Finite Group Databasebeta

← #29835446

Subgroups — #29835446

Order 15616.

814 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840205C211yesmaximal, characteristic, complemented
2C2×#16840203C211yesmaximal, characteristic, complemented
3C2×#16840202C211yesmaximal, characteristic, complemented
4#21456243C211yesmaximal, characteristic
5C2×#16840140C211yesmaximal, characteristic, complemented
6#21448089C212yesmaximal, complemented
7#21448087C212yesmaximal, complemented
8#21448073C212yesmaximal, complemented
9#21448070C212yesmaximal, complemented
10M32×Dic61C211yesmaximal, characteristic, complemented
11C2×C61×#510C211yesmaximal, characteristic, complemented
12C2×#14519799C2211yescharacteristic
13C2×#14519797—21no—
14C22×Dic244C2211yescharacteristic
15C22×#11758202—21no—
16#16840308—21no—
17C2×#14519775—21no—
18#16840205—24no—
19#16840203—24no—
20#16840202—24no—
21C2×Dic488C2212yes—
22C2×#14519723C2212yes—
23C2×#14519723C2212yes—
24C2×#14519722C2212yes—
25C2×#14519721C2212yes—
26C2×#14519720C2212yes—
27C2×#14519717C2211yescharacteristic
28#16840147C2211yescharacteristic
29#16840145C2211yescharacteristic
30#16840143—21no—
31#16840140—21no—
32#16840140—21no—
33#16840140C2212yes—
34#16840140C2212yes—
35#16840140C2211yescharacteristic
36#16840137C2211yescharacteristic
37#16840133—21no—
38C2×C8×Dic61C2211yescharacteristic
39#16839537C2212yes—
40C16×Dic61C2212yes—
41C2×#135×C61C2211yescharacteristic
42C22×Q16×C61C2211yescharacteristic
43C61×#510—24no—
44C2×Q32×C61C2212yes—
45C2×SD32×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—22no—
58C2×#11758202—22no—
59C2×#11758202—21no—
60C2×#11758202—44no—
61C2×#11758190—21no—
62C2×#11758200—21no—
63C2×#11758201—21no—
64C2×D488C2311yescharacteristic
65C2×#11758199—21no—
66C22×#92418D411yescharacteristic
67#14519775—22no—
68#14519775—22no—
69#14519775—21no—
70#14519775—42no—
71#14519773—21no—
72#14519766—21no—
73#14519764—21no—
74#14519749—42no—
75C4×C22⋊D61—42no—
76C2×#11758196—42no—
77C2×C4×Dic61D411yescharacteristic
78Dic488—24no—
79#14519723—24no—
80#14519723—24no—
81#14519722—24no—
82#14519721—24no—
83#14519720—24no—
84#14519717D414yescomplemented
85C2×#11758185C2312yes—
86C8×Dic61D411yescharacteristic
87C8×Dic61D411yescharacteristic
88C8×Dic61C2311yescharacteristic
89C8×Dic61C2311yescharacteristic
90C2×#41×C61—21no—
91C22×Q8×C61—21no—
92#135×C61—22no—
93#135×C61—22no—
94C2×Q16×C61—22no—
95C2×Q16×C61—22no—
96C2×Q16×C61—21no—
97C2×Q16×C61C2312yes—
98C2×Q16×C61C2311yescharacteristic
99C2×SD16×C61—21no—
100C2×D8×C61C2311yescharacteristic
101C22×C8×C61C2311yescharacteristic
102Q32×C61—24no—
103SD32×C61—24no—
104M32×C61D414yescomplemented
105C2×C16×C61C2312yes—
106#41×C61—42no—
107#41×C61—42no—
108C2×Q8×C61—21no—
109C2×Q8×C61—22no—
110C2×Q8×C61—41no—
111C2×Q8×C61—42no—
112C2×Q8×C61—42no—
113C2×D4×C61—21no—
114C2×D4×C61—41no—
115C22×C4×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—22no—
132Q16×C61—21no—
133Q16×C61—22no—
134Q16×C61—21no—
135Q16×C61—42no—
136SD16×C61—42no—
137D8×C61—21no—
138D8×C61—21no—
139C2×C8×C61C2×D411yescharacteristic
140C2×C8×C61C2×D412yes—
141C2×C8×C61C2×D412yes—
142C2×C8×C61C2411yescharacteristic
143C22⋊Dic61—42no—
144#11758202—44no—
145#11758202—44no—
146#11758202—42no—
147#11758201—42no—
148#11758200—42no—
149#11758199—42no—
150#11758198—42no—
151#11758196—44no—
152#11758196—44no—
153#11758196—42no—
154C4×Dic61D812yes—
155C4×Dic61C2×D411yescharacteristic
156C4×Dic61C2×D411yescharacteristic
157C2×#92418D814yes—
158C2×#92418C2×D411yescharacteristic
159C2×#92418C2×D411yescharacteristic
160Dic244—22no—
161Dic244—22no—
162Dic244—21no—
163Dic244—21no—
164Dic244—42no—
165D488—21no—
166D488—21no—
167#11758190—42no—
168C16×C61C2×D414yes—
169#11758185C2×D414yes—
170C23×C61—41no—
171C23×C61C2×D811yescharacteristic
172C22×D61—41no—
173Q8×C61—41no—
174Q8×C61—42no—
175Q8×C61—41no—
176Q8×C61—42no—
177Q8×C61—82no—
178D4×C61—41no—
179D4×C61—41no—
180D4×C61—82no—
181C2×C4×C61—42no—
182C2×C4×C61—41no—
183C2×C4×C61—82no—
184C2×C4×C61—82no—
185C2×C4×C61#12712yes—
186C2×C4×C61C2×D812yes—
187C2×C4×C61C2×D811yescharacteristic
188C2×C4×C61C22×D411yescharacteristic
189C22⋊D61—82no—
190C22⋊D61—84no—
191C2×Dic61—24no—
192C2×Dic61—42no—
193C2×Dic61—41no—
194C2×Dic61—82no—
195C2×Dic61C2×D812yes—
196C2×Dic61C2×D812yes—
197D244—41no—
198D244—41no—
199C4×D61—82no—
200Dic122—41no—
201Dic122—42no—
202Dic122—41no—
203Dic122—42no—
204Dic122—82no—
205C8×C61#12712yes—
206C8×C61C22×D411yescharacteristic
207C8×C61C22×D411yescharacteristic
208#92418C2×D812yes—
209#92418C2×D812yes—
210#33718—611nomaximal, Sylow
211C22×C61—22no—
212C22×C61—81no—
213C22×C61—82no—
214C22×C61#49312yes—
215C22×C61C22×D811yescharacteristic
216C2×D61—81no—
217C2×D61—82no—
218C4×C61—81no—
219C4×C61—82no—
220C4×C61#49312yes—
221C4×C61C2×#12711yescharacteristic
222C4×C61C22×D811yescharacteristic
223Dic61—22no—
224Dic61—82no—
225Dic61—81no—
226Dic61#51014yescomplemented
227C2×#510—611no—
228C2×#510—611no—
229C2×#510—611no—
230C2×#510D6111yescharacteristic, complemented
231#3147—611no—
232C2×#494—611no—
233#2251—612no—
234#2251—612no—
235#2248—612no—
236#2247—612no—
237C4×M32—611no—
238C2×C61—22no—
239C2×C61—161no—
240C2×C61C2×#49311yescharacteristic
241C2×C61C2×#51012yes—
242D61—161no—
243C2×#135—611no—
244C2×#135—1221no—
245C2×#135C2×D6111yescharacteristic
246C22×Q16—611no—
247C22×Q16—1221no—
248C22×Q16C2×D6111yescharacteristic
249#532—1221no—
250C2×C4⋊Q8—1221no—
251#510—24no—
252#510—1224no—
253#510—1224no—
254#510—1224no—
255C2×Q32—612no—
256C2×Q32—612no—
257C2×Q32—612no—
258C2×Q32C2×D6112yes—
259C2×SD32—612no—
260C2×SD32—612no—
261C2×SD32—612no—
262C2×SD32C2×D6112yes—
263C2×M32—611no—
264C2×M32C2×D6111yescharacteristic
265#495—611no—
266#495—611no—
267#495—1221no—
268#494—612no—
269#494—612no—
270#494—611no—
271#494—1221no—
272#494—1221no—
273#493—611no—
274#492—1221no—
275C2×C4×C8—611no—
276#346—612no—
277C4×C16—612no—
278C61#3371811yescharacteristic, complemented, Sylow
279C2×#41—21no—
280C2×#41—1221no—
281C22×Q8—21no—
282C22×Q8—1221no—
283#135—22no—
284#135—22no—
285#135—1222no—
286#135—1222no—
287#135—2444no—
288C2×Q16—22no—
289C2×Q16—21no—
290C2×Q16—22no—
291C2×Q16—612no—
292C2×Q16—611no—
293C2×Q16—1222no—
294C2×Q16—1222no—
295C2×Q16—1221no—
296C2×Q16—1222no—
297C2×Q16—1221no—
298C2×Q16—1222no—
299C2×Q16—2444no—
300C2×Q16C22×D6112yes—
301C2×Q16C22×D6111yescharacteristic
302C2×SD16—21no—
303C2×SD16—1221no—
304C2×SD16—1221no—
305C2×SD16—1221no—
306C2×D8—611no—
307C2×D8—1221no—
308C2×D8C22×D6111yescharacteristic
309C22×C8—611no—
310C22×C8C22×D6111yescharacteristic
311C4⋊Q8—1222no—
312C4⋊Q8—1222no—
313C4⋊Q8—1221no—
314C4⋊Q8—2442no—
315#127—1221no—
316#124—1221no—
317#124—1221no—
318#121—2442no—
319C4×D4—2442no—
320C2×C4⋊C4—2442no—
321C2×C42—611no—
322Q32—24no—
323Q32—1224no—
324Q32—1224no—
325Q32—1224no—
326SD32—24no—
327SD32—1224no—
328SD32—1224no—
329SD32—1224no—
330M32—614no—
331M32C22⋊D6114yescomplemented
332C2×C16—612no—
333C2×C16C22×D6112yes—
334C4×C8—611no—
335C4×C8—611no—
336C4×C8—611no—
337C4×C8—611no—
338#41—42no—
339#41—42no—
340#41—2442no—
341#41—2442no—
342C2×Q8—21no—
343C2×Q8—22no—
344C2×Q8—41no—
345C2×Q8—42no—
346C2×Q8—42no—
347C2×Q8—1221no—
348C2×Q8—1222no—
349C2×Q8—2442no—
350C2×Q8—2442no—
351C2×Q8—2441no—
352C2×D4—21no—
353C2×D4—41no—
354C2×D4—1221no—
355C2×D4—2442no—
356C2×D4—2441no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—612no—
360C22×C4—2441no—
361C22×C4—2441no—
362C22×C4D4×D6111yescharacteristic
363Q16—22no—
364Q16—21no—
365Q16—22no—
366Q16—21no—
367Q16—42no—
368Q16—1222no—
369Q16—1221no—
370Q16—1222no—
371Q16—1221no—
372Q16—2444no—
373Q16—2444no—
374Q16—2442no—
375Q16—2442no—
376SD16—42no—
377SD16—2442no—
378SD16—2442no—
379SD16—2442no—
380D8—21no—
381D8—21no—
382D8—1221no—
383D8—1221no—
384D8—2442no—
385C2×C8—611no—
386C2×C8—611no—
387C2×C8—614no—
388C2×C8D4×D6112yes—
389C2×C8D4×D6111yescharacteristic
390C2×C8C2×C22⋊D6112yes—
391C2×C8C23×D6111yescharacteristic
392C4⋊C4—2444no—
393C4⋊C4—2444no—
394C4⋊C4—2442no—
395C22⋊C4—2442no—
396C22⋊C4—2442no—
397C42—611no—
398C42—611no—
399C42—612no—
400C16—614no—
401C16C2×C22⋊D6114yes—
402C23—41no—
403C23—2441no—
404C23D8×D6111yescharacteristic
405Q8—41no—
406Q8—42no—
407Q8—41no—
408Q8—42no—
409Q8—82no—
410Q8—4882no—
411Q8—2441no—
412Q8—2442no—
413Q8—2441no—
414Q8—2442no—
415D4—41no—
416D4—41no—
417D4—82no—
418D4—4884no—
419D4—4882no—
420D4—2441no—
421D4—2441no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—612no—
427C2×C4—612no—
428C2×C4—1224no—
429C2×C4—4882no—
430C2×C4—4882no—
431C2×C4—2441no—
432C2×C4—2442no—
433C2×C4#1451977312yes—
434C2×C4D8×D6112yes—
435C2×C4D8×D6111yescharacteristic
436C2×C4C2×D4×D6111yescharacteristic
437C8—612no—
438C8—612no—
439C8#1451977312yes—
440C8C2×D4×D6111yescharacteristic
441C8C22×C22⋊D6111yescharacteristic
442C22—22no—
443C22—82no—
444C22—81no—
445C22—4882no—
446C22—4881no—
447C22#1684013712yes—
448C22C2×D8×D6111yescharacteristic
449C4—81no—
450C4—82no—
451C4—614no—
452C4—1222no—
453C4—4881no—
454C4—4882no—
455C4#1684013712yes—
456C4C2×#1451977311yescharacteristic
457C4C2×D8×D6111yescharacteristic
458C2—22no—
459C2—161no—
460C2—9761no—
461C2C2×#1684013711yescharacteristic
462C2#510×D6112yes—
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC2×C61×#510C2
socleC22×C61C22×D8
prime radical#21456243C2
hypercenterC2×#510D61
hypocenterC61#33718
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#510

Complements

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

NH ≅ G/Ntype
1. C2×#16840205C2semidirect
2. C2×#16840203C2semidirect
3. C2×#16840202C2semidirect
5. C2×#16840140C2semidirect
6. #21448089C2semidirect
7. #21448087C2semidirect
8. #21448073C2semidirect
9. #21448070C2semidirect
10. M32×Dic61C2semidirect
11. C2×C61×#510C2semidirect
84. #14519717D4semidirect
104. M32×C61D4semidirect
226. Dic61#510semidirect
230. C2×#510D61semidirect
278. C61#33718semidirect
331. 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×#168402032
C2×#168402022
#214562432
C2×#168401402
#214480892
#214480872
#214480732
#214480702
M32×Dic612
C2×C61×#5102
#3371861