Finite Group Databasebeta

← #29662841

Subgroups — #29662841

Order 15616.

696 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455815C211yesmaximal, characteristic
2C2×#16840049C211yesmaximal, characteristic, complemented
3M16×D244C211yesmaximal, characteristic, complemented
4#21445391C212yesmaximal, complemented
5#21445349C212yesmaximal, complemented
6#21445267C212yesmaximal, complemented
7#21445225C212yesmaximal, complemented
8#21444571C211yesmaximal, characteristic, complemented
9#21444500C211yesmaximal, characteristic, complemented
10#21444496C211yesmaximal, characteristic, complemented
11C61×#3141C211yesmaximal, characteristic, complemented
12C2×M16×D61—21no—
13C2×C4×D244C2211yescharacteristic
14#16840073—21no—
15#16840049—21no—
16#16840049—21no—
17#16840049C2212yes—
18#16840049C2212yes—
19#16840039C2211yescharacteristic
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×#14519708—21no—
31#16839786C2211yescharacteristic
32C2×C4⋊#92418C2211yescharacteristic
33C2×#14519697—21no—
34#16839725—21no—
35#16839515C2212yescomplemented
36#16839513C2212yescomplemented
37#16839511C2212yescomplemented
38#16839511C2212yescomplemented
39#16839508C2212yescomplemented
40#16839506C2212yescomplemented
41C61×#532C2211yescharacteristic
42C2×C4⋊Q8×C61C2211yescharacteristic
43C61×#489—22no—
44C61×#487C2212yescomplemented
45C61×#486C2212yescomplemented
46C61×#417—21no—
47C61×#417—21no—
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—21no—
67#14519710—21no—
68#14519708—22no—
69#14519708—22no—
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—
88C2×M16×C61D411yescharacteristic
89C2×M16×C61D411yescharacteristic
90C4⋊Q8×C61—21no—
91C4⋊Q8×C61—21no—
92C4⋊Q8×C61—21no—
93C4⋊Q8×C61C2312yes—
94C4⋊Q8×C61C2311yescharacteristic
95#127×C61C2311yescharacteristic
96#124×C61—21no—
97#121×C61—21no—
98#121×C61—21no—
99C4×D4×C61—21no—
100C4×D4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C42×C61C2311yescharacteristic
104#103×C61—22no—
105#103×C61—22no—
106#102×C61—22no—
107#102×C61—22no—
108#97×C61D412yes—
109C4×C8×C61C2312yes—
110C23×D61—21no—
111#41×C61—42no—
112#41×C61—42no—
113C2×Q8×C61—21no—
114C2×Q8×C61—22no—
115C2×Q8×C61—41no—
116C2×Q8×C61—82no—
117C2×D4×C61—21no—
118C2×D4×C61—21no—
119C2×D4×C61—21no—
120C2×D4×C61—41no—
121C22×C4×C61—41no—
122C22×C4×C61—41no—
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61#4111yescharacteristic
126C22×Dic61—21no—
127C2×D244—22no—
128C2×D244—21no—
129C2×D244—41no—
130C2×D244C2×D411yescharacteristic
131C2×D244C2×D411yescharacteristic
132C2×C4×D61—21no—
133C2×C4×D61—21no—
134C2×C4×D61—21no—
135C2×C4×D61—21no—
136C2×C4×D61—42no—
137C2×C4×D61—42no—
138M16×C61—22no—
139M16×C61—22no—
140C2×C8×C61C2×D412yes—
141C2×C8×C61C2×D412yes—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—41no—
147C4⋊C4×C61—41no—
148C22⋊C4×C61—41no—
149C22⋊C4×C61—41no—
150C42×C61C2×D411yescharacteristic
151C42×C61C2×D411yescharacteristic
152C42×C61#4111yescharacteristic
153C42×C61C2411yescharacteristic
154#11758199—42no—
155#11758199—42no—
156#11758198—21no—
157#11758198—21no—
158#11758198—21no—
159#11758198—21no—
160C4⋊Dic61—21no—
161C4⋊Dic61C2×D411yescharacteristic
162C4⋊Dic61C2×D411yescharacteristic
163#11758194—42no—
164C2×#92418—21no—
165C2×#92418—21no—
166C2×#92418—22no—
167C2×#92418—22no—
168C2×#92418—42no—
169#11758189—44no—
170C8×D61—44no—
171C23×C61—41no—
172C23×C61#12111yescharacteristic
173C22×D61—21no—
174C22×D61—21no—
175C22×D61—42no—
176C22×D61—42no—
177C22×D61—81no—
178Q8×C61—41no—
179Q8×C61—41no—
180Q8×C61—82no—
181Q8×C61—82no—
182D4×C61—41no—
183D4×C61—41no—
184D4×C61—41no—
185D4×C61—41no—
186D4×C61—81no—
187D4×C61—82no—
188C2×C4×C61—21no—
189C2×C4×C61—21no—
190C2×C4×C61—41no—
191C2×C4×C61—41no—
192C2×C4×C61—42no—
193C2×C4×C61—82no—
194C2×C4×C61—82no—
195C2×C4×C61#12111yescharacteristic
196C2×C4×C61#12111yescharacteristic
197C2×C4×C61#12111yescharacteristic
198C2×C4×C61#13512yes—
199C2×C4×C61AGL(1,ℤ/8)12yes—
200C2×C4×C61C22×D411yescharacteristic
201C2×C4×C61C22×D411yescharacteristic
202C2×C4×C61C2×#4111yescharacteristic
203C2×Dic61—21no—
204C2×Dic61—21no—
205C2×Dic61—42no—
206D244—21no—
207D244—21no—
208D244—21no—
209D244—42no—
210D244#13712yes—
211C4×D61—41no—
212C4×D61—41no—
213C4×D61—41no—
214C4×D61—41no—
215C4×D61—42no—
216C4×D61—42no—
217C8×C61—22no—
218C8×C61—22no—
219#92418—42no—
220#92418—41no—
221#92418—42no—
222#92418—41no—
223#19168—611nomaximal, Sylow
224C22×C61—21no—
225C22×C61—21no—
226C22×C61—81no—
227C22×C61—81no—
228C22×C61—81no—
229C22×C61#46312yes—
230C22×C61C2×#12111yescharacteristic
231C2×D61—22no—
232C2×D61—41no—
233C2×D61—41no—
234C2×D61—41no—
235C2×D61—41no—
236C2×D61—41no—
237C2×D61—84no—
238C4×C61—21no—
239C4×C61—21no—
240C4×C61—21no—
241C4×C61—81no—
242C4×C61—82no—
243C4×C61#46312yes—
244C4×C61C2×#13511yescharacteristic
245C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
246C4×C61C2×#13711yescharacteristic
247C4×C61#57511yescharacteristic
248Dic61—41no—
249Dic61—41no—
250#3141D6111yescharacteristic, complemented
251#3100—611no—
252C2×#476—611no—
253D4×M16—611no—
254#1703—612no—
255#1697—612no—
256#1683—612no—
257#1677—612no—
258#1565—611no—
259#1548—611no—
260#1547—611no—
261C2×C61—22no—
262C2×C61—161no—
263C2×C61C2×#46311yescharacteristic
264C2×C61#305611yescharacteristic
265C2×C61#306811yescharacteristic
266D61—42no—
267D61—81no—
268C22×M16—1221no—
269#532C2×D6111yescharacteristic
270C2×C4⋊Q8C2×D6111yescharacteristic
271C2×C4×D4—611no—
272#489—22no—
273#487C2×D6112yescomplemented
274#486C2×D6112yescomplemented
275#480—1221no—
276#476—612no—
277#476—612no—
278#476—1221no—
279#476—1221no—
280#474—611no—
281#474—611no—
282#459—611no—
283#459—611no—
284#447—1221no—
285C4×D8—1221no—
286C4×D8—1221no—
287#436—612no—
288#435—1222no—
289C8×D4—612no—
290#431—611no—
291C2×#106—1221no—
292#423—611no—
293C2×C4⋊C8—611no—
294#417—21no—
295#417—21no—
296C2×#102—1221no—
297#407—1221no—
298C4×M16C2×D6111yescharacteristic
299#335—612no—
300#334—612no—
301#333—612no—
302#333—612no—
303#332—612no—
304#331—612no—
305C61#1916811yescharacteristic, complemented, Sylow
306C2×#41—21no—
307C22×Q8—21no—
308C22×D4—611no—
309C23×C4—1221no—
310C2×D8—1221no—
311C2×D8—1221no—
312C2×M16—1221no—
313C2×M16—1222no—
314C2×M16—2444no—
315C2×M16C22⋊D6111yescharacteristic
316C2×M16C22⋊D6111yescharacteristic
317C22×C8—1221no—
318C22×C8—1222no—
319C4⋊Q8—21no—
320C4⋊Q8—21no—
321C4⋊Q8—21no—
322C4⋊Q8C22×D6112yes—
323C4⋊Q8C22×D6111yescharacteristic
324#127C22×D6111yescharacteristic
325#124—21no—
326#121—21no—
327#121—21no—
328C4×D4—21no—
329C4×D4—21no—
330C4×D4—611no—
331C4×D4—611no—
332C4×D4—611no—
333C4×D4—611no—
334C4×D4—1221no—
335C4×D4—1221no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—611no—
339C2×C22⋊C4—1221no—
340C2×C42C22×D6111yescharacteristic
341#107—1221no—
342#107—1221no—
343#106—1222no—
344#106—1222no—
345C4⋊C8—612no—
346C4⋊C8—612no—
347C4⋊C8—611no—
348C4⋊C8—611no—
349C4⋊C8—1221no—
350C4⋊C8—1221no—
351#103—22no—
352#103—22no—
353#102—22no—
354#102—22no—
355#102—1222no—
356#102—1222no—
357#102—1221no—
358#102—1221no—
359#102—1221no—
360#102—1221no—
361C22⋊C8—1221no—
362C22⋊C8—1222no—
363C22⋊C8—1222no—
364#97C22⋊D6112yes—
365C4×C8—1221no—
366C4×C8C22×D6112yes—
367C24—1221no—
368#41—42no—
369#41—42no—
370C2×Q8—21no—
371C2×Q8—22no—
372C2×Q8—41no—
373C2×Q8—82no—
374C2×D4—21no—
375C2×D4—21no—
376C2×D4—21no—
377C2×D4—41no—
378C2×D4—611no—
379C2×D4—611no—
380C2×D4—1222no—
381C2×D4—1221no—
382C2×D4—2441no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—1221no—
386C22×C4—1221no—
387C22×C4—1221no—
388C22×C4—1221no—
389C22×C4—1221no—
390C22×C4—2442no—
391C22×C4—2442no—
392C22×C4D4×D6111yescharacteristic
393C22×C4#1175822411yescharacteristic
394C22×C4C2×C22⋊D6111yescharacteristic
395D8—2442no—
396D8—2442no—
397M16—22no—
398M16—22no—
399M16—2444no—
400M16—2442no—
401C2×C8—1221no—
402C2×C8—1221no—
403C2×C8—1222no—
404C2×C8—1222no—
405C2×C8—2444no—
406C2×C8—2442no—
407C2×C8C2×C22⋊D6112yes—
408C2×C8C2×C22⋊D6112yes—
409C4⋊C4—22no—
410C4⋊C4—21no—
411C4⋊C4—22no—
412C4⋊C4—21no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—611no—
416C4⋊C4—611no—
417C4⋊C4—1221no—
418C22⋊C4—41no—
419C22⋊C4—41no—
420C22⋊C4—1221no—
421C22⋊C4—1221no—
422C22⋊C4—1221no—
423C22⋊C4—1221no—
424C42D4×D6111yescharacteristic
425C42#1175822411yescharacteristic
426C42C2×C22⋊D6111yescharacteristic
427C42C23×D6111yescharacteristic
428C23—41no—
429C23—1221no—
430C23—1221no—
431C23—4881no—
432C23—2442no—
433C23—2442no—
434C23#1451974811yescharacteristic
435Q8—41no—
436Q8—41no—
437Q8—82no—
438Q8—82no—
439D4—41no—
440D4—41no—
441D4—41no—
442D4—41no—
443D4—81no—
444D4—82no—
445D4—612no—
446D4—1221no—
447D4—1221no—
448D4—1221no—
449D4—2442no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—1221no—
458C2×C4—1221no—
459C2×C4—2442no—
460C2×C4—2442no—
461C2×C4—2442no—
462C2×C4—2441no—
463C2×C4—2441no—
464C2×C4—2441no—
465C2×C4—2441no—
466C2×C4#1451974811yescharacteristic
467C2×C4#1451974811yescharacteristic
468C2×C4#1451974811yescharacteristic
469C2×C4#1451979712yes—
470C2×C4#1451980212yes—
471C2×C4C2×D4×D6111yescharacteristic
472C2×C4C22×C22⋊D6111yescharacteristic
473C2×C4C2×#1175822411yescharacteristic
474C8—22no—
475C8—22no—
476C8—2442no—
477C8—2441no—
478C8—2442no—
479C8—2441no—
480C22—21no—
481C22—21no—
482C22—81no—
483C22—81no—
484C22—81no—
485C22—1222no—
486C22—4884no—
487C22—2441no—
488C22—2441no—
489C22—2441no—
490C22—2441no—
491C22—2441no—
492C22#1683997412yes—
493C22C2×#1451974811yescharacteristic
494C4—21no—
495C4—21no—
496C4—21no—
497C4—81no—
498C4—82no—
499C4—2441no—
500C4—2441no—
501C4#1683997412yes—
502C4C2×#1451979711yescharacteristic
503C4C2×#1451980211yescharacteristic
504C4#137×D6111yescharacteristic
505C4#1684056111yescharacteristic
506C2—22no—
507C2—161no—
508C2—4881no—
509C2—2442no—
510C2C2×#1683997411yescharacteristic
511C2#2145544311yescharacteristic
512C2#2145554311yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3141C2
socleC22×C61C2×#121
prime radical#21455815C2
hypercenter#3141D61
hypocenterC61#19168
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3141

Complements

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

NH ≅ G/Ntype
2. C2×#16840049C2semidirect
3. M16×D244C2semidirect
4. #21445391C2semidirect
5. #21445349C2semidirect
6. #21445267C2semidirect
7. #21445225C2semidirect
8. #21444571C2semidirect
9. #21444500C2semidirect
10. #21444496C2semidirect
11. C61×#3141C2semidirect
35. #16839515C22semidirect
36. #16839513C22semidirect
37. #16839511C22semidirect
38. #16839511C22semidirect
39. #16839508C22semidirect
40. #16839506C22semidirect
44. C61×#487C22semidirect
45. C61×#486C22semidirect
250. #3141D61semidirect
273. #487C2×D61semidirect
274. #486C2×D61semidirect
305. C61#19168semidirect

Maximal subgroups

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

subgroupindex
#214558152
C2×#168400492
M16×D2442
#214453912
#214453492
#214452672
#214452252
#214445712
#214445002
#214444962
C61×#31412
#1916861