Finite Group Databasebeta

← #29736675

Subgroups — #29736675

Order 15616.

1092 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840317C211yesmaximal, characteristic, complemented
2C2×#16840121C211yesmaximal, characteristic, complemented
3#21454115C211yesmaximal, characteristic
4#21446968C212yesmaximal, complemented
5#21446963C212yesmaximal, complemented
6#21446793C211yesmaximal, characteristic, complemented
7#21446792C211yesmaximal, characteristic, complemented
8#21446138C212yesmaximal
9#21446128C212yesmaximal, complemented
10#21445582C211yesmaximal, characteristic, complemented
11C2×C61×#489C211yesmaximal, characteristic, complemented
12#16840317—24no—
13#16840317—24no—
14C2×#14519777C2211yescharacteristic
15C2×#14519768C2212yes—
16C2×#14519757—21no—
17C2×#14519753—21no—
18C2×#14519753C2211yescharacteristic
19C2×#14519748C2211yescharacteristic
20C2×Q8×Dic61—21no—
21C2×#14519742—21no—
22C2×#14519735C2212yes—
23C2×C42×D61C2211yescharacteristic
24#16840121—24no—
25C2×#14519701C2212yes—
26C2×#14519698C2212yes—
27C2×#14519689C2212yes—
28C2×#14519688C2211yescharacteristic
29#16839669C2211yescharacteristic
30#16839665C2211yescharacteristic
31#16839650—21no—
32#16839650—21no—
33#16839638—21no—
34#16839638—21no—
35#16839628C2211yescharacteristic
36#16839525C2212yes—
37#16839525C2212yes—
38#16839524C2212yes—
39#16839524C2212yes—
40#16839516C2212yes—
41C2×C4⋊Q8×C61C2211yescharacteristic
42C2×#124×C61C2211yescharacteristic
43C61×#489—24no—
44C2×#103×C61C2212yes—
45C2×#102×C61C2212yes—
46C2×#97×C61C2211yescharacteristic
47C22×Dic122—21no—
48C22×C22⋊D61—21no—
49C22×C4×D61D411yescharacteristic
50C22×C4×D61D411yescharacteristic
51C22×C4×D61C2311yescharacteristic
52C22×#92418C2312yes—
53#14519777—22no—
54#14519777D414yescomplemented
55#14519768—24no—
56#14519768—24no—
57#14519757—28no—
58#14519753—28no—
59#14519753—24no—
60#14519748—24no—
61Q8×Dic61—28no—
62#14519742—28no—
63#14519735—24no—
64#14519735—24no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—22no—
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2312yes—
70C2×#11758196—22no—
71C2×#11758196—21no—
72C2×C22⋊Dic61—21no—
73C2×C22⋊Dic61—21no—
74C2×C22⋊Dic61—21no—
75C2×#11758198—22no—
76C2×#11758198—21no—
77C2×C4×Dic61—22no—
78C2×C4×Dic61D411yescharacteristic
79C2×C4×Dic61D411yescharacteristic
80C2×C4×Dic61C2311yescharacteristic
81C42×D61—22no—
82C42×D61—22no—
83#14519701—24no—
84#14519698—24no—
85#14519689—24no—
86#14519688—22no—
87#14519684—21no—
88#14519684—21no—
89#14519684—21no—
90#14519684—21no—
91C22×Q8×C61—21no—
92C22×Q8×C61C2311yescharacteristic
93C22×D4×C61C2311yescharacteristic
94C22×C8×C61C2312yes—
95C4⋊Q8×C61—22no—
96C4⋊Q8×C61D414yescomplemented
97#124×C61—22no—
98#124×C61—22no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61C2312yes—
101C2×C22⋊C4×C61—21no—
102C2×C22⋊C4×C61—21no—
103C2×C42×C61C2311yescharacteristic
104#103×C61—24no—
105#102×C61—24no—
106#97×C61—22no—
107C24×C61—21no—
108C23×D61C2×D411yescharacteristic
109C2×Q8×C61—24no—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—22no—
113C2×Q8×C61—22no—
114C2×Q8×C61—42no—
115C2×Q8×C61—42no—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61—22no—
119C2×D4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—21no—
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2411yescharacteristic
125C2×C22⋊D61—42no—
126C2×C22⋊D61—42no—
127C2×C22⋊D61—42no—
128C22×Dic61—22no—
129C22×Dic61—21no—
130C22×Dic61—21no—
131C22×Dic61C2×D411yescharacteristic
132C22×Dic61C2×D411yescharacteristic
133C22×Dic61C2×D411yescharacteristic
134C2×C4×D61—21no—
135C2×C4×D61—21no—
136C2×C4×D61—22no—
137C2×C4×D61—22no—
138C2×C4×D61—22no—
139C2×C4×D61—41no—
140C2×C4×D61—41no—
141C2×C4×D61—41no—
142C2×C4×D61—41no—
143C2×C4×D61#4114yes—
144C2×Dic122—24no—
145C2×Dic122—42no—
146C2×Dic122—42no—
147C2×C8×C61—22no—
148C2×C8×C61—22no—
149C2×C8×C61—22no—
150C4⋊C4×C61—24no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61C2×D414yes—
153C22⋊C4×C61—42no—
154C22⋊C4×C61—42no—
155C42×C61C2×D412yes—
156C42×C61C2×D412yes—
157C22⋊Dic61—42no—
158C22⋊Dic61—42no—
159C22⋊Dic61—42no—
160#11758198—28no—
161#11758198—42no—
162C4⋊Dic61—28no—
163C4⋊Dic61—24no—
164C4⋊Dic61—22no—
165C4⋊Dic61—22no—
166C4⋊Dic61—22no—
167C4⋊Dic61C2×D414yes—
168#11758196—28no—
169#11758196—42no—
170C4×Dic61—28no—
171C4×Dic61—22no—
172C4×Dic61—22no—
173C4×Dic61—21no—
174C4×Dic61—21no—
175C2×#92418—22no—
176C2×#92418—22no—
177C2×#92418—22no—
178C23×C61—41no—
179C23×C61—41no—
180C23×C61—41no—
181C23×C61—41no—
182C23×C61—41no—
183C23×C61—41no—
184C23×C61—41no—
185C23×C61C22×D411yescharacteristic
186C22×D61—21no—
187C22×D61—21no—
188C22×D61—21no—
189C22×D61—21no—
190C22×D61—21no—
191C22×D61—21no—
192C22×D61#12412yes—
193Q8×C61—48no—
194Q8×C61—42no—
195Q8×C61—42no—
196D4×C61—42no—
197D4×C61—42no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—24no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61#12012yes—
209C2×C4×C61#12012yes—
210C2×C4×C61C22×D411yescharacteristic
211C2×C4×C61C22×D411yescharacteristic
212C2×C4×C61C2×#4112yes—
213C2×C4×C612−1+412yes—
214C22⋊D61—84no—
215C2×Dic61—21no—
216C2×Dic61—21no—
217C2×Dic61—21no—
218C2×Dic61—21no—
219C2×Dic61—24no—
220C2×Dic61—21no—
221C2×Dic61—21no—
222C2×Dic61—21no—
223C2×Dic61—21no—
224C2×Dic61—42no—
225C2×Dic61—42no—
226C2×Dic61—41no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61#12412yes—
233C4×D61—24no—
234C4×D61—24no—
235C4×D61—44no—
236C4×D61—44no—
237Dic122—48no—
238C8×C61—42no—
239#92418—42no—
240#25909—611nomaximal, Sylow
241C22×C61—81no—
242C22×C61—81no—
243C22×C61—81no—
244C22×C61—81no—
245C22×C61—81no—
246C22×C61—81no—
247C22×C61—81no—
248C22×C61C2×#12011yescharacteristic
249C22×C61C2×#12411yescharacteristic
250C22×C61#53611yescharacteristic
251C22×C61#57511yescharacteristic
252C22×C61#57511yescharacteristic
253C22×C61#57511yescharacteristic
254C22×C61#57511yescharacteristic
255C2×D61—22no—
256C2×D61—22no—
257C2×D61—22no—
258C2×D61—41no—
259C2×D61—41no—
260C2×D61—41no—
261C2×D61—41no—
262C4×C61—22no—
263C4×C61—22no—
264C4×C61—44no—
265C4×C61—81no—
266C4×C61#54812yes—
267C4×C61#54812yes—
268Dic61—22no—
269Dic61—44no—
270Dic61—41no—
271Dic61—41no—
272Dic61—81no—
273Dic61—81no—
274C2×#533—611no—
275C2×#489—611no—
276C2×#489D6111yescharacteristic, complemented
277#2850—611no—
278#2026—612no—
279#2025—612no—
280#1993—611no—
281#1993—611no—
282#1875—612no—
283#1873—612no—
284#1749—611no—
285C2×C61—161no—
286C2×C61#243611yescharacteristic
287C2×C61#300511yescharacteristic
288C2×C61#300511yescharacteristic
289C2×C61#313511yescharacteristic
290C2×C61#314011yescharacteristic
291C2×C61#324011yescharacteristic
292C2×C61#324011yescharacteristic
293D61—42no—
294#533—1224no—
295#533—1224no—
296C2×C4⋊Q8—611no—
297C2×C4⋊Q8C2×D6111yescharacteristic
298C2×#125—612no—
299C2×#124C2×D6111yescharacteristic
300C2×C22⋊Q8—611no—
301C2×C22⋊Q8—1221no—
302C2×C22⋊Q8—1221no—
303C2×#121—611no—
304C2×C4×Q8—1221no—
305C2×C4×Q8—1221no—
306C2×#117—612no—
307C22×C42—611no—
308#489—24no—
309#489—1224no—
310C2×#103—612no—
311C2×#103C2×D6112yes—
312C2×#102—612no—
313C2×#102C2×D6112yes—
314C2×C22⋊C8—612no—
315C2×#97—611no—
316C2×#97C2×D6111yescharacteristic
317#391—611no—
318#390—611no—
319#388—1221no—
320#388—1221no—
321#386—1221no—
322#386—1221no—
323#382—611no—
324#340—612no—
325#340—612no—
326#340—612no—
327#340—612no—
328#336—612no—
329C61#2590911yescharacteristic, complemented, Sylow
330C22×Q8—21no—
331C22×Q8—1221no—
332C22×Q8C22×D6111yescharacteristic
333C22×D4—1221no—
334C22×D4C22×D6111yescharacteristic
335C23×C4—611no—
336C23×C4—611no—
337C23×C4—611no—
338C22×C8—612no—
339C22×C8C22×D6112yes—
340C4⋊Q8—22no—
341C4⋊Q8—614no—
342C4⋊Q8—1222no—
343C4⋊Q8C22⋊D6114yescomplemented
344#125—1224no—
345#125—1224no—
346#124—22no—
347#124—22no—
348C22⋊Q8—1228no—
349C22⋊Q8—1228no—
350C22⋊Q8—1224no—
351#121—1224no—
352C4×Q8—1228no—
353C4×Q8—1228no—
354#117—1224no—
355#117—1224no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—612no—
358C2×C4⋊C4—611no—
359C2×C4⋊C4—611no—
360C2×C4⋊C4—1222no—
361C2×C4⋊C4—1222no—
362C2×C4⋊C4—1221no—
363C2×C4⋊C4—1221no—
364C2×C4⋊C4C22×D6112yes—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—1222no—
368C2×C22⋊C4—1221no—
369C2×C22⋊C4—1221no—
370C2×C22⋊C4—1221no—
371C2×C22⋊C4—1221no—
372C2×C42—611no—
373C2×C42—611no—
374C2×C42—611no—
375C2×C42—1222no—
376C2×C42—1222no—
377C2×C42—1222no—
378C2×C42C22×D6111yescharacteristic
379#103—24no—
380#103—1224no—
381#102—24no—
382#102—1224no—
383C22⋊C8—1224no—
384#97—22no—
385#97—1222no—
386#95—1221no—
387#95—1221no—
388#95—1221no—
389#95—1221no—
390C24—21no—
391C24—611no—
392C2×Q8—24no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8—1224no—
400C2×Q8—2442no—
401C2×Q8—2442no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—2442no—
407C2×D4—2442no—
408C2×D4—2442no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—611no—
412C22×C4—614no—
413C22×C4—611no—
414C22×C4—611no—
415C22×C4—1221no—
416C22×C4—1221no—
417C22×C4—1222no—
418C22×C4—1222no—
419C22×C4—1222no—
420C22×C4—1222no—
421C22×C4—1221no—
422C22×C4—1221no—
423C22×C4—2441no—
424C22×C4—2441no—
425C22×C4—2441no—
426C22×C4—2441no—
427C22×C4D4×D6111yescharacteristic
428C22×C4D4×D6111yescharacteristic
429C22×C4C23×D6111yescharacteristic
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—1222no—
434C2×C8—1222no—
435C2×C8—1222no—
436C4⋊C4—22no—
437C4⋊C4—24no—
438C4⋊C4—614no—
439C4⋊C4—1228no—
440C4⋊C4—1228no—
441C4⋊C4—1222no—
442C4⋊C4—1224no—
443C4⋊C4—1222no—
444C4⋊C4—1222no—
445C4⋊C4—2442no—
446C4⋊C4C2×C22⋊D6114yes—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C22⋊C4—1228no—
450C22⋊C4—2442no—
451C22⋊C4—2442no—
452C22⋊C4—2442no—
453C22⋊C4—2442no—
454C42—1228no—
455C42—1221no—
456C42—1221no—
457C42—1222no—
458C42—1222no—
459C42D4×D6112yes—
460C42C2×C22⋊D6112yes—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—612no—
469C23—1221no—
470C23—1221no—
471C23—1221no—
472C23—1221no—
473C23—1221no—
474C23—1221no—
475C23C2×D4×D6111yescharacteristic
476Q8—48no—
477Q8—42no—
478Q8—42no—
479Q8—2448no—
480D4—42no—
481D4—42no—
482D4—4884no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—24no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—612no—
494C2×C4—1221no—
495C2×C4—1221no—
496C2×C4—1224no—
497C2×C4—1224no—
498C2×C4—1221no—
499C2×C4—1221no—
500C2×C4—1224no—
501C2×C4—1221no—
502C2×C4—1221no—
503C2×C4—1221no—
504C2×C4—1221no—
505C2×C4—2444no—
506C2×C4—2444no—
507C2×C4—2441no—
508C2×C4—2441no—
509C2×C4—2441no—
510C2×C4—2441no—
511C2×C4—2441no—
512C2×C4—2441no—
513C2×C4—2442no—
514C2×C4—2442no—
515C2×C4#1451974512yes—
516C2×C4#1451974512yes—
517C2×C4C2×D4×D6111yescharacteristic
518C2×C4C22×C22⋊D6111yescharacteristic
519C2×C4#41×D6112yes—
520C2×C4#1451982412yes—
521C8—42no—
522C8—2442no—
523C22—81no—
524C22—81no—
525C22—81no—
526C22—81no—
527C22—81no—
528C22—81no—
529C22—81no—
530C22—1222no—
531C22—1222no—
532C22—1222no—
533C22—2441no—
534C22—2441no—
535C22—2441no—
536C22—2441no—
537C22C2×#1451974511yescharacteristic
538C22#124×D6111yescharacteristic
539C22#1684033211yescharacteristic
540C22#1684055811yescharacteristic
541C22#1684055811yescharacteristic
542C22#1684056111yescharacteristic
543C22#1684056111yescharacteristic
544C4—22no—
545C4—22no—
546C4—44no—
547C4—81no—
548C4—1222no—
549C4—4881no—
550C4—4881no—
551C4—2441no—
552C4—2441no—
553C4—2444no—
554C4#1684041912yes—
555C4#1684042012yes—
556C2—161no—
557C2—2442no—
558C2#2144947411yescharacteristic
559C2#2145505811yescharacteristic
560C2#2145506011yescharacteristic
561C2#2145616111yescharacteristic
562C2#2145620411yescharacteristic
563C2#2145705411yescharacteristic
564C2#2145705611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#489C2
socleC23×C61C22×D4
prime radicalC2×#14519748C22
hypercenterC2×#489D61
hypocenterC61#25909
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#489

Complements

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

NH ≅ G/Ntype
1. C2×#16840317C2semidirect
2. C2×#16840121C2semidirect
4. #21446968C2semidirect
5. #21446963C2semidirect
6. #21446793C2semidirect
7. #21446792C2semidirect
9. #21446128C2semidirect
10. #21445582C2semidirect
11. C2×C61×#489C2semidirect
54. #14519777D4semidirect
96. C4⋊Q8×C61D4semidirect
276. C2×#489D61semidirect
329. C61#25909semidirect
343. C4⋊Q8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403172
C2×#168401212
#214541152
#214469682
#214469632
#214467932
#214467922
#214461382
#214461282
#214455822
C2×C61×#4892
#2590961