Finite Group Databasebeta

← #29708172

Subgroups — #29708172

Order 15616.

1240 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840317C211yesmaximal, characteristic, complemented
2C2×SD16×Dic61C212yesmaximal, complemented
3#21448335C211yesmaximal, characteristic
4#21447153C211yesmaximal, characteristic, complemented
5#21447062C211yesmaximal, characteristic, complemented
6#21447014C212yesmaximal, complemented
7#21446955C211yesmaximal, characteristic, complemented
8#21446819C212yesmaximal
9#103×Dic61C212yesmaximal, complemented
10#21445576C211yesmaximal, characteristic, complemented
11C61×#2013C211yesmaximal, characteristic, complemented
12#16840317—28no—
13C2×#14519777C2211yescharacteristic
14C2×#14519768—21no—
15C2×#14519757—21no—
16C2×#14519753C2212yes—
17C2×#14519748C2211yescharacteristic
18C2×Q8×Dic61C2212yes—
19C2×#14519742—21no—
20C2×D4×Dic61C2211yescharacteristic
21C2×#14519737—21no—
22C2×#14519735—21no—
23C2×C42×D61C2211yescharacteristic
24SD16×Dic61—28no—
25C2×#14519709C2212yes—
26C2×#14519701C2212yes—
27C2×#14519701C2212yes—
28C2×#14519698C2212yes—
29C2×#14519689C2212yes—
30C2×C8×Dic61C2212yes—
31#16839660C2211yescharacteristic
32#16839656—21no—
33#16839638—21no—
34#16839619C2211yescharacteristic
35C4⋊C4×Dic61C2211yescharacteristic
36C22⋊C4×Dic61—21no—
37#16839525C2212yes—
38#16839524C2212yes—
39#16839517C2211yescharacteristic
40C22×SD16×C61C2212yes—
41C2×C4⋊Q8×C61C2211yescharacteristic
42C2×#103×C61C2212yes—
43C61×#390C2211yescharacteristic
44C61×#336C2211yescharacteristic
45C22×Dic122—21no—
46C22×C22⋊D61—21no—
47C23×Dic61—21no—
48C22×C4×D61D411yescharacteristic
49C22×C4×D61D411yescharacteristic
50C22×C4×D61C2311yescharacteristic
51C22×#92418C2312yes—
52#14519777—24no—
53#14519768—28no—
54#14519757—28no—
55#14519753—24no—
56#14519753—24no—
57#14519748—22no—
58#14519748—22no—
59Q8×Dic61—24no—
60Q8×Dic61D418yescomplemented
61#14519742—28no—
62D4×Dic61—24no—
63#14519737—44no—
64#14519735—28no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—22no—
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2312yes—
69C2×C4⋊Dic61C2311yescharacteristic
70C2×#11758196—22no—
71C2×#11758196—21no—
72C2×C22⋊Dic61—21no—
73C2×C22⋊Dic61—21no—
74C2×#11758198—22no—
75C2×#11758198—21no—
76C2×C4×Dic61—22no—
77C2×C4×Dic61—21no—
78C2×C4×Dic61D411yescharacteristic
79C2×C4×Dic61D411yescharacteristic
80C2×C4×Dic61C2311yescharacteristic
81C42×D61—24no—
82#14519709—24no—
83#14519701—24no—
84#14519701—24no—
85#14519698—24no—
86#14519689—24no—
87C8×Dic61—24no—
88#14519684—21no—
89#14519684—21no—
90#14519684—21no—
91C22×Q8×C61C2312yes—
92C22×D4×C61C2311yescharacteristic
93C2×SD16×C61—24no—
94C2×SD16×C61—24no—
95C2×SD16×C61—24no—
96C22×C8×C61C2312yes—
97C4⋊Q8×C61—24no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61C2311yescharacteristic
101C2×C22⋊C4×C61—21no—
102C2×C22⋊C4×C61—21no—
103C2×C42×C61C2311yescharacteristic
104#103×C61—24no—
105C24×C61—21no—
106C23×D61C2×D411yescharacteristic
107C2×Q8×C61—22no—
108C2×Q8×C61—22no—
109C2×Q8×C61—22no—
110C2×Q8×C61—22no—
111C2×Q8×C61—22no—
112C2×Q8×C61C2×D414yes—
113C2×D4×C61—22no—
114C2×D4×C61—22no—
115C2×D4×C61—21no—
116C2×D4×C61#4112yes—
117C22×C4×C61—22no—
118C22×C4×C61—21no—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2411yescharacteristic
122C2×C22⋊D61—24no—
123C2×C22⋊D61—42no—
124C2×C22⋊D61—42no—
125C22×Dic61—21no—
126C22×Dic61—22no—
127C22×Dic61—21no—
128C22×Dic61—42no—
129C22×Dic61—42no—
130C22×Dic61—42no—
131C22×Dic61C2×D411yescharacteristic
132C22×Dic61C2×D411yescharacteristic
133C22×Dic61C2×D411yescharacteristic
134C2×C4×D61—22no—
135C2×C4×D61—22no—
136C2×C4×D61—22no—
137C2×C4×D61—22no—
138C2×C4×D61—22no—
139C2×C4×D61—21no—
140C2×C4×D61—21no—
141C2×C4×D61—22no—
142C2×C4×D61—41no—
143C2×C4×D61SD1614yescomplemented
144C2×Dic122—24no—
145C2×Dic122—42no—
146C2×Dic122—42no—
147SD16×C61—216no—
148C2×C8×C61—22no—
149C2×C8×C61—22no—
150C2×C8×C61#4114yes—
151C4⋊C4×C61—24no—
152C4⋊C4×C61—24no—
153C4⋊C4×C61—22no—
154C22⋊C4×C61—42no—
155C22⋊C4×C61—42no—
156C42×C61—22no—
157C22⋊Dic61—24no—
158C22⋊Dic61—42no—
159#11758198—28no—
160#11758198—42no—
161C4⋊Dic61—24no—
162C4⋊Dic61—22no—
163C4⋊Dic61—21no—
164C4⋊Dic61—28no—
165C4⋊Dic61—22no—
166C4⋊Dic61C2×D414yes—
167C4⋊Dic61#4112yes—
168#11758196—28no—
169#11758196—42no—
170C4×Dic61—22no—
171C4×Dic61—22no—
172C4×Dic61—28no—
173C4×Dic61—42no—
174C4×Dic61C2×D414yes—
175C2×#92418—22no—
176C2×#92418—22no—
177C2×#92418—22no—
178C23×C61—22no—
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×D61C2×SD1612yes—
193Q8×C61—24no—
194Q8×C61—24no—
195Q8×C61—42no—
196Q8×C61—42no—
197D4×C61—24no—
198D4×C61—42no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—24no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61C2×SD1612yes—
210C2×C4×C61#13512yes—
211C2×C4×C61C22×D412yes—
212C2×C4×C61C2×#4111yescharacteristic
213C2×C4×C61C2×#4112yes—
214C2×C4×C61C2×#4111yescharacteristic
215C22⋊D61—48no—
216C2×Dic61—21no—
217C2×Dic61—21no—
218C2×Dic61—21no—
219C2×Dic61—24no—
220C2×Dic61—22no—
221C2×Dic61—21no—
222C2×Dic61—21no—
223C2×Dic61—41no—
224C2×Dic61—42no—
225C2×Dic61—42no—
226C2×Dic61—41no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—84no—
231C2×Dic61#12012yes—
232C2×Dic61#12012yes—
233C2×Dic61C2×SD1612yes—
234C2×Dic61#13712yes—
235C4×D61—28no—
236C4×D61—44no—
237C4×D61—42no—
238C4×D61—42no—
239Dic122—48no—
240C8×C61—24no—
241#92418—42no—
242#23568—611nomaximal, Sylow
243C22×C61—42no—
244C22×C61—42no—
245C22×C61—42no—
246C22×C61—81no—
247C22×C61—81no—
248C22×C61—81no—
249C22×C61—81no—
250C22×C61C2×#12011yescharacteristic
251C22×C61#52511yescharacteristic
252C22×C61#53211yescharacteristic
253C22×C61C22×SD1611yescharacteristic
254C22×C61C2×#13511yescharacteristic
255C22×C61C2×#13711yescharacteristic
256C22×C61#57511yescharacteristic
257C2×D61—22no—
258C2×D61—22no—
259C2×D61—22no—
260C2×D61—41no—
261C2×D61—41no—
262C2×D61—41no—
263C2×D61—41no—
264C4×C61—22no—
265C4×C61—22no—
266C4×C61—44no—
267C4×C61—81no—
268C4×C61#52512yes—
269C4×C61#53212yes—
270Dic61—22no—
271Dic61—42no—
272Dic61—41no—
273Dic61—44no—
274Dic61—81no—
275Dic61#44814yes—
276C2×#533—611no—
277C2×C4×SD16—612no—
278C4×#121—611no—
279#2058—611no—
280#2043—611no—
281#2035—612no—
282#2023—611no—
283#2013D6111yescharacteristic, complemented
284#2000—612no—
285C4×#103—612no—
286C4×C22⋊C8—611no—
287C2×C61—82no—
288C2×C61#238811yescharacteristic
289C2×C61C2×#44811yescharacteristic
290C2×C61#300211yescharacteristic
291C2×C61#310111yescharacteristic
292C2×C61#310711yescharacteristic
293C2×C61#316711yescharacteristic
294C2×C61#317211yescharacteristic
295D61—42no—
296C22×SD16C2×D6112yes—
297#533—1228no—
298C2×C4⋊Q8—611no—
299C2×C4⋊Q8C2×D6111yescharacteristic
300C2×#125—1221no—
301C2×C22⋊Q8—612no—
302C2×C22⋊Q8—1221no—
303C2×#121—611no—
304C2×C4×Q8—612no—
305C2×C4×Q8—1221no—
306C2×C4×D4—611no—
307C2×C4×D4—1221no—
308C2×#117—1221no—
309C22×C42—611no—
310C4×SD16—1228no—
311C2×#106—612no—
312C2×#103—612no—
313C2×#103—612no—
314C2×#103C2×D6112yes—
315C2×#102—612no—
316C2×C22⋊C8—612no—
317C2×C4×C8—612no—
318#390C2×D6111yescharacteristic
319#389—611no—
320#388—1221no—
321#386—1221no—
322#380—611no—
323C4×C4⋊C4—611no—
324C4×C22⋊C4—1221no—
325#340—612no—
326#340—612no—
327#336—611no—
328#336C2×D6111yescharacteristic
329C61#2356811yescharacteristic, complemented, Sylow
330C22×Q8—1221no—
331C22×Q8C22×D6112yes—
332C22×D4—1221no—
333C22×D4C22×D6111yescharacteristic
334C23×C4—611no—
335C23×C4—611no—
336C23×C4—611no—
337C23×C4—1221no—
338C2×SD16—24no—
339C2×SD16—24no—
340C2×SD16—24no—
341C22×C8—612no—
342C22×C8C22×D6112yes—
343C4⋊Q8—24no—
344C4⋊Q8—1224no—
345#125—1228no—
346C22⋊Q8—1228no—
347C22⋊Q8—1224no—
348C22⋊Q8—1224no—
349#121—1222no—
350#121—1222no—
351C4×Q8—618no—
352C4×Q8—1224no—
353C4×Q8—1228no—
354C4×D4—1224no—
355C4×D4—2444no—
356#117—1228no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—612no—
360C2×C4⋊C4—611no—
361C2×C4⋊C4—611no—
362C2×C4⋊C4—1222no—
363C2×C4⋊C4—1222no—
364C2×C4⋊C4—1221no—
365C2×C4⋊C4—1221no—
366C2×C4⋊C4C22×D6111yescharacteristic
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—1222no—
370C2×C22⋊C4—1221no—
371C2×C22⋊C4—1221no—
372C2×C22⋊C4—1221no—
373C2×C42—611no—
374C2×C42—611no—
375C2×C42—611no—
376C2×C42—1224no—
377C2×C42—1222no—
378C2×C42—1221no—
379C2×C42C22×D6111yescharacteristic
380#106—1224no—
381#103—24no—
382#103—1224no—
383#103—1224no—
384#102—1224no—
385C22⋊C8—1224no—
386C4×C8—1224no—
387#95—1221no—
388#95—1221no—
389#95—1221no—
390C24—21no—
391C24—611no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—1224no—
398C2×Q8—2442no—
399C2×Q8—2442no—
400C2×Q8D4×D6114yes—
401C2×D4—22no—
402C2×D4—21no—
403C2×D4—22no—
404C2×D4—1224no—
405C2×D4—2442no—
406C2×D4—2442no—
407C2×D4#1175822412yes—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—611no—
411C22×C4—611no—
412C22×C4—614no—
413C22×C4—611no—
414C22×C4—1222no—
415C22×C4—1222no—
416C22×C4—1222no—
417C22×C4—1221no—
418C22×C4—1222no—
419C22×C4—1222no—
420C22×C4—1221no—
421C22×C4—1222no—
422C22×C4—1221no—
423C22×C4—1222no—
424C22×C4—1221no—
425C22×C4—2441no—
426C22×C4—2442no—
427C22×C4—2442no—
428C22×C4—2442no—
429C22×C4D4×D6111yescharacteristic
430C22×C4D4×D6111yescharacteristic
431C22×C4C23×D6111yescharacteristic
432SD16—216no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—1222no—
436C2×C8—1222no—
437C2×C8—1222no—
438C2×C8#1175822414yes—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—22no—
442C4⋊C4—614no—
443C4⋊C4—612no—
444C4⋊C4—1228no—
445C4⋊C4—1228no—
446C4⋊C4—1224no—
447C4⋊C4—1222no—
448C4⋊C4—1221no—
449C4⋊C4—1222no—
450C4⋊C4—2442no—
451C22⋊C4—42no—
452C22⋊C4—42no—
453C22⋊C4—1228no—
454C22⋊C4—1224no—
455C22⋊C4—2442no—
456C22⋊C4—2442no—
457C42—22no—
458C42—614no—
459C42—1222no—
460C42—1222no—
461C42—1228no—
462C42—2442no—
463C23—22no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—612no—
471C23—1221no—
472C23—1221no—
473C23—1221no—
474C23—1221no—
475C23—1221no—
476C23—1221no—
477C23C2×D4×D6111yescharacteristic
478Q8—24no—
479Q8—24no—
480Q8—42no—
481Q8—42no—
482Q8—2448no—
483D4—24no—
484D4—42no—
485D4—2448no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—24no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—612no—
497C2×C4—612no—
498C2×C4—612no—
499C2×C4—612no—
500C2×C4—1228no—
501C2×C4—1221no—
502C2×C4—1221no—
503C2×C4—1221no—
504C2×C4—1224no—
505C2×C4—1221no—
506C2×C4—1222no—
507C2×C4—1221no—
508C2×C4—4884no—
509C2×C4—2444no—
510C2×C4—2441no—
511C2×C4—2441no—
512C2×C4—2441no—
513C2×C4—2441no—
514C2×C4—2442no—
515C2×C4—2441no—
516C2×C4—2442no—
517C2×C4—2442no—
518C2×C4—2442no—
519C2×C4SD16×D6112yes—
520C2×C4#1451979812yes—
521C2×C4C2×D4×D6112yes—
522C2×C4#41×D6111yescharacteristic
523C2×C4C2×#1175822411yescharacteristic
524C2×C4C2×#1175822412yes—
525C8—24no—
526C8—2442no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—81no—
531C22—81no—
532C22—81no—
533C22—81no—
534C22—1222no—
535C22—1222no—
536C22—1222no—
537C22—2441no—
538C22—2441no—
539C22—2441no—
540C22—2441no—
541C22#120×D6111yescharacteristic
542C22#1684027211yescharacteristic
543C22#1684031011yescharacteristic
544C22C2×SD16×D6111yescharacteristic
545C22C2×#1451979811yescharacteristic
546C22#137×D6111yescharacteristic
547C22#1684056111yescharacteristic
548C4—22no—
549C4—22no—
550C4—44no—
551C4—81no—
552C4—614no—
553C4—1222no—
554C4—4881no—
555C4—2441no—
556C4—2444no—
557C4—2442no—
558C4#1684026812yes—
559C4#1684030712yes—
560C2—82no—
561C2—2442no—
562C2#2144903111yescharacteristic
563C2#448×D6111yescharacteristic
564C2#2145503611yescharacteristic
565C2#2145582611yescharacteristic
566C2#2145588711yescharacteristic
567C2#2145637611yescharacteristic
568C2#2145642211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2013C2
socleC23×C61C22×D4
prime radicalC2×#14519748C22
hypercenter#2013D61
hypocenterC61#23568
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2013

Complements

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

NH ≅ G/Ntype
1. C2×#16840317C2semidirect
2. C2×SD16×Dic61C2semidirect
4. #21447153C2semidirect
5. #21447062C2semidirect
6. #21447014C2semidirect
7. #21446955C2semidirect
9. #103×Dic61C2semidirect
10. #21445576C2semidirect
11. C61×#2013C2semidirect
60. Q8×Dic61D4semidirect
143. C2×C4×D61SD16semidirect
283. #2013D61semidirect
329. C61#23568semidirect

Maximal subgroups

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

subgroupindex
C2×#168403172
C2×SD16×Dic612
#214483352
#214471532
#214470622
#214470142
#214469552
#214468192
#103×Dic612
#214455762
C61×#20132
#2356861