Finite Group Databasebeta

← #29662872

Subgroups — #29662872

Order 15616.

766 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456272C211yesmaximal, characteristic
2#21455846C211yesmaximal, characteristic, complemented
3C2×#16839963C211yesmaximal, characteristic, complemented
4#21454770C211yesmaximal, characteristic, complemented
5#21445302C212yesmaximal, complemented
6#21445294C212yesmaximal, complemented
7#21445178C212yesmaximal, complemented
8#21445170C212yesmaximal, complemented
9#21444569C211yesmaximal, characteristic, complemented
10#21444488C211yesmaximal, characteristic, complemented
11C61×#1547C211yesmaximal, characteristic, complemented
12C2×#14519802—21no—
13C2×#14519802—21no—
14C22×#11758190—21no—
15C2×#14519774C2211yescharacteristic
16C2×C4×Dic122C2211yescharacteristic
17#16840152—22no—
18#16840138C2212yes—
19#16840136C2212yes—
20#16840078—21no—
21#16840044C2211yescharacteristic, complemented
22#16840044C2211yescharacteristic, complemented
23#16839963—21no—
24#16839963—21no—
25#16839963C2212yes—
26#16839963C2212yes—
27#16839959C2211yescharacteristic, complemented
28#16839959C2211yescharacteristic, complemented
29#16839900—21no—
30#16839891—22no—
31#16839889C2212yescomplemented
32#16839886C2212yescomplemented
33C4×#11758200—21no—
34C4×#11758200—21no—
35#16839815C2211yescharacteristic
36#16839785—21no—
37C2×#14519699—21no—
38C4×#11758194C2211yescharacteristic
39M16×Dic61—21no—
40#16839514C2212yescomplemented
41#16839512C2212yescomplemented
42#16839505C2212yes—
43#16839505C2212yes—
44C61×#532C2211yescharacteristic
45C61×#423C2211yescharacteristic
46C2×C4⋊C8×C61C2211yescharacteristic
47C61×#332C2212yescomplemented
48C61×#331C2212yescomplemented
49C22×Dic122D411yescharacteristic
50C22×D244—21no—
51C22×D244—21no—
52C22×D244—21no—
53#14519802—44no—
54#14519802—44no—
55C2×#11758190—22no—
56C2×#11758190—22no—
57C2×#11758190—44no—
58C2×#11758200—21no—
59C2×#11758200—21no—
60C2×#11758201—22no—
61C2×#11758201—22no—
62C2×#11758199—22no—
63C2×#11758199—22no—
64C2×#11758194D411yescharacteristic
65C2×#11758194D411yescharacteristic
66#14519774—21no—
67#14519774—21no—
68#14519774—21no—
69#14519774C2312yes—
70C4×Dic122—21no—
71C4×Dic122—21no—
72C4×Dic122D411yescharacteristic
73C4×Dic122D411yescharacteristic
74C4×Dic122C2311yescharacteristic
75C4×Dic122C2311yescharacteristic
76C2×C4⋊Dic61D411yescharacteristic
77C2×#11758196—21no—
78C2×C4×Dic61—21no—
79#14519708—21no—
80#14519708—21no—
81C4⋊#92418C2311yescharacteristic
82C4⋊#92418C2311yescharacteristic
83#14519706—22no—
84#14519706—22no—
85#14519700—21no—
86#14519700—21no—
87#14519699—22no—
88#14519699—22no—
89#14519698—21no—
90#14519698—21no—
91C22⋊#92418—21no—
92#14519688D412yes—
93#14519687—22no—
94C4×#92418—21no—
95C4×#92418C2312yes—
96C8×Dic61—22no—
97C2×#41×C61—21no—
98C2×M16×C61—21no—
99C22×C8×C61—21no—
100C4⋊Q8×C61C2311yescharacteristic
101#127×C61C2311yescharacteristic
102#124×C61—21no—
103#121×C61—21no—
104#121×C61—21no—
105C4×D4×C61—21no—
106C4×D4×C61—21no—
107C2×C42×C61C2311yescharacteristic
108C4⋊C8×C61—21no—
109C4⋊C8×C61—21no—
110C4⋊C8×C61C2312yes—
111C4⋊C8×C61C2312yes—
112C23×D61—41no—
113#41×C61—42no—
114#41×C61—42no—
115C2×Q8×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61—21no—
119C2×D4×C61—41no—
120C22×C4×C61—41no—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61#4111yescharacteristic
124C22×Dic61—21no—
125C22×Dic61—21no—
126C2×D244—22no—
127C2×D244—22no—
128C2×D244—22no—
129C2×D244—42no—
130C2×D244—42no—
131C2×D244—42no—
132C2×D244—42no—
133C2×D244—41no—
134C2×D244—41no—
135C2×D244—41no—
136C2×D244—82no—
137C2×Dic122—22no—
138C2×Dic122—21no—
139C2×Dic122—41no—
140C2×Dic122C2×D411yescharacteristic
141C2×Dic122C2×D411yescharacteristic
142M16×C61—42no—
143C2×C8×C61—22no—
144C2×C8×C61—22no—
145C2×C8×C61—42no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C22⋊C4×C61—41no—
149C22⋊C4×C61—41no—
150C42×C61C2×D411yescharacteristic
151C42×C61C2×D411yescharacteristic
152C42×C61#4111yescharacteristic
153C42×C61C2411yescharacteristic
154#11758201—44no—
155#11758201—44no—
156#11758200—42no—
157#11758200—42no—
158#11758199—44no—
159#11758199—44no—
160C4⋊Dic61—21no—
161C4⋊Dic61C2×D411yescharacteristic
162C4⋊Dic61C2×D411yescharacteristic
163#11758196—21no—
164#11758196—21no—
165#11758196—21no—
166#11758196—21no—
167C4×Dic61—21no—
168C4×Dic61—21no—
169C4×Dic61—21no—
170C4×Dic61—21no—
171#11758194—22no—
172#11758194SD1614yescomplemented
173C2×#92418—21no—
174C2×#92418—21no—
175C2×#92418C2×D412yes—
176C2×#92418C2×D412yes—
177#11758190—44no—
178#11758190—44no—
179C23×C61—41no—
180C23×C61#12111yescharacteristic
181C22×D61—42no—
182C22×D61—82no—
183C22×D61—82no—
184C22×D61—82no—
185Q8×C61—41no—
186Q8×C61—41no—
187D4×C61—41no—
188D4×C61—41no—
189D4×C61—41no—
190D4×C61—41no—
191D4×C61—81no—
192D4×C61—82no—
193C2×C4×C61—21no—
194C2×C4×C61—21no—
195C2×C4×C61—41no—
196C2×C4×C61—41no—
197C2×C4×C61—82no—
198C2×C4×C61#12111yescharacteristic
199C2×C4×C61#12111yescharacteristic
200C2×C4×C61#12111yescharacteristic
201C2×C4×C61C2×SD1612yes—
202C2×C4×C61AGL(1,ℤ/8)12yes—
203C2×C4×C61C22×D411yescharacteristic
204C2×C4×C61C22×D411yescharacteristic
205C2×C4×C61C2×#4111yescharacteristic
206C2×Dic61—21no—
207C2×Dic61—21no—
208C2×Dic61—21no—
209C2×Dic61—21no—
210C2×Dic61—42no—
211C2×Dic61—81no—
212D244—42no—
213D244—42no—
214D244—42no—
215D244—42no—
216D244—82no—
217D244—82no—
218D244—82no—
219D244—82no—
220Dic122—21no—
221Dic122—21no—
222Dic122—21no—
223Dic122—42no—
224Dic122AGL(1,ℤ/8)12yescomplemented
225C8×C61—42no—
226C8×C61—42no—
227#92418—22no—
228#92418—41no—
229#92418—41no—
230#92418C2×SD1614yescomplemented
231#19170—611nomaximal, Sylow
232C22×C61—21no—
233C22×C61—21no—
234C22×C61—81no—
235C22×C61—81no—
236C22×C61—81no—
237C22×C61#46012yes—
238C22×C61C2×#12111yescharacteristic
239C2×D61—82no—
240C2×D61—82no—
241C2×D61—82no—
242C2×D61—164no—
243C4×C61—21no—
244C4×C61—21no—
245C4×C61—21no—
246C4×C61—81no—
247C4×C61#46012yes—
248C4×C61C22×SD1611yescharacteristic
249C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
250C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
251C4×C61#57511yescharacteristic
252Dic61—22no—
253Dic61—41no—
254Dic61—41no—
255Dic61—41no—
256#3151—611no—
257#3103—611no—
258C2×#460—611no—
259Q8×M16—611no—
260#1688—612no—
261#1686—612no—
262#1668—612no—
263#1666—612no—
264#1565—611no—
265#1547D6111yescharacteristic, complemented
266#1545—611no—
267C2×C61—22no—
268C2×C61—161no—
269C2×C61C2×#46011yescharacteristic
270C2×C61#305511yescharacteristic
271C2×C61#306811yescharacteristic
272D61—162no—
273C2×AGL(1,ℤ/8)—1221no—
274C2×AGL(1,ℤ/8)—1221no—
275C22×SD16—1221no—
276#532C2×D6111yescharacteristic
277C2×#127—611no—
278C2×C4×Q8—611no—
279#496—1222no—
280#493—612no—
281#492—612no—
282#481—1221no—
283#475—611no—
284#475—611no—
285#460—611no—
286#460—611no—
287#460—612no—
288#460—612no—
289#460—1221no—
290#460—1221no—
291#448—1221no—
292#446—612no—
293#446—1222no—
294C8×Q8—612no—
295C4×SD16—1221no—
296C4×SD16—1221no—
297#431—611no—
298#423—1221no—
299#423C2×D6111yescharacteristic
300C2×C4⋊C8C2×D6111yescharacteristic
301C2×#102—1221no—
302C4×M16—611no—
303C4×M16—1221no—
304#335—612no—
305#334—612no—
306#332C2×D6112yescomplemented
307#331—612no—
308#331—612no—
309#331C2×D6112yescomplemented
310C61#1917011yescharacteristic, complemented, Sylow
311C2×#41—21no—
312C22×Q8—611no—
313C22×D4—1221no—
314C22×D4—1221no—
315C22×D4—1221no—
316AGL(1,ℤ/8)—2444no—
317AGL(1,ℤ/8)—2444no—
318C2×SD16—1222no—
319C2×SD16—1222no—
320C2×SD16—1222no—
321C2×SD16—1222no—
322C2×SD16—1221no—
323C2×SD16—1221no—
324C2×SD16—2444no—
325C2×D8—1222no—
326C2×D8—1222no—
327C2×M16—21no—
328C2×M16—611no—
329C2×M16—611no—
330C22×C8—21no—
331C4⋊Q8C22×D6111yescharacteristic
332#127—612no—
333#127—1221no—
334#127—1221no—
335#127—1221no—
336#127C22×D6111yescharacteristic
337#124—21no—
338#121—21no—
339#121—21no—
340C4×Q8—611no—
341C4×Q8—611no—
342C4×Q8—611no—
343C4×Q8—611no—
344C4×Q8—1221no—
345C4×Q8—1221no—
346C4×D4—21no—
347C4×D4—21no—
348C2×C4⋊C4—611no—
349C2×C4⋊C4—1221no—
350C2×C42—1221no—
351C2×C42C22×D6111yescharacteristic
352#106—1221no—
353#106—1221no—
354C4⋊C8—21no—
355C4⋊C8—21no—
356C4⋊C8—611no—
357C4⋊C8—611no—
358C4⋊C8—1222no—
359C4⋊C8—1222no—
360C4⋊C8C22×D6112yes—
361C4⋊C8C22×D6112yes—
362#103—1221no—
363#103—1221no—
364#102—1222no—
365#102—1222no—
366#102—1221no—
367#102—1221no—
368C22⋊C8—1221no—
369#97—612no—
370#97—1222no—
371C4×C8—612no—
372C4×C8—1222no—
373C4×C8—1221no—
374C24—2441no—
375#41—42no—
376#41—42no—
377C2×Q8—21no—
378C2×Q8—611no—
379C2×Q8—611no—
380C2×Q8—1222no—
381C2×Q8—1221no—
382C2×Q8—2441no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—41no—
387C2×D4—1222no—
388C2×D4—1222no—
389C2×D4—1222no—
390C2×D4—4882no—
391C2×D4—2442no—
392C2×D4—2442no—
393C2×D4—2441no—
394C2×D4—2441no—
395C2×D4—2441no—
396C2×D4—2442no—
397C2×D4—2442no—
398C22×C4—41no—
399C22×C4—1221no—
400C22×C4—1221no—
401C22×C4#1175822211yescharacteristic
402C22×C4D4×D6111yescharacteristic
403C22×C4D4×D6111yescharacteristic
404SD16—2444no—
405SD16—2444no—
406SD16—2444no—
407SD16—2444no—
408SD16—2442no—
409SD16—2442no—
410D8—2444no—
411D8—2444no—
412M16—42no—
413M16—614no—
414M16—1222no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—42no—
418C2×C8—612no—
419C2×C8—612no—
420C2×C8—1221no—
421C2×C8—1221no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—611no—
425C4⋊C4—611no—
426C4⋊C4—1221no—
427C4⋊C4—1221no—
428C4⋊C4—1221no—
429C4⋊C4—1221no—
430C4⋊C4—1221no—
431C22⋊C4—41no—
432C22⋊C4—41no—
433C42—1221no—
434C42—1221no—
435C42—1221no—
436C42—1221no—
437C42#1175822211yescharacteristic
438C42D4×D6111yescharacteristic
439C42D4×D6111yescharacteristic
440C42C23×D6111yescharacteristic
441C23—41no—
442C23—4882no—
443C23—4882no—
444C23—4882no—
445C23—2442no—
446C23#1451975011yescharacteristic
447Q8—41no—
448Q8—41no—
449Q8—612no—
450Q8—1221no—
451Q8—1221no—
452Q8—1221no—
453Q8—2442no—
454D4—41no—
455D4—41no—
456D4—41no—
457D4—41no—
458D4—81no—
459D4—82no—
460D4—4882no—
461D4—4882no—
462D4—4882no—
463D4—4882no—
464D4—2442no—
465D4—2442no—
466D4—2442no—
467D4—2442no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—82no—
473C2×C4—1221no—
474C2×C4—1221no—
475C2×C4—1221no—
476C2×C4—1221no—
477C2×C4—4881no—
478C2×C4—2442no—
479C2×C4#1451975011yescharacteristic
480C2×C4#1451975011yescharacteristic
481C2×C4#1451975011yescharacteristic
482C2×C4SD16×D6112yes—
483C2×C4#1451980112yes—
484C2×C4C2×D4×D6111yescharacteristic
485C2×C4C2×D4×D6111yescharacteristic
486C2×C4C2×#1175822211yescharacteristic
487C8—42no—
488C8—42no—
489C8—614no—
490C8—1222no—
491C8—2441no—
492C8—2441no—
493C22—21no—
494C22—21no—
495C22—81no—
496C22—81no—
497C22—81no—
498C22—9764no—
499C22—4882no—
500C22—4882no—
501C22—4882no—
502C22#1683996112yes—
503C22C2×#1451975011yescharacteristic
504C4—21no—
505C4—21no—
506C4—21no—
507C4—81no—
508C4—1222no—
509C4—2441no—
510C4—2441no—
511C4—2441no—
512C4#1683996112yes—
513C4C2×SD16×D6111yescharacteristic
514C4AGL(1,ℤ/8)×D6111yescharacteristic
515C4C2×#1451980111yescharacteristic
516C4#1684056111yescharacteristic
517C2—22no—
518C2—161no—
519C2—9762no—
520C2C2×#1683996111yescharacteristic
521C2#2145544111yescharacteristic
522C2#2145555011yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1547C2
socleC22×C61C2×#121
prime radical#21456272C2
hypercenter#1547D61
hypocenterC61#19170
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1547

Complements

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

NH ≅ G/Ntype
2. #21455846C2semidirect
3. C2×#16839963C2semidirect
4. #21454770C2semidirect
5. #21445302C2semidirect
6. #21445294C2semidirect
7. #21445178C2semidirect
8. #21445170C2semidirect
9. #21444569C2semidirect
10. #21444488C2semidirect
11. C61×#1547C2semidirect
21. #16840044C22semidirect
22. #16840044C22semidirect
27. #16839959C22semidirect
28. #16839959C22semidirect
31. #16839889C22semidirect
32. #16839886C22semidirect
40. #16839514C22semidirect
41. #16839512C22semidirect
47. C61×#332C22semidirect
48. C61×#331C22semidirect
172. #11758194SD16semidirect
224. Dic122AGL(1,ℤ/8)semidirect
230. #92418C2×SD16semidirect
265. #1547D61semidirect
306. #332C2×D61semidirect
309. #331C2×D61semidirect
310. C61#19170semidirect

Maximal subgroups

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

subgroupindex
#214562722
#214558462
C2×#168399632
#214547702
#214453022
#214452942
#214451782
#214451702
#214445692
#214444882
C61×#15472
#1917061