Finite Group Databasebeta

← #29733313

Subgroups — #29733313

Order 15616.

1226 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840477C211yesmaximal, characteristic, complemented
2C2×#16840121C211yesmaximal, characteristic, complemented
3#21450759C211yesmaximal, characteristic, complemented
4#21446819C212yesmaximal, complemented
5#21446814C212yesmaximal, complemented
6#21446770C211yesmaximal, characteristic, complemented
7#21446616C211yesmaximal, characteristic, complemented
8#21446138C212yesmaximal, complemented
9#21446128C212yesmaximal, complemented
10#21445644C211yesmaximal, characteristic, complemented
11C2×C61×#463C211yesmaximal, characteristic, complemented
12#16840477—24no—
13#16840477—24no—
14C2×#14519777C2211yescharacteristic
15C2×#14519763—22no—
16C2×#14519753C2212yes—
17C2×#14519751—21no—
18C2×#14519748C2212yes—
19C2×#14519748C2212yes—
20C2×C4×D244C2211yescharacteristic
21C2×#14519736—21no—
22C2×#14519735C2212yes—
23#16840121—24no—
24C2×#14519701C2212yes—
25C2×#14519698C2212yes—
26C2×#14519698C2212yes—
27C2×#14519689C2212yes—
28C2×#14519689C2212yes—
29C2×#14519688C2211yescharacteristic
30#16839698—21no—
31#16839671—21no—
32#16839662—22no—
33C4×C4⋊Dic61C2211yescharacteristic
34#16839525C2212yes—
35#16839525C2212yes—
36#16839524C2212yes—
37C22×SD16×C61C2211yescharacteristic
38C22×D8×C61C2211yescharacteristic
39C2×#124×C61C2211yescharacteristic
40C2×C4×D4×C61C2211yescharacteristic
41C61×#463—24no—
42C2×C4⋊C8×C61C2211yescharacteristic
43C2×#103×C61C2211yescharacteristic
44C2×#102×C61C2211yescharacteristic
45C22×Dic122—21no—
46C22×D244D411yescharacteristic
47C22×C22⋊D61—22no—
48C22×C22⋊D61—22no—
49C22×C4×D61C2312yes—
50C22×#92418C2312yes—
51#14519777—22no—
52#14519777D414yescomplemented
53#14519763—216no—
54#14519753—24no—
55#14519753—42no—
56#14519751—28no—
57#14519748—24no—
58#14519748—24no—
59#14519748—24no—
60#14519748—42no—
61C4×D244—22no—
62C4×D244—22no—
63#14519736—28no—
64#14519735—24no—
65#14519735—24no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61D411yescharacteristic
68C2×C4⋊Dic61C2312yes—
69C2×C4⋊Dic61C2312yes—
70C2×#11758196—22no—
71C2×C22⋊Dic61—22no—
72C2×C22⋊Dic61—22no—
73C2×#11758198—22no—
74C2×#11758198—22no—
75C2×#11758198D412yes—
76C2×C4×Dic61—22no—
77C2×C4×Dic61—21no—
78#14519701—24no—
79#14519698—24no—
80#14519698—24no—
81#14519689—42no—
82#14519689—42no—
83#14519688—22no—
84#14519684—21no—
85C22×Q8×C61C2311yescharacteristic
86C22×D4×C61C2311yescharacteristic
87C22×D4×C61C2311yescharacteristic
88C23×C4×C61—21no—
89C2×SD16×C61—22no—
90C2×SD16×C61—42no—
91C2×D8×C61—22no—
92C2×D8×C61—42no—
93C22×C8×C61C2311yescharacteristic
94C22×C8×C61C2311yescharacteristic
95#124×C61—22no—
96#124×C61—22no—
97C4×D4×C61—22no—
98C4×D4×C61D414yescomplemented
99C2×C4⋊C4×C61C2311yescharacteristic
100C2×C22⋊C4×C61—21no—
101C2×C22⋊C4×C61—21no—
102C2×C22⋊C4×C61—21no—
103C2×C42×C61C2311yescharacteristic
104C4⋊C8×C61—22no—
105#103×C61—22no—
106#102×C61—22no—
107C24×C61—21no—
108C24×C61—21no—
109C23×D61C2×D412yes—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—22no—
113C2×Q8×C61—22no—
114C2×D4×C61—21no—
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—22no—
118C2×D4×C61—22no—
119C2×D4×C61—22no—
120C2×D4×C61—22no—
121C2×D4×C61C2×D412yes—
122C22×C4×C61—21no—
123C22×C4×C61—24no—
124C22×C4×C61—21no—
125C22×C4×C61—44no—
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2411yescharacteristic
129C2×C22⋊D61—28no—
130C2×C22⋊D61—44no—
131C2×C22⋊D61—44no—
132C2×C22⋊D61—82no—
133C2×C22⋊D61—82no—
134C2×C22⋊D61—82no—
135C22×Dic61—22no—
136C22×Dic61—22no—
137C22×Dic61C2×D412yes—
138C2×D244—22no—
139C2×D244—42no—
140C2×C4×D61—22no—
141C2×C4×D61—22no—
142C2×C4×D61—22no—
143C2×C4×D61—22no—
144C2×C4×D61—22no—
145C2×C4×D61—22no—
146C2×Dic122—24no—
147C2×Dic122—81no—
148C2×Dic122—81no—
149SD16×C61—44no—
150D8×C61—44no—
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61—22no—
154C2×C8×C61—22no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61C2×D412yes—
157C22⋊C4×C61—24no—
158C22⋊C4×C61—42no—
159C22⋊C4×C61—42no—
160C42×C61C2×D412yes—
161C42×C61C2×D412yes—
162C22⋊Dic61—28no—
163C22⋊Dic61—44no—
164#11758198—28no—
165#11758198—24no—
166#11758198—44no—
167C4⋊Dic61—24no—
168C4⋊Dic61—22no—
169C4⋊Dic61—22no—
170C4⋊Dic61—22no—
171C4⋊Dic61C2×D414yes—
172C4⋊Dic61#4114yes—
173#11758196—44no—
174C4×Dic61—28no—
175C4×Dic61—81no—
176C2×#92418—22no—
177C2×#92418—22no—
178C2×#92418—22no—
179C23×C61—22no—
180C23×C61—42no—
181C23×C61—42no—
182C23×C61—41no—
183C23×C61—41no—
184C23×C61—41no—
185C23×C61—42no—
186C23×C61—42no—
187C23×C61—41no—
188C23×C61—41no—
189C23×C61C22×D411yescharacteristic
190C22×D61—22no—
191C22×D61—22no—
192C22×D61—22no—
193C22×D61—22no—
194C22×D61—22no—
195C22×D61—22no—
196C22×D61—22no—
197Q8×C61—41no—
198Q8×C61—41no—
199Q8×C61—42no—
200D4×C61—22no—
201D4×C61—22no—
202D4×C61—41no—
203D4×C61—41no—
204D4×C61—42no—
205D4×C61—42no—
206C2×C4×C61—22no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—22no—
210C2×C4×C61—48no—
211C2×C4×C61—42no—
212C2×C4×C61—41no—
213C2×C4×C61—42no—
214C2×C4×C61#12012yes—
215C2×C4×C61#12012yes—
216C2×C4×C61C22×D411yescharacteristic
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C2×#4112yes—
219C2×C4×C61H(5,𝔽2)12yes—
220C22⋊D61—88no—
221C22⋊D61—88no—
222C2×Dic61—22no—
223C2×Dic61—24no—
224C2×Dic61—22no—
225C2×Dic61—24no—
226C2×Dic61—22no—
227C2×Dic61—42no—
228C2×Dic61—42no—
229C2×Dic61—42no—
230C2×Dic61—42no—
231D244—44no—
232C4×D61—42no—
233C4×D61—42no—
234C4×D61—42no—
235C4×D61—42no—
236Dic122—84no—
237C8×C61—41no—
238C8×C61—41no—
239#92418—42no—
240#25578—611nomaximal, Sylow
241C22×C61—42no—
242C22×C61—42no—
243C22×C61—42no—
244C22×C61—82no—
245C22×C61—82no—
246C22×C61—81no—
247C22×C61—81no—
248C22×C61—81no—
249C22×C61—82no—
250C22×C61—82no—
251C22×C61C2×#12011yescharacteristic
252C22×C61#54612yes—
253C22×C61#57512yes—
254C22×C61#57611yescharacteristic
255C22×C61#57711yescharacteristic
256C2×D61—42no—
257C2×D61—42no—
258C2×D61—42no—
259C2×D61—42no—
260C2×D61—42no—
261C2×D61—42no—
262C2×D61—42no—
263C4×C61—22no—
264C4×C61—22no—
265C4×C61—42no—
266C4×C61—81no—
267C4×C61#53211yescharacteristic
268C4×C61#53511yescharacteristic
269C4×C61#55012yes—
270Dic61—42no—
271Dic61—44no—
272Dic61—44no—
273C2×#555—611no—
274C2×#489—611no—
275C2×#463D6111yescharacteristic, complemented
276#2540—611no—
277#2000—612no—
278#1999—612no—
279#1988—611no—
280#1955—611no—
281#1875—612no—
282#1873—612no—
283#1762—611no—
284C2×C61—82no—
285C2×C61—161no—
286C2×C61#243111yescharacteristic
287C2×C61#300511yescharacteristic
288C2×C61#301211yescharacteristic
289C2×C61#331912yes—
290C2×C61#332112yes—
291D61—82no—
292C22×SD16C2×D6111yescharacteristic
293C22×D8C2×D6111yescharacteristic
294#555—1224no—
295#555—1224no—
296C2×C4⋊Q8—611no—
297C2×#124—1222no—
298C2×#124C2×D6111yescharacteristic
299C2×C22⋊Q8—612no—
300C2×#121—612no—
301C2×#121—612no—
302C2×#121—1221no—
303C2×C4×D4—611no—
304C2×C4×D4C2×D6111yescharacteristic
305C2×#117—612no—
306C2×#117—1221no—
307#489—1224no—
308#463—24no—
309C2×C4⋊C8C2×D6111yescharacteristic
310C2×#103—612no—
311C2×#103C2×D6111yescharacteristic
312C2×#102—612no—
313C2×#102—612no—
314C2×#102C2×D6111yescharacteristic
315C2×C22⋊C8—612no—
316C2×C22⋊C8—612no—
317C2×#97—611no—
318#399—1221no—
319#392—1221no—
320#390—1222no—
321C4×C4⋊C4—611no—
322#340—612no—
323#340—612no—
324#340—612no—
325C61#2557811yescharacteristic, complemented, Sylow
326C22×Q8—1221no—
327C22×Q8C22×D6111yescharacteristic
328C22×D4—611no—
329C22×D4—1222no—
330C22×D4—1222no—
331C22×D4C22×D6111yescharacteristic
332C22×D4C22×D6111yescharacteristic
333C23×C4—21no—
334C23×C4—612no—
335C2×SD16—22no—
336C2×SD16—42no—
337C2×D8—22no—
338C2×D8—42no—
339C22×C8—612no—
340C22×C8C22×D6111yescharacteristic
341C22×C8C22×D6111yescharacteristic
342C4⋊Q8—614no—
343C4⋊Q8—1222no—
344#124—22no—
345#124—22no—
346#124—12216no—
347C22⋊Q8—1224no—
348C22⋊Q8—2442no—
349#121—1224no—
350#121—1228no—
351#121—1224no—
352#121—1224no—
353#121—2442no—
354C4×D4—22no—
355C4×D4—1222no—
356C4×D4—1222no—
357C4×D4C22⋊D6114yescomplemented
358#117—1224no—
359#117—1228no—
360#117—1224no—
361C2×C4⋊C4—612no—
362C2×C4⋊C4—612no—
363C2×C4⋊C4—611no—
364C2×C4⋊C4—1221no—
365C2×C4⋊C4—1222no—
366C2×C4⋊C4C22×D6111yescharacteristic
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—612no—
371C2×C22⋊C4—1222no—
372C2×C22⋊C4—1222no—
373C2×C22⋊C4—1222no—
374C2×C22⋊C4—1222no—
375C2×C42—1222no—
376C2×C42—1221no—
377C2×C42C22×D6111yescharacteristic
378C4⋊C8—22no—
379#103—22no—
380#103—1224no—
381#102—22no—
382#102—1224no—
383#102—1224no—
384C22⋊C8—2442no—
385C22⋊C8—2442no—
386#97—1222no—
387#95—1221no—
388C24—21no—
389C24—21no—
390C24—612no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—1224no—
396C2×Q8—4881no—
397C2×Q8—4881no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—1228no—
406C2×D4—1222no—
407C2×D4—4882no—
408C2×D4—4882no—
409C2×D4—4882no—
410C2×D4—2442no—
411C2×D4—2444no—
412C2×D4—2444no—
413C2×D4C2×C22⋊D6112yes—
414C22×C4—21no—
415C22×C4—24no—
416C22×C4—21no—
417C22×C4—44no—
418C22×C4—612no—
419C22×C4—1222no—
420C22×C4—1222no—
421C22×C4—1222no—
422C22×C4—1222no—
423C22×C4—1222no—
424C22×C4—1222no—
425C22×C4—1222no—
426C22×C4—1222no—
427C22×C4D4×D6111yescharacteristic
428C22×C4D4×D6111yescharacteristic
429C22×C4C23×D6111yescharacteristic
430SD16—44no—
431D8—44no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—1222no—
437C2×C8—1222no—
438C2×C8—1222no—
439C4⋊C4—21no—
440C4⋊C4—614no—
441C4⋊C4—614no—
442C4⋊C4—1222no—
443C4⋊C4—1222no—
444C4⋊C4—1222no—
445C4⋊C4—1224no—
446C4⋊C4—2444no—
447C4⋊C4C2×C22⋊D6112yes—
448C22⋊C4—24no—
449C22⋊C4—42no—
450C22⋊C4—42no—
451C22⋊C4—1228no—
452C22⋊C4—1228no—
453C22⋊C4—1224no—
454C22⋊C4—2444no—
455C22⋊C4—2444no—
456C42—1228no—
457C42—4881no—
458C42D4×D6112yes—
459C42C2×C22⋊D6112yes—
460C23—22no—
461C23—42no—
462C23—41no—
463C23—41no—
464C23—42no—
465C23—41no—
466C23—42no—
467C23—41no—
468C23—41no—
469C23—42no—
470C23—1222no—
471C23—1222no—
472C23—1222no—
473C23—1222no—
474C23—1222no—
475C23—1222no—
476C23—1222no—
477C23C2×D4×D6111yescharacteristic
478Q8—41no—
479Q8—41no—
480Q8—42no—
481Q8—4884no—
482D4—22no—
483D4—22no—
484D4—41no—
485D4—41no—
486D4—42no—
487D4—42no—
488D4—4888no—
489D4—4888no—
490D4—2444no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—48no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—1222no—
500C2×C4—1224no—
501C2×C4—1224no—
502C2×C4—1222no—
503C2×C4—1222no—
504C2×C4—2442no—
505C2×C4—2442no—
506C2×C4—2442no—
507C2×C4—2442no—
508C2×C4—2442no—
509C2×C4—2442no—
510C2×C4—2442no—
511C2×C4—2442no—
512C2×C4#1451974512yes—
513C2×C4#1451974512yes—
514C2×C4C2×D4×D6111yescharacteristic
515C2×C4C22×C22⋊D6111yescharacteristic
516C2×C4#41×D6112yes—
517C2×C4#1451982212yes—
518C8—41no—
519C8—41no—
520C8—2442no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—82no—
525C22—82no—
526C22—81no—
527C22—82no—
528C22—82no—
529C22—81no—
530C22—81no—
531C22—2442no—
532C22—2442no—
533C22—2442no—
534C22—2442no—
535C22—2442no—
536C22—2442no—
537C22—2442no—
538C22C2×#1451974511yescharacteristic
539C22#1684040912yes—
540C22#1684055812yes—
541C22#1684056411yescharacteristic
542C22#1684056911yescharacteristic
543C4—22no—
544C4—22no—
545C4—42no—
546C4—81no—
547C4—2444no—
548C4—2444no—
549C4—2442no—
550C4#1684030811yescharacteristic
551C4#1684032911yescharacteristic
552C4#1684043012yes—
553C2—82no—
554C2—161no—
555C2—4882no—
556C2#2144943911yescharacteristic
557C2#2145505811yescharacteristic
558C2#2145511911yescharacteristic
559C2#2145798212yes—
560C2#2145801212yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#463C2
socleC23×C61C22×D4
hypercenterC2×#463D61
hypocenterC61#25578
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#463

Complements

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

NH ≅ G/Ntype
1. C2×#16840477C2semidirect
2. C2×#16840121C2semidirect
3. #21450759C2semidirect
4. #21446819C2semidirect
5. #21446814C2semidirect
6. #21446770C2semidirect
7. #21446616C2semidirect
8. #21446138C2semidirect
9. #21446128C2semidirect
10. #21445644C2semidirect
11. C2×C61×#463C2semidirect
52. #14519777D4semidirect
98. C4×D4×C61D4semidirect
275. C2×#463D61semidirect
325. C61#25578semidirect
357. C4×D4C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168404772
C2×#168401212
#214507592
#214468192
#214468142
#214467702
#214466162
#214461382
#214461282
#214456442
C2×C61×#4632
#2557861