Finite Group Databasebeta

← #29662777

Subgroups — #29662777

Order 15616.

844 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456269C211yesmaximal, characteristic, complemented
2#21455819C211yesmaximal, characteristic, complemented
3D4×#11758194C211yesmaximal, characteristic, complemented
4#21445296C212yesmaximal, complemented
5#21445282C212yesmaximal, complemented
6#21445172C212yesmaximal, complemented
7#21445158C212yesmaximal, complemented
8#21444569C211yesmaximal, characteristic, complemented
9#21444494C211yesmaximal, characteristic, complemented
10#21444487C211yesmaximal, characteristic, complemented
11C2×C61×#459C211yesmaximal, characteristic, complemented
12C2×#14519803—21no—
13C2×#14519803—21no—
14C22×#11758194—21no—
15#16840312C2211yescharacteristic
16#16840149—22no—
17#16840138C2212yescomplemented
18#16840134C2212yescomplemented
19#16840075—21no—
20#16840042C2211yescharacteristic, complemented
21#16840042C2211yescharacteristic, complemented
22#16839957C2211yescharacteristic, complemented
23#16839957C2211yescharacteristic, complemented
24#16839896—21no—
25C4×#11758199—21no—
26C4×#11758199—21no—
27#16839840C2212yescomplemented
28#16839834—22no—
29D4×#92418C2212yescomplemented
30#16839815C2211yescharacteristic
31#16839786C2211yescharacteristic
32#16839726—21no—
33C4×#11758194C2211yescharacteristic
34#16839514C2212yescomplemented
35#16839512C2212yescomplemented
36#16839507C2212yescomplemented
37#16839506C2212yescomplemented
38#16839506C2212yescomplemented
39#16839505C2212yescomplemented
40C22×D8×C61—21no—
41C2×#127×C61C2211yescharacteristic
42C2×C4×D4×C61C2211yescharacteristic
43C61×#459—21no—
44C61×#459—21no—
45C61×#459C2212yes—
46C61×#459C2212yes—
47C2×C4⋊C8×C61C2211yescharacteristic
48C2×#102×C61—21no—
49C2×#11758222—21no—
50#14519803—44no—
51#14519803—44no—
52C2×#11758200—22no—
53C2×#11758200—22no—
54C2×#11758199—21no—
55C2×#11758199—21no—
56C2×#11758199—22no—
57C2×#11758199—22no—
58C2×#11758194—21no—
59C2×#11758194—22no—
60C2×#11758194—44no—
61C2×#11758194D411yescharacteristic
62C2×#11758194D411yescharacteristic
63C22×#92418—22no—
64#14519777C2311yescharacteristic
65#14519774C2311yescharacteristic
66#14519765—21no—
67#14519751—21no—
68#14519751—21no—
69C4×D244—21no—
70C4×D244—21no—
71#14519710—21no—
72#14519710—21no—
73C4⋊#92418—21no—
74C4⋊#92418C2311yescharacteristic
75C4⋊#92418C2311yescharacteristic
76C4⋊#92418C2312yes—
77#14519698—21no—
78#14519698—21no—
79#14519697—21no—
80#14519697—21no—
81C22⋊#92418—21no—
82C22⋊#92418—22no—
83C22⋊#92418—22no—
84#14519688D412yes—
85C4×#92418—21no—
86C4×#92418C2312yes—
87C22×D4×C61—21no—
88C22×D4×C61—21no—
89C22×D4×C61—21no—
90C22×D4×C61D411yescharacteristic
91C23×C4×C61—21no—
92C2×D8×C61—22no—
93C2×D8×C61—22no—
94C2×D8×C61—44no—
95C22×C8×C61—21no—
96#127×C61—21no—
97#127×C61—21no—
98#127×C61—21no—
99#127×C61C2312yes—
100C4×D4×C61—21no—
101C4×D4×C61—21no—
102C4×D4×C61D411yescharacteristic
103C4×D4×C61D411yescharacteristic
104C4×D4×C61C2311yescharacteristic
105C4×D4×C61C2311yescharacteristic
106C2×C4⋊C4×C61D411yescharacteristic
107C2×C22⋊C4×C61—21no—
108C2×C42×C61C2311yescharacteristic
109C4⋊C8×C61—21no—
110C4⋊C8×C61C2312yes—
111#102×C61—22no—
112#102×C61—22no—
113C24×C61—21no—
114C24×C61—41no—
115C2×D4×C61—22no—
116C2×D4×C61—21no—
117C2×D4×C61—22no—
118C2×D4×C61—22no—
119C2×D4×C61—22no—
120C2×D4×C61—41no—
121C2×D4×C61—42no—
122C2×D4×C61—42no—
123C2×D4×C61—41no—
124C2×D4×C61—41no—
125C2×D4×C61—41no—
126C2×D4×C61—42no—
127C2×D4×C61—42no—
128C2×D4×C61—82no—
129C2×D4×C61C2×D411yescharacteristic
130C2×D4×C61C2×D411yescharacteristic
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—42no—
137C22×C4×C61—42no—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61#4111yescharacteristic
141C2×C22⋊D61—41no—
142#11758222—42no—
143#11758222—42no—
144C2×D244—21no—
145C2×D244—21no—
146C2×D244—21no—
147C2×C4×D61—41no—
148C2×Dic122—21no—
149D8×C61—44no—
150D8×C61—44no—
151C2×C8×C61—22no—
152C2×C8×C61—42no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61C2×D411yescharacteristic
155C4⋊C4×C61C2×D411yescharacteristic
156C22⋊C4×C61—21no—
157C22⋊C4×C61—21no—
158C22⋊C4×C61—21no—
159C22⋊C4×C61—21no—
160C42×C61C2×D411yescharacteristic
161C42×C61C2×D411yescharacteristic
162C42×C61#4111yescharacteristic
163C42×C61C2411yescharacteristic
164#11758200—44no—
165#11758200—44no—
166#11758199—44no—
167#11758199—42no—
168#11758199—42no—
169#11758199—44no—
170#11758198—41no—
171#11758198—41no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174#11758194—22no—
175#11758194—42no—
176#11758194—44no—
177#11758194D814yescomplemented
178C2×#92418—21no—
179C2×#92418—21no—
180C2×#92418—22no—
181C2×#92418—44no—
182C2×#92418C2×D412yes—
183C2×#92418C2×D412yes—
184C23×C61—21no—
185C23×C61—21no—
186C23×C61—42no—
187C23×C61—42no—
188C23×C61—42no—
189C23×C61—81no—
190C23×C61—82no—
191C23×C61—82no—
192C23×C61—82no—
193C23×C61#12111yescharacteristic
194C22×D61—41no—
195D4×C61—21no—
196D4×C61—21no—
197D4×C61—21no—
198D4×C61—42no—
199D4×C61—42no—
200D4×C61—42no—
201D4×C61—42no—
202D4×C61—42no—
203D4×C61—82no—
204D4×C61—82no—
205D4×C61—82no—
206D4×C61—82no—
207D4×C61AGL(1,ℤ/8)12yescomplemented
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—41no—
213C2×C4×C61—42no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61#12111yescharacteristic
221C2×C4×C61#12111yescharacteristic
222C2×C4×C61#12111yescharacteristic
223C2×C4×C61C2×D812yes—
224C2×C4×C61AGL(1,ℤ/8)12yes—
225C2×C4×C61C22×D411yescharacteristic
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C2×#4111yescharacteristic
228C22⋊D61—82no—
229C2×Dic61—41no—
230D244—41no—
231D244—41no—
232D244—41no—
233D244—41no—
234D244—81no—
235C4×D61—82no—
236Dic122—41no—
237Dic122—41no—
238C8×C61—42no—
239#92418—22no—
240#92418—42no—
241#92418—41no—
242#92418—41no—
243#92418C2×D814yescomplemented
244#19162—611nomaximal, Sylow
245C22×C61—21no—
246C22×C61—21no—
247C22×C61—22no—
248C22×C61—41no—
249C22×C61—41no—
250C22×C61—41no—
251C22×C61—41no—
252C22×C61—41no—
253C22×C61—84no—
254C22×C61—82no—
255C22×C61—82no—
256C22×C61—82no—
257C22×C61—164no—
258C22×C61#45912yes—
259C22×C61C2×#12111yescharacteristic
260C2×D61—81no—
261C2×D61—81no—
262C2×D61—81no—
263C4×C61—21no—
264C4×C61—21no—
265C4×C61—21no—
266C4×C61—41no—
267C4×C61—41no—
268C4×C61#45912yes—
269C4×C61C22×D811yescharacteristic
270C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
271C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
272C4×C61#57511yescharacteristic
273Dic61—81no—
274#3151—611no—
275#3100—611no—
276C2×#459D6111yescharacteristic, complemented
277D4×M16—611no—
278#1687—612no—
279#1685—612no—
280#1667—612no—
281#1665—612no—
282#1565—611no—
283#1547—611no—
284#1545—611no—
285C2×C61—22no—
286C2×C61—42no—
287C2×C61—81no—
288C2×C61—162no—
289C2×C61C2×#45911yescharacteristic
290C2×C61#305511yescharacteristic
291C2×C61#306811yescharacteristic
292D61—161no—
293C2×AGL(1,ℤ/8)—1221no—
294C2×AGL(1,ℤ/8)—1221no—
295C22×D8—21no—
296C22×M16—1221no—
297#532—611no—
298C2×#127C2×D6111yescharacteristic
299C2×C4×D4C2×D6111yescharacteristic
300#496—1222no—
301#493—612no—
302#492—612no—
303#480—1221no—
304#474—611no—
305#474—611no—
306#459—21no—
307#459—21no—
308#459—611no—
309#459—611no—
310#459C2×D6112yes—
311#459C2×D6112yes—
312#447—1221no—
313C4×D8—1221no—
314C4×D8—1221no—
315#436—612no—
316#435—1222no—
317C8×D4—612no—
318#431—611no—
319#423—611no—
320C2×C4⋊C8C2×D6111yescharacteristic
321C2×#102—21no—
322#407—1221no—
323C4×M16—611no—
324#335—612no—
325#334—612no—
326#332—612no—
327#331—612no—
328#331—612no—
329#331—612no—
330C61#1916211yescharacteristic, complemented, Sylow
331C2×#41—1221no—
332C22×D4—21no—
333C22×D4—21no—
334C22×D4—21no—
335C22×D4C22⋊D6111yescharacteristic
336C23×C4—21no—
337AGL(1,ℤ/8)—2444no—
338AGL(1,ℤ/8)—2444no—
339C2×SD16—1222no—
340C2×SD16—1222no—
341C2×D8—22no—
342C2×D8—22no—
343C2×D8—44no—
344C2×D8—1222no—
345C2×D8—1222no—
346C2×D8—1221no—
347C2×D8—1221no—
348C2×M16—611no—
349C2×M16—611no—
350C2×M16—1222no—
351C2×M16—1221no—
352C2×M16—2444no—
353C22×C8—21no—
354C22×C8—1222no—
355C4⋊Q8—611no—
356#127—21no—
357#127—21no—
358#127—21no—
359#127—611no—
360#127C22×D6112yes—
361#124—1221no—
362#121—1221no—
363#121—1221no—
364C4×D4—21no—
365C4×D4—21no—
366C4×D4—1221no—
367C4×D4—1221no—
368C4×D4C22⋊D6111yescharacteristic
369C4×D4C22⋊D6111yescharacteristic
370C4×D4C22×D6111yescharacteristic
371C4×D4C22×D6111yescharacteristic
372C2×C4⋊C4C22⋊D6111yescharacteristic
373C2×C22⋊C4—21no—
374C2×C42C22×D6111yescharacteristic
375#107—1221no—
376#107—1221no—
377C4⋊C8—21no—
378C4⋊C8—612no—
379C4⋊C8—611no—
380C4⋊C8—611no—
381C4⋊C8—1221no—
382C4⋊C8C22×D6112yes—
383#102—22no—
384#102—22no—
385#102—1221no—
386#102—1221no—
387#102—1221no—
388#102—1221no—
389C22⋊C8—1222no—
390C22⋊C8—1222no—
391C22⋊C8—1221no—
392#97—612no—
393C4×C8—612no—
394C4×C8—1221no—
395C24—21no—
396C24—41no—
397#41—2442no—
398#41—2442no—
399C2×Q8—1221no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—21no—
405C2×D4—41no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—41no—
409C2×D4—41no—
410C2×D4—41no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—82no—
414C2×D4—1221no—
415C2×D4—1221no—
416C2×D4—1221no—
417C2×D4—2441no—
418C2×D4C2×C22⋊D6111yescharacteristic
419C2×D4C2×C22⋊D6111yescharacteristic
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—2441no—
428C22×C4D4×D6111yescharacteristic
429C22×C4#1175822411yescharacteristic
430C22×C4C2×C22⋊D6111yescharacteristic
431SD16—2444no—
432SD16—2444no—
433D8—44no—
434D8—44no—
435D8—2444no—
436D8—2444no—
437D8—2442no—
438D8—2442no—
439M16—614no—
440M16—1222no—
441M16—2444no—
442M16—2442no—
443C2×C8—22no—
444C2×C8—42no—
445C2×C8—612no—
446C2×C8—612no—
447C2×C8—1222no—
448C2×C8—1221no—
449C2×C8—1221no—
450C2×C8—2444no—
451C4⋊C4—21no—
452C4⋊C4—1221no—
453C4⋊C4—1221no—
454C4⋊C4C2×C22⋊D6111yescharacteristic
455C4⋊C4C2×C22⋊D6111yescharacteristic
456C22⋊C4—21no—
457C22⋊C4—21no—
458C22⋊C4—21no—
459C22⋊C4—21no—
460C22⋊C4—2441no—
461C22⋊C4—2441no—
462C42D4×D6111yescharacteristic
463C42#1175822411yescharacteristic
464C42C2×C22⋊D6111yescharacteristic
465C42C23×D6111yescharacteristic
466C23—21no—
467C23—21no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—81no—
472C23—82no—
473C23—82no—
474C23—82no—
475C23—2441no—
476C23#1451974911yescharacteristic
477Q8—2441no—
478Q8—2441no—
479D4—21no—
480D4—21no—
481D4—21no—
482D4—42no—
483D4—42no—
484D4—42no—
485D4—42no—
486D4—42no—
487D4—82no—
488D4—82no—
489D4—82no—
490D4—82no—
491D4—4881no—
492D4—4882no—
493D4—2441no—
494D4—2441no—
495D4—2441no—
496D4—2441no—
497D4#1451980312yescomplemented
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—4882no—
511C2×C4—2441no—
512C2×C4#1451974911yescharacteristic
513C2×C4#1451974911yescharacteristic
514C2×C4#1451974911yescharacteristic
515C2×C4D8×D6112yes—
516C2×C4#1451980112yes—
517C2×C4C2×D4×D6111yescharacteristic
518C2×C4C22×C22⋊D6111yescharacteristic
519C2×C4C2×#1175822411yescharacteristic
520C8—42no—
521C8—614no—
522C8—1222no—
523C8—2442no—
524C8—2441no—
525C8—2441no—
526C22—21no—
527C22—21no—
528C22—22no—
529C22—41no—
530C22—41no—
531C22—41no—
532C22—41no—
533C22—41no—
534C22—84no—
535C22—82no—
536C22—82no—
537C22—82no—
538C22—164no—
539C22—4881no—
540C22—4881no—
541C22—4881no—
542C22#1683995512yes—
543C22C2×#1451974911yescharacteristic
544C4—21no—
545C4—21no—
546C4—21no—
547C4—41no—
548C4—41no—
549C4—4881no—
550C4#1683995512yes—
551C4C2×D8×D6111yescharacteristic
552C4C2×#1451980111yescharacteristic
553C4C2×#1451980311yescharacteristic
554C4#1684056011yescharacteristic
555C2—22no—
556C2—42no—
557C2—81no—
558C2—162no—
559C2—9761no—
560C2C2×#1683995511yescharacteristic
561C2#2145543411yescharacteristic
562C2#2145554411yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#459C2
socleC22×C61C2×#121
hypercenterC2×#459D61
hypocenterC61#19162
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#459

Complements

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

NH ≅ G/Ntype
1. #21456269C2semidirect
2. #21455819C2semidirect
3. D4×#11758194C2semidirect
4. #21445296C2semidirect
5. #21445282C2semidirect
6. #21445172C2semidirect
7. #21445158C2semidirect
8. #21444569C2semidirect
9. #21444494C2semidirect
10. #21444487C2semidirect
11. C2×C61×#459C2semidirect
17. #16840138C22semidirect
18. #16840134C22semidirect
20. #16840042C22semidirect
21. #16840042C22semidirect
22. #16839957C22semidirect
23. #16839957C22semidirect
27. #16839840C22semidirect
29. D4×#92418C22semidirect
34. #16839514C22semidirect
35. #16839512C22semidirect
36. #16839507C22semidirect
37. #16839506C22semidirect
38. #16839506C22semidirect
39. #16839505C22semidirect
177. #11758194D8semidirect
207. D4×C61AGL(1,ℤ/8)semidirect
243. #92418C2×D8semidirect
276. C2×#459D61semidirect
330. C61#19162semidirect
497. D4#14519803semidirect

Maximal subgroups

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

subgroupindex
#214562692
#214558192
D4×#117581942
#214452962
#214452822
#214451722
#214451582
#214445692
#214444942
#214444872
C2×C61×#4592
#1916261