Finite Group Databasebeta

← #29713252

Subgroups — #29713252

Order 15616.

1392 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#21454138C212yesmaximal, complemented
2#21448310C211yesmaximal, characteristic, complemented
3#21447067C214yesmaximal, complemented
4#21446278C212yesmaximal, complemented
5#21446100C212yesmaximal, complemented
6#21445767C212yesmaximal, complemented
7#21445737C211yesmaximal, characteristic, complemented
8C61×#3485C211yesmaximal, characteristic
9C2×#14519736C2211yescharacteristic, complemented
10C22×C4×Dic61C2211yescharacteristic, complemented
11C2×#14519692C2212yes—
12C2×#14519692C2214yes—
13C2×#14519692C2212yescomplemented
14#16839669—21no—
15#16839665—21no—
16#16839650—24no—
17#16839643—24no—
18#16839642C2214yes—
19#16839642C2212yes—
20#16839642C2212yescomplemented
21#16839634C2214yescomplemented
22#16839625C2212yescomplemented
23C4×C22⋊Dic61—21no—
24C4×C22⋊Dic61—21no—
25#16839529C2214yescomplemented
26C2×H(5,𝔽2)×C61C2211yescharacteristic
27C22×#41×C61C2211yescharacteristic
28C61×#547—24no—
29C61×#546—24no—
30C61×#536—21no—
31C61×#534—21no—
32C61×#534C2212yes—
33C61×#525—21no—
34C2×#123×C61—21no—
35C2×#123×C61C2212yes—
36C61×#518C2211yescharacteristic
37C23×Dic61C2312yes—
38#14519736—42no—
39C2×C4⋊Dic61—21no—
40C2×#11758196—24no—
41C2×C22⋊Dic61—22no—
42C2×C22⋊Dic61—22no—
43C2×C22⋊Dic61—24no—
44C2×C22⋊Dic61C2312yes—
45C2×C22⋊Dic61C2314yes—
46C2×C4×Dic61—21no—
47C2×C4×Dic61—24no—
48C2×C4×Dic61—21no—
49C2×C4×Dic61—21no—
50#14519692—24no—
51#14519692—44no—
52#14519692—42no—
53#14519684—22no—
54#14519684—22no—
55#14519684—24no—
56#14519684—24no—
57#14519684—24no—
58H(5,𝔽2)×C61—24no—
59H(5,𝔽2)×C61—42no—
60C2×#41×C61—24no—
61C2×#41×C61—21no—
62C2×#41×C61—24no—
63C2×#41×C61—22no—
64C2×#41×C61—42no—
65C22×Q8×C61D411yescharacteristic
66C22×D4×C61—21no—
67C22×D4×C61—22no—
68C22×D4×C61D411yescharacteristic
69C22×D4×C61C2311yescharacteristic
70C22×D4×C61C2312yes—
71C22×D4×C61C2311yescharacteristic
72C23×C4×C61D411yescharacteristic
73C23×C4×C61D411yescharacteristic
74C23×C4×C61C2311yescharacteristic
75C4⋊Q8×C61—22no—
76#124×C61—22no—
77#123×C61—22no—
78#123×C61—24no—
79#123×C61—22no—
80#123×C61—22no—
81#123×C61—22no—
82#123×C61—24no—
83#123×C61—22no—
84C22⋊Q8×C61—24no—
85C22⋊Q8×C61—22no—
86#121×C61—24no—
87#121×C61—22no—
88#121×C61—24no—
89#120×C61—22no—
90#120×C61—22no—
91#120×C61—22no—
92C4×D4×C61—24no—
93C4×D4×C61—24no—
94#117×C61—21no—
95C2×C4⋊C4×C61—22no—
96C2×C4⋊C4×C61D414yes—
97C2×C22⋊C4×C61—22no—
98C2×C22⋊C4×C61—21no—
99C2×C22⋊C4×C61D414yes—
100C2×C22⋊C4×C61C2312yes—
101C24×C61—21no—
102C24×C61C2×D412yes—
103C24×C61C2×D411yescharacteristic
104C24×C61C2×D411yescharacteristic
105C24×C61C2411yescharacteristic
106#41×C61—416no—
107#41×C61—48no—
108#41×C61—42no—
109#41×C61—42no—
110C2×Q8×C61—21no—
111C2×Q8×C61—22no—
112C2×Q8×C61—22no—
113C2×Q8×C61—41no—
114C2×D4×C61—22no—
115C2×D4×C61—22no—
116C2×D4×C61—22no—
117C2×D4×C61—21no—
118C2×D4×C61—22no—
119C2×D4×C61—22no—
120C2×D4×C61—22no—
121C2×D4×C61—22no—
122C2×D4×C61—22no—
123C2×D4×C61—24no—
124C2×D4×C61—22no—
125C2×D4×C61—24no—
126C2×D4×C61—24no—
127C2×D4×C61—22no—
128C2×D4×C61—41no—
129C2×D4×C61—48no—
130C2×D4×C61—42no—
131C2×D4×C61—42no—
132C2×D4×C61#4114yes—
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—21no—
136C22×C4×C61—22no—
137C22×C4×C61—22no—
138C22×C4×C61—22no—
139C22×C4×C61—22no—
140C22×C4×C61—22no—
141C22×C4×C61—22no—
142C22×C4×C61—41no—
143C22×C4×C61—41no—
144C22×C4×C61C2×D412yes—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D414yes—
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D414yes—
151C22×Dic61—22no—
152C22×Dic61—24no—
153C22×Dic61—24no—
154C22×Dic61—22no—
155C22×Dic61—44no—
156C4⋊C4×C61—24no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—24no—
159C4⋊C4×C61—24no—
160C22⋊C4×C61—22no—
161C22⋊C4×C61—24no—
162C22⋊C4×C61—24no—
163C22⋊C4×C61—22no—
164C22⋊C4×C61—22no—
165C22⋊C4×C61—24no—
166C22⋊C4×C61—24no—
167C42×C61—22no—
168C22⋊Dic61—24no—
169C22⋊Dic61—24no—
170C22⋊Dic61—24no—
171C22⋊Dic61—48no—
172C22⋊Dic61—44no—
173C22⋊Dic61—44no—
174C4⋊Dic61—42no—
175#11758196—48no—
176C4×Dic61—42no—
177C4×Dic61—42no—
178C4×Dic61—42no—
179C23×C61—21no—
180C23×C61—21no—
181C23×C61—22no—
182C23×C61—21no—
183C23×C61—21no—
184C23×C61—21no—
185C23×C61—24no—
186C23×C61—42no—
187C23×C61—42no—
188C23×C61—42no—
189C23×C61—42no—
190C23×C61—44no—
191C23×C61—82no—
192C23×C61#12312yes—
193C23×C61C22×D411yescharacteristic
194C23×C61C22×D412yes—
195C23×C61C2×#4112yes—
196C23×C61H(5,𝔽2)12yes—
197Q8×C61—44no—
198Q8×C61—42no—
199D4×C61—28no—
200D4×C61—44no—
201D4×C61—42no—
202D4×C61—44no—
203D4×C61—42no—
204D4×C61—42no—
205D4×C61—44no—
206D4×C61—44no—
207D4×C61—84no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—28no—
215C2×C4×C61—24no—
216C2×C4×C61—21no—
217C2×C4×C61—22no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—24no—
221C2×C4×C61—44no—
222C2×C4×C61—44no—
223C2×C4×C61—44no—
224C2×C4×C61—42no—
225C2×C4×C61—44no—
226C2×C4×C61—82no—
227C2×C4×C61#12012yes—
228C2×C4×C61#12012yes—
229C2×C4×C61#12312yes—
230C2×Dic61—28no—
231C2×Dic61—24no—
232C2×Dic61—44no—
233C2×Dic61—42no—
234C2×Dic61—42no—
235C2×Dic61—44no—
236C2×Dic61—44no—
237C2×Dic61—42no—
238C2×Dic61—42no—
239C2×Dic61—42no—
240C2×Dic61—44no—
241#24057—611nomaximal, Sylow
242C22×C61—24no—
243C22×C61—22no—
244C22×C61—22no—
245C22×C61—42no—
246C22×C61—42no—
247C22×C61—42no—
248C22×C61—42no—
249C22×C61—42no—
250C22×C61—41no—
251C22×C61—42no—
252C22×C61—42no—
253C22×C61—44no—
254C22×C61—81no—
255C22×C61—81no—
256C22×C61—84no—
257C22×C61—162no—
258C22×C61C2×#12011yescharacteristic
259C22×C61C2×#12311yescharacteristic
260C22×C61#53411yescharacteristic
261C22×C61#54614yes—
262C4×C61—22no—
263C4×C61—22no—
264C4×C61—22no—
265C4×C61—42no—
266C4×C61—41no—
267C4×C61—44no—
268Dic61—44no—
269Dic61—82no—
270Dic61—84no—
271#3485D6111yescharacteristic
272#2853—612no—
273#2297—611no—
274#2044—614no—
275#1899—612no—
276#1867—612no—
277#1790—612no—
278#1782—611no—
279C2×C61—21no—
280C2×C61—24no—
281C2×C61—21no—
282C2×C61—42no—
283C2×C61—81no—
284C2×C61—82no—
285C2×C61#240511yescharacteristic
286C2×C61#302512yes—
287C2×H(5,𝔽2)C2×D6111yescharacteristic
288C22×#41C2×D6111yescharacteristic
289#547—24no—
290#546—24no—
291#536—21no—
292#534—21no—
293#534C2×D6112yes—
294#525—21no—
295C2×#123—21no—
296C2×#123C2×D6112yes—
297#518C2×D6111yescharacteristic
298C2×#117—611no—
299C22×C42—611no—
300C2×#99—612no—
301C2×#99—614no—
302C2×#99—612no—
303#391—1221no—
304#390—1221no—
305#388—1224no—
306#387—1224no—
307#386—614no—
308#386—612no—
309#386—612no—
310#385—614no—
311#381—612no—
312C4×C22⋊C4—1221no—
313C4×C22⋊C4—1221no—
314#342—614no—
315C61#2405711yescharacteristic, complemented, Sylow
316H(5,𝔽2)—24no—
317H(5,𝔽2)—42no—
318C2×#41—24no—
319C2×#41—21no—
320C2×#41—24no—
321C2×#41—22no—
322C2×#41—42no—
323C22×Q8C22⋊D6111yescharacteristic
324C22×D4—21no—
325C22×D4—22no—
326C22×D4C22⋊D6111yescharacteristic
327C22×D4C22×D6112yes—
328C22×D4C22×D6111yescharacteristic
329C22×D4C22×D6111yescharacteristic
330C23×C4—612no—
331C23×C4C22⋊D6111yescharacteristic
332C23×C4C22⋊D6111yescharacteristic
333C23×C4C22×D6111yescharacteristic
334C4⋊Q8—22no—
335#124—22no—
336#123—24no—
337#123—22no—
338#123—22no—
339#123—22no—
340#123—22no—
341#123—24no—
342#123—22no—
343C22⋊Q8—22no—
344C22⋊Q8—24no—
345#121—22no—
346#121—24no—
347#121—24no—
348#120—22no—
349#120—22no—
350#120—22no—
351C4×D4—24no—
352C4×D4—24no—
353#117—21no—
354#117—2442no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—1221no—
357C2×C4⋊C4—1224no—
358C2×C4⋊C4C22⋊D6114yes—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—612no—
362C2×C22⋊C4—614no—
363C2×C22⋊C4—1222no—
364C2×C22⋊C4—1222no—
365C2×C22⋊C4—1224no—
366C2×C22⋊C4C22⋊D6114yes—
367C2×C22⋊C4C22×D6112yes—
368C2×C42—1221no—
369C2×C42—1221no—
370C2×C42—1224no—
371C2×C42—1221no—
372#99—1224no—
373#99—2442no—
374#99—2444no—
375#95—1222no—
376#95—1222no—
377#95—1224no—
378#95—1224no—
379#95—1224no—
380C24—21no—
381C24C2×C22⋊D6112yes—
382C24C2×C22⋊D6111yescharacteristic
383C24C2×C22⋊D6111yescharacteristic
384C24C23×D6111yescharacteristic
385#41—416no—
386#41—48no—
387#41—42no—
388#41—42no—
389C2×Q8—21no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—41no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—21no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—24no—
402C2×D4—22no—
403C2×D4—24no—
404C2×D4—22no—
405C2×D4—24no—
406C2×D4—22no—
407C2×D4—41no—
408C2×D4—48no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4#1175822414yes—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—1222no—
424C22×C4—1224no—
425C22×C4—1224no—
426C22×C4—1222no—
427C22×C4—2444no—
428C22×C4C2×C22⋊D6112yes—
429C22×C4C2×C22⋊D6111yescharacteristic
430C22×C4C2×C22⋊D6114yes—
431C22×C4C2×C22⋊D6111yescharacteristic
432C22×C4C2×C22⋊D6111yescharacteristic
433C22×C4C2×C22⋊D6111yescharacteristic
434C22×C4C2×C22⋊D6114yes—
435C4⋊C4—24no—
436C4⋊C4—22no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—2442no—
440C4⋊C4—2448no—
441C22⋊C4—22no—
442C22⋊C4—24no—
443C22⋊C4—24no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—24no—
447C22⋊C4—24no—
448C22⋊C4—1224no—
449C22⋊C4—1224no—
450C22⋊C4—1224no—
451C22⋊C4—2444no—
452C22⋊C4—2444no—
453C22⋊C4—2448no—
454C42—22no—
455C42—2442no—
456C42—2442no—
457C42—2442no—
458C23—21no—
459C23—21no—
460C23—22no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—24no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—44no—
470C23—82no—
471C23#1451975812yes—
472C23C22×C22⋊D6111yescharacteristic
473C23C22×C22⋊D6112yes—
474C23C2×#1175822412yes—
475C23#1451982212yes—
476Q8—44no—
477Q8—42no—
478D4—28no—
479D4—44no—
480D4—42no—
481D4—44no—
482D4—42no—
483D4—42no—
484D4—44no—
485D4—44no—
486D4—84no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—28no—
494C2×C4—24no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—24no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—82no—
506C2×C4—1224no—
507C2×C4—1228no—
508C2×C4—2444no—
509C2×C4—2442no—
510C2×C4—2442no—
511C2×C4—2444no—
512C2×C4—2442no—
513C2×C4—2444no—
514C2×C4—2444no—
515C2×C4—2442no—
516C2×C4—2442no—
517C2×C4#1451974612yes—
518C2×C4#1451974612yes—
519C2×C4#1451975812yes—
520C22—24no—
521C22—22no—
522C22—22no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—44no—
530C22—41no—
531C22—42no—
532C22—81no—
533C22—81no—
534C22—84no—
535C22—162no—
536C22C2×#1451974611yescharacteristic
537C22C2×#1451975811yescharacteristic
538C22#1684032111yescharacteristic
539C22#1684040614yes—
540C4—22no—
541C4—22no—
542C4—22no—
543C4—44no—
544C4—42no—
545C4—41no—
546C4—4882no—
547C4—4884no—
548C4—2444no—
549C2—21no—
550C2—24no—
551C2—21no—
552C2—42no—
553C2—81no—
554C2—82no—
555C2#2144914211yescharacteristic
556C2#2145521112yes—
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC24×C61C24
FrattiniC24C23×D61
Fitting, prime radicalC61×#3485C2
socleC22×C61C2×#120
hypercenter#3485D61
hypocenterC61#24057
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3485

Complements

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

NH ≅ G/Ntype
1. #21454138C2semidirect
2. #21448310C2semidirect
3. #21447067C2semidirect
4. #21446278C2semidirect
5. #21446100C2semidirect
6. #21445767C2semidirect
7. #21445737C2semidirect
9. C2×#14519736C22semidirect
10. C22×C4×Dic61C22semidirect
13. C2×#14519692C22semidirect
20. #16839642C22semidirect
21. #16839634C22semidirect
22. #16839625C22semidirect
25. #16839529C22semidirect
315. C61#24057semidirect

Maximal subgroups

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

subgroupindex
#214541382
#214483102
#214470672
#214462782
#214461002
#214457672
#214457372
C61×#34852
#2405761