Finite Group Databasebeta

← #29834286

Subgroups — #29834286

Order 15616.

770 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×#16840198C211yesmaximal, characteristic, complemented
2#21456243C211yesmaximal, characteristic, complemented
3C2×D8×Dic61C211yesmaximal, characteristic, complemented
4#21447876C212yesmaximal, complemented
5#21447872C212yesmaximal, complemented
6#21447862C212yesmaximal, complemented
7#21447852C212yesmaximal, complemented
8#21447574C211yesmaximal, characteristic, complemented
9#21447532C211yesmaximal, characteristic, complemented
10#21447531C211yesmaximal, characteristic, complemented
11C2×C61×#509C211yesmaximal, characteristic, complemented
12C2×#14519799C2211yescharacteristic
13C2×#14519797—21no—
14#16840308—21no—
15C2×D4×Dic61—21no—
16#16840198—24no—
17C2×#14519721C2212yes—
18C2×#14519718C2212yes—
19C2×#14519717C2211yescharacteristic
20#16840147C2211yescharacteristic, complemented
21#16840145C2211yescharacteristic, complemented
22#16840143—21no—
23#16840140C2211yescharacteristic, complemented
24#16840137C2211yescharacteristic, complemented
25#16840133—21no—
26D8×Dic61—21no—
27D8×Dic61—21no—
28D8×Dic61C2212yes—
29D8×Dic61C2212yes—
30C2×#14519711C2211yescharacteristic
31C2×#14519698—21no—
32C2×C8×Dic61C2211yescharacteristic
33#16839580C2212yescomplemented
34#16839580C2212yescomplemented
35#16839567C2212yescomplemented
36#16839566C2212yescomplemented
37#16839564C2212yescomplemented
38#16839563C2212yescomplemented
39C2×#135×C61C2211yescharacteristic
40C22×D8×C61C2211yescharacteristic
41C61×#509—24no—
42C2×SD32×C61C2212yes—
43C2×D16×C61C2212yes—
44C2×M32×C61C2211yescharacteristic
45C2×#11758222—21no—
46C23×Dic61—41no—
47#14519799—22no—
48#14519799—22no—
49#14519797—44no—
50C2×Dic244C2311yescharacteristic
51C2×#11758202—21no—
52C2×#11758190—21no—
53C2×#11758200—21no—
54C2×#11758201—21no—
55C2×D488C2311yescharacteristic
56C2×#11758199—21no—
57C22×#92418D411yescharacteristic
58#14519775—21no—
59#14519773—21no—
60#14519766—21no—
61#14519764—21no—
62#14519749—41no—
63#14519749—41no—
64D4×Dic61—22no—
65D4×Dic61—22no—
66D4×Dic61—41no—
67D4×Dic61—41no—
68C4×C22⋊D61—41no—
69C4×C22⋊D61—41no—
70C2×C4⋊Dic61—21no—
71C2×C22⋊Dic61—41no—
72C2×C4×Dic61D411yescharacteristic
73#14519721—24no—
74#14519718—24no—
75#14519717—22no—
76C2×#11758185C2312yes—
77#14519711—21no—
78#14519711C2312yes—
79#14519698—22no—
80#14519698—22no—
81C8×Dic61D411yescharacteristic
82C8×Dic61D411yescharacteristic
83C8×Dic61C2311yescharacteristic
84C8×Dic61C2311yescharacteristic
85C2×#41×C61—21no—
86C22×D4×C61—21no—
87#135×C61—22no—
88#135×C61—22no—
89C2×Q16×C61C2311yescharacteristic
90C2×SD16×C61—21no—
91C2×D8×C61—21no—
92C2×D8×C61—22no—
93C2×D8×C61D414yescomplemented
94C2×D8×C61C2311yescharacteristic
95C2×D8×C61C2312yes—
96C22×C8×C61C2311yescharacteristic
97SD32×C61—24no—
98D16×C61—24no—
99M32×C61—22no—
100C2×C16×C61C2312yes—
101C24×C61—41no—
102#41×C61—42no—
103#41×C61—42no—
104C2×Q8×C61—21no—
105C2×D4×C61—24no—
106C2×D4×C61—22no—
107C2×D4×C61—21no—
108C2×D4×C61—41no—
109C2×D4×C61—41no—
110C2×D4×C61—42no—
111C22×C4×C61—41no—
112C22×C4×C61C2×D411yescharacteristic
113C2×C22⋊D61—41no—
114C2×C22⋊D61—41no—
115C2×C22⋊D61—41no—
116C22×Dic61—41no—
117C22×Dic61—42no—
118C22×Dic61—42no—
119C22×Dic61—84no—
120C22×Dic61D811yescharacteristic
121C22×Dic61D811yescharacteristic
122#11758222—42no—
123#11758222—42no—
124C2×D244—21no—
125C2×C4×D61—41no—
126C2×Dic122—21no—
127Q16×C61—21no—
128Q16×C61—21no—
129SD16×C61—42no—
130D8×C61—24no—
131D8×C61—21no—
132D8×C61—22no—
133D8×C61—21no—
134D8×C61C2×D414yes—
135C2×C8×C61C2×D411yescharacteristic
136C2×C8×C61C2×D412yes—
137C2×C8×C61#4112yes—
138C2×C8×C61C2411yescharacteristic
139C22⋊Dic61—42no—
140C22⋊Dic61—42no—
141C22⋊Dic61—41no—
142C22⋊Dic61—41no—
143#11758202—42no—
144#11758201—42no—
145#11758200—42no—
146#11758199—42no—
147#11758198—41no—
148#11758198—41no—
149C4⋊Dic61—22no—
150C4⋊Dic61—41no—
151#11758196—41no—
152#11758196—41no—
153C4×Dic61D811yescharacteristic
154C4×Dic61D811yescharacteristic
155C4×Dic61C2×D411yescharacteristic
156C4×Dic61C2×D411yescharacteristic
157C2×#92418—22no—
158C2×#92418C2×D411yescharacteristic
159C2×#92418C2×D411yescharacteristic
160Dic244—21no—
161Dic244—21no—
162D488—21no—
163D488—21no—
164#11758190—42no—
165C16×C61—22no—
166#11758185—22no—
167C23×C61—42no—
168C23×C61—44no—
169C23×C61—41no—
170C23×C61—82no—
171C23×C61—82no—
172C23×C61C2×D811yescharacteristic
173C22×D61—41no—
174Q8×C61—41no—
175Q8×C61—41no—
176D4×C61—24no—
177D4×C61—41no—
178D4×C61—42no—
179D4×C61—41no—
180D4×C61—44no—
181D4×C61—82no—
182C2×C4×C61—41no—
183C2×C4×C61—82no—
184C2×C4×C61#12112yes—
185C2×C4×C61C2×D811yescharacteristic
186C2×C4×C61AGL(1,ℤ/8)12yes—
187C2×C4×C61C22×D411yescharacteristic
188C22⋊D61—82no—
189C22⋊D61—82no—
190C22⋊D61—82no—
191C2×Dic61—22no—
192C2×Dic61—41no—
193C2×Dic61—41no—
194C2×Dic61—42no—
195C2×Dic61—84no—
196C2×Dic61—82no—
197C2×Dic61—84no—
198C2×Dic61C2×D811yescharacteristic
199C2×Dic61C2×D811yescharacteristic
200C2×Dic61C2×D811yescharacteristic
201C2×Dic61C2×D811yescharacteristic
202D244—41no—
203D244—41no—
204C4×D61—82no—
205Dic122—41no—
206Dic122—41no—
207C8×C61#12112yes—
208C8×C61C22×D411yescharacteristic
209C8×C61C2×#4111yescharacteristic
210#92418—21no—
211#92418—21no—
212#33580—611nomaximal, Sylow
213C22×C61—21no—
214C22×C61—21no—
215C22×C61—44no—
216C22×C61—88no—
217C22×C61—82no—
218C22×C61—82no—
219C22×C61—81no—
220C22×C61—81no—
221C22×C61—81no—
222C22×C61#45912yes—
223C22×C61C22×D811yescharacteristic
224C2×D61—81no—
225C2×D61—81no—
226C2×D61—81no—
227C4×C61—81no—
228C4×C61#45912yes—
229C4×C61C2×#12111yescharacteristic
230C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
231Dic61—21no—
232Dic61—21no—
233Dic61—21no—
234Dic61—82no—
235Dic61—81no—
236Dic61#50912yescomplemented
237C2×#509—611no—
238C2×#509D6111yescharacteristic, complemented
239#3147—611no—
240C2×C4×D8—611no—
241#2210—612no—
242#2210—612no—
243#2208—612no—
244#2206—612no—
245#2150—611no—
246#2139—611no—
247#2139—611no—
248C2×C61—22no—
249C2×C61—84no—
250C2×C61—161no—
251C2×C61C2×#45911yescharacteristic
252C2×C61C2×#50911yescharacteristic
253C2×C61#341411yescharacteristic
254D61—161no—
255C2×#135—611no—
256C2×#135—1221no—
257C2×#135C2×D6111yescharacteristic
258C22×D8C2×D6111yescharacteristic
259#532—1221no—
260C2×C4×D4—1221no—
261#509—24no—
262#509—1224no—
263C2×SD32—612no—
264C2×SD32C2×D6112yes—
265C2×D16—612no—
266C2×D16C2×D6112yes—
267C2×M32—611no—
268C2×M32C2×D6111yescharacteristic
269#495—611no—
270#495—611no—
271#495—1221no—
272#494—611no—
273#493—611no—
274#492—1221no—
275C4×D8—612no—
276C4×D8—612no—
277C4×D8—1221no—
278C4×D8—1221no—
279C2×#107—611no—
280C2×#102—1221no—
281C2×C4×C8—611no—
282C4⋊C16—612no—
283C4⋊C16—612no—
284#358—612no—
285#358—612no—
286#357—612no—
287#357—612no—
288C61#3358011yescharacteristic, complemented, Sylow
289C2×#41—21no—
290C2×#41—1221no—
291C22×D4—21no—
292C23×C4—2441no—
293#135—22no—
294#135—22no—
295#135—1222no—
296#135—1222no—
297#135—2444no—
298C2×Q16—611no—
299C2×Q16—1221no—
300C2×Q16C22×D6111yescharacteristic
301C2×SD16—21no—
302C2×SD16—1221no—
303C2×SD16—1221no—
304C2×SD16—1221no—
305C2×D8—21no—
306C2×D8—22no—
307C2×D8—611no—
308C2×D8—1221no—
309C2×D8C22⋊D6114yescomplemented
310C2×D8C22×D6111yescharacteristic
311C2×D8C22×D6112yes—
312C22×C8—611no—
313C22×C8C22×D6111yescharacteristic
314C4⋊Q8—1221no—
315#127—1221no—
316#124—1221no—
317#124—1221no—
318#121—2441no—
319#121—2441no—
320C4×D4—1222no—
321C4×D4—1222no—
322C4×D4—2441no—
323C4×D4—2441no—
324C4×D4—2441no—
325C4×D4—2441no—
326C2×C4⋊C4—1221no—
327C2×C22⋊C4—2441no—
328C2×C42—611no—
329SD32—24no—
330SD32—1224no—
331D16—24no—
332D16—1224no—
333M32—22no—
334M32—1222no—
335C2×C16—612no—
336C2×C16C22×D6112yes—
337#107—612no—
338#107—1221no—
339#102—1222no—
340#102—1222no—
341C4×C8—611no—
342C4×C8—611no—
343C4×C8—611no—
344C4×C8—611no—
345C24—41no—
346#41—42no—
347#41—42no—
348#41—2442no—
349#41—2442no—
350C2×Q8—21no—
351C2×Q8—1221no—
352C2×D4—22no—
353C2×D4—21no—
354C2×D4—24no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—42no—
358C2×D4—1221no—
359C2×D4—2441no—
360C2×D4—2441no—
361C2×D4—2441no—
362C22×C4—41no—
363C22×C4—611no—
364C22×C4—611no—
365C22×C4—4884no—
366C22×C4—2442no—
367C22×C4—2442no—
368C22×C4—2441no—
369C22×C4—2441no—
370C22×C4D4×D6111yescharacteristic
371Q16—21no—
372Q16—21no—
373Q16—1221no—
374Q16—1221no—
375Q16—2442no—
376SD16—42no—
377SD16—2442no—
378SD16—2442no—
379SD16—2442no—
380D8—21no—
381D8—22no—
382D8—21no—
383D8—24no—
384D8—1221no—
385D8—1221no—
386D8—2442no—
387D8C2×C22⋊D6114yes—
388C2×C8—611no—
389C2×C8—611no—
390C2×C8—1222no—
391C2×C8D4×D6111yescharacteristic
392C2×C8#1175822412yes—
393C2×C8C2×C22⋊D6112yes—
394C2×C8C23×D6111yescharacteristic
395C4⋊C4—1222no—
396C4⋊C4—2441no—
397C4⋊C4—2441no—
398C4⋊C4—2441no—
399C22⋊C4—2441no—
400C22⋊C4—2441no—
401C22⋊C4—2442no—
402C22⋊C4—2442no—
403C22⋊C4—2441no—
404C22⋊C4—2441no—
405C42—611no—
406C42—611no—
407C42—611no—
408C42—611no—
409C16—22no—
410C16—1222no—
411C23—44no—
412C23—42no—
413C23—41no—
414C23—82no—
415C23—82no—
416C23—2441no—
417C23D8×D6111yescharacteristic
418Q8—41no—
419Q8—41no—
420Q8—2441no—
421Q8—2441no—
422D4—24no—
423D4—44no—
424D4—41no—
425D4—42no—
426D4—41no—
427D4—82no—
428D4—4882no—
429D4—4882no—
430D4—4882no—
431D4—2441no—
432D4—2441no—
433C2×C4—41no—
434C2×C4—82no—
435C2×C4—611no—
436C2×C4—611no—
437C2×C4—611no—
438C2×C4—611no—
439C2×C4—1222no—
440C2×C4—4884no—
441C2×C4—4884no—
442C2×C4—4882no—
443C2×C4—4882no—
444C2×C4—2441no—
445C2×C4—2442no—
446C2×C4—2441no—
447C2×C4#1451974912yes—
448C2×C4D8×D6111yescharacteristic
449C2×C4#1451980112yes—
450C2×C4C2×D4×D6111yescharacteristic
451C8—1221no—
452C8—1221no—
453C8#1451974912yes—
454C8C22×C22⋊D6111yescharacteristic
455C8C2×#1175822411yescharacteristic
456C22—21no—
457C22—21no—
458C22—44no—
459C22—88no—
460C22—82no—
461C22—82no—
462C22—81no—
463C22—81no—
464C22—81no—
465C22—4881no—
466C22—4881no—
467C22—4881no—
468C22#1683995512yes—
469C22C2×D8×D6111yescharacteristic
470C4—81no—
471C4—612no—
472C4—1221no—
473C4—1221no—
474C4—1221no—
475C4—4881no—
476C4—4882no—
477C4#1683995512yes—
478C4C2×#1451974911yescharacteristic
479C4C2×#1451980111yescharacteristic
480C2—22no—
481C2—84no—
482C2—161no—
483C2—9761no—
484C2C2×#1683995511yescharacteristic
485C2#509×D6111yescharacteristic
486C2#2145895111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC2×C61×#509C2
socleC22×C61C22×D8
hypercenterC2×#509D61
hypocenterC61#33580
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×#16840198C2semidirect
2. #21456243C2semidirect
3. C2×D8×Dic61C2semidirect
4. #21447876C2semidirect
5. #21447872C2semidirect
6. #21447862C2semidirect
7. #21447852C2semidirect
8. #21447574C2semidirect
9. #21447532C2semidirect
10. #21447531C2semidirect
11. C2×C61×#509C2semidirect
20. #16840147C22semidirect
21. #16840145C22semidirect
23. #16840140C22semidirect
24. #16840137C22semidirect
33. #16839580C22semidirect
34. #16839580C22semidirect
35. #16839567C22semidirect
36. #16839566C22semidirect
37. #16839564C22semidirect
38. #16839563C22semidirect
93. C2×D8×C61D4semidirect
236. Dic61#509semidirect
238. C2×#509D61semidirect
288. C61#33580semidirect
309. C2×D8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168401982
#214562432
C2×D8×Dic612
#214478762
#214478722
#214478622
#214478522
#214475742
#214475322
#214475312
C2×C61×#5092
#3358061