Finite Group Databasebeta

← #29714892

Subgroups — #29714892

Order 15616.

1428 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839954C211yesmaximal, characteristic, complemented
2C2×#16839872C211yesmaximal, characteristic, complemented
3#21451177C211yesmaximal, characteristic, complemented
4#21447050C212yesmaximal, complemented
5#21446890C212yesmaximal, complemented
6#21446760C211yesmaximal, characteristic, complemented
7#21446644C211yesmaximal, characteristic, complemented
8#21446167C212yesmaximal, complemented
9#21446080C212yesmaximal, complemented
10#21445829C211yesmaximal, characteristic, complemented
11C2×C61×#550C211yesmaximal, characteristic, complemented
12C22×#11758199C2211yescharacteristic
13C22×#11758199C2212yes—
14C2×C4×D244C2211yescharacteristic
15#16839954—24no—
16#16839872—24no—
17C2×#14519708C2211yescharacteristic
18C2×C4⋊#92418C2211yescharacteristic
19C2×#14519698C2211yescharacteristic
20C2×#14519698C2212yes—
21C2×#14519698C2212yes—
22C2×#14519698C2212yes—
23C2×#14519698C2212yes—
24C2×#14519697C2211yescharacteristic
25C2×#14519688C2211yescharacteristic
26#16839697—21no—
27#16839673—21no—
28#16839664C2212yes—
29#16839660C2212yes—
30#16839633—22no—
31#16839628C2211yescharacteristic
32#16839524C2212yes—
33#16839517C2212yes—
34#16839517C2212yes—
35C22×#41×C61—21no—
36C61×#550—24no—
37C61×#550—24no—
38C2×#127×C61C2212yes—
39C2×#127×C61C2211yescharacteristic
40C2×#121×C61—21no—
41C2×#121×C61—22no—
42C2×C4×Q8×C61C2211yescharacteristic
43C2×C4×D4×C61C2212yes—
44C2×C4×D4×C61C2211yescharacteristic
45C22×D244C2311yescharacteristic
46C22×C4×D61—21no—
47C2×#11758199—22no—
48C2×#11758199—22no—
49C2×#11758199—24no—
50C2×#11758199—24no—
51C2×#11758199—22no—
52C2×#11758199—42no—
53C22×#92418C2311yescharacteristic
54C22×#92418C2312yes—
55C22×#92418C2311yescharacteristic
56C4×D244—22no—
57C4×D244—22no—
58C2×C4⋊Dic61C2311yescharacteristic
59C2×C4⋊Dic61C2312yes—
60C2×#11758196—22no—
61C2×#11758198—21no—
62C2×#11758198—22no—
63C2×#11758198—22no—
64#14519708—22no—
65C4⋊#92418—22no—
66#14519698—24no—
67#14519698—22no—
68#14519698—24no—
69#14519698—42no—
70#14519698—42no—
71#14519697—22no—
72#14519688—22no—
73#14519684—21no—
74#14519684—21no—
75#14519684—22no—
76C2×#41×C61—28no—
77C2×#41×C61—48no—
78C22×Q8×C61D411yescharacteristic
79C22×D4×C61—21no—
80C22×D4×C61—22no—
81C22×D4×C61—22no—
82C22×D4×C61—21no—
83C22×D4×C61C2312yes—
84C22×D4×C61C2311yescharacteristic
85C23×C4×C61—22no—
86C23×C4×C61—21no—
87#127×C61—24no—
88#127×C61—24no—
89#127×C61—22no—
90#127×C61D414yes—
91#121×C61—28no—
92#121×C61—216no—
93C4×Q8×C61—22no—
94C4×Q8×C61—22no—
95C4×D4×C61—24no—
96C4×D4×C61—24no—
97C4×D4×C61—22no—
98C4×D4×C61D414yes—
99C2×C4⋊C4×C61D411yescharacteristic
100C2×C4⋊C4×C61D412yes—
101C2×C4⋊C4×C61C2311yescharacteristic
102C2×C22⋊C4×C61—22no—
103C2×C22⋊C4×C61—21no—
104C2×C42×C61C2312yes—
105C2×C42×C61C2311yescharacteristic
106C24×C61—22no—
107C24×C61—21no—
108C23×D61—21no—
109#41×C61—432no—
110C2×Q8×C61—22no—
111C2×Q8×C61—24no—
112C2×D4×C61—28no—
113C2×D4×C61—28no—
114C2×D4×C61—24no—
115C2×D4×C61—24no—
116C2×D4×C61—22no—
117C2×D4×C61—21no—
118C2×D4×C61—22no—
119C2×D4×C61—21no—
120C2×D4×C61—22no—
121C2×D4×C61—21no—
122C2×D4×C61—21no—
123C2×D4×C61—22no—
124C2×D4×C61—21no—
125C2×D4×C61—22no—
126C2×D4×C61—44no—
127C2×D4×C61—44no—
128C2×D4×C61—44no—
129C2×D4×C61—82no—
130C2×D4×C61—82no—
131C2×D4×C61—82no—
132C2×D4×C61C2×D414yes—
133C2×D4×C61C2×D412yes—
134C22×C4×C61—28no—
135C22×C4×C61—21no—
136C22×C4×C61—24no—
137C22×C4×C61—44no—
138C22×C4×C61—44no—
139C22×C4×C61—44no—
140C22×C4×C61C2×D412yes—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D412yes—
144C22×C4×C61C2411yescharacteristic
145C22×Dic61—21no—
146C22×Dic61—22no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244#4112yescomplemented
153C2×C4×D61—42no—
154C2×C4×D61—82no—
155C4⋊C4×C61—24no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61C2×D412yes—
159C22⋊C4×C61—28no—
160C22⋊C4×C61—24no—
161C42×C61—22no—
162C42×C61—22no—
163C42×C61C2×D412yes—
164C42×C61C2×D412yes—
165#11758199—28no—
166#11758199—48no—
167#11758198—42no—
168#11758198—44no—
169#11758198—82no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—22no—
173C4⋊Dic61—22no—
174#11758196—82no—
175C2×#92418—21no—
176C2×#92418—21no—
177C2×#92418—21no—
178C2×#92418—22no—
179C2×#92418—22no—
180C2×#92418—21no—
181C2×#92418—22no—
182C2×#92418—21no—
183C2×#92418#4112yescomplemented
184C23×C61—24no—
185C23×C61—22no—
186C23×C61—41no—
187C23×C61—42no—
188C23×C61—42no—
189C23×C61—41no—
190C23×C61—42no—
191C23×C61—42no—
192C23×C61—42no—
193C23×C61—42no—
194C23×C61—42no—
195C23×C61—42no—
196C23×C61C22×D411yescharacteristic
197C22×D61—22no—
198C22×D61—41no—
199C22×D61—41no—
200C22×D61—42no—
201C22×D61—42no—
202Q8×C61—28no—
203D4×C61—22no—
204D4×C61—22no—
205D4×C61—24no—
206D4×C61—24no—
207D4×C61—48no—
208D4×C61—416no—
209D4×C61—42no—
210D4×C61—42no—
211D4×C61—41no—
212D4×C61—41no—
213D4×C61—84no—
214D4×C61—88no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—48no—
224C2×C4×C61—416no—
225C2×C4×C61—42no—
226C2×C4×C61#12012yes—
227C2×C4×C61#12012yes—
228C2×C4×C61AGL(1,ℤ/8)14yes—
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×C4×C61H(5,𝔽2)11yescharacteristic
234C2×C4×C612−1+411yescharacteristic
235C2×Dic61—41no—
236C2×Dic61—42no—
237C2×Dic61—42no—
238C2×Dic61—42no—
239C2×Dic61—42no—
240D244—22no—
241D244—22no—
242D244—41no—
243D244—41no—
244C4×D61—84no—
245#92418—22no—
246#92418—41no—
247#92418—42no—
248#24196—611nomaximal, Sylow
249C22×C61—42no—
250C22×C61—44no—
251C22×C61—42no—
252C22×C61—44no—
253C22×C61—44no—
254C22×C61—42no—
255C22×C61—82no—
256C22×C61—82no—
257C22×C61—82no—
258C22×C61—82no—
259C22×C61—82no—
260C22×C61—82no—
261C22×C61C2×#12011yescharacteristic
262C22×C61#54811yescharacteristic
263C22×C61#55011yescharacteristic
264C22×C61C2×AGL(1,ℤ/8)12yes—
265C22×C61#57612yes—
266C2×D61—44no—
267C2×D61—42no—
268C2×D61—82no—
269C2×D61—81no—
270C2×D61—81no—
271C4×C61—22no—
272C4×C61—24no—
273C4×C61—22no—
274C4×C61—42no—
275C4×C61—42no—
276C4×C61#52511yescharacteristic
277C4×C61#53711yescharacteristic
278C4×C61#54611yescharacteristic
279C4×C61#54711yescharacteristic
280Dic61—82no—
281Dic61—81no—
282C2×#550D6111yescharacteristic, complemented
283C2×#459—611no—
284C2×#442—611no—
285#2580—611no—
286#2041—612no—
287#2011—612no—
288#1985—611no—
289#1961—611no—
290#1879—612no—
291#1864—612no—
292#1804—611no—
293C2×C61—82no—
294C2×C61—84no—
295C2×C61#240711yescharacteristic
296C2×C61#300311yescharacteristic
297C2×C61#300811yescharacteristic
298C2×C61#336612yes—
299C2×C61#339912yes—
300D61—82no—
301C22×#41—21no—
302C22×D8—611no—
303C22×D8—612no—
304#550—24no—
305#550—24no—
306C2×#127C2×D6112yes—
307C2×#127C2×D6111yescharacteristic
308C2×#121—22no—
309C2×#121—21no—
310C2×C4×Q8C2×D6111yescharacteristic
311C2×C4×D4—611no—
312C2×C4×D4C2×D6112yes—
313C2×C4×D4C2×D6111yescharacteristic
314#459—1224no—
315#442—1224no—
316C2×#106—611no—
317C2×C4⋊C8—611no—
318C2×#102—612no—
319C2×#102—612no—
320C2×#102—611no—
321C2×#102—611no—
322C2×#102—612no—
323C2×#102—612no—
324C2×#97—611no—
325#399—1221no—
326#393—1221no—
327#390—612no—
328#389—612no—
329#385—1222no—
330#382—611no—
331#340—612no—
332#336—612no—
333#336—612no—
334C61#2419611yescharacteristic, complemented, Sylow
335C2×#41—28no—
336C2×#41—48no—
337C22×Q8C22⋊D6111yescharacteristic
338C22×D4—21no—
339C22×D4—22no—
340C22×D4—22no—
341C22×D4—21no—
342C22×D4—611no—
343C22×D4C22×D6112yes—
344C22×D4C22×D6111yescharacteristic
345C23×C4—22no—
346C23×C4—21no—
347C23×C4—1221no—
348C2×D8—1222no—
349C2×D8—1222no—
350C2×D8—1222no—
351C2×D8—1224no—
352C2×D8—1224no—
353C2×D8—2442no—
354C22×C8—611no—
355C22×C8—612no—
356C22×C8—611no—
357#127—24no—
358#127—24no—
359#127—22no—
360#127C22⋊D6114yes—
361#121—28no—
362#121—216no—
363C4×Q8—22no—
364C4×Q8—22no—
365C4×D4—24no—
366C4×D4—24no—
367C4×D4—22no—
368C4×D4—1222no—
369C4×D4—1222no—
370C4×D4C22⋊D6114yes—
371C2×C4⋊C4—611no—
372C2×C4⋊C4—612no—
373C2×C4⋊C4—1222no—
374C2×C4⋊C4C22⋊D6112yes—
375C2×C4⋊C4C22⋊D6111yescharacteristic
376C2×C4⋊C4C22×D6111yescharacteristic
377C2×C22⋊C4—22no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—1222no—
380C2×C22⋊C4—1221no—
381C2×C22⋊C4—1222no—
382C2×C42C22×D6112yes—
383C2×C42C22×D6111yescharacteristic
384#106—1222no—
385C4⋊C8—1222no—
386#102—1224no—
387#102—1222no—
388#102—1224no—
389#102—1222no—
390#102—2442no—
391#102—2442no—
392#97—1222no—
393#95—1221no—
394#95—1221no—
395#95—1222no—
396C24—22no—
397C24—21no—
398C24—1221no—
399#41—432no—
400C2×Q8—22no—
401C2×Q8—24no—
402C2×D4—28no—
403C2×D4—28no—
404C2×D4—22no—
405C2×D4—21no—
406C2×D4—22no—
407C2×D4—21no—
408C2×D4—22no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—22no—
412C2×D4—21no—
413C2×D4—22no—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—82no—
420C2×D4—82no—
421C2×D4—82no—
422C2×D4—612no—
423C2×D4—1221no—
424C2×D4—1221no—
425C2×D4—1221no—
426C2×D4—1221no—
427C2×D4—1221no—
428C2×D4C2×C22⋊D6114yes—
429C2×D4C2×C22⋊D6112yes—
430C22×C4—28no—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—1222no—
437C22×C4—1221no—
438C22×C4—4882no—
439C22×C4—2442no—
440C22×C4C2×C22⋊D6112yes—
441C22×C4C2×C22⋊D6111yescharacteristic
442C22×C4C2×C22⋊D6111yescharacteristic
443C22×C4C2×C22⋊D6112yes—
444C22×C4C23×D6111yescharacteristic
445D8—1228no—
446D8—2448no—
447C2×C8—612no—
448C2×C8—1221no—
449C2×C8—1221no—
450C2×C8—1221no—
451C2×C8—1222no—
452C2×C8—1222no—
453C2×C8—1221no—
454C2×C8—1222no—
455C2×C8—1221no—
456C4⋊C4—21no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—1221no—
460C4⋊C4—1221no—
461C4⋊C4—1222no—
462C4⋊C4—1222no—
463C4⋊C4—4882no—
464C4⋊C4C2×C22⋊D6112yes—
465C22⋊C4—28no—
466C22⋊C4—24no—
467C22⋊C4—4882no—
468C22⋊C4—2442no—
469C22⋊C4—2444no—
470C42—22no—
471C42—22no—
472C42C2×C22⋊D6112yes—
473C42C2×C22⋊D6112yes—
474C23—24no—
475C23—22no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—41no—
483C23—42no—
484C23—42no—
485C23—41no—
486C23—1222no—
487C23—2441no—
488C23—2442no—
489C23—2442no—
490C23—2441no—
491C23C22×C22⋊D6111yescharacteristic
492Q8—28no—
493D4—22no—
494D4—22no—
495D4—24no—
496D4—24no—
497D4—48no—
498D4—416no—
499D4—42no—
500D4—42no—
501D4—41no—
502D4—41no—
503D4—84no—
504D4—88no—
505D4—1222no—
506D4—1222no—
507D4—2441no—
508D4—2441no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—48no—
518C2×C4—416no—
519C2×C4—42no—
520C2×C4—4884no—
521C2×C4—2441no—
522C2×C4—2442no—
523C2×C4—2442no—
524C2×C4—2442no—
525C2×C4—2442no—
526C2×C4#1451974612yes—
527C2×C4#1451974612yes—
528C2×C4#1451980314yes—
529C2×C4C22×C22⋊D6111yescharacteristic
530C2×C4C22×C22⋊D6111yescharacteristic
531C2×C4#41×D6111yescharacteristic
532C2×C4#41×D6111yescharacteristic
533C2×C4#1451982311yescharacteristic
534C2×C4#1451982411yescharacteristic
535C8—1222no—
536C8—2441no—
537C8—2442no—
538C22—42no—
539C22—44no—
540C22—42no—
541C22—44no—
542C22—44no—
543C22—42no—
544C22—82no—
545C22—82no—
546C22—82no—
547C22—82no—
548C22—82no—
549C22—82no—
550C22—4882no—
551C22—4881no—
552C22—4881no—
553C22—2444no—
554C22—2442no—
555C22C2×#1451974611yescharacteristic
556C22#1684041911yescharacteristic
557C22#1684043111yescharacteristic
558C22C2×#1451980312yes—
559C22#1684056512yes—
560C4—22no—
561C4—24no—
562C4—22no—
563C4—42no—
564C4—42no—
565C4—4882no—
566C4—4881no—
567C4#1684026911yescharacteristic
568C4#1684034011yescharacteristic
569C4#1684040011yescharacteristic
570C4#1684041111yescharacteristic
571C2—82no—
572C2—84no—
573C2—4882no—
574C2#2144915811yescharacteristic
575C2#2145504611yescharacteristic
576C2#2145508711yescharacteristic
577C2#2145852212yes—
578C2#2145886112yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#550C2
socleC23×C61C22×D4
hypercenterC2×#550D61
hypocenterC61#24196
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#550

Complements

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

NH ≅ G/Ntype
1. C2×#16839954C2semidirect
2. C2×#16839872C2semidirect
3. #21451177C2semidirect
4. #21447050C2semidirect
5. #21446890C2semidirect
6. #21446760C2semidirect
7. #21446644C2semidirect
8. #21446167C2semidirect
9. #21446080C2semidirect
10. #21445829C2semidirect
11. C2×C61×#550C2semidirect
152. C2×D244#41semidirect
183. C2×#92418#41semidirect
282. C2×#550D61semidirect
334. C61#24196semidirect

Maximal subgroups

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

subgroupindex
C2×#168399542
C2×#168398722
#214511772
#214470502
#214468902
#214467602
#214466442
#214461672
#214460802
#214458292
C2×C61×#5502
#2419661