Finite Group Databasebeta

← #29662592

Subgroups — #29662592

Order 15616.

684 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456141C211yesmaximal, characteristic, complemented
2#21455827C211yesmaximal, characteristic, complemented
3C2×#16839840C211yesmaximal, characteristic, complemented
4#21445392C212yesmaximal, complemented
5#21445378C212yesmaximal, complemented
6#21445364C212yesmaximal, complemented
7#21445350C212yesmaximal, complemented
8#21444565C211yesmaximal, characteristic, complemented
9#21444474C211yesmaximal, characteristic, complemented
10#21444468C211yesmaximal, characteristic, complemented
11C61×#3100C211yesmaximal, characteristic
12C22×#11758194—21no—
13#16840319C2211yescharacteristic
14#16840116—21no—
15#16840116—21no—
16#16840112C2211yescharacteristic, complemented
17#16840112C2211yescharacteristic, complemented
18#16840108C2211yescharacteristic, complemented
19#16840108C2211yescharacteristic, complemented
20#16840092—21no—
21#16840052C2212yes—
22#16839967C2212yes—
23#16839913—21no—
24C4×#11758200—22no—
25#16839840—21no—
26#16839840—21no—
27#16839840C2212yes—
28#16839840C2212yes—
29#16839815C2211yescharacteristic
30C2×C4⋊#92418C2211yescharacteristic
31#16839758—21no—
32#16839758—21no—
33C2×C22⋊#92418—21no—
34C2×C4×#92418C2211yescharacteristic
35#16839514C2212yescomplemented
36#16839512C2212yescomplemented
37#16839510C2212yescomplemented
38#16839509C2212yescomplemented
39#16839508C2212yescomplemented
40#16839506C2212yescomplemented
41C61×#532C2211yescharacteristic
42C2×C4×D4×C61C2211yescharacteristic
43C61×#480—21no—
44C61×#474C2212yes—
45C61×#459C2212yes—
46C61×#447—21no—
47C4×D8×C61—22no—
48C61×#431C2211yescharacteristic
49C2×#11758200—22no—
50C2×#11758194—22no—
51C2×#11758194—22no—
52C2×#11758194—44no—
53C22×#92418—21no—
54C22×#92418D411yescharacteristic
55C22×#92418D411yescharacteristic
56#14519777C2311yescharacteristic
57#14519777C2311yescharacteristic
58#14519769—21no—
59#14519756—21no—
60#14519756—21no—
61C4×Dic122—21no—
62C4×Dic122—21no—
63#14519736—21no—
64#14519708—22no—
65C4⋊#92418—21no—
66C4⋊#92418C2312yes—
67C4⋊#92418C2312yes—
68#14519701—21no—
69#14519701—21no—
70#14519701—21no—
71#14519701—21no—
72#14519700—22no—
73#14519698—22no—
74#14519698—21no—
75#14519698—21no—
76#14519698—21no—
77#14519698—21no—
78C22⋊#92418—21no—
79C22⋊#92418—22no—
80C22⋊#92418—22no—
81C4×#92418—21no—
82C4×#92418D411yescharacteristic
83C4×#92418D411yescharacteristic
84C4×#92418C2311yescharacteristic
85C4×#92418C2311yescharacteristic
86C2×#41×C61—21no—
87C22×D4×C61D411yescharacteristic
88C23×C4×C61—21no—
89C2×D8×C61—22no—
90C4⋊Q8×C61C2311yescharacteristic
91#127×C61C2311yescharacteristic
92#124×C61—21no—
93#121×C61—21no—
94#121×C61—21no—
95C4×D4×C61—21no—
96C4×D4×C61—21no—
97C4×D4×C61—21no—
98C4×D4×C61—21no—
99C4×D4×C61D412yes—
100C4×D4×C61C2312yes—
101C2×C4⋊C4×C61D411yescharacteristic
102C2×C22⋊C4×C61—21no—
103C2×C42×C61C2311yescharacteristic
104#107×C61—22no—
105C4⋊C8×C61C2312yes—
106#102×C61—22no—
107#102×C61—22no—
108C22⋊C8×C61—21no—
109C4×C8×C61—21no—
110C24×C61—21no—
111#41×C61—42no—
112#41×C61—42no—
113C2×Q8×C61—21no—
114C2×D4×C61—21no—
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61—42no—
119C2×D4×C61—41no—
120C2×D4×C61C2×D412yes—
121C22×C4×C61—21no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—42no—
125C22×C4×C61—42no—
126C22×C4×C61—41no—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61#4111yescharacteristic
130C2×Dic122—21no—
131C2×Dic122—21no—
132D8×C61—44no—
133C2×C8×C61—22no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—21no—
137C4⋊C4×C61C2×D412yes—
138C22⋊C4×C61—22no—
139C22⋊C4×C61—22no—
140C22⋊C4×C61—41no—
141C22⋊C4×C61—41no—
142C42×C61C2×D411yescharacteristic
143C42×C61C2×D411yescharacteristic
144C42×C61#4111yescharacteristic
145C42×C61C2411yescharacteristic
146C22⋊Dic61—41no—
147#11758200—44no—
148C4⋊Dic61—21no—
149C4⋊Dic61—21no—
150C4⋊Dic61—21no—
151C4⋊Dic61—21no—
152#11758196—41no—
153#11758196—41no—
154C4×Dic61—41no—
155#11758194—44no—
156#11758194—44no—
157C2×#92418—22no—
158C2×#92418—22no—
159C2×#92418—22no—
160C2×#92418—22no—
161C2×#92418—42no—
162C2×#92418C2×D411yescharacteristic
163C2×#92418C2×D411yescharacteristic
164C2×#92418C2×D411yescharacteristic
165C2×#92418C2×D411yescharacteristic
166C23×C61—22no—
167C23×C61—42no—
168C23×C61—42no—
169C23×C61—42no—
170C23×C61—41no—
171C23×C61#12111yescharacteristic
172Q8×C61—41no—
173Q8×C61—41no—
174D4×C61—22no—
175D4×C61—22no—
176D4×C61—42no—
177D4×C61—41no—
178D4×C61—41no—
179D4×C61—42no—
180D4×C61—81no—
181D4×C61—82no—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—41no—
185C2×C4×C61—42no—
186C2×C4×C61—42no—
187C2×C4×C61—42no—
188C2×C4×C61—44no—
189C2×C4×C61—41no—
190C2×C4×C61—82no—
191C2×C4×C61#12111yescharacteristic
192C2×C4×C61#12111yescharacteristic
193C2×C4×C61#12111yescharacteristic
194C2×C4×C61#13512yes—
195C2×C4×C61AGL(1,ℤ/8)12yes—
196C2×C4×C61C22×D411yescharacteristic
197C2×C4×C61C22×D411yescharacteristic
198C2×C4×C61C2×#4111yescharacteristic
199C2×Dic61—41no—
200C2×Dic61—41no—
201Dic122—42no—
202Dic122—81no—
203C8×C61—42no—
204#92418—21no—
205#92418—21no—
206#92418—21no—
207#92418—21no—
208#92418—42no—
209#92418—42no—
210#19142—611nomaximal, Sylow
211C22×C61—21no—
212C22×C61—21no—
213C22×C61—42no—
214C22×C61—42no—
215C22×C61—42no—
216C22×C61—84no—
217C22×C61—81no—
218C22×C61—81no—
219C22×C61—81no—
220C22×C61#46312yes—
221C22×C61C2×#12111yescharacteristic
222C4×C61—21no—
223C4×C61—21no—
224C4×C61—21no—
225C4×C61—42no—
226C4×C61—81no—
227C4×C61#46312yes—
228C4×C61C2×#13511yescharacteristic
229C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
230C4×C61#57512yes—
231Dic61—81no—
232Dic61—81no—
233#3131—611no—
234#3101—611no—
235#3100D6111yescharacteristic
236C2×#436—611no—
237#1703—612no—
238#1701—612no—
239#1699—612no—
240#1697—612no—
241#1564—611no—
242#1542—611no—
243#1540—611no—
244C2×C61—22no—
245C2×C61—82no—
246C2×C61—161no—
247C2×C61C2×#46311yescharacteristic
248C2×C61#305411yescharacteristic
249C2×C61#306711yescharacteristic
250C22×M16—1221no—
251#533—611no—
252#532C2×D6111yescharacteristic
253C2×C4×D4C2×D6111yescharacteristic
254#488—1221no—
255#488—1221no—
256#487—611no—
257#487—611no—
258#486—611no—
259#486—611no—
260#483—1221no—
261#480—21no—
262#476—612no—
263#474C2×D6112yes—
264#461—612no—
265#459C2×D6112yes—
266#450—1221no—
267#447—21no—
268C4×SD16—1222no—
269C4×D8—22no—
270#436—612no—
271#436—612no—
272#436—1221no—
273#436—1221no—
274#431—611no—
275#431C2×D6111yescharacteristic
276C2×C4⋊C8—611no—
277#416—1221no—
278#416—1221no—
279C2×C22⋊C8—1221no—
280C2×C4×C8—611no—
281#335—612no—
282#334—612no—
283#333—612no—
284#332—612no—
285#332—612no—
286#331—612no—
287C61#1914211yescharacteristic, complemented, Sylow
288C2×#41—21no—
289C22×D4C22⋊D6111yescharacteristic
290C23×C4—21no—
291C2×SD16—1222no—
292C2×D8—22no—
293C2×M16—1222no—
294C2×M16—1222no—
295C2×M16—2444no—
296C22×C8—611no—
297C22×C8—611no—
298C22×C8—1221no—
299C4⋊Q8—611no—
300C4⋊Q8—611no—
301C4⋊Q8C22×D6111yescharacteristic
302#127C22×D6111yescharacteristic
303#125—1221no—
304#124—21no—
305C22⋊Q8—1221no—
306C22⋊Q8—1221no—
307#121—21no—
308#121—21no—
309C4×Q8—1221no—
310C4×Q8—1221no—
311C4×D4—21no—
312C4×D4—21no—
313C4×D4—21no—
314C4×D4—21no—
315C4×D4C22⋊D6112yes—
316C4×D4C22×D6112yes—
317#117—1221no—
318C2×C4⋊C4C22⋊D6111yescharacteristic
319C2×C22⋊C4—21no—
320C2×C42C22×D6111yescharacteristic
321#107—22no—
322#106—1222no—
323C4⋊C8—612no—
324C4⋊C8—612no—
325C4⋊C8—1221no—
326C4⋊C8C22×D6112yes—
327#103—1221no—
328#103—1221no—
329#103—1221no—
330#103—1221no—
331#103—1222no—
332#102—22no—
333#102—22no—
334#102—1221no—
335#102—1221no—
336#102—1221no—
337#102—1221no—
338#102—1222no—
339C22⋊C8—21no—
340C22⋊C8—1222no—
341C22⋊C8—1222no—
342C22⋊C8—1221no—
343C4×C8—21no—
344C4×C8—611no—
345C4×C8—611no—
346C4×C8—611no—
347C4×C8—611no—
348C4×C8—1221no—
349C24—21no—
350#41—42no—
351#41—42no—
352C2×Q8—21no—
353C2×Q8—1221no—
354C2×Q8—1221no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—42no—
360C2×D4—41no—
361C2×D4C2×C22⋊D6112yes—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4D4×D6111yescharacteristic
369C22×C4#1175822411yescharacteristic
370C22×C4C2×C22⋊D6111yescharacteristic
371SD16—2444no—
372D8—44no—
373M16—2444no—
374M16—2444no—
375C2×C8—22no—
376C2×C8—611no—
377C2×C8—611no—
378C2×C8—611no—
379C2×C8—611no—
380C2×C8—1222no—
381C2×C8—1222no—
382C2×C8—1222no—
383C2×C8—1222no—
384C2×C8—2442no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—1221no—
389C4⋊C4—1221no—
390C4⋊C4—1221no—
391C4⋊C4—1221no—
392C4⋊C4—2441no—
393C4⋊C4—2441no—
394C4⋊C4C2×C22⋊D6112yes—
395C22⋊C4—22no—
396C22⋊C4—22no—
397C22⋊C4—41no—
398C22⋊C4—41no—
399C22⋊C4—2441no—
400C42—2441no—
401C42D4×D6111yescharacteristic
402C42#1175822411yescharacteristic
403C42C2×C22⋊D6111yescharacteristic
404C42C23×D6111yescharacteristic
405C23—22no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—41no—
410C23#1451974911yescharacteristic
411Q8—41no—
412Q8—41no—
413Q8—4881no—
414Q8—2442no—
415D4—22no—
416D4—22no—
417D4—42no—
418D4—41no—
419D4—41no—
420D4—42no—
421D4—81no—
422D4—82no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—82no—
432C2×C4—2441no—
433C2×C4—2441no—
434C2×C4#1451974911yescharacteristic
435C2×C4#1451974911yescharacteristic
436C2×C4#1451974911yescharacteristic
437C2×C4#1451979612yes—
438C2×C4#1451980112yes—
439C2×C4C2×D4×D6111yescharacteristic
440C2×C4C22×C22⋊D6111yescharacteristic
441C2×C4C2×#1175822411yescharacteristic
442C8—42no—
443C8—1221no—
444C8—1221no—
445C8—1221no—
446C8—1221no—
447C8—2442no—
448C8—2442no—
449C22—21no—
450C22—21no—
451C22—42no—
452C22—42no—
453C22—42no—
454C22—84no—
455C22—81no—
456C22—81no—
457C22—81no—
458C22#1683997512yes—
459C22C2×#1451974911yescharacteristic
460C4—21no—
461C4—21no—
462C4—21no—
463C4—42no—
464C4—81no—
465C4—4881no—
466C4—4881no—
467C4#1683997512yes—
468C4C2×#1451979611yescharacteristic
469C4C2×#1451980111yescharacteristic
470C4#1684056012yes—
471C2—22no—
472C2—82no—
473C2—161no—
474C2C2×#1683997511yescharacteristic
475C2#2145542511yescharacteristic
476C2#2145553611yescharacteristic
subgroupG/H
centerC2×C4#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3100C2
socleC22×C61C2×#121
hypercenter#3100D61
hypocenterC61#19142
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3100

Complements

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

NH ≅ G/Ntype
1. #21456141C2semidirect
2. #21455827C2semidirect
3. C2×#16839840C2semidirect
4. #21445392C2semidirect
5. #21445378C2semidirect
6. #21445364C2semidirect
7. #21445350C2semidirect
8. #21444565C2semidirect
9. #21444474C2semidirect
10. #21444468C2semidirect
16. #16840112C22semidirect
17. #16840112C22semidirect
18. #16840108C22semidirect
19. #16840108C22semidirect
35. #16839514C22semidirect
36. #16839512C22semidirect
37. #16839510C22semidirect
38. #16839509C22semidirect
39. #16839508C22semidirect
40. #16839506C22semidirect
287. C61#19142semidirect

Maximal subgroups

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

subgroupindex
#214561412
#214558272
C2×#168398402
#214453922
#214453782
#214453642
#214453502
#214445652
#214444742
#214444682
C61×#31002
#1914261