Finite Group Databasebeta

← #29662781

Subgroups — #29662781

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#21456271C211yesmaximal, characteristic, complemented
2#21455819C211yesmaximal, characteristic, complemented
3C2×#16839957C211yesmaximal, characteristic, complemented
4#21445298C212yesmaximal, complemented
5#21445284C212yesmaximal, complemented
6#21445174C212yesmaximal, complemented
7#21445160C212yesmaximal, complemented
8#21444571C211yesmaximal, characteristic, complemented
9#21444497C211yesmaximal, characteristic, complemented
10#21444488C211yesmaximal, characteristic, complemented
11D4×M16×C61C211yesmaximal, characteristic, complemented
12C2×#14519805—21no—
13C2×#14519805—21no—
14C22×#11758199—21no—
15#16840312C2211yescharacteristic
16C2×#14519774C2211yescharacteristic
17#16840151—22no—
18#16840139C2212yescomplemented
19#16840135C2212yescomplemented
20#16840075—21no—
21#16840042C2211yescharacteristic, complemented
22#16840042C2211yescharacteristic, complemented
23#16839957—21no—
24#16839957—21no—
25#16839957C2212yes—
26#16839957C2212yes—
27#16839957C2211yescharacteristic, complemented
28#16839957C2211yescharacteristic, complemented
29#16839896—21no—
30C4×#11758199—21no—
31C4×#11758199—21no—
32#16839815C2211yescharacteristic
33C2×C4⋊#92418C2211yescharacteristic
34C2×#14519697—21no—
35#16839515C2212yescomplemented
36#16839513C2212yescomplemented
37#16839509C2212yescomplemented
38#16839505C2212yescomplemented
39#16839505C2212yescomplemented
40#16839505C2212yescomplemented
41C22×M16×C61—21no—
42C2×C4×D4×C61C2211yescharacteristic
43C61×#436C2212yescomplemented
44C61×#435—22no—
45C8×D4×C61C2212yescomplemented
46C61×#423C2211yescharacteristic
47C61×#407—21no—
48C4×M16×C61C2211yescharacteristic
49C2×#11758222—21no—
50C22×D244—21no—
51C22×D244—21no—
52C22×D244—21no—
53#14519805—44no—
54#14519805—44no—
55C2×#11758190—22no—
56C2×#11758190—22no—
57C2×D488—22no—
58C2×D488—22no—
59C2×#11758199—22no—
60C2×#11758199—22no—
61C2×#11758199—21no—
62C2×#11758199—21no—
63C2×#11758199—44no—
64C22×#92418—21no—
65#14519777C2311yescharacteristic
66#14519774—21no—
67#14519774—21no—
68#14519774—21no—
69#14519774C2312yes—
70#14519774C2311yescharacteristic
71#14519765—21no—
72#14519751—21no—
73#14519751—21no—
74C4×D244—21no—
75C4×D244—21no—
76#14519710—21no—
77#14519710—21no—
78C4⋊#92418—21no—
79C4⋊#92418C2312yes—
80C4⋊#92418C2311yescharacteristic
81C4⋊#92418C2311yescharacteristic
82#14519698—21no—
83#14519698—21no—
84#14519697—22no—
85#14519697—22no—
86#14519697—21no—
87#14519697—21no—
88C22⋊#92418—21no—
89C4×#92418—21no—
90C22×D4×C61D411yescharacteristic
91C23×C4×C61—21no—
92C2×M16×C61—21no—
93C2×M16×C61—22no—
94C2×M16×C61—44no—
95C2×M16×C61D411yescharacteristic
96C2×M16×C61D411yescharacteristic
97C22×C8×C61—22no—
98C4×D4×C61—21no—
99C4×D4×C61—21no—
100C4×D4×C61D411yescharacteristic
101C4×D4×C61D411yescharacteristic
102C4×D4×C61C2311yescharacteristic
103C4×D4×C61C2311yescharacteristic
104C2×C4⋊C4×C61D411yescharacteristic
105C2×C22⋊C4×C61—21no—
106C2×C42×C61C2311yescharacteristic
107C4⋊C8×C61—21no—
108C4⋊C8×C61C2312yes—
109C22⋊C8×C61—22no—
110C22⋊C8×C61—22no—
111#97×C61D412yes—
112C4×C8×C61C2312yes—
113C24×C61—21no—
114C23×D61—41no—
115C2×D4×C61—22no—
116C2×D4×C61—21no—
117C2×D4×C61—41no—
118C2×D4×C61C2×D411yescharacteristic
119C2×D4×C61C2×D411yescharacteristic
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61—42no—
126C22×C4×C61—42no—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61#4111yescharacteristic
130C2×C22⋊D61—41no—
131#11758222—42no—
132#11758222—42no—
133C2×D244—22no—
134C2×D244—22no—
135C2×D244—21no—
136C2×D244—21no—
137C2×D244—22no—
138C2×D244—21no—
139C2×D244—42no—
140C2×D244—42no—
141C2×D244—42no—
142C2×D244—42no—
143C2×D244—41no—
144C2×D244—41no—
145C2×D244—41no—
146C2×D244—82no—
147C2×C4×D61—41no—
148C2×Dic122—21no—
149M16×C61—22no—
150M16×C61—42no—
151M16×C61—44no—
152M16×C61D814yescomplemented
153C2×C8×C61—22no—
154C2×C8×C61—44no—
155C2×C8×C61C2×D412yes—
156C2×C8×C61C2×D412yes—
157C4⋊C4×C61—21no—
158C4⋊C4×C61C2×D411yescharacteristic
159C4⋊C4×C61C2×D411yescharacteristic
160C22⋊C4×C61—21no—
161C22⋊C4×C61—21no—
162C22⋊C4×C61—21no—
163C22⋊C4×C61—21no—
164C42×C61C2×D411yescharacteristic
165C42×C61C2×D411yescharacteristic
166C42×C61#4111yescharacteristic
167C42×C61C2411yescharacteristic
168#11758199—42no—
169#11758199—44no—
170#11758199—42no—
171#11758199—44no—
172#11758198—41no—
173#11758198—41no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C2×#92418—21no—
177C2×#92418—21no—
178C2×#92418—22no—
179C2×#92418—42no—
180D488—44no—
181D488—44no—
182#11758190—44no—
183#11758190—44no—
184C23×C61—21no—
185C23×C61—21no—
186C23×C61—42no—
187C23×C61—42no—
188C23×C61—81no—
189C23×C61#12111yescharacteristic
190C22×D61—41no—
191C22×D61—42no—
192C22×D61—82no—
193C22×D61—82no—
194C22×D61—82no—
195D4×C61—21no—
196D4×C61—21no—
197D4×C61—21no—
198D4×C61—42no—
199D4×C61AGL(1,ℤ/8)12yescomplemented
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—41no—
205C2×C4×C61—42no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61—41no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61#12111yescharacteristic
213C2×C4×C61#12111yescharacteristic
214C2×C4×C61#12111yescharacteristic
215C2×C4×C61C2×D812yes—
216C2×C4×C61AGL(1,ℤ/8)12yes—
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C22×D411yescharacteristic
219C2×C4×C61C2×#4111yescharacteristic
220C22⋊D61—82no—
221C2×Dic61—41no—
222D244—42no—
223D244—41no—
224D244—42no—
225D244—41no—
226D244—42no—
227D244—41no—
228D244—41no—
229D244—42no—
230D244—81no—
231D244—82no—
232D244—82no—
233D244—82no—
234D244—82no—
235C4×D61—82no—
236Dic122—41no—
237Dic122—41no—
238C8×C61—22no—
239C8×C61—42no—
240C8×C61C2×D814yescomplemented
241#92418—42no—
242#92418—41no—
243#92418—41no—
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#45912yes—
255C22×C61C2×#12111yescharacteristic
256C2×D61—81no—
257C2×D61—82no—
258C2×D61—82no—
259C2×D61—81no—
260C2×D61—82no—
261C2×D61—81no—
262C2×D61—164no—
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×#459—611no—
277D4×M16D6111yescharacteristic, complemented
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×C61C2×#45911yescharacteristic
289C2×C61#305511yescharacteristic
290C2×C61#306811yescharacteristic
291D61—161no—
292D61—162no—
293C2×AGL(1,ℤ/8)—1221no—
294C2×AGL(1,ℤ/8)—1221no—
295C22×D8—1221no—
296C22×M16—21no—
297#532—611no—
298C2×#127—611no—
299C2×C4×D4C2×D6111yescharacteristic
300#496—1222no—
301#493—612no—
302#492—612no—
303#480—1221no—
304#474—611no—
305#474—611no—
306#459—612no—
307#459—612no—
308#459—611no—
309#459—611no—
310#459—1221no—
311#459—1221no—
312#447—1221no—
313C4×D8—1221no—
314C4×D8—1221no—
315#436C2×D6112yescomplemented
316#435—22no—
317C8×D4C2×D6112yescomplemented
318#431—611no—
319#423C2×D6111yescharacteristic
320C2×C4⋊C8—611no—
321C2×#102—1221no—
322#407—21no—
323C4×M16C2×D6111yescharacteristic
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—1221no—
333C22×D4—1221no—
334C22×D4—1221no—
335C22×D4D24411yescharacteristic
336C23×C4—21no—
337AGL(1,ℤ/8)—2444no—
338AGL(1,ℤ/8)—2444no—
339C2×SD16—1222no—
340C2×SD16—1222no—
341C2×D8—1222no—
342C2×D8—1222no—
343C2×D8—1221no—
344C2×D8—1221no—
345C2×D8—1222no—
346C2×D8—1222no—
347C2×D8—2444no—
348C2×M16—21no—
349C2×M16—22no—
350C2×M16—44no—
351C2×M16C22⋊D6111yescharacteristic
352C2×M16C22⋊D6111yescharacteristic
353C22×C8—22no—
354C22×C8—1221no—
355C4⋊Q8—611no—
356#127—612no—
357#127—611no—
358#127—1221no—
359#127—1221no—
360#127—1221no—
361#124—1221no—
362#121—1221no—
363#121—1221no—
364C4×D4—21no—
365C4×D4—21no—
366C4×D4—1221no—
367C4×D4—1221no—
368C4×D4D24411yescharacteristic
369C4×D4D24411yescharacteristic
370C4×D4C22×D6111yescharacteristic
371C4×D4C22×D6111yescharacteristic
372C2×C4⋊C4D24411yescharacteristic
373C2×C22⋊C4—21no—
374C2×C42C22×D6111yescharacteristic
375#107—1221no—
376#107—1221no—
377C4⋊C8—21no—
378C4⋊C8—611no—
379C4⋊C8—611no—
380C4⋊C8—612no—
381C4⋊C8—1221no—
382C4⋊C8C22×D6112yes—
383#102—1221no—
384#102—1221no—
385#102—1222no—
386#102—1222no—
387#102—1221no—
388#102—1221no—
389C22⋊C8—22no—
390C22⋊C8—22no—
391C22⋊C8—1221no—
392#97C22⋊D6112yes—
393C4×C8—1221no—
394C4×C8C22×D6112yes—
395C24—21no—
396C24—2441no—
397#41—2442no—
398#41—2442no—
399C2×Q8—1221no—
400C2×D4—22no—
401C2×D4—21no—
402C2×D4—41no—
403C2×D4—1221no—
404C2×D4—1222no—
405C2×D4—1221no—
406C2×D4—1222no—
407C2×D4—1221no—
408C2×D4—1222no—
409C2×D4—4882no—
410C2×D4—2442no—
411C2×D4—2442no—
412C2×D4—2441no—
413C2×D4—2441no—
414C2×D4—2441no—
415C2×D4—2441no—
416C2×D4—2442no—
417C2×D4—2442no—
418C2×D4C2×D24411yescharacteristic
419C2×D4C2×D24411yescharacteristic
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×C4C2×D24411yescharacteristic
429C22×C4#1175822211yescharacteristic
430C22×C4C2×C22⋊D6111yescharacteristic
431SD16—2444no—
432SD16—2444no—
433D8—2444no—
434D8—2442no—
435D8—2444no—
436D8—2442no—
437D8—2444no—
438D8—2444no—
439M16—22no—
440M16—42no—
441M16—44no—
442M16#1175819914yescomplemented
443C2×C8—22no—
444C2×C8—44no—
445C2×C8—1221no—
446C2×C8—1221no—
447C2×C8—1222no—
448C2×C8—2442no—
449C2×C8C2×C22⋊D6112yes—
450C2×C8C2×C22⋊D6112yes—
451C4⋊C4—21no—
452C4⋊C4—1221no—
453C4⋊C4—1221no—
454C4⋊C4C2×D24411yescharacteristic
455C4⋊C4C2×D24411yescharacteristic
456C22⋊C4—21no—
457C22⋊C4—21no—
458C22⋊C4—21no—
459C22⋊C4—21no—
460C22⋊C4—2441no—
461C22⋊C4—2441no—
462C42C2×D24411yescharacteristic
463C42#1175822211yescharacteristic
464C42C2×C22⋊D6111yescharacteristic
465C42C23×D6111yescharacteristic
466C23—21no—
467C23—21no—
468C23—42no—
469C23—42no—
470C23—81no—
471C23—4882no—
472C23—4882no—
473C23—4882no—
474C23—2441no—
475C23—2442no—
476C23#1451975111yescharacteristic
477Q8—2441no—
478Q8—2441no—
479D4—21no—
480D4—21no—
481D4—21no—
482D4—42no—
483D4—4881no—
484D4—4882no—
485D4—4882no—
486D4—4882no—
487D4—4882no—
488D4—4882no—
489D4—2441no—
490D4—2442no—
491D4—2441no—
492D4—2442no—
493D4—2442no—
494D4—2441no—
495D4—2441no—
496D4—2442no—
497D4#1451980512yescomplemented
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#1451975111yescharacteristic
513C2×C4#1451975111yescharacteristic
514C2×C4#1451975111yescharacteristic
515C2×C4C2×#1175819912yes—
516C2×C4#1451980212yes—
517C2×C4C22×C22⋊D6111yescharacteristic
518C2×C4C22×D24411yescharacteristic
519C2×C4C2×#1175822211yescharacteristic
520C8—22no—
521C8—42no—
522C8—2442no—
523C8—2441no—
524C8—2441no—
525C8C2×#1175819914yescomplemented
526C22—21no—
527C22—21no—
528C22—22no—
529C22—41no—
530C22—41no—
531C22—41no—
532C22—41no—
533C22—41no—
534C22—84no—
535C22—9764no—
536C22—4882no—
537C22—4881no—
538C22—4881no—
539C22—4882no—
540C22—4882no—
541C22—4881no—
542C22#1683995712yes—
543C22C2×#1451975111yescharacteristic
544C4—21no—
545C4—21no—
546C4—21no—
547C4—41no—
548C4—41no—
549C4—4881no—
550C4#1683995712yes—
551C4C22×#1175819911yescharacteristic
552C4C2×#1451980211yescharacteristic
553C4C2×#1451980511yescharacteristic
554C4#1684056211yescharacteristic
555C2—22no—
556C2—42no—
557C2—81no—
558C2—9762no—
559C2—9761no—
560C2C2×#1683995711yescharacteristic
561C2#2145543811yescharacteristic
562C2#2145554711yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingD4×M16×C61C2
socleC22×C61C2×#121
hypercenterD4×M16D61
hypocenterC61#19162
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ D4×M16

Complements

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

NH ≅ G/Ntype
1. #21456271C2semidirect
2. #21455819C2semidirect
3. C2×#16839957C2semidirect
4. #21445298C2semidirect
5. #21445284C2semidirect
6. #21445174C2semidirect
7. #21445160C2semidirect
8. #21444571C2semidirect
9. #21444497C2semidirect
10. #21444488C2semidirect
11. D4×M16×C61C2semidirect
18. #16840139C22semidirect
19. #16840135C22semidirect
21. #16840042C22semidirect
22. #16840042C22semidirect
27. #16839957C22semidirect
28. #16839957C22semidirect
35. #16839515C22semidirect
36. #16839513C22semidirect
37. #16839509C22semidirect
38. #16839505C22semidirect
39. #16839505C22semidirect
40. #16839505C22semidirect
43. C61×#436C22semidirect
45. C8×D4×C61C22semidirect
152. M16×C61D8semidirect
199. D4×C61AGL(1,ℤ/8)semidirect
240. C8×C61C2×D8semidirect
277. D4×M16D61semidirect
315. #436C2×D61semidirect
317. C8×D4C2×D61semidirect
330. C61#19162semidirect
442. M16#11758199semidirect
497. D4#14519805semidirect
525. C8C2×#11758199semidirect

Maximal subgroups

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

subgroupindex
#214562712
#214558192
C2×#168399572
#214452982
#214452842
#214451742
#214451602
#214445712
#214444972
#214444882
D4×M16×C612
#1916261