Finite Group Databasebeta

← #29662598

Subgroups — #29662598

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#21456144C211yesmaximal, characteristic, complemented
2#21455819C211yesmaximal, characteristic
3C2×#16839840C211yesmaximal, characteristic, complemented
4#21445396C212yesmaximal, complemented
5#21445382C212yesmaximal, complemented
6#21445368C212yesmaximal, complemented
7#21445354C212yesmaximal, complemented
8#21444565C211yesmaximal, characteristic, complemented
9#21444476C211yesmaximal, characteristic, complemented
10#21444466C211yesmaximal, characteristic, complemented
11C61×#3101C211yesmaximal, characteristic, complemented
12C22×#11758194—21no—
13#16840312C2211yescharacteristic
14#16840119—21no—
15#16840119—21no—
16#16840114C2211yescharacteristic, complemented
17#16840114C2211yescharacteristic, complemented
18#16840110C2211yescharacteristic, complemented
19#16840110C2211yescharacteristic, complemented
20#16840075—21no—
21#16840042C2212yes—
22#16839957C2212yes—
23#16839896—21no—
24C4×#11758199—22no—
25#16839840—21no—
26#16839840—21no—
27#16839840C2212yes—
28#16839840C2212yes—
29#16839815C2211yescharacteristic
30C2×C4⋊#92418C2211yescharacteristic
31#16839756—21no—
32#16839756—21no—
33C2×C22⋊#92418—21no—
34C2×C4×#92418C2211yescharacteristic
35#16839514C2212yescomplemented
36#16839512C2212yescomplemented
37#16839511C2212yescomplemented
38#16839508C2212yescomplemented
39#16839507C2212yescomplemented
40#16839505C2212yescomplemented
41C61×#533C2211yescharacteristic
42C2×C4×D4×C61C2211yescharacteristic
43C61×#483—21no—
44C61×#476C2212yes—
45C61×#461C2212yes—
46C61×#450—21no—
47C4×SD16×C61—22no—
48C61×#431C2211yescharacteristic
49C2×#11758222—21no—
50C2×#11758199—22no—
51C2×#11758194—22no—
52C2×#11758194—22no—
53C2×#11758194—44no—
54C22×#92418—21no—
55C22×#92418D411yescharacteristic
56C22×#92418D411yescharacteristic
57#14519777C2311yescharacteristic
58#14519774C2311yescharacteristic
59#14519765—21no—
60#14519751—21no—
61#14519751—21no—
62C4×D244—21no—
63C4×D244—21no—
64#14519710—22no—
65C4⋊#92418—21no—
66C4⋊#92418C2312yes—
67C4⋊#92418C2312yes—
68#14519700—21no—
69#14519700—21no—
70#14519700—21no—
71#14519700—21no—
72#14519698—22no—
73#14519697—22no—
74#14519697—21no—
75#14519697—21no—
76#14519697—21no—
77#14519697—21no—
78C22⋊#92418—21no—
79C22⋊#92418—22no—
80C22⋊#92418—22no—
81C4×#92418—21no—
82C4×#92418D411yescharacteristic
83C4×#92418D411yescharacteristic
84C4×#92418C2311yescharacteristic
85C4×#92418C2311yescharacteristic
86C22×D4×C61D411yescharacteristic
87C23×C4×C61—21no—
88C2×SD16×C61—22no—
89C4⋊Q8×C61C2311yescharacteristic
90C4⋊Q8×C61C2311yescharacteristic
91#125×C61—21no—
92C22⋊Q8×C61—21no—
93C22⋊Q8×C61—21no—
94C4×Q8×C61—21no—
95C4×Q8×C61—21no—
96C4×D4×C61—21no—
97C4×D4×C61—21no—
98C4×D4×C61D412yes—
99C4×D4×C61C2312yes—
100#117×C61—21no—
101C2×C4⋊C4×C61D411yescharacteristic
102C2×C22⋊C4×C61—21no—
103C2×C42×C61C2311yescharacteristic
104#106×C61—22no—
105C4⋊C8×C61C2312yes—
106#103×C61—22no—
107#102×C61—22no—
108C22⋊C8×C61—21no—
109C4×C8×C61—21no—
110C24×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×D4×C61—21no—
114C2×D4×C61—42no—
115C2×D4×C61C2×D412yes—
116C22×C4×C61—21no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—42no—
120C22×C4×C61—42no—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61#4111yescharacteristic
124C2×C22⋊D61—41no—
125#11758222—42no—
126#11758222—42no—
127C2×D244—21no—
128C2×D244—21no—
129C2×D244—21no—
130C2×C4×D61—41no—
131C2×Dic122—21no—
132SD16×C61—44no—
133C2×C8×C61—22no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—21no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—41no—
140C4⋊C4×C61—41no—
141C4⋊C4×C61C2×D412yes—
142C22⋊C4×C61—22no—
143C22⋊C4×C61—22no—
144C22⋊C4×C61—41no—
145C42×C61—41no—
146C42×C61C2×D411yescharacteristic
147C42×C61C2×D411yescharacteristic
148C42×C61#4111yescharacteristic
149C42×C61C2411yescharacteristic
150#11758199—44no—
151#11758198—41no—
152#11758198—41no—
153C4⋊Dic61—21no—
154C4⋊Dic61—21no—
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#12111yescharacteristic
171C22×D61—41no—
172Q8×C61—42no—
173Q8×C61—81no—
174D4×C61—22no—
175D4×C61—22no—
176D4×C61—42no—
177C2×C4×C61—22no—
178C2×C4×C61—22no—
179C2×C4×C61—41no—
180C2×C4×C61—42no—
181C2×C4×C61—42no—
182C2×C4×C61—42no—
183C2×C4×C61—44no—
184C2×C4×C61—41no—
185C2×C4×C61—41no—
186C2×C4×C61#12111yescharacteristic
187C2×C4×C61#12111yescharacteristic
188C2×C4×C61#12111yescharacteristic
189C2×C4×C61#13512yes—
190C2×C4×C61AGL(1,ℤ/8)12yes—
191C2×C4×C61C22×D411yescharacteristic
192C2×C4×C61C22×D411yescharacteristic
193C2×C4×C61C2×#4111yescharacteristic
194C22⋊D61—82no—
195C2×Dic61—41no—
196D244—41no—
197D244—41no—
198D244—42no—
199D244—81no—
200C4×D61—82no—
201Dic122—41no—
202Dic122—41no—
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#46312yes—
218C22×C61C2×#12111yescharacteristic
219C2×D61—81no—
220C2×D61—81no—
221C2×D61—81no—
222C4×C61—21no—
223C4×C61—21no—
224C4×C61—21no—
225C4×C61—42no—
226C4×C61—81no—
227C4×C61—81no—
228C4×C61#46312yes—
229C4×C61C2×#13511yescharacteristic
230C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
231C4×C61#57512yes—
232Dic61—81no—
233#3131—611no—
234#3101D6111yescharacteristic, complemented
235#3100—611no—
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×C61C2×#46311yescharacteristic
247C2×C61#305411yescharacteristic
248C2×C61#306711yescharacteristic
249D61—161no—
250C22×M16—1221no—
251#533C2×D6111yescharacteristic
252#532—611no—
253C2×C4×D4C2×D6111yescharacteristic
254#488—1221no—
255#488—1221no—
256#487—611no—
257#487—611no—
258#486—611no—
259#486—611no—
260#483—21no—
261#480—1221no—
262#476C2×D6112yes—
263#474—612no—
264#461C2×D6112yes—
265#459—612no—
266#450—21no—
267#447—1221no—
268C4×SD16—22no—
269C4×D8—1222no—
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—1221no—
289C22×D4C22⋊D6111yescharacteristic
290C23×C4—21no—
291C2×SD16—22no—
292C2×D8—1222no—
293C2×M16—1222no—
294C2×M16—1222no—
295C2×M16—2444no—
296C22×C8—611no—
297C22×C8—611no—
298C22×C8—1221no—
299C4⋊Q8—611no—
300C4⋊Q8C22×D6111yescharacteristic
301C4⋊Q8C22×D6111yescharacteristic
302#127—611no—
303#125—21no—
304#124—1221no—
305C22⋊Q8—21no—
306C22⋊Q8—21no—
307#121—1221no—
308#121—1221no—
309C4×Q8—21no—
310C4×Q8—21no—
311C4×D4—21no—
312C4×D4—21no—
313C4×D4—1221no—
314C4×D4—1221no—
315C4×D4C22⋊D6112yes—
316C4×D4C22×D6112yes—
317#117—21no—
318C2×C4⋊C4C22⋊D6111yescharacteristic
319C2×C22⋊C4—21no—
320C2×C42C22×D6111yescharacteristic
321#107—1222no—
322#106—22no—
323C4⋊C8—612no—
324C4⋊C8—612no—
325C4⋊C8—1221no—
326C4⋊C8C22×D6112yes—
327#103—22no—
328#103—1221no—
329#103—1221no—
330#103—1221no—
331#103—1221no—
332#102—22no—
333#102—1221no—
334#102—1221no—
335#102—1221no—
336#102—1221no—
337#102—1222no—
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—2442no—
351#41—2442no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—1221no—
355C2×D4—21no—
356C2×D4—42no—
357C2×D4—1221no—
358C2×D4—1221no—
359C2×D4—1221no—
360C2×D4—2441no—
361C2×D4C2×C22⋊D6112yes—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—2441no—
368C22×C4D4×D6111yescharacteristic
369C22×C4#1175822411yescharacteristic
370C22×C4C2×C22⋊D6111yescharacteristic
371SD16—44no—
372D8—2444no—
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—21no—
389C4⋊C4—21no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—1221no—
393C4⋊C4—1221no—
394C4⋊C4C2×C22⋊D6112yes—
395C22⋊C4—22no—
396C22⋊C4—22no—
397C22⋊C4—41no—
398C22⋊C4—2441no—
399C22⋊C4—2441no—
400C42—41no—
401C42D4×D6111yescharacteristic
402C42#1175822411yescharacteristic
403C42C2×C22⋊D6111yescharacteristic
404C42C23×D6111yescharacteristic
405C23—22no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—2441no—
410C23#1451974911yescharacteristic
411Q8—42no—
412Q8—81no—
413Q8—2441no—
414Q8—2441no—
415D4—22no—
416D4—22no—
417D4—42no—
418D4—4881no—
419D4—4882no—
420D4—2441no—
421D4—2441no—
422D4—2442no—
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—41no—
432C2×C4—4882no—
433C2×C4—2441no—
434C2×C4#1451974911yescharacteristic
435C2×C4#1451974911yescharacteristic
436C2×C4#1451974911yescharacteristic
437C2×C4#1451979812yes—
438C2×C4#1451980412yes—
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—4881no—
456C22—4881no—
457C22—4881no—
458C22#1683997912yes—
459C22C2×#1451974911yescharacteristic
460C4—21no—
461C4—21no—
462C4—21no—
463C4—42no—
464C4—81no—
465C4—81no—
466C4—4881no—
467C4#1683997912yes—
468C4C2×#1451979811yescharacteristic
469C4C2×#1451980411yescharacteristic
470C4#1684056012yes—
471C2—22no—
472C2—82no—
473C2—9761no—
474C2C2×#1683997911yescharacteristic
475C2#2145542511yescharacteristic
476C2#2145553611yescharacteristic
subgroupG/H
centerC2×C4#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3101C2
socleC22×C61C2×#121
prime radical#21455819C2
hypercenter#3101D61
hypocenterC61#19142
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3101

Complements

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

NH ≅ G/Ntype
1. #21456144C2semidirect
3. C2×#16839840C2semidirect
4. #21445396C2semidirect
5. #21445382C2semidirect
6. #21445368C2semidirect
7. #21445354C2semidirect
8. #21444565C2semidirect
9. #21444476C2semidirect
10. #21444466C2semidirect
11. C61×#3101C2semidirect
16. #16840114C22semidirect
17. #16840114C22semidirect
18. #16840110C22semidirect
19. #16840110C22semidirect
35. #16839514C22semidirect
36. #16839512C22semidirect
37. #16839511C22semidirect
38. #16839508C22semidirect
39. #16839507C22semidirect
40. #16839505C22semidirect
234. #3101D61semidirect
287. C61#19142semidirect

Maximal subgroups

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

subgroupindex
#214561442
#214558192
C2×#168398402
#214453962
#214453822
#214453682
#214453542
#214445652
#214444762
#214444662
C61×#31012
#1914261