Finite Group Databasebeta

← #29718370

Subgroups — #29718370

Order 15616.

1232 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840080C211yesmaximal, characteristic, complemented
2C22×#14519697C211yesmaximal, characteristic, complemented
3#21449456C211yesmaximal, characteristic
4#21447190C212yesmaximal, complemented
5#21447053C212yesmaximal, complemented
6#21446746C212yesmaximal, complemented
7#21446446C212yesmaximal, complemented
8#21446245C211yesmaximal, characteristic, complemented
9#21445999C211yesmaximal, characteristic, complemented
10#21445749C211yesmaximal, characteristic, complemented
11C61×#2345C211yesmaximal, characteristic, complemented
12C23×D244C2211yescharacteristic
13C23×#92418C2211yescharacteristic
14C2×#14519774C2212yes—
15C2×#14519751C2211yescharacteristic
16C2×#14519747—21no—
17C2×#14519747—21no—
18#16840080—24no—
19C2×#14519708C2211yescharacteristic, complemented
20C2×#14519708C2211yescharacteristic, complemented
21C2×C4⋊#92418C2212yescomplemented
22C2×#14519697—24no—
23C2×#14519697C2212yes—
24C2×#14519697C2212yes—
25C2×#14519697C2212yes—
26C2×#14519697C2212yes—
27C2×#14519697C2211yescharacteristic, complemented
28C2×#14519697C2211yescharacteristic, complemented
29C2×#14519697C2212yescomplemented
30C2×C22⋊#92418C2211yescharacteristic
31#16839672—22no—
32#16839664C2212yes—
33#16839524C2212yescomplemented
34#16839517C2212yescomplemented
35C22×C4⋊C4×C61C2211yescharacteristic
36C61×#389C2212yescomplemented
37C61×#387—22no—
38C61×#380—21no—
39C61×#380C2211yescharacteristic
40C4×C4⋊C4×C61C2212yescomplemented
41C4×C22⋊C4×C61—21no—
42C4×C22⋊C4×C61C2211yescharacteristic
43C24×D61—21no—
44C22×D244—22no—
45C22×D244—24no—
46C22×D244—22no—
47C22×D244—21no—
48C22×D244—22no—
49C22×D244—22no—
50C22×D244—42no—
51C22×D244C2311yescharacteristic
52C22×D244C2312yes—
53C22×C22⋊D61—21no—
54C22×#92418—22no—
55C22×#92418—42no—
56C22×#92418C2311yescharacteristic
57C22×#92418C2311yescharacteristic
58C22×#92418C2312yes—
59#14519774—24no—
60#14519774—42no—
61#14519751—22no—
62#14519751—22no—
63#14519747—44no—
64#14519747—44no—
65C2×C4⋊Dic61C2311yescharacteristic
66C2×#11758198—22no—
67C2×#11758198—22no—
68C2×#11758198—21no—
69#14519708—22no—
70#14519708—22no—
71C4⋊#92418—42no—
72#14519697—24no—
73#14519697—24no—
74#14519697—24no—
75#14519697—22no—
76#14519697—22no—
77#14519697—42no—
78#14519697—42no—
79C22⋊#92418—22no—
80C23×C4×C61—21no—
81C23×C4×C61C2311yescharacteristic
82C2×C4⋊C4×C61—21no—
83C2×C4⋊C4×C61—22no—
84C2×C4⋊C4×C61—22no—
85C2×C4⋊C4×C61D414yescomplemented
86C2×C4⋊C4×C61C2312yes—
87C2×C4⋊C4×C61C2311yescharacteristic
88C2×C4⋊C4×C61C2311yescharacteristic
89C2×C22⋊C4×C61—21no—
90C2×C22⋊C4×C61D411yescharacteristic
91C2×C22⋊C4×C61D411yescharacteristic
92C2×C42×C61—22no—
93C2×C42×C61C2312yes—
94#95×C61—22no—
95#95×C61—22no—
96#95×C61—21no—
97#95×C61D412yes—
98C24×C61C2×D411yescharacteristic
99C23×D61—22no—
100C23×D61—21no—
101C23×D61—44no—
102C23×D61—42no—
103C23×D61—42no—
104C23×D61—42no—
105C23×D61—82no—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—21no—
109C22×C4×C61—21no—
110C22×C4×C61—24no—
111C22×C4×C61—22no—
112C22×C4×C61—82no—
113C22×C4×C61C2×D412yes—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D412yes—
116C22×C4×C61C2×D412yes—
117C22×C4×C61#4112yes—
118C22×C4×C61C2411yescharacteristic
119C2×C22⋊D61—42no—
120C2×C22⋊D61—42no—
121C2×C22⋊D61—42no—
122C22×Dic61—21no—
123C2×D244—28no—
124C2×D244—22no—
125C2×D244—22no—
126C2×D244—22no—
127C2×D244—22no—
128C2×D244—21no—
129C2×D244—21no—
130C2×D244—22no—
131C2×D244—22no—
132C2×D244—22no—
133C2×D244—48no—
134C2×D244—44no—
135C2×D244—44no—
136C2×D244—42no—
137C2×D244—44no—
138C2×D244—44no—
139C2×D244—44no—
140C2×D244—44no—
141C2×D244—82no—
142C2×D244—82no—
143C2×D244#4114yes—
144C4⋊C4×C61—24no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—82no—
150C4⋊C4×C61C2×D412yes—
151C4⋊C4×C61C2×D412yes—
152C22⋊C4×C61—22no—
153C22⋊C4×C61—42no—
154C22⋊C4×C61SD1614yes—
155C42×C61—22no—
156C42×C61—22no—
157C42×C61—82no—
158#11758198—44no—
159#11758198—42no—
160#11758198—44no—
161C4⋊Dic61—21no—
162C4⋊Dic61—21no—
163C2×#92418—21no—
164C2×#92418—21no—
165C2×#92418—22no—
166C2×#92418—22no—
167C2×#92418—22no—
168C2×#92418—22no—
169C2×#92418—22no—
170C2×#92418—44no—
171C23×C61—22no—
172C23×C61—22no—
173C23×C61#12112yes—
174C23×C61C2×SD1612yes—
175C23×C61AGL(1,ℤ/8)12yes—
176C23×C61C22×D411yescharacteristic
177C22×D61—24no—
178C22×D61—48no—
179C22×D61—42no—
180C22×D61—42no—
181C22×D61—42no—
182C22×D61—42no—
183C22×D61—42no—
184C22×D61—42no—
185C22×D61—41no—
186C22×D61—42no—
187C22×D61—41no—
188C22×D61—41no—
189C22×D61—42no—
190C22×D61—88no—
191C22×D61—88no—
192C22×D61—84no—
193C22×D61—84no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—24no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—22no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61—84no—
211C2×C4×C61#12112yes—
212C2×C4×C61C2×SD1614yes—
213C2×C4×C61C22×D411yescharacteristic
214C2×C4×C61C2×#4112yes—
215C2×C4×C61C2×#4111yescharacteristic
216C2×C4×C61H(5,𝔽2)12yes—
217C22⋊D61—84no—
218C2×Dic61—41no—
219C2×Dic61—41no—
220C2×Dic61—41no—
221D244—24no—
222D244—24no—
223D244—48no—
224D244—41no—
225D244—42no—
226D244—41no—
227D244—42no—
228D244—42no—
229D244—88no—
230D244—88no—
231#92418—41no—
232#92418—41no—
233#92418—42no—
234#24478—611nomaximal, Sylow
235C22×C61—22no—
236C22×C61—24no—
237C22×C61—22no—
238C22×C61—24no—
239C22×C61#46014yes—
240C22×C61C2×#12111yescharacteristic
241C22×C61#54612yes—
242C22×C61C22×SD1611yescharacteristic
243C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
244C22×C61#57612yes—
245C2×D61—44no—
246C2×D61—44no—
247C2×D61—44no—
248C2×D61—816no—
249C2×D61—82no—
250C2×D61—82no—
251C2×D61—82no—
252C2×D61—82no—
253C2×D61—81no—
254C2×D61—81no—
255C2×D61—82no—
256C2×D61—81no—
257C2×D61—82no—
258C4×C61—22no—
259C4×C61—24no—
260C4×C61—42no—
261C4×C61—42no—
262C4×C61—42no—
263C4×C61—82no—
264C4×C61#52511yescharacteristic
265C4×C61#53511yescharacteristic
266C4×C61#55112yes—
267Dic61—81no—
268C2×#481—611no—
269C22×#102—611no—
270#2433—611no—
271#2345D6111yescharacteristic, complemented
272#2065—612no—
273#2041—612no—
274#1982—612no—
275#1926—612no—
276#1893—611no—
277#1850—611no—
278#1786—611no—
279C2×C61—24no—
280C2×C61#241111yescharacteristic
281C2×C61C2×#46011yescharacteristic
282C2×C61#304111yescharacteristic
283C2×C61#329512yes—
284C2×C61#331812yes—
285D61—84no—
286D61—161no—
287C23×D4—611no—
288C23×C8—611no—
289C2×#127—612no—
290C2×#121—611no—
291C2×#120—1221no—
292C2×#120—1221no—
293C22×C4⋊C4C2×D6111yescharacteristic
294#481—1224no—
295C2×#106—611no—
296C2×#106—611no—
297C2×C4⋊C8—612no—
298C2×#102—612no—
299C2×#102—611no—
300C2×#102—611no—
301C2×#102—612no—
302C2×#102—612no—
303C2×#102—612no—
304C2×#102—612no—
305C2×#102—1224no—
306C2×C22⋊C8—611no—
307#392—1222no—
308#390—612no—
309#389C2×D6112yescomplemented
310#387—22no—
311#380—21no—
312#380C2×D6111yescharacteristic
313C4×C4⋊C4C2×D6112yescomplemented
314C4×C22⋊C4—21no—
315C4×C22⋊C4C2×D6111yescharacteristic
316#340—612no—
317#336—612no—
318C61#2447811yescharacteristic, complemented, Sylow
319C25—1221no—
320C22×D4—612no—
321C22×D4—611no—
322C22×D4—1224no—
323C22×D4—1221no—
324C22×D4—1222no—
325C22×D4—1222no—
326C22×D4—1222no—
327C22×D4—1222no—
328C22×D4—1221no—
329C22×D4—2442no—
330C23×C4—21no—
331C23×C4C22×D6111yescharacteristic
332C22×C8—612no—
333C22×C8—611no—
334C22×C8—611no—
335C22×C8—1222no—
336C22×C8—2442no—
337#127—1224no—
338#127—2442no—
339#121—1222no—
340#121—1222no—
341#120—2444no—
342#120—2444no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—611no—
347C2×C4⋊C4D24414yescomplemented
348C2×C4⋊C4C22×D6112yes—
349C2×C4⋊C4C22×D6111yescharacteristic
350C2×C4⋊C4C22×D6111yescharacteristic
351C2×C22⋊C4—21no—
352C2×C22⋊C4—1222no—
353C2×C22⋊C4—1222no—
354C2×C22⋊C4—1221no—
355C2×C22⋊C4C22⋊D6111yescharacteristic
356C2×C22⋊C4C22⋊D6111yescharacteristic
357C2×C42—22no—
358C2×C42C22×D6112yes—
359#106—1222no—
360#106—1222no—
361C4⋊C8—2442no—
362#102—1224no—
363#102—1224no—
364#102—1224no—
365#102—1222no—
366#102—1222no—
367#102—2442no—
368#102—2442no—
369C22⋊C8—1222no—
370#95—22no—
371#95—22no—
372#95—21no—
373#95C22⋊D6112yes—
374C24—1222no—
375C24—1221no—
376C24—4882no—
377C24—2444no—
378C24—2442no—
379C24—2442no—
380C24—2442no—
381C24C2×C22⋊D6111yescharacteristic
382C2×D4—614no—
383C2×D4—1228no—
384C2×D4—1222no—
385C2×D4—1221no—
386C2×D4—1222no—
387C2×D4—1222no—
388C2×D4—1222no—
389C2×D4—1221no—
390C2×D4—1222no—
391C2×D4—1222no—
392C2×D4—1222no—
393C2×D4—4882no—
394C2×D4—4882no—
395C2×D4—2448no—
396C2×D4—2444no—
397C2×D4—2444no—
398C2×D4—2444no—
399C2×D4—2444no—
400C2×D4—2444no—
401C2×D4—2442no—
402C2×D4—2442no—
403C2×D4—2442no—
404C2×D4—2442no—
405C2×D4—2444no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—82no—
413C22×C4—1221no—
414C22×C4C2×D24412yes—
415C22×C4#1175822212yes—
416C22×C4C2×C22⋊D6112yes—
417C22×C4C2×C22⋊D6111yescharacteristic
418C22×C4C2×C22⋊D6112yes—
419C22×C4C23×D6111yescharacteristic
420C2×C8—1221no—
421C2×C8—1221no—
422C2×C8—1222no—
423C2×C8—1222no—
424C2×C8—1222no—
425C2×C8—1222no—
426C2×C8—1222no—
427C2×C8—2444no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—24no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—82no—
434C4⋊C4—1221no—
435C4⋊C4—1221no—
436C4⋊C4C2×D24412yes—
437C4⋊C4C2×D24412yes—
438C22⋊C4—22no—
439C22⋊C4—42no—
440C22⋊C4—2444no—
441C22⋊C4—2442no—
442C22⋊C4—2444no—
443C22⋊C4#1175820114yes—
444C42—22no—
445C42—22no—
446C42—82no—
447C23—22no—
448C23—22no—
449C23—1224no—
450C23—4884no—
451C23—4888no—
452C23—4888no—
453C23—4884no—
454C23—2448no—
455C23—2441no—
456C23—2441no—
457C23—2441no—
458C23—2442no—
459C23—2442no—
460C23—2442no—
461C23—2442no—
462C23—2442no—
463C23—2442no—
464C23—2442no—
465C23—2442no—
466C23#1451975112yes—
467C23C2×#1175820112yes—
468C23#1451980212yes—
469C23C22×C22⋊D6111yescharacteristic
470D4—1224no—
471D4—1224no—
472D4—4888no—
473D4—4888no—
474D4—4884no—
475D4—2448no—
476D4—2441no—
477D4—2441no—
478D4—2442no—
479D4—2442no—
480D4—2442no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—84no—
498C2×C4—2441no—
499C2×C4—2441no—
500C2×C4—2441no—
501C2×C4#1451975112yes—
502C2×C4C2×#1175820114yes—
503C2×C4C22×D24411yescharacteristic
504C2×C4#41×D6112yes—
505C2×C4C2×#1175822211yescharacteristic
506C2×C4#1451982312yes—
507C8—2441no—
508C8—2441no—
509C8—2442no—
510C22—24no—
511C22—22no—
512C22—24no—
513C22—22no—
514C22—48816no—
515C22—4882no—
516C22—4881no—
517C22—4881no—
518C22—4882no—
519C22—4881no—
520C22—4882no—
521C22—4882no—
522C22—4882no—
523C22—4882no—
524C22—2444no—
525C22—2444no—
526C22—2444no—
527C22#1683996214yes—
528C22C2×#1451975111yescharacteristic
529C22#1684040012yes—
530C22C22×#1175820111yescharacteristic
531C22C2×#1451980211yescharacteristic
532C22#1684056512yes—
533C4—24no—
534C4—22no—
535C4—42no—
536C4—42no—
537C4—42no—
538C4—82no—
539C4—4881no—
540C4#1684027411yescharacteristic
541C4#1684033111yescharacteristic
542C4#1684043612yes—
543C2—24no—
544C2—9761no—
545C2—4884no—
546C2#2144919811yescharacteristic
547C2C2×#1683996211yescharacteristic
548C2#2145532211yescharacteristic
549C2#2145763712yes—
550C2#2145795412yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2345C2
socleC23×C61C22×D4
prime radical#21449456C2
hypercenter#2345D61
hypocenterC61#24478
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2345

Complements

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

NH ≅ G/Ntype
1. C2×#16840080C2semidirect
2. C22×#14519697C2semidirect
4. #21447190C2semidirect
5. #21447053C2semidirect
6. #21446746C2semidirect
7. #21446446C2semidirect
8. #21446245C2semidirect
9. #21445999C2semidirect
10. #21445749C2semidirect
11. C61×#2345C2semidirect
19. C2×#14519708C22semidirect
20. C2×#14519708C22semidirect
21. C2×C4⋊#92418C22semidirect
27. C2×#14519697C22semidirect
28. C2×#14519697C22semidirect
29. C2×#14519697C22semidirect
33. #16839524C22semidirect
34. #16839517C22semidirect
36. C61×#389C22semidirect
40. C4×C4⋊C4×C61C22semidirect
85. C2×C4⋊C4×C61D4semidirect
271. #2345D61semidirect
309. #389C2×D61semidirect
313. C4×C4⋊C4C2×D61semidirect
318. C61#24478semidirect
347. C2×C4⋊C4D244semidirect

Maximal subgroups

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

subgroupindex
C2×#168400802
C22×#145196972
#214494562
#214471902
#214470532
#214467462
#214464462
#214462452
#214459992
#214457492
C61×#23452
#2447861