Finite Group Databasebeta

← #29657284

Subgroups — #29657284

Order 15616.

1004 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#21456764C211yesmaximal, characteristic, complemented
2C2×#16840108C211yesmaximal, characteristic, complemented
3#21455634C211yesmaximal, characteristic, complemented
4#21445012C214yesmaximal, complemented
5#21445005C214yesmaximal, complemented
6#21444487C211yesmaximal, characteristic, complemented
7#21444274C211yesmaximal, characteristic, complemented
8#21444201C211yesmaximal, characteristic, complemented
9C61×#3508C211yesmaximal, characteristic
10C2×#14519777C2211yescharacteristic
11#16840141—21no—
12#16840114—21no—
13#16840108—22no—
14#16840108C2214yes—
15#16840057—21no—
16#16840052—21no—
17#16840047C2212yescomplemented
18#16840042C2212yescomplemented
19#16839972C2214yescomplemented
20#16839917—22no—
21#16839815C2211yescharacteristic
22#16839805—21no—
23#16839786C2211yescharacteristic
24#16839763—21no—
25C2×#14519698—22no—
26C2×C4×#92418C2211yescharacteristic
27#16839506C2214yes—
28Q8⋊#92418C2212yescomplemented
29Q8⋊#92418C2214yescomplemented
30D4⋊#92418C2212yescomplemented
31C61×#559—21no—
32C61×#559C2212yes—
33C61×#550—21no—
34C61×#550C2212yes—
35C61×#546—22no—
36C61×#546—22no—
37C61×#540—22no—
38C61×#535—21no—
39C61×#533C2211yescharacteristic
40C61×#532—21no—
41C61×#532—21no—
42C2×#127×C61C2211yescharacteristic
43C61×#518—21no—
44C4×#41×C61C2211yescharacteristic
45C22×Dic122—21no—
46C2×#11758202—24no—
47C2×#11758194—21no—
48C22×#92418—21no—
49#14519777—21no—
50#14519777—21no—
51#14519777—21no—
52#14519777C2312yes—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61—21no—
55#14519710—22no—
56#14519708—22no—
57C4⋊#92418—21no—
58C4⋊#92418C2312yes—
59C4⋊#92418C2312yes—
60#14519701—22no—
61#14519700—24no—
62#14519698—28no—
63#14519698—22no—
64C22⋊#92418—21no—
65C4×#92418—21no—
66C4×#92418—21no—
67C4×#92418C2312yes—
68C2×#41×C61—22no—
69C2×#41×C61—22no—
70C2×#41×C61D411yescharacteristic
71C22×D4×C61—21no—
72C22×D4×C61—21no—
73C22×D4×C61D412yes—
74C4⋊Q8×C61D411yescharacteristic
75C4⋊Q8×C61D411yescharacteristic
76#127×C61—22no—
77#127×C61—21no—
78#127×C61—21no—
79#127×C61D412yes—
80#127×C61C2312yes—
81#125×C61—21no—
82#124×C61—22no—
83#124×C61—22no—
84#124×C61—22no—
85#123×C61—24no—
86#123×C61—24no—
87C22⋊Q8×C61—21no—
88C22⋊Q8×C61D412yes—
89#121×C61—22no—
90#121×C61—22no—
91#121×C61—22no—
92#121×C61—24no—
93#121×C61—24no—
94#121×C61—22no—
95#120×C61—24no—
96C4×Q8×C61—21no—
97C4×Q8×C61D411yescharacteristic
98C4×Q8×C61C2312yes—
99C4×Q8×C61C2311yescharacteristic
100C4×D4×C61—22no—
101C4×D4×C61—22no—
102C4×D4×C61—22no—
103C4×D4×C61—21no—
104C4×D4×C61—21no—
105C4×D4×C61—22no—
106C4×D4×C61—22no—
107C4×D4×C61—22no—
108C4×D4×C61D411yescharacteristic
109C4×D4×C61C2311yescharacteristic
110#117×C61—21no—
111#117×C61—21no—
112#117×C61D411yescharacteristic
113C2×C22⋊C4×C61—24no—
114C2×C42×C61—21no—
115C2×C42×C61C2311yescharacteristic
116C24×C61—22no—
117#41×C61—22no—
118#41×C61—22no—
119#41×C61—84no—
120#41×C61—84no—
121C2×Q8×C61—21no—
122C2×Q8×C61C2×D411yescharacteristic
123C2×Q8×C61C2×D412yes—
124C2×D4×C61—21no—
125C2×D4×C61—24no—
126C2×D4×C61—24no—
127C2×D4×C61—22no—
128C2×D4×C61—22no—
129C2×D4×C61—22no—
130C2×D4×C61—22no—
131C2×D4×C61—22no—
132C2×D4×C61—44no—
133C2×D4×C61—42no—
134C2×D4×C61—42no—
135C2×D4×C61—82no—
136C2×D4×C61C2×D411yescharacteristic
137C2×D4×C61C2×D414yes—
138C22×C4×C61—24no—
139C22×C4×C61—24no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×Dic61—41no—
147C2×Dic122—22no—
148C2×Dic122—41no—
149C2×Dic122—82no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61C2×D411yescharacteristic
158C4⋊C4×C61C2×D411yescharacteristic
159C4⋊C4×C61C2×D412yes—
160C22⋊C4×C61—24no—
161C22⋊C4×C61—24no—
162C22⋊C4×C61—24no—
163C22⋊C4×C61—24no—
164C22⋊C4×C61—21no—
165C22⋊C4×C61—22no—
166C22⋊C4×C61—21no—
167C22⋊C4×C61—21no—
168C42×C61—21no—
169C42×C61—21no—
170C42×C61—21no—
171C42×C61—22no—
172C42×C61C2×D411yescharacteristic
173C42×C61C2×D411yescharacteristic
174C42×C61C2×D411yescharacteristic
175C42×C61C2411yescharacteristic
176#11758202—84no—
177C4⋊Dic61—22no—
178C4⋊Dic61—22no—
179C4⋊Dic61—41no—
180C4⋊Dic61—41no—
181#11758194—42no—
182C2×#92418—22no—
183C2×#92418—22no—
184C2×#92418—22no—
185C2×#92418—42no—
186C23×C61—24no—
187C23×C61—21no—
188C23×C61—44no—
189C23×C61—82no—
190C23×C61—82no—
191C23×C61#12011yescharacteristic
192Q8×C61—21no—
193Q8×C61—21no—
194Q8×C61—24no—
195Q8×C61—42no—
196D4×C61—21no—
197D4×C61—21no—
198D4×C61—44no—
199D4×C61—42no—
200D4×C61—42no—
201D4×C61—42no—
202D4×C61—44no—
203D4×C61—44no—
204D4×C61—44no—
205D4×C61—82no—
206D4×C61—84no—
207D4×C61—84no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61—84no—
219C2×C4×C61—84no—
220C2×C4×C61#12011yescharacteristic
221C2×C4×C61#12011yescharacteristic
222C2×C4×C61#12011yescharacteristic
223C2×C4×C61#13512yes—
224C2×C4×C61#13712yes—
225C2×C4×C61C22×D411yescharacteristic
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C22×D411yescharacteristic
228C2×Dic61—42no—
229C2×Dic61—82no—
230Dic122—44no—
231Dic122—82no—
232#92418—42no—
233#92418—42no—
234#92418—42no—
235#18635—611nomaximal, Sylow
236C22×C61—21no—
237C22×C61—21no—
238C22×C61—41no—
239C22×C61—44no—
240C22×C61—41no—
241C22×C61—44no—
242C22×C61—41no—
243C22×C61—44no—
244C22×C61—164no—
245C22×C61#45212yes—
246C22×C61C2×#12011yescharacteristic
247C4×C61—21no—
248C4×C61—21no—
249C4×C61—41no—
250C4×C61—41no—
251C4×C61—42no—
252C4×C61—41no—
253C4×C61—42no—
254C4×C61#45212yes—
255C4×C61#45512yes—
256C4×C61C2×#13511yescharacteristic
257C4×C61C2×#13711yescharacteristic
258C4×C61#57612yes—
259Dic61—82no—
260#3508D6111yescharacteristic
261#3209—611no—
262C2×#486—611no—
263#3079—611no—
264#1641—614no—
265#1640—614no—
266#1545—611no—
267#1493—611no—
268#1476—611no—
269C2×C61—22no—
270C2×C61—81no—
271C2×C61—84no—
272C2×C61C2×#45211yescharacteristic
273C2×C61#300811yescharacteristic
274C2×C61C2×#45511yescharacteristic
275#559—21no—
276#559C2×D6112yes—
277#550—21no—
278#550C2×D6112yes—
279#546—22no—
280#546—22no—
281#540—22no—
282#535—21no—
283#533C2×D6111yescharacteristic
284#532—21no—
285#532—21no—
286C2×C4⋊Q8—611no—
287C2×#127C2×D6111yescharacteristic
288#518—21no—
289C4×#41C2×D6111yescharacteristic
290#494—1221no—
291#487—1221no—
292#486—614no—
293#486—1222no—
294#477—1221no—
295#476—1221no—
296#475—612no—
297#474—612no—
298#462—614no—
299#451—1222no—
300#431—611no—
301#428—1221no—
302#423—611no—
303#417—1221no—
304C2×#102—1222no—
305C2×C4×C8—611no—
306#331—614no—
307Q8⋊C8—612no—
308Q8⋊C8—614no—
309D4⋊C8—612no—
310C61#1863511yescharacteristic, complemented, Sylow
311C2×#41—22no—
312C2×#41—22no—
313C2×#41C22⋊D6111yescharacteristic
314C22×Q8—1221no—
315C22×D4—21no—
316C22×D4—21no—
317C22×D4C22⋊D6112yes—
318C2×Q16—1224no—
319C2×M16—1221no—
320C22×C8—1221no—
321C4⋊Q8—612no—
322C4⋊Q8—1221no—
323C4⋊Q8—1221no—
324C4⋊Q8—1221no—
325C4⋊Q8C22⋊D6111yescharacteristic
326C4⋊Q8C22⋊D6111yescharacteristic
327#127—22no—
328#127—21no—
329#127—21no—
330#127C22⋊D6112yes—
331#127C22×D6112yes—
332#125—21no—
333#124—22no—
334#124—22no—
335#124—22no—
336#123—24no—
337#123—24no—
338C22⋊Q8—21no—
339C22⋊Q8C22⋊D6112yes—
340#121—22no—
341#121—22no—
342#121—22no—
343#121—24no—
344#121—24no—
345#121—22no—
346#120—24no—
347C4×Q8—21no—
348C4×Q8C22⋊D6111yescharacteristic
349C4×Q8C22×D6111yescharacteristic
350C4×Q8C22×D6112yes—
351C4×D4—22no—
352C4×D4—22no—
353C4×D4—22no—
354C4×D4—21no—
355C4×D4—21no—
356C4×D4—22no—
357C4×D4—22no—
358C4×D4—22no—
359C4×D4C22⋊D6111yescharacteristic
360C4×D4C22×D6111yescharacteristic
361#117—21no—
362#117—21no—
363#117C22⋊D6111yescharacteristic
364C2×C4⋊C4—1221no—
365C2×C4⋊C4—1221no—
366C2×C22⋊C4—24no—
367C2×C42—21no—
368C2×C42C22×D6111yescharacteristic
369#107—1222no—
370#106—1222no—
371C4⋊C8—612no—
372C4⋊C8—612no—
373C4⋊C8—1221no—
374#103—1222no—
375#103—1224no—
376#102—1228no—
377#102—1222no—
378C22⋊C8—1221no—
379C4×C8—612no—
380C4×C8—1221no—
381C4×C8—1221no—
382C24—22no—
383#41—22no—
384#41—22no—
385#41—84no—
386#41—84no—
387C2×Q8—21no—
388C2×Q8—1222no—
389C2×Q8—4882no—
390C2×Q8—2441no—
391C2×Q8C2×C22⋊D6111yescharacteristic
392C2×Q8C2×C22⋊D6112yes—
393C2×D4—21no—
394C2×D4—24no—
395C2×D4—24no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—44no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—82no—
405C2×D4C2×C22⋊D6111yescharacteristic
406C2×D4C2×C22⋊D6114yes—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—2441no—
413C22×C4C2×C22⋊D6111yescharacteristic
414C22×C4C2×C22⋊D6111yescharacteristic
415C22×C4C2×C22⋊D6111yescharacteristic
416Q16—4884no—
417M16—2442no—
418C2×C8—1222no—
419C2×C8—1222no—
420C2×C8—1222no—
421C2×C8—2442no—
422C4⋊C4—21no—
423C4⋊C4—22no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—22no—
429C4⋊C4—1222no—
430C4⋊C4—1222no—
431C4⋊C4—2441no—
432C4⋊C4—2441no—
433C4⋊C4C2×C22⋊D6112yes—
434C4⋊C4C2×C22⋊D6111yescharacteristic
435C4⋊C4C2×C22⋊D6111yescharacteristic
436C22⋊C4—24no—
437C22⋊C4—24no—
438C22⋊C4—24no—
439C22⋊C4—24no—
440C22⋊C4—21no—
441C22⋊C4—22no—
442C22⋊C4—21no—
443C22⋊C4—21no—
444C42—21no—
445C42—22no—
446C42—21no—
447C42—21no—
448C42C2×C22⋊D6111yescharacteristic
449C42C2×C22⋊D6111yescharacteristic
450C42C2×C22⋊D6111yescharacteristic
451C42C23×D6111yescharacteristic
452C23—24no—
453C23—21no—
454C23—44no—
455C23—82no—
456C23—82no—
457C23#1451974611yescharacteristic
458Q8—21no—
459Q8—21no—
460Q8—24no—
461Q8—42no—
462Q8—4882no—
463Q8—2444no—
464D4—21no—
465D4—21no—
466D4—44no—
467D4—42no—
468D4—44no—
469D4—42no—
470D4—42no—
471D4—44no—
472D4—44no—
473D4—82no—
474D4—84no—
475D4—84no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—84no—
487C2×C4—84no—
488C2×C4—4882no—
489C2×C4—2442no—
490C2×C4#1451974611yescharacteristic
491C2×C4#1451974611yescharacteristic
492C2×C4#1451974611yescharacteristic
493C2×C4#1451979712yes—
494C2×C4#1451980712yes—
495C2×C4C22×C22⋊D6111yescharacteristic
496C2×C4C22×C22⋊D6111yescharacteristic
497C2×C4C22×C22⋊D6111yescharacteristic
498C8—2442no—
499C8—2442no—
500C8—2442no—
501C22—21no—
502C22—21no—
503C22—41no—
504C22—44no—
505C22—41no—
506C22—44no—
507C22—41no—
508C22—44no—
509C22—164no—
510C22#1683992412yes—
511C22C2×#1451974611yescharacteristic
512C4—21no—
513C4—21no—
514C4—41no—
515C4—42no—
516C4—41no—
517C4—41no—
518C4—42no—
519C4—4882no—
520C4#1683992412yes—
521C4#1683994012yes—
522C4C2×#1451979711yescharacteristic
523C4C2×#1451980711yescharacteristic
524C4#1684056512yes—
525C2—22no—
526C2—84no—
527C2—81no—
528C2C2×#1683992411yescharacteristic
529C2#2145508711yescharacteristic
530C2C2×#1683994011yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3508C2
socleC22×C61C2×#120
hypercenter#3508D61
hypocenterC61#18635
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3508

Complements

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

NH ≅ G/Ntype
1. #21456764C2semidirect
2. C2×#16840108C2semidirect
3. #21455634C2semidirect
4. #21445012C2semidirect
5. #21445005C2semidirect
6. #21444487C2semidirect
7. #21444274C2semidirect
8. #21444201C2semidirect
17. #16840047C22semidirect
18. #16840042C22semidirect
19. #16839972C22semidirect
28. Q8⋊#92418C22semidirect
29. Q8⋊#92418C22semidirect
30. D4⋊#92418C22semidirect
310. C61#18635semidirect

Maximal subgroups

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

subgroupindex
#214567642
C2×#168401082
#214556342
#214450122
#214450052
#214444872
#214442742
#214442012
C61×#35082
#1863561