
525C8—2442no—
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#1683995714yes—
548C22C2×#1451975111yescharacteristic
549C22#1684040612yes—
550C22C22×#1175819911yescharacteristic
551C22C2×#1451980211yescharacteristic
552C22#1684057312yes—
553C4—24no—
554C4—22no—
555C4—42no—
556C4—82no—
557C4—4882no—
558C4—4881no—
559C4C2×#1451976111yescharacteristic
560C4#1684032411yescharacteristic
561C4#1684043812yes—
562C2—24no—
563C2—44no—
564C2—82no—
565C2—9761no—
566C2#2144920911yescharacteristic
567C2C2×#1683995711yescharacteristic
568C2#2145533111yescharacteristic
569C2#2145763212yes—
570C2#2145778012yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingD4×C22⋊C4×C61C2
socleC23×C61C22×D4
prime radicalC2×#16839896C2
hypercenterD4×C22⋊C4D61
hypocenterC61#24526
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ D4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×#14519698C2semidirect
3. #21450290C2semidirect
4. #21447222C2semidirect
5. #21447129C2semidirect
6. #21446245C2semidirect
7. #21446167C2semidirect
8. #21445999C2semidirect
9. #21445903C2semidirect
10. #21445819C2semidirect
11. D4×C22⋊C4×C61C2semidirect
20. C2×C4⋊#92418C22semidirect
26. C2×#14519697C22semidirect
29. #16839664C22semidirect
30. #16839632C22semidirect
31. #16839524C22semidirect
32. #16839524C22semidirect
33. #16839524C22semidirect
34. #16839517C22semidirect
152. C22⋊C4×C61D8semidirect
207. D4×C61#123semidirect
280. D4×C22⋊C4D61semidirect
328. C61#24526semidirect
458. C22⋊C4#11758199semidirect
497. D4#14519761semidirect

Maximal subgroups

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

subgroupindex
C2×#168398962
C22×#145196982
#214502902
#214472222
#214471292
#214462452
#214461672
#214459992
#214459032
#214458192
D4×C22⋊C4×C612
#2452661