Finite Group Databasebeta

← #29647758

Subgroups — #29647758

Order 15616.

1202 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459585C211yesmaximal, characteristic
2#21454899C211yesmaximal, characteristic, complemented
3#21448739C211yesmaximal, characteristic, complemented
4#21444380C211yesmaximal, characteristic, complemented
5#21444379C211yesmaximal, characteristic, complemented
6#21444377C214yesmaximal, complemented
7#21444313C211yesmaximal, characteristic, complemented
8#21444312C214yesmaximal, complemented
9C61×#1502C211yesmaximal, characteristic, complemented
10C2×#14519824—21no—
11#16840433C412yescomplemented
12#16840432C412yescomplemented
13D4×Dic122C412yescomplemented
14#16840427C412yescomplemented
15#16840421—22no—
16#16840420—22no—
17#16840340—21no—
18#16840339—22no—
19#16840312C2211yescharacteristic
20#16840307C2214yes—
21C2×C4×Dic122—21no—
22C2×C4×Dic122C2211yescharacteristic
23C8×Dic122—22no—
24#16839886—22no—
25#16839815C2211yescharacteristic
26#16839812C2214yes—
27#16839660—21no—
28#16839657—22no—
29#16839628C2211yescharacteristic
30#16839626C2212yes—
31#16839613C2212yes—
32#16839497C2214yes—
33#16839496C2214yes—
34#16839495C2212yes—
35#16839495C2212yes—
36C61×#532C2211yescharacteristic
37C61×#431C2211yescharacteristic
38C4×C4⋊C4×C61C2211yescharacteristic
39C61×#327C2212yes—
40C61×#327C2212yes—
41#14519824—44no—
42#14519824—44no—
43C2×#11758222—21no—
44C2×#11758222—22no—
45C2×#11758224—24no—
46C2×#11758224—22no—
47C22×Dic122—22no—
48C22×Dic122D411yescharacteristic
49C2×Q8×D61—22no—
50#14519777C2×C411yescharacteristic
51#14519776C2×C414yes—
52#14519774C2×C411yescharacteristic
53#14519773C2×C414yes—
54#14519765—21no—
55#14519763—24no—
56#14519756—22no—
57#14519755—24no—
58#14519754—24no—
59#14519753—22no—
60#14519751—21no—
61#14519751—22no—
62#14519751C2312yes—
63#14519750—24no—
64#14519749—24no—
65#14519749—24no—
66#14519748—22no—
67C4×Dic122—44no—
68C4×Dic122C2×C412yes—
69C4×Dic122C2×C412yes—
70C4×Dic122D412yes—
71C4×Dic122D412yes—
72C4×D244—21no—
73C4×D244C2×C412yes—
74D4×Dic61C2×C414yes—
75#14519739C2×C414yes—
76#14519737—24no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61D411yescharacteristic
79C2×#11758196—22no—
80C2×#11758196—22no—
81C2×#11758196D412yes—
82C2×C4×Dic61—22no—
83C2×C4×Dic61C2312yes—
84C4⋊#92418—21no—
85#14519706—24no—
86C22⋊#92418C2312yes—
87C4×#92418—21no—
88C8×Dic61—24no—
89#14519684—21no—
90#14519684—22no—
91#14519684C2312yes—
92C2×#41×C61—21no—
93C4⋊Q8×C61C2×C411yescharacteristic
94#127×C61C2×C411yescharacteristic
95#124×C61—21no—
96#121×C61—21no—
97#121×C61C2312yes—
98C4×D4×C61—21no—
99C4×D4×C61C2×C412yes—
100C2×C4⋊C4×C61—21no—
101C2×C42×C61—21no—
102C2×C42×C61C2311yescharacteristic
103C4⋊C8×C61—21no—
104C22⋊C8×C61C2312yes—
105C4×C8×C61—21no—
106#95×C61C2311yescharacteristic
107#95×C61C2311yescharacteristic
108#41×C61—42no—
109#41×C61—42no—
110C2×Q8×C61—21no—
111C2×D4×C61—22no—
112C2×D4×C61—21no—
113C2×D4×C61—21no—
114C2×D4×C61C22×C412yes—
115C22×C4×C61—21no—
116C22×C4×C61—22no—
117C22×C4×C61—21no—
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2411yescharacteristic
121C2×C22⋊D61—22no—
122C2×C22⋊D61—24no—
123C2×C22⋊D61—24no—
124C22×Dic61—24no—
125C22×Dic61—22no—
126C22×Dic61C2×D412yes—
127C22×Dic61C2×D412yes—
128Q8×D61—48no—
129#11758224—48no—
130#11758224—48no—
131#11758224—48no—
132#11758222—42no—
133#11758222—42no—
134#11758222—48no—
135C2×D244—21no—
136C2×D244—21no—
137C2×D244C22×C412yes—
138C2×C4×D61—22no—
139C2×C4×D61—24no—
140C2×Dic122—21no—
141C2×Dic122—24no—
142C2×Dic122—48no—
143C2×Dic122C22⋊C412yes—
144C2×Dic122C22⋊C412yes—
145C2×Dic122D814yes—
146C2×Dic122SD1614yes—
147C2×C8×C61—22no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—42no—
150C4⋊C4×C61C22×C412yes—
151C22⋊C4×C61—22no—
152C22⋊C4×C61—22no—
153C42×C61—42no—
154C42×C61C22×C411yescharacteristic
155C42×C61C22×C411yescharacteristic
156C42×C61C2×D411yescharacteristic
157C42×C61C2×D411yescharacteristic
158C22⋊Dic61—24no—
159#11758198—22no—
160#11758198—22no—
161#11758198—24no—
162C4⋊Dic61—21no—
163C4⋊Dic61—42no—
164C4⋊Dic61C22⋊C412yes—
165C4⋊Dic61C22⋊C412yes—
166C4⋊Dic61C22×C412yes—
167#11758196—24no—
168#11758196—44no—
169#11758196—44no—
170#11758196C22⋊C414yes—
171#11758196C22⋊C414yes—
172C4×Dic61—44no—
173C4×Dic61C22×C414yes—
174C4×Dic61C2×D414yes—
175C2×#92418—22no—
176C23×C61—22no—
177C23×C61C22×D411yescharacteristic
178C22×D61—22no—
179Q8×C61—42no—
180D4×C61—42no—
181D4×C61—42no—
182D4×C61—42no—
183D4×C61—42no—
184D4×C61—81no—
185C2×C4×C61—22no—
186C2×C4×C61—22no—
187C2×C4×C61—41no—
188C2×C4×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61—44no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61—41no—
195C2×C4×C61C2×C22⋊C411yescharacteristic
196C2×C4×C61C2×C22⋊C411yescharacteristic
197C2×C4×C61C2×C22⋊C411yescharacteristic
198C2×C4×C61C2×C22⋊C411yescharacteristic
199C2×C4×C61C2×D812yes—
200C2×C4×C61C2×SD1612yes—
201C2×C4×C61C23×C411yescharacteristic
202C2×C4×C61C22×D411yescharacteristic
203C22⋊D61—42no—
204C22⋊D61—42no—
205C22⋊D61—48no—
206C22⋊D61—84no—
207C2×Dic61—22no—
208C2×Dic61—42no—
209C2×Dic61—48no—
210C2×Dic61—42no—
211C2×Dic61—42no—
212C2×Dic61—42no—
213C2×Dic61C2×C22⋊C414yes—
214C2×Dic61C2×C22⋊C414yes—
215C2×Dic61C2×D814yes—
216C2×Dic61C2×SD1614yes—
217D244—42no—
218D244—42no—
219D244—81no—
220C4×D61—44no—
221C4×D61—48no—
222Dic122—42no—
223Dic122—48no—
224Dic122#10218yes—
225Dic122#10218yes—
226C8×C61—42no—
227#92418—42no—
228#17618—611nomaximal, Sylow
229C22×C61—21no—
230C22×C61—21no—
231C22×C61—42no—
232C22×C61—42no—
233C22×C61—42no—
234C22×C61C22×C22⋊C411yescharacteristic
235C22×C61C22×D811yescharacteristic
236C22×C61C22×SD1611yescharacteristic
237C2×D61—42no—
238C2×D61—42no—
239C2×D61—42no—
240C4×C61—21no—
241C4×C61—21no—
242C4×C61—21no—
243C4×C61—21no—
244C4×C61—42no—
245C4×C61—42no—
246C4×C61—81no—
247C4×C61C2×#10212yes—
248C4×C61C2×#10212yes—
249Dic61—24no—
250Dic61—42no—
251Dic61—82no—
252Dic61C2×#10218yes—
253#3497—611no—
254#2982—611no—
255#2351—611no—
256#1517—611no—
257#1517—614no—
258#1517—611no—
259#1502—611no—
260#1502—614no—
261#1502D6111yescharacteristic, complemented
262C2×C61—21no—
263C2×C61—21no—
264C2×C61—82no—
265C2×C61#288411yescharacteristic
266C2×C61C22×#10211yescharacteristic
267C2×C61#290111yescharacteristic
268D61—82no—
269C2×2−1+4—1221no—
270#550—612no—
271#550—612no—
272D4×Q8—612no—
273D4×Q8—612no—
274#548—1222no—
275#548—1222no—
276#537—1222no—
277#537—1221no—
278#532—614no—
279#532—611no—
280#532C2×D6111yescharacteristic
281C2×C4×Q8—611no—
282C2×C4×Q8—1221no—
283C8×Q8—1222no—
284C8×Q8—1222no—
285#431—611no—
286#431—614no—
287#431C2×D6111yescharacteristic
288#389—1221no—
289#389—1222no—
290#382—611no—
291#382—612no—
292C4×C4⋊C4—612no—
293C4×C4⋊C4C2×D6111yescharacteristic
294#327—614no—
295#327—614no—
296#327—612no—
297#327—612no—
298#327C2×D6112yes—
299#327C2×D6112yes—
300C61#1761811yescharacteristic, complemented, Sylow
3012−1+4—2444no—
3022−1+4—2444no—
303C2×#41—21no—
304C2×#41—1222no—
305C2×#41—1224no—
306C2×#41—1222no—
307C2×#41—1221no—
308C22×Q8—611no—
309C22×Q8—1222no—
310C22×Q8—1222no—
311C4⋊Q8—614no—
312C4⋊Q8—611no—
313C4⋊Q8C4×D6111yescharacteristic
314#127—614no—
315#127—611no—
316#127C4×D6111yescharacteristic
317#124—21no—
318#124—1224no—
319#124—1221no—
320C22⋊Q8—1224no—
321C22⋊Q8—1224no—
322C22⋊Q8—1222no—
323C22⋊Q8—1222no—
324#121—21no—
325#121—612no—
326#121—1224no—
327#121—1222no—
328#121—1222no—
329#121—1224no—
330#121—1224no—
331#121—1221no—
332#121C22×D6112yes—
333C4×Q8—612no—
334C4×Q8—612no—
335C4×Q8—612no—
336C4×Q8—612no—
337C4×Q8—2444no—
338C4×D4—21no—
339C4×D4—614no—
340C4×D4—614no—
341C4×D4—612no—
342C4×D4—1224no—
343C4×D4—1221no—
344C4×D4C4×D6112yes—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—612no—
347C2×C4⋊C4—611no—
348C2×C4⋊C4—1222no—
349C2×C4⋊C4—1222no—
350C2×C4⋊C4—1221no—
351C2×C42—21no—
352C2×C42—612no—
353C2×C42—1222no—
354C2×C42C22×D6111yescharacteristic
355C4⋊C8—21no—
356C4⋊C8—1224no—
357C4⋊C8—1221no—
358C22⋊C8—612no—
359C22⋊C8C22×D6112yes—
360C4×C8—21no—
361C4×C8—1224no—
362C4×C8—1221no—
363#95—612no—
364#95—1222no—
365#95—1221no—
366#95C22×D6111yescharacteristic
367#95C22×D6111yescharacteristic
368#41—42no—
369#41—42no—
370#41—2448no—
371#41—2448no—
372#41—2448no—
373#41—2448no—
374#41—2442no—
375#41—2442no—
376C2×Q8—21no—
377C2×Q8—612no—
378C2×Q8—612no—
379C2×Q8—614no—
380C2×Q8—614no—
381C2×Q8—1224no—
382C2×Q8—1221no—
383C2×Q8—2448no—
384C2×Q8—2448no—
385C2×D4—22no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—612no—
389C2×D4—1224no—
390C2×D4—1224no—
391C2×D4—1222no—
392C2×D4—1221no—
393C2×D4—1221no—
394C2×D4C2×C4×D6112yes—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—612no—
399C22×C4—612no—
400C22×C4—1224no—
401C22×C4—1224no—
402C22×C4—1222no—
403C22×C4—1222no—
404C22×C4D4×D6111yescharacteristic
405C22×C4D4×D6111yescharacteristic
406C22×C4C23×D6111yescharacteristic
407C2×C8—22no—
408C2×C8—1222no—
409C4⋊C4—21no—
410C4⋊C4—42no—
411C4⋊C4—614no—
412C4⋊C4—614no—
413C4⋊C4—612no—
414C4⋊C4—612no—
415C4⋊C4—612no—
416C4⋊C4—1224no—
417C4⋊C4—1221no—
418C4⋊C4—2444no—
419C4⋊C4—2444no—
420C4⋊C4—2442no—
421C4⋊C4C2×C4×D6112yes—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—1224no—
425C22⋊C4—1224no—
426C22⋊C4—1222no—
427C22⋊C4—1222no—
428C42—42no—
429C42—614no—
430C42—614no—
431C42—2444no—
432C42C2×C4×D6111yescharacteristic
433C42C2×C4×D6111yescharacteristic
434C42D4×D6111yescharacteristic
435C42D4×D6111yescharacteristic
436C23—22no—
437C23—1222no—
438C23C2×D4×D6111yescharacteristic
439Q8—42no—
440Q8—618no—
441Q8—618no—
442Q8—2448no—
443Q8—2442no—
444D4—42no—
445D4—42no—
446D4—42no—
447D4—42no—
448D4—81no—
449D4—4881no—
450D4—4884no—
451D4—2442no—
452D4—2448no—
453D4—2442no—
454D4—2442no—
455D4—2442no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—44no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—614no—
467C2×C4—614no—
468C2×C4—614no—
469C2×C4—614no—
470C2×C4—1222no—
471C2×C4—2448no—
472C2×C4—2448no—
473C2×C4—2442no—
474C2×C4—2442no—
475C2×C4—2442no—
476C2×C4—2442no—
477C2×C4—2444no—
478C2×C4C22⋊C4×D6111yescharacteristic
479C2×C4C22⋊C4×D6111yescharacteristic
480C2×C4C22⋊C4×D6111yescharacteristic
481C2×C4C22⋊C4×D6111yescharacteristic
482C2×C4D8×D6112yes—
483C2×C4SD16×D6112yes—
484C2×C4C22×C4×D6111yescharacteristic
485C2×C4C2×D4×D6111yescharacteristic
486C8—42no—
487C8—2442no—
488C22—21no—
489C22—21no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—2442no—
494C22—2442no—
495C22—2442no—
496C22C2×C22⋊C4×D6111yescharacteristic
497C22C2×D8×D6111yescharacteristic
498C22C2×SD16×D6111yescharacteristic
499C4—21no—
500C4—21no—
501C4—21no—
502C4—21no—
503C4—42no—
504C4—42no—
505C4—81no—
506C4—618no—
507C4—1224no—
508C4—4882no—
509C4—2442no—
510C4#102×D6112yes—
511C4#102×D6112yes—
512C2—21no—
513C2—21no—
514C2—82no—
515C2—4882no—
516C2#2145428111yescharacteristic
517C2C2×#102×D6111yescharacteristic
518C2#2145436311yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1502C2
socleC22×C61C22×C22⋊C4
prime radical#21459585C2
cosocleC22×C4×C61C24
hypercenter#1502D61
hypocenterC61#17618
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1502

Complements

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

NH ≅ G/Ntype
2. #21454899C2semidirect
3. #21448739C2semidirect
4. #21444380C2semidirect
5. #21444379C2semidirect
6. #21444377C2semidirect
7. #21444313C2semidirect
8. #21444312C2semidirect
9. C61×#1502C2semidirect
11. #16840433C4semidirect
12. #16840432C4semidirect
13. D4×Dic122C4semidirect
14. #16840427C4semidirect
261. #1502D61semidirect
300. C61#17618semidirect

Maximal subgroups

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

subgroupindex
#214595852
#214548992
#214487392
#214443802
#214443792
#214443772
#214443132
#214443122
C61×#15022
#1761861