Finite Group Databasebeta

← #29652734

Subgroups — #29652734

Order 15616.

1242 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459583C211yesmaximal, characteristic
2#21455759C212yesmaximal, complemented
3#21444855C212yesmaximal, complemented
4#21444838C212yesmaximal, complemented
5#21444821C212yesmaximal, complemented
6#21444804C212yesmaximal, complemented
7#21444334C212yesmaximal, complemented
8#21444157C211yesmaximal, characteristic, complemented
9C61×#2395C211yesmaximal, characteristic, complemented
10C2×#14519822—21no—
11#16840416—22no—
12#16840407—24no—
13D4×D244—22no—
14#16840343C2212yes—
15#16840324—21no—
16#16840324C2212yes—
17#16840312—21no—
18C2×#14519751—21no—
19C2×#14519751C2212yes—
20#16840086—24no—
21#16840080C2214yescomplemented
22#16840075C2214yescomplemented
23#16839805—22no—
24#16839766—22no—
25#16839726C2212yescomplemented
26#16839495C2214yescomplemented
27#16839495C2214yescomplemented
28#16839488C2212yescomplemented
29#16839486C2212yescomplemented
30C2×C4×D4×C61C2211yescharacteristic
31C2×#117×C61C2211yescharacteristic, complemented
32C61×#400C2211yescharacteristic, complemented
33C61×#399C2211yescharacteristic, complemented
34C61×#398C2211yescharacteristic, complemented
35C61×#397—21no—
36C61×#397C2211yescharacteristic, complemented
37C61×#394—21no—
38C4×C4⋊C4×C61—21no—
39C4×C22⋊C4×C61—21no—
40C4×C22⋊C4×C61C2211yescharacteristic, complemented
41#14519822—44no—
42#14519822—44no—
43C2×#11758222—22no—
44C2×#11758224—24no—
45C22×D244—21no—
46C22×D244D412yes—
47C22×C22⋊D61—22no—
48C22×C22⋊D61—22no—
49C2×D4×D61—24no—
50C2×#11758194—22no—
51#14519777—21no—
52#14519774—21no—
53#14519765—22no—
54#14519761—24no—
55#14519761—24no—
56#14519757—24no—
57#14519752—24no—
58#14519751—22no—
59#14519751—22no—
60#14519751—22no—
61#14519751—22no—
62#14519751—22no—
63#14519751—22no—
64#14519751D414yes—
65#14519751C2314yes—
66#14519747—24no—
67#14519747—24no—
68C4×D244—22no—
69C4×D244—22no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61D412yes—
72C2×#11758198—22no—
73C2×#11758198—22no—
74#14519710—24no—
75#14519708—24no—
76#14519697—24no—
77#14519697—24no—
78C22⋊#92418—22no—
79C22⋊#92418C2314yes—
80C22×D4×C61D411yescharacteristic
81C23×C4×C61D411yescharacteristic
82C23×C4×C61C2311yescharacteristic
83C4×D4×C61—22no—
84C4×D4×C61—22no—
85#117×C61—22no—
86#117×C61—22no—
87C2×C4⋊C4×C61—21no—
88C2×C4⋊C4×C61—21no—
89C2×C4⋊C4×C61C2311yescharacteristic
90C2×C4⋊C4×C61C2311yescharacteristic
91C2×C4⋊C4×C61C2311yescharacteristic
92C2×C22⋊C4×C61—21no—
93C2×C22⋊C4×C61—21no—
94C2×C22⋊C4×C61—21no—
95C2×C22⋊C4×C61—21no—
96C2×C22⋊C4×C61D411yescharacteristic
97C2×C22⋊C4×C61C2311yescharacteristic
98C2×C42×C61—21no—
99C2×C42×C61—21no—
100C2×C42×C61D411yescharacteristic
101#95×C61—21no—
102#95×C61—21no—
103#95×C61—21no—
104#95×C61—21no—
105#95×C61C2311yescharacteristic
106#95×C61C2311yescharacteristic
107C24×C61C2×D411yescharacteristic
108C24×C61C2×D411yescharacteristic
109C23×D61—22no—
110C23×D61—22no—
111C2×D4×C61—22no—
112C2×D4×C61—22no—
113C2×D4×C61—22no—
114C22×C4×C61—21no—
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61—42no—
121C22×C4×C61—42no—
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D412yes—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D412yes—
128C22×C4×C61C2411yescharacteristic
129C2×C22⋊D61—24no—
130C2×C22⋊D61—24no—
131C2×C22⋊D61—24no—
132C2×C22⋊D61—44no—
133C2×C22⋊D61—44no—
134C2×C22⋊D61—44no—
135C2×C22⋊D61—44no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138#11758224—416no—
139D4×D61—416no—
140#11758222—44no—
141#11758222—44no—
142C2×D244—22no—
143C2×D244—22no—
144C2×D244—22no—
145C2×D244—22no—
146C2×D244—44no—
147C2×D244—82no—
148C2×D244C2×D414yes—
149C2×C4×D61—24no—
150C2×Dic122—22no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—42no—
155C4⋊C4×C61—42no—
156C22⋊C4×C61—22no—
157C22⋊C4×C61—22no—
158C22⋊C4×C61—42no—
159C22⋊C4×C61—42no—
160C22⋊C4×C61—42no—
161C22⋊C4×C61—42no—
162C42×C61—21no—
163C42×C61—21no—
164C42×C61—42no—
165C42×C61—41no—
166C42×C61—41no—
167#11758198—24no—
168#11758198—24no—
169#11758198—24no—
170#11758198—24no—
171C4⋊Dic61—22no—
172C4⋊Dic61—22no—
173C4⋊Dic61—22no—
174C4⋊Dic61C2×D414yes—
175#11758194—44no—
176C2×#92418—24no—
177C23×C61—21no—
178C23×C61—21no—
179C23×C61—21no—
180C23×C61—21no—
181C23×C61—41no—
182C23×C61—41no—
183C23×C61—41no—
184C23×C61—41no—
185C23×C61#12012yes—
186C23×C61AGL(1,ℤ/8)12yes—
187C23×C61C22×D411yescharacteristic
188C22×D61—24no—
189C22×D61—44no—
190C22×D61—44no—
191C22×D61—82no—
192C22×D61—82no—
193C22×D61—82no—
194C22×D61—82no—
195D4×C61—28no—
196C2×C4×C61—21no—
197C2×C4×C61—21no—
198C2×C4×C61—24no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—42no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61—42no—
208C2×C4×C61—41no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—81no—
220C2×C4×C61#12011yescharacteristic
221C2×C4×C61#12011yescharacteristic
222C2×C4×C61AGL(1,ℤ/8)12yes—
223C2×C4×C61C22×D412yes—
224C22⋊D61—48no—
225C22⋊D61—48no—
226C22⋊D61—44no—
227C22⋊D61—44no—
228C2×Dic61—24no—
229C2×Dic61—44no—
230C2×Dic61—44no—
231D244—44no—
232D244—44no—
233D244—44no—
234D244—82no—
235D244—82no—
236C4×D61—48no—
237Dic122—44no—
238#92418—44no—
239#18173—611nomaximal, Sylow
240C22×C61—21no—
241C22×C61—21no—
242C22×C61—22no—
243C22×C61—22no—
244C22×C61—22no—
245C22×C61—22no—
246C22×C61—42no—
247C22×C61—41no—
248C22×C61—41no—
249C22×C61—81no—
250C22×C61—81no—
251C22×C61—81no—
252C22×C61C2×#12011yescharacteristic
253C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
254C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
255C2×D61—44no—
256C2×D61—44no—
257C2×D61—44no—
258C2×D61—164no—
259C2×D61—164no—
260C4×C61—22no—
261C4×C61—22no—
262C4×C61—21no—
263C4×C61—42no—
264C4×C61—41no—
265C4×C61—41no—
266C4×C61—81no—
267C4×C61—81no—
268C4×C61—81no—
269C4×C61#45812yes—
270Dic61—44no—
271#3496—611no—
272#3094—612no—
273#2395D6111yescharacteristic, complemented
274#1616—612no—
275#1613—612no—
276#1610—612no—
277#1607—612no—
278#1507—612no—
279#1465—611no—
280C2×C61—21no—
281C2×C61—21no—
282C2×C61—42no—
283C2×C61—42no—
284C2×C61#300311yescharacteristic
285C2×C61#301811yescharacteristic
286C2×C61C2×#45811yescharacteristic
287D61—84no—
288C2×H(5,𝔽2)—1221no—
289#547—1222no—
290#546—1224no—
291D42—1222no—
292#537—612no—
293#534—612no—
294#534—1221no—
295#532—1221no—
296C2×#121—612no—
297C2×#121—1221no—
298C2×C4×D4C2×D6111yescharacteristic
299C2×#117C2×D6111yescharacteristic, complemented
300#482—1224no—
301#481—614no—
302#480—614no—
303#428—1222no—
304#418—1222no—
305#407—612no—
306#400C2×D6111yescharacteristic, complemented
307#399C2×D6111yescharacteristic, complemented
308#398C2×D6111yescharacteristic, complemented
309#397—21no—
310#397C2×D6111yescharacteristic, complemented
311#394—21no—
312C4×C4⋊C4—21no—
313C4×C22⋊C4—21no—
314C4×C22⋊C4C2×D6111yescharacteristic, complemented
315#327—614no—
316#327—614no—
317#324—612no—
318#323—612no—
319C61#1817311yescharacteristic, complemented, Sylow
320H(5,𝔽2)—2444no—
321H(5,𝔽2)—2444no—
322C2×#41—1224no—
323C2×#41—1222no—
324C22×D4—612no—
325C22×D4—1221no—
326C22×D4—1224no—
327C22×D4—1222no—
328C22×D4—1222no—
329C22×D4C22⋊D6111yescharacteristic
330C23×C4C22⋊D6111yescharacteristic
331C23×C4C22×D6111yescharacteristic
332C2×M16—1222no—
333C4⋊Q8—1221no—
334#127—1221no—
335#124—1222no—
336#123—1224no—
337#123—1224no—
338C22⋊Q8—1224no—
339#121—614no—
340#121—614no—
341#121—1224no—
342#121—1222no—
343#121—1222no—
344#121—1222no—
345#121—1222no—
346#121—1222no—
347#121—1222no—
348#120—1224no—
349#120—1224no—
350C4×D4—22no—
351C4×D4—22no—
352C4×D4—1222no—
353C4×D4—1222no—
354#117—22no—
355#117—22no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—612no—
359C2×C4⋊C4—1221no—
360C2×C4⋊C4C22×D6111yescharacteristic
361C2×C4⋊C4C22×D6111yescharacteristic
362C2×C4⋊C4C22×D6111yescharacteristic
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—1222no—
368C2×C22⋊C4—1222no—
369C2×C22⋊C4C22⋊D6111yescharacteristic
370C2×C22⋊C4C22×D6111yescharacteristic
371C2×C42—21no—
372C2×C42—21no—
373C2×C42C22⋊D6111yescharacteristic
374#107—1224no—
375#106—1224no—
376#102—1224no—
377#102—1224no—
378C22⋊C8—614no—
379C22⋊C8—1222no—
380#95—21no—
381#95—21no—
382#95—21no—
383#95—21no—
384#95C22×D6111yescharacteristic
385#95C22×D6111yescharacteristic
386C24—1222no—
387C24—1222no—
388C24C2×C22⋊D6111yescharacteristic
389C24C2×C22⋊D6111yescharacteristic
390#41—2444no—
391#41—24416no—
392#41—2444no—
393C2×Q8—1222no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—614no—
398C2×D4—1222no—
399C2×D4—1222no—
400C2×D4—1222no—
401C2×D4—1224no—
402C2×D4—1224no—
403C2×D4—1224no—
404C2×D4—1222no—
405C2×D4—4882no—
406C2×D4—2444no—
407C2×D4—24416no—
408C2×D4—2444no—
409C2×D4—2444no—
410C2×D4—2444no—
411C2×D4—2444no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—1224no—
421C22×C4—1222no—
422C22×C4—1222no—
423C22×C4D4×D6112yes—
424C22×C4D4×D6112yes—
425C22×C4C2×C22⋊D6111yescharacteristic
426C22×C4C2×C22⋊D6111yescharacteristic
427C22×C4C2×C22⋊D6111yescharacteristic
428C22×C4C2×C22⋊D6111yescharacteristic
429C22×C4C23×D6111yescharacteristic
430M16—2444no—
431C2×C8—1224no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—614no—
438C4⋊C4—1222no—
439C4⋊C4—1222no—
440C4⋊C4—1222no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—42no—
446C22⋊C4—42no—
447C22⋊C4—1224no—
448C22⋊C4—1224no—
449C22⋊C4—1224no—
450C22⋊C4—1224no—
451C42—21no—
452C42—21no—
453C42—42no—
454C42—41no—
455C42—41no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—1224no—
465C23—4882no—
466C23—4882no—
467C23—4882no—
468C23—4882no—
469C23—2444no—
470C23—2444no—
471C23#1451974512yes—
472C23#1451980212yes—
473C23C22×C22⋊D6111yescharacteristic
474Q8—2444no—
475D4—28no—
476D4—4882no—
477D4—4882no—
478D4—2444no—
479D4—2444no—
480D4—2448no—
481D4—2448no—
482D4—2444no—
483D4—2444no—
484D4—2444no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—81no—
509C2×C4—1224no—
510C2×C4—2448no—
511C2×C4—2444no—
512C2×C4—2444no—
513C2×C4#1451974511yescharacteristic
514C2×C4#1451974511yescharacteristic
515C2×C4#1451980212yes—
516C2×C4C2×D4×D6112yes—
517C8—2444no—
518C22—21no—
519C22—21no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—42no—
525C22—41no—
526C22—41no—
527C22—81no—
528C22—81no—
529C22—81no—
530C22—9764no—
531C22—9764no—
532C22—2444no—
533C22—2444no—
534C22—2444no—
535C22C2×#1451974511yescharacteristic
536C22C2×#1451980211yescharacteristic
537C22C2×#1451980211yescharacteristic
538C4—22no—
539C4—21no—
540C4—22no—
541C4—42no—
542C4—41no—
543C4—41no—
544C4—81no—
545C4—81no—
546C4—81no—
547C4—2444no—
548C4#1683995112yes—
549C2—21no—
550C2—21no—
551C2—42no—
552C2—42no—
553C2—4884no—
554C2#2145504511yescharacteristic
555C2#2145516711yescharacteristic
556C2C2×#1683995111yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2395C2
socleC22×C61C2×#120
prime radical#21459583C2
hypercenter#2395D61
hypocenterC61#18173
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2395

Complements

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

NH ≅ G/Ntype
2. #21455759C2semidirect
3. #21444855C2semidirect
4. #21444838C2semidirect
5. #21444821C2semidirect
6. #21444804C2semidirect
7. #21444334C2semidirect
8. #21444157C2semidirect
9. C61×#2395C2semidirect
21. #16840080C22semidirect
22. #16840075C22semidirect
25. #16839726C22semidirect
26. #16839495C22semidirect
27. #16839495C22semidirect
28. #16839488C22semidirect
29. #16839486C22semidirect
31. C2×#117×C61C22semidirect
32. C61×#400C22semidirect
33. C61×#399C22semidirect
34. C61×#398C22semidirect
36. C61×#397C22semidirect
40. C4×C22⋊C4×C61C22semidirect
273. #2395D61semidirect
299. C2×#117C2×D61semidirect
306. #400C2×D61semidirect
307. #399C2×D61semidirect
308. #398C2×D61semidirect
310. #397C2×D61semidirect
314. C4×C22⋊C4C2×D61semidirect
319. C61#18173semidirect

Maximal subgroups

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

subgroupindex
#214595832
#214557592
#214448552
#214448382
#214448212
#214448042
#214443342
#214441572
C61×#23952
#1817361