Finite Group Databasebeta

← #29689758

Subgroups — #29689758

Order 15616.

1118 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×#16840291C211yesmaximal, characteristic
2C2×#16839817C211yesmaximal, characteristic, complemented
3#21448224C211yesmaximal, characteristic, complemented
4#21446788C211yesmaximal, characteristic, complemented
5#21446787C211yesmaximal, characteristic, complemented
6#21446636C211yesmaximal, characteristic, complemented
7#21446138C212yesmaximal
8#21446128C212yesmaximal, complemented
9#21445832C212yesmaximal, complemented
10#21445827C212yesmaximal, complemented
11C61×#1959C211yesmaximal, characteristic, complemented
12#16840291—24no—
13#16840291C418yescomplemented
14C2×#14519770—21no—
15C2×#14519769C2211yescharacteristic
16C2×#14519759—21no—
17C2×#14519753C2211yescharacteristic
18C2×#14519748C2211yescharacteristic
19C2×Q8×Dic61C2211yescharacteristic
20C2×D4×Dic61C2211yescharacteristic
21C2×#14519737—21no—
22C2×#14519735C2212yes—
23C2×#14519733—21no—
24C2×C42×D61C2211yescharacteristic
25#16839817—24no—
26C2×#14519706C2212yes—
27C2×#14519701C2212yes—
28C2×#14519698C2212yes—
29C2×#14519689C2212yes—
30C2×#14519688C2211yescharacteristic
31#16839632C2211yescharacteristic
32#16839627C2212yes—
33#16839619—21no—
34#16839619—21no—
35C4⋊C4×Dic61—21no—
36C4⋊C4×Dic61—21no—
37#16839525C2212yes—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×#124×C61C2211yescharacteristic
42C2×#103×C61C2212yes—
43C2×#102×C61C2212yes—
44C2×#97×C61C2211yescharacteristic
45C61×#384C2211yescharacteristic
46C22×C22⋊D61—21no—
47C23×Dic61—21no—
48C22×C4×D61D411yescharacteristic
49C22×C4×D61D411yescharacteristic
50C22×C4×D61C2311yescharacteristic
51C22×#92418C2312yes—
52#14519770—28no—
53#14519769—22no—
54#14519769C2×C414yes—
55#14519759—28no—
56#14519753—22no—
57#14519753C2×C414yes—
58#14519748—22no—
59#14519748C2×C414yes—
60Q8×Dic61—22no—
61Q8×Dic61C2×C414yes—
62D4×Dic61—22no—
63D4×Dic61C2×C414yes—
64#14519737—28no—
65#14519735—42no—
66#14519735—42no—
67#14519733—28no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2312yes—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61—21no—
79C2×#11758198—21no—
80C2×#11758198—22no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—22no—
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61D411yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86C42×D61C2×C414yes—
87C42×D61D414yes—
88#14519706—24no—
89#14519701—42no—
90#14519698—42no—
91#14519689—24no—
92#14519688—22no—
93#14519684—22no—
94#14519684—22no—
95C22×Q8×C61C2311yescharacteristic
96C22×D4×C61C2311yescharacteristic
97C22×C8×C61C2312yes—
98#124×C61—22no—
99#124×C61C2×C414yes—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61C2312yes—
103C2×C22⋊C4×C61—21no—
104C2×C22⋊C4×C61—21no—
105C2×C42×C61C2311yescharacteristic
106#103×C61—42no—
107#102×C61—42no—
108#97×C61—22no—
109C24×C61—21no—
110C23×D61C2×D411yescharacteristic
111C2×Q8×C61—21no—
112C2×Q8×C61—22no—
113C2×Q8×C61—22no—
114C2×Q8×C61C22×C412yes—
115C2×D4×C61—21no—
116C2×D4×C61—22no—
117C2×D4×C61—22no—
118C2×D4×C61C22×C412yes—
119C22×C4×C61—21no—
120C22×C4×C61—22no—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2411yescharacteristic
124C2×C22⋊D61—24no—
125C2×C22⋊D61—42no—
126C2×C22⋊D61—42no—
127C22×Dic61—21no—
128C22×Dic61—21no—
129C22×Dic61—24no—
130C22×Dic61—22no—
131C22×Dic61—42no—
132C22×Dic61—42no—
133C22×Dic61C2×D411yescharacteristic
134C22×Dic61C2×D411yescharacteristic
135C22×Dic61C2×D411yescharacteristic
136C2×C4×D61—22no—
137C2×C4×D61—22no—
138C2×C4×D61—41no—
139C2×C4×D61—41no—
140C2×C4×D61—41no—
141C2×C4×D61—41no—
142C2×C4×D61C22⋊C414yes—
143C2×C4×D61C22⋊C414yes—
144C2×C4×D61C22×C412yes—
145C2×C4×D61C2×D412yes—
146C2×C8×C61—22no—
147C2×C8×C61—22no—
148C2×C8×C61—22no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—42no—
152C4⋊C4×C61—42no—
153C22⋊C4×C61—24no—
154C22⋊C4×C61—24no—
155C42×C61C22×C412yes—
156C42×C61C2×D412yes—
157C22⋊Dic61—24no—
158C22⋊Dic61—24no—
159#11758198—24no—
160#11758198—44no—
161C4⋊Dic61—21no—
162C4⋊Dic61—21no—
163C4⋊Dic61—24no—
164C4⋊Dic61—22no—
165C4⋊Dic61—22no—
166C4⋊Dic61—42no—
167C4⋊Dic61—42no—
168C4⋊Dic61C22×C412yes—
169C4⋊Dic61C22×C412yes—
170#11758196—24no—
171#11758196—24no—
172#11758196—24no—
173C4×Dic61—24no—
174C4×Dic61—44no—
175C4×Dic61C22⋊C414yes—
176C4×Dic61C22⋊C414yes—
177C4×Dic61C22×C412yes—
178C4×Dic61C2×D412yes—
179C2×#92418—22no—
180C2×#92418—22no—
181C2×#92418—22no—
182C23×C61—22no—
183C23×C61—41no—
184C23×C61—41no—
185C23×C61—41no—
186C23×C61—41no—
187C23×C61—41no—
188C23×C61—41no—
189C23×C61C22×D411yescharacteristic
190C22×D61—21no—
191C22×D61—21no—
192C22×D61—21no—
193C22×D61—21no—
194C22×D61—21no—
195C22×D61—21no—
196C22×D61C2×C22⋊C412yes—
197Q8×C61—24no—
198Q8×C61—42no—
199D4×C61—24no—
200D4×C61—42no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—22no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61—42no—
211C2×C4×C61C2×C22⋊C412yes—
212C2×C4×C61C2×C22⋊C412yes—
213C2×C4×C61C23×C411yescharacteristic
214C2×C4×C61C22×D411yescharacteristic
215C2×C4×C61H(5,𝔽2)12yes—
216C2×C4×C612−1+412yes—
217C22⋊D61—48no—
218C2×Dic61—21no—
219C2×Dic61—21no—
220C2×Dic61—21no—
221C2×Dic61—21no—
222C2×Dic61—21no—
223C2×Dic61—21no—
224C2×Dic61—22no—
225C2×Dic61—22no—
226C2×Dic61—48no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—42no—
232C2×Dic61—42no—
233C2×Dic61—42no—
234C2×Dic61C2×C22⋊C412yes—
235C2×Dic61C2×C22⋊C412yes—
236C2×Dic61C2×C22⋊C412yes—
237C4×D61—24no—
238C4×D61—24no—
239C4×D61—44no—
240C4×D61—44no—
241C8×C61—42no—
242#92418—42no—
243#21868—611nomaximal, Sylow
244C22×C61—42no—
245C22×C61—42no—
246C22×C61—42no—
247C22×C61—81no—
248C22×C61—81no—
249C22×C61—81no—
250C22×C61—81no—
251C22×C61C22×C22⋊C411yescharacteristic
252C22×C61#53511yescharacteristic
253C22×C61#53611yescharacteristic
254C22×C61#57511yescharacteristic
255C22×C61#57511yescharacteristic
256C22×C61#57511yescharacteristic
257C22×C61#57511yescharacteristic
258C2×D61—22no—
259C2×D61—22no—
260C2×D61—22no—
261C2×D61—41no—
262C2×D61—41no—
263C2×D61—41no—
264C2×D61—41no—
265C4×C61—22no—
266C4×C61—22no—
267C4×C61—42no—
268C4×C61—82no—
269C4×C61#52012yes—
270C4×C61#53712yes—
271Dic61—22no—
272Dic61—22no—
273Dic61—22no—
274Dic61—42no—
275Dic61—42no—
276Dic61—82no—
277C2×#529—611no—
278C2×#432—611no—
279#2285—611no—
280#1992—611no—
281#1992—611no—
282#1959—611no—
283#1959D6111yescharacteristic, complemented
284#1875—612no—
285#1873—612no—
286#1805—612no—
287#1804—612no—
288C2×C61—82no—
289C2×C61#231111yescharacteristic
290C2×C61#289611yescharacteristic
291C2×C61#289611yescharacteristic
292C2×C61#321911yescharacteristic
293C2×C61#321911yescharacteristic
294C2×C61#324011yescharacteristic
295C2×C61#324011yescharacteristic
296D61—42no—
297#529—618no—
298#529—1224no—
299C2×#126—1221no—
300C2×#125—611no—
301C2×#124C2×D6111yescharacteristic
302C2×#123—1221no—
303C2×C22⋊Q8—611no—
304C2×#121—611no—
305C2×C4×Q8—611no—
306C2×C4×D4—611no—
307C2×C4×D4—1221no—
308C2×#117—612no—
309C2×#117—1221no—
310C22×C42—611no—
311#432—1224no—
312C2×C4⋊C8—612no—
313C2×#103—612no—
314C2×#103C2×D6112yes—
315C2×#102—612no—
316C2×#102C2×D6112yes—
317C2×C22⋊C8—612no—
318C2×#97—611no—
319C2×#97C2×D6111yescharacteristic
320#384—611no—
321#384C2×D6111yescharacteristic
322#382—612no—
323#380—1221no—
324#380—1221no—
325C4×C4⋊C4—1221no—
326C4×C4⋊C4—1221no—
327#340—612no—
328#340—612no—
329#340—612no—
330#340—612no—
331C61#2186811yescharacteristic, complemented, Sylow
332C22×Q8C22×D6111yescharacteristic
333C22×D4—1221no—
334C22×D4C22×D6111yescharacteristic
335C23×C4—611no—
336C23×C4—611no—
337C23×C4—611no—
338C23×C4—1221no—
339C22×C8—612no—
340C22×C8C22×D6112yes—
341#126—1228no—
342#125—614no—
343#125—1222no—
344#124—22no—
345#124C4×D6114yes—
346#123—1228no—
347C22⋊Q8—614no—
348C22⋊Q8—1222no—
349#121—614no—
350#121—1222no—
351C4×Q8—614no—
352C4×Q8—1222no—
353C4×D4—614no—
354C4×D4—1222no—
355C4×D4—1228no—
356#117—1228no—
357#117—2442no—
358#117—2442no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—611no—
362C2×C4⋊C4—611no—
363C2×C4⋊C4—612no—
364C2×C4⋊C4—1221no—
365C2×C4⋊C4—1221no—
366C2×C4⋊C4—1221no—
367C2×C4⋊C4—1221no—
368C2×C4⋊C4—1221no—
369C2×C4⋊C4—1221no—
370C2×C4⋊C4C22×D6112yes—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—1221no—
374C2×C22⋊C4—1221no—
375C2×C22⋊C4—1221no—
376C2×C22⋊C4—1222no—
377C2×C42—611no—
378C2×C42—611no—
379C2×C42—611no—
380C2×C42—614no—
381C2×C42—614no—
382C2×C42—1221no—
383C2×C42—1222no—
384C2×C42C22×D6111yescharacteristic
385C4⋊C8—1224no—
386#103—42no—
387#103—2442no—
388#102—42no—
389#102—2442no—
390C22⋊C8—1224no—
391#97—22no—
392#97—1222no—
393#95—1222no—
394#95—1222no—
395C24—21no—
396C24—611no—
397C2×Q8—21no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8C2×C4×D6112yes—
401C2×D4—21no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—1224no—
405C2×D4—2442no—
406C2×D4—2442no—
407C2×D4C2×C4×D6112yes—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—611no—
411C22×C4—611no—
412C22×C4—611no—
413C22×C4—612no—
414C22×C4—612no—
415C22×C4—614no—
416C22×C4—614no—
417C22×C4—1224no—
418C22×C4—1222no—
419C22×C4—1222no—
420C22×C4—1221no—
421C22×C4—1221no—
422C22×C4—1222no—
423C22×C4—2441no—
424C22×C4—2441no—
425C22×C4—2441no—
426C22×C4—2441no—
427C22×C4—2442no—
428C22×C4—2442no—
429C22×C4D4×D6111yescharacteristic
430C22×C4D4×D6111yescharacteristic
431C22×C4C23×D6111yescharacteristic
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—1222no—
436C2×C8—1222no—
437C2×C8—1222no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—612no—
443C4⋊C4—612no—
444C4⋊C4—1224no—
445C4⋊C4—1224no—
446C4⋊C4—1221no—
447C4⋊C4—1221no—
448C4⋊C4—1224no—
449C4⋊C4—1224no—
450C4⋊C4—1222no—
451C4⋊C4—1222no—
452C4⋊C4—2442no—
453C4⋊C4—2442no—
454C22⋊C4—24no—
455C22⋊C4—24no—
456C22⋊C4—1224no—
457C22⋊C4—1224no—
458C22⋊C4—1224no—
459C22⋊C4—2444no—
460C42—612no—
461C42—612no—
462C42—614no—
463C42—614no—
464C42—1224no—
465C42—2444no—
466C42C2×C4×D6112yes—
467C42D4×D6112yes—
468C23—22no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—612no—
476C23—1221no—
477C23—1221no—
478C23—1221no—
479C23—1221no—
480C23—1221no—
481C23—1221no—
482C23C2×D4×D6111yescharacteristic
483Q8—24no—
484Q8—42no—
485D4—24no—
486D4—42no—
487D4—2448no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—612no—
499C2×C4—612no—
500C2×C4—612no—
501C2×C4—1221no—
502C2×C4—1221no—
503C2×C4—1221no—
504C2×C4—1221no—
505C2×C4—1221no—
506C2×C4—1221no—
507C2×C4—1224no—
508C2×C4—1224no—
509C2×C4—1222no—
510C2×C4—1222no—
511C2×C4—2444no—
512C2×C4—2444no—
513C2×C4—2448no—
514C2×C4—2442no—
515C2×C4—2442no—
516C2×C4—2442no—
517C2×C4—2441no—
518C2×C4—2441no—
519C2×C4—2441no—
520C2×C4—2441no—
521C2×C4C22⋊C4×D6112yes—
522C2×C4C22⋊C4×D6112yes—
523C2×C4C22×C4×D6111yescharacteristic
524C2×C4C2×D4×D6111yescharacteristic
525C2×C4#1451982212yes—
526C2×C4#1451982412yes—
527C8—42no—
528C8—2442no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—81no—
533C22—81no—
534C22—81no—
535C22—81no—
536C22—1222no—
537C22—1222no—
538C22—1222no—
539C22—2441no—
540C22—2441no—
541C22—2441no—
542C22—2441no—
543C22C2×C22⋊C4×D6111yescharacteristic
544C22#1684032511yescharacteristic
545C22#1684033211yescharacteristic
546C22#1684055811yescharacteristic
547C22#1684055811yescharacteristic
548C22#1684056111yescharacteristic
549C22#1684056111yescharacteristic
550C4—22no—
551C4—22no—
552C4—42no—
553C4—82no—
554C4—1222no—
555C4—1222no—
556C4—1222no—
557C4—4882no—
558C4—2442no—
559C4—2442no—
560C4#1684024112yes—
561C4#1684034112yes—
562C2—82no—
563C2—2442no—
564C2#2144840911yescharacteristic
565C2#2145433511yescharacteristic
566C2#2145433511yescharacteristic
567C2#2145685111yescharacteristic
568C2#2145685311yescharacteristic
569C2#2145705411yescharacteristic
570C2#2145705611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1959C2
socleC23×C61C22×D4
prime radicalC2×#14519748C22
cosocleC22×C4×C61C24
hypercenter#1959D61
hypocenterC61#21868
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1959

Complements

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

NH ≅ G/Ntype
2. C2×#16839817C2semidirect
3. #21448224C2semidirect
4. #21446788C2semidirect
5. #21446787C2semidirect
6. #21446636C2semidirect
8. #21446128C2semidirect
9. #21445832C2semidirect
10. #21445827C2semidirect
11. C61×#1959C2semidirect
13. #16840291C4semidirect
283. #1959D61semidirect
331. C61#21868semidirect

Maximal subgroups

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

subgroupindex
C2×#168402912
C2×#168398172
#214482242
#214467882
#214467872
#214466362
#214461382
#214461282
#214458322
#214458272
C61×#19592
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