Finite Group Databasebeta

← #29647874

Subgroups — #29647874

Order 15616.

1138 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459585C211yesmaximal, characteristic
2#21454902C211yesmaximal, characteristic, complemented
3#21448782C211yesmaximal, characteristic, complemented
4#21444380C211yesmaximal, characteristic, complemented
5#21444379C211yesmaximal, characteristic, complemented
6#21444364C214yesmaximal, complemented
7#21444325C214yesmaximal, complemented
8#21444313C211yesmaximal, characteristic, 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×Dic122C2211yescharacteristic
22#16839892—22no—
23#16839889—22no—
24#16839816C2214yes—
25#16839815C2211yescharacteristic
26#16839668—21no—
27#16839643—22no—
28#16839629C2212yes—
29#16839628C2211yescharacteristic
30C4×C4⋊Dic61—21no—
31#16839607C2212yes—
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×Dic122C2×C412yes—
68C4×Dic122C2×C412yes—
69C4×Dic122D412yes—
70C4×Dic122D412yes—
71C4×D244—21no—
72C4×D244C2×C412yes—
73D4×Dic61C2×C414yes—
74#14519739C2×C414yes—
75#14519737—24no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61D411yescharacteristic
78C2×#11758196D412yes—
79C2×C4×Dic61—22no—
80C2×C4×Dic61C2312yes—
81C4⋊#92418—21no—
82#14519706—24no—
83C22⋊#92418C2312yes—
84#14519687—24no—
85C4×#92418—21no—
86#14519684—21no—
87#14519684—22no—
88#14519684—22no—
89#14519684—22no—
90#14519684C2312yes—
91C2×#41×C61—21no—
92C4⋊Q8×C61C2×C411yescharacteristic
93#127×C61C2×C411yescharacteristic
94#124×C61—21no—
95#121×C61—21no—
96#121×C61C2312yes—
97C4×D4×C61—21no—
98C4×D4×C61C2×C412yes—
99C2×C4⋊C4×C61—21no—
100C2×C42×C61—21no—
101C2×C42×C61C2311yescharacteristic
102C4⋊C8×C61—21no—
103C22⋊C8×C61C2312yes—
104C4×C8×C61—21no—
105#95×C61C2311yescharacteristic
106#95×C61C2311yescharacteristic
107#41×C61—42no—
108#41×C61—42no—
109C2×Q8×C61—21no—
110C2×D4×C61—22no—
111C2×D4×C61—21no—
112C2×D4×C61—21no—
113C2×D4×C61C22×C412yes—
114C22×C4×C61—21no—
115C22×C4×C61—22no—
116C22×C4×C61—21no—
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2411yescharacteristic
120C2×C22⋊D61—22no—
121C2×C22⋊D61—24no—
122C2×C22⋊D61—24no—
123C22×Dic61—24no—
124C22×Dic61—22no—
125C22×Dic61C2×D412yes—
126C22×Dic61C2×D412yes—
127Q8×D61—48no—
128#11758224—48no—
129#11758224—48no—
130#11758224—48no—
131#11758222—48no—
132#11758222—42no—
133#11758222—42no—
134C2×D244—21no—
135C2×D244—21no—
136C2×D244C22×C412yes—
137C2×C4×D61—22no—
138C2×C4×D61—24no—
139C2×Dic122—22no—
140C2×Dic122—22no—
141C2×Dic122—21no—
142C2×Dic122—24no—
143C2×Dic122—48no—
144C2×Dic122C22⋊C412yes—
145C2×Dic122C22⋊C412yes—
146C2×C8×C61—22no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—42no—
149C4⋊C4×C61C22×C412yes—
150C22⋊C4×C61—22no—
151C22⋊C4×C61—22no—
152C42×C61—42no—
153C42×C61C22×C411yescharacteristic
154C42×C61C22×C411yescharacteristic
155C42×C61C2×D411yescharacteristic
156C42×C61C2×D411yescharacteristic
157C22⋊Dic61—24no—
158#11758198—22no—
159#11758198—22no—
160#11758198—24no—
161C4⋊Dic61—21no—
162C4⋊Dic61—42no—
163C4⋊Dic61C22⋊C412yes—
164C4⋊Dic61C22⋊C412yes—
165C4⋊Dic61C22×C412yes—
166#11758196—24no—
167#11758196C22⋊C414yes—
168#11758196C22⋊C414yes—
169C4×Dic61—44no—
170C4×Dic61C22×C414yes—
171C4×Dic61C2×D414yes—
172C2×#92418—22no—
173C23×C61—22no—
174C23×C61C22×D411yescharacteristic
175C22×D61—22no—
176Q8×C61—42no—
177D4×C61—42no—
178D4×C61—42no—
179D4×C61—42no—
180D4×C61—42no—
181D4×C61—81no—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—41no—
185C2×C4×C61—41no—
186C2×C4×C61—41no—
187C2×C4×C61—41no—
188C2×C4×C61—44no—
189C2×C4×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61—41no—
192C2×C4×C61C2×C22⋊C411yescharacteristic
193C2×C4×C61C2×C22⋊C411yescharacteristic
194C2×C4×C61C2×C22⋊C411yescharacteristic
195C2×C4×C61C2×C22⋊C411yescharacteristic
196C2×C4×C61AGL(1,ℤ/8)12yes—
197C2×C4×C61AGL(1,ℤ/8)12yes—
198C2×C4×C61C23×C411yescharacteristic
199C2×C4×C61C22×D411yescharacteristic
200C22⋊D61—48no—
201C22⋊D61—42no—
202C22⋊D61—42no—
203C22⋊D61—84no—
204C2×Dic61—22no—
205C2×Dic61—22no—
206C2×Dic61—22no—
207C2×Dic61—42no—
208C2×Dic61—48no—
209C2×Dic61—42no—
210C2×Dic61—42no—
211C2×Dic61—42no—
212C2×Dic61C2×C22⋊C414yes—
213C2×Dic61C2×C22⋊C414yes—
214D244—42no—
215D244—42no—
216D244—81no—
217C4×D61—48no—
218C4×D61—44no—
219Dic122—24no—
220Dic122—24no—
221Dic122—48no—
222Dic122—42no—
223C8×C61—42no—
224#92418—42no—
225#17631—611nomaximal, Sylow
226C22×C61—21no—
227C22×C61—21no—
228C22×C61—42no—
229C22×C61—42no—
230C22×C61—42no—
231C22×C61C22×C22⋊C411yescharacteristic
232C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
233C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
234C2×D61—42no—
235C2×D61—42no—
236C2×D61—42no—
237C4×C61—21no—
238C4×C61—21no—
239C4×C61—21no—
240C4×C61—21no—
241C4×C61—42no—
242C4×C61—42no—
243C4×C61—81no—
244C4×C61#41812yes—
245C4×C61#41812yes—
246Dic61—24no—
247Dic61—24no—
248Dic61—42no—
249Dic61—82no—
250#3497—611no—
251#2983—611no—
252#2356—611no—
253#1517—611no—
254#1517—611no—
255#1514—614no—
256#1505—614no—
257#1502—611no—
258#1502D6111yescharacteristic, complemented
259C2×C61—21no—
260C2×C61—21no—
261C2×C61—82no—
262C2×C61#288411yescharacteristic
263C2×C61C2×#41811yescharacteristic
264C2×C61#290111yescharacteristic
265D61—82no—
266C2×2−1+4—1221no—
267#550—612no—
268#550—612no—
269D4×Q8—612no—
270D4×Q8—612no—
271#548—1222no—
272#548—1222no—
273#537—1222no—
274#537—1221no—
275#532—614no—
276#532—611no—
277#532C2×D6111yescharacteristic
278C2×C4×Q8—611no—
279#446—1222no—
280#446—1222no—
281#432—614no—
282#431—611no—
283#431C2×D6111yescharacteristic
284#391—1221no—
285#387—1222no—
286#383—612no—
287#382—611no—
288C4×C4⋊C4—1221no—
289C4×C4⋊C4C2×D6111yescharacteristic
290#376—612no—
291#327—614no—
292#327—614no—
293#327—612no—
294#327—612no—
295#327C2×D6112yes—
296#327C2×D6112yes—
297C61#1763111yescharacteristic, complemented, Sylow
2982−1+4—2444no—
2992−1+4—2444no—
300C2×#41—21no—
301C2×#41—1222no—
302C2×#41—1224no—
303C2×#41—1222no—
304C2×#41—1221no—
305C22×Q8—611no—
306C22×Q8—1222no—
307C22×Q8—1222no—
308C4⋊Q8—614no—
309C4⋊Q8—611no—
310C4⋊Q8C4×D6111yescharacteristic
311#127—614no—
312#127—611no—
313#127C4×D6111yescharacteristic
314#124—21no—
315#124—1224no—
316#124—1221no—
317C22⋊Q8—1224no—
318C22⋊Q8—1224no—
319C22⋊Q8—1222no—
320C22⋊Q8—1222no—
321#121—21no—
322#121—612no—
323#121—1224no—
324#121—1222no—
325#121—1222no—
326#121—1224no—
327#121—1224no—
328#121—1221no—
329#121C22×D6112yes—
330C4×Q8—612no—
331C4×Q8—612no—
332C4×Q8—612no—
333C4×Q8—612no—
334C4×D4—21no—
335C4×D4—614no—
336C4×D4—614no—
337C4×D4—612no—
338C4×D4—1224no—
339C4×D4—1221no—
340C4×D4C4×D6112yes—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—611no—
343C2×C4⋊C4—612no—
344C2×C4⋊C4—1221no—
345C2×C42—21no—
346C2×C42—612no—
347C2×C42—1222no—
348C2×C42C22×D6111yescharacteristic
349C4⋊C8—21no—
350C4⋊C8—1224no—
351C4⋊C8—1221no—
352C22⋊C8—612no—
353C22⋊C8C22×D6112yes—
354#97—1224no—
355C4×C8—21no—
356C4×C8—1221no—
357#95—612no—
358#95—1222no—
359#95—1222no—
360#95—1222no—
361#95—1221no—
362#95C22×D6111yescharacteristic
363#95C22×D6111yescharacteristic
364#41—42no—
365#41—42no—
366#41—2448no—
367#41—2448no—
368#41—2448no—
369#41—2448no—
370#41—2442no—
371#41—2442no—
372C2×Q8—21no—
373C2×Q8—612no—
374C2×Q8—612no—
375C2×Q8—1224no—
376C2×Q8—1221no—
377C2×Q8—1222no—
378C2×Q8—1222no—
379C2×Q8—2448no—
380C2×Q8—2448no—
381C2×D4—22no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—612no—
385C2×D4—1224no—
386C2×D4—1224no—
387C2×D4—1222no—
388C2×D4—1221no—
389C2×D4—1221no—
390C2×D4C2×C4×D6112yes—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—612no—
395C22×C4—612no—
396C22×C4—1224no—
397C22×C4—1224no—
398C22×C4—1222no—
399C22×C4—1222no—
400C22×C4D4×D6111yescharacteristic
401C22×C4D4×D6111yescharacteristic
402C22×C4C23×D6111yescharacteristic
403C2×C8—22no—
404C2×C8—1222no—
405C4⋊C4—21no—
406C4⋊C4—42no—
407C4⋊C4—614no—
408C4⋊C4—614no—
409C4⋊C4—612no—
410C4⋊C4—612no—
411C4⋊C4—612no—
412C4⋊C4—1224no—
413C4⋊C4—1221no—
414C4⋊C4—2442no—
415C4⋊C4C2×C4×D6112yes—
416C22⋊C4—22no—
417C22⋊C4—22no—
418C22⋊C4—1224no—
419C22⋊C4—1224no—
420C22⋊C4—1222no—
421C22⋊C4—1222no—
422C42—42no—
423C42—614no—
424C42—614no—
425C42—2444no—
426C42C2×C4×D6111yescharacteristic
427C42C2×C4×D6111yescharacteristic
428C42D4×D6111yescharacteristic
429C42D4×D6111yescharacteristic
430C23—22no—
431C23—1222no—
432C23C2×D4×D6111yescharacteristic
433Q8—42no—
434Q8—1224no—
435Q8—1224no—
436Q8—2448no—
437Q8—2442no—
438D4—42no—
439D4—42no—
440D4—42no—
441D4—42no—
442D4—81no—
443D4—4881no—
444D4—4884no—
445D4—2442no—
446D4—2448no—
447D4—2442no—
448D4—2442no—
449D4—2442no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—44no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—614no—
461C2×C4—614no—
462C2×C4—1222no—
463C2×C4—1222no—
464C2×C4—1222no—
465C2×C4—2448no—
466C2×C4—2442no—
467C2×C4—2448no—
468C2×C4—2442no—
469C2×C4—2442no—
470C2×C4—2442no—
471C2×C4—2444no—
472C2×C4C22⋊C4×D6111yescharacteristic
473C2×C4C22⋊C4×D6111yescharacteristic
474C2×C4C22⋊C4×D6111yescharacteristic
475C2×C4C22⋊C4×D6111yescharacteristic
476C2×C4#1451980112yes—
477C2×C4#1451980412yes—
478C2×C4C22×C4×D6111yescharacteristic
479C2×C4C2×D4×D6111yescharacteristic
480C8—42no—
481C8—2442no—
482C22—21no—
483C22—21no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—2442no—
488C22—2442no—
489C22—2442no—
490C22C2×C22⋊C4×D6111yescharacteristic
491C22C2×#1451980111yescharacteristic
492C22C2×#1451980411yescharacteristic
493C4—21no—
494C4—21no—
495C4—21no—
496C4—21no—
497C4—42no—
498C4—42no—
499C4—81no—
500C4—1224no—
501C4—1224no—
502C4—4882no—
503C4—2442no—
504C4#1683976512yes—
505C4#1683976512yes—
506C2—21no—
507C2—21no—
508C2—82no—
509C2—4882no—
510C2#2145428111yescharacteristic
511C2C2×#1683976511yescharacteristic
512C2#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#17631
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. #21454902C2semidirect
3. #21448782C2semidirect
4. #21444380C2semidirect
5. #21444379C2semidirect
6. #21444364C2semidirect
7. #21444325C2semidirect
8. #21444313C2semidirect
9. C61×#1502C2semidirect
11. #16840433C4semidirect
12. #16840432C4semidirect
13. D4×Dic122C4semidirect
14. #16840427C4semidirect
258. #1502D61semidirect
297. C61#17631semidirect

Maximal subgroups

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

subgroupindex
#214595852
#214549022
#214487822
#214443802
#214443792
#214443642
#214443252
#214443132
C61×#15022
#1763161