Finite Group Databasebeta

← #29656159

Subgroups — #29656159

Order 15616.

1168 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×D244C211yesmaximal, characteristic, complemented
2#21456212C211yesmaximal, characteristic, complemented
3#21455906C211yesmaximal, characteristic
4#21444982C214yesmaximal, complemented
5#21444920C214yesmaximal, complemented
6#21444537C211yesmaximal, characteristic, complemented
7#21444266C211yesmaximal, characteristic, complemented
8#21444261C211yesmaximal, characteristic, complemented
9C61×#3112C211yesmaximal, characteristic, complemented
10C22×Q8×D61—21no—
11Q8×D244—22no—
12Q8×D244—22no—
13Q8×D244—22no—
14Q8×D244C2214yes—
15C2×#14519777—21no—
16C2×#14519777C2211yescharacteristic
17C2×#14519757—22no—
18C2×#14519757—21no—
19C2×C4×D244—21no—
20C2×C4×D244C2211yescharacteristic
21#16840121—22no—
22#16840112C2212yescomplemented
23#16840108C2212yescomplemented
24#16840078—21no—
25#16840049C2212yes—
26#16839954C2212yes—
27#16839895—21no—
28#16839872—22no—
29#16839819C2211yescharacteristic
30#16839763—22no—
31C4×#11758194C2211yescharacteristic
32#16839509C2214yescomplemented
33Q8⋊#92418C2214yescomplemented
34#16839489C2214yes—
35C61×#532C2211yescharacteristic
36C2×C4×Q8×C61C2211yescharacteristic
37C61×#480—21no—
38C61×#477C2212yescomplemented
39C61×#460C2212yescomplemented
40C61×#448—21no—
41C61×#440—22no—
42C61×#432C2211yescharacteristic
43C22×Dic122—22no—
44C22×Dic122—21no—
45C2×Q8×D61—24no—
46C2×Q8×D61—24no—
47C2×Q8×D61—48no—
48C22×D244D411yescharacteristic
49C22×C4×D61—22no—
50C22×C4×D61—21no—
51C2×#11758199—22no—
52C2×#11758194—21no—
53#14519777—24no—
54#14519777—21no—
55#14519777—21no—
56#14519777—24no—
57#14519777D412yes—
58#14519777C2312yes—
59#14519757—24no—
60#14519757—28no—
61#14519757—24no—
62#14519757—28no—
63C4×D244—24no—
64C4×D244—24no—
65C4×D244—21no—
66C4×D244—21no—
67C4×D244D412yes—
68C4×D244C2312yes—
69C2×C4⋊Dic61—22no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61D411yescharacteristic
72C2×C4⋊Dic61D412yes—
73C2×#11758198—22no—
74C2×#11758198—21no—
75#14519708—22no—
76C4⋊#92418C2312yes—
77#14519701—24no—
78#14519698—22no—
79#14519698—24no—
80#14519697—22no—
81C22⋊#92418—21no—
82#14519688—21no—
83#14519688—21no—
84C4×#92418C2312yes—
85C2×#41×C61—21no—
86C22×Q8×C61D411yescharacteristic
87C2×SD16×C61—22no—
88C4⋊Q8×C61C2311yescharacteristic
89#127×C61C2311yescharacteristic
90#124×C61—21no—
91#121×C61—21no—
92#121×C61—21no—
93C4×Q8×C61—21no—
94C4×Q8×C61—21no—
95C4×Q8×C61D412yes—
96C4×Q8×C61C2312yes—
97C4×D4×C61—21no—
98C4×D4×C61—21no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61D411yescharacteristic
101C2×C42×C61—21no—
102C2×C42×C61C2311yescharacteristic
103#107×C61—22no—
104C4⋊C8×C61C2312yes—
105#103×C61—22no—
106#102×C61—22no—
107C22⋊C8×C61—21no—
108#97×C61—21no—
109C23×D61—21no—
110#41×C61—44no—
111C2×Q8×C61—24no—
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C2×Q8×C61C2×D412yes—
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61—41no—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—41no—
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×Dic61—22no—
126C22×Dic61—21no—
127Q8×D61—416no—
128Q8×D61—48no—
129Q8×D61—48no—
130C2×D244—21no—
131C2×D244—24no—
132C2×D244C2×D412yes—
133C2×C4×D61—28no—
134C2×C4×D61—22no—
135C2×C4×D61—22no—
136C2×C4×D61—48no—
137C2×C4×D61—44no—
138C2×Dic122—28no—
139C2×Dic122—22no—
140C2×Dic122—22no—
141C2×Dic122—48no—
142C2×Dic122—44no—
143SD16×C61—44no—
144C2×C8×C61—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61C2×D412yes—
151C22⋊C4×C61—41no—
152C22⋊C4×C61—41no—
153C42×C61—22no—
154C42×C61—22no—
155C42×C61C2×D411yescharacteristic
156C42×C61C2×D411yescharacteristic
157C42×C61C2×D411yescharacteristic
158C42×C61C2411yescharacteristic
159#11758199—44no—
160#11758198—28no—
161#11758198—22no—
162#11758198—22no—
163C4⋊Dic61—28no—
164C4⋊Dic61—22no—
165C4⋊Dic61—21no—
166C4⋊Dic61—22no—
167C4⋊Dic61—22no—
168C4⋊Dic61C2×D414yes—
169C4⋊Dic61C2×D412yes—
170#11758194—42no—
171C2×#92418—22no—
172C2×#92418—22no—
173C23×C61—41no—
174C23×C61#12011yescharacteristic
175C22×D61—24no—
176C22×D61—22no—
177C22×D61—42no—
178C22×D61—42no—
179Q8×C61—24no—
180Q8×C61—22no—
181Q8×C61—42no—
182Q8×C61AGL(1,ℤ/8)14yes—
183D4×C61—42no—
184D4×C61—42no—
185D4×C61—42no—
186D4×C61—82no—
187C2×C4×C61—24no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—42no—
194C2×C4×C61—41no—
195C2×C4×C61—82no—
196C2×C4×C61#12011yescharacteristic
197C2×C4×C61#12011yescharacteristic
198C2×C4×C61#12011yescharacteristic
199C2×C4×C61C22×D411yescharacteristic
200C2×C4×C61C22×D411yescharacteristic
201C2×C4×C61C22×D411yescharacteristic
202C2×Dic61—22no—
203C2×Dic61—24no—
204C2×Dic61—42no—
205C2×Dic61—44no—
206D244—24no—
207D244—22no—
208D244#13714yes—
209C4×D61—416no—
210C4×D61—44no—
211C4×D61—44no—
212Dic122—416no—
213Dic122—44no—
214Dic122—44no—
215C8×C61—42no—
216#92418—42no—
217#92418—42no—
218#18534—611nomaximal, Sylow
219C22×C61—21no—
220C22×C61—21no—
221C22×C61—81no—
222C22×C61—81no—
223C22×C61—81no—
224C22×C61#45412yes—
225C22×C61C2×#12011yescharacteristic
226C2×D61—24no—
227C2×D61—48no—
228C2×D61—42no—
229C2×D61—42no—
230C4×C61—24no—
231C4×C61—21no—
232C4×C61—21no—
233C4×C61—21no—
234C4×C61—21no—
235C4×C61—42no—
236C4×C61—81no—
237C4×C61C2×AGL(1,ℤ/8)12yes—
238C4×C61C2×#13712yes—
239Dic61—42no—
240Dic61—44no—
241C2×D4×Q8—611no—
242#3141—611no—
243#3112D6111yescharacteristic, complemented
244#3109—611no—
245#1637—614no—
246#1627—614no—
247#1556—611no—
248#1491—611no—
249#1490—611no—
250C2×C61—22no—
251C2×C61—161no—
252C2×C61#300311yescharacteristic
253C2×C61#300411yescharacteristic
254C2×C61C2×#45411yescharacteristic
255D61—44no—
256C23×Q8—1221no—
257D4×Q8—614no—
258D4×Q8—1222no—
259D4×Q8—1222no—
260D4×Q8—1222no—
261#532C2×D6111yescharacteristic
262C2×C4⋊Q8—611no—
263C2×C4⋊Q8—1221no—
264C2×C22⋊Q8—1222no—
265C2×C22⋊Q8—1221no—
266C2×C4×Q8C2×D6111yescharacteristic
267C2×C4×D4—611no—
268C2×C4×D4—1221no—
269#489—1222no—
270#487—612no—
271#486—612no—
272#481—1221no—
273#480—21no—
274#477C2×D6112yescomplemented
275#476—612no—
276#460C2×D6112yescomplemented
277#459—612no—
278#448—21no—
279#447—1221no—
280#442—1222no—
281#440—22no—
282#432—611no—
283#432C2×D6111yescharacteristic
284#417—1222no—
285C4×M16—611no—
286#332—614no—
287Q8⋊C8—614no—
288D4⋊C8—614no—
289C61#1853411yescharacteristic, complemented, Sylow
290C2×#41—21no—
291C22×Q8—1222no—
292C22×Q8—1224no—
293C22×Q8—1224no—
294C22×Q8—1221no—
295C22×Q8—2448no—
296C22×Q8C22⋊D6111yescharacteristic
297C22×D4—611no—
298C23×C4—1222no—
299C23×C4—1221no—
300C2×SD16—22no—
301C2×D8—1222no—
302C2×M16—1221no—
303C4⋊Q8—612no—
304C4⋊Q8—612no—
305C4⋊Q8—1224no—
306C4⋊Q8—1224no—
307C4⋊Q8—1221no—
308C4⋊Q8—1221no—
309C4⋊Q8C22×D6111yescharacteristic
310#127C22×D6111yescharacteristic
311#124—21no—
312C22⋊Q8—1228no—
313C22⋊Q8—1224no—
314C22⋊Q8—1228no—
315C22⋊Q8—1224no—
316#121—21no—
317#121—21no—
318C4×Q8—21no—
319C4×Q8—21no—
320C4×Q8C22⋊D6112yes—
321C4×Q8C22×D6112yes—
322C4×D4—21no—
323C4×D4—21no—
324C4×D4—612no—
325C4×D4—612no—
326C4×D4—1224no—
327C4×D4—1224no—
328C4×D4—1221no—
329C4×D4—1221no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—611no—
332C2×C4⋊C4—612no—
333C2×C4⋊C4—1222no—
334C2×C4⋊C4—1221no—
335C2×C4⋊C4C22⋊D6111yescharacteristic
336C2×C22⋊C4—1222no—
337C2×C22⋊C4—1221no—
338C2×C42—21no—
339C2×C42C22×D6111yescharacteristic
340#107—22no—
341#106—1222no—
342C4⋊C8—612no—
343C4⋊C8C22×D6112yes—
344#103—22no—
345#103—1224no—
346#102—22no—
347#102—1222no—
348#102—1224no—
349#102—1222no—
350C22⋊C8—21no—
351C22⋊C8—1221no—
352#97—21no—
353#97—1221no—
354#97—1221no—
355C4×C8—612no—
356C24—1221no—
357#41—44no—
358C2×Q8—24no—
359C2×Q8—21no—
360C2×Q8—21no—
361C2×Q8—1228no—
362C2×Q8—1222no—
363C2×Q8—1222no—
364C2×Q8—2448no—
365C2×Q8—2444no—
366C2×Q8—24416no—
367C2×Q8—2448no—
368C2×Q8—2448no—
369C2×Q8C2×C22⋊D6112yes—
370C2×D4—21no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—41no—
374C2×D4—612no—
375C2×D4—1224no—
376C2×D4—1221no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—41no—
380C22×C4—1228no—
381C22×C4—1222no—
382C22×C4—1221no—
383C22×C4—1222no—
384C22×C4—1222no—
385C22×C4—2448no—
386C22×C4—2444no—
387C22×C4D4×D6111yescharacteristic
388C22×C4D4×D6111yescharacteristic
389C22×C4C2×C22⋊D6111yescharacteristic
390SD16—44no—
391D8—2444no—
392M16—2442no—
393C2×C8—22no—
394C2×C8—1222no—
395C2×C8—1222no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—614no—
402C4⋊C4—612no—
403C4⋊C4—1228no—
404C4⋊C4—1222no—
405C4⋊C4—1221no—
406C4⋊C4—1222no—
407C4⋊C4—1222no—
408C4⋊C4C2×C22⋊D6112yes—
409C22⋊C4—41no—
410C22⋊C4—41no—
411C22⋊C4—1228no—
412C22⋊C4—1222no—
413C22⋊C4—1222no—
414C42—22no—
415C42—22no—
416C42D4×D6111yescharacteristic
417C42D4×D6111yescharacteristic
418C42C2×C22⋊D6111yescharacteristic
419C42C23×D6111yescharacteristic
420C23—41no—
421C23—1224no—
422C23—1222no—
423C23—2442no—
424C23—2442no—
425C23#1451974511yescharacteristic
426Q8—24no—
427Q8—22no—
428Q8—42no—
429Q8—24416no—
430Q8—2444no—
431Q8—2444no—
432Q8#1451980214yes—
433D4—42no—
434D4—42no—
435D4—42no—
436D4—82no—
437D4—614no—
438D4—1224no—
439D4—1222no—
440C2×C4—24no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—82no—
449C2×C4—1224no—
450C2×C4—1222no—
451C2×C4—24416no—
452C2×C4—2444no—
453C2×C4—2444no—
454C2×C4—2442no—
455C2×C4—2444no—
456C2×C4#1451974511yescharacteristic
457C2×C4#1451974511yescharacteristic
458C2×C4#1451974511yescharacteristic
459C2×C4C2×D4×D6111yescharacteristic
460C2×C4C2×D4×D6111yescharacteristic
461C2×C4C22×C22⋊D6111yescharacteristic
462C8—42no—
463C8—2442no—
464C8—2442no—
465C22—21no—
466C22—21no—
467C22—81no—
468C22—81no—
469C22—81no—
470C22—1224no—
471C22—2448no—
472C22—2442no—
473C22—2442no—
474C22#1683993312yes—
475C22C2×#1451974511yescharacteristic
476C4—21no—
477C4—21no—
478C4—24no—
479C4—21no—
480C4—21no—
481C4—42no—
482C4—81no—
483C4—2444no—
484C4—2442no—
485C4C2×#1451980212yes—
486C4#137×D6112yes—
487C2—22no—
488C2—161no—
489C2—2444no—
490C2#2145504511yescharacteristic
491C2#2145505111yescharacteristic
492C2C2×#1683993311yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3112C2
socleC22×C61C2×#120
prime radical#21455906C2
hypercenter#3112D61
hypocenterC61#18534
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3112

Complements

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

NH ≅ G/Ntype
1. C2×Q8×D244C2semidirect
2. #21456212C2semidirect
4. #21444982C2semidirect
5. #21444920C2semidirect
6. #21444537C2semidirect
7. #21444266C2semidirect
8. #21444261C2semidirect
9. C61×#3112C2semidirect
22. #16840112C22semidirect
23. #16840108C22semidirect
32. #16839509C22semidirect
33. Q8⋊#92418C22semidirect
38. C61×#477C22semidirect
39. C61×#460C22semidirect
243. #3112D61semidirect
274. #477C2×D61semidirect
276. #460C2×D61semidirect
289. C61#18534semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×D2442
#214562122
#214559062
#214449822
#214449202
#214445372
#214442662
#214442612
C61×#31122
#1853461