Finite Group Databasebeta

← #29737014

Subgroups — #29737014

Order 15616.

1100 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840488C211yesmaximal, characteristic, complemented
2C2×#16840126C211yesmaximal, characteristic
3#21450822C211yesmaximal, characteristic
4#21446963C212yesmaximal, complemented
5#21446953C212yesmaximal, complemented
6#21446774C211yesmaximal, characteristic, complemented
7#21446758C211yesmaximal, characteristic, complemented
8#21446176C212yesmaximal, complemented
9#21446176C212yesmaximal, complemented
10#21445649C211yesmaximal, characteristic, complemented
11C2×C61×#460C211yesmaximal, characteristic
12#16840488—24no—
13#16840488—24no—
14C2×#14519776—21no—
15C2×#14519776C2211yescharacteristic
16C2×#14519769—21no—
17C2×#14519769C2211yescharacteristic
18C2×#14519768—21no—
19C2×#14519768C2211yescharacteristic
20C2×#14519764—21no—
21C2×Q8×Dic61C2211yescharacteristic
22C2×Q8×Dic61C2211yescharacteristic
23C2×C4×Dic122—21no—
24C2×C4×Dic122C2211yescharacteristic
25#16840126—24no—
26C2×#14519701C2212yes—
27C2×#14519698C2212yes—
28C2×#14519698C2212yes—
29C2×#14519688C2211yescharacteristic
30#16839676—21no—
31#16839676—21no—
32#16839665—21no—
33#16839665C2212yes—
34#16839665C2212yes—
35C4×C4⋊Dic61C2211yescharacteristic
36#16839525C2212yes—
37#16839525C2212yes—
38#16839524C2212yes—
39#16839516C2212yes—
40#16839516C2212yes—
41C22×SD16×C61C2212yes—
42C2×#127×C61C2211yescharacteristic
43C2×C4×Q8×C61C2211yescharacteristic
44C61×#460—24no—
45C2×C4⋊C8×C61C2211yescharacteristic
46C2×#102×C61C2212yes—
47C22×Dic122—21no—
48C22×Dic122D411yescharacteristic
49C22×#92418C2312yes—
50#14519776—28no—
51#14519776—22no—
52#14519776—22no—
53#14519769—28no—
54#14519769—22no—
55#14519769D414yes—
56#14519768—28no—
57#14519768—22no—
58#14519768—22no—
59#14519764—44no—
60Q8×Dic61—22no—
61Q8×Dic61—22no—
62Q8×Dic61—22no—
63Q8×Dic61—22no—
64C4×Dic122—28no—
65C4×Dic122—22no—
66C4×Dic122—22no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61D411yescharacteristic
71C2×C4⋊Dic61C2312yes—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196D411yescharacteristic
79C2×#11758196D411yescharacteristic
80C2×C22⋊Dic61—22no—
81C2×C22⋊Dic61—22no—
82C2×C22⋊Dic61—22no—
83C2×C22⋊Dic61—22no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61C2311yescharacteristic
88C2×C4×Dic61C2311yescharacteristic
89#14519701—24no—
90#14519698—24no—
91#14519698—24no—
92#14519688—22no—
93#14519684—21no—
94C22×Q8×C61C2311yescharacteristic
95C22×D4×C61—21no—
96C22×D4×C61—21no—
97C22×D4×C61C2312yes—
98C2×SD16×C61—24no—
99C2×SD16×C61—24no—
100C2×SD16×C61—24no—
101C22×C8×C61C2312yes—
102#127×C61—22no—
103#127×C61—22no—
104C4×Q8×C61—22no—
105C4×Q8×C61D414yes—
106C2×C4⋊C4×C61—21no—
107C2×C4⋊C4×C61C2311yescharacteristic
108C2×C42×C61—21no—
109C2×C42×C61C2311yescharacteristic
110C4⋊C8×C61—22no—
111#102×C61—24no—
112C24×C61—22no—
113C2×Q8×C61—21no—
114C2×Q8×C61—22no—
115C2×Q8×C61C2×D412yes—
116C2×Q8×C61#4114yes—
117C2×D4×C61—22no—
118C2×D4×C61—22no—
119C2×D4×C61—22no—
120C2×D4×C61—22no—
121C2×D4×C61—22no—
122C2×D4×C61—22no—
123C2×D4×C61—42no—
124C2×D4×C61—42no—
125C2×D4×C61—42no—
126C2×D4×C61—42no—
127C2×D4×C61—82no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2411yescharacteristic
133C22×Dic61—21no—
134C22×Dic61—21no—
135C22×Dic61—22no—
136C22×Dic61C2×D411yescharacteristic
137C22×Dic61C2×D411yescharacteristic
138C22×Dic61C2×D411yescharacteristic
139C22×Dic61C2×D411yescharacteristic
140C2×Dic122—24no—
141C2×Dic122—22no—
142C2×Dic122—22no—
143C2×Dic122—41no—
144C2×Dic122—42no—
145C2×Dic122—42no—
146SD16×C61—48no—
147C2×C8×C61—22no—
148C2×C8×C61—22no—
149C2×C8×C61—22no—
150C4⋊C4×C61—24no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61C2×D412yes—
153C42×C61—24no—
154C42×C61C2×D412yes—
155C42×C61C2×D412yes—
156C22⋊Dic61—44no—
157C22⋊Dic61—44no—
158C22⋊Dic61—82no—
159C22⋊Dic61—82no—
160C4⋊Dic61—24no—
161C4⋊Dic61—24no—
162C4⋊Dic61—24no—
163C4⋊Dic61—22no—
164C4⋊Dic61—21no—
165C4⋊Dic61—21no—
166C4⋊Dic61—22no—
167C4⋊Dic61—22no—
168C4⋊Dic61C2×D412yes—
169C4⋊Dic61C2×D412yes—
170#11758196—24no—
171#11758196—24no—
172#11758196—24no—
173#11758196—24no—
174#11758196—22no—
175#11758196—22no—
176C4×Dic61—24no—
177C4×Dic61—24no—
178C4×Dic61—21no—
179C4×Dic61—21no—
180C4×Dic61—21no—
181C4×Dic61—21no—
182C4×Dic61—42no—
183C2×#92418—22no—
184C2×#92418—22no—
185C2×#92418—22no—
186C23×C61—42no—
187C23×C61—42no—
188C23×C61—42no—
189C23×C61—42no—
190C23×C61—42no—
191C23×C61—42no—
192C23×C61—42no—
193C23×C61C22×D411yescharacteristic
194Q8×C61—22no—
195Q8×C61—24no—
196Q8×C61#12414yescomplemented
197D4×C61—42no—
198D4×C61—42no—
199D4×C61—42no—
200D4×C61—42no—
201D4×C61—84no—
202D4×C61—84no—
203C2×C4×C61—22no—
204C2×C4×C61—24no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61#12012yes—
212C2×C4×C61#12012yes—
213C2×C4×C61C22×D411yescharacteristic
214C2×C4×C61C22×D411yescharacteristic
215C2×C4×C61C2×#4112yes—
216C2×C4×C612−1+412yes—
217C2×Dic61—21no—
218C2×Dic61—21no—
219C2×Dic61—21no—
220C2×Dic61—21no—
221C2×Dic61—22no—
222C2×Dic61—21no—
223C2×Dic61—22no—
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—22no—
227C2×Dic61—22no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—42no—
231C2×Dic61—42no—
232C2×Dic61—42no—
233C2×Dic61—42no—
234C2×Dic61AGL(1,ℤ/8)12yes—
235Dic122—44no—
236Dic122—48no—
237C8×C61—42no—
238#92418—42no—
239#25944—611nomaximal, Sylow
240C22×C61—82no—
241C22×C61—82no—
242C22×C61—82no—
243C22×C61—82no—
244C22×C61—82no—
245C22×C61—82no—
246C22×C61—82no—
247C22×C61C2×#12011yescharacteristic
248C22×C61#54812yes—
249C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
250C22×C61#57511yescharacteristic
251C22×C61#57712yes—
252C4×C61—24no—
253C4×C61—22no—
254C4×C61—22no—
255C4×C61—42no—
256C4×C61C2×#12411yescharacteristic
257C4×C61#53611yescharacteristic
258C4×C61#54812yes—
259Dic61—22no—
260Dic61—41no—
261Dic61—42no—
262Dic61—41no—
263Dic61—41no—
264Dic61—42no—
265Dic61—82no—
266C2×#557—611no—
267C2×#490—611no—
268C2×#460D6111yescharacteristic
269#2548—611no—
270#2025—612no—
271#2023—612no—
272#1989—611no—
273#1985—611no—
274#1880—612no—
275#1880—612no—
276#1763—611no—
277C2×C61—162no—
278C2×C61#243611yescharacteristic
279C2×C61#300611yescharacteristic
280C2×C61#301011yescharacteristic
281C2×C61#336812yes—
282C2×C61#337112yes—
283C22×SD16C2×D6112yes—
284#557—1224no—
285#557—1224no—
286C2×C4⋊Q8—611no—
287C2×C4⋊Q8—1221no—
288C2×#127C2×D6111yescharacteristic
289C2×#125—611no—
290C2×#125—611no—
291C2×#125—1221no—
292C2×#125—1221no—
293C2×#124—1221no—
294C2×C4×Q8—611no—
295C2×C4×Q8—611no—
296C2×C4×Q8—611no—
297C2×C4×Q8—1221no—
298C2×C4×Q8C2×D6111yescharacteristic
299#490—1224no—
300#460—24no—
301C2×C4⋊C8C2×D6111yescharacteristic
302C2×#103—612no—
303C2×#102—612no—
304C2×#102—612no—
305C2×#102C2×D6112yes—
306C2×#97—611no—
307#394—1221no—
308#394—1221no—
309#390—612no—
310#390—612no—
311#390—1221no—
312C4×C4⋊C4—611no—
313#340—612no—
314#340—612no—
315#340—612no—
316#336—612no—
317#336—612no—
318C61#2594411yescharacteristic, complemented, Sylow
319C22×Q8—611no—
320C22×Q8—1221no—
321C22×Q8C22×D6111yescharacteristic
322C22×D4—21no—
323C22×D4—21no—
324C22×D4C22×D6112yes—
325C2×SD16—24no—
326C2×SD16—24no—
327C2×SD16—24no—
328C22×C8—612no—
329C22×C8C22×D6112yes—
330C4⋊Q8—1222no—
331C4⋊Q8—1228no—
332C4⋊Q8—1222no—
333#127—22no—
334#127—22no—
335#125—614no—
336#125—1228no—
337#125—1222no—
338#125—1228no—
339#125—1222no—
340#125—1222no—
341#124—2444no—
342C4×Q8—22no—
343C4×Q8—1228no—
344C4×Q8—1222no—
345C4×Q8—1222no—
346C4×Q8—1222no—
347C4×Q8—1222no—
348C4×Q8—1222no—
349C4×Q8—1222no—
350C4×Q8C22⋊D6114yes—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—611no—
353C2×C4⋊C4—611no—
354C2×C4⋊C4—611no—
355C2×C4⋊C4—611no—
356C2×C4⋊C4—611no—
357C2×C4⋊C4—612no—
358C2×C4⋊C4—1221no—
359C2×C4⋊C4—1221no—
360C2×C4⋊C4—1221no—
361C2×C4⋊C4—1221no—
362C2×C4⋊C4—1221no—
363C2×C4⋊C4—1221no—
364C2×C4⋊C4—1221no—
365C2×C4⋊C4C22×D6111yescharacteristic
366C2×C22⋊C4—1222no—
367C2×C22⋊C4—1222no—
368C2×C22⋊C4—1222no—
369C2×C22⋊C4—1222no—
370C2×C42—21no—
371C2×C42—611no—
372C2×C42—611no—
373C2×C42—1221no—
374C2×C42—1221no—
375C2×C42—1221no—
376C2×C42C22×D6111yescharacteristic
377C4⋊C8—22no—
378#103—1224no—
379#102—24no—
380#102—1224no—
381#102—1224no—
382#97—1222no—
383#95—1221no—
384C24—22no—
385C2×Q8—21no—
386C2×Q8—22no—
387C2×Q8—1222no—
388C2×Q8—1224no—
389C2×Q8—1222no—
390C2×Q8—2441no—
391C2×Q8—2442no—
392C2×Q8—2442no—
393C2×Q8#1175822414yes—
394C2×Q8C2×C22⋊D6112yes—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—82no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—611no—
409C22×C4—611no—
410C22×C4—611no—
411C22×C4—611no—
412C22×C4—1221no—
413C22×C4—1221no—
414C22×C4—1222no—
415C22×C4D4×D6111yescharacteristic
416C22×C4D4×D6111yescharacteristic
417C22×C4C23×D6111yescharacteristic
418SD16—48no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—1222no—
423C2×C8—1222no—
424C2×C8—1222no—
425C4⋊C4—21no—
426C4⋊C4—24no—
427C4⋊C4—612no—
428C4⋊C4—612no—
429C4⋊C4—1224no—
430C4⋊C4—1224no—
431C4⋊C4—1224no—
432C4⋊C4—1224no—
433C4⋊C4—1224no—
434C4⋊C4—1221no—
435C4⋊C4—1221no—
436C4⋊C4—1222no—
437C4⋊C4—1222no—
438C4⋊C4—1222no—
439C4⋊C4—1224no—
440C4⋊C4—1224no—
441C4⋊C4—1222no—
442C4⋊C4—1222no—
443C4⋊C4C2×C22⋊D6112yes—
444C22⋊C4—4882no—
445C22⋊C4—4882no—
446C22⋊C4—2444no—
447C22⋊C4—2444no—
448C42—24no—
449C42—1224no—
450C42—1224no—
451C42—1221no—
452C42—1221no—
453C42—1221no—
454C42—1221no—
455C42—2442no—
456C42D4×D6112yes—
457C42C2×C22⋊D6112yes—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23C2×D4×D6111yescharacteristic
466Q8—22no—
467Q8—24no—
468Q8—2444no—
469Q8—2448no—
470Q8#1451976414yescomplemented
471D4—42no—
472D4—42no—
473D4—42no—
474D4—42no—
475D4—84no—
476D4—84no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—612no—
486C2×C4—1221no—
487C2×C4—1221no—
488C2×C4—1221no—
489C2×C4—1222no—
490C2×C4—1222no—
491C2×C4—1221no—
492C2×C4—1221no—
493C2×C4—1221no—
494C2×C4—1221no—
495C2×C4—1222no—
496C2×C4—1222no—
497C2×C4—2442no—
498C2×C4—2442no—
499C2×C4—2442no—
500C2×C4—2441no—
501C2×C4—2441no—
502C2×C4—2442no—
503C2×C4#1451974512yes—
504C2×C4#1451974512yes—
505C2×C4C2×D4×D6111yescharacteristic
506C2×C4C22×C22⋊D6111yescharacteristic
507C2×C4C2×#1175822412yes—
508C2×C4#1451982412yes—
509C8—42no—
510C8—2442no—
511C22—82no—
512C22—82no—
513C22—82no—
514C22—82no—
515C22—82no—
516C22—82no—
517C22—82no—
518C22C2×#1451974511yescharacteristic
519C22#1684041812yes—
520C22AGL(1,ℤ/8)×D6111yescharacteristic
521C22#1684055811yescharacteristic
522C22#1684056812yes—
523C4—24no—
524C4—22no—
525C4—22no—
526C4—42no—
527C4—1222no—
528C4—4882no—
529C4—2441no—
530C4—2442no—
531C4—2441no—
532C4—2441no—
533C4—2442no—
534C4C2×#1451976411yescharacteristic
535C4#1684033311yescharacteristic
536C4#1684041812yes—
537C2—162no—
538C2#2144947811yescharacteristic
539C2#2145506711yescharacteristic
540C2#2145510111yescharacteristic
541C2#2145854312yes—
542C2#2145858012yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#460C2
socleC23×C61C22×D4
prime radicalC2×#127×C61C22
hypercenterC2×#460D61
hypocenterC61#25944
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#460

Complements

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

NH ≅ G/Ntype
1. C2×#16840488C2semidirect
4. #21446963C2semidirect
5. #21446953C2semidirect
6. #21446774C2semidirect
7. #21446758C2semidirect
8. #21446176C2semidirect
9. #21446176C2semidirect
10. #21445649C2semidirect
196. Q8×C61#124semidirect
318. C61#25944semidirect
470. Q8#14519764semidirect

Maximal subgroups

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

subgroupindex
C2×#168404882
C2×#168401262
#214508222
#214469632
#214469532
#214467742
#214467582
#214461762
#214456492
C2×C61×#4602
#2594461