Finite Group Databasebeta

← #29833908

Subgroups — #29833908

Order 15616.

1132 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840562C211yesmaximal, characteristic, complemented
2#21458953C212yesmaximal, complemented
3#21447769C214yesmaximal, complemented
4#21447764C214yesmaximal, complemented
5#21447554C212yesmaximal, complemented
6C2×#16839544C211yesmaximal, characteristic, complemented
7C61×#3054C211yesmaximal, characteristic, complemented
8C22×#11758222—21no—
9#16840562—24no—
10#16840562—22no—
11#16840562C2214yes—
12C2×#14519810—22no—
13C2×#14519805—22no—
14C2×#14519799—21no—
15C2×#14519799C2212yes—
16#16840203C2214yescomplemented
17#16840202—22no—
18#16840198C2214yescomplemented
19#16840197—22no—
20#16840192—24no—
21C2×#14519717C2212yescomplemented
22#16839571C2218yescomplemented
23#16839544C2214yescomplemented
24C2×#135×C61C2211yescharacteristic, complemented
25C22×M16×C61C2211yescharacteristic
26C61×#525—21no—
27C61×#470C2212yescomplemented
28C61×#469C2212yescomplemented
29C61×#468—22no—
30C61×#427C2211yescharacteristic, complemented
31C61×#416—21no—
32C2×#11758222—22no—
33C2×#11758222—24no—
34C2×#11758222—22no—
35C2×#11758222—24no—
36C2×#11758222—42no—
37C2×#11758222—41no—
38C2×#11758222—42no—
39C2×#11758222—41no—
40C22×Dic122—21no—
41C22×D244—21no—
42C22×C22⋊D61—41no—
43C22×C4×D61—41no—
44#14519810—28no—
45#14519810—28no—
46#14519805—28no—
47#14519805—28no—
48#14519799—24no—
49#14519799—24no—
50#14519799—22no—
51#14519799—22no—
52#14519799D414yescomplemented
53#14519799C2314yes—
54C2×Dic244—21no—
55C2×Dic244D412yescomplemented
56C2×#11758190—22no—
57C2×#11758190—22no—
58C2×D488—21no—
59C2×D488D412yescomplemented
60#14519723—24no—
61#14519721—24no—
62#14519720—24no—
63#14519718—24no—
64#14519717—22no—
65#14519717C2314yescomplemented
66C2×#11758185—22no—
67C2×#41×C61—21no—
68C23×C4×C61D411yescharacteristic
69#135×C61—22no—
70#135×C61—22no—
71C2×Q16×C61C2311yescharacteristic, complemented
72C2×SD16×C61—21no—
73C2×D8×C61C2311yescharacteristic, complemented
74C2×M16×C61—22no—
75C2×M16×C61—22no—
76C2×M16×C61D412yes—
77C2×M16×C61C2312yes—
78C22×C8×C61D411yescharacteristic
79C22×C8×C61C2311yescharacteristic
80#123×C61—41no—
81C22⋊Q8×C61—22no—
82#121×C61—22no—
83#120×C61—41no—
84C4×D4×C61—42no—
85#117×C61—21no—
86#107×C61—21no—
87#106×C61C2312yescomplemented
88#103×C61—22no—
89#102×C61—22no—
90C24×C61D811yescharacteristic
91C23×D61—41no—
92#41×C61—41no—
93#41×C61—41no—
94#41×C61—41no—
95#41×C61—41no—
96C2×Q8×C61—21no—
97C2×D4×C61—21no—
98C2×D4×C61—42no—
99C2×D4×C61—41no—
100C22×C4×C61—21no—
101C22×C4×C61—21no—
102C22×C4×C61—21no—
103C22×C4×C61—21no—
104C22×C4×C61—41no—
105C22×C4×C61D812yes—
106C22×C4×C61D811yescharacteristic
107C22×C4×C61C2×D412yes—
108C22×C4×C61C2×D411yescharacteristic
109C2×C22⋊D61—44no—
110C2×C22⋊D61—42no—
111C2×C22⋊D61—44no—
112C2×C22⋊D61—42no—
113C22×Dic61—41no—
114#11758222—28no—
115#11758222—24no—
116#11758222—24no—
117#11758222—44no—
118#11758222—44no—
119#11758222—44no—
120#11758222—44no—
121#11758222—44no—
122#11758222—42no—
123#11758222—42no—
124C2×D244—24no—
125C2×D244—22no—
126C2×D244—41no—
127C2×D244—42no—
128C2×C4×D61—42no—
129C2×C4×D61—44no—
130C2×C4×D61—44no—
131C2×C4×D61—42no—
132C2×Dic122—24no—
133C2×Dic122—22no—
134C2×Dic122—41no—
135C2×Dic122—42no—
136Q16×C61—22no—
137SD16×C61—42no—
138D8×C61—22no—
139M16×C61—24no—
140M16×C61—24no—
141C2×C8×C61—21no—
142C2×C8×C61—21no—
143C2×C8×C61C2×D412yes—
144C2×C8×C61C2×D411yescharacteristic
145C2×C8×C61C2×D412yes—
146C2×C8×C61C2×D412yes—
147C2×C8×C61C2411yescharacteristic
148C4⋊C4×C61—22no—
149C4⋊C4×C61—42no—
150C22⋊C4×C61—41no—
151C22⋊C4×C61—42no—
152C22⋊C4×C61—42no—
153C42×C61—41no—
154Dic244—24no—
155Dic244—22no—
156Dic244C2×D414yes—
157D488—24no—
158D488—22no—
159D488C2×D414yes—
160#11758190—28no—
161#11758190—24no—
162#11758190—24no—
163#11758185—24no—
164C23×C61—21no—
165C23×C61—21no—
166C23×C61—41no—
167C23×C61—81no—
168C23×C61C2×D812yes—
169C23×C61C2×D811yescharacteristic
170C22×D61—44no—
171C22×D61—42no—
172C22×D61—84no—
173Q8×C61—41no—
174Q8×C61—41no—
175D4×C61—41no—
176D4×C61—41no—
177D4×C61—84no—
178D4×C61—81no—
179D4×C61—81no—
180C2×C4×C61—21no—
181C2×C4×C61—21no—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—22no—
185C2×C4×C61—22no—
186C2×C4×C61—42no—
187C2×C4×C61—41no—
188C2×C4×C61—81no—
189C2×C4×C61—81no—
190C2×C4×C61#12011yescharacteristic
191C2×C4×C61#12011yescharacteristic
192C2×C4×C61C2×D812yes—
193C2×C4×C61C2×D811yescharacteristic
194C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
195C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
196C2×C4×C61C22×D411yescharacteristic
197C22⋊D61—48no—
198C22⋊D61—44no—
199C22⋊D61—44no—
200C2×Dic61—44no—
201C2×Dic61—42no—
202D244—24no—
203D244—44no—
204D244—42no—
205C4×D61—48no—
206C4×D61—44no—
207C4×D61—44no—
208Dic122—24no—
209Dic122—44no—
210Dic122—42no—
211C8×C61—22no—
212C8×C61#12012yes—
213C8×C61C22×D412yes—
214#33530—611nomaximal, Sylow
215C22×C61—22no—
216C22×C61—41no—
217C22×C61—42no—
218C22×C61—42no—
219C22×C61—81no—
220C22×C61—82no—
221C22×C61—81no—
222C22×C61#44711yescharacteristic
223C22×C61#44711yescharacteristic
224C22×C61C22×D811yescharacteristic
225C2×D61—44no—
226C2×D61—84no—
227C2×D61—88no—
228C4×C61—22no—
229C4×C61—82no—
230C4×C61—81no—
231C4×C61#44711yescharacteristic
232C4×C61#44711yescharacteristic
233C4×C61C2×#12011yescharacteristic
234C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
235Dic61—44no—
236C2×#575—611no—
237#3414—612no—
238#3054D6111yescharacteristic, complemented
239#2191—614no—
240#2190—614no—
241#2145—612no—
242C2×#349—611no—
243C2×C61—21no—
244C2×C61—21no—
245C2×C61—21no—
246C2×C61—42no—
247C2×C61—161no—
248C2×C61C2×#44711yescharacteristic
249D61—84no—
250C22×#41—1221no—
251#575—614no—
252#575—1224no—
253#575—1222no—
254C2×#137—1222no—
255C2×AGL(1,ℤ/8)—1222no—
256C2×#135—612no—
257C2×#135—1221no—
258C2×#135C2×D6111yescharacteristic, complemented
259C22×M16C2×D6111yescharacteristic
260#525—21no—
261#510—614no—
262#510—1222no—
263#509—614no—
264#509—1222no—
265#508—1224no—
266C2×M32—612no—
267#470C2×D6112yescomplemented
268#469C2×D6112yescomplemented
269#468—22no—
270#427C2×D6111yescharacteristic, complemented
271#416—21no—
272#360—618no—
273#349—614no—
274C61#3353011yescharacteristic, complemented, Sylow
275C2×#41—21no—
276C2×#41—1222no—
277C2×#41—1224no—
278C2×#41—1224no—
279C2×#41—1222no—
280C2×#41—2441no—
281C2×#41—2442no—
282C2×#41—2442no—
283C2×#41—2441no—
284C22×Q8—1221no—
285C22×D4—1221no—
286C22×D4—2441no—
287C23×C4—2441no—
288C23×C4C22⋊D6111yescharacteristic
289#137—1228no—
290#137—1228no—
291AGL(1,ℤ/8)—1228no—
292AGL(1,ℤ/8)—1228no—
293#135—22no—
294#135—22no—
295#135—614no—
296#135—614no—
297#135—1224no—
298#135—1222no—
299#135—1222no—
300#135—1224no—
301C2×Q16—612no—
302C2×Q16—1221no—
303C2×Q16C22×D6111yescharacteristic, complemented
304C2×SD16—21no—
305C2×SD16—1222no—
306C2×SD16—1222no—
307C2×D8—612no—
308C2×D8—1221no—
309C2×D8C22×D6111yescharacteristic, complemented
310C2×M16—22no—
311C2×M16—22no—
312C2×M16C22⋊D6112yes—
313C2×M16C22×D6112yes—
314C22×C8C22⋊D6111yescharacteristic
315C22×C8C22×D6111yescharacteristic
316#123—41no—
317C22⋊Q8—22no—
318#121—22no—
319#120—41no—
320C4×D4—42no—
321#117—21no—
322Q32—1224no—
323SD32—1224no—
324SD32—1224no—
325D16—1224no—
326M32—614no—
327M32—1222no—
328C2×C16—1222no—
329#107—21no—
330#106C22×D6112yescomplemented
331#103—22no—
332#102—22no—
333C24—2441no—
334C24#1175819911yescharacteristic
335#41—41no—
336#41—41no—
337#41—41no—
338#41—41no—
339#41—1228no—
340#41—1224no—
341#41—1224no—
342#41—2444no—
343#41—2444no—
344#41—2444no—
345#41—2444no—
346#41—2444no—
347#41—2442no—
348#41—2442no—
349C2×Q8—21no—
350C2×Q8—1224no—
351C2×Q8—1222no—
352C2×Q8—2441no—
353C2×Q8—2442no—
354C2×D4—21no—
355C2×D4—42no—
356C2×D4—41no—
357C2×D4—1224no—
358C2×D4—1222no—
359C2×D4—2441no—
360C2×D4—2442no—
361C2×D4—2444no—
362C2×D4—2442no—
363C2×D4—2444no—
364C2×D4—2442no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—41no—
370C22×C4—2442no—
371C22×C4—2444no—
372C22×C4—2444no—
373C22×C4—2441no—
374C22×C4—2442no—
375C22×C4#1175819912yes—
376C22×C4#1175819911yescharacteristic
377C22×C4C2×C22⋊D6112yes—
378C22×C4C2×C22⋊D6111yescharacteristic
379Q16—22no—
380Q16—614no—
381Q16—1224no—
382Q16—1222no—
383SD16—42no—
384SD16—1228no—
385SD16—1224no—
386SD16—1224no—
387D8—22no—
388D8—614no—
389D8—1224no—
390D8—1222no—
391M16—24no—
392M16—24no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8D4×D6112yes—
396C2×C8D4×D6112yes—
397C2×C8C2×C22⋊D6112yes—
398C2×C8C2×C22⋊D6111yescharacteristic
399C2×C8C23×D6111yescharacteristic
400C4⋊C4—22no—
401C4⋊C4—42no—
402C22⋊C4—41no—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C42—41no—
406C16—1224no—
407C23—21no—
408C23—21no—
409C23—41no—
410C23—81no—
411C23—4884no—
412C23—2444no—
413C23—2442no—
414C23C2×#1175819912yes—
415C23C2×#1175819911yescharacteristic
416Q8—41no—
417Q8—41no—
418Q8—1224no—
419Q8—2444no—
420Q8—2442no—
421D4—41no—
422D4—41no—
423D4—84no—
424D4—81no—
425D4—81no—
426D4—1224no—
427D4—2444no—
428D4—2448no—
429D4—2444no—
430D4—2444no—
431D4—2442no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—2448no—
443C2×C4—2444no—
444C2×C4—2442no—
445C2×C4—2444no—
446C2×C4—2444no—
447C2×C4#1451974511yescharacteristic
448C2×C4#1451974511yescharacteristic
449C2×C4C2×#1175819912yes—
450C2×C4C2×#1175819911yescharacteristic
451C2×C4#1451980211yescharacteristic
452C2×C4#1451980211yescharacteristic
453C2×C4C22×C22⋊D6111yescharacteristic
454C8—22no—
455C8#1451974512yes—
456C8C2×D4×D6112yes—
457C22—22no—
458C22—41no—
459C22—42no—
460C22—42no—
461C22—81no—
462C22—82no—
463C22—81no—
464C22—4884no—
465C22—4888no—
466C22—2444no—
467C22#1683989511yescharacteristic
468C22#1683989511yescharacteristic
469C22C22×#1175819911yescharacteristic
470C4—22no—
471C4—82no—
472C4—81no—
473C4—2444no—
474C4#1683989511yescharacteristic
475C4#1683989511yescharacteristic
476C4C2×#1451974511yescharacteristic
477C4C2×#1451980211yescharacteristic
478C2—21no—
479C2—21no—
480C2—21no—
481C2—42no—
482C2—161no—
483C2—4884no—
484C2C2×#1683989511yescharacteristic
subgroupG/H
centerC4#16839895
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#3054C2
socleC2×C61C2×#447
hypercenter#3054D61
hypocenterC61#33530
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×M16 ◁ #3054

Complements

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

NH ≅ G/Ntype
1. C2×#16840562C2semidirect
2. #21458953C2semidirect
3. #21447769C2semidirect
4. #21447764C2semidirect
5. #21447554C2semidirect
6. C2×#16839544C2semidirect
7. C61×#3054C2semidirect
16. #16840203C22semidirect
18. #16840198C22semidirect
21. C2×#14519717C22semidirect
22. #16839571C22semidirect
23. #16839544C22semidirect
24. C2×#135×C61C22semidirect
27. C61×#470C22semidirect
28. C61×#469C22semidirect
30. C61×#427C22semidirect
52. #14519799D4semidirect
55. C2×Dic244D4semidirect
59. C2×D488D4semidirect
65. #14519717C23semidirect
71. C2×Q16×C61C23semidirect
73. C2×D8×C61C23semidirect
87. #106×C61C23semidirect
238. #3054D61semidirect
258. C2×#135C2×D61semidirect
267. #470C2×D61semidirect
268. #469C2×D61semidirect
270. #427C2×D61semidirect
274. C61#33530semidirect
303. C2×Q16C22×D61semidirect
309. C2×D8C22×D61semidirect
330. #106C22×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168405622
#214589532
#214477692
#214477642
#214475542
C2×#168395442
C61×#30542
#3353061