Finite Group Databasebeta

← #29654431

Subgroups — #29654431

Order 15616.

1226 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456625C212yesmaximal, complemented
2#21451736C211yesmaximal, characteristic, complemented
3#21444822C212yesmaximal, complemented
4#21444805C212yesmaximal, complemented
5#21444788C212yesmaximal, complemented
6#21444771C212yesmaximal, complemented
7#21444348C212yesmaximal, complemented
8#21444157C211yesmaximal, characteristic, complemented
9C61×#3496C211yesmaximal, characteristic
10C2×#14519756C2211yescharacteristic, complemented
11#16840097—22no—
12#16840073C2214yescomplemented
13#16840006—24no—
14#16839913—22no—
15#16839900C2214yescomplemented
16#16839726C2212yescomplemented
17#16839692C2211yescharacteristic, complemented
18#16839690—21no—
19#16839687C2211yescharacteristic, complemented
20#16839682C2211yescharacteristic, complemented
21#16839676—21no—
22#16839674C2211yescharacteristic, complemented
23#16839646—21no—
24#16839635—21no—
25#16839625C2211yescharacteristic, complemented
26#16839496C2214yescomplemented
27#16839496C2214yescomplemented
28#16839488C2212yescomplemented
29#16839486C2212yescomplemented
30C2×H(5,𝔽2)×C61—21no—
31C61×#547—22no—
32C61×#546—24no—
33D42×C61—22no—
34C61×#537C2212yes—
35C61×#534—21no—
36C61×#534C2212yes—
37C61×#532—21no—
38C2×#121×C61—21no—
39C2×#121×C61C2212yes—
40C2×C4×D4×C61C2211yescharacteristic
41C22×Dic122C2311yescharacteristic
42C2×#11758200—24no—
43C2×#11758194—22no—
44#14519756—22no—
45#14519756—22no—
46C2×C4⋊Dic61C2311yescharacteristic
47C2×#11758196—21no—
48C2×#11758196—21no—
49C2×#11758196—21no—
50C2×C22⋊Dic61—21no—
51C2×C22⋊Dic61—21no—
52C2×C22⋊Dic61—21no—
53C2×C22⋊Dic61—21no—
54#14519710—24no—
55#14519700—24no—
56#14519698—24no—
57C22⋊#92418—22no—
58C22⋊#92418C2314yes—
59#14519684—21no—
60#14519684—21no—
61#14519684—21no—
62#14519684—21no—
63#14519684—21no—
64#14519684C2311yescharacteristic
65#14519684C2311yescharacteristic
66H(5,𝔽2)×C61—44no—
67H(5,𝔽2)×C61—44no—
68C2×#41×C61—24no—
69C2×#41×C61—22no—
70C22×D4×C61—24no—
71C22×D4×C61—22no—
72C22×D4×C61—22no—
73C22×D4×C61—21no—
74C22×D4×C61D411yescharacteristic
75C22×D4×C61D412yes—
76C23×C4×C61D411yescharacteristic
77C23×C4×C61C2311yescharacteristic
78C4⋊Q8×C61—21no—
79#127×C61—21no—
80#124×C61—22no—
81#123×C61—24no—
82#123×C61—24no—
83C22⋊Q8×C61—24no—
84#121×C61—22no—
85#121×C61—22no—
86#121×C61—24no—
87#121×C61—22no—
88#121×C61—22no—
89#121×C61—22no—
90#121×C61—22no—
91#121×C61D414yes—
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×C61—22no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61D412yes—
101C2×C4⋊C4×C61C2311yescharacteristic
102C2×C22⋊C4×C61—22no—
103C2×C22⋊C4×C61—22no—
104C2×C22⋊C4×C61D411yescharacteristic
105C2×C22⋊C4×C61C2311yescharacteristic
106C2×C42×C61D411yescharacteristic
107C24×C61—22no—
108C24×C61—22no—
109C24×C61C2×D411yescharacteristic
110C24×C61C2×D411yescharacteristic
111#41×C61—44no—
112#41×C61—416no—
113#41×C61—44no—
114C2×Q8×C61—22no—
115C2×D4×C61—22no—
116C2×D4×C61—22no—
117C2×D4×C61—24no—
118C2×D4×C61—22no—
119C2×D4×C61—24no—
120C2×D4×C61—24no—
121C2×D4×C61—22no—
122C2×D4×C61—22no—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61—416no—
126C2×D4×C61—44no—
127C2×D4×C61—44no—
128C2×D4×C61—44no—
129C2×D4×C61—44no—
130C2×D4×C61—82no—
131C2×D4×C61—82no—
132C2×D4×C61C2×D414yes—
133C22×C4×C61—22no—
134C22×C4×C61—24no—
135C22×C4×C61—22no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—81no—
139C22×C4×C61—81no—
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D412yes—
145C22×C4×C61C2×D412yes—
146C22×C4×C61C2411yescharacteristic
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C2×Dic122—22no—
152C2×Dic122—81no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61C2×D414yes—
158C22⋊C4×C61—24no—
159C22⋊C4×C61—24no—
160C22⋊C4×C61—22no—
161C22⋊C4×C61—24no—
162C22⋊C4×C61—24no—
163C22⋊C4×C61—22no—
164C42×C61—21no—
165C42×C61—21no—
166C22⋊Dic61—42no—
167C22⋊Dic61—42no—
168C22⋊Dic61—81no—
169C22⋊Dic61—81no—
170#11758200—84no—
171C4⋊Dic61—22no—
172#11758196—41no—
173#11758196—41no—
174#11758196—42no—
175#11758196—81no—
176#11758194—82no—
177C2×#92418—24no—
178C23×C61—21no—
179C23×C61—21no—
180C23×C61—21no—
181C23×C61—21no—
182C23×C61—24no—
183C23×C61—41no—
184C23×C61—41no—
185C23×C61—41no—
186C23×C61—41no—
187C23×C61—44no—
188C23×C61—44no—
189C23×C61—82no—
190C23×C61—82no—
191C23×C61—82no—
192C23×C61—82no—
193C23×C61#12012yes—
194C23×C61AGL(1,ℤ/8)12yes—
195C23×C61C22×D411yescharacteristic
196Q8×C61—44no—
197D4×C61—28no—
198D4×C61—44no—
199D4×C61—44no—
200D4×C61—44no—
201D4×C61—48no—
202D4×C61—48no—
203D4×C61—44no—
204D4×C61—44no—
205D4×C61—82no—
206D4×C61—82no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—24no—
212C2×C4×C61—42no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61—48no—
218C2×C4×C61—44no—
219C2×C4×C61—44no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—81no—
223C2×C4×C61#12011yescharacteristic
224C2×C4×C61#12011yescharacteristic
225C2×C4×C61AGL(1,ℤ/8)12yes—
226C2×C4×C61C22×D412yes—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61—41no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239Dic122—44no—
240#92418—44no—
241#18351—611nomaximal, Sylow
242C22×C61—21no—
243C22×C61—21no—
244C22×C61—22no—
245C22×C61—22no—
246C22×C61—22no—
247C22×C61—22no—
248C22×C61—42no—
249C22×C61—41no—
250C22×C61—41no—
251C22×C61—44no—
252C22×C61—44no—
253C22×C61—44no—
254C22×C61—81no—
255C22×C61—81no—
256C22×C61—81no—
257C22×C61—164no—
258C22×C61—164no—
259C22×C61C2×#12011yescharacteristic
260C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
261C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
262C4×C61—22no—
263C4×C61—22no—
264C4×C61—21no—
265C4×C61—21no—
266C4×C61—44no—
267C4×C61—41no—
268C4×C61—41no—
269Dic61—81no—
270Dic61—81no—
271Dic61—81no—
272Dic61—81no—
273#3496D6111yescharacteristic
274#3193—612no—
275#2638—611no—
276#1610—612no—
277#1607—612no—
278#1604—612no—
279#1601—612no—
280#1510—612no—
281#1465—611no—
282C2×C61—21no—
283C2×C61—21no—
284C2×C61—42no—
285C2×C61—42no—
286C2×C61—84no—
287C2×C61#300311yescharacteristic
288C2×C61#302211yescharacteristic
289C2×C61#302611yescharacteristic
290C2×H(5,𝔽2)—21no—
291#547—22no—
292#546—24no—
293D42—22no—
294#537C2×D6112yes—
295#534—21no—
296#534C2×D6112yes—
297#532—21no—
298C2×C22⋊Q8—611no—
299C2×#121—21no—
300C2×#121C2×D6112yes—
301C2×C4×D4C2×D6111yescharacteristic
302#484—1222no—
303#480—614no—
304#468—1224no—
305#450—1222no—
306#448—614no—
307#407—612no—
308#398—611no—
309#397—611no—
310#397—1221no—
311#395—611no—
312#394—1221no—
313#393—611no—
314#387—1221no—
315#385—1221no—
316#381—611no—
317#327—614no—
318#327—614no—
319#324—612no—
320#323—612no—
321C61#1835111yescharacteristic, complemented, Sylow
322H(5,𝔽2)—44no—
323H(5,𝔽2)—44no—
324C2×#41—24no—
325C2×#41—22no—
326C22×Q8—611no—
327C22×D4—24no—
328C22×D4—22no—
329C22×D4—22no—
330C22×D4—21no—
331C22×D4C22⋊D6111yescharacteristic
332C22×D4C22⋊D6112yes—
333C23×C4C22⋊D6111yescharacteristic
334C23×C4C22×D6111yescharacteristic
335C2×SD16—1224no—
336C2×M16—1222no—
337C4⋊Q8—21no—
338#127—21no—
339#124—22no—
340#123—24no—
341#123—24no—
342C22⋊Q8—24no—
343C22⋊Q8—1222no—
344C22⋊Q8—1222no—
345#121—22no—
346#121—22no—
347#121—24no—
348#121—22no—
349#121—22no—
350#121—22no—
351#121—22no—
352#121C22⋊D6114yes—
353#121C22×D6114yes—
354#120—24no—
355#120—24no—
356C4×D4—22no—
357C4×D4—22no—
358C4×D4—22no—
359C4×D4—22no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—611no—
362C2×C4⋊C4—1221no—
363C2×C4⋊C4—1221no—
364C2×C4⋊C4—1221no—
365C2×C4⋊C4C22⋊D6112yes—
366C2×C4⋊C4C22×D6111yescharacteristic
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—1221no—
370C2×C22⋊C4—1221no—
371C2×C22⋊C4—1221no—
372C2×C22⋊C4—1221no—
373C2×C22⋊C4C22⋊D6111yescharacteristic
374C2×C22⋊C4C22×D6111yescharacteristic
375C2×C42C22⋊D6111yescharacteristic
376#107—1224no—
377#103—1224no—
378#102—1224no—
379C22⋊C8—614no—
380C22⋊C8—1222no—
381#95—611no—
382#95—611no—
383#95—1221no—
384#95—1221no—
385#95—1221no—
386#95—1221no—
387#95—1221no—
388C24—22no—
389C24—22no—
390C24C2×C22⋊D6111yescharacteristic
391C24C2×C22⋊D6111yescharacteristic
392#41—44no—
393#41—416no—
394#41—44no—
395C2×Q8—22no—
396C2×Q8—1222no—
397C2×Q8—4881no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—24no—
401C2×D4—22no—
402C2×D4—24no—
403C2×D4—24no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—416no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—82no—
414C2×D4—82no—
415C2×D4C2×C22⋊D6114yes—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—81no—
422C22×C4—81no—
423C22×C4—1221no—
424C22×C4—1221no—
425C22×C4—1221no—
426C22×C4—1221no—
427C22×C4C2×C22⋊D6111yescharacteristic
428C22×C4C2×C22⋊D6111yescharacteristic
429C22×C4C2×C22⋊D6111yescharacteristic
430C22×C4C2×C22⋊D6112yes—
431C22×C4C2×C22⋊D6112yes—
432C22×C4C2×C22⋊D6111yescharacteristic
433C22×C4C23×D6111yescharacteristic
434SD16—4884no—
435M16—4882no—
436C2×C8—1224no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—1222no—
442C4⋊C4—4881no—
443C4⋊C4—2441no—
444C4⋊C4—2441no—
445C4⋊C4—2442no—
446C4⋊C4C2×C22⋊D6114yes—
447C22⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—22no—
450C22⋊C4—24no—
451C22⋊C4—22no—
452C22⋊C4—24no—
453C22⋊C4—4881no—
454C22⋊C4—4881no—
455C22⋊C4—2442no—
456C22⋊C4—2442no—
457C42—21no—
458C42—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—24no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—44no—
469C23—44no—
470C23—82no—
471C23—82no—
472C23—82no—
473C23—82no—
474C23#1451974612yes—
475C23#1451980312yes—
476C23C22×C22⋊D6111yescharacteristic
477Q8—44no—
478Q8—2444no—
479D4—28no—
480D4—44no—
481D4—44no—
482D4—44no—
483D4—48no—
484D4—48no—
485D4—44no—
486D4—44no—
487D4—82no—
488D4—82no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—24no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—48no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—81no—
505C2×C4—2441no—
506C2×C4—2441no—
507C2×C4—2441no—
508C2×C4—2441no—
509C2×C4—2441no—
510C2×C4—2441no—
511C2×C4—2441no—
512C2×C4—2441no—
513C2×C4—2441no—
514C2×C4—2441no—
515C2×C4—2441no—
516C2×C4—2441no—
517C2×C4#1451974611yescharacteristic
518C2×C4#1451974611yescharacteristic
519C2×C4#1451980312yes—
520C2×C4C22×C22⋊D6112yes—
521C8—2444no—
522C22—21no—
523C22—21no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—42no—
529C22—41no—
530C22—41no—
531C22—44no—
532C22—44no—
533C22—44no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22—164no—
538C22—164no—
539C22C2×#1451974611yescharacteristic
540C22C2×#1451980311yescharacteristic
541C22C2×#1451980311yescharacteristic
542C4—22no—
543C4—21no—
544C4—21no—
545C4—22no—
546C4—41no—
547C4—41no—
548C4—44no—
549C4—4881no—
550C4—4881no—
551C4—4881no—
552C4—4881no—
553C2—21no—
554C2—21no—
555C2—42no—
556C2—42no—
557C2—84no—
558C2#2145504611yescharacteristic
559C2#2145519811yescharacteristic
560C2#2145521611yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3496C2
socleC22×C61C2×#120
hypercenter#3496D61
hypocenterC61#18351
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3496

Complements

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

NH ≅ G/Ntype
1. #21456625C2semidirect
2. #21451736C2semidirect
3. #21444822C2semidirect
4. #21444805C2semidirect
5. #21444788C2semidirect
6. #21444771C2semidirect
7. #21444348C2semidirect
8. #21444157C2semidirect
10. C2×#14519756C22semidirect
12. #16840073C22semidirect
15. #16839900C22semidirect
16. #16839726C22semidirect
17. #16839692C22semidirect
19. #16839687C22semidirect
20. #16839682C22semidirect
22. #16839674C22semidirect
25. #16839625C22semidirect
26. #16839496C22semidirect
27. #16839496C22semidirect
28. #16839488C22semidirect
29. #16839486C22semidirect
321. C61#18351semidirect

Maximal subgroups

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

subgroupindex
#214566252
#214517362
#214448222
#214448052
#214447882
#214447712
#214443482
#214441572
C61×#34962
#1835161