Finite Group Databasebeta

← #29662786

Subgroups — #29662786

Order 15616.

760 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455815C211yesmaximal, characteristic
2C2×#16840039C211yesmaximal, characteristic, complemented
3M16×D244C211yesmaximal, characteristic, complemented
4#21445305C212yesmaximal, complemented
5#21445281C212yesmaximal, complemented
6#21445181C212yesmaximal, complemented
7#21445157C212yesmaximal, complemented
8#21444571C211yesmaximal, characteristic, complemented
9#21444496C211yesmaximal, characteristic, complemented
10#21444491C211yesmaximal, characteristic, complemented
11C61×#3152C211yesmaximal, characteristic, complemented
12C2×M16×D61—21no—
13C2×C4×D244C2211yescharacteristic
14#16840073—21no—
15#16840039—21no—
16#16840039—21no—
17#16840039C2212yes—
18#16840039C2211yescharacteristic
19#16840039C2212yes—
20#16840039C2211yescharacteristic
21#16839954C2211yescharacteristic
22#16839954C2211yescharacteristic
23#16839895—21no—
24C4×#11758199—21no—
25C4×#11758199—21no—
26#16839841C2212yes—
27#16839835—22no—
28C8×D244C2212yes—
29#16839815C2211yescharacteristic
30C2×#14519710—21no—
31#16839786C2211yescharacteristic
32C2×C4⋊#92418C2211yescharacteristic
33C2×#14519697—21no—
34#16839725—21no—
35#16839515C2212yescomplemented
36#16839513C2212yescomplemented
37#16839508C2212yescomplemented
38#16839508C2212yescomplemented
39#16839508C2212yescomplemented
40#16839506C2212yescomplemented
41C2×#137×C61—21no—
42C2×#137×C61—21no—
43C61×#532C2211yescharacteristic
44C2×C4⋊Q8×C61C2211yescharacteristic
45C61×#497—22no—
46C61×#494C2212yescomplemented
47C61×#492C2212yescomplemented
48C4×M16×C61C2211yescharacteristic
49C22×D244D411yescharacteristic
50C22×C4×D61—21no—
51C2×#11758199—21no—
52C2×#11758199—21no—
53C2×#11758194—21no—
54C2×#11758189—22no—
55M16×D61—44no—
56C22×#92418—21no—
57C2×C8×D61—22no—
58C4×D244—21no—
59C4×D244—21no—
60C4×D244D411yescharacteristic
61C4×D244D411yescharacteristic
62C4×D244C2311yescharacteristic
63C4×D244C2311yescharacteristic
64C2×C4⋊Dic61D411yescharacteristic
65C2×#11758198—21no—
66#14519710—22no—
67#14519710—22no—
68#14519710—21no—
69#14519710—21no—
70C4⋊#92418—21no—
71C4⋊#92418—21no—
72C4⋊#92418C2311yescharacteristic
73C4⋊#92418C2311yescharacteristic
74C4⋊#92418C2312yes—
75C4⋊#92418C2312yes—
76#14519698—21no—
77#14519698—21no—
78#14519697—22no—
79#14519697—22no—
80#14519697—21no—
81#14519697—21no—
82C22⋊#92418—21no—
83#14519689—22no—
84#14519689—22no—
85C4×#92418—21no—
86C2×#41×C61—21no—
87C22×Q8×C61—21no—
88#137×C61—44no—
89#137×C61—44no—
90C2×Q16×C61—22no—
91C2×Q16×C61—22no—
92C2×SD16×C61—22no—
93C2×SD16×C61—22no—
94C2×M16×C61D411yescharacteristic
95C2×M16×C61D411yescharacteristic
96C4⋊Q8×C61—21no—
97C4⋊Q8×C61—21no—
98C4⋊Q8×C61—21no—
99C4⋊Q8×C61C2312yes—
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×C4⋊C4×C61—21no—
108C2×C4⋊C4×C61—21no—
109C2×C42×C61C2311yescharacteristic
110#97×C61D412yes—
111C4×C8×C61C2312yes—
112C23×D61—21no—
113#41×C61—42no—
114#41×C61—42no—
115C2×Q8×C61—21no—
116C2×Q8×C61—22no—
117C2×Q8×C61—41no—
118C2×Q8×C61—42no—
119C2×Q8×C61—42no—
120C2×D4×C61—21no—
121C2×D4×C61—21no—
122C2×D4×C61—21no—
123C2×D4×C61—41no—
124C22×C4×C61—41no—
125C22×C4×C61—41no—
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61#4111yescharacteristic
129C22×Dic61—21no—
130C2×D244—22no—
131C2×D244—21no—
132C2×D244—41no—
133C2×D244C2×D411yescharacteristic
134C2×D244C2×D411yescharacteristic
135C2×C4×D61—21no—
136C2×C4×D61—21no—
137C2×C4×D61—21no—
138C2×C4×D61—21no—
139C2×C4×D61—42no—
140C2×C4×D61—42no—
141Q16×C61—44no—
142Q16×C61—44no—
143SD16×C61—44no—
144SD16×C61—44no—
145M16×C61—22no—
146M16×C61D814yescomplemented
147C2×C8×C61C2×D412yes—
148C2×C8×C61C2×D412yes—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—41no—
154C4⋊C4×C61—41no—
155C22⋊C4×C61—41no—
156C22⋊C4×C61—41no—
157C42×C61C2×D411yescharacteristic
158C42×C61C2×D411yescharacteristic
159C42×C61#4111yescharacteristic
160C42×C61C2411yescharacteristic
161#11758199—42no—
162#11758199—42no—
163#11758198—21no—
164#11758198—21no—
165#11758198—21no—
166#11758198—21no—
167C4⋊Dic61—21no—
168C4⋊Dic61C2×D411yescharacteristic
169C4⋊Dic61C2×D411yescharacteristic
170#11758194—42no—
171C2×#92418—21no—
172C2×#92418—21no—
173C2×#92418—22no—
174C2×#92418—22no—
175C2×#92418—42no—
176#11758189—44no—
177C8×D61—44no—
178C23×C61—41no—
179C23×C61#12111yescharacteristic
180C22×D61—21no—
181C22×D61—21no—
182C22×D61—42no—
183C22×D61—42no—
184C22×D61—81no—
185Q8×C61—42no—
186Q8×C61—41no—
187Q8×C61—42no—
188Q8×C61—41no—
189Q8×C61—82no—
190D4×C61—41no—
191D4×C61—41no—
192D4×C61—41no—
193D4×C61—41no—
194D4×C61—81no—
195D4×C61—82no—
196C2×C4×C61—21no—
197C2×C4×C61—21no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61—42no—
201C2×C4×C61—82no—
202C2×C4×C61—82no—
203C2×C4×C61#12111yescharacteristic
204C2×C4×C61#12111yescharacteristic
205C2×C4×C61#12111yescharacteristic
206C2×C4×C61C2×D812yes—
207C2×C4×C61AGL(1,ℤ/8)12yes—
208C2×C4×C61C22×D411yescharacteristic
209C2×C4×C61C22×D411yescharacteristic
210C2×C4×C61C2×#4111yescharacteristic
211C2×Dic61—21no—
212C2×Dic61—21no—
213C2×Dic61—42no—
214D244—21no—
215D244—21no—
216D244—21no—
217D244—42no—
218D244#13712yescomplemented
219C4×D61—41no—
220C4×D61—41no—
221C4×D61—41no—
222C4×D61—41no—
223C4×D61—42no—
224C4×D61—42no—
225C8×C61—22no—
226C8×C61C2×D814yes—
227#92418—42no—
228#92418—41no—
229#92418—42no—
230#92418—41no—
231#19163—611nomaximal, Sylow
232C22×C61—21no—
233C22×C61—21no—
234C22×C61—81no—
235C22×C61—81no—
236C22×C61—81no—
237C22×C61#45912yes—
238C22×C61C2×#12111yescharacteristic
239C2×D61—22no—
240C2×D61—41no—
241C2×D61—41no—
242C2×D61—41no—
243C2×D61—41no—
244C2×D61—41no—
245C2×D61—84no—
246C4×C61—21no—
247C4×C61—21no—
248C4×C61—21no—
249C4×C61—81no—
250C4×C61—82no—
251C4×C61#45912yes—
252C4×C61C22×D811yescharacteristic
253C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
254C4×C61C2×#13711yescharacteristic
255C4×C61#57511yescharacteristic
256Dic61—41no—
257Dic61—41no—
258#3152D6111yescharacteristic, complemented
259#3100—611no—
260C2×#474—611no—
261D4×M16—611no—
262#1689—612no—
263#1685—612no—
264#1669—612no—
265#1665—612no—
266#1565—611no—
267#1547—611no—
268#1546—611no—
269C2×C61—22no—
270C2×C61—161no—
271C2×C61C2×#45911yescharacteristic
272C2×C61#305611yescharacteristic
273C2×C61#306911yescharacteristic
274D61—42no—
275D61—81no—
276C2×#137—21no—
277C2×#137—21no—
278C22×M16—1221no—
279#532C2×D6111yescharacteristic
280C2×C4⋊Q8C2×D6111yescharacteristic
281C2×C4×D4—611no—
282#497—22no—
283#494C2×D6112yescomplemented
284#492C2×D6112yescomplemented
285#480—1221no—
286#474—611no—
287#474—611no—
288#474—612no—
289#474—612no—
290#474—1221no—
291#474—1221no—
292#459—611no—
293#459—611no—
294#447—1221no—
295C4×D8—1221no—
296C4×D8—1221no—
297#436—612no—
298#435—1222no—
299C8×D4—612no—
300#431—611no—
301C2×#107—1221no—
302#423—611no—
303C2×C4⋊C8—611no—
304C2×#102—1221no—
305#407—1221no—
306C4×M16C2×D6111yescharacteristic
307#335—612no—
308#334—612no—
309#332—612no—
310#332—612no—
311#332—612no—
312#331—612no—
313C61#1916311yescharacteristic, complemented, Sylow
314C2×#41—21no—
315C22×Q8—21no—
316C22×D4—611no—
317C23×C4—1221no—
318#137—44no—
319#137—44no—
320C2×Q16—22no—
321C2×Q16—22no—
322C2×SD16—22no—
323C2×SD16—22no—
324C2×D8—1221no—
325C2×D8—1221no—
326C2×M16—1221no—
327C2×M16—1222no—
328C2×M16—2444no—
329C2×M16C22⋊D6111yescharacteristic
330C2×M16C22⋊D6111yescharacteristic
331C22×C8—1221no—
332C22×C8—1222no—
333C4⋊Q8—21no—
334C4⋊Q8—21no—
335C4⋊Q8—21no—
336C4⋊Q8C22×D6112yes—
337C4⋊Q8C22×D6111yescharacteristic
338#127C22×D6111yescharacteristic
339#124—21no—
340#121—21no—
341#121—21no—
342C4×D4—21no—
343C4×D4—21no—
344C4×D4—611no—
345C4×D4—611no—
346C4×D4—611no—
347C4×D4—611no—
348C4×D4—1221no—
349C4×D4—1221no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—611no—
353C2×C22⋊C4—1221no—
354C2×C42C22×D6111yescharacteristic
355#107—1222no—
356#107—1222no—
357#107—1221no—
358#107—1221no—
359C4⋊C8—612no—
360C4⋊C8—612no—
361C4⋊C8—611no—
362C4⋊C8—611no—
363C4⋊C8—1221no—
364C4⋊C8—1221no—
365#102—1222no—
366#102—1222no—
367#102—1221no—
368#102—1221no—
369#102—1221no—
370#102—1221no—
371C22⋊C8—1221no—
372C22⋊C8—1222no—
373C22⋊C8—1222no—
374#97C22⋊D6112yes—
375C4×C8—1221no—
376C4×C8C22×D6112yes—
377C24—1221no—
378#41—42no—
379#41—42no—
380C2×Q8—21no—
381C2×Q8—22no—
382C2×Q8—41no—
383C2×Q8—42no—
384C2×Q8—42no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—41no—
389C2×D4—611no—
390C2×D4—611no—
391C2×D4—1222no—
392C2×D4—1221no—
393C2×D4—2441no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—1221no—
397C22×C4—1221no—
398C22×C4—1221no—
399C22×C4—1221no—
400C22×C4—1221no—
401C22×C4—2442no—
402C22×C4—2442no—
403C22×C4D4×D6111yescharacteristic
404C22×C4#1175822411yescharacteristic
405C22×C4C2×C22⋊D6111yescharacteristic
406Q16—44no—
407Q16—44no—
408SD16—44no—
409SD16—44no—
410D8—2442no—
411D8—2442no—
412M16—22no—
413M16—2442no—
414M16—2444no—
415M16#1175819914yescomplemented
416C2×C8—1221no—
417C2×C8—1221no—
418C2×C8—1222no—
419C2×C8—1222no—
420C2×C8—2442no—
421C2×C8—2444no—
422C2×C8C2×C22⋊D6112yes—
423C2×C8C2×C22⋊D6112yes—
424C4⋊C4—22no—
425C4⋊C4—21no—
426C4⋊C4—22no—
427C4⋊C4—21no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—611no—
431C4⋊C4—611no—
432C4⋊C4—1221no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—1221no—
436C22⋊C4—1221no—
437C22⋊C4—1221no—
438C22⋊C4—1221no—
439C42D4×D6111yescharacteristic
440C42#1175822411yescharacteristic
441C42C2×C22⋊D6111yescharacteristic
442C42C23×D6111yescharacteristic
443C23—41no—
444C23—1221no—
445C23—1221no—
446C23—4881no—
447C23—2442no—
448C23—2442no—
449C23#1451974811yescharacteristic
450Q8—42no—
451Q8—41no—
452Q8—42no—
453Q8—41no—
454Q8—82no—
455D4—41no—
456D4—41no—
457D4—41no—
458D4—41no—
459D4—81no—
460D4—82no—
461D4—612no—
462D4—1221no—
463D4—1221no—
464D4—1221no—
465D4—2442no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—1221no—
474C2×C4—1221no—
475C2×C4—2442no—
476C2×C4—2442no—
477C2×C4—2442no—
478C2×C4—2441no—
479C2×C4—2441no—
480C2×C4—2441no—
481C2×C4—2441no—
482C2×C4#1451974811yescharacteristic
483C2×C4#1451974811yescharacteristic
484C2×C4#1451974811yescharacteristic
485C2×C4C2×#1175819912yes—
486C2×C4#1451980312yes—
487C2×C4C2×D4×D6111yescharacteristic
488C2×C4C22×C22⋊D6111yescharacteristic
489C2×C4C2×#1175822411yescharacteristic
490C8—22no—
491C8—2442no—
492C8—2441no—
493C8—2442no—
494C8—2441no—
495C8C2×#1175819914yes—
496C22—21no—
497C22—21no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22—1222no—
502C22—4884no—
503C22—2441no—
504C22—2441no—
505C22—2441no—
506C22—2441no—
507C22—2441no—
508C22#1683995412yes—
509C22C2×#1451974811yescharacteristic
510C4—21no—
511C4—21no—
512C4—21no—
513C4—81no—
514C4—82no—
515C4—2441no—
516C4—2441no—
517C4#1683995412yes—
518C4C22×#1175819911yescharacteristic
519C4C2×#1451980311yescharacteristic
520C4#137×D6111yescharacteristic
521C4#1684056111yescharacteristic
522C2—22no—
523C2—161no—
524C2—4881no—
525C2—2442no—
526C2C2×#1683995411yescharacteristic
527C2#2145544311yescharacteristic
528C2#2145555211yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3152C2
socleC22×C61C2×#121
prime radical#21455815C2
hypercenter#3152D61
hypocenterC61#19163
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3152

Complements

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

NH ≅ G/Ntype
2. C2×#16840039C2semidirect
3. M16×D244C2semidirect
4. #21445305C2semidirect
5. #21445281C2semidirect
6. #21445181C2semidirect
7. #21445157C2semidirect
8. #21444571C2semidirect
9. #21444496C2semidirect
10. #21444491C2semidirect
11. C61×#3152C2semidirect
35. #16839515C22semidirect
36. #16839513C22semidirect
37. #16839508C22semidirect
38. #16839508C22semidirect
39. #16839508C22semidirect
40. #16839506C22semidirect
46. C61×#494C22semidirect
47. C61×#492C22semidirect
146. M16×C61D8semidirect
218. D244#137semidirect
258. #3152D61semidirect
283. #494C2×D61semidirect
284. #492C2×D61semidirect
313. C61#19163semidirect
415. M16#11758199semidirect

Maximal subgroups

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

subgroupindex
#214558152
C2×#168400392
M16×D2442
#214453052
#214452812
#214451812
#214451572
#214445712
#214444962
#214444912
C61×#31522
#1916361