Finite Group Databasebeta

← #29663586

Subgroups — #29663586

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#21456762C211yesmaximal, characteristic, complemented
3#21456271C211yesmaximal, characteristic
4#21454766C211yesmaximal, characteristic, complemented
5#21445304C212yesmaximal, complemented
6#21445292C212yesmaximal, complemented
7#21445180C212yesmaximal, complemented
8#21445168C212yesmaximal, complemented
9#21444466C211yesmaximal, characteristic, complemented
10C2×#16839505C211yesmaximal, characteristic, complemented
11C61×#1565C211yesmaximal, characteristic, complemented
12C2×#14519805—21no—
13C2×#14519805—21no—
14#16840319C2211yescharacteristic
15#16840312C2211yescharacteristic
16C2×#14519774C2211yescharacteristic
17#16840151—22no—
18#16840139C2212yes—
19#16840135—21no—
20#16840135C2212yes—
21#16840109—21no—
22#16840053—21no—
23#16840048C2212yescomplemented
24#16839991—22no—
25#16839968—21no—
26#16839963C2212yes—
27#16839963C2212yes—
28#16839963C2212yescomplemented
29#16839915—22no—
30#16839892C2212yescomplemented
31#16839890—22no—
32C8×Dic122C2212yescomplemented
33#16839790—21no—
34C2×C4⋊#92418C2211yescharacteristic
35#16839720—21no—
36#16839507C2211yescharacteristic, complemented
37#16839507C2211yescharacteristic, complemented
38#16839505C2212yes—
39#16839505C2212yes—
40#16839505C2211yescharacteristic, complemented
41#16839505C2211yescharacteristic, complemented
42C61×#431C2211yescharacteristic
43C2×C4⋊C8×C61C2211yescharacteristic
44C4×M16×C61C2211yescharacteristic
45C61×#335C2212yescomplemented
46C61×#334C2212yescomplemented
47C2×#11758222—21no—
48C22×D244—21no—
49C22×D244—21no—
50C22×D244—21no—
51#14519805—44no—
52#14519805—44no—
53C2×#11758190—22no—
54C2×#11758190—22no—
55C2×#11758190—22no—
56C2×#11758190—22no—
57C2×#11758190—22no—
58C2×D488—22no—
59C2×D488—22no—
60C22×#92418—21no—
61#14519777D411yescharacteristic
62#14519777D411yescharacteristic
63#14519777C2311yescharacteristic
64#14519774—21no—
65#14519774—21no—
66#14519774—21no—
67#14519774C2312yes—
68#14519774C2311yescharacteristic
69#14519769—21no—
70#14519765—21no—
71#14519756—21no—
72#14519756D412yes—
73#14519751—21no—
74#14519751—21no—
75C4×Dic122—21no—
76C4×Dic122C2312yes—
77C4×D244—21no—
78C4×D244—21no—
79#14519736—21no—
80#14519709—22no—
81C4⋊#92418—21no—
82C4⋊#92418C2311yescharacteristic
83C4⋊#92418C2311yescharacteristic
84#14519706—22no—
85#14519706—22no—
86#14519702—22no—
87#14519699—22no—
88#14519699—22no—
89#14519699—22no—
90#14519687—22no—
91C8×Dic61—22no—
92C2×M16×C61D411yescharacteristic
93C2×M16×C61D411yescharacteristic
94C22×C8×C61—21no—
95C2×C42×C61C2311yescharacteristic
96C4⋊C8×C61—21no—
97C4⋊C8×C61C2311yescharacteristic
98C4⋊C8×C61C2311yescharacteristic
99C4⋊C8×C61C2312yes—
100C22⋊C8×C61—21no—
101#97×C61D412yes—
102C4×C8×C61—21no—
103C4×C8×C61C2312yes—
104C23×D61—41no—
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61#4111yescharacteristic
108C2×C22⋊D61—41no—
109#11758222—42no—
110#11758222—42no—
111C2×D244—22no—
112C2×D244—22no—
113C2×D244—21no—
114C2×D244—21no—
115C2×D244—22no—
116C2×D244—21no—
117C2×D244—42no—
118C2×D244—42no—
119C2×D244—41no—
120C2×D244—41no—
121C2×D244—41no—
122C2×D244—81no—
123C2×D244—81no—
124C2×D244—82no—
125C2×C4×D61—41no—
126C2×Dic122—21no—
127C2×Dic122—21no—
128C2×Dic122C2×D412yes—
129M16×C61—22no—
130M16×C61SD1614yescomplemented
131C2×C8×C61—21no—
132C2×C8×C61—21no—
133C2×C8×C61—22no—
134C2×C8×C61—81no—
135C2×C8×C61C2×D412yes—
136C2×C8×C61C2×D412yes—
137C42×C61C2×D411yescharacteristic
138C42×C61C2×D411yescharacteristic
139C42×C61#4111yescharacteristic
140C42×C61C2411yescharacteristic
141C22⋊Dic61—22no—
142#11758198—41no—
143#11758198—41no—
144C4⋊Dic61—21no—
145C4⋊Dic61—21no—
146C4⋊Dic61—21no—
147C4⋊Dic61—21no—
148C4⋊Dic61—21no—
149C4⋊Dic61C2×D412yes—
150#11758196—22no—
151#11758196—22no—
152C4×Dic61—22no—
153C2×#92418—21no—
154C2×#92418—21no—
155C2×#92418—42no—
156D488—44no—
157D488—44no—
158#11758190—44no—
159#11758190—44no—
160#11758190—82no—
161#11758190—82no—
162#11758190—82no—
163C23×C61#12111yescharacteristic
164C22×D61—41no—
165C22×D61—42no—
166C22×D61—82no—
167C22×D61—82no—
168C22×D61—82no—
169C2×C4×C61—21no—
170C2×C4×C61—21no—
171C2×C4×C61—41no—
172C2×C4×C61#12111yescharacteristic
173C2×C4×C61#12111yescharacteristic
174C2×C4×C61#12111yescharacteristic
175C2×C4×C61C2×SD1612yes—
176C2×C4×C61AGL(1,ℤ/8)12yes—
177C2×C4×C61C22×D411yescharacteristic
178C2×C4×C61C22×D411yescharacteristic
179C2×C4×C61C2×#4111yescharacteristic
180C22⋊D61—82no—
181C2×Dic61—22no—
182C2×Dic61—22no—
183C2×Dic61—41no—
184D244—42no—
185D244—41no—
186D244—42no—
187D244—41no—
188D244—41no—
189D244—42no—
190D244—42no—
191D244—41no—
192D244—81no—
193D244—82no—
194D244—82no—
195D244—82no—
196D244—82no—
197C4×D61—82no—
198Dic122—22no—
199Dic122—22no—
200Dic122—41no—
201Dic122—41no—
202Dic122—81no—
203C8×C61—22no—
204C8×C61—41no—
205C8×C61—42no—
206C8×C61—41no—
207C8×C61C2×SD1614yes—
208#92418—41no—
209#92418—41no—
210#19238—611nomaximal, Sylow
211C22×C61—21no—
212C22×C61—21no—
213C22×C61#46012yes—
214C22×C61C2×#12111yescharacteristic
215C2×D61—81no—
216C2×D61—82no—
217C2×D61—82no—
218C2×D61—81no—
219C2×D61—82no—
220C2×D61—81no—
221C2×D61—164no—
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#1565D6111yescharacteristic, complemented
242#1540—611no—
243C2×#331—611no—
244C2×C61—22no—
245C2×C61C2×#46011yescharacteristic
246C2×C61#306211yescharacteristic
247C2×C61#306811yescharacteristic
248D61—161no—
249D61—162no—
250C2×AGL(1,ℤ/8)—1221no—
251C2×AGL(1,ℤ/8)—1221no—
252#533—611no—
253#532—611no—
254C2×#127—611no—
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#431C2×D6111yescharacteristic
272#424—1221no—
273C2×C4⋊C8—611no—
274C2×C4⋊C8C2×D6111yescharacteristic
275#405—1221no—
276C4×M16C2×D6111yescharacteristic
277#335C2×D6112yescomplemented
278#334C2×D6112yescomplemented
279#332—611no—
280#332—611no—
281#331—612no—
282#331—612no—
283#331—611no—
284#331—611no—
285C61#1923811yescharacteristic, complemented, Sylow
286C2×#41—1221no—
287C22×D4—1221no—
288C22×D4—1221no—
289C22×D4—1221no—
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×M16D24411yescharacteristic
300C2×M16D24411yescharacteristic
301C22×C8—21no—
302C22×C8—1221no—
303C4⋊Q8—611no—
304C4⋊Q8—611no—
305C4⋊Q8—611no—
306#127—612no—
307#127—611no—
308#127—1221no—
309#127—1221no—
310#127—1221no—
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—611no—
326C4⋊C8—611no—
327C4⋊C8—1222no—
328C4⋊C8—1222no—
329C4⋊C8—1221no—
330C4⋊C8C22×D6112yes—
331C4⋊C8C22×D6111yescharacteristic
332C4⋊C8C22×D6111yescharacteristic
333#103—1222no—
334#102—1222no—
335#102—1222no—
336#102—1222no—
337C22⋊C8—21no—
338#97—1222no—
339#97D24412yes—
340C4×C8—21no—
341C4×C8—1222no—
342C4×C8C22×D6112yes—
343C24—2441no—
344#41—2442no—
345#41—2442no—
346C2×Q8—612no—
347C2×Q8—1221no—
348C2×Q8—1221no—
349C2×D4—1221no—
350C2×D4—1222no—
351C2×D4—1221no—
352C2×D4—1222no—
353C2×D4—1221no—
354C2×D4—1222no—
355C2×D4—4881no—
356C2×D4—4881no—
357C2×D4—4882no—
358C2×D4—2441no—
359C2×D4—2441no—
360C2×D4—2441no—
361C2×D4—2441no—
362C2×D4—2442no—
363C2×D4—2442no—
364C22×C4—2441no—
365C22×C4C2×D24411yescharacteristic
366C22×C4D4×D6111yescharacteristic
367C22×C4#1175822611yescharacteristic
368SD16—4882no—
369SD16—4882no—
370SD16—4882no—
371SD16—2444no—
372SD16—2444no—
373D8—2444no—
374D8—2444no—
375M16—22no—
376M16#1175819014yescomplemented
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—22no—
380C2×C8—81no—
381C2×C8—1221no—
382C2×C8—1221no—
383C2×C8—2442no—
384C2×C8C2×D24412yes—
385C2×C8C2×D24412yes—
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—
398C42C2×D24411yescharacteristic
399C42D4×D6111yescharacteristic
400C42#1175822611yescharacteristic
401C42C23×D6111yescharacteristic
402C23—4882no—
403C23—4882no—
404C23—4882no—
405C23—2441no—
406C23—2442no—
407C23#1451975211yescharacteristic
408Q8—1222no—
409Q8—1222no—
410Q8—4881no—
411Q8—2441no—
412Q8—2441no—
413D4—4881no—
414D4—4882no—
415D4—4882no—
416D4—4882no—
417D4—4882no—
418D4—4882no—
419D4—2441no—
420D4—2442no—
421D4—2441no—
422D4—2442no—
423D4—2441no—
424D4—2442no—
425D4—2442no—
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#1451975211yescharacteristic
435C2×C4#1451975211yescharacteristic
436C2×C4#1451975211yescharacteristic
437C2×C4C2×#1175819012yes—
438C2×C4#1451980512yes—
439C2×C4C2×D4×D6111yescharacteristic
440C2×C4C22×D24411yescharacteristic
441C2×C4C2×#1175822611yescharacteristic
442C8—22no—
443C8—42no—
444C8—41no—
445C8—41no—
446C8—2441no—
447C8—2441no—
448C8C2×#1175819014yes—
449C22—21no—
450C22—21no—
451C22—9764no—
452C22—4882no—
453C22—4881no—
454C22—4881no—
455C22—4882no—
456C22—4882no—
457C22—4881no—
458C22#1683996312yes—
459C22C2×#1451975211yescharacteristic
460C4—21no—
461C4—21no—
462C4—21no—
463C4—4881no—
464C4—2442no—
465C4—2442no—
466C4#1683996312yes—
467C4C22×#1175819011yescharacteristic
468C4C2×#1451980511yescharacteristic
469C4#1684056611yescharacteristic
470C4#1684057111yescharacteristic
471C2—22no—
472C2—9762no—
473C2—9761no—
474C2C2×#1683996311yescharacteristic
475C2#2145550011yescharacteristic
476C2#2145554811yescharacteristic
subgroupG/H
centerC22C2×#14519752
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1565C2
socleC22×C61C2×#121
prime radical#21456271C2
hypercenter#1565D61
hypocenterC61#19238
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1565

Complements

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

NH ≅ G/Ntype
1. #21457136C2semidirect
2. #21456762C2semidirect
4. #21454766C2semidirect
5. #21445304C2semidirect
6. #21445292C2semidirect
7. #21445180C2semidirect
8. #21445168C2semidirect
9. #21444466C2semidirect
10. C2×#16839505C2semidirect
11. C61×#1565C2semidirect
23. #16840048C22semidirect
28. #16839963C22semidirect
30. #16839892C22semidirect
32. C8×Dic122C22semidirect
36. #16839507C22semidirect
37. #16839507C22semidirect
40. #16839505C22semidirect
41. #16839505C22semidirect
45. C61×#335C22semidirect
46. C61×#334C22semidirect
130. M16×C61SD16semidirect
241. #1565D61semidirect
277. #335C2×D61semidirect
278. #334C2×D61semidirect
285. C61#19238semidirect
376. M16#11758190semidirect

Maximal subgroups

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

subgroupindex
#214571362
#214567622
#214562712
#214547662
#214453042
#214452922
#214451802
#214451682
#214444662
C2×#168395052
C61×#15652
#1923861