Finite Group Databasebeta

← #29652735

Subgroups — #29652735

Order 15616.

1242 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455755C212yesmaximal, complemented
2#21449078C211yesmaximal, characteristic, complemented
3#21444856C212yesmaximal, complemented
4#21444839C212yesmaximal, complemented
5#21444822C212yesmaximal, complemented
6#21444805C212yesmaximal, complemented
7#21444335C212yesmaximal, complemented
8#21444157C211yesmaximal, characteristic, complemented
9C61×#3496C211yesmaximal, characteristic
10C2×#14519736C2211yescharacteristic, complemented
11#16840083—24no—
12#16840078C2214yescomplemented
13#16840073C2214yescomplemented
14#16839802—22no—
15#16839767—22no—
16#16839726C2212yescomplemented
17#16839701C2211yescharacteristic, complemented
18#16839698C2211yescharacteristic, complemented
19#16839692C2211yescharacteristic, complemented
20#16839690—21no—
21#16839687C2211yescharacteristic, complemented
22#16839676—21no—
23C4×#11758196—21no—
24C4×C22⋊Dic61—21no—
25C4×C22⋊Dic61C2211yescharacteristic, 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
41C2×#11758194—22no—
42#14519736—22no—
43#14519736—22no—
44C2×C4⋊Dic61C2311yescharacteristic
45C2×C4⋊Dic61C2311yescharacteristic
46C2×#11758196—21no—
47C2×#11758196—21no—
48C2×C22⋊Dic61—21no—
49C2×C22⋊Dic61—21no—
50C2×C22⋊Dic61—21no—
51C2×C22⋊Dic61—21no—
52C2×C4×Dic61—21no—
53C2×C4×Dic61—21no—
54#14519710—24no—
55#14519708—24no—
56#14519698—24no—
57#14519698—24no—
58C22⋊#92418—22no—
59C22⋊#92418C2314yes—
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—44no—
126C2×D4×C61—416no—
127C2×D4×C61—44no—
128C2×D4×C61—44no—
129C2×D4×C61—44no—
130C2×D4×C61—44no—
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—42no—
139C22×C4×C61—42no—
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D412yes—
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D412yes—
146C22×C4×C61C2411yescharacteristic
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61C2×D414yes—
156C22⋊C4×C61—24no—
157C22⋊C4×C61—24no—
158C22⋊C4×C61—22no—
159C22⋊C4×C61—24no—
160C22⋊C4×C61—24no—
161C22⋊C4×C61—22no—
162C42×C61—21no—
163C42×C61—21no—
164C22⋊Dic61—42no—
165C22⋊Dic61—42no—
166C22⋊Dic61—42no—
167C22⋊Dic61—42no—
168C4⋊Dic61—22no—
169C4⋊Dic61—22no—
170#11758196—42no—
171#11758196—42no—
172C4×Dic61—42no—
173C4×Dic61—41no—
174C4×Dic61—41no—
175#11758194—44no—
176C2×#92418—24no—
177C23×C61—21no—
178C23×C61—21no—
179C23×C61—21no—
180C23×C61—21no—
181C23×C61—24no—
182C23×C61—41no—
183C23×C61—41no—
184C23×C61—41no—
185C23×C61—41no—
186C23×C61—44no—
187C23×C61—44no—
188C23×C61—82no—
189C23×C61—82no—
190C23×C61—82no—
191C23×C61—82no—
192C23×C61#12012yes—
193C23×C61AGL(1,ℤ/8)12yes—
194C23×C61C22×D411yescharacteristic
195Q8×C61—44no—
196D4×C61—28no—
197D4×C61—44no—
198D4×C61—44no—
199D4×C61—48no—
200D4×C61—48no—
201D4×C61—44no—
202D4×C61—44no—
203D4×C61—44no—
204D4×C61—82no—
205D4×C61—82no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—24no—
209C2×C4×C61—22no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—24no—
213C2×C4×C61—42no—
214C2×C4×C61—41no—
215C2×C4×C61—48no—
216C2×C4×C61—44no—
217C2×C4×C61—44no—
218C2×C4×C61—42no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—81no—
222C2×C4×C61#12011yescharacteristic
223C2×C4×C61#12011yescharacteristic
224C2×C4×C61AGL(1,ℤ/8)12yes—
225C2×C4×C61C22×D412yes—
226C2×Dic61—22no—
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—
238#92418—44no—
239#18173—611nomaximal, Sylow
240C22×C61—21no—
241C22×C61—21no—
242C22×C61—22no—
243C22×C61—22no—
244C22×C61—22no—
245C22×C61—22no—
246C22×C61—42no—
247C22×C61—41no—
248C22×C61—41no—
249C22×C61—44no—
250C22×C61—44no—
251C22×C61—44no—
252C22×C61—81no—
253C22×C61—81no—
254C22×C61—81no—
255C22×C61—164no—
256C22×C61—164no—
257C22×C61C2×#12011yescharacteristic
258C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
259C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
260C4×C61—22no—
261C4×C61—22no—
262C4×C61—21no—
263C4×C61—44no—
264C4×C61—41no—
265C4×C61—41no—
266C4×C61#45812yes—
267Dic61—42no—
268Dic61—81no—
269Dic61—81no—
270Dic61—81no—
271#3496D6111yescharacteristic
272#3094—612no—
273#2395—611no—
274#1616—612no—
275#1613—612no—
276#1610—612no—
277#1607—612no—
278#1507—612no—
279#1465—611no—
280C2×C61—21no—
281C2×C61—21no—
282C2×C61—42no—
283C2×C61—42no—
284C2×C61—84no—
285C2×C61#300311yescharacteristic
286C2×C61#301811yescharacteristic
287C2×C61C2×#45811yescharacteristic
288C2×H(5,𝔽2)—21no—
289#547—22no—
290#546—24no—
291D42—22no—
292#537C2×D6112yes—
293#534—21no—
294#534C2×D6112yes—
295#532—21no—
296C2×#121—21no—
297C2×#121C2×D6112yes—
298C2×C4×D4C2×D6111yescharacteristic
299C2×#117—611no—
300#482—1224no—
301#481—614no—
302#480—614no—
303#428—1222no—
304#418—1222no—
305#407—612no—
306#400—611no—
307#399—611no—
308#398—611no—
309#397—611no—
310#397—1221no—
311#394—1221no—
312C4×C4⋊C4—1221no—
313C4×C22⋊C4—611no—
314C4×C22⋊C4—1221no—
315#327—614no—
316#327—614no—
317#324—612no—
318#323—612no—
319C61#1817311yescharacteristic, complemented, Sylow
320H(5,𝔽2)—44no—
321H(5,𝔽2)—44no—
322C2×#41—24no—
323C2×#41—22no—
324C22×D4—24no—
325C22×D4—22no—
326C22×D4—22no—
327C22×D4—21no—
328C22×D4C22⋊D6111yescharacteristic
329C22×D4C22⋊D6112yes—
330C23×C4C22⋊D6111yescharacteristic
331C23×C4C22×D6111yescharacteristic
332C2×M16—1222no—
333C4⋊Q8—21no—
334#127—21no—
335#124—22no—
336#123—24no—
337#123—24no—
338C22⋊Q8—24no—
339#121—22no—
340#121—22no—
341#121—24no—
342#121—22no—
343#121—22no—
344#121—22no—
345#121—22no—
346#121C22⋊D6114yes—
347#121C22×D6114yes—
348#120—24no—
349#120—24no—
350C4×D4—22no—
351C4×D4—22no—
352C4×D4—22no—
353C4×D4—22no—
354#117—1222no—
355#117—1222no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—611no—
358C2×C4⋊C4—611no—
359C2×C4⋊C4—1221no—
360C2×C4⋊C4—1221no—
361C2×C4⋊C4C22⋊D6112yes—
362C2×C4⋊C4C22×D6111yescharacteristic
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—1221no—
366C2×C22⋊C4—1221no—
367C2×C22⋊C4—1221no—
368C2×C22⋊C4—1221no—
369C2×C22⋊C4C22⋊D6111yescharacteristic
370C2×C22⋊C4C22×D6111yescharacteristic
371C2×C42—1221no—
372C2×C42—1221no—
373C2×C42C22⋊D6111yescharacteristic
374#107—1224no—
375#106—1224no—
376#102—1224no—
377#102—1224no—
378C22⋊C8—614no—
379C22⋊C8—1222no—
380#95—611no—
381#95—611no—
382#95—1221no—
383#95—1221no—
384#95—1221no—
385#95—1221no—
386C24—22no—
387C24—22no—
388C24C2×C22⋊D6111yescharacteristic
389C24C2×C22⋊D6111yescharacteristic
390#41—44no—
391#41—416no—
392#41—44no—
393C2×Q8—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—24no—
397C2×D4—22no—
398C2×D4—24no—
399C2×D4—24no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—44no—
405C2×D4—416no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—82no—
411C2×D4C2×C22⋊D6114yes—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—1221no—
420C22×C4—1221no—
421C22×C4—1221no—
422C22×C4—1221no—
423C22×C4C2×C22⋊D6112yes—
424C22×C4C2×C22⋊D6111yescharacteristic
425C22×C4C2×C22⋊D6111yescharacteristic
426C22×C4C2×C22⋊D6111yescharacteristic
427C22×C4C2×C22⋊D6111yescharacteristic
428C22×C4C2×C22⋊D6112yes—
429C22×C4C23×D6111yescharacteristic
430M16—2444no—
431C2×C8—1224no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—1222no—
437C4⋊C4—1222no—
438C4⋊C4—2442no—
439C4⋊C4—2442no—
440C4⋊C4C2×C22⋊D6114yes—
441C22⋊C4—24no—
442C22⋊C4—24no—
443C22⋊C4—22no—
444C22⋊C4—24no—
445C22⋊C4—22no—
446C22⋊C4—24no—
447C22⋊C4—2442no—
448C22⋊C4—2442no—
449C22⋊C4—2442no—
450C22⋊C4—2442no—
451C42—21no—
452C42—21no—
453C42—2442no—
454C42—2441no—
455C42—2441no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—24no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—44no—
466C23—44no—
467C23—82no—
468C23—82no—
469C23—82no—
470C23—82no—
471C23#1451974612yes—
472C23#1451980312yes—
473C23C22×C22⋊D6111yescharacteristic
474Q8—44no—
475D4—28no—
476D4—44no—
477D4—44no—
478D4—48no—
479D4—48no—
480D4—44no—
481D4—44no—
482D4—44no—
483D4—82no—
484D4—82no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—24no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—48no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—81no—
501C2×C4—1222no—
502C2×C4—2441no—
503C2×C4—2441no—
504C2×C4—2441no—
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#1451974611yescharacteristic
514C2×C4#1451974611yescharacteristic
515C2×C4#1451980312yes—
516C2×C4C22×C22⋊D6112yes—
517C8—2444no—
518C22—21no—
519C22—21no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—42no—
525C22—41no—
526C22—41no—
527C22—44no—
528C22—44no—
529C22—44no—
530C22—81no—
531C22—81no—
532C22—81no—
533C22—164no—
534C22—164no—
535C22C2×#1451974611yescharacteristic
536C22C2×#1451980311yescharacteristic
537C22C2×#1451980311yescharacteristic
538C4—22no—
539C4—21no—
540C4—22no—
541C4—41no—
542C4—41no—
543C4—44no—
544C4—4881no—
545C4—4881no—
546C4—4881no—
547C4—2442no—
548C4#1683995212yes—
549C2—21no—
550C2—21no—
551C2—42no—
552C2—42no—
553C2—84no—
554C2#2145504611yescharacteristic
555C2#2145516811yescharacteristic
556C2C2×#1683995211yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3496C2
socleC22×C61C2×#120
hypercenter#3496D61
hypocenterC61#18173
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. #21455755C2semidirect
2. #21449078C2semidirect
3. #21444856C2semidirect
4. #21444839C2semidirect
5. #21444822C2semidirect
6. #21444805C2semidirect
7. #21444335C2semidirect
8. #21444157C2semidirect
10. C2×#14519736C22semidirect
12. #16840078C22semidirect
13. #16840073C22semidirect
16. #16839726C22semidirect
17. #16839701C22semidirect
18. #16839698C22semidirect
19. #16839692C22semidirect
21. #16839687C22semidirect
25. C4×C22⋊Dic61C22semidirect
26. #16839496C22semidirect
27. #16839496C22semidirect
28. #16839488C22semidirect
29. #16839486C22semidirect
319. C61#18173semidirect

Maximal subgroups

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

subgroupindex
#214557552
#214490782
#214448562
#214448392
#214448222
#214448052
#214443352
#214441572
C61×#34962
#1817361