Finite Group Databasebeta

← #29779545

Subgroups — #29779545

Order 15616.

1122 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454169C211yesmaximal, characteristic, complemented
2C2×#16839663C211yesmaximal, characteristic
3#21447226C212yesmaximal, complemented
4#21446954C212yesmaximal, complemented
5#21446517C212yesmaximal, complemented
6#21446514C212yesmaximal, complemented
7#21445909C211yesmaximal, characteristic, complemented
8#21445751C211yesmaximal, characteristic
9#21445749C211yesmaximal, characteristic
10#21445583C211yesmaximal, characteristic, complemented
11C61×#1824C211yesmaximal, characteristic, complemented
12C23×D244C2211yescharacteristic
13C2×#14519776C2211yescharacteristic, complemented
14C2×#14519754—21no—
15C22×#11758198—21no—
16C22×#11758198—21no—
17C22×C4×Dic61C2211yescharacteristic
18C2×#14519699C2212yes—
19C2×#14519699C2212yes—
20C2×#14519697C2212yes—
21C2×#14519697C2212yes—
22C2×C22⋊#92418C2211yescharacteristic
23C2×#14519687C2212yescomplemented
24#16839666—21no—
25#16839663—21no—
26#16839663—21no—
27#16839663C2212yes—
28#16839663C2212yes—
29#16839661C2211yescharacteristic, complemented
30#16839659—21no—
31#16839658C2211yescharacteristic, complemented
32#16839631C2211yescharacteristic, complemented
33#16839622C2211yescharacteristic
34#16839620—21no—
35#16839524C2212yescomplemented
36#16839524C2212yescomplemented
37#16839516C2212yescomplemented
38C22×C4⋊C4×C61C2211yescharacteristic
39C2×C22⋊C8×C61C2211yescharacteristic
40C61×#380C2211yescharacteristic
41C61×#340C2212yescomplemented
42C61×#340C2212yescomplemented
43C24×D61—21no—
44C22×Dic122—21no—
45C22×D244—21no—
46C22×D244—24no—
47C22×D244—22no—
48C22×D244—22no—
49C22×D244—42no—
50C22×D244C2312yes—
51C23×Dic61D411yescharacteristic
52C23×Dic61D411yescharacteristic
53C22×#92418C2312yes—
54#14519776—42no—
55#14519754—44no—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61C2311yescharacteristic
58C2×C4⋊Dic61C2311yescharacteristic
59C2×#11758196—21no—
60C2×#11758196—21no—
61C2×#11758196—21no—
62C2×#11758196—21no—
63C2×C22⋊Dic61—21no—
64C2×#11758198—22no—
65C2×#11758198—22no—
66C2×#11758198—22no—
67C2×#11758198—22no—
68C2×#11758198—44no—
69C2×#11758198—44no—
70C2×C4×Dic61—42no—
71C2×C4×Dic61D411yescharacteristic
72C2×C4×Dic61D411yescharacteristic
73C2×C4×Dic61C2311yescharacteristic
74C2×C4×Dic61C2311yescharacteristic
75#14519699—24no—
76#14519699—42no—
77#14519697—24no—
78#14519697—42no—
79C22⋊#92418—22no—
80#14519687—42no—
81#14519684—21no—
82#14519684—21no—
83#14519684—21no—
84C23×C4×C61—21no—
85C23×C4×C61C2311yescharacteristic
86C22×C8×C61C2312yes—
87C2×C4⋊C4×C61—22no—
88C2×C4⋊C4×C61—22no—
89C2×C4⋊C4×C61—21no—
90C2×C4⋊C4×C61C2311yescharacteristic
91C2×C4⋊C4×C61C2311yescharacteristic
92C2×C4⋊C4×C61C2312yes—
93C2×C22⋊C4×C61—21no—
94C22⋊C8×C61—22no—
95#95×C61—21no—
96C24×C61C2×D411yescharacteristic
97C23×D61—24no—
98C23×D61—22no—
99C23×D61—42no—
100C23×D61—42no—
101C23×D61—42no—
102C23×D61—42no—
103C23×D61—42no—
104C23×D61—42no—
105C22×C4×C61—22no—
106C22×C4×C61—22no—
107C22×C4×C61—21no—
108C22×C4×C61—21no—
109C22×C4×C61—22no—
110C22×C4×C61—42no—
111C22×C4×C61—42no—
112C22×C4×C61—42no—
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61#4112yes—
115C22×C4×C61#4112yes—
116C22×C4×C61C2411yescharacteristic
117C22×Dic61—22no—
118C22×Dic61—22no—
119C22×Dic61—22no—
120C22×Dic61—22no—
121C22×Dic61—21no—
122C22×Dic61—21no—
123C22×Dic61—41no—
124C22×Dic61—41no—
125C22×Dic61C2×D411yescharacteristic
126C22×Dic61C2×D411yescharacteristic
127C22×Dic61C2×D411yescharacteristic
128C22×Dic61C2×D411yescharacteristic
129C2×D244—22no—
130C2×D244—22no—
131C2×D244—22no—
132C2×D244—22no—
133C2×D244—24no—
134C2×D244—22no—
135C2×D244—48no—
136C2×D244—48no—
137C2×D244—42no—
138C2×D244—42no—
139C2×D244#4114yes—
140C2×Dic122—42no—
141C2×Dic122—42no—
142C2×Dic122—42no—
143C2×C8×C61—22no—
144C2×C8×C61—22no—
145C2×C8×C61—22no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—42no—
151C22⋊C4×C61—42no—
152C22⋊Dic61—42no—
153#11758198—44no—
154#11758198—44no—
155#11758198—44no—
156#11758198—44no—
157C4⋊Dic61—22no—
158C4⋊Dic61—22no—
159C4⋊Dic61—42no—
160#11758196—42no—
161#11758196—42no—
162#11758196—42no—
163#11758196—42no—
164C4×Dic61—22no—
165C4×Dic61—22no—
166C4×Dic61—41no—
167C4×Dic61—41no—
168C2×#92418—22no—
169C2×#92418—22no—
170C2×#92418—22no—
171C23×C61—21no—
172C23×C61—21no—
173C23×C61—21no—
174C23×C61—21no—
175C23×C61#12412yes—
176C23×C61AGL(1,ℤ/8)12yes—
177C23×C61#13712yes—
178C23×C61C22×D411yescharacteristic
179C22×D61—24no—
180C22×D61—48no—
181C22×D61—48no—
182C22×D61—48no—
183C22×D61—42no—
184C22×D61—42no—
185C22×D61—42no—
186C22×D61—42no—
187C22×D61—42no—
188C22×D61—42no—
189C22×D61—84no—
190C22×D61—84no—
191C22×D61—84no—
192C22×D61—84no—
193C2×C4×C61—21no—
194C2×C4×C61—21no—
195C2×C4×C61—24no—
196C2×C4×C61—24no—
197C2×C4×C61—41no—
198C2×C4×C61—42no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61—41no—
206C2×C4×C61—84no—
207C2×C4×C61#12412yes—
208C2×C4×C61C2×#4112yes—
209C2×C4×C61C2×#4111yescharacteristic
210C2×C4×C61C2×#4111yescharacteristic
211C2×C4×C61H(5,𝔽2)12yes—
212C2×Dic61—21no—
213C2×Dic61—21no—
214C2×Dic61—21no—
215C2×Dic61—21no—
216C2×Dic61—21no—
217C2×Dic61—21no—
218C2×Dic61—21no—
219C2×Dic61—21no—
220C2×Dic61—21no—
221C2×Dic61—21no—
222C2×Dic61—44no—
223C2×Dic61—44no—
224C2×Dic61—41no—
225C2×Dic61—41no—
226C2×Dic61—41no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61AGL(1,ℤ/8)12yes—
231C2×Dic61AGL(1,ℤ/8)12yes—
232D244—24no—
233D244—24no—
234D244—48no—
235D244—42no—
236D244—42no—
237Dic122—84no—
238C8×C61—42no—
239#92418—42no—
240#29109—611nomaximal, Sylow
241C22×C61—22no—
242C22×C61—22no—
243C22×C61—22no—
244C22×C61—22no—
245C22×C61—22no—
246C22×C61—22no—
247C22×C61#48914yes—
248C22×C61#52511yescharacteristic
249C22×C61C2×#12411yescharacteristic
250C22×C61#53411yescharacteristic
251C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
252C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
253C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
254C22×C61C2×#13711yescharacteristic
255C2×D61—44no—
256C2×D61—44no—
257C2×D61—44no—
258C2×D61—816no—
259C2×D61—82no—
260C2×D61—82no—
261C2×D61—82no—
262C2×D61—82no—
263C4×C61—22no—
264C4×C61—81no—
265C4×C61—82no—
266C4×C61—81no—
267C4×C61#55112yes—
268C4×C61#55112yes—
269Dic61—22no—
270Dic61—22no—
271Dic61—41no—
272Dic61—41no—
273Dic61—81no—
274Dic61—81no—
275#2857—611no—
276C2×#390—611no—
277#2071—612no—
278#2023—612no—
279#1936—612no—
280#1936—612no—
281#1824—611no—
282#1824D6111yescharacteristic, complemented
283#1786—611no—
284#1786—611no—
285#1749—611no—
286C2×C61—24no—
287C2×C61#256111yescharacteristic
288C2×C61#310911yescharacteristic
289C2×C61#311011yescharacteristic
290C2×C61C2×#48911yescharacteristic
291C2×C61#313811yescharacteristic
292C2×C61#319211yescharacteristic
293C2×C61#319311yescharacteristic
294D61—84no—
295C23×D4—611no—
296C2×C4⋊Q8—611no—
297C2×C22⋊Q8—1221no—
298C22×C4⋊C4C2×D6111yescharacteristic
299C22×C22⋊C4—1221no—
300C22×C22⋊C4—1221no—
301C22×C42—611no—
302C2×#102—612no—
303C2×#102—612no—
304C2×#102—612no—
305C2×#102—612no—
306C2×C22⋊C8—611no—
307C2×C22⋊C8C2×D6111yescharacteristic
308C2×#97—612no—
309#391—1221no—
310#390—612no—
311#390—612no—
312#390—1221no—
313#390—1221no—
314#389—611no—
315#389—611no—
316#389—1221no—
317#384—611no—
318#380—611no—
319#380—1221no—
320#380C2×D6111yescharacteristic
321#340—612no—
322#340—612no—
323#340C2×D6112yescomplemented
324#340C2×D6112yescomplemented
325#336—612no—
326C61#2910911yescharacteristic, complemented, Sylow
327C25—1221no—
328C22×Q8—1221no—
329C22×D4—612no—
330C22×D4—1224no—
331C22×D4—1222no—
332C22×D4—1221no—
333C22×D4—1222no—
334C22×D4—2442no—
335C23×C4—21no—
336C23×C4—611no—
337C23×C4—611no—
338C23×C4C22×D6111yescharacteristic
339C22×C8—612no—
340C22×C8C22×D6112yes—
341C4⋊Q8—2442no—
342C22⋊Q8—2444no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—611no—
347C2×C4⋊C4—611no—
348C2×C4⋊C4—1221no—
349C2×C4⋊C4—1221no—
350C2×C4⋊C4—1221no—
351C2×C4⋊C4—1221no—
352C2×C4⋊C4—1221no—
353C2×C4⋊C4C22×D6112yes—
354C2×C4⋊C4C22×D6111yescharacteristic
355C2×C4⋊C4C22×D6111yescharacteristic
356C2×C22⋊C4—21no—
357C2×C22⋊C4—1222no—
358C2×C22⋊C4—1222no—
359C2×C22⋊C4—1222no—
360C2×C22⋊C4—1222no—
361C2×C22⋊C4—1221no—
362C2×C22⋊C4—2444no—
363C2×C22⋊C4—2444no—
364C2×C42—611no—
365C2×C42—611no—
366C2×C42—611no—
367C2×C42—611no—
368C2×C42—2442no—
369#102—1224no—
370#102—1224no—
371#102—2442no—
372#102—2442no—
373C22⋊C8—22no—
374C22⋊C8—1222no—
375#97—2442no—
376#95—21no—
377#95—1221no—
378#95—1221no—
379#95—1221no—
380C24—1224no—
381C24—1222no—
382C24—2442no—
383C24—2442no—
384C24—2442no—
385C24—2442no—
386C24—2442no—
387C24—2442no—
388C24D4×D6111yescharacteristic
389C2×Q8—2442no—
390C2×Q8—2442no—
391C2×Q8—2442no—
392C2×D4—614no—
393C2×D4—1224no—
394C2×D4—1222no—
395C2×D4—1222no—
396C2×D4—1222no—
397C2×D4—1222no—
398C2×D4—1222no—
399C2×D4—2448no—
400C2×D4—2448no—
401C2×D4—2442no—
402C2×D4—2442no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—611no—
412C22×C4—611no—
413C22×C4—611no—
414C22×C4—611no—
415C22×C4—1222no—
416C22×C4—1222no—
417C22×C4—1222no—
418C22×C4—1222no—
419C22×C4—1221no—
420C22×C4—1221no—
421C22×C4—2441no—
422C22×C4—2441no—
423C22×C4#1175822212yes—
424C22×C4D4×D6111yescharacteristic
425C22×C4#1175822412yes—
426C22×C4C23×D6111yescharacteristic
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—1222no—
431C2×C8—1222no—
432C2×C8—1222no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—42no—
438C4⋊C4—1222no—
439C4⋊C4—1222no—
440C4⋊C4—2442no—
441C4⋊C4—2442no—
442C4⋊C4—2442no—
443C4⋊C4—2442no—
444C4⋊C4—2442no—
445C22⋊C4—42no—
446C22⋊C4—2444no—
447C22⋊C4—2444no—
448C22⋊C4—2444no—
449C22⋊C4—2444no—
450C22⋊C4—2442no—
451C42—1222no—
452C42—1222no—
453C42—2441no—
454C42—2441no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—1224no—
460C23—4884no—
461C23—4884no—
462C23—4884no—
463C23—4884no—
464C23—2448no—
465C23—2448no—
466C23—2448no—
467C23—2442no—
468C23—2442no—
469C23—2442no—
470C23—2442no—
471C23—2442no—
472C23—2442no—
473C23#1451976312yes—
474C23#1451980412yes—
475C23#1451980912yes—
476C23C2×D4×D6111yescharacteristic
477Q8—4884no—
478D4—1224no—
479D4—1224no—
480D4—2448no—
481D4—2442no—
482D4—2442no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—84no—
497C2×C4—612no—
498C2×C4—612no—
499C2×C4—1221no—
500C2×C4—1221no—
501C2×C4—1221no—
502C2×C4—1221no—
503C2×C4—1221no—
504C2×C4—1221no—
505C2×C4—1221no—
506C2×C4—1221no—
507C2×C4—1221no—
508C2×C4—1221no—
509C2×C4—2444no—
510C2×C4—2444no—
511C2×C4—2441no—
512C2×C4—2441no—
513C2×C4—2441no—
514C2×C4—2441no—
515C2×C4—2441no—
516C2×C4—2441no—
517C2×C4#1451976312yes—
518C2×C4#41×D6112yes—
519C2×C4C2×#1175822411yescharacteristic
520C2×C4C2×#1175822211yescharacteristic
521C2×C4#1451982312yes—
522C8—42no—
523C8—2442no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—48816no—
531C22—4882no—
532C22—4882no—
533C22—4882no—
534C22—4882no—
535C22—2444no—
536C22—2444no—
537C22—2444no—
538C22#1684012314yes—
539C22#1684027011yescharacteristic
540C22C2×#1451976311yescharacteristic
541C22#1684032311yescharacteristic
542C22AGL(1,ℤ/8)×D6111yescharacteristic
543C22AGL(1,ℤ/8)×D6111yescharacteristic
544C22C2×#1451980411yescharacteristic
545C22C2×#1451980911yescharacteristic
546C4—22no—
547C4—81no—
548C4—82no—
549C4—81no—
550C4—1222no—
551C4—1222no—
552C4—4881no—
553C4—4881no—
554C4—2441no—
555C4—2441no—
556C4#1684043512yes—
557C4#1684043612yes—
558C2—24no—
559C2—4884no—
560C2#2145097611yescharacteristic
561C2#2145590711yescharacteristic
562C2#2145591511yescharacteristic
563C2C2×#1684012311yescharacteristic
564C2#2145619111yescharacteristic
565C2#2145661811yescharacteristic
566C2#2145662411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1824C2
socleC23×C61C22×D4
prime radicalC23×D244C22
hypercenter#1824D61
hypocenterC61#29109
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1824

Complements

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

NH ≅ G/Ntype
1. #21454169C2semidirect
3. #21447226C2semidirect
4. #21446954C2semidirect
5. #21446517C2semidirect
6. #21446514C2semidirect
7. #21445909C2semidirect
10. #21445583C2semidirect
11. C61×#1824C2semidirect
13. C2×#14519776C22semidirect
23. C2×#14519687C22semidirect
29. #16839661C22semidirect
31. #16839658C22semidirect
32. #16839631C22semidirect
35. #16839524C22semidirect
36. #16839524C22semidirect
37. #16839516C22semidirect
41. C61×#340C22semidirect
42. C61×#340C22semidirect
282. #1824D61semidirect
323. #340C2×D61semidirect
324. #340C2×D61semidirect
326. C61#29109semidirect

Maximal subgroups

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

subgroupindex
#214541692
C2×#168396632
#214472262
#214469542
#214465172
#214465142
#214459092
#214457512
#214457492
#214455832
C61×#18242
#2910961