Finite Group Databasebeta

← #29714768

Subgroups — #29714768

Order 15616.

1214 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840498C211yesmaximal, characteristic, complemented
2C2×C4×#11758200C211yesmaximal, characteristic, complemented
3#21450741C211yesmaximal, characteristic, complemented
4#21447058C212yesmaximal, complemented
5#21446894C212yesmaximal, complemented
6#21446754C211yesmaximal, characteristic, complemented
7#21446626C211yesmaximal, characteristic, complemented
8#21446169C212yesmaximal, complemented
9#21446078C212yesmaximal, complemented
10#21445825C211yesmaximal, characteristic, complemented
11C2×C61×#459C211yesmaximal, characteristic
12C22×#11758200C2211yescharacteristic
13#16840498—24no—
14#16840498—24no—
15C2×#14519764—21no—
16C2×#14519764C2212yes—
17C2×#14519758—22no—
18C2×#14519756—21no—
19C2×C4×Dic122C2211yescharacteristic
20C2×C4×Dic122C2211yescharacteristic
21C2×D4×Dic61—21no—
22C2×D4×Dic61C2212yes—
23C4×#11758200—24no—
24C2×#14519708C2211yescharacteristic
25C2×#14519702C2212yes—
26C2×#14519700C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×#14519698C2212yes—
29C2×#14519698C2212yes—
30C2×C4×#92418C2211yescharacteristic
31#16839685—22no—
32#16839665C2212yes—
33#16839658C2212yes—
34#16839632C2211yescharacteristic
35#16839622—21no—
36#16839525C2212yes—
37#16839516C2212yes—
38#16839516C2212yes—
39C22×D8×C61C2212yes—
40C2×#127×C61C2211yescharacteristic
41C2×C4×D4×C61C2211yescharacteristic
42C61×#459—24no—
43C2×C4⋊C8×C61C2211yescharacteristic
44C2×#102×C61C2212yes—
45C22×Dic122D411yescharacteristic
46C22×Dic122C2311yescharacteristic
47C23×Dic61—22no—
48C2×#11758200—22no—
49C2×#11758200—22no—
50C2×#11758200—22no—
51C22×#92418C2311yescharacteristic
52C22×#92418C2311yescharacteristic
53#14519764—28no—
54#14519764—24no—
55#14519764—24no—
56#14519758—216no—
57#14519756—44no—
58C4×Dic122—22no—
59C4×Dic122—22no—
60C4×Dic122—22no—
61C4×Dic122D414yes—
62D4×Dic61—28no—
63D4×Dic61—24no—
64D4×Dic61—24no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61D411yescharacteristic
68C2×C4⋊Dic61C2312yes—
69C2×C4⋊Dic61C2311yescharacteristic
70C2×#11758196—21no—
71C2×#11758196—22no—
72C2×#11758196D412yes—
73C2×C22⋊Dic61—22no—
74C2×C22⋊Dic61—22no—
75C2×C22⋊Dic61—22no—
76C2×C22⋊Dic61—22no—
77C2×C22⋊Dic61—22no—
78C2×C4×Dic61—21no—
79C2×C4×Dic61C2312yes—
80#14519708—22no—
81#14519702—42no—
82#14519700—22no—
83#14519698—22no—
84#14519698—24no—
85#14519698—42no—
86C4×#92418—22no—
87#14519684—22no—
88C22×D4×C61—21no—
89C22×D4×C61—21no—
90C22×D4×C61C2312yes—
91C22×D4×C61C2311yescharacteristic
92C23×C4×C61—21no—
93C2×D8×C61—24no—
94C2×D8×C61—24no—
95C2×D8×C61—42no—
96C22×C8×C61C2312yes—
97#127×C61—22no—
98#127×C61D414yes—
99C4×D4×C61—22no—
100C4×D4×C61—22no—
101C2×C4⋊C4×C61C2311yescharacteristic
102C2×C22⋊C4×C61—21no—
103C2×C42×C61C2311yescharacteristic
104C4⋊C8×C61—22no—
105#102×C61—24no—
106C24×C61—22no—
107C24×C61—21no—
108C2×D4×C61—24no—
109C2×D4×C61—24no—
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61—22no—
113C2×D4×C61—22no—
114C2×D4×C61—22no—
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—22no—
118C2×D4×C61—22no—
119C2×D4×C61—21no—
120C2×D4×C61—42no—
121C2×D4×C61—42no—
122C2×D4×C61—81no—
123C2×D4×C61—81no—
124C2×D4×C61C2×D414yes—
125C2×D4×C61#4112yes—
126C22×C4×C61—21no—
127C22×C4×C61—42no—
128C22×C4×C61—42no—
129C22×C4×C61—42no—
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2411yescharacteristic
133C22×Dic61—21no—
134C22×Dic61—21no—
135C22×Dic61—28no—
136C22×Dic61—22no—
137C22×Dic61—82no—
138C22×Dic61—82no—
139C22×Dic61C2×D412yes—
140C22×Dic61C2×D412yes—
141C2×Dic122—21no—
142C2×Dic122—21no—
143C2×Dic122—21no—
144C2×Dic122—21no—
145C2×Dic122—22no—
146C2×Dic122—22no—
147C2×Dic122—22no—
148C2×Dic122—21no—
149C2×Dic122C2×D412yes—
150D8×C61—48no—
151C2×C8×C61—22no—
152C2×C8×C61—22no—
153C2×C8×C61—22no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C22⋊C4×C61—42no—
157C42×C61C2×D412yes—
158C42×C61C2×D412yes—
159C22⋊Dic61—28no—
160C22⋊Dic61—28no—
161C22⋊Dic61—28no—
162C22⋊Dic61—44no—
163C22⋊Dic61—82no—
164#11758200—28no—
165C4⋊Dic61—22no—
166C4⋊Dic61—21no—
167C4⋊Dic61—22no—
168C4⋊Dic61—22no—
169C4⋊Dic61—42no—
170C4⋊Dic61—81no—
171C4⋊Dic61C2×D412yes—
172#11758196—24no—
173#11758196—24no—
174#11758196—44no—
175C4×Dic61—24no—
176C4×Dic61—22no—
177C4×Dic61—22no—
178C2×#92418—21no—
179C2×#92418—21no—
180C2×#92418—21no—
181C2×#92418—21no—
182C2×#92418—21no—
183C2×#92418#4112yes—
184C23×C61—24no—
185C23×C61—22no—
186C23×C61—41no—
187C23×C61—42no—
188C23×C61—42no—
189C23×C61—41no—
190C23×C61—41no—
191C23×C61—42no—
192C23×C61—41no—
193C23×C61—42no—
194C23×C61—41no—
195C23×C61—42no—
196C23×C61—42no—
197C23×C61—41no—
198C23×C61C22×D411yescharacteristic
199D4×C61—22no—
200D4×C61—22no—
201D4×C61—24no—
202D4×C61—24no—
203D4×C61—42no—
204D4×C61—41no—
205D4×C61—42no—
206D4×C61—41no—
207D4×C61—48no—
208D4×C61—84no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—84no—
215C2×C4×C61#12012yes—
216C2×C4×C61#12012yes—
217C2×C4×C61#13512yes—
218C2×C4×C61AGL(1,ℤ/8)12yes—
219C2×C4×C61C22×D411yescharacteristic
220C2×C4×C61C22×D411yescharacteristic
221C2×C4×C61C2×#4111yescharacteristic
222C2×C4×C61C2×#4111yescharacteristic
223C2×C4×C61H(5,𝔽2)11yescharacteristic
224C2×C4×C612−1+411yescharacteristic
225C2×Dic61—22no—
226C2×Dic61—22no—
227C2×Dic61—22no—
228C2×Dic61—22no—
229C2×Dic61—22no—
230C2×Dic61—22no—
231C2×Dic61—22no—
232C2×Dic61—22no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61—42no—
238C2×Dic61—42no—
239C2×Dic61—42no—
240C2×Dic61—88no—
241Dic122—22no—
242Dic122—22no—
243Dic122—41no—
244Dic122—41no—
245Dic122—44no—
246C8×C61—42no—
247#92418—22no—
248#92418—41no—
249#24186—611nomaximal, Sylow
250C22×C61—42no—
251C22×C61—42no—
252C22×C61—42no—
253C22×C61—44no—
254C22×C61—44no—
255C22×C61—44no—
256C22×C61—82no—
257C22×C61—82no—
258C22×C61—81no—
259C22×C61—82no—
260C22×C61—81no—
261C22×C61—82no—
262C22×C61—81no—
263C22×C61—81no—
264C22×C61C2×#12011yescharacteristic
265C22×C61#54811yescharacteristic
266C22×C61#55011yescharacteristic
267C22×C61C2×#13511yescharacteristic
268C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
269C22×C61#57611yescharacteristic
270C22×C61#57711yescharacteristic
271C4×C61—22no—
272C4×C61—81no—
273C4×C61#44914yes—
274C4×C61#52511yescharacteristic
275C4×C61#53711yescharacteristic
276C4×C61#54611yescharacteristic
277C4×C61#54711yescharacteristic
278Dic61—42no—
279Dic61—42no—
280Dic61—42no—
281Dic61—42no—
282Dic61—82no—
283C2×#559—611no—
284C2×#459D6111yescharacteristic
285C2×C4×SD16—611no—
286#2538—611no—
287#2042—612no—
288#2012—612no—
289#1984—611no—
290#1957—611no—
291#1879—612no—
292#1864—612no—
293#1803—611no—
294C2×C61—82no—
295C2×C61—84no—
296C2×C61#240711yescharacteristic
297C2×C61C2×#44911yescharacteristic
298C2×C61#301211yescharacteristic
299C2×C61#336111yescharacteristic
300C2×C61#337311yescharacteristic
301C2×C61#339411yescharacteristic
302C2×C61#339911yescharacteristic
303C22×SD16—611no—
304C22×D8C2×D6112yes—
305#559—1224no—
306#559—1224no—
307C2×#127C2×D6111yescharacteristic
308C2×#124—612no—
309C2×#124—1221no—
310C2×#123—1222no—
311C2×C22⋊Q8—1221no—
312C2×C4×Q8—611no—
313C2×C4×Q8—611no—
314C2×C4×D4—612no—
315C2×C4×D4—1221no—
316C2×C4×D4C2×D6111yescharacteristic
317#459—24no—
318C4×SD16—1224no—
319C2×#106—611no—
320C2×C4⋊C8C2×D6111yescharacteristic
321C2×#103—611no—
322C2×#103—612no—
323C2×#102—612no—
324C2×#102—611no—
325C2×#102—612no—
326C2×#102C2×D6112yes—
327C2×C4×C8—611no—
328#396—1222no—
329#390—612no—
330#389—612no—
331#384—611no—
332#380—1221no—
333#340—612no—
334#336—612no—
335#336—612no—
336C61#2418611yescharacteristic, complemented, Sylow
337C22×Q8—611no—
338C22×Q8—611no—
339C22×D4—21no—
340C22×D4—21no—
341C22×D4C22×D6111yescharacteristic
342C22×D4C22×D6112yes—
343C23×C4—21no—
344C23×C4—1222no—
345C2×SD16—1222no—
346C2×SD16—1222no—
347C2×SD16—1222no—
348C2×D8—24no—
349C2×D8—24no—
350C2×D8—42no—
351C22×C8—611no—
352C22×C8—611no—
353C22×C8C22×D6112yes—
354#127—22no—
355#127C22⋊D6114yes—
356#124—1224no—
357#124—1228no—
358#124—1224no—
359#123—12216no—
360C22⋊Q8—2444no—
361C4×Q8—614no—
362C4×Q8—1222no—
363C4×Q8—1222no—
364C4×Q8—1222no—
365C4×D4—22no—
366C4×D4—22no—
367C4×D4—1224no—
368C4×D4—1228no—
369C4×D4—1224no—
370C2×C4⋊C4—611no—
371C2×C4⋊C4—611no—
372C2×C4⋊C4—612no—
373C2×C4⋊C4—612no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4—1222no—
377C2×C4⋊C4—1221no—
378C2×C4⋊C4C22×D6111yescharacteristic
379C2×C22⋊C4—21no—
380C2×C22⋊C4—1222no—
381C2×C22⋊C4—1222no—
382C2×C22⋊C4—1222no—
383C2×C22⋊C4—1222no—
384C2×C22⋊C4—1222no—
385C2×C42—612no—
386C2×C42—1221no—
387C2×C42C22×D6111yescharacteristic
388#106—1222no—
389C4⋊C8—22no—
390#103—1222no—
391#103—2442no—
392#102—24no—
393#102—1222no—
394#102—1224no—
395#102—2442no—
396C4×C8—1222no—
397#95—1222no—
398C24—21no—
399C24—22no—
400C2×Q8—612no—
401C2×Q8—1221no—
402C2×Q8—1222no—
403C2×Q8—1222no—
404C2×Q8—1222no—
405C2×Q8—1221no—
406C2×Q8—1221no—
407C2×Q8—1221no—
408C2×Q8—1221no—
409C2×D4—21no—
410C2×D4—22no—
411C2×D4—21no—
412C2×D4—22no—
413C2×D4—21no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—21no—
417C2×D4—22no—
418C2×D4—21no—
419C2×D4—24no—
420C2×D4—24no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—81no—
424C2×D4—81no—
425C2×D4#1175822412yes—
426C2×D4C2×C22⋊D6114yes—
427C22×C4—21no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—612no—
432C22×C4—612no—
433C22×C4—1228no—
434C22×C4—1221no—
435C22×C4—1221no—
436C22×C4—1222no—
437C22×C4—4882no—
438C22×C4—4882no—
439C22×C4D4×D6111yescharacteristic
440C22×C4D4×D6111yescharacteristic
441C22×C4C23×D6111yescharacteristic
442SD16—1228no—
443D8—48no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—612no—
448C2×C8—1221no—
449C2×C8—1221no—
450C2×C8—1221no—
451C2×C8—1221no—
452C2×C8—1221no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—612no—
456C4⋊C4—1221no—
457C4⋊C4—1224no—
458C4⋊C4—1222no—
459C4⋊C4—1224no—
460C4⋊C4—1222no—
461C4⋊C4—1222no—
462C4⋊C4—4881no—
463C4⋊C4—2444no—
464C4⋊C4—2442no—
465C22⋊C4—42no—
466C22⋊C4—1228no—
467C22⋊C4—1228no—
468C22⋊C4—1228no—
469C22⋊C4—4882no—
470C22⋊C4—2444no—
471C42—1222no—
472C42—1222no—
473C42—1224no—
474C42D4×D6112yes—
475C42C2×C22⋊D6112yes—
476C23—24no—
477C23—22no—
478C23—41no—
479C23—41no—
480C23—42no—
481C23—41no—
482C23—42no—
483C23—41no—
484C23—42no—
485C23—42no—
486C23—41no—
487C23—42no—
488C23—42no—
489C23—41no—
490C23C2×D4×D6111yescharacteristic
491Q8—1222no—
492Q8—1222no—
493Q8—2441no—
494Q8—2441no—
495Q8—2444no—
496D4—22no—
497D4—22no—
498D4—24no—
499D4—24no—
500D4—42no—
501D4—41no—
502D4—42no—
503D4—41no—
504D4—48no—
505D4—84no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—84no—
512C2×C4—1222no—
513C2×C4—1222no—
514C2×C4—1222no—
515C2×C4—1222no—
516C2×C4—1222no—
517C2×C4—1222no—
518C2×C4—1222no—
519C2×C4—1222no—
520C2×C4—4888no—
521C2×C4—2442no—
522C2×C4—2442no—
523C2×C4—2442no—
524C2×C4—2441no—
525C2×C4—2441no—
526C2×C4—2441no—
527C2×C4—2441no—
528C2×C4#1451974512yes—
529C2×C4#1451974512yes—
530C2×C4#1451979612yes—
531C2×C4#1451980112yes—
532C2×C4C2×D4×D6111yescharacteristic
533C2×C4C22×C22⋊D6111yescharacteristic
534C2×C4#41×D6111yescharacteristic
535C2×C4C2×#1175822411yescharacteristic
536C2×C4#1451982211yescharacteristic
537C2×C4#1451982411yescharacteristic
538C8—42no—
539C8—1222no—
540C8—2441no—
541C22—44no—
542C22—42no—
543C22—44no—
544C22—42no—
545C22—42no—
546C22—44no—
547C22—81no—
548C22—82no—
549C22—81no—
550C22—82no—
551C22—81no—
552C22—81no—
553C22—82no—
554C22—82no—
555C22C2×#1451974511yescharacteristic
556C22#1684041811yescharacteristic
557C22#1684043011yescharacteristic
558C22C2×#1451979611yescharacteristic
559C22C2×#1451980111yescharacteristic
560C22#1684056611yescharacteristic
561C22#1684057211yescharacteristic
562C4—22no—
563C4—81no—
564C4—4882no—
565C4—2442no—
566C4—2442no—
567C4—2442no—
568C4—2442no—
569C4#1683990414yes—
570C4#1684027211yescharacteristic
571C4#1684034111yescharacteristic
572C4#1684040611yescharacteristic
573C4#1684041111yescharacteristic
574C2—84no—
575C2—82no—
576C2#2144916011yescharacteristic
577C2C2×#1683990411yescharacteristic
578C2#2145512211yescharacteristic
579C2#2145848311yescharacteristic
580C2#2145861511yescharacteristic
581C2#2145881911yescharacteristic
582C2#2145886511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC2×C61×#459C2
socleC23×C61C22×D4
hypercenterC2×#459D61
hypocenterC61#24186
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#459

Complements

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

NH ≅ G/Ntype
1. C2×#16840498C2semidirect
2. C2×C4×#11758200C2semidirect
3. #21450741C2semidirect
4. #21447058C2semidirect
5. #21446894C2semidirect
6. #21446754C2semidirect
7. #21446626C2semidirect
8. #21446169C2semidirect
9. #21446078C2semidirect
10. #21445825C2semidirect
336. C61#24186semidirect

Maximal subgroups

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

subgroupindex
C2×#168404982
C2×C4×#117582002
#214507412
#214470582
#214468942
#214467542
#214466262
#214461692
#214460782
#214458252
C2×C61×#4592
#2418661