Finite Group Databasebeta

← #29662845

Subgroups — #29662845

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#21456213C211yesmaximal, characteristic, complemented
2#21455819C211yesmaximal, characteristic
3C2×#16840052C211yesmaximal, characteristic, complemented
4#21445394C212yesmaximal, complemented
5#21445352C212yesmaximal, complemented
6#21445270C212yesmaximal, complemented
7#21445228C212yesmaximal, complemented
8#21444571C211yesmaximal, characteristic, complemented
9#21444501C211yesmaximal, characteristic, complemented
10#21444497C211yesmaximal, characteristic, complemented
11D4×M16×C61C211yesmaximal, characteristic, complemented
12#16840312C2211yescharacteristic
13C2×#14519777C2211yescharacteristic
14#16840122—22no—
15#16840113C2212yescomplemented
16#16840109C2212yescomplemented
17#16840075—21no—
18#16840052—21no—
19#16840052—21no—
20#16840052C2212yes—
21#16840052C2212yes—
22#16840042C2211yescharacteristic
23#16840042C2211yescharacteristic
24#16839957C2211yescharacteristic
25#16839957C2211yescharacteristic
26#16839896—21no—
27C4×#11758199—21no—
28C4×#11758199—21no—
29#16839815C2211yescharacteristic
30C2×#14519708—21no—
31C2×C4⋊#92418C2211yescharacteristic
32#16839764—21no—
33#16839764—21no—
34C2×#14519698—21no—
35#16839515C2212yescomplemented
36#16839513C2212yescomplemented
37#16839510C2212yescomplemented
38#16839510C2212yescomplemented
39#16839509C2212yescomplemented
40#16839505C2212yescomplemented
41C22×M16×C61—21no—
42C2×C4×D4×C61C2211yescharacteristic
43C61×#436C2212yes—
44C61×#435—22no—
45C8×D4×C61C2212yes—
46C61×#423C2211yescharacteristic
47C61×#407—21no—
48C4×M16×C61C2211yescharacteristic
49C2×#11758222—21no—
50C22×Dic122—21no—
51C2×#11758199—21no—
52C2×#11758199—21no—
53C22×#92418—21no—
54#14519777—21no—
55#14519777—21no—
56#14519777—21no—
57#14519777C2311yescharacteristic
58#14519777C2312yes—
59#14519774C2311yescharacteristic
60#14519765—21no—
61#14519751—21no—
62#14519751—21no—
63C4×D244—21no—
64C4×D244—21no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—21no—
67#14519710—21no—
68#14519710—21no—
69#14519708—22no—
70#14519708—22no—
71C4⋊#92418—21no—
72C4⋊#92418C2311yescharacteristic
73C4⋊#92418C2311yescharacteristic
74C4⋊#92418C2312yes—
75#14519702—22no—
76#14519702—22no—
77#14519699—22no—
78#14519699—22no—
79#14519698—22no—
80#14519698—22no—
81#14519698—21no—
82#14519698—21no—
83#14519697—21no—
84#14519697—21no—
85C22⋊#92418—21no—
86C4×#92418—21no—
87C22×D4×C61D411yescharacteristic
88C23×C4×C61—21no—
89C2×M16×C61—21no—
90C2×M16×C61—22no—
91C2×M16×C61—44no—
92C2×M16×C61D411yescharacteristic
93C2×M16×C61D411yescharacteristic
94C22×C8×C61—22no—
95C4×D4×C61—21no—
96C4×D4×C61—21no—
97C4×D4×C61D411yescharacteristic
98C4×D4×C61D411yescharacteristic
99C4×D4×C61C2311yescharacteristic
100C4×D4×C61C2311yescharacteristic
101C2×C4⋊C4×C61D411yescharacteristic
102C2×C22⋊C4×C61—21no—
103C2×C42×C61C2311yescharacteristic
104C4⋊C8×C61—21no—
105C4⋊C8×C61C2312yes—
106C22⋊C8×C61—22no—
107C22⋊C8×C61—22no—
108#97×C61D412yes—
109C4×C8×C61C2312yes—
110C24×C61—21no—
111C2×D4×C61—22no—
112C2×D4×C61—21no—
113C2×D4×C61—41no—
114C2×D4×C61C2×D411yescharacteristic
115C2×D4×C61C2×D411yescharacteristic
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—42no—
122C22×C4×C61—42no—
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61#4111yescharacteristic
126C2×C22⋊D61—41no—
127C22×Dic61—41no—
128#11758222—42no—
129#11758222—42no—
130C2×D244—21no—
131C2×D244—21no—
132C2×D244—21no—
133C2×C4×D61—41no—
134C2×Dic122—21no—
135C2×Dic122—22no—
136C2×Dic122—41no—
137C2×Dic122—82no—
138M16×C61—22no—
139M16×C61—22no—
140M16×C61—44no—
141M16×C61—42no—
142C2×C8×C61—22no—
143C2×C8×C61—44no—
144C2×C8×C61C2×D412yes—
145C2×C8×C61C2×D412yes—
146C4⋊C4×C61—21no—
147C4⋊C4×C61C2×D411yescharacteristic
148C4⋊C4×C61C2×D411yescharacteristic
149C22⋊C4×C61—21no—
150C22⋊C4×C61—21no—
151C22⋊C4×C61—21no—
152C22⋊C4×C61—21no—
153C42×C61C2×D411yescharacteristic
154C42×C61C2×D411yescharacteristic
155C42×C61#4111yescharacteristic
156C42×C61C2411yescharacteristic
157#11758199—42no—
158#11758199—42no—
159#11758198—41no—
160#11758198—41no—
161C4⋊Dic61—22no—
162C4⋊Dic61—21no—
163C4⋊Dic61—22no—
164C4⋊Dic61—21no—
165C4⋊Dic61—41no—
166C4⋊Dic61—41no—
167C2×#92418—21no—
168C2×#92418—21no—
169C2×#92418—22no—
170C2×#92418—42no—
171C23×C61—21no—
172C23×C61—21no—
173C23×C61—42no—
174C23×C61—42no—
175C23×C61—81no—
176C23×C61#12111yescharacteristic
177C22×D61—41no—
178D4×C61—21no—
179D4×C61—21no—
180D4×C61—21no—
181D4×C61—42no—
182D4×C61#13712yes—
183C2×C4×C61—21no—
184C2×C4×C61—21no—
185C2×C4×C61—21no—
186C2×C4×C61—21no—
187C2×C4×C61—41no—
188C2×C4×C61—42no—
189C2×C4×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61—42no—
194C2×C4×C61—42no—
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
203C22⋊D61—82no—
204C2×Dic61—42no—
205C2×Dic61—41no—
206C2×Dic61—82no—
207D244—41no—
208D244—41no—
209D244—41no—
210D244—41no—
211D244—81no—
212C4×D61—82no—
213Dic122—41no—
214Dic122—41no—
215Dic122—82no—
216Dic122—82no—
217C8×C61—22no—
218C8×C61—22no—
219C8×C61—42no—
220#92418—41no—
221#92418—42no—
222#92418—41no—
223#19168—611nomaximal, Sylow
224C22×C61—21no—
225C22×C61—21no—
226C22×C61—22no—
227C22×C61—41no—
228C22×C61—41no—
229C22×C61—41no—
230C22×C61—41no—
231C22×C61—41no—
232C22×C61—84no—
233C22×C61#46312yes—
234C22×C61C2×#12111yescharacteristic
235C2×D61—81no—
236C2×D61—81no—
237C2×D61—81no—
238C4×C61—21no—
239C4×C61—21no—
240C4×C61—21no—
241C4×C61—41no—
242C4×C61—41no—
243C4×C61#46312yes—
244C4×C61C2×#13511yescharacteristic
245C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
246C4×C61C2×#13711yescharacteristic
247C4×C61#57511yescharacteristic
248Dic61—81no—
249Dic61—82no—
250#3141—611no—
251#3100—611no—
252C2×#476—611no—
253D4×M16D6111yescharacteristic, complemented
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—42no—
263C2×C61—81no—
264C2×C61C2×#46311yescharacteristic
265C2×C61#305611yescharacteristic
266C2×C61#306811yescharacteristic
267D61—161no—
268C22×M16—21no—
269#532—611no—
270C2×C4⋊Q8—611no—
271C2×C4×D4C2×D6111yescharacteristic
272#489—1222no—
273#487—612no—
274#486—612no—
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#436C2×D6112yes—
288#435—22no—
289C8×D4C2×D6112yes—
290#431—611no—
291C2×#106—1221no—
292#423C2×D6111yescharacteristic
293C2×C4⋊C8—611no—
294#417—1221no—
295#417—1221no—
296C2×#102—1221no—
297#407—21no—
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—1221no—
307C22×Q8—1221no—
308C22×D4D24411yescharacteristic
309C23×C4—21no—
310C2×D8—1221no—
311C2×D8—1221no—
312C2×M16—21no—
313C2×M16—22no—
314C2×M16—44no—
315C2×M16C22⋊D6111yescharacteristic
316C2×M16C22⋊D6111yescharacteristic
317C22×C8—22no—
318C22×C8—1221no—
319C4⋊Q8—612no—
320C4⋊Q8—611no—
321C4⋊Q8—1221no—
322C4⋊Q8—1221no—
323C4⋊Q8—1221no—
324#127—611no—
325#124—1221no—
326#121—1221no—
327#121—1221no—
328C4×D4—21no—
329C4×D4—21no—
330C4×D4—1221no—
331C4×D4—1221no—
332C4×D4D24411yescharacteristic
333C4×D4D24411yescharacteristic
334C4×D4C22×D6111yescharacteristic
335C4×D4C22×D6111yescharacteristic
336C2×C4⋊C4—1221no—
337C2×C4⋊C4—1221no—
338C2×C4⋊C4D24411yescharacteristic
339C2×C22⋊C4—21no—
340C2×C42C22×D6111yescharacteristic
341#107—1221no—
342#107—1221no—
343#106—1222no—
344#106—1222no—
345C4⋊C8—21no—
346C4⋊C8—612no—
347C4⋊C8—611no—
348C4⋊C8—611no—
349C4⋊C8—1221no—
350C4⋊C8C22×D6112yes—
351#103—1222no—
352#103—1222no—
353#102—1222no—
354#102—1222no—
355#102—1221no—
356#102—1221no—
357#102—1222no—
358#102—1222no—
359#102—1221no—
360#102—1221no—
361C22⋊C8—22no—
362C22⋊C8—22no—
363C22⋊C8—1221no—
364#97C22⋊D6112yes—
365C4×C8—1221no—
366C4×C8C22×D6112yes—
367C24—21no—
368#41—2442no—
369#41—2442no—
370C2×Q8—1221no—
371C2×Q8—1222no—
372C2×Q8—4882no—
373C2×Q8—2441no—
374C2×D4—22no—
375C2×D4—21no—
376C2×D4—41no—
377C2×D4—1221no—
378C2×D4—1221no—
379C2×D4—1221no—
380C2×D4—2441no—
381C2×D4C2×D24411yescharacteristic
382C2×D4C2×D24411yescharacteristic
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—2441no—
391C22×C4—2441no—
392C22×C4C2×D24411yescharacteristic
393C22×C4#1175822211yescharacteristic
394C22×C4C2×C22⋊D6111yescharacteristic
395D8—2442no—
396D8—2442no—
397M16—22no—
398M16—22no—
399M16—44no—
400M16—42no—
401C2×C8—22no—
402C2×C8—44no—
403C2×C8—1221no—
404C2×C8—1221no—
405C2×C8—1222no—
406C2×C8—2442no—
407C2×C8C2×C22⋊D6112yes—
408C2×C8C2×C22⋊D6112yes—
409C4⋊C4—21no—
410C4⋊C4—1222no—
411C4⋊C4—1222no—
412C4⋊C4—1221no—
413C4⋊C4—1221no—
414C4⋊C4—2441no—
415C4⋊C4—2441no—
416C4⋊C4C2×D24411yescharacteristic
417C4⋊C4C2×D24411yescharacteristic
418C22⋊C4—21no—
419C22⋊C4—21no—
420C22⋊C4—21no—
421C22⋊C4—21no—
422C22⋊C4—2441no—
423C22⋊C4—2441no—
424C42C2×D24411yescharacteristic
425C42#1175822211yescharacteristic
426C42C2×C22⋊D6111yescharacteristic
427C42C23×D6111yescharacteristic
428C23—21no—
429C23—21no—
430C23—42no—
431C23—42no—
432C23—81no—
433C23—2441no—
434C23#1451975111yescharacteristic
435Q8—4882no—
436Q8—4882no—
437Q8—2441no—
438Q8—2441no—
439D4—21no—
440D4—21no—
441D4—21no—
442D4—42no—
443D4—4881no—
444D4—4882no—
445D4—2441no—
446D4—2441no—
447D4—2441no—
448D4—2441no—
449D4#1451981012yes—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—4882no—
463C2×C4—4882no—
464C2×C4—2441no—
465C2×C4—2442no—
466C2×C4#1451975111yescharacteristic
467C2×C4#1451975111yescharacteristic
468C2×C4#1451975111yescharacteristic
469C2×C4#1451979712yes—
470C2×C4#1451980312yes—
471C2×C4C22×C22⋊D6111yescharacteristic
472C2×C4C22×D24411yescharacteristic
473C2×C4C2×#1175822211yescharacteristic
474C8—22no—
475C8—22no—
476C8—42no—
477C8—2441no—
478C8—2442no—
479C8—2441no—
480C22—21no—
481C22—21no—
482C22—22no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—84no—
489C22—4881no—
490C22—4881no—
491C22—4881no—
492C22#1683997712yes—
493C22C2×#1451975111yescharacteristic
494C4—21no—
495C4—21no—
496C4—21no—
497C4—41no—
498C4—41no—
499C4—4881no—
500C4—4882no—
501C4#1683997712yes—
502C4C2×#1451979711yescharacteristic
503C4C2×#1451980311yescharacteristic
504C4C2×#1451981011yescharacteristic
505C4#1684056211yescharacteristic
506C2—22no—
507C2—42no—
508C2—81no—
509C2—9761no—
510C2C2×#1683997711yescharacteristic
511C2#2145544911yescharacteristic
512C2#2145554711yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingD4×M16×C61C2
socleC22×C61C2×#121
prime radical#21455819C2
hypercenterD4×M16D61
hypocenterC61#19168
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. #21456213C2semidirect
3. C2×#16840052C2semidirect
4. #21445394C2semidirect
5. #21445352C2semidirect
6. #21445270C2semidirect
7. #21445228C2semidirect
8. #21444571C2semidirect
9. #21444501C2semidirect
10. #21444497C2semidirect
11. D4×M16×C61C2semidirect
15. #16840113C22semidirect
16. #16840109C22semidirect
35. #16839515C22semidirect
36. #16839513C22semidirect
37. #16839510C22semidirect
38. #16839510C22semidirect
39. #16839509C22semidirect
40. #16839505C22semidirect
253. D4×M16D61semidirect
305. C61#19168semidirect

Maximal subgroups

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

subgroupindex
#214562132
#214558192
C2×#168400522
#214453942
#214453522
#214452702
#214452282
#214445712
#214445012
#214444972
D4×M16×C612
#1916861