Finite Group Databasebeta

← #29658694

Subgroups — #29658694

Order 15616.

1234 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840433C211yesmaximal, characteristic
2#21456757C211yesmaximal, characteristic, complemented
3#21455620C211yesmaximal, characteristic, complemented
4#21445017C214yesmaximal, complemented
5#21444945C214yesmaximal, complemented
6#21444486C211yesmaximal, characteristic, complemented
7#21444244C211yesmaximal, characteristic, complemented
8#21444213C211yesmaximal, characteristic, complemented
9C61×#3101C211yesmaximal, characteristic, complemented
10C22×#11758222—21no—
11#16840433—22no—
12#16840433—22no—
13#16840433—22no—
14#16840433C2214yes—
15C2×#14519774C2211yescharacteristic
16C2×#14519773—21no—
17C2×#14519751—21no—
18C2×#14519750—22no—
19C2×C4×Dic122C2211yescharacteristic
20C2×#14519739—21no—
21#16840136C2212yescomplemented
22#16840110C2212yescomplemented
23#16840066—22no—
24#16840054C2212yescomplemented
25#16840044C2212yescomplemented
26#16839962—22no—
27#16839912—22no—
28#16839815C2211yescharacteristic
29#16839805—21no—
30#16839786C2211yescharacteristic
31#16839770—21no—
32#16839505C2214yescomplemented
33#16839492C2214yescomplemented
34D4⋊#92418C2214yescomplemented
35C61×#533C2211yescharacteristic
36C2×C4×D4×C61C2211yescharacteristic
37C61×#483—21no—
38C61×#476C2212yescomplemented
39C61×#461C2212yescomplemented
40C61×#450—21no—
41C4×SD16×C61—22no—
42C61×#431C2211yescharacteristic
43C2×#11758222—24no—
44C2×#11758222—24no—
45C2×#11758222—44no—
46C2×#11758222—44no—
47C22×Dic122D411yescharacteristic
48C22×D244—21no—
49C22×D244—21no—
50C22×D244D412yes—
51C22×C22⋊D61—22no—
52C22×C22⋊D61—22no—
53C22×C4×D61—22no—
54C2×#11758201—24no—
55C2×#11758194—21no—
56#14519774—21no—
57#14519774—21no—
58#14519774D412yes—
59#14519774C2312yes—
60#14519773—24no—
61#14519773—24no—
62#14519751—24no—
63#14519751—24no—
64#14519750—28no—
65#14519750—28no—
66C4×Dic122—21no—
67C4×Dic122—21no—
68C4×Dic122D412yes—
69C4×Dic122C2312yes—
70#14519739—24no—
71#14519739—24no—
72C2×C4⋊Dic61D411yescharacteristic
73C2×#11758196—21no—
74C2×#11758198—22no—
75C2×C4×Dic61—21no—
76#14519710—22no—
77#14519708—22no—
78C4⋊#92418—21no—
79C4⋊#92418—21no—
80C4⋊#92418C2312yes—
81#14519700—22no—
82#14519700—22no—
83#14519697—24no—
84C22⋊#92418—21no—
85C4×#92418C2312yes—
86C22×D4×C61D411yescharacteristic
87C23×C4×C61—21no—
88C2×SD16×C61—22no—
89C4⋊Q8×C61C2311yescharacteristic
90C4⋊Q8×C61C2311yescharacteristic
91#125×C61—21no—
92C22⋊Q8×C61—21no—
93C22⋊Q8×C61—21no—
94C4×Q8×C61—21no—
95C4×Q8×C61—21no—
96C4×D4×C61—21no—
97C4×D4×C61—21no—
98C4×D4×C61D412yes—
99C4×D4×C61C2312yes—
100#117×C61—21no—
101C2×C4⋊C4×C61D411yescharacteristic
102C2×C22⋊C4×C61—21no—
103C2×C42×C61C2311yescharacteristic
104#106×C61—22no—
105C4⋊C8×C61C2312yes—
106#103×C61—22no—
107#102×C61—22no—
108C22⋊C8×C61—21no—
109C4×C8×C61—21no—
110C24×C61—21no—
111C23×D61—22no—
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C2×D4×C61—21no—
115C2×D4×C61—42no—
116C2×D4×C61C2×D412yes—
117C22×C4×C61—21no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—42no—
121C22×C4×C61—42no—
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C2×C22⋊D61—28no—
126C2×C22⋊D61—28no—
127C2×C22⋊D61—48no—
128C2×C22⋊D61—84no—
129C22×Dic61—21no—
130C22×Dic61—21no—
131#11758222—48no—
132#11758222—416no—
133#11758222—48no—
134C2×D244—22no—
135C2×D244—22no—
136C2×D244—22no—
137C2×D244—22no—
138C2×D244—22no—
139C2×D244—44no—
140C2×D244—42no—
141C2×D244—42no—
142C2×D244—82no—
143C2×D244C2×D414yes—
144C2×C4×D61—28no—
145C2×C4×D61—48no—
146C2×Dic122—21no—
147C2×Dic122—22no—
148C2×Dic122SD1614yescomplemented
149C2×Dic122C2×D412yes—
150SD16×C61—44no—
151C2×C8×C61—22no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—41no—
158C4⋊C4×C61—41no—
159C4⋊C4×C61C2×D412yes—
160C22⋊C4×C61—22no—
161C22⋊C4×C61—22no—
162C22⋊C4×C61—41no—
163C42×C61—41no—
164C42×C61C2×D411yescharacteristic
165C42×C61C2×D411yescharacteristic
166C42×C61C2×D411yescharacteristic
167C42×C61C2411yescharacteristic
168#11758201—84no—
169#11758198—28no—
170C4⋊Dic61—21no—
171C4⋊Dic61C2×D412yes—
172#11758196—22no—
173#11758196—22no—
174C4×Dic61—22no—
175C4×Dic61—22no—
176#11758194—42no—
177C2×#92418—22no—
178C2×#92418—22no—
179C23×C61—22no—
180C23×C61—42no—
181C23×C61—42no—
182C23×C61—42no—
183C23×C61#12011yescharacteristic
184C22×D61—24no—
185C22×D61—44no—
186C22×D61—44no—
187C22×D61—44no—
188Q8×C61—42no—
189Q8×C61—42no—
190D4×C61—22no—
191D4×C61—22no—
192D4×C61—42no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—41no—
196C2×C4×C61—42no—
197C2×C4×C61—42no—
198C2×C4×C61—42no—
199C2×C4×C61—44no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61#12011yescharacteristic
203C2×C4×C61#12011yescharacteristic
204C2×C4×C61#12011yescharacteristic
205C2×C4×C61C2×SD1612yes—
206C2×C4×C61AGL(1,ℤ/8)12yes—
207C2×C4×C61C22×D411yescharacteristic
208C2×C4×C61C22×D411yescharacteristic
209C2×C4×C61C22×D411yescharacteristic
210C22⋊D61—416no—
211C22⋊D61—88no—
212C2×Dic61—22no—
213C2×Dic61—24no—
214C2×Dic61—22no—
215C2×Dic61—42no—
216D244—44no—
217D244—44no—
218D244—42no—
219D244—42no—
220D244—44no—
221D244—44no—
222D244—82no—
223C4×D61—416no—
224Dic122—22no—
225Dic122—24no—
226Dic122C2×SD1614yes—
227C8×C61—42no—
228#92418—42no—
229#92418—42no—
230#18767—611nomaximal, Sylow
231C22×C61—21no—
232C22×C61—21no—
233C22×C61—42no—
234C22×C61—42no—
235C22×C61—42no—
236C22×C61—84no—
237C22×C61#45012yes—
238C22×C61C2×#12011yescharacteristic
239C2×D61—44no—
240C2×D61—44no—
241C2×D61—44no—
242C2×D61—88no—
243C4×C61—21no—
244C4×C61—21no—
245C4×C61—21no—
246C4×C61—42no—
247C4×C61—81no—
248C4×C61—81no—
249C4×C61#45012yes—
250C4×C61C22×SD1611yescharacteristic
251C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
252C4×C61#57512yes—
253Dic61—24no—
254Dic61—42no—
255C2×#550—611no—
256#3208—611no—
257#3101D6111yescharacteristic, complemented
258#3078—611no—
259#1642—614no—
260#1631—614no—
261#1545—611no—
262#1486—611no—
263#1479—611no—
264C2×C61—22no—
265C2×C61—82no—
266C2×C61C2×#45011yescharacteristic
267C2×C61#300511yescharacteristic
268C2×C61#301911yescharacteristic
269D61—84no—
270C22×#41—1221no—
271#550—614no—
272#550—1222no—
273#550—1222no—
274#550—1222no—
275#533C2×D6111yescharacteristic
276C2×#127—611no—
277C2×#127—1221no—
278C2×#121—1222no—
279C2×#121—1221no—
280C2×C4×Q8—611no—
281C2×C4×D4—1221no—
282C2×C4×D4C2×D6111yescharacteristic
283#492—612no—
284#486—612no—
285#483—21no—
286#479—1222no—
287#477—612no—
288#476C2×D6112yescomplemented
289#475—612no—
290#461C2×D6112yescomplemented
291#460—1222no—
292#450—21no—
293#450—1222no—
294C4×SD16—22no—
295#431—611no—
296#431C2×D6111yescharacteristic
297#428—1221no—
298#423—611no—
299#419—1221no—
300#331—614no—
301Q8⋊C8—614no—
302D4⋊C8—614no—
303C61#1876711yescharacteristic, complemented, Sylow
304C2×#41—1224no—
305C2×#41—1224no—
306C2×#41—2444no—
307C2×#41—2444no—
308C22×Q8—611no—
309C22×D4—612no—
310C22×D4—1222no—
311C22×D4—1222no—
312C22×D4—1221no—
313C22×D4—1221no—
314C22×D4C22⋊D6111yescharacteristic
315C23×C4—21no—
316C23×C4—1222no—
317C2×SD16—22no—
318C2×SD16—1224no—
319C2×M16—1221no—
320C4⋊Q8C22×D6111yescharacteristic
321C4⋊Q8C22×D6111yescharacteristic
322#127—612no—
323#127—612no—
324#127—1224no—
325#127—1224no—
326#127—1221no—
327#127—1221no—
328#125—21no—
329C22⋊Q8—21no—
330C22⋊Q8—21no—
331#121—1228no—
332#121—1228no—
333#121—1224no—
334#121—1224no—
335C4×Q8—21no—
336C4×Q8—21no—
337C4×Q8—612no—
338C4×Q8—612no—
339C4×Q8—1221no—
340C4×Q8—1221no—
341C4×D4—21no—
342C4×D4—21no—
343C4×D4—1224no—
344C4×D4—1224no—
345C4×D4C22⋊D6112yes—
346C4×D4C22×D6112yes—
347#117—21no—
348C2×C4⋊C4—611no—
349C2×C4⋊C4—1221no—
350C2×C4⋊C4C22⋊D6111yescharacteristic
351C2×C22⋊C4—21no—
352C2×C22⋊C4—1222no—
353C2×C42—1221no—
354C2×C42C22×D6111yescharacteristic
355#107—1222no—
356#106—22no—
357#106—1222no—
358C4⋊C8—612no—
359C4⋊C8—1221no—
360C4⋊C8—1221no—
361C4⋊C8C22×D6112yes—
362#103—22no—
363#103—1222no—
364#103—1222no—
365#102—22no—
366#102—1224no—
367C22⋊C8—21no—
368C22⋊C8—1221no—
369C4×C8—21no—
370C4×C8—612no—
371C24—21no—
372C24—1222no—
373#41—2448no—
374#41—2448no—
375#41—24416no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—612no—
379C2×Q8—614no—
380C2×Q8—1221no—
381C2×Q8—1222no—
382C2×D4—21no—
383C2×D4—42no—
384C2×D4—614no—
385C2×D4—1228no—
386C2×D4—1228no—
387C2×D4—1222no—
388C2×D4—1222no—
389C2×D4—1222no—
390C2×D4—1222no—
391C2×D4—1222no—
392C2×D4—4882no—
393C2×D4—4884no—
394C2×D4—2444no—
395C2×D4—2448no—
396C2×D4—2442no—
397C2×D4—2442no—
398C2×D4C2×C22⋊D6112yes—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—1228no—
405C22×C4—1221no—
406C22×C4—1221no—
407C22×C4—2448no—
408C22×C4D4×D6111yescharacteristic
409C22×C4D4×D6111yescharacteristic
410C22×C4C2×C22⋊D6111yescharacteristic
411SD16—44no—
412SD16—4884no—
413M16—2442no—
414C2×C8—22no—
415C2×C8—1222no—
416C2×C8—1222no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—612no—
425C4⋊C4—1222no—
426C4⋊C4—1221no—
427C4⋊C4—1222no—
428C4⋊C4C2×C22⋊D6112yes—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—41no—
432C22⋊C4—1228no—
433C42—41no—
434C42—1222no—
435C42—1222no—
436C42D4×D6111yescharacteristic
437C42D4×D6111yescharacteristic
438C42C2×C22⋊D6111yescharacteristic
439C42C23×D6111yescharacteristic
440C23—22no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—1224no—
445C23—2444no—
446C23—2444no—
447C23—2444no—
448C23#1451974511yescharacteristic
449Q8—42no—
450Q8—42no—
451Q8—614no—
452Q8—1222no—
453Q8—1224no—
454D4—22no—
455D4—22no—
456D4—42no—
457D4—4882no—
458D4—4888no—
459D4—2444no—
460D4—2444no—
461D4—24416no—
462D4—2442no—
463D4—2442no—
464D4—2444no—
465D4—2444no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—1222no—
476C2×C4—1224no—
477C2×C4—1222no—
478C2×C4—24416no—
479C2×C4—2442no—
480C2×C4#1451974511yescharacteristic
481C2×C4#1451974511yescharacteristic
482C2×C4#1451974511yescharacteristic
483C2×C4SD16×D6112yes—
484C2×C4#1451980412yes—
485C2×C4C2×D4×D6111yescharacteristic
486C2×C4C2×D4×D6111yescharacteristic
487C2×C4C22×C22⋊D6111yescharacteristic
488C8—42no—
489C8—2442no—
490C8—2442no—
491C22—21no—
492C22—21no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—84no—
497C22—4888no—
498C22—2444no—
499C22—2444no—
500C22—2444no—
501C22#1683991112yes—
502C22C2×#1451974511yescharacteristic
503C4—21no—
504C4—21no—
505C4—21no—
506C4—42no—
507C4—81no—
508C4—81no—
509C4—1224no—
510C4—2442no—
511C4#1683991112yes—
512C4C2×SD16×D6111yescharacteristic
513C4C2×#1451980411yescharacteristic
514C4#1684056012yes—
515C2—22no—
516C2—82no—
517C2—4884no—
518C2C2×#1683991111yescharacteristic
519C2#2145505911yescharacteristic
520C2#2145517411yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3101C2
socleC22×C61C2×#120
prime radicalC2×#16840433C2
hypercenter#3101D61
hypocenterC61#18767
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3101

Complements

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

NH ≅ G/Ntype
2. #21456757C2semidirect
3. #21455620C2semidirect
4. #21445017C2semidirect
5. #21444945C2semidirect
6. #21444486C2semidirect
7. #21444244C2semidirect
8. #21444213C2semidirect
9. C61×#3101C2semidirect
21. #16840136C22semidirect
22. #16840110C22semidirect
24. #16840054C22semidirect
25. #16840044C22semidirect
32. #16839505C22semidirect
33. #16839492C22semidirect
34. D4⋊#92418C22semidirect
38. C61×#476C22semidirect
39. C61×#461C22semidirect
148. C2×Dic122SD16semidirect
257. #3101D61semidirect
288. #476C2×D61semidirect
290. #461C2×D61semidirect
303. C61#18767semidirect

Maximal subgroups

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

subgroupindex
C2×#168404332
#214567572
#214556202
#214450172
#214449452
#214444862
#214442442
#214442132
C61×#31012
#1876761