Finite Group Databasebeta

← #29663583

Subgroups — #29663583

Order 15616.

722 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#21457136C211yesmaximal, characteristic, complemented
2#21456760C211yesmaximal, characteristic, complemented
3#21456269C211yesmaximal, characteristic
4#21454762C211yesmaximal, characteristic, complemented
5#21445302C212yesmaximal, complemented
6#21445290C212yesmaximal, complemented
7#21445178C212yesmaximal, complemented
8#21445166C212yesmaximal, complemented
9#21444569C211yesmaximal, characteristic, complemented
10#21444469C211yesmaximal, characteristic, complemented
11C2×C61×#331C211yesmaximal, characteristic, complemented
12C2×#14519803—21no—
13C2×#14519803—21no—
14#16840319C2211yescharacteristic
15#16840312C2211yescharacteristic
16#16840149—22no—
17#16840138C2212yes—
18#16840134—21no—
19#16840134C2212yes—
20#16840108—21no—
21#16840053—21no—
22#16840048C2212yescomplemented
23#16839991—22no—
24#16839968—21no—
25#16839963C2212yescomplemented
26#16839959C2212yes—
27#16839959C2212yes—
28#16839913—22no—
29#16839891—22no—
30#16839889C2212yescomplemented
31#16839886C2212yescomplemented
32#16839815C2211yescharacteristic
33#16839788—21no—
34#16839717—21no—
35C4×#11758194C2211yescharacteristic
36#16839514C2212yescomplemented
37#16839512C2212yescomplemented
38#16839509C2211yescharacteristic, complemented
39#16839509C2211yescharacteristic, complemented
40#16839505C2211yescharacteristic, complemented
41#16839505C2211yescharacteristic, complemented
42C2×#127×C61C2211yescharacteristic
43C2×C4⋊C8×C61C2211yescharacteristic
44C2×C4⋊C8×C61C2211yescharacteristic
45C61×#331C2212yes—
46C61×#331C2212yes—
47C2×#11758222—21no—
48#14519803—44no—
49#14519803—44no—
50C2×#11758190—22no—
51C2×#11758200—22no—
52C2×#11758200—22no—
53C2×#11758200—22no—
54C2×#11758200—22no—
55C2×#11758199—22no—
56C2×#11758199—22no—
57C2×#11758194D411yescharacteristic
58C2×#11758194D411yescharacteristic
59#14519777D411yescharacteristic
60#14519777D411yescharacteristic
61#14519777C2311yescharacteristic
62#14519774C2311yescharacteristic
63#14519769—21no—
64#14519765—21no—
65#14519756—21no—
66#14519756D412yes—
67#14519751—21no—
68#14519751—21no—
69C4×Dic122—21no—
70C4×Dic122C2312yes—
71C4×D244—21no—
72C4×D244—21no—
73#14519736—21no—
74#14519709—22no—
75C4⋊#92418C2311yescharacteristic
76C4⋊#92418C2311yescharacteristic
77#14519706—22no—
78#14519706—22no—
79#14519702—22no—
80#14519699—22no—
81#14519698—22no—
82#14519698—22no—
83C22⋊#92418—21no—
84#14519688D412yes—
85#14519687—22no—
86C4×#92418—21no—
87C4×#92418C2312yes—
88C8×Dic61—22no—
89C22×D4×C61—21no—
90C22×D4×C61—21no—
91C22×D4×C61—21no—
92C22×C8×C61—21no—
93C22×C8×C61—21no—
94#127×C61—21no—
95#127×C61—21no—
96#127×C61—21no—
97#127×C61C2312yes—
98C2×C42×C61C2311yescharacteristic
99C4⋊C8×C61—21no—
100C4⋊C8×C61—21no—
101C4⋊C8×C61C2312yes—
102C4⋊C8×C61C2311yescharacteristic
103C4⋊C8×C61C2311yescharacteristic
104C24×C61—41no—
105C2×D4×C61—22no—
106C2×D4×C61—22no—
107C2×D4×C61—22no—
108C2×D4×C61—41no—
109C2×D4×C61—41no—
110C2×D4×C61—41no—
111C2×D4×C61—42no—
112C2×D4×C61—42no—
113C2×D4×C61—81no—
114C2×D4×C61—81no—
115C2×D4×C61—82no—
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61#4111yescharacteristic
119C2×C22⋊D61—41no—
120#11758222—42no—
121#11758222—42no—
122C2×D244—21no—
123C2×D244—21no—
124C2×D244—21no—
125C2×C4×D61—41no—
126C2×Dic122—21no—
127C2×Dic122—21no—
128C2×Dic122C2×D412yes—
129C2×C8×C61—21no—
130C2×C8×C61—21no—
131C2×C8×C61—22no—
132C2×C8×C61—42no—
133C2×C8×C61—81no—
134C42×C61C2×D411yescharacteristic
135C42×C61C2×D411yescharacteristic
136C42×C61#4111yescharacteristic
137C42×C61C2411yescharacteristic
138C22⋊Dic61—22no—
139#11758200—44no—
140#11758200—44no—
141#11758200—82no—
142#11758200—82no—
143#11758199—44no—
144#11758199—44no—
145#11758198—41no—
146#11758198—41no—
147C4⋊Dic61—21no—
148C4⋊Dic61—21no—
149C4⋊Dic61—21no—
150C4⋊Dic61—21no—
151C4⋊Dic61—21no—
152C4⋊Dic61C2×D412yes—
153#11758196—22no—
154#11758196—22no—
155C4×Dic61—22no—
156#11758194—22no—
157#11758194SD1614yescomplemented
158C2×#92418—21no—
159C2×#92418—21no—
160C2×#92418C2×D412yes—
161C2×#92418C2×D412yes—
162#11758190—82no—
163C23×C61—42no—
164C23×C61—82no—
165C23×C61—82no—
166C23×C61—82no—
167C23×C61#12111yescharacteristic
168C22×D61—41no—
169D4×C61—42no—
170D4×C61—42no—
171D4×C61—42no—
172D4×C61—42no—
173D4×C61—82no—
174D4×C61—82no—
175D4×C61—82no—
176D4×C61—82no—
177C2×C4×C61—21no—
178C2×C4×C61—21no—
179C2×C4×C61—41no—
180C2×C4×C61#12111yescharacteristic
181C2×C4×C61#12111yescharacteristic
182C2×C4×C61#12111yescharacteristic
183C2×C4×C61C2×SD1612yes—
184C2×C4×C61AGL(1,ℤ/8)12yes—
185C2×C4×C61C22×D411yescharacteristic
186C2×C4×C61C22×D411yescharacteristic
187C2×C4×C61C2×#4111yescharacteristic
188C22⋊D61—82no—
189C2×Dic61—22no—
190C2×Dic61—22no—
191C2×Dic61—41no—
192D244—41no—
193D244—41no—
194D244—41no—
195D244—41no—
196D244—81no—
197C4×D61—82no—
198Dic122—22no—
199Dic122—22no—
200Dic122—41no—
201Dic122—41no—
202Dic122—81no—
203C8×C61—41no—
204C8×C61—41no—
205C8×C61—42no—
206#92418—22no—
207#92418—41no—
208#92418—41no—
209#92418C2×SD1614yes—
210#19238—611nomaximal, Sylow
211C22×C61—21no—
212C22×C61—21no—
213C22×C61—82no—
214C22×C61—82no—
215C22×C61—82no—
216C22×C61—164no—
217C22×C61#46012yes—
218C22×C61C2×#12111yescharacteristic
219C2×D61—81no—
220C2×D61—81no—
221C2×D61—81no—
222C4×C61—21no—
223C4×C61—21no—
224C4×C61—21no—
225C4×C61#46012yes—
226C4×C61C22×SD1611yescharacteristic
227C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
228C4×C61#57611yescharacteristic
229C4×C61#57711yescharacteristic
230Dic61—42no—
231Dic61—42no—
232Dic61—81no—
233#3246—611no—
234#3208—611no—
235#3151—611no—
236#2962—611no—
237#1688—612no—
238#1686—612no—
239#1668—612no—
240#1666—612no—
241#1565—611no—
242#1540—611no—
243C2×#331D6111yescharacteristic, complemented
244C2×C61—22no—
245C2×C61—162no—
246C2×C61C2×#46011yescharacteristic
247C2×C61#306211yescharacteristic
248C2×C61#306811yescharacteristic
249D61—161no—
250C2×AGL(1,ℤ/8)—1221no—
251C2×AGL(1,ℤ/8)—1221no—
252#533—611no—
253#532—611no—
254C2×#127C2×D6111yescharacteristic
255#496—1222no—
256#493—612no—
257#492—612no—
258#492—1221no—
259#486—1221no—
260#476—1221no—
261#475—612no—
262#465—1222no—
263#461—1221no—
264#460—612no—
265#460—612no—
266#460—612no—
267#450—1222no—
268#446—612no—
269#446—1222no—
270C8×Q8—612no—
271#431—611no—
272#424—1221no—
273C2×C4⋊C8C2×D6111yescharacteristic
274C2×C4⋊C8C2×D6111yescharacteristic
275#405—1221no—
276C4×M16—611no—
277#335—612no—
278#334—612no—
279#332—611no—
280#332—611no—
281#331—611no—
282#331—611no—
283#331C2×D6112yes—
284#331C2×D6112yes—
285C61#1923811yescharacteristic, complemented, Sylow
286C2×#41—1221no—
287C22×D4—21no—
288C22×D4—21no—
289C22×D4—21no—
290AGL(1,ℤ/8)—2444no—
291AGL(1,ℤ/8)—2444no—
292C2×SD16—1222no—
293C2×SD16—1222no—
294C2×SD16—1222no—
295C2×SD16—1222no—
296C2×SD16—1222no—
297C2×D8—1222no—
298C2×D8—1222no—
299C2×M16—611no—
300C2×M16—611no—
301C22×C8—21no—
302C22×C8—21no—
303C4⋊Q8—611no—
304C4⋊Q8—611no—
305C4⋊Q8—611no—
306#127—21no—
307#127—21no—
308#127—21no—
309#127—611no—
310#127C22×D6112yes—
311#125—1221no—
312#124—1221no—
313C22⋊Q8—612no—
314C22⋊Q8—1221no—
315#121—1221no—
316#121—1221no—
317C4×Q8—612no—
318C4×Q8—1221no—
319C4×D4—1221no—
320C4×D4—1221no—
321#117—1221no—
322C2×C42C22×D6111yescharacteristic
323#106—1222no—
324C4⋊C8—21no—
325C4⋊C8—21no—
326C4⋊C8—611no—
327C4⋊C8—611no—
328C4⋊C8—1222no—
329C4⋊C8—1222no—
330C4⋊C8C22×D6111yescharacteristic
331C4⋊C8C22×D6111yescharacteristic
332C4⋊C8C22×D6112yes—
333#103—1222no—
334#102—1222no—
335#102—1222no—
336#102—1222no—
337C22⋊C8—1221no—
338#97—612no—
339#97—1222no—
340C4×C8—612no—
341C4×C8—1222no—
342C4×C8—1221no—
343C24—41no—
344#41—2442no—
345#41—2442no—
346C2×Q8—612no—
347C2×Q8—1221no—
348C2×Q8—1221no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D4—81no—
358C2×D4—81no—
359C2×D4—82no—
360C2×D4—1221no—
361C2×D4—1221no—
362C2×D4—1221no—
363C2×D4—2441no—
364C22×C4—2441no—
365C22×C4#1175822211yescharacteristic
366C22×C4D4×D6111yescharacteristic
367C22×C4D4×D6111yescharacteristic
368SD16—4882no—
369SD16—4882no—
370SD16—4882no—
371SD16—2444no—
372SD16—2444no—
373D8—2444no—
374D8—2444no—
375M16—614no—
376M16—1222no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—22no—
380C2×C8—42no—
381C2×C8—81no—
382C2×C8—612no—
383C2×C8—612no—
384C2×C8—1221no—
385C2×C8—1221no—
386C4⋊C4—612no—
387C4⋊C4—1222no—
388C4⋊C4—1222no—
389C4⋊C4—1221no—
390C4⋊C4—1221no—
391C4⋊C4—1221no—
392C4⋊C4—1221no—
393C4⋊C4—1221no—
394C22⋊C4—1222no—
395C22⋊C4—2441no—
396C22⋊C4—2441no—
397C42—1222no—
398C42#1175822211yescharacteristic
399C42D4×D6111yescharacteristic
400C42D4×D6111yescharacteristic
401C42C23×D6111yescharacteristic
402C23—42no—
403C23—82no—
404C23—82no—
405C23—82no—
406C23—2441no—
407C23#1451975011yescharacteristic
408Q8—1222no—
409Q8—1222no—
410Q8—4881no—
411Q8—2441no—
412Q8—2441no—
413D4—42no—
414D4—42no—
415D4—42no—
416D4—42no—
417D4—82no—
418D4—82no—
419D4—82no—
420D4—82no—
421D4—4881no—
422D4—4882no—
423D4—2441no—
424D4—2441no—
425D4—2441no—
426D4—2441no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—41no—
430C2×C4—1222no—
431C2×C4—1222no—
432C2×C4—4882no—
433C2×C4—2441no—
434C2×C4#1451975011yescharacteristic
435C2×C4#1451975011yescharacteristic
436C2×C4#1451975011yescharacteristic
437C2×C4SD16×D6112yes—
438C2×C4#1451980112yes—
439C2×C4C2×D4×D6111yescharacteristic
440C2×C4C2×D4×D6111yescharacteristic
441C2×C4C2×#1175822211yescharacteristic
442C8—41no—
443C8—41no—
444C8—42no—
445C8—614no—
446C8—1222no—
447C8—2441no—
448C8—2441no—
449C22—21no—
450C22—21no—
451C22—82no—
452C22—82no—
453C22—82no—
454C22—164no—
455C22—4881no—
456C22—4881no—
457C22—4881no—
458C22#1683996112yes—
459C22C2×#1451975011yescharacteristic
460C4—21no—
461C4—21no—
462C4—21no—
463C4—4881no—
464C4—2442no—
465C4—2442no—
466C4#1683996112yes—
467C4C2×SD16×D6111yescharacteristic
468C4C2×#1451980111yescharacteristic
469C4#1684056311yescharacteristic
470C4#1684056811yescharacteristic
471C2—22no—
472C2—162no—
473C2—9761no—
474C2C2×#1683996111yescharacteristic
475C2#2145549711yescharacteristic
476C2#2145554611yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#331C2
socleC22×C61C2×#121
prime radical#21456269C2
hypercenterC2×#331D61
hypocenterC61#19238
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#331

Complements

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

NH ≅ G/Ntype
1. #21457136C2semidirect
2. #21456760C2semidirect
4. #21454762C2semidirect
5. #21445302C2semidirect
6. #21445290C2semidirect
7. #21445178C2semidirect
8. #21445166C2semidirect
9. #21444569C2semidirect
10. #21444469C2semidirect
11. C2×C61×#331C2semidirect
22. #16840048C22semidirect
25. #16839963C22semidirect
30. #16839889C22semidirect
31. #16839886C22semidirect
36. #16839514C22semidirect
37. #16839512C22semidirect
38. #16839509C22semidirect
39. #16839509C22semidirect
40. #16839505C22semidirect
41. #16839505C22semidirect
157. #11758194SD16semidirect
243. C2×#331D61semidirect
285. C61#19238semidirect

Maximal subgroups

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

subgroupindex
#214571362
#214567602
#214562692
#214547622
#214453022
#214452902
#214451782
#214451662
#214445692
#214444692
C2×C61×#3312
#1923861