Finite Group Databasebeta

← #29652549

Subgroups — #29652549

Order 15616.

1184 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455937C212yesmaximal, complemented
2#21451396C211yesmaximal, characteristic, complemented
3#21444822C212yesmaximal, complemented
4#21444805C212yesmaximal, complemented
5#21444788C212yesmaximal, complemented
6#21444771C212yesmaximal, complemented
7#21444366C212yesmaximal, complemented
8#21444153C211yesmaximal, characteristic, complemented
9C61×#3449C211yesmaximal, characteristic
10C2×C4×Dic122C2211yescharacteristic, complemented
11#16840073C2214yescomplemented
12#16840054—22no—
13#16839959—22no—
14#16839900C2214yescomplemented
15#16839862—24no—
16#16839819C2212yescomplemented
17#16839692C2211yescharacteristic, complemented
18#16839687C2211yescharacteristic
19#16839682C2211yescharacteristic, complemented
20#16839674C2211yescharacteristic
21#16839650—21no—
22#16839646—21no—
23#16839643—21no—
24#16839635—21no—
25#16839630C2211yescharacteristic, complemented
26#16839496C2214yescomplemented
27#16839496C2214yescomplemented
28#16839488C2212yescomplemented
29#16839486C2212yes—
30C22×#41×C61—21no—
31C61×#537—21no—
32C61×#537C2212yes—
33C61×#536—22no—
34C61×#535—22no—
35C61×#534—21no—
36C61×#534C2212yes—
37C61×#532—21no—
38C61×#532C2212yes—
39C61×#525—24no—
40C2×#117×C61C2211yescharacteristic
41C22×Dic122C2311yescharacteristic
42C2×#11758200—24no—
43C4×Dic122—22no—
44C4×Dic122—22no—
45C2×C4⋊Dic61C2311yescharacteristic
46C2×#11758196—21no—
47C2×#11758196—21no—
48C2×#11758196—21no—
49C2×C22⋊Dic61—21no—
50C2×C22⋊Dic61—21no—
51C2×C4×Dic61—21no—
52#14519710—24no—
53C4⋊#92418—22no—
54#14519700—24no—
55#14519698—24no—
56C22⋊#92418C2314yes—
57#14519688—22no—
58#14519684—21no—
59#14519684—21no—
60#14519684—21no—
61#14519684—21no—
62#14519684—21no—
63#14519684—21no—
64#14519684C2311yescharacteristic
65#14519684C2311yescharacteristic
66C2×#41×C61—22no—
67C2×#41×C61—22no—
68C2×#41×C61—22no—
69C2×#41×C61—22no—
70C2×#41×C61—44no—
71C2×#41×C61—44no—
72C22×Q8×C61—21no—
73C22×D4×C61—21no—
74C22×D4×C61—22no—
75C23×C4×C61—22no—
76C23×C4×C61D411yescharacteristic
77C4⋊Q8×C61—21no—
78C4⋊Q8×C61D412yes—
79#127×C61—21no—
80#127×C61D412yes—
81#124×C61—22no—
82#124×C61—22no—
83#123×C61—24no—
84#123×C61—24no—
85C22⋊Q8×C61—24no—
86C22⋊Q8×C61—24no—
87#121×C61—24no—
88#121×C61—24no—
89#121×C61—22no—
90#121×C61—22no—
91#121×C61—22no—
92#121×C61C2314yes—
93#120×C61—24no—
94#120×C61—24no—
95C4×D4×C61—22no—
96C4×D4×C61—22no—
97C4×D4×C61—22no—
98C4×D4×C61D414yes—
99#117×C61—22no—
100#117×C61—22no—
101C2×C4⋊C4×C61D411yescharacteristic
102C2×C4⋊C4×C61C2311yescharacteristic
103C2×C22⋊C4×C61D411yescharacteristic
104C2×C22⋊C4×C61C2311yescharacteristic
105C2×C42×C61D411yescharacteristic
106C2×C42×C61C2311yescharacteristic
107C24×C61—22no—
108C24×C61C2×D411yescharacteristic
109#41×C61—48no—
110#41×C61—48no—
111#41×C61—44no—
112#41×C61—44no—
113#41×C61—44no—
114#41×C61—44no—
115C2×Q8×C61—22no—
116C2×Q8×C61—22no—
117C2×Q8×C61—44no—
118C2×D4×C61—24no—
119C2×D4×C61—22no—
120C2×D4×C61—24no—
121C2×D4×C61—22no—
122C2×D4×C61—22no—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61—44no—
126C2×D4×C61—44no—
127C2×D4×C61—44no—
128C2×D4×C61C2×D414yes—
129C22×C4×C61—22no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—22no—
133C22×C4×C61—24no—
134C22×C4×C61—22no—
135C22×C4×C61—24no—
136C22×C4×C61—48no—
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2411yescharacteristic
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C2×Dic122—22no—
148C2×Dic122—42no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61C2×D414yes—
155C22⋊C4×C61—24no—
156C22⋊C4×C61—24no—
157C22⋊C4×C61—22no—
158C22⋊C4×C61—24no—
159C22⋊C4×C61—22no—
160C22⋊C4×C61—24no—
161C42×C61—21no—
162C42×C61—21no—
163C42×C61C2×D412yes—
164C42×C61C2×D412yes—
165C22⋊Dic61—42no—
166C22⋊Dic61—42no—
167#11758200—48no—
168C4⋊Dic61—22no—
169#11758196—42no—
170#11758196—41no—
171#11758196—41no—
172#11758196—42no—
173C4×Dic61—42no—
174C2×#92418—24no—
175C23×C61—21no—
176C23×C61—21no—
177C23×C61—24no—
178C23×C61—41no—
179C23×C61—41no—
180C23×C61—44no—
181C23×C61—82no—
182C23×C61—82no—
183C23×C61AGL(1,ℤ/8)12yes—
184C23×C61C22×D411yescharacteristic
185Q8×C61—44no—
186Q8×C61—44no—
187D4×C61—44no—
188D4×C61—44no—
189D4×C61—44no—
190D4×C61—48no—
191D4×C61—44no—
192D4×C61—44no—
193D4×C61—82no—
194D4×C61—82no—
195D4×C61—82no—
196C2×C4×C61—24no—
197C2×C4×C61—24no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—24no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61—48no—
208C2×C4×C61—48no—
209C2×C4×C61—44no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61#12012yes—
213C2×C4×C61#12011yescharacteristic
214C2×C4×C61#12011yescharacteristic
215C2×C4×C61AGL(1,ℤ/8)12yes—
216C2×C4×C61C22×D412yes—
217C2×Dic61—22no—
218C2×Dic61—41no—
219C2×Dic61—41no—
220C2×Dic61—41no—
221C2×Dic61—41no—
222C2×Dic61—41no—
223C2×Dic61—41no—
224C2×Dic61—41no—
225C2×Dic61—41no—
226C2×Dic61—41no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229Dic122—24no—
230Dic122—42no—
231#92418—44no—
232#18153—611nomaximal, Sylow
233C22×C61—21no—
234C22×C61—21no—
235C22×C61—22no—
236C22×C61—42no—
237C22×C61—41no—
238C22×C61—41no—
239C22×C61—44no—
240C22×C61—44no—
241C22×C61—44no—
242C22×C61—81no—
243C22×C61—88no—
244C22×C61C2×#12011yescharacteristic
245C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
246C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
247C4×C61—21no—
248C4×C61—22no—
249C4×C61—21no—
250C4×C61—22no—
251C4×C61—22no—
252C4×C61—44no—
253C4×C61—41no—
254C4×C61—41no—
255Dic61—42no—
256Dic61—81no—
257Dic61—81no—
258Dic61—81no—
259#3449D6111yescharacteristic
260#3112—612no—
261#2599—611no—
262#1610—612no—
263#1607—612no—
264#1604—612no—
265#1601—612no—
266#1514—612no—
267#1464—611no—
268C2×C61—21no—
269C2×C61—21no—
270C2×C61—42no—
271C2×C61—84no—
272C2×C61#300311yescharacteristic
273C2×C61#302211yescharacteristic
274C2×C61#302511yescharacteristic
275C22×#41—21no—
276#537—21no—
277#537C2×D6112yes—
278#536—22no—
279#535—22no—
280#534—21no—
281#534C2×D6112yes—
282#532—21no—
283#532C2×D6112yes—
284#525—24no—
285C2×C4×Q8—611no—
286C2×#117C2×D6111yescharacteristic
287#480—614no—
288#477—1222no—
289#460—1222no—
290#448—614no—
291#440—1224no—
292#432—612no—
293#398—611no—
294#397—611no—
295#395—611no—
296#393—611no—
297#388—1221no—
298#387—1221no—
299#387—1221no—
300#385—1221no—
301#383—611no—
302#327—614no—
303#327—614no—
304#324—612no—
305#323—612no—
306C61#1815311yescharacteristic, complemented, Sylow
307C2×#41—22no—
308C2×#41—22no—
309C2×#41—22no—
310C2×#41—22no—
311C2×#41—44no—
312C2×#41—44no—
313C22×Q8—21no—
314C22×Q8—611no—
315C22×D4—21no—
316C22×D4—22no—
317C23×C4—22no—
318C23×C4C22⋊D6111yescharacteristic
319C2×SD16—1224no—
320C4⋊Q8—21no—
321C4⋊Q8C22⋊D6112yes—
322#127—21no—
323#127C22⋊D6112yes—
324#124—22no—
325#124—22no—
326#123—24no—
327#123—24no—
328C22⋊Q8—24no—
329C22⋊Q8—24no—
330#121—24no—
331#121—24no—
332#121—22no—
333#121—22no—
334#121—22no—
335#121C22×D6114yes—
336#120—24no—
337#120—24no—
338C4×Q8—1222no—
339C4×Q8—1222no—
340C4×D4—22no—
341C4×D4—22no—
342C4×D4—22no—
343C4×D4C22⋊D6114yes—
344#117—22no—
345#117—22no—
346C2×C4⋊C4—611no—
347C2×C4⋊C4—1221no—
348C2×C4⋊C4—1221no—
349C2×C4⋊C4—1221no—
350C2×C4⋊C4C22⋊D6111yescharacteristic
351C2×C4⋊C4C22×D6111yescharacteristic
352C2×C22⋊C4—1221no—
353C2×C22⋊C4—1221no—
354C2×C22⋊C4C22⋊D6111yescharacteristic
355C2×C22⋊C4C22×D6111yescharacteristic
356C2×C42—1221no—
357C2×C42C22⋊D6111yescharacteristic
358C2×C42C22×D6111yescharacteristic
359#107—1224no—
360C4⋊C8—1222no—
361#103—1224no—
362#102—1224no—
363C22⋊C8—614no—
364#97—1222no—
365#95—611no—
366#95—611no—
367#95—1221no—
368#95—1221no—
369#95—1221no—
370#95—1221no—
371#95—1221no—
372#95—1221no—
373C24—22no—
374C24C2×C22⋊D6111yescharacteristic
375#41—48no—
376#41—48no—
377#41—44no—
378#41—44no—
379#41—44no—
380#41—44no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—44no—
384C2×Q8—1222no—
385C2×Q8—2442no—
386C2×D4—24no—
387C2×D4—22no—
388C2×D4—24no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4C2×C22⋊D6114yes—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—48no—
405C22×C4—1221no—
406C22×C4—1221no—
407C22×C4—1221no—
408C22×C4—1221no—
409C22×C4C2×C22⋊D6111yescharacteristic
410C22×C4C2×C22⋊D6111yescharacteristic
411C22×C4C2×C22⋊D6111yescharacteristic
412C22×C4C2×C22⋊D6111yescharacteristic
413C22×C4C2×C22⋊D6111yescharacteristic
414C22×C4C23×D6111yescharacteristic
415SD16—2448no—
416C2×C8—1224no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—1222no—
423C4⋊C4—2442no—
424C4⋊C4—2441no—
425C4⋊C4—2441no—
426C4⋊C4—2442no—
427C4⋊C4C2×C22⋊D6114yes—
428C22⋊C4—24no—
429C22⋊C4—24no—
430C22⋊C4—22no—
431C22⋊C4—24no—
432C22⋊C4—22no—
433C22⋊C4—24no—
434C22⋊C4—2442no—
435C22⋊C4—2442no—
436C42—21no—
437C42—21no—
438C42—2442no—
439C42C2×C22⋊D6112yes—
440C42C2×C22⋊D6112yes—
441C23—21no—
442C23—21no—
443C23—24no—
444C23—41no—
445C23—41no—
446C23—44no—
447C23—82no—
448C23—82no—
449C23#1451980312yes—
450C23C22×C22⋊D6111yescharacteristic
451Q8—44no—
452Q8—44no—
453Q8—1224no—
454Q8—2442no—
455D4—44no—
456D4—44no—
457D4—44no—
458D4—48no—
459D4—44no—
460D4—44no—
461D4—82no—
462D4—82no—
463D4—82no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—24no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—48no—
476C2×C4—48no—
477C2×C4—44no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—1222no—
481C2×C4—2441no—
482C2×C4—2441no—
483C2×C4—2441no—
484C2×C4—2441no—
485C2×C4—2441no—
486C2×C4—2441no—
487C2×C4—2441no—
488C2×C4—2441no—
489C2×C4—2441no—
490C2×C4—2441no—
491C2×C4—2441no—
492C2×C4#1451974611yescharacteristic
493C2×C4#1451974611yescharacteristic
494C2×C4#1451974612yes—
495C2×C4#1451980312yes—
496C2×C4C22×C22⋊D6112yes—
497C8—2444no—
498C22—21no—
499C22—21no—
500C22—22no—
501C22—42no—
502C22—41no—
503C22—41no—
504C22—44no—
505C22—44no—
506C22—44no—
507C22—81no—
508C22—88no—
509C22C2×#1451974611yescharacteristic
510C22C2×#1451980311yescharacteristic
511C22C2×#1451980311yescharacteristic
512C4—22no—
513C4—22no—
514C4—21no—
515C4—22no—
516C4—21no—
517C4—41no—
518C4—41no—
519C4—44no—
520C4—4881no—
521C4—4881no—
522C4—4881no—
523C4—2442no—
524C2—21no—
525C2—21no—
526C2—42no—
527C2—84no—
528C2#2145504611yescharacteristic
529C2#2145519811yescharacteristic
530C2#2145521111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3449C2
socleC22×C61C2×#120
hypercenter#3449D61
hypocenterC61#18153
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3449

Complements

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

NH ≅ G/Ntype
1. #21455937C2semidirect
2. #21451396C2semidirect
3. #21444822C2semidirect
4. #21444805C2semidirect
5. #21444788C2semidirect
6. #21444771C2semidirect
7. #21444366C2semidirect
8. #21444153C2semidirect
10. C2×C4×Dic122C22semidirect
11. #16840073C22semidirect
14. #16839900C22semidirect
16. #16839819C22semidirect
17. #16839692C22semidirect
19. #16839682C22semidirect
25. #16839630C22semidirect
26. #16839496C22semidirect
27. #16839496C22semidirect
28. #16839488C22semidirect
306. C61#18153semidirect

Maximal subgroups

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

subgroupindex
#214559372
#214513962
#214448222
#214448052
#214447882
#214447712
#214443662
#214441532
C61×#34492
#1815361