Finite Group Databasebeta

← #29656086

Subgroups — #29656086

Order 15616.

1048 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840472C211yesmaximal, characteristic, complemented
2#21456279C211yesmaximal, characteristic, complemented
3#21455906C211yesmaximal, characteristic
4#21444952C214yesmaximal, complemented
5#21444890C214yesmaximal, complemented
6#21444544C211yesmaximal, characteristic, complemented
7#21444266C211yesmaximal, characteristic, complemented
8#21444261C211yesmaximal, characteristic, complemented
9C61×#3112C211yesmaximal, characteristic, complemented
10C2×#14519807—22no—
11#16840472—22no—
12#16840472—22no—
13#16840472—22no—
14#16840472C2214yes—
15C2×#14519777C2211yescharacteristic
16C2×#14519769—21no—
17C2×#14519755—22no—
18C2×#14519753—21no—
19C2×C4×D244—21no—
20C2×C4×D244C2211yescharacteristic
21C2×C4⋊C4×D61—21no—
22#16840157—22no—
23#16840141C2212yescomplemented
24#16840134C2212yescomplemented
25#16840078—21no—
26#16840049C2212yes—
27#16839954C2212yes—
28#16839895—21no—
29#16839872—22no—
30#16839819C2211yescharacteristic
31C4×#11758194C2211yescharacteristic
32#16839510C2214yescomplemented
33Q8⋊#92418C2214yescomplemented
34#16839489C2214yes—
35C61×#532C2211yescharacteristic
36C2×C4×Q8×C61C2211yescharacteristic
37C61×#480—21no—
38C61×#477C2212yescomplemented
39C61×#460C2212yescomplemented
40C61×#448—21no—
41C61×#440—22no—
42C61×#432C2211yescharacteristic
43C22×Dic122D412yes—
44C22×D244D411yescharacteristic
45C22×C4×D61—21no—
46C22×C4×D61—22no—
47#14519807—48no—
48C2×#11758202—24no—
49C2×#11758200—24no—
50C2×#11758199—22no—
51C2×#11758194—21no—
52#14519777—21no—
53#14519777—21no—
54#14519777D412yes—
55#14519777C2312yes—
56#14519769—24no—
57#14519769—24no—
58#14519755—28no—
59#14519755—28no—
60#14519753—24no—
61#14519753—24no—
62C4×D244—24no—
63C4×D244—24no—
64C4×D244—21no—
65C4×D244—21no—
66C4×D244D412yes—
67C4×D244C2312yes—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61D411yescharacteristic
71C2×#11758196—22no—
72C2×#11758196—22no—
73C4⋊C4×D61—24no—
74C4⋊C4×D61—24no—
75C2×#11758198—22no—
76C2×#11758198—21no—
77#14519708—22no—
78C4⋊#92418C2312yes—
79#14519698—22no—
80#14519697—22no—
81C22⋊#92418—21no—
82#14519688—21no—
83#14519688—21no—
84C4×#92418C2312yes—
85C2×#41×C61—21no—
86C22×Q8×C61D411yescharacteristic
87C2×SD16×C61—22no—
88C4⋊Q8×C61C2311yescharacteristic
89#127×C61C2311yescharacteristic
90#124×C61—21no—
91#121×C61—21no—
92#121×C61—21no—
93C4×Q8×C61—21no—
94C4×Q8×C61—21no—
95C4×Q8×C61D412yes—
96C4×Q8×C61C2312yes—
97C4×D4×C61—21no—
98C4×D4×C61—21no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61D411yescharacteristic
101C2×C42×C61—21no—
102C2×C42×C61C2311yescharacteristic
103#107×C61—22no—
104C4⋊C8×C61C2312yes—
105#103×C61—22no—
106#102×C61—22no—
107C22⋊C8×C61—21no—
108#97×C61—21no—
109C23×D61—21no—
110#41×C61—44no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×Q8×C61—42no—
114C2×Q8×C61C2×D412yes—
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61—41no—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—41no—
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×Dic61—22no—
126C22×Dic61—21no—
127C2×D244—21no—
128C2×D244—24no—
129C2×D244C2×D412yes—
130C2×C4×D61—22no—
131C2×C4×D61—22no—
132C2×C4×D61—28no—
133C2×C4×D61—44no—
134C2×C4×D61—84no—
135C2×Dic122—22no—
136C2×Dic122—44no—
137C2×Dic122C2×D414yes—
138SD16×C61—44no—
139C2×C8×C61—22no—
140C4⋊C4×C61—22no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61C2×D412yes—
146C22⋊C4×C61—41no—
147C22⋊C4×C61—41no—
148C42×C61—22no—
149C42×C61—22no—
150C42×C61C2×D411yescharacteristic
151C42×C61C2×D411yescharacteristic
152C42×C61C2×D411yescharacteristic
153C42×C61C2411yescharacteristic
154#11758202—48no—
155#11758200—48no—
156#11758199—44no—
157#11758198—28no—
158#11758198—22no—
159#11758198—22no—
160C4⋊Dic61—22no—
161C4⋊Dic61—22no—
162C4⋊Dic61—21no—
163C4⋊Dic61—22no—
164C4⋊Dic61—22no—
165C4⋊Dic61C2×D412yes—
166#11758196—28no—
167#11758196—28no—
168#11758194—24no—
169C2×#92418—22no—
170C2×#92418—22no—
171C23×C61—41no—
172C23×C61#12011yescharacteristic
173C22×D61—24no—
174C22×D61—22no—
175C22×D61—42no—
176C22×D61—42no—
177Q8×C61—22no—
178Q8×C61—22no—
179Q8×C61—42no—
180Q8×C61—42no—
181D4×C61—42no—
182D4×C61—42no—
183D4×C61—42no—
184D4×C61—82no—
185C2×C4×C61—22no—
186C2×C4×C61—22no—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—42no—
191C2×C4×C61—42no—
192C2×C4×C61—41no—
193C2×C4×C61—82no—
194C2×C4×C61#12011yescharacteristic
195C2×C4×C61#12011yescharacteristic
196C2×C4×C61#12011yescharacteristic
197C2×C4×C61C22×D411yescharacteristic
198C2×C4×C61C22×D411yescharacteristic
199C2×C4×C61C22×D411yescharacteristic
200C2×Dic61—22no—
201C2×Dic61—24no—
202C2×Dic61—42no—
203C2×Dic61—44no—
204D244—24no—
205D244—22no—
206D244#13714yescomplemented
207C4×D61—44no—
208C4×D61—44no—
209C4×D61—88no—
210Dic122—44no—
211Dic122—82no—
212C8×C61—42no—
213#92418—24no—
214#92418—42no—
215#18525—611nomaximal, Sylow
216C22×C61—21no—
217C22×C61—21no—
218C22×C61—81no—
219C22×C61—81no—
220C22×C61—81no—
221C22×C61#45312yes—
222C22×C61C2×#12011yescharacteristic
223C2×D61—24no—
224C2×D61—48no—
225C2×D61—42no—
226C2×D61—42no—
227C4×C61—21no—
228C4×C61—21no—
229C4×C61—21no—
230C4×C61—21no—
231C4×C61—42no—
232C4×C61—42no—
233C4×C61—81no—
234C4×C61C2×#13712yes—
235C4×C61#57512yes—
236Dic61—42no—
237Dic61—44no—
238C2×#554—611no—
239#3152—611no—
240#3112D6111yescharacteristic, complemented
241#3109—611no—
242#1632—614no—
243#1622—614no—
244#1558—611no—
245#1491—611no—
246#1490—611no—
247C2×C61—22no—
248C2×C61—161no—
249C2×C61#300411yescharacteristic
250C2×C61#300511yescharacteristic
251C2×C61C2×#45311yescharacteristic
252D61—44no—
253C2×#137—1222no—
254#554—614no—
255#554—1222no—
256#554—1222no—
257#554—1222no—
258#532C2×D6111yescharacteristic
259C2×C4⋊Q8—611no—
260C2×#125—1221no—
261C2×C22⋊Q8—1222no—
262C2×C22⋊Q8—1221no—
263C2×C4×Q8C2×D6111yescharacteristic
264C2×C4×D4—611no—
265C2×C4×D4—1221no—
266C22×C4⋊C4—1221no—
267#497—1222no—
268#494—612no—
269#492—612no—
270#481—1221no—
271#480—21no—
272#477C2×D6112yescomplemented
273#476—612no—
274#460C2×D6112yescomplemented
275#459—612no—
276#448—21no—
277#447—1221no—
278#442—1222no—
279#440—22no—
280#432—611no—
281#432C2×D6111yescharacteristic
282C4×M16—611no—
283#333—614no—
284Q8⋊C8—614no—
285D4⋊C8—614no—
286C61#1852511yescharacteristic, complemented, Sylow
287C2×#41—21no—
288C22×Q8—612no—
289C22×Q8C22⋊D6111yescharacteristic
290C22×D4—611no—
291C23×C4—1222no—
292C23×C4—1221no—
293#137—2448no—
294C2×Q16—1224no—
295C2×SD16—22no—
296C2×SD16—1224no—
297C2×D8—1222no—
298C2×M16—1221no—
299C4⋊Q8—612no—
300C4⋊Q8—612no—
301C4⋊Q8—1221no—
302C4⋊Q8—1221no—
303C4⋊Q8C22×D6111yescharacteristic
304#127C22×D6111yescharacteristic
305#125—1224no—
306#125—1224no—
307#124—21no—
308C22⋊Q8—1228no—
309C22⋊Q8—1228no—
310C22⋊Q8—1224no—
311C22⋊Q8—1224no—
312#121—21no—
313#121—21no—
314C4×Q8—21no—
315C4×Q8—21no—
316C4×Q8C22⋊D6112yes—
317C4×Q8C22×D6112yes—
318C4×D4—21no—
319C4×D4—21no—
320C4×D4—612no—
321C4×D4—612no—
322C4×D4—1224no—
323C4×D4—1224no—
324C4×D4—1221no—
325C4×D4—1221no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—611no—
328C2×C4⋊C4—1224no—
329C2×C4⋊C4—1222no—
330C2×C4⋊C4—1222no—
331C2×C4⋊C4—1221no—
332C2×C4⋊C4—1221no—
333C2×C4⋊C4—1224no—
334C2×C4⋊C4C22⋊D6111yescharacteristic
335C2×C22⋊C4—1222no—
336C2×C22⋊C4—1221no—
337C2×C42—21no—
338C2×C42C22×D6111yescharacteristic
339#107—22no—
340#106—1222no—
341C4⋊C8—612no—
342C4⋊C8C22×D6112yes—
343#103—22no—
344#102—22no—
345#102—1222no—
346#102—1222no—
347C22⋊C8—21no—
348C22⋊C8—1221no—
349#97—21no—
350#97—1221no—
351#97—1221no—
352C4×C8—612no—
353C24—1221no—
354#41—44no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—42no—
358C2×Q8—614no—
359C2×Q8—1222no—
360C2×Q8—2444no—
361C2×Q8C2×C22⋊D6112yes—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—41no—
366C2×D4—612no—
367C2×D4—1224no—
368C2×D4—1221no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—41no—
372C22×C4—1228no—
373C22×C4—1221no—
374C22×C4—1222no—
375C22×C4—1222no—
376C22×C4—1222no—
377C22×C4—4884no—
378C22×C4—2444no—
379C22×C4D4×D6111yescharacteristic
380C22×C4D4×D6111yescharacteristic
381C22×C4C2×C22⋊D6111yescharacteristic
382Q16—2448no—
383SD16—44no—
384SD16—2448no—
385D8—2444no—
386M16—1224no—
387C2×C8—22no—
388C2×C8—1222no—
389C2×C8—1222no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—612no—
396C4⋊C4—1228no—
397C4⋊C4—1228no—
398C4⋊C4—1222no—
399C4⋊C4—1222no—
400C4⋊C4—1221no—
401C4⋊C4—1222no—
402C4⋊C4—1222no—
403C4⋊C4C2×C22⋊D6112yes—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C22⋊C4—1228no—
407C22⋊C4—1222no—
408C22⋊C4—1222no—
409C42—22no—
410C42—22no—
411C42D4×D6111yescharacteristic
412C42D4×D6111yescharacteristic
413C42C2×C22⋊D6111yescharacteristic
414C42C23×D6111yescharacteristic
415C23—41no—
416C23—1224no—
417C23—1222no—
418C23—2442no—
419C23—2442no—
420C23#1451974511yescharacteristic
421Q8—22no—
422Q8—22no—
423Q8—42no—
424Q8—42no—
425Q8—4882no—
426Q8—2444no—
427D4—42no—
428D4—42no—
429D4—42no—
430D4—82no—
431D4—614no—
432D4—1224no—
433D4—1222no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—82no—
443C2×C4—1222no—
444C2×C4—1224no—
445C2×C4—4888no—
446C2×C4—2444no—
447C2×C4—2442no—
448C2×C4—2444no—
449C2×C4—2444no—
450C2×C4#1451974511yescharacteristic
451C2×C4#1451974511yescharacteristic
452C2×C4#1451974511yescharacteristic
453C2×C4C2×D4×D6111yescharacteristic
454C2×C4C2×D4×D6111yescharacteristic
455C2×C4C22×C22⋊D6111yescharacteristic
456C8—42no—
457C8—1224no—
458C8—2442no—
459C22—21no—
460C22—21no—
461C22—81no—
462C22—81no—
463C22—81no—
464C22—1224no—
465C22—2448no—
466C22—2442no—
467C22—2442no—
468C22#1683992812yes—
469C22C2×#1451974511yescharacteristic
470C4—21no—
471C4—21no—
472C4—21no—
473C4—21no—
474C4—42no—
475C4—42no—
476C4—81no—
477C4—2444no—
478C4—2442no—
479C4#137×D6112yes—
480C4#1684056012yes—
481C2—22no—
482C2—161no—
483C2—2444no—
484C2#2145505111yescharacteristic
485C2#2145505911yescharacteristic
486C2C2×#1683992811yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3112C2
socleC22×C61C2×#120
prime radical#21455906C2
hypercenter#3112D61
hypocenterC61#18525
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3112

Complements

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

NH ≅ G/Ntype
1. C2×#16840472C2semidirect
2. #21456279C2semidirect
4. #21444952C2semidirect
5. #21444890C2semidirect
6. #21444544C2semidirect
7. #21444266C2semidirect
8. #21444261C2semidirect
9. C61×#3112C2semidirect
23. #16840141C22semidirect
24. #16840134C22semidirect
32. #16839510C22semidirect
33. Q8⋊#92418C22semidirect
38. C61×#477C22semidirect
39. C61×#460C22semidirect
206. D244#137semidirect
240. #3112D61semidirect
272. #477C2×D61semidirect
274. #460C2×D61semidirect
286. C61#18525semidirect

Maximal subgroups

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

subgroupindex
C2×#168404722
#214562792
#214559062
#214449522
#214448902
#214445442
#214442662
#214442612
C61×#31122
#1852561