Finite Group Databasebeta

← #29658786

Subgroups — #29658786

Order 15616.

1192 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
1C2×Q8×D244C211yesmaximal, characteristic, complemented
2#21456937C211yesmaximal, characteristic, complemented
3#21455815C211yesmaximal, characteristic
4#21445066C214yesmaximal, complemented
5#21444920C214yesmaximal, complemented
6#21444492C211yesmaximal, characteristic, complemented
7#21444243C211yesmaximal, characteristic, complemented
8#21444214C211yesmaximal, characteristic, complemented
9C61×#3037C211yesmaximal, characteristic, complemented
10C22×Q8×D61—21no—
11Q8×D244—22no—
12Q8×D244—22no—
13Q8×D244—22no—
14Q8×D244C2214yes—
15C2×#14519777—21no—
16C2×#14519777C2211yescharacteristic
17C2×#14519757—22no—
18C2×#14519757—21no—
19C2×C4×D244—21no—
20C2×C4×D244C2211yescharacteristic
21#16840165C2212yescomplemented
22#16840108C2212yescomplemented
23#16840073—22no—
24#16840073—21no—
25#16840042—22no—
26#16840039C2212yes—
27#16839954C2212yes—
28#16839895—21no—
29C4×#11758199—22no—
30#16839815C2211yescharacteristic
31#16839815C2211yescharacteristic
32#16839509C2214yescomplemented
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×Dic122—22no—
44C22×Dic122—21no—
45C2×Q8×D61—24no—
46C2×Q8×D61—24no—
47C2×Q8×D61—44no—
48C2×Q8×D61—44no—
49C22×D244D411yescharacteristic
50C22×C4×D61—22no—
51C22×C4×D61—21no—
52C2×#11758199—22no—
53#14519777—24no—
54#14519777—21no—
55#14519777—21no—
56#14519777—24no—
57#14519777D412yes—
58#14519777C2312yes—
59#14519757—24no—
60#14519757—28no—
61#14519757—24no—
62#14519757—28no—
63C4×D244—24no—
64C4×D244—24no—
65C4×D244—21no—
66C4×D244—21no—
67C4×D244D412yes—
68C4×D244C2312yes—
69C2×C4⋊Dic61—22no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61D411yescharacteristic
72C2×C4⋊Dic61D412yes—
73C2×#11758198—22no—
74C2×#11758198—21no—
75#14519710—22no—
76#14519710—24no—
77C4⋊#92418—21no—
78C4⋊#92418C2312yes—
79#14519698—22no—
80#14519698—24no—
81#14519697—22no—
82C22⋊#92418—21no—
83C22⋊#92418—21no—
84C4×#92418—21no—
85C4×#92418C2312yes—
86C2×#41×C61—21no—
87C22×Q8×C61D411yescharacteristic
88#137×C61—44no—
89C2×Q16×C61—22no—
90C2×SD16×C61—22no—
91C2×M16×C61—21no—
92C4⋊Q8×C61C2311yescharacteristic
93#127×C61C2311yescharacteristic
94#124×C61—21no—
95#121×C61—21no—
96#121×C61—21no—
97C4×Q8×C61—21no—
98C4×Q8×C61—21no—
99C4×Q8×C61D412yes—
100C4×Q8×C61C2312yes—
101C4×D4×C61—21no—
102C4×D4×C61—21no—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61D411yescharacteristic
105C2×C42×C61—21no—
106C2×C42×C61C2311yescharacteristic
107C4⋊C8×C61—21no—
108C4⋊C8×C61C2312yes—
109#103×C61—22no—
110#102×C61—22no—
111C23×D61—21no—
112#41×C61—42no—
113#41×C61—42no—
114C2×Q8×C61—21no—
115C2×Q8×C61—22no—
116C2×Q8×C61—21no—
117C2×Q8×C61D814yescomplemented
118C2×Q8×C61C2×D412yes—
119C2×D4×C61—21no—
120C2×D4×C61—21no—
121C2×D4×C61—21no—
122C2×D4×C61—41no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61—41no—
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×Dic61—22no—
130C22×Dic61—21no—
131Q8×D61—48no—
132Q8×D61—416no—
133Q8×D61—48no—
134C2×D244—21no—
135C2×D244—24no—
136C2×D244C2×D412yes—
137C2×C4×D61—28no—
138C2×C4×D61—22no—
139C2×C4×D61—22no—
140C2×C4×D61—48no—
141C2×C4×D61—42no—
142C2×C4×D61—42no—
143C2×Dic122—28no—
144C2×Dic122—22no—
145C2×Dic122—22no—
146C2×Dic122—48no—
147C2×Dic122—42no—
148C2×Dic122—42no—
149Q16×C61—44no—
150SD16×C61—44no—
151M16×C61—42no—
152C2×C8×C61—22no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61C2×D412yes—
159C22⋊C4×C61—41no—
160C22⋊C4×C61—41no—
161C42×C61—22no—
162C42×C61—22no—
163C42×C61C2×D411yescharacteristic
164C42×C61C2×D411yescharacteristic
165C42×C61C2×D411yescharacteristic
166C42×C61C2411yescharacteristic
167#11758199—44no—
168#11758198—28no—
169#11758198—22no—
170#11758198—22no—
171C4⋊Dic61—28no—
172C4⋊Dic61—22no—
173C4⋊Dic61—21no—
174C4⋊Dic61—22no—
175C4⋊Dic61—22no—
176C4⋊Dic61C2×D412yes—
177C4⋊Dic61C2×D414yes—
178C2×#92418—22no—
179C2×#92418—22no—
180C23×C61—41no—
181C23×C61#12011yescharacteristic
182C22×D61—24no—
183C22×D61—22no—
184C22×D61—42no—
185C22×D61—42no—
186Q8×C61—22no—
187Q8×C61—24no—
188Q8×C61—41no—
189Q8×C61—41no—
190Q8×C61C2×D814yes—
191D4×C61—42no—
192D4×C61—41no—
193D4×C61—41no—
194D4×C61—42no—
195D4×C61—82no—
196C2×C4×C61—22no—
197C2×C4×C61—24no—
198C2×C4×C61—22no—
199C2×C4×C61—22no—
200C2×C4×C61—41no—
201C2×C4×C61—42no—
202C2×C4×C61—41no—
203C2×C4×C61—82no—
204C2×C4×C61#12011yescharacteristic
205C2×C4×C61#12011yescharacteristic
206C2×C4×C61#12011yescharacteristic
207C2×C4×C61C2×D812yes—
208C2×C4×C61AGL(1,ℤ/8)12yes—
209C2×C4×C61C22×D411yescharacteristic
210C2×C4×C61C22×D411yescharacteristic
211C2×C4×C61C22×D411yescharacteristic
212C2×Dic61—22no—
213C2×Dic61—24no—
214C2×Dic61—42no—
215C2×Dic61—44no—
216D244—24no—
217D244—22no—
218D244#13714yescomplemented
219C4×D61—416no—
220C4×D61—42no—
221C4×D61—42no—
222C4×D61—44no—
223Dic122—416no—
224Dic122—42no—
225Dic122—42no—
226Dic122—44no—
227C8×C61—42no—
228#92418—42no—
229#92418—42no—
230#18776—611nomaximal, Sylow
231C22×C61—21no—
232C22×C61—21no—
233C22×C61—81no—
234C22×C61—81no—
235C22×C61—81no—
236C22×C61#44712yes—
237C22×C61C2×#12011yescharacteristic
238C2×D61—24no—
239C2×D61—48no—
240C2×D61—42no—
241C2×D61—42no—
242C4×C61—24no—
243C4×C61—21no—
244C4×C61—21no—
245C4×C61—21no—
246C4×C61—42no—
247C4×C61—81no—
248C4×C61#44712yes—
249C4×C61C22×D811yescharacteristic
250C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
251C4×C61C2×#13712yes—
252Dic61—42no—
253Dic61—44no—
254C2×D4×Q8—611no—
255#3226—611no—
256#3100—611no—
257#3037D6111yescharacteristic, complemented
258#1649—614no—
259#1627—614no—
260#1546—611no—
261#1486—611no—
262#1479—611no—
263C2×C61—22no—
264C2×C61—161no—
265C2×C61C2×#44711yescharacteristic
266C2×C61#300411yescharacteristic
267C2×C61#302011yescharacteristic
268D61—44no—
269C23×Q8—1221no—
270C2×#137—21no—
271D4×Q8—614no—
272D4×Q8—1222no—
273D4×Q8—1222no—
274D4×Q8—1222no—
275#532C2×D6111yescharacteristic
276C2×C4⋊Q8—611no—
277C2×C4⋊Q8—1221no—
278C2×C22⋊Q8—1222no—
279C2×C22⋊Q8—1221no—
280C2×C4×Q8C2×D6111yescharacteristic
281C2×C4×D4—611no—
282C2×C4×D4—1221no—
283#500—612no—
284#486—612no—
285#480—1221no—
286#480—1222no—
287#474—612no—
288#474—1222no—
289#464—22no—
290#462C2×D6112yescomplemented
291#460C2×D6112yescomplemented
292#459—612no—
293#447—1221no—
294C4×D8—1222no—
295#431—611no—
296#431—611no—
297#423C2×D6111yescharacteristic
298#417—21no—
299#332—614no—
300Q8⋊C8—614no—
301D4⋊C8—614no—
302C61#1877611yescharacteristic, complemented, Sylow
303C2×#41—21no—
304C22×Q8—1222no—
305C22×Q8—1224no—
306C22×Q8—1224no—
307C22×Q8—1221no—
308C22×Q8—2444no—
309C22×Q8—2444no—
310C22×Q8C22⋊D6111yescharacteristic
311C22×D4—611no—
312C23×C4—1222no—
313C23×C4—1221no—
314#137—44no—
315C2×Q16—22no—
316C2×SD16—22no—
317C2×D8—1222no—
318C2×M16—21no—
319C4⋊Q8—612no—
320C4⋊Q8—612no—
321C4⋊Q8—1224no—
322C4⋊Q8—1224no—
323C4⋊Q8—1221no—
324C4⋊Q8—1221no—
325C4⋊Q8C22×D6111yescharacteristic
326#127C22×D6111yescharacteristic
327#124—21no—
328C22⋊Q8—1228no—
329C22⋊Q8—1224no—
330C22⋊Q8—1228no—
331C22⋊Q8—1224no—
332#121—21no—
333#121—21no—
334C4×Q8—21no—
335C4×Q8—21no—
336C4×Q8C22⋊D6112yes—
337C4×Q8C22×D6112yes—
338C4×D4—21no—
339C4×D4—21no—
340C4×D4—612no—
341C4×D4—612no—
342C4×D4—1224no—
343C4×D4—1224no—
344C4×D4—1221no—
345C4×D4—1221no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—611no—
348C2×C4⋊C4—612no—
349C2×C4⋊C4—1222no—
350C2×C4⋊C4—1221no—
351C2×C4⋊C4C22⋊D6111yescharacteristic
352C2×C22⋊C4—1222no—
353C2×C22⋊C4—1221no—
354C2×C42—21no—
355C2×C42C22×D6111yescharacteristic
356#107—1222no—
357#107—1224no—
358C4⋊C8—21no—
359C4⋊C8—612no—
360C4⋊C8—1221no—
361C4⋊C8C22×D6112yes—
362#103—22no—
363#102—22no—
364#102—1222no—
365#102—1222no—
366#102—1224no—
367C22⋊C8—1221no—
368C22⋊C8—1221no—
369C4×C8—612no—
370C4×C8—1221no—
371C24—1221no—
372#41—42no—
373#41—42no—
374C2×Q8—21no—
375C2×Q8—22no—
376C2×Q8—21no—
377C2×Q8—1228no—
378C2×Q8—1222no—
379C2×Q8—1222no—
380C2×Q8—2448no—
381C2×Q8—2448no—
382C2×Q8—24416no—
383C2×Q8—2448no—
384C2×Q8—2442no—
385C2×Q8—2442no—
386C2×Q8#1175819914yescomplemented
387C2×Q8C2×C22⋊D6112yes—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—41no—
392C2×D4—612no—
393C2×D4—1224no—
394C2×D4—1221no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—41no—
398C22×C4—1228no—
399C22×C4—1222no—
400C22×C4—1221no—
401C22×C4—1222no—
402C22×C4—1222no—
403C22×C4—2448no—
404C22×C4—2442no—
405C22×C4—2442no—
406C22×C4D4×D6111yescharacteristic
407C22×C4D4×D6111yescharacteristic
408C22×C4C2×C22⋊D6111yescharacteristic
409Q16—44no—
410SD16—44no—
411D8—2444no—
412M16—42no—
413C2×C8—22no—
414C2×C8—1222no—
415C2×C8—1222no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—612no—
422C4⋊C4—614no—
423C4⋊C4—1228no—
424C4⋊C4—1222no—
425C4⋊C4—1221no—
426C4⋊C4—1222no—
427C4⋊C4—1222no—
428C4⋊C4C2×C22⋊D6112yes—
429C22⋊C4—41no—
430C22⋊C4—41no—
431C22⋊C4—1228no—
432C22⋊C4—1222no—
433C22⋊C4—1222no—
434C42—22no—
435C42—22no—
436C42D4×D6111yescharacteristic
437C42D4×D6111yescharacteristic
438C42C2×C22⋊D6111yescharacteristic
439C42C23×D6111yescharacteristic
440C23—41no—
441C23—1224no—
442C23—1222no—
443C23—2442no—
444C23—2442no—
445C23#1451974511yescharacteristic
446Q8—22no—
447Q8—24no—
448Q8—41no—
449Q8—41no—
450Q8—24416no—
451Q8—2442no—
452Q8—2442no—
453Q8—2444no—
454Q8C2×#1175819914yes—
455D4—42no—
456D4—41no—
457D4—41no—
458D4—42no—
459D4—82no—
460D4—614no—
461D4—1224no—
462D4—1222no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—82no—
471C2×C4—1224no—
472C2×C4—1222no—
473C2×C4—24416no—
474C2×C4—2444no—
475C2×C4—2442no—
476C2×C4—2442no—
477C2×C4—2442no—
478C2×C4—2444no—
479C2×C4#1451974511yescharacteristic
480C2×C4#1451974511yescharacteristic
481C2×C4#1451974511yescharacteristic
482C2×C4C2×#1175819912yes—
483C2×C4#1451980212yes—
484C2×C4C2×D4×D6111yescharacteristic
485C2×C4C2×D4×D6111yescharacteristic
486C2×C4C22×C22⋊D6111yescharacteristic
487C8—42no—
488C8—2442no—
489C8—2442no—
490C22—21no—
491C22—21no—
492C22—81no—
493C22—81no—
494C22—81no—
495C22—1224no—
496C22—2448no—
497C22—2442no—
498C22—2442no—
499C22#1683989512yes—
500C22C2×#1451974511yescharacteristic
501C4—21no—
502C4—21no—
503C4—24no—
504C4—21no—
505C4—42no—
506C4—81no—
507C4—2444no—
508C4—2442no—
509C4#1683989512yes—
510C4C22×#1175819911yescharacteristic
511C4C2×#1451980211yescharacteristic
512C4#137×D6112yes—
513C2—22no—
514C2—161no—
515C2—2444no—
516C2C2×#1683989511yescharacteristic
517C2#2145505111yescharacteristic
518C2#2145518111yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3037C2
socleC22×C61C2×#120
prime radical#21455815C2
hypercenter#3037D61
hypocenterC61#18776
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×Q8×D244C2semidirect
2. #21456937C2semidirect
4. #21445066C2semidirect
5. #21444920C2semidirect
6. #21444492C2semidirect
7. #21444243C2semidirect
8. #21444214C2semidirect
9. C61×#3037C2semidirect
21. #16840165C22semidirect
22. #16840108C22semidirect
32. #16839509C22semidirect
33. Q8⋊#92418C22semidirect
39. C61×#462C22semidirect
40. C61×#460C22semidirect
117. C2×Q8×C61D8semidirect
218. D244#137semidirect
257. #3037D61semidirect
290. #462C2×D61semidirect
291. #460C2×D61semidirect
302. C61#18776semidirect
386. C2×Q8#11758199semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×D2442
#214569372
#214558152
#214450662
#214449202
#214444922
#214442432
#214442142
C61×#30372
#1877661