Finite Group Databasebeta

← #29736647

Subgroups — #29736647

Order 15616.

1104 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840317C211yesmaximal, characteristic, complemented
2C2×#16840115C212yesmaximal, complemented
3#21454119C211yesmaximal, characteristic
4#21446961C211yesmaximal, characteristic, complemented
5#21446957C211yesmaximal, characteristic, complemented
6#21446955C211yesmaximal, characteristic, complemented
7#21446782C212yesmaximal, complemented
8#21446138C212yesmaximal
9#21446130C212yesmaximal, complemented
10#21445576C211yesmaximal, characteristic, complemented
11C61×#2023C211yesmaximal, characteristic, complemented
12#16840317—28no—
13C2×#14519777C2211yescharacteristic
14C2×#14519768C2212yes—
15C2×#14519764C2211yescharacteristic
16C2×#14519757—21no—
17C2×#14519757C2211yescharacteristic
18C2×#14519753—21no—
19C2×#14519748C2211yescharacteristic
20C2×Q8×Dic61—21no—
21C2×#14519742—21no—
22C2×#14519735C2212yes—
23C2×C42×D61C2211yescharacteristic
24#16840115—28no—
25C2×#14519702C2212yes—
26C2×#14519700C2212yes—
27C2×#14519698C2212yes—
28C2×#14519689C2212yes—
29C2×C8×Dic61C2212yes—
30#16839668C2211yescharacteristic
31#16839655—21no—
32#16839655—21no—
33#16839638—21no—
34#16839638—21no—
35#16839627C2211yescharacteristic
36#16839525C2212yes—
37#16839524C2212yes—
38#16839524C2212yes—
39#16839517C2211yescharacteristic
40C2×C4⋊Q8×C61C2211yescharacteristic
41C2×#102×C61C2212yes—
42C61×#390C2211yescharacteristic
43C61×#340C2212yes—
44C61×#336C2211yescharacteristic
45C22×Dic122—21no—
46C22×Dic122C2311yescharacteristic
47C22×C22⋊D61—21no—
48C22×C4×D61D411yescharacteristic
49C22×C4×D61D411yescharacteristic
50C22×C4×D61C2311yescharacteristic
51C22×#92418C2312yes—
52#14519777—24no—
53#14519768—24no—
54#14519768D418yescomplemented
55#14519764—24no—
56#14519757—28no—
57#14519757—24no—
58#14519753—28no—
59#14519748—24no—
60Q8×Dic61—28no—
61#14519742—28no—
62#14519735—24no—
63#14519735—24no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61—22no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2312yes—
69C2×#11758196—22no—
70C2×C22⋊Dic61—21no—
71C2×C22⋊Dic61—21no—
72C2×C22⋊Dic61—21no—
73C2×#11758198—22no—
74C2×#11758198—21no—
75C2×C4×Dic61—22no—
76C2×C4×Dic61D411yescharacteristic
77C2×C4×Dic61D411yescharacteristic
78C2×C4×Dic61C2311yescharacteristic
79C42×D61—24no—
80#14519702—24no—
81#14519700—24no—
82#14519698—24no—
83#14519689—24no—
84C8×Dic61—24no—
85#14519684—21no—
86#14519684—21no—
87#14519684—21no—
88#14519684—21no—
89C22×Q8×C61—21no—
90C22×D4×C61C2311yescharacteristic
91C22×C8×C61C2312yes—
92C4⋊Q8×C61—24no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61C2311yescharacteristic
95C2×C4⋊C4×C61C2312yes—
96C2×C22⋊C4×C61—21no—
97C2×C22⋊C4×C61—21no—
98C2×C42×C61C2311yescharacteristic
99#102×C61—24no—
100C24×C61—21no—
101C23×D61C2×D411yescharacteristic
102C2×Q8×C61—24no—
103C2×Q8×C61—42no—
104C2×Q8×C61—42no—
105C2×D4×C61—22no—
106C2×D4×C61—22no—
107C2×D4×C61—21no—
108C2×D4×C61—21no—
109C22×C4×C61—22no—
110C22×C4×C61—21no—
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2411yescharacteristic
114C2×C22⋊D61—42no—
115C2×C22⋊D61—42no—
116C2×C22⋊D61—42no—
117C22×Dic61—22no—
118C22×Dic61—21no—
119C22×Dic61—21no—
120C22×Dic61C2×D411yescharacteristic
121C22×Dic61C2×D411yescharacteristic
122C22×Dic61C2×D411yescharacteristic
123C2×C4×D61—22no—
124C2×C4×D61—21no—
125C2×C4×D61—21no—
126C2×C4×D61—22no—
127C2×C4×D61—22no—
128C2×C4×D61—22no—
129C2×C4×D61—22no—
130C2×C4×D61—41no—
131C2×C4×D61—41no—
132C2×C4×D61#4114yes—
133C2×Dic122—24no—
134C2×Dic122—22no—
135C2×Dic122—22no—
136C2×Dic122—21no—
137C2×Dic122—21no—
138C2×Dic122—42no—
139C2×Dic122—42no—
140C2×C8×C61—22no—
141C2×C8×C61—22no—
142C2×C8×C61—22no—
143C4⋊C4×C61—24no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61C2×D414yes—
147C22⋊C4×C61—42no—
148C22⋊C4×C61—42no—
149C42×C61—22no—
150C22⋊Dic61—42no—
151C22⋊Dic61—42no—
152C22⋊Dic61—42no—
153#11758198—28no—
154#11758198—42no—
155C4⋊Dic61—24no—
156C4⋊Dic61—22no—
157C4⋊Dic61—28no—
158C4⋊Dic61—22no—
159C4⋊Dic61—42no—
160C4⋊Dic61C2×D414yes—
161#11758196—28no—
162C4×Dic61—28no—
163C4×Dic61—22no—
164C4×Dic61—22no—
165C4×Dic61C2×D414yes—
166C2×#92418—22no—
167C2×#92418—22no—
168C2×#92418—22no—
169C23×C61—41no—
170C23×C61—41no—
171C23×C61—41no—
172C23×C61—41no—
173C23×C61—41no—
174C23×C61—41no—
175C23×C61—41no—
176C23×C61C22×D411yescharacteristic
177C22×D61—21no—
178C22×D61—21no—
179C22×D61—21no—
180C22×D61—21no—
181C22×D61—21no—
182C22×D61—21no—
183C22×D61#12412yes—
184Q8×C61—48no—
185D4×C61—42no—
186D4×C61—42no—
187C2×C4×C61—21no—
188C2×C4×C61—21no—
189C2×C4×C61—21no—
190C2×C4×C61—21no—
191C2×C4×C61—24no—
192C2×C4×C61—42no—
193C2×C4×C61—42no—
194C2×C4×C61—41no—
195C2×C4×C61—41no—
196C2×C4×C61—41no—
197C2×C4×C61#13512yes—
198C2×C4×C61#13512yes—
199C2×C4×C61C22×D412yes—
200C2×C4×C61C2×#4112yes—
201C2×C4×C612−1+411yescharacteristic
202C2×C4×C612−1+411yescharacteristic
203C22⋊D61—84no—
204C2×Dic61—21no—
205C2×Dic61—21no—
206C2×Dic61—21no—
207C2×Dic61—21no—
208C2×Dic61—21no—
209C2×Dic61—21no—
210C2×Dic61—24no—
211C2×Dic61—42no—
212C2×Dic61—42no—
213C2×Dic61—41no—
214C2×Dic61—41no—
215C2×Dic61—41no—
216C2×Dic61—41no—
217C2×Dic61—41no—
218C2×Dic61—41no—
219C2×Dic61#12012yes—
220C2×Dic61#12012yes—
221C2×Dic61#12412yes—
222C4×D61—24no—
223C4×D61—24no—
224C4×D61—44no—
225C4×D61—44no—
226Dic122—48no—
227Dic122—42no—
228Dic122—42no—
229C8×C61—42no—
230#92418—42no—
231#25906—611nomaximal, Sylow
232C22×C61—81no—
233C22×C61—81no—
234C22×C61—81no—
235C22×C61—81no—
236C22×C61—81no—
237C22×C61—81no—
238C22×C61—81no—
239C22×C61C2×#12011yescharacteristic
240C22×C61C2×#12411yescharacteristic
241C22×C61#53611yescharacteristic
242C22×C61C2×#13511yescharacteristic
243C22×C61C2×#13511yescharacteristic
244C22×C61#57511yescharacteristic
245C22×C61#57511yescharacteristic
246C2×D61—22no—
247C2×D61—22no—
248C2×D61—22no—
249C2×D61—41no—
250C2×D61—41no—
251C2×D61—41no—
252C2×D61—41no—
253C4×C61—22no—
254C4×C61—22no—
255C4×C61—44no—
256C4×C61—81no—
257C4×C61#54812yes—
258C4×C61#54812yes—
259Dic61—22no—
260Dic61—22no—
261Dic61—22no—
262Dic61—44no—
263Dic61—81no—
264Dic61—81no—
265C2×#533—611no—
266C2×#488—612no—
267#2850—611no—
268#2024—611no—
269#2024—611no—
270#2023—611no—
271#2023D6111yescharacteristic, complemented
272#1991—612no—
273#1875—612no—
274#1873—612no—
275C4×C22⋊C8—611no—
276C2×C61—161no—
277C2×C61#243611yescharacteristic
278C2×C61#300211yescharacteristic
279C2×C61#300211yescharacteristic
280C2×C61#313111yescharacteristic
281C2×C61#314011yescharacteristic
282C2×C61#323311yescharacteristic
283C2×C61#323311yescharacteristic
284D61—42no—
285#533—1228no—
286C2×C4⋊Q8—611no—
287C2×C4⋊Q8C2×D6111yescharacteristic
288C2×#125—612no—
289C2×#124—611no—
290C2×C22⋊Q8—611no—
291C2×C22⋊Q8—1221no—
292C2×C22⋊Q8—1221no—
293C2×#121—611no—
294C2×C4×Q8—1221no—
295C2×C4×Q8—1221no—
296C2×#117—612no—
297C22×C42—611no—
298#488—1228no—
299C2×#103—612no—
300C2×#103—612no—
301C2×#102—612no—
302C2×#102C2×D6112yes—
303C2×C22⋊C8—612no—
304C2×C4×C8—612no—
305#391—611no—
306#390C2×D6111yescharacteristic
307#388—1221no—
308#388—1221no—
309#386—1221no—
310#386—1221no—
311#382—611no—
312#340—612no—
313#340—612no—
314#340—612no—
315#340C2×D6112yes—
316#336—611no—
317#336C2×D6111yescharacteristic
318C61#2590611yescharacteristic, complemented, Sylow
319C22×Q8—21no—
320C22×Q8—611no—
321C22×Q8—1221no—
322C22×D4—1221no—
323C22×D4C22×D6111yescharacteristic
324C23×C4—611no—
325C23×C4—611no—
326C23×C4—611no—
327C22×C8—612no—
328C22×C8C22×D6112yes—
329C4⋊Q8—24no—
330C4⋊Q8—1224no—
331#125—618no—
332#125—1224no—
333#124—1224no—
334C22⋊Q8—1228no—
335C22⋊Q8—1228no—
336C22⋊Q8—1224no—
337#121—1224no—
338C4×Q8—1228no—
339C4×Q8—1228no—
340#117—1224no—
341#117—1224no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—612no—
344C2×C4⋊C4—611no—
345C2×C4⋊C4—1222no—
346C2×C4⋊C4—1222no—
347C2×C4⋊C4—1221no—
348C2×C4⋊C4—1221no—
349C2×C4⋊C4C22×D6111yescharacteristic
350C2×C4⋊C4C22×D6112yes—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—1222no—
354C2×C22⋊C4—1221no—
355C2×C22⋊C4—1221no—
356C2×C22⋊C4—1221no—
357C2×C22⋊C4—1221no—
358C2×C42—611no—
359C2×C42—611no—
360C2×C42—611no—
361C2×C42—1224no—
362C2×C42—1222no—
363C2×C42C22×D6111yescharacteristic
364#103—1224no—
365#103—1224no—
366#102—24no—
367#102—1224no—
368C22⋊C8—1224no—
369C4×C8—1224no—
370#95—1221no—
371#95—1221no—
372#95—1221no—
373#95—1221no—
374C24—21no—
375C24—611no—
376C2×Q8—24no—
377C2×Q8—42no—
378C2×Q8—42no—
379C2×Q8—1224no—
380C2×Q8—1222no—
381C2×Q8—1222no—
382C2×Q8—1221no—
383C2×Q8—1221no—
384C2×Q8—2442no—
385C2×Q8—2442no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—2442no—
391C2×D4—2442no—
392C2×D4—2442no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—611no—
396C22×C4—611no—
397C22×C4—611no—
398C22×C4—614no—
399C22×C4—1222no—
400C22×C4—1222no—
401C22×C4—1222no—
402C22×C4—1221no—
403C22×C4—1221no—
404C22×C4—1222no—
405C22×C4—1222no—
406C22×C4—1222no—
407C22×C4—1221no—
408C22×C4—1221no—
409C22×C4—2441no—
410C22×C4—2441no—
411C22×C4D4×D6111yescharacteristic
412C22×C4D4×D6111yescharacteristic
413C22×C4C23×D6111yescharacteristic
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—1222no—
418C2×C8—1222no—
419C2×C8—1222no—
420C4⋊C4—24no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—614no—
424C4⋊C4—1228no—
425C4⋊C4—1228no—
426C4⋊C4—1224no—
427C4⋊C4—1222no—
428C4⋊C4—1222no—
429C4⋊C4—2442no—
430C4⋊C4D4×D6114yes—
431C22⋊C4—42no—
432C22⋊C4—42no—
433C22⋊C4—1228no—
434C22⋊C4—2442no—
435C22⋊C4—2442no—
436C22⋊C4—2442no—
437C22⋊C4—2442no—
438C42—22no—
439C42—614no—
440C42—1222no—
441C42—1222no—
442C42—1228no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—612no—
451C23—1221no—
452C23—1221no—
453C23—1221no—
454C23—1221no—
455C23—1221no—
456C23—1221no—
457C23C2×D4×D6111yescharacteristic
458Q8—48no—
459Q8—2448no—
460Q8—2442no—
461Q8—2442no—
462D4—42no—
463D4—42no—
464D4—4884no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—24no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—612no—
476C2×C4—612no—
477C2×C4—612no—
478C2×C4—1221no—
479C2×C4—1221no—
480C2×C4—1221no—
481C2×C4—1221no—
482C2×C4—1221no—
483C2×C4—1221no—
484C2×C4—1224no—
485C2×C4—1224no—
486C2×C4—1224no—
487C2×C4—2444no—
488C2×C4—2444no—
489C2×C4—2441no—
490C2×C4—2441no—
491C2×C4—2441no—
492C2×C4—2441no—
493C2×C4—2441no—
494C2×C4—2441no—
495C2×C4—2442no—
496C2×C4—2442no—
497C2×C4#1451979612yes—
498C2×C4#1451979812yes—
499C2×C4C2×D4×D6112yes—
500C2×C4#41×D6112yes—
501C2×C4#1451982411yescharacteristic
502C2×C4#1451982511yescharacteristic
503C8—42no—
504C8—2442no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22—1222no—
513C22—1222no—
514C22—1222no—
515C22—2441no—
516C22—2441no—
517C22—2441no—
518C22—2441no—
519C22#120×D6111yescharacteristic
520C22#124×D6111yescharacteristic
521C22#1684033511yescharacteristic
522C22C2×#1451979611yescharacteristic
523C22C2×#1451979811yescharacteristic
524C22#1684055811yescharacteristic
525C22#1684056111yescharacteristic
526C4—22no—
527C4—22no—
528C4—44no—
529C4—81no—
530C4—1222no—
531C4—1222no—
532C4—1222no—
533C4—4881no—
534C4—4881no—
535C4—2444no—
536C4#1684042012yes—
537C4#1684042212yes—
538C2—161no—
539C2—2442no—
540C2#2144947211yescharacteristic
541C2#2145503211yescharacteristic
542C2#2145503611yescharacteristic
543C2#2145613811yescharacteristic
544C2#2145620411yescharacteristic
545C2#2145699611yescharacteristic
546C2#2145699711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2023C2
socleC23×C61C22×D4
prime radicalC2×#14519748C22
hypercenter#2023D61
hypocenterC61#25906
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2023

Complements

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

NH ≅ G/Ntype
1. C2×#16840317C2semidirect
2. C2×#16840115C2semidirect
4. #21446961C2semidirect
5. #21446957C2semidirect
6. #21446955C2semidirect
7. #21446782C2semidirect
9. #21446130C2semidirect
10. #21445576C2semidirect
11. C61×#2023C2semidirect
54. #14519768D4semidirect
271. #2023D61semidirect
318. C61#25906semidirect

Maximal subgroups

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

subgroupindex
C2×#168403172
C2×#168401152
#214541192
#214469612
#214469572
#214469552
#214467822
#214461382
#214461302
#214455762
C61×#20232
#2590661