Finite Group Databasebeta

← #29733463

Subgroups — #29733463

Order 15616.

1234 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456208C211yesmaximal, characteristic, complemented
2#21455313C211yesmaximal, characteristic, complemented
3#21451926C211yesmaximal, characteristic, complemented
4#21446846C212yesmaximal, complemented
5#21446832C212yesmaximal, complemented
6#21446801C211yesmaximal, characteristic, complemented
7#21446660C211yesmaximal, characteristic, complemented
8#21446217C212yesmaximal, complemented
9#21446207C212yesmaximal, complemented
10#21445629C211yesmaximal, characteristic, complemented
11C61×#3508C211yesmaximal, characteristic, complemented
12C2×#14519807C2211yescharacteristic
13C2×#14519802C2211yescharacteristic
14C2×#14519773C2212yes—
15C2×#14519765C2211yescharacteristic
16C2×#14519764C2212yes—
17#16840124—22no—
18#16840119—22no—
19#16839984—22no—
20#16839977—22no—
21#16839786C2211yescharacteristic
22C2×#14519704C2212yes—
23#16839770C2211yescharacteristic
24#16839770C2212yes—
25#16839766C2211yescharacteristic
26#16839766C2212yes—
27C2×#14519694C2212yes—
28C2×#14519694C2212yes—
29C4×#11758194C2211yescharacteristic
30#16839670—24no—
31#16839608C2211yescharacteristic
32#16839521C2212yes—
33#16839521C2212yes—
34#16839521C2212yes—
35C61×#559—22no—
36C61×#550—22no—
37C61×#546—24no—
38C61×#540—22no—
39C61×#535C2212yes—
40C61×#533C2211yescharacteristic
41C61×#532—22no—
42C2×#127×C61C2211yescharacteristic
43C61×#518C2212yes—
44C4×#41×C61C2211yescharacteristic
45C22×Dic122C2311yescharacteristic
46C22×D244C2311yescharacteristic
47C22×C22⋊D61—24no—
48#14519807—42no—
49#14519802—42no—
50C2×#11758202—21no—
51C2×#11758200—21no—
52C2×#11758201—21no—
53C2×#11758199—21no—
54C2×#11758194C2312yes—
55C2×#11758194C2311yescharacteristic
56C2×#11758194C2311yescharacteristic
57#14519773—24no—
58#14519773—42no—
59#14519765—22no—
60#14519765—21no—
61#14519765—21no—
62#14519764—24no—
63#14519764—42no—
64C2×C22⋊Dic61—24no—
65C2×#11758198—22no—
66C2×C4×Dic61C2312yes—
67C4⋊#92418—22no—
68#14519704—24no—
69#14519700—22no—
70#14519700—24no—
71#14519697—22no—
72#14519697—24no—
73#14519694—42no—
74#14519694—42no—
75#14519688—21no—
76C4×#92418—21no—
77#14519684—22no—
78C2×#41×C61—24no—
79C2×#41×C61C2311yescharacteristic
80C22×D4×C61D414yes—
81C22×D4×C61C2312yes—
82C4⋊Q8×C61D412yescomplemented
83#127×C61—22no—
84#127×C61—21no—
85#127×C61—21no—
86#127×C61—22no—
87#125×C61D412yescomplemented
88#124×C61—24no—
89#124×C61—22no—
90#123×C61—28no—
91C22⋊Q8×C61—22no—
92#121×C61—24no—
93#121×C61—28no—
94#121×C61—24no—
95#120×C61—24no—
96C4×Q8×C61—22no—
97C4×Q8×C61D412yescomplemented
98C4×D4×C61—24no—
99C4×D4×C61—22no—
100C4×D4×C61—24no—
101C4×D4×C61—24no—
102C4×D4×C61D412yescomplemented
103#117×C61—21no—
104#117×C61C2312yes—
105#117×C61C2311yescharacteristic
106C2×C22⋊C4×C61—24no—
107C2×C42×C61—21no—
108C2×C42×C61C2311yescharacteristic
109C24×C61C2×D414yes—
110C23×D61—21no—
111#41×C61—22no—
112#41×C61—22no—
113#41×C61—88no—
114C2×Q8×C61—22no—
115C2×Q8×C61C2×D411yescharacteristic
116C2×D4×C61—21no—
117C2×D4×C61—28no—
118C2×D4×C61—24no—
119C2×D4×C61—24no—
120C2×D4×C61—22no—
121C2×D4×C61—22no—
122C2×D4×C61—22no—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61—22no—
126C2×D4×C61—84no—
127C2×D4×C61C2×D411yescharacteristic
128C22×C4×C61—28no—
129C22×C4×C61—22no—
130C22×C4×C61—21no—
131C22×C4×C61C2×D412yes—
132C22×C4×C61C2411yescharacteristic
133C2×C22⋊D61—48no—
134C2×C22⋊D61—88no—
135C22×Dic61—22no—
136C22×Dic61—21no—
137C2×D244—21no—
138C2×D244—21no—
139C2×D244—22no—
140C2×D244—22no—
141C2×Dic122—21no—
142C2×Dic122—21no—
143C2×Dic122—22no—
144C2×Dic122—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—24no—
148C4⋊C4×C61C2×D412yes—
149C4⋊C4×C61C2×D414yes—
150C22⋊C4×C61—28no—
151C22⋊C4×C61—28no—
152C22⋊C4×C61—21no—
153C22⋊C4×C61—22no—
154C22⋊C4×C61—22no—
155C42×C61—22no—
156C42×C61—21no—
157C42×C61—22no—
158C42×C61C2×D411yescharacteristic
159C42×C61C2×D411yescharacteristic
160C42×C61C2×D412yes—
161C22⋊Dic61—48no—
162#11758202—42no—
163#11758201—42no—
164#11758200—42no—
165#11758199—42no—
166#11758198—42no—
167#11758198—42no—
168C4×Dic61—22no—
169C4×Dic61#4114yescomplemented
170#11758194—22no—
171#11758194—22no—
172#11758194—22no—
173#11758194—22no—
174C2×#92418—22no—
175C2×#92418—21no—
176C2×#92418—21no—
177C23×C61—24no—
178C23×C61—21no—
179C23×C61—24no—
180C23×C61—24no—
181C23×C61—48no—
182C23×C61C22×D411yescharacteristic
183C22×D61—44no—
184C22×D61—41no—
185C22×D61—42no—
186Q8×C61—21no—
187Q8×C61—21no—
188Q8×C61—44no—
189D4×C61—21no—
190D4×C61—21no—
191D4×C61—48no—
192D4×C61—42no—
193D4×C61—42no—
194D4×C61—42no—
195D4×C61—42no—
196D4×C61—42no—
197D4×C61—84no—
198D4×C61—88no—
199C2×C4×C61—21no—
200C2×C4×C61—22no—
201C2×C4×C61—24no—
202C2×C4×C61—42no—
203C2×C4×C61—44no—
204C2×C4×C61—42no—
205C2×C4×C61—88no—
206C2×C4×C61#12012yes—
207C2×C4×C61#12012yes—
208C2×C4×C61C22×D411yescharacteristic
209C2×C4×C61C22×D411yescharacteristic
210C2×C4×C61C2×#4112yes—
211C2×C4×C61H(5,𝔽2)12yes—
212C22⋊D61—816no—
213C2×Dic61—24no—
214C2×Dic61—44no—
215C2×Dic61—41no—
216C2×Dic61—42no—
217D244—41no—
218D244—41no—
219D244—42no—
220Dic122—41no—
221Dic122—41no—
222Dic122—42no—
223#92418—41no—
224#92418—41no—
225#92418—42no—
226#25593—611nomaximal, Sylow
227C22×C61—22no—
228C22×C61—41no—
229C22×C61—44no—
230C22×C61—42no—
231C22×C61—48no—
232C22×C61—84no—
233C22×C61—84no—
234C22×C61C2×#12011yescharacteristic
235C22×C61#54612yes—
236C2×D61—84no—
237C2×D61—81no—
238C2×D61—82no—
239C4×C61—22no—
240C4×C61—22no—
241C4×C61—41no—
242C4×C61—42no—
243C4×C61—44no—
244C4×C61#53211yescharacteristic
245C4×C61#53511yescharacteristic
246C4×C61#55012yes—
247Dic61—44no—
248Dic61—81no—
249#3508D6111yescharacteristic, complemented
250#3140—611no—
251#3040—611no—
252#2657—611no—
253#2004—612no—
254#2002—612no—
255#1995—611no—
256#1965—611no—
257#1887—612no—
258#1885—612no—
259#1759—611no—
260C2×C61—22no—
261C2×C61—81no—
262C2×C61—84no—
263C2×C61#243111yescharacteristic
264C2×C61#301912yes—
265D61—161no—
266C2×#137—611no—
267C2×AGL(1,ℤ/8)—611no—
268#559—22no—
269#550—22no—
270#546—24no—
271#540—22no—
272#535C2×D6112yes—
273#533C2×D6111yescharacteristic
274#532—22no—
275C2×#127—612no—
276C2×#127C2×D6111yescharacteristic
277C2×#124—612no—
278C2×#124—611no—
279#518C2×D6112yes—
280C4×#41C2×D6111yescharacteristic
281#489—1222no—
282#488—1222no—
283#464—1222no—
284#463—1222no—
285#423—611no—
286C2×#104—612no—
287#419—612no—
288#419—611no—
289#418—612no—
290#418—611no—
291C2×#100—612no—
292C2×#100—612no—
293C4×M16—611no—
294#392—1224no—
295#376—611no—
296#339—612no—
297#339—612no—
298#339—612no—
299C61#2559311yescharacteristic, complemented, Sylow
300C2×#41—24no—
301C2×#41C22×D6111yescharacteristic
302C22×Q8—611no—
303C22×D4—611no—
304C22×D4—1224no—
305C22×D4C22⋊D6114yes—
306C22×D4C22×D6112yes—
307#137—2442no—
308AGL(1,ℤ/8)—2442no—
309C2×Q16—1221no—
310C2×SD16—1221no—
311C2×SD16—1221no—
312C2×D8—1221no—
313C2×M16—611no—
314C2×M16—611no—
315C2×M16—612no—
316C4⋊Q8C22⋊D6112yescomplemented
317#127—22no—
318#127—21no—
319#127—21no—
320#127—22no—
321#127—1224no—
322#127—2442no—
323#125C22⋊D6112yescomplemented
324#124—24no—
325#124—22no—
326#124—1224no—
327#124—1222no—
328#124—1221no—
329#124—1221no—
330#124—2442no—
331#123—28no—
332C22⋊Q8—22no—
333#121—24no—
334#121—28no—
335#121—24no—
336#120—24no—
337C4×Q8—22no—
338C4×Q8C22⋊D6112yescomplemented
339C4×D4—24no—
340C4×D4—22no—
341C4×D4—24no—
342C4×D4—24no—
343C4×D4C22⋊D6112yescomplemented
344#117—21no—
345#117C22×D6111yescharacteristic
346#117C22×D6112yes—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—1224no—
349C2×C22⋊C4—1222no—
350C2×C42—21no—
351C2×C42—612no—
352C2×C42C22×D6111yescharacteristic
353C4⋊C8—1222no—
354#104—1224no—
355#103—1222no—
356#103—1224no—
357#102—1222no—
358#102—1224no—
359#100—2442no—
360#100—2442no—
361#97—1221no—
362C4×C8—1221no—
363#95—1222no—
364C24—1221no—
365C24C2×C22⋊D6114yes—
366#41—22no—
367#41—22no—
368#41—88no—
369C2×Q8—22no—
370C2×Q8—1221no—
371C2×Q8—1221no—
372C2×Q8—1222no—
373C2×Q8—1222no—
374C2×Q8C2×C22⋊D6111yescharacteristic
375C2×D4—21no—
376C2×D4—28no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—24no—
384C2×D4—24no—
385C2×D4—84no—
386C2×D4—1221no—
387C2×D4—1221no—
388C2×D4—1222no—
389C2×D4—1222no—
390C2×D4—4888no—
391C2×D4—2448no—
392C2×D4C2×C22⋊D6111yescharacteristic
393C22×C4—28no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—1222no—
397C22×C4—1221no—
398C22×C4C2×C22⋊D6112yes—
399C22×C4C23×D6111yescharacteristic
400Q16—2442no—
401SD16—2442no—
402SD16—2442no—
403D8—2442no—
404M16—1222no—
405M16—1222no—
406M16—1222no—
407M16—1222no—
408C2×C8—1222no—
409C2×C8—1221no—
410C2×C8—1221no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—24no—
414C4⋊C4C2×C22⋊D6112yes—
415C4⋊C4C2×C22⋊D6114yes—
416C22⋊C4—28no—
417C22⋊C4—28no—
418C22⋊C4—21no—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C22⋊C4—2448no—
422C22⋊C4—2442no—
423C22⋊C4—2442no—
424C42—21no—
425C42—22no—
426C42—22no—
427C42—614no—
428C42—1222no—
429C42C2×C22⋊D6111yescharacteristic
430C42C2×C22⋊D6111yescharacteristic
431C42C2×C22⋊D6112yes—
432C23—24no—
433C23—21no—
434C23—24no—
435C23—24no—
436C23—48no—
437C23—2444no—
438C23—2441no—
439C23—2442no—
440C23C22×C22⋊D6111yescharacteristic
441Q8—21no—
442Q8—21no—
443Q8—44no—
444Q8—2441no—
445Q8—2441no—
446Q8—2442no—
447D4—21no—
448D4—21no—
449D4—48no—
450D4—42no—
451D4—42no—
452D4—42no—
453D4—42no—
454D4—42no—
455D4—84no—
456D4—88no—
457D4—48816no—
458D4—2441no—
459D4—2441no—
460D4—2442no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—88no—
468C2×C4—1224no—
469C2×C4—2441no—
470C2×C4—2442no—
471C2×C4—2444no—
472C2×C4#1451974612yes—
473C2×C4#1451974612yes—
474C2×C4C22×C22⋊D6111yescharacteristic
475C2×C4C22×C22⋊D6111yescharacteristic
476C2×C4#41×D6112yes—
477C2×C4#1451982312yes—
478C8—2441no—
479C8—2441no—
480C8—2442no—
481C22—22no—
482C22—42no—
483C22—48no—
484C22—41no—
485C22—44no—
486C22—84no—
487C22—84no—
488C22—4884no—
489C22—4882no—
490C22—4881no—
491C22C2×#1451974611yescharacteristic
492C22#1684040012yes—
493C4—22no—
494C4—22no—
495C4—42no—
496C4—44no—
497C4—41no—
498C4—4881no—
499C4—2444no—
500C4#1684030811yescharacteristic
501C4#1684032611yescharacteristic
502C4#1684043112yes—
503C2—22no—
504C2—84no—
505C2—81no—
506C2—9761no—
507C2#2144943411yescharacteristic
508C2#2145517612yes—
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3508C2
socleC22×C61C2×#120
hypercenter#3508D61
hypocenterC61#25593
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3508

Complements

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

NH ≅ G/Ntype
1. #21456208C2semidirect
2. #21455313C2semidirect
3. #21451926C2semidirect
4. #21446846C2semidirect
5. #21446832C2semidirect
6. #21446801C2semidirect
7. #21446660C2semidirect
8. #21446217C2semidirect
9. #21446207C2semidirect
10. #21445629C2semidirect
11. C61×#3508C2semidirect
82. C4⋊Q8×C61D4semidirect
87. #125×C61D4semidirect
97. C4×Q8×C61D4semidirect
102. C4×D4×C61D4semidirect
169. C4×Dic61#41semidirect
249. #3508D61semidirect
299. C61#25593semidirect
316. C4⋊Q8C22⋊D61semidirect
323. #125C22⋊D61semidirect
338. C4×Q8C22⋊D61semidirect
343. C4×D4C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214562082
#214553132
#214519262
#214468462
#214468322
#214468012
#214466602
#214462172
#214462072
#214456292
C61×#35082
#2559361