Finite Group Databasebeta

← #29835453

Subgroups — #29835453

Order 15616.

834 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840205C211yesmaximal, characteristic, complemented
2C2×#16840202C211yesmaximal, characteristic, complemented
3C2×#16840198C211yesmaximal, characteristic, complemented
4#21456243C211yesmaximal, characteristic, complemented
5C2×#16840145C211yesmaximal, characteristic, complemented
6#21448089C212yesmaximal, complemented
7#21448082C212yesmaximal, complemented
8#21448079C212yesmaximal, complemented
9#21448077C212yesmaximal, complemented
10M32×Dic61C211yesmaximal, characteristic, complemented
11C2×C61×#509C211yesmaximal, characteristic, complemented
12C2×#14519799C2211yescharacteristic
13C2×#14519797—21no—
14C22×Dic244C2211yescharacteristic
15C22×#11758200—21no—
16#16840308—21no—
17C2×#14519764—21no—
18#16840205—24no—
19#16840202—24no—
20#16840198—24no—
21C2×Dic488C2212yescomplemented
22C2×#14519723C2212yescomplemented
23C2×#14519722C2212yescomplemented
24C2×#14519721C2212yes—
25C2×#14519721C2212yescomplemented
26C2×#14519718C2212yes—
27C2×#14519717C2211yescharacteristic
28#16840147C2211yescharacteristic, complemented
29#16840145—21no—
30#16840145—21no—
31#16840145C2212yes—
32#16840145C2212yes—
33#16840145C2211yescharacteristic, complemented
34#16840143—21no—
35#16840140C2211yescharacteristic, complemented
36#16840137C2211yescharacteristic, complemented
37#16840133—21no—
38C2×C8×Dic61C2211yescharacteristic
39#16839537C2212yescomplemented
40C16×Dic61C2212yescomplemented
41C2×#135×C61C2211yescharacteristic
42C22×D8×C61C2211yescharacteristic
43C61×#509—24no—
44C2×SD32×C61C2212yes—
45C2×D16×C61C2212yes—
46C2×M32×C61C2211yescharacteristic
47C2×#11758222—21no—
48C22×Dic122—21no—
49#14519799—22no—
50#14519799—22no—
51#14519797—44no—
52C2×Dic244—22no—
53C2×Dic244—22no—
54C2×Dic244—21no—
55C2×Dic244C2312yes—
56C2×Dic244C2311yescharacteristic
57C2×#11758202—21no—
58C2×#11758190—21no—
59C2×#11758200—22no—
60C2×#11758200—22no—
61C2×#11758200—21no—
62C2×#11758200—44no—
63C2×#11758201—21no—
64C2×D488C2311yescharacteristic
65C2×#11758199—21no—
66C22×#92418D411yescharacteristic
67#14519775—21no—
68#14519773—21no—
69#14519766—21no—
70#14519764—22no—
71#14519764—22no—
72#14519764—21no—
73#14519764—42no—
74#14519749—42no—
75C4×C22⋊D61—42no—
76C2×C22⋊Dic61—42no—
77C2×C4×Dic61D411yescharacteristic
78Dic488—24no—
79#14519723—24no—
80#14519722—24no—
81#14519721—24no—
82#14519721—24no—
83#14519718—24no—
84#14519717D414yescomplemented
85C2×#11758185C2312yes—
86C8×Dic61D411yescharacteristic
87C8×Dic61D411yescharacteristic
88C8×Dic61C2311yescharacteristic
89C8×Dic61C2311yescharacteristic
90C2×#41×C61—21no—
91C22×D4×C61—21no—
92#135×C61—22no—
93#135×C61—22no—
94C2×Q16×C61C2311yescharacteristic
95C2×SD16×C61—21no—
96C2×D8×C61—22no—
97C2×D8×C61—22no—
98C2×D8×C61—21no—
99C2×D8×C61C2312yes—
100C2×D8×C61C2311yescharacteristic
101C22×C8×C61C2311yescharacteristic
102SD32×C61—24no—
103D16×C61—24no—
104M32×C61D414yescomplemented
105C2×C16×C61C2312yes—
106C24×C61—41no—
107#41×C61—42no—
108#41×C61—42no—
109C2×Q8×C61—21no—
110C2×D4×C61—22no—
111C2×D4×C61—21no—
112C2×D4×C61—41no—
113C2×D4×C61—42no—
114C2×D4×C61—42no—
115C2×D4×C61—41no—
116C22×C4×C61—41no—
117C22×C4×C61C2×D411yescharacteristic
118C2×C22⋊D61—41no—
119C2×C22⋊D61—42no—
120C22×Dic61—41no—
121C22×Dic61D812yes—
122#11758222—42no—
123#11758222—42no—
124C2×D244—21no—
125C2×C4×D61—41no—
126C2×Dic122—22no—
127C2×Dic122—21no—
128C2×Dic122—42no—
129C2×Dic122—42no—
130C2×Dic122—41no—
131Q16×C61—21no—
132Q16×C61—21no—
133SD16×C61—42no—
134D8×C61—21no—
135D8×C61—22no—
136D8×C61—22no—
137D8×C61—21no—
138D8×C61—42no—
139C2×C8×C61C2×D411yescharacteristic
140C2×C8×C61C2×D412yes—
141C2×C8×C61C2×D412yes—
142C2×C8×C61C2411yescharacteristic
143C22⋊Dic61—44no—
144C22⋊Dic61—44no—
145C22⋊Dic61—42no—
146#11758202—42no—
147#11758201—42no—
148#11758200—44no—
149#11758200—44no—
150#11758200—42no—
151#11758199—42no—
152#11758198—42no—
153#11758196—42no—
154C4×Dic61D812yes—
155C4×Dic61C2×D411yescharacteristic
156C4×Dic61C2×D411yescharacteristic
157C2×#92418D814yes—
158C2×#92418C2×D411yescharacteristic
159C2×#92418C2×D411yescharacteristic
160Dic244—21no—
161Dic244—22no—
162Dic244—22no—
163Dic244—21no—
164Dic244—42no—
165D488—21no—
166D488—21no—
167#11758190—42no—
168C16×C61C2×D414yes—
169#11758185C2×D414yes—
170C23×C61—42no—
171C23×C61—41no—
172C23×C61—84no—
173C23×C61—82no—
174C23×C61C2×D811yescharacteristic
175C22×D61—41no—
176Q8×C61—41no—
177Q8×C61—41no—
178D4×C61—42no—
179D4×C61—41no—
180D4×C61—41no—
181D4×C61—42no—
182D4×C61—82no—
183D4×C61—82no—
184C2×C4×C61—41no—
185C2×C4×C61—82no—
186C2×C4×C61#12712yes—
187C2×C4×C61C2×D812yes—
188C2×C4×C61C2×D811yescharacteristic
189C2×C4×C61C22×D411yescharacteristic
190C22⋊D61—84no—
191C22⋊D61—82no—
192C2×Dic61—42no—
193C2×Dic61—41no—
194C2×Dic61—42no—
195C2×Dic61—82no—
196C2×Dic61C2×D812yes—
197C2×Dic61C2×D812yes—
198D244—41no—
199D244—41no—
200C4×D61—82no—
201Dic122—41no—
202Dic122—42no—
203Dic122—42no—
204Dic122—41no—
205Dic122—82no—
206C8×C61#12712yes—
207C8×C61C22×D411yescharacteristic
208C8×C61C22×D411yescharacteristic
209#92418C2×D812yes—
210#92418C2×D812yes—
211#33719—611nomaximal, Sylow
212C22×C61—22no—
213C22×C61—81no—
214C22×C61—82no—
215C22×C61—82no—
216C22×C61—84no—
217C22×C61—164no—
218C22×C61#49312yes—
219C22×C61C22×D811yescharacteristic
220C2×D61—81no—
221C2×D61—82no—
222C4×C61—81no—
223C4×C61#49312yes—
224C4×C61C2×#12711yescharacteristic
225C4×C61C22×D811yescharacteristic
226Dic61—22no—
227Dic61—22no—
228Dic61—82no—
229Dic61—81no—
230C2×#510—611no—
231C2×#510—611no—
232C2×#509—611no—
233C2×#509D6111yescharacteristic, complemented
234#3147—611no—
235C2×#495—611no—
236#2251—612no—
237#2250—612no—
238#2249—612no—
239#2249—612no—
240C4×M32—611no—
241C2×C61—22no—
242C2×C61—161no—
243C2×C61—162no—
244C2×C61C2×#49311yescharacteristic
245C2×C61#341412yes—
246D61—161no—
247C2×#135—611no—
248C2×#135—1221no—
249C2×#135C2×D6111yescharacteristic
250C22×Q16—611no—
251C22×SD16—1221no—
252C22×D8C2×D6111yescharacteristic
253#532—1221no—
254C2×#124—1221no—
255#510—1224no—
256#510—1224no—
257#509—24no—
258#509—1224no—
259C2×Q32—612no—
260C2×Q32—612no—
261C2×SD32—612no—
262C2×SD32—612no—
263C2×SD32—612no—
264C2×SD32C2×D6112yes—
265C2×D16—612no—
266C2×D16C2×D6112yes—
267C2×M32—611no—
268C2×M32C2×D6111yescharacteristic
269#495—611no—
270#495—612no—
271#495—612no—
272#495—611no—
273#495—1221no—
274#495—1221no—
275#495—1221no—
276#494—611no—
277#493—611no—
278#492—1221no—
279C2×C4×C8—611no—
280#346—612no—
281C4×C16—612no—
282C61#3371911yescharacteristic, complemented, Sylow
283C2×#41—21no—
284C2×#41—1221no—
285C22×Q8—1221no—
286C22×D4—21no—
287#135—22no—
288#135—22no—
289#135—1222no—
290#135—1222no—
291#135—2444no—
292C2×Q16—612no—
293C2×Q16—611no—
294C2×Q16—1221no—
295C2×Q16—1222no—
296C2×Q16—1221no—
297C2×Q16—1222no—
298C2×Q16C22×D6111yescharacteristic
299C2×SD16—21no—
300C2×SD16—1222no—
301C2×SD16—1222no—
302C2×SD16—1221no—
303C2×SD16—1221no—
304C2×SD16—1221no—
305C2×SD16—2444no—
306C2×D8—22no—
307C2×D8—21no—
308C2×D8—22no—
309C2×D8—611no—
310C2×D8—1221no—
311C2×D8C22×D6112yes—
312C2×D8C22×D6111yescharacteristic
313C22×C8—611no—
314C22×C8C22×D6111yescharacteristic
315C4⋊Q8—1221no—
316#127—1221no—
317#124—1222no—
318#124—1222no—
319#124—1221no—
320#124—1221no—
321#124—2442no—
322#121—2442no—
323C4×D4—2442no—
324C2×C22⋊C4—2442no—
325C2×C42—611no—
326Q32—1224no—
327Q32—1224no—
328SD32—24no—
329SD32—1224no—
330SD32—1224no—
331SD32—1224no—
332D16—24no—
333D16—1224no—
334M32—614no—
335M32C22⋊D6114yescomplemented
336C2×C16—612no—
337C2×C16C22×D6112yes—
338C4×C8—611no—
339C4×C8—611no—
340C4×C8—611no—
341C4×C8—611no—
342C24—41no—
343#41—42no—
344#41—42no—
345#41—2442no—
346#41—2442no—
347C2×Q8—21no—
348C2×Q8—1222no—
349C2×Q8—1221no—
350C2×Q8—2442no—
351C2×Q8—2442no—
352C2×Q8—2441no—
353C2×D4—22no—
354C2×D4—21no—
355C2×D4—41no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4—41no—
359C2×D4—1221no—
360C2×D4—2442no—
361C2×D4—2441no—
362C22×C4—41no—
363C22×C4—612no—
364C22×C4—2441no—
365C22×C4—2441no—
366C22×C4D4×D6111yescharacteristic
367Q16—21no—
368Q16—21no—
369Q16—1221no—
370Q16—1222no—
371Q16—1222no—
372Q16—1221no—
373Q16—2442no—
374Q16—2442no—
375SD16—42no—
376SD16—2444no—
377SD16—2444no—
378SD16—2442no—
379SD16—2442no—
380SD16—2442no—
381D8—22no—
382D8—21no—
383D8—21no—
384D8—22no—
385D8—42no—
386D8—1221no—
387D8—1221no—
388D8—2442no—
389C2×C8—611no—
390C2×C8—611no—
391C2×C8—614no—
392C2×C8D4×D6112yes—
393C2×C8D4×D6111yescharacteristic
394C2×C8C2×C22⋊D6112yes—
395C2×C8C23×D6111yescharacteristic
396C4⋊C4—2442no—
397C22⋊C4—2442no—
398C22⋊C4—2444no—
399C22⋊C4—2444no—
400C22⋊C4—2442no—
401C42—611no—
402C42—611no—
403C42—612no—
404C16—614no—
405C16C2×C22⋊D6114yes—
406C23—42no—
407C23—41no—
408C23—84no—
409C23—82no—
410C23—2441no—
411C23D8×D6111yescharacteristic
412Q8—41no—
413Q8—41no—
414Q8—4882no—
415Q8—2442no—
416Q8—2441no—
417Q8—2441no—
418Q8—2442no—
419D4—41no—
420D4—42no—
421D4—42no—
422D4—41no—
423D4—82no—
424D4—82no—
425D4—4884no—
426D4—4882no—
427D4—2441no—
428D4—2441no—
429C2×C4—41no—
430C2×C4—82no—
431C2×C4—612no—
432C2×C4—612no—
433C2×C4—4882no—
434C2×C4—4882no—
435C2×C4—2442no—
436C2×C4—2442no—
437C2×C4—2441no—
438C2×C4#1451977312yes—
439C2×C4D8×D6112yes—
440C2×C4D8×D6111yescharacteristic
441C2×C4C2×D4×D6111yescharacteristic
442C8—612no—
443C8—612no—
444C8#1451977312yes—
445C8C2×D4×D6111yescharacteristic
446C8C22×C22⋊D6111yescharacteristic
447C22—22no—
448C22—82no—
449C22—84no—
450C22—81no—
451C22—82no—
452C22—164no—
453C22—4882no—
454C22—4881no—
455C22#1684013712yes—
456C22C2×D8×D6111yescharacteristic
457C4—81no—
458C4—1222no—
459C4—1222no—
460C4—4881no—
461C4—4882no—
462C4#1684013712yes—
463C4C2×#1451977311yescharacteristic
464C4C2×D8×D6111yescharacteristic
465C2—22no—
466C2—162no—
467C2—161no—
468C2—9761no—
469C2C2×#1684013711yescharacteristic
470C2#2145895112yes—
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC2×C61×#509C2
socleC22×C61C22×D8
hypercenterC2×#509D61
hypocenterC61#33719
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#509

Complements

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

NH ≅ G/Ntype
1. C2×#16840205C2semidirect
2. C2×#16840202C2semidirect
3. C2×#16840198C2semidirect
4. #21456243C2semidirect
5. C2×#16840145C2semidirect
6. #21448089C2semidirect
7. #21448082C2semidirect
8. #21448079C2semidirect
9. #21448077C2semidirect
10. M32×Dic61C2semidirect
11. C2×C61×#509C2semidirect
21. C2×Dic488C22semidirect
22. C2×#14519723C22semidirect
23. C2×#14519722C22semidirect
25. C2×#14519721C22semidirect
28. #16840147C22semidirect
33. #16840145C22semidirect
35. #16840140C22semidirect
36. #16840137C22semidirect
39. #16839537C22semidirect
40. C16×Dic61C22semidirect
84. #14519717D4semidirect
104. M32×C61D4semidirect
233. C2×#509D61semidirect
282. C61#33719semidirect
335. M32C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168402052
C2×#168402022
C2×#168401982
#214562432
C2×#168401452
#214480892
#214480822
#214480792
#214480772
M32×Dic612
C2×C61×#5092
#3371961