Finite Group Databasebeta

← #29658498

Subgroups — #29658498

Order 15616.

1020 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840469C211yesmaximal, characteristic, complemented
2#21456741C211yesmaximal, characteristic, complemented
3#21455846C211yesmaximal, characteristic, complemented
4#21445024C214yesmaximal, complemented
5#21444889C214yesmaximal, complemented
6#21444501C211yesmaximal, characteristic, complemented
7#21444247C211yesmaximal, characteristic, complemented
8#21444210C211yesmaximal, characteristic, complemented
9C61×#3036C211yesmaximal, characteristic
10#16840469—22no—
11#16840469—22no—
12#16840469—22no—
13#16840469C2214yes—
14C2×#14519777C2211yescharacteristic
15C2×#14519768—21no—
16C2×#14519756—21no—
17C2×#14519754—22no—
18C2×C4×Dic122C2211yescharacteristic
19C2×D4×Dic61—21no—
20C22×C4⋊Dic61—21no—
21#16840134C2212yescomplemented
22#16840114C2212yescomplemented
23#16840078—21no—
24#16840044C2212yescomplemented
25#16839967—22no—
26#16839959C2212yescomplemented
27#16839900—22no—
28#16839900—21no—
29C4×#11758200—22no—
30#16839815C2211yescharacteristic
31#16839815C2211yescharacteristic
32#16839510C2214yescomplemented
33#16839492C2214yescomplemented
34D4⋊#92418C2214yes—
35C2×AGL(1,ℤ/8)×C61—21no—
36C61×#532C2211yescharacteristic
37C2×C4×D4×C61C2211yescharacteristic
38C61×#463—22no—
39C61×#461C2212yes—
40C61×#459C2212yes—
41C61×#423C2211yescharacteristic
42C61×#417—21no—
43C22×Dic122D411yescharacteristic
44C22×Dic122D412yes—
45C23×Dic61—22no—
46C2×#11758200—22no—
47C2×#11758200—24no—
48#14519777—21no—
49#14519777—21no—
50#14519777D412yes—
51#14519777C2312yes—
52#14519768—24no—
53#14519768—24no—
54#14519756—24no—
55#14519756—24no—
56#14519754—28no—
57#14519754—28no—
58C4×Dic122—21no—
59C4×Dic122—21no—
60C4×Dic122D412yes—
61C4×Dic122C2312yes—
62D4×Dic61—24no—
63D4×Dic61—24no—
64C2×C4⋊Dic61—24no—
65C2×C4⋊Dic61—22no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—24no—
69C2×C4⋊Dic61D411yescharacteristic
70C2×#11758196—22no—
71C2×#11758196—21no—
72C2×C22⋊Dic61—22no—
73C2×C4×Dic61—21no—
74#14519708—22no—
75C4⋊#92418—21no—
76C4⋊#92418C2312yes—
77#14519700—24no—
78#14519700—22no—
79#14519698—22no—
80C22⋊#92418—21no—
81C22⋊#92418—21no—
82C4×#92418—21no—
83C4×#92418C2312yes—
84C2×#41×C61—21no—
85C22×D4×C61D411yescharacteristic
86C23×C4×C61—21no—
87AGL(1,ℤ/8)×C61—44no—
88C2×SD16×C61—22no—
89C2×D8×C61—22no—
90C2×M16×C61—21no—
91C4⋊Q8×C61C2311yescharacteristic
92#127×C61C2311yescharacteristic
93#124×C61—21no—
94#121×C61—21no—
95#121×C61—21no—
96C4×D4×C61—21no—
97C4×D4×C61—21no—
98C4×D4×C61—21no—
99C4×D4×C61—21no—
100C4×D4×C61D412yes—
101C4×D4×C61C2312yes—
102C2×C4⋊C4×C61D411yescharacteristic
103C2×C22⋊C4×C61—21no—
104C2×C42×C61C2311yescharacteristic
105C4⋊C8×C61—21no—
106C4⋊C8×C61C2312yes—
107#103×C61—22no—
108#102×C61—22no—
109C24×C61—21no—
110#41×C61—42no—
111#41×C61—42no—
112C2×Q8×C61—21no—
113C2×D4×C61—21no—
114C2×D4×C61—22no—
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61—41no—
119C2×D4×C61SD1614yescomplemented
120C2×D4×C61C2×D412yes—
121C22×C4×C61—21no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—44no—
125C22×C4×C61—41no—
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×Dic61—21no—
130C22×Dic61—22no—
131C22×Dic61—21no—
132C22×Dic61—28no—
133C22×Dic61—84no—
134C2×Dic122—21no—
135C2×Dic122—22no—
136C2×Dic122—42no—
137C2×Dic122—82no—
138C2×Dic122C2×D412yes—
139C2×Dic122C2×D414yes—
140SD16×C61—44no—
141D8×C61—44no—
142M16×C61—42no—
143C2×C8×C61—22no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61C2×D412yes—
148C22⋊C4×C61—22no—
149C22⋊C4×C61—22no—
150C22⋊C4×C61—41no—
151C22⋊C4×C61—41no—
152C42×C61C2×D411yescharacteristic
153C42×C61C2×D411yescharacteristic
154C42×C61C2×D411yescharacteristic
155C42×C61C2411yescharacteristic
156C22⋊Dic61—28no—
157#11758200—44no—
158#11758200—84no—
159C4⋊Dic61—28no—
160C4⋊Dic61—21no—
161C4⋊Dic61—22no—
162C4⋊Dic61—22no—
163C4⋊Dic61—22no—
164C4⋊Dic61—22no—
165C4⋊Dic61C2×D412yes—
166#11758196—28no—
167#11758196—22no—
168#11758196—22no—
169C4×Dic61—22no—
170C4×Dic61—22no—
171C2×#92418—22no—
172C2×#92418—22no—
173C23×C61—22no—
174C23×C61—24no—
175C23×C61—42no—
176C23×C61—42no—
177C23×C61—41no—
178C23×C61#12011yescharacteristic
179Q8×C61—41no—
180Q8×C61—41no—
181D4×C61—22no—
182D4×C61—24no—
183D4×C61—42no—
184D4×C61—41no—
185D4×C61—41no—
186D4×C61—42no—
187D4×C61—82no—
188D4×C61C2×SD1614yes—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—41no—
192C2×C4×C61—44no—
193C2×C4×C61—42no—
194C2×C4×C61—44no—
195C2×C4×C61—41no—
196C2×C4×C61—82no—
197C2×C4×C61#12011yescharacteristic
198C2×C4×C61#12011yescharacteristic
199C2×C4×C61#12011yescharacteristic
200C2×C4×C61C2×SD1612yes—
201C2×C4×C61#13712yes—
202C2×C4×C61C22×D411yescharacteristic
203C2×C4×C61C22×D411yescharacteristic
204C2×C4×C61C22×D411yescharacteristic
205C2×Dic61—22no—
206C2×Dic61—22no—
207C2×Dic61—24no—
208C2×Dic61—42no—
209C2×Dic61—44no—
210C2×Dic61—42no—
211C2×Dic61—88no—
212Dic122—22no—
213Dic122—22no—
214Dic122—42no—
215Dic122—44no—
216Dic122—82no—
217C8×C61—42no—
218#92418—42no—
219#92418—42no—
220#18748—611nomaximal, Sylow
221C22×C61—21no—
222C22×C61—21no—
223C22×C61—24no—
224C22×C61—48no—
225C22×C61—42no—
226C22×C61—42no—
227C22×C61—81no—
228C22×C61—81no—
229C22×C61—81no—
230C22×C61#44812yes—
231C22×C61C2×#12011yescharacteristic
232C4×C61—21no—
233C4×C61—21no—
234C4×C61—21no—
235C4×C61—42no—
236C4×C61—81no—
237C4×C61#44812yes—
238C4×C61C22×SD1611yescharacteristic
239C4×C61C2×#13711yescharacteristic
240C4×C61#57512yes—
241Dic61—42no—
242Dic61—44no—
243Dic61—42no—
244C2×#554—611no—
245#3206—611no—
246#3103—611no—
247#3036D6111yescharacteristic
248#1643—614no—
249#1622—614no—
250#1548—611no—
251#1487—611no—
252#1478—611no—
253C2×C61—22no—
254C2×C61—44no—
255C2×C61—161no—
256C2×C61C2×#44811yescharacteristic
257C2×C61#300511yescharacteristic
258C2×C61#301911yescharacteristic
259C2×AGL(1,ℤ/8)—21no—
260#554—614no—
261#554—1222no—
262#554—1222no—
263#554—1222no—
264#532C2×D6111yescharacteristic
265C2×C4⋊Q8—611no—
266C2×#125—1221no—
267C2×C22⋊Q8—1222no—
268C2×C22⋊Q8—1221no—
269C2×C4×Q8—611no—
270C2×C4×D4—1221no—
271C2×C4×D4C2×D6111yescharacteristic
272C22×C4⋊C4—1221no—
273#492—612no—
274#487—612no—
275#481—1221no—
276#475—612no—
277#463—22no—
278#461—1222no—
279#461C2×D6112yes—
280#460—612no—
281#459C2×D6112yes—
282#448—1221no—
283#448—1222no—
284C4×SD16—1222no—
285#431—611no—
286#431—611no—
287#423C2×D6111yescharacteristic
288#417—21no—
289#333—614no—
290Q8⋊C8—614no—
291D4⋊C8—614no—
292C61#1874811yescharacteristic, complemented, Sylow
293C2×#41—21no—
294C22×Q8—612no—
295C22×Q8—611no—
296C22×D4C22⋊D6111yescharacteristic
297C23×C4—21no—
298C23×C4—1222no—
299AGL(1,ℤ/8)—44no—
300C2×SD16—22no—
301C2×SD16—1224no—
302C2×SD16—1222no—
303C2×D8—22no—
304C2×M16—21no—
305C4⋊Q8—612no—
306C4⋊Q8—612no—
307C4⋊Q8—1221no—
308C4⋊Q8—1221no—
309C4⋊Q8C22×D6111yescharacteristic
310#127C22×D6111yescharacteristic
311#125—1224no—
312#125—1224no—
313#124—21no—
314C22⋊Q8—1228no—
315C22⋊Q8—1228no—
316C22⋊Q8—1224no—
317C22⋊Q8—1224no—
318#121—21no—
319#121—21no—
320C4×Q8—612no—
321C4×Q8—612no—
322C4×Q8—1221no—
323C4×Q8—1221no—
324C4×D4—21no—
325C4×D4—21no—
326C4×D4—21no—
327C4×D4—21no—
328C4×D4—1224no—
329C4×D4—1224no—
330C4×D4C22⋊D6112yes—
331C4×D4C22×D6112yes—
332C2×C4⋊C4—611no—
333C2×C4⋊C4—1224no—
334C2×C4⋊C4—1222no—
335C2×C4⋊C4—1224no—
336C2×C4⋊C4—1222no—
337C2×C4⋊C4—1221no—
338C2×C4⋊C4—1221no—
339C2×C4⋊C4—1221no—
340C2×C4⋊C4C22⋊D6111yescharacteristic
341C2×C22⋊C4—21no—
342C2×C22⋊C4—1222no—
343C2×C42—1221no—
344C2×C42C22×D6111yescharacteristic
345#106—1222no—
346C4⋊C8—21no—
347C4⋊C8—612no—
348C4⋊C8—1221no—
349C4⋊C8C22×D6112yes—
350#103—22no—
351#103—1222no—
352#103—1224no—
353#102—22no—
354#102—1222no—
355C22⋊C8—1221no—
356C22⋊C8—1221no—
357C4×C8—612no—
358C4×C8—1221no—
359C24—21no—
360#41—42no—
361#41—42no—
362C2×Q8—21no—
363C2×Q8—612no—
364C2×Q8—614no—
365C2×Q8—1222no—
366C2×Q8—1221no—
367C2×Q8—4882no—
368C2×Q8—2442no—
369C2×D4—21no—
370C2×D4—21no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—22no—
374C2×D4—41no—
375C2×D4#1175820014yescomplemented
376C2×D4C2×C22⋊D6112yes—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—44no—
381C22×C4—41no—
382C22×C4—1228no—
383C22×C4—1221no—
384C22×C4—1221no—
385C22×C4—1222no—
386C22×C4—4884no—
387C22×C4D4×D6111yescharacteristic
388C22×C4D4×D6111yescharacteristic
389C22×C4C2×C22⋊D6111yescharacteristic
390SD16—44no—
391SD16—4884no—
392SD16—2444no—
393D8—44no—
394M16—42no—
395C2×C8—22no—
396C2×C8—1222no—
397C2×C8—1222no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—612no—
402C4⋊C4—1228no—
403C4⋊C4—1228no—
404C4⋊C4—1222no—
405C4⋊C4—1222no—
406C4⋊C4—1222no—
407C4⋊C4—1221no—
408C4⋊C4—1222no—
409C4⋊C4—1222no—
410C4⋊C4—1222no—
411C4⋊C4C2×C22⋊D6112yes—
412C22⋊C4—22no—
413C22⋊C4—22no—
414C22⋊C4—41no—
415C22⋊C4—41no—
416C22⋊C4—1228no—
417C42—1222no—
418C42—1222no—
419C42D4×D6111yescharacteristic
420C42D4×D6111yescharacteristic
421C42C2×C22⋊D6111yescharacteristic
422C42C23×D6111yescharacteristic
423C23—22no—
424C23—24no—
425C23—42no—
426C23—42no—
427C23—41no—
428C23#1451974511yescharacteristic
429Q8—41no—
430Q8—41no—
431Q8—1222no—
432Q8—1222no—
433Q8—4882no—
434Q8—2444no—
435Q8—2442no—
436D4—22no—
437D4—24no—
438D4—42no—
439D4—41no—
440D4—41no—
441D4—42no—
442D4—82no—
443D4C2×#1175820014yes—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—41no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—41no—
451C2×C4—82no—
452C2×C4—1222no—
453C2×C4—1222no—
454C2×C4—1224no—
455C2×C4—4888no—
456C2×C4—2442no—
457C2×C4—2444no—
458C2×C4—2442no—
459C2×C4#1451974511yescharacteristic
460C2×C4#1451974511yescharacteristic
461C2×C4#1451974511yescharacteristic
462C2×C4C2×#1175820012yes—
463C2×C4#1451980712yes—
464C2×C4C2×D4×D6111yescharacteristic
465C2×C4C2×D4×D6111yescharacteristic
466C2×C4C22×C22⋊D6111yescharacteristic
467C8—42no—
468C8—2442no—
469C8—2442no—
470C22—21no—
471C22—21no—
472C22—24no—
473C22—48no—
474C22—42no—
475C22—42no—
476C22—81no—
477C22—81no—
478C22—81no—
479C22#1683990012yes—
480C22C2×#1451974511yescharacteristic
481C4—21no—
482C4—21no—
483C4—21no—
484C4—42no—
485C4—81no—
486C4—2442no—
487C4—2444no—
488C4—2442no—
489C4#1683990012yes—
490C4C22×#1175820011yescharacteristic
491C4C2×#1451980711yescharacteristic
492C4#1684055812yes—
493C2—22no—
494C2—44no—
495C2—161no—
496C2C2×#1683990011yescharacteristic
497C2#2145505811yescharacteristic
498C2#2145517411yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3036C2
socleC22×C61C2×#120
hypercenter#3036D61
hypocenterC61#18748
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3036

Complements

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

NH ≅ G/Ntype
1. C2×#16840469C2semidirect
2. #21456741C2semidirect
3. #21455846C2semidirect
4. #21445024C2semidirect
5. #21444889C2semidirect
6. #21444501C2semidirect
7. #21444247C2semidirect
8. #21444210C2semidirect
21. #16840134C22semidirect
22. #16840114C22semidirect
24. #16840044C22semidirect
26. #16839959C22semidirect
32. #16839510C22semidirect
33. #16839492C22semidirect
119. C2×D4×C61SD16semidirect
292. C61#18748semidirect
375. C2×D4#11758200semidirect

Maximal subgroups

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

subgroupindex
C2×#168404692
#214567412
#214558462
#214450242
#214448892
#214445012
#214442472
#214442102
C61×#30362
#1874861