Finite Group Databasebeta

← #29733297

Subgroups — #29733297

Order 15616.

1422 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×#16840416C211yesmaximal, characteristic, complemented
2C2×#16840121C211yesmaximal, characteristic, complemented
3C2×#16839974C211yesmaximal, characteristic, complemented
4#21446820C212yesmaximal, complemented
5#21446815C212yesmaximal, complemented
6#21446767C211yesmaximal, characteristic, complemented
7#21446618C211yesmaximal, characteristic, complemented
8#21446137C212yesmaximal, complemented
9#21446127C212yesmaximal, complemented
10#21445647C211yesmaximal, characteristic, complemented
11C61×#2540C211yesmaximal, characteristic, complemented
12C22×#11758224—21no—
13C22×#11758201C2211yescharacteristic
14C22×#11758199C2211yescharacteristic
15#16840416—24no—
16#16840416—24no—
17C2×#14519777C2211yescharacteristic
18C2×#14519761—22no—
19C2×#14519757—21no—
20C2×#14519751C2212yes—
21C2×C4×D244C2211yescharacteristic
22C22×C4⋊Dic61C2212yes—
23#16840121—24no—
24#16839974—24no—
25C2×C4⋊#92418C2211yescharacteristic
26C2×#14519701C2212yes—
27C2×#14519701C2211yescharacteristic
28C2×#14519698C2212yes—
29C2×#14519698C2211yescharacteristic
30C2×#14519697C2212yes—
31C2×C22⋊#92418C2212yes—
32C2×C22⋊#92418C2212yes—
33C2×#14519688C2211yescharacteristic
34#16839524C2212yes—
35#16839524C2212yes—
36#16839524C2212yes—
37C2×#124×C61C2211yescharacteristic
38C2×C22⋊Q8×C61C2212yes—
39C2×#121×C61C2212yes—
40C2×C4×D4×C61C2211yescharacteristic
41C61×#399—21no—
42C61×#392—21no—
43C61×#390—22no—
44C4×C4⋊C4×C61C2211yescharacteristic
45C2×#11758224—28no—
46C2×#11758224—48no—
47C22×Dic122—21no—
48C22×D244C2311yescharacteristic
49C22×C22⋊D61—22no—
50C23×Dic61—22no—
51C22×C4×D61—21no—
52C2×#11758201—22no—
53C2×#11758201—42no—
54C2×#11758199—22no—
55C2×#11758199—42no—
56C22×#92418C2312yes—
57C22×#92418C2311yescharacteristic
58C22×#92418C2311yescharacteristic
59#14519777—22no—
60#14519777D414yescomplemented
61#14519761—216no—
62#14519757—28no—
63#14519751—24no—
64#14519751—24no—
65C4×D244—22no—
66C4×D244D414yescomplemented
67C2×C4⋊Dic61—22no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—24no—
70C2×C4⋊Dic61—24no—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2312yes—
73C2×#11758198—21no—
74C2×#11758198—22no—
75C4⋊#92418—22no—
76#14519701—22no—
77#14519701—24no—
78#14519698—22no—
79#14519698—24no—
80#14519697—24no—
81C22⋊#92418—42no—
82C22⋊#92418—42no—
83#14519688—22no—
84C22×Q8×C61C2311yescharacteristic
85C22×D4×C61—22no—
86C22×D4×C61D411yescharacteristic
87C22×D4×C61C2311yescharacteristic
88C23×C4×C61C2312yes—
89#124×C61—22no—
90#124×C61—22no—
91C22⋊Q8×C61—24no—
92C22⋊Q8×C61—42no—
93#121×C61—24no—
94#121×C61—42no—
95C4×D4×C61—22no—
96C4×D4×C61—22no—
97C2×C4⋊C4×C61—22no—
98C2×C4⋊C4×C61D411yescharacteristic
99C2×C4⋊C4×C61C2312yes—
100C2×C22⋊C4×C61—22no—
101C2×C22⋊C4×C61—21no—
102C2×C22⋊C4×C61—22no—
103C2×C22⋊C4×C61—21no—
104C2×C22⋊C4×C61D412yes—
105C2×C42×C61—21no—
106C2×C42×C61C2311yescharacteristic
107#95×C61—21no—
108C24×C61—21no—
109C24×C61C2×D412yes—
110C23×D61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×Q8×C61—22no—
114C2×Q8×C61—22no—
115C2×D4×C61—24no—
116C2×D4×C61—22no—
117C2×D4×C61—21no—
118C2×D4×C61—21no—
119C2×D4×C61—22no—
120C2×D4×C61—22no—
121C2×D4×C61—44no—
122C2×D4×C61—44no—
123C2×D4×C61—82no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61—21no—
132C22×C4×C61C2×D412yes—
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2411yescharacteristic
136C2×C22⋊D61—28no—
137C2×C22⋊D61—44no—
138C2×C22⋊D61—44no—
139C22×Dic61—28no—
140C22×Dic61—21no—
141C22×Dic61—22no—
142C22×Dic61—44no—
143C22×Dic61—44no—
144#11758224—432no—
145C2×D244—22no—
146C2×D244—22no—
147C2×D244—21no—
148C2×D244C2×D412yes—
149C2×C4×D61—24no—
150C2×C4×D61—44no—
151C2×Dic122—24no—
152C2×Dic122—44no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—44no—
156C4⋊C4×C61#4114yescomplemented
157C22⋊C4×C61—24no—
158C22⋊C4×C61—44no—
159C22⋊C4×C61—44no—
160C22⋊C4×C61—42no—
161C22⋊C4×C61—42no—
162C42×C61—81no—
163C42×C61C2×D412yes—
164C42×C61C2×D412yes—
165#11758201—44no—
166#11758199—44no—
167#11758198—28no—
168#11758198—24no—
169C4⋊Dic61—28no—
170C4⋊Dic61—24no—
171C4⋊Dic61—21no—
172C4⋊Dic61—22no—
173C4⋊Dic61C2×D412yes—
174C4⋊Dic61C2×D414yes—
175C2×#92418—22no—
176C2×#92418—21no—
177C2×#92418—21no—
178C2×#92418—22no—
179C2×#92418—22no—
180C2×#92418—22no—
181C2×#92418—22no—
182C23×C61—22no—
183C23×C61—22no—
184C23×C61—22no—
185C23×C61—22no—
186C23×C61—22no—
187C23×C61—22no—
188C23×C61—41no—
189C23×C61—42no—
190C23×C61—42no—
191C23×C61—41no—
192C23×C61—41no—
193C23×C61AGL(1,ℤ/8)14yes—
194C23×C61C22×D411yescharacteristic
195C22×D61—22no—
196C22×D61—42no—
197C22×D61—42no—
198C22×D61—41no—
199C22×D61—41no—
200Q8×C61—41no—
201Q8×C61—41no—
202Q8×C61—42no—
203D4×C61—28no—
204D4×C61—41no—
205D4×C61—41no—
206D4×C61—42no—
207D4×C61—88no—
208C2×C4×C61—22no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—22no—
212C2×C4×C61—24no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61—42no—
221C2×C4×C61—41no—
222C2×C4×C61—42no—
223C2×C4×C61#12012yes—
224C2×C4×C61#12012yes—
225C2×C4×C61C22×D411yescharacteristic
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C2×#4112yes—
228C2×C4×C61H(5,𝔽2)12yes—
229C22⋊D61—416no—
230C2×Dic61—24no—
231C2×Dic61—22no—
232C2×Dic61—416no—
233C2×Dic61—42no—
234C2×Dic61—42no—
235C2×Dic61—42no—
236D244—22no—
237D244—22no—
238D244—42no—
239C4×D61—48no—
240Dic122—48no—
241#92418—41no—
242#92418—41no—
243#92418—42no—
244#25577—611nomaximal, Sylow
245C22×C61—24no—
246C22×C61—24no—
247C22×C61—24no—
248C22×C61—42no—
249C22×C61—42no—
250C22×C61—42no—
251C22×C61—42no—
252C22×C61—81no—
253C22×C61—81no—
254C22×C61—81no—
255C22×C61—82no—
256C22×C61—82no—
257C22×C61C2×#12011yescharacteristic
258C22×C61#54612yes—
259C22×C61C2×AGL(1,ℤ/8)12yes—
260C22×C61#57711yescharacteristic
261C22×C61#57811yescharacteristic
262C2×D61—42no—
263C2×D61—42no—
264C2×D61—42no—
265C2×D61—82no—
266C2×D61—82no—
267C4×C61—22no—
268C4×C61—22no—
269C4×C61—42no—
270C4×C61—44no—
271C4×C61—81no—
272C4×C61#53211yescharacteristic
273C4×C61#53511yescharacteristic
274C4×C61#55012yes—
275Dic61—44no—
276Dic61—42no—
277C2×#547—611no—
278C2×#489—611no—
279C2×#463—611no—
280#2540D6111yescharacteristic, complemented
281#2000—612no—
282#1999—612no—
283#1987—611no—
284#1955—611no—
285#1874—612no—
286#1872—612no—
287#1762—611no—
288C2×C61—44no—
289C2×C61—161no—
290C2×C61#243111yescharacteristic
291C2×C61#300311yescharacteristic
292C2×C61#301311yescharacteristic
293C2×C61#330612yes—
294C2×C61#330812yes—
295D61—82no—
296C22×#41—1221no—
297C22×SD16—611no—
298C22×D8—611no—
299#547—1224no—
300#547—1224no—
301C2×C4⋊Q8—611no—
302C2×#124C2×D6111yescharacteristic
303C2×#123—1222no—
304C2×C22⋊Q8—1221no—
305C2×C22⋊Q8C2×D6112yes—
306C2×#121—612no—
307C2×#121C2×D6112yes—
308C2×C4×D4—611no—
309C2×C4×D4C2×D6111yescharacteristic
310C22×C4⋊C4—612no—
311#489—1224no—
312#463—1224no—
313C2×C4⋊C8—611no—
314C2×#103—612no—
315C2×#103—611no—
316C2×#102—612no—
317C2×#102—611no—
318C2×#102—612no—
319C2×C22⋊C8—612no—
320C2×C22⋊C8—612no—
321C2×#97—611no—
322#399—21no—
323#392—21no—
324#390—22no—
325C4×C4⋊C4C2×D6111yescharacteristic
326#340—612no—
327#340—612no—
328#340—612no—
329C61#2557711yescharacteristic, complemented, Sylow
330C2×#41—1228no—
331C2×#41—2448no—
332C22×Q8—1221no—
333C22×Q8C22×D6111yescharacteristic
334C22×D4—22no—
335C22×D4—611no—
336C22×D4—1222no—
337C22×D4C22⋊D6111yescharacteristic
338C22×D4C22×D6111yescharacteristic
339C23×C4—1222no—
340C23×C4—1221no—
341C23×C4C22×D6112yes—
342C2×SD16—1222no—
343C2×SD16—2442no—
344C2×D8—1222no—
345C2×D8—2442no—
346C22×C8—611no—
347C22×C8—611no—
348C22×C8—612no—
349C4⋊Q8—614no—
350C4⋊Q8—1222no—
351#124—22no—
352#124—22no—
353#123—12216no—
354C22⋊Q8—24no—
355C22⋊Q8—42no—
356C22⋊Q8—1228no—
357#121—24no—
358#121—42no—
359#121—1224no—
360#121—1224no—
361C4×D4—22no—
362C4×D4—22no—
363C4×D4—614no—
364C4×D4—1222no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—611no—
367C2×C4⋊C4—612no—
368C2×C4⋊C4—1224no—
369C2×C4⋊C4—1224no—
370C2×C4⋊C4—1222no—
371C2×C4⋊C4—1221no—
372C2×C4⋊C4C22⋊D6111yescharacteristic
373C2×C4⋊C4C22×D6112yes—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—1221no—
379C2×C22⋊C4—1222no—
380C2×C22⋊C4C22⋊D6112yes—
381C2×C42—21no—
382C2×C42C22×D6111yescharacteristic
383C4⋊C8—1222no—
384#103—1222no—
385#103—1224no—
386#102—1222no—
387#102—1224no—
388#102—1224no—
389C22⋊C8—2442no—
390C22⋊C8—2442no—
391#97—1222no—
392#95—21no—
393C24—21no—
394C24—1221no—
395C24C2×C22⋊D6112yes—
396#41—24432no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—1224no—
402C2×Q8—2444no—
403C2×D4—24no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—82no—
412C2×D4—612no—
413C2×D4—1228no—
414C2×D4—1221no—
415C2×D4—1222no—
416C2×D4—1222no—
417C2×D4—2444no—
418C2×D4—2444no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—1228no—
428C22×C4—1221no—
429C22×C4—1222no—
430C22×C4—1224no—
431C22×C4—2444no—
432C22×C4—2444no—
433C22×C4—2444no—
434C22×C4C2×C22⋊D6112yes—
435C22×C4C2×C22⋊D6111yescharacteristic
436C22×C4C2×C22⋊D6111yescharacteristic
437C22×C4C23×D6111yescharacteristic
438SD16—2444no—
439D8—2444no—
440C2×C8—1222no—
441C2×C8—1221no—
442C2×C8—1221no—
443C2×C8—1222no—
444C2×C8—1222no—
445C2×C8—1222no—
446C2×C8—1222no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—44no—
450C4⋊C4—612no—
451C4⋊C4—614no—
452C4⋊C4—1228no—
453C4⋊C4—1221no—
454C4⋊C4—1222no—
455C4⋊C4—1224no—
456C4⋊C4#1175822414yescomplemented
457C22⋊C4—24no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C22⋊C4—1228no—
463C22⋊C4—1224no—
464C42—81no—
465C42D4×D6112yes—
466C42D4×D6112yes—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—41no—
474C23—42no—
475C23—41no—
476C23—41no—
477C23—42no—
478C23—1222no—
479C23—2442no—
480C23—2441no—
481C23—2441no—
482C23—2442no—
483C23#1451980214yes—
484C23C22×C22⋊D6111yescharacteristic
485Q8—41no—
486Q8—41no—
487Q8—42no—
488Q8—2448no—
489D4—28no—
490D4—41no—
491D4—41no—
492D4—42no—
493D4—88no—
494D4—1222no—
495D4—1222no—
496D4—24416no—
497D4—2442no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—1222no—
514C2×C4—1224no—
515C2×C4—2448no—
516C2×C4—24416no—
517C2×C4—2442no—
518C2×C4—2442no—
519C2×C4—2442no—
520C2×C4#1451974512yes—
521C2×C4#1451974512yes—
522C2×C4C2×D4×D6111yescharacteristic
523C2×C4C2×D4×D6111yescharacteristic
524C2×C4C2×#1175822412yes—
525C2×C4#1451982212yes—
526C8—2441no—
527C8—2441no—
528C8—2442no—
529C22—24no—
530C22—24no—
531C22—24no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—81no—
537C22—82no—
538C22—82no—
539C22—81no—
540C22—81no—
541C22—4882no—
542C22—4882no—
543C22—2442no—
544C22—2442no—
545C22—2442no—
546C22C2×#1451974511yescharacteristic
547C22#1684040612yes—
548C22C2×#1451980212yes—
549C22#1684057311yescharacteristic
550C22#1684057711yescharacteristic
551C4—22no—
552C4—22no—
553C4—44no—
554C4—42no—
555C4—81no—
556C4—2442no—
557C4—2444no—
558C4#1684031011yescharacteristic
559C4#1684032511yescharacteristic
560C4#1684043212yes—
561C2—44no—
562C2—161no—
563C2—4882no—
564C2#2144943511yescharacteristic
565C2#2145504511yescharacteristic
566C2#2145513111yescharacteristic
567C2#2145778012yes—
568C2#2145781012yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2540C2
socleC23×C61C22×D4
hypercenter#2540D61
hypocenterC61#25577
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2540

Complements

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

NH ≅ G/Ntype
1. C2×#16840416C2semidirect
2. C2×#16840121C2semidirect
3. C2×#16839974C2semidirect
4. #21446820C2semidirect
5. #21446815C2semidirect
6. #21446767C2semidirect
7. #21446618C2semidirect
8. #21446137C2semidirect
9. #21446127C2semidirect
10. #21445647C2semidirect
11. C61×#2540C2semidirect
60. #14519777D4semidirect
66. C4×D244D4semidirect
156. C4⋊C4×C61#41semidirect
280. #2540D61semidirect
329. C61#25577semidirect
456. C4⋊C4#11758224semidirect

Maximal subgroups

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

subgroupindex
C2×#168404162
C2×#168401212
C2×#168399742
#214468202
#214468152
#214467672
#214466182
#214461372
#214461272
#214456472
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