Finite Group Databasebeta

← #29658534

Subgroups — #29658534

Order 15616.

1020 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840472C211yesmaximal, characteristic, complemented
2#21456741C211yesmaximal, characteristic, complemented
3#21455815C211yesmaximal, characteristic
4#21445031C214yesmaximal, complemented
5#21444890C214yesmaximal, complemented
6#21444501C211yesmaximal, characteristic, complemented
7#21444243C211yesmaximal, characteristic, complemented
8#21444214C211yesmaximal, characteristic, complemented
9C61×#3037C211yesmaximal, characteristic, complemented
10#16840472—22no—
11#16840472—22no—
12#16840472—22no—
13#16840472C2214yes—
14C2×#14519777C2211yescharacteristic
15C2×#14519769—21no—
16C2×#14519755—22no—
17C2×#14519753—21no—
18C2×C4×D244—21no—
19C2×C4×D244C2211yescharacteristic
20C2×C4⋊C4×D61—21no—
21#16840134C2212yescomplemented
22#16840114C2212yescomplemented
23#16840073—21no—
24#16840039C2212yes—
25#16839967—22no—
26#16839954C2212yes—
27#16839900—22no—
28#16839895—21no—
29C4×#11758199—22no—
30#16839815C2211yescharacteristic
31#16839815C2211yescharacteristic
32#16839510C2214yescomplemented
33Q8⋊#92418C2214yescomplemented
34#16839489C2214yes—
35C2×#137×C61—21no—
36C61×#532C2211yescharacteristic
37C2×C4×Q8×C61C2211yescharacteristic
38C61×#464—22no—
39C61×#462C2212yescomplemented
40C61×#460C2212yescomplemented
41C61×#423C2211yescharacteristic
42C61×#417—21no—
43C22×Dic122D412yes—
44C22×D244D411yescharacteristic
45C22×C4×D61—21no—
46C22×C4×D61—22no—
47C2×#11758200—24no—
48C2×#11758199—22no—
49#14519777—21no—
50#14519777—21no—
51#14519777D412yes—
52#14519777C2312yes—
53#14519769—24no—
54#14519769—24no—
55#14519755—28no—
56#14519755—28no—
57#14519753—24no—
58#14519753—24no—
59C4×D244—24no—
60C4×D244—24no—
61C4×D244—21no—
62C4×D244—21no—
63C4×D244D412yes—
64C4×D244C2312yes—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61D411yescharacteristic
68C2×#11758196—22no—
69C2×#11758196—22no—
70C4⋊C4×D61—24no—
71C4⋊C4×D61—24no—
72C2×#11758198—22no—
73C2×#11758198—21no—
74#14519710—22no—
75C4⋊#92418—21no—
76C4⋊#92418C2312yes—
77#14519700—24no—
78#14519698—22no—
79#14519697—22no—
80C22⋊#92418—21no—
81C22⋊#92418—21no—
82C4×#92418—21no—
83C4×#92418C2312yes—
84C2×#41×C61—21no—
85C22×Q8×C61D411yescharacteristic
86#137×C61—44no—
87C2×Q16×C61—22no—
88C2×SD16×C61—22no—
89C2×M16×C61—21no—
90C4⋊Q8×C61C2311yescharacteristic
91#127×C61C2311yescharacteristic
92#124×C61—21no—
93#121×C61—21no—
94#121×C61—21no—
95C4×Q8×C61—21no—
96C4×Q8×C61—21no—
97C4×Q8×C61D412yes—
98C4×Q8×C61C2312yes—
99C4×D4×C61—21no—
100C4×D4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61D411yescharacteristic
103C2×C42×C61—21no—
104C2×C42×C61C2311yescharacteristic
105C4⋊C8×C61—21no—
106C4⋊C8×C61C2312yes—
107#103×C61—22no—
108#102×C61—22no—
109C23×D61—21no—
110#41×C61—42no—
111#41×C61—42no—
112C2×Q8×C61—21no—
113C2×Q8×C61—22no—
114C2×Q8×C61—22no—
115C2×Q8×C61—21no—
116C2×Q8×C61C2×D412yes—
117C2×D4×C61—21no—
118C2×D4×C61—21no—
119C2×D4×C61—21no—
120C2×D4×C61—41no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—41no—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×Dic61—22no—
128C22×Dic61—21no—
129C2×D244—21no—
130C2×D244—24no—
131C2×D244C2×D412yes—
132C2×C4×D61—22no—
133C2×C4×D61—22no—
134C2×C4×D61—28no—
135C2×C4×D61—42no—
136C2×C4×D61—42no—
137C2×C4×D61—84no—
138C2×Dic122—22no—
139C2×Dic122—82no—
140C2×Dic122C2×D414yes—
141Q16×C61—44no—
142SD16×C61—44no—
143M16×C61—42no—
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#11758200—84no—
160#11758199—44no—
161#11758198—28no—
162#11758198—22no—
163#11758198—22no—
164C4⋊Dic61—22no—
165C4⋊Dic61—22no—
166C4⋊Dic61—21no—
167C4⋊Dic61—22no—
168C4⋊Dic61—22no—
169C4⋊Dic61C2×D412yes—
170#11758196—28no—
171#11758196—28no—
172C2×#92418—22no—
173C2×#92418—22no—
174C23×C61—41no—
175C23×C61#12011yescharacteristic
176C22×D61—24no—
177C22×D61—22no—
178C22×D61—42no—
179C22×D61—42no—
180Q8×C61—22no—
181Q8×C61—22no—
182Q8×C61—42no—
183Q8×C61—41no—
184Q8×C61—41no—
185D4×C61—42no—
186D4×C61—41no—
187D4×C61—41no—
188D4×C61—42no—
189D4×C61—82no—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—41no—
194C2×C4×C61—42no—
195C2×C4×C61—42no—
196C2×C4×C61—41no—
197C2×C4×C61—82no—
198C2×C4×C61#12011yescharacteristic
199C2×C4×C61#12011yescharacteristic
200C2×C4×C61#12011yescharacteristic
201C2×C4×C61#13512yes—
202C2×C4×C61AGL(1,ℤ/8)12yes—
203C2×C4×C61C22×D411yescharacteristic
204C2×C4×C61C22×D411yescharacteristic
205C2×C4×C61C22×D411yescharacteristic
206C2×Dic61—22no—
207C2×Dic61—24no—
208C2×Dic61—44no—
209C2×Dic61—42no—
210D244—24no—
211D244—22no—
212D244#13714yescomplemented
213C4×D61—42no—
214C4×D61—42no—
215C4×D61—44no—
216C4×D61—88no—
217Dic122—44no—
218Dic122—82no—
219C8×C61—42no—
220#92418—42no—
221#92418—42no—
222#18752—611nomaximal, Sylow
223C22×C61—21no—
224C22×C61—21no—
225C22×C61—81no—
226C22×C61—81no—
227C22×C61—81no—
228C22×C61#44912yes—
229C22×C61C2×#12011yescharacteristic
230C2×D61—24no—
231C2×D61—48no—
232C2×D61—42no—
233C2×D61—42no—
234C4×C61—21no—
235C4×C61—21no—
236C4×C61—21no—
237C4×C61—42no—
238C4×C61—42no—
239C4×C61—81no—
240C4×C61#44912yes—
241C4×C61C2×#13511yescharacteristic
242C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
243C4×C61C2×#13712yes—
244Dic61—42no—
245Dic61—44no—
246C2×#554—611no—
247#3206—611no—
248#3100—611no—
249#3037D6111yescharacteristic, complemented
250#1644—614no—
251#1622—614no—
252#1548—611no—
253#1486—611no—
254#1479—611no—
255C2×C61—22no—
256C2×C61—161no—
257C2×C61C2×#44911yescharacteristic
258C2×C61#300411yescharacteristic
259C2×C61#301911yescharacteristic
260D61—44no—
261C2×#137—21no—
262#554—614no—
263#554—1222no—
264#554—1222no—
265#554—1222no—
266#532C2×D6111yescharacteristic
267C2×C4⋊Q8—611no—
268C2×#125—1221no—
269C2×C22⋊Q8—1222no—
270C2×C22⋊Q8—1221no—
271C2×C4×Q8C2×D6111yescharacteristic
272C2×C4×D4—611no—
273C2×C4×D4—1221no—
274C22×C4⋊C4—1221no—
275#492—612no—
276#487—612no—
277#480—1221no—
278#474—612no—
279#464—22no—
280#462C2×D6112yescomplemented
281#461—1222no—
282#460C2×D6112yescomplemented
283#459—612no—
284#448—1222no—
285#447—1221no—
286C4×D8—1222no—
287#431—611no—
288#431—611no—
289#423C2×D6111yescharacteristic
290#417—21no—
291#333—614no—
292Q8⋊C8—614no—
293D4⋊C8—614no—
294C61#1875211yescharacteristic, complemented, Sylow
295C2×#41—21no—
296C22×Q8—612no—
297C22×Q8C22⋊D6111yescharacteristic
298C22×D4—611no—
299C23×C4—1222no—
300C23×C4—1221no—
301#137—44no—
302C2×Q16—22no—
303C2×SD16—22no—
304C2×SD16—1224no—
305C2×D8—1222no—
306C2×M16—21no—
307C4⋊Q8—612no—
308C4⋊Q8—612no—
309C4⋊Q8—1221no—
310C4⋊Q8—1221no—
311C4⋊Q8C22×D6111yescharacteristic
312#127C22×D6111yescharacteristic
313#125—1224no—
314#125—1224no—
315#124—21no—
316C22⋊Q8—1228no—
317C22⋊Q8—1228no—
318C22⋊Q8—1224no—
319C22⋊Q8—1224no—
320#121—21no—
321#121—21no—
322C4×Q8—21no—
323C4×Q8—21no—
324C4×Q8C22⋊D6112yes—
325C4×Q8C22×D6112yes—
326C4×D4—21no—
327C4×D4—21no—
328C4×D4—612no—
329C4×D4—612no—
330C4×D4—1224no—
331C4×D4—1224no—
332C4×D4—1221no—
333C4×D4—1221no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—611no—
336C2×C4⋊C4—1224no—
337C2×C4⋊C4—1222no—
338C2×C4⋊C4—1222no—
339C2×C4⋊C4—1221no—
340C2×C4⋊C4—1221no—
341C2×C4⋊C4—1224no—
342C2×C4⋊C4C22⋊D6111yescharacteristic
343C2×C22⋊C4—1222no—
344C2×C22⋊C4—1221no—
345C2×C42—21no—
346C2×C42C22×D6111yescharacteristic
347#107—1222no—
348C4⋊C8—21no—
349C4⋊C8—612no—
350C4⋊C8—1221no—
351C4⋊C8C22×D6112yes—
352#103—22no—
353#103—1224no—
354#102—22no—
355#102—1222no—
356#102—1222no—
357C22⋊C8—1221no—
358C22⋊C8—1221no—
359C4×C8—612no—
360C4×C8—1221no—
361C24—1221no—
362#41—42no—
363#41—42no—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—614no—
369C2×Q8—1222no—
370C2×Q8—4882no—
371C2×Q8C2×C22⋊D6112yes—
372C2×D4—21no—
373C2×D4—21no—
374C2×D4—21no—
375C2×D4—41no—
376C2×D4—612no—
377C2×D4—1224no—
378C2×D4—1221no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—41no—
382C22×C4—1228no—
383C22×C4—1221no—
384C22×C4—1222no—
385C22×C4—1222no—
386C22×C4—1222no—
387C22×C4—4884no—
388C22×C4—2442no—
389C22×C4—2442no—
390C22×C4D4×D6111yescharacteristic
391C22×C4D4×D6111yescharacteristic
392C22×C4C2×C22⋊D6111yescharacteristic
393Q16—44no—
394SD16—44no—
395SD16—4884no—
396D8—2444no—
397M16—42no—
398C2×C8—22no—
399C2×C8—1222no—
400C2×C8—1222no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—612no—
407C4⋊C4—1228no—
408C4⋊C4—1228no—
409C4⋊C4—1222no—
410C4⋊C4—1222no—
411C4⋊C4—1221no—
412C4⋊C4—1222no—
413C4⋊C4—1222no—
414C4⋊C4C2×C22⋊D6112yes—
415C22⋊C4—41no—
416C22⋊C4—41no—
417C22⋊C4—1228no—
418C22⋊C4—1222no—
419C22⋊C4—1222no—
420C42—22no—
421C42—22no—
422C42D4×D6111yescharacteristic
423C42D4×D6111yescharacteristic
424C42C2×C22⋊D6111yescharacteristic
425C42C23×D6111yescharacteristic
426C23—41no—
427C23—1224no—
428C23—1222no—
429C23—2442no—
430C23—2442no—
431C23#1451974511yescharacteristic
432Q8—22no—
433Q8—22no—
434Q8—42no—
435Q8—41no—
436Q8—41no—
437Q8—4882no—
438Q8—2444no—
439D4—42no—
440D4—41no—
441D4—41no—
442D4—42no—
443D4—82no—
444D4—614no—
445D4—1224no—
446D4—1222no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—82no—
455C2×C4—1222no—
456C2×C4—1224no—
457C2×C4—4888no—
458C2×C4—2444no—
459C2×C4—2442no—
460C2×C4—2442no—
461C2×C4—2442no—
462C2×C4—2444no—
463C2×C4#1451974511yescharacteristic
464C2×C4#1451974511yescharacteristic
465C2×C4#1451974511yescharacteristic
466C2×C4#1451979712yes—
467C2×C4#1451980312yes—
468C2×C4C2×D4×D6111yescharacteristic
469C2×C4C2×D4×D6111yescharacteristic
470C2×C4C22×C22⋊D6111yescharacteristic
471C8—42no—
472C8—2442no—
473C8—2442no—
474C22—21no—
475C22—21no—
476C22—81no—
477C22—81no—
478C22—81no—
479C22—1224no—
480C22—2448no—
481C22—2442no—
482C22—2442no—
483C22#1683990512yes—
484C22C2×#1451974511yescharacteristic
485C4—21no—
486C4—21no—
487C4—21no—
488C4—42no—
489C4—42no—
490C4—81no—
491C4—2444no—
492C4—2442no—
493C4#1683990512yes—
494C4C2×#1451979711yescharacteristic
495C4C2×#1451980311yescharacteristic
496C4#137×D6112yes—
497C2—22no—
498C2—161no—
499C2—2444no—
500C2C2×#1683990511yescharacteristic
501C2#2145505111yescharacteristic
502C2#2145517411yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3037C2
socleC22×C61C2×#120
prime radical#21455815C2
hypercenter#3037D61
hypocenterC61#18752
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3037

Complements

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

NH ≅ G/Ntype
1. C2×#16840472C2semidirect
2. #21456741C2semidirect
4. #21445031C2semidirect
5. #21444890C2semidirect
6. #21444501C2semidirect
7. #21444243C2semidirect
8. #21444214C2semidirect
9. C61×#3037C2semidirect
21. #16840134C22semidirect
22. #16840114C22semidirect
32. #16839510C22semidirect
33. Q8⋊#92418C22semidirect
39. C61×#462C22semidirect
40. C61×#460C22semidirect
212. D244#137semidirect
249. #3037D61semidirect
280. #462C2×D61semidirect
282. #460C2×D61semidirect
294. C61#18752semidirect

Maximal subgroups

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

subgroupindex
C2×#168404722
#214567412
#214558152
#214450312
#214448902
#214445012
#214442432
#214442142
C61×#30372
#1875261