Finite Group Databasebeta

← #29713986

Subgroups — #29713986

Order 15616.

1090 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840486C211yesmaximal, characteristic
2C2×#16840044C211yesmaximal, characteristic, complemented
3C2×C4×Dic244C211yesmaximal, characteristic, complemented
4#21450742C211yesmaximal, characteristic
5#21447018C212yesmaximal, complemented
6#21446830C212yesmaximal, complemented
7#21446613C211yesmaximal, characteristic
8#21446130C212yesmaximal, complemented
9#21446021C212yesmaximal, complemented
10#21445821C211yesmaximal, characteristic, complemented
11C61×#1990C211yesmaximal, characteristic, complemented
12C22×Dic244C2211yescharacteristic
13#16840486—24no—
14#16840486—24no—
15C2×#14519772—22no—
16C2×#14519766—21no—
17C2×#14519757C2212yes—
18C2×#14519757C2212yes—
19C2×Q8×Dic61—21no—
20C2×C4×Dic122C2211yescharacteristic
21C2×C4×Dic122C2211yescharacteristic
22C2×C4×D244C2211yescharacteristic
23C2×#14519735C2212yes—
24#16840044—24no—
25C4×Dic244—24no—
26C2×#14519711C2211yescharacteristic
27C2×#14519708C2212yes—
28C2×C4⋊#92418C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519702C2211yescharacteristic
31C2×#14519700C2212yes—
32C2×#14519700C2212yes—
33C2×#14519689C2212yes—
34C2×#14519689C2212yes—
35#16839684—22no—
36#16839663—21no—
37#16839661—21no—
38#16839619C2212yes—
39#16839524C2212yes—
40C2×C4⋊Q8×C61C2211yescharacteristic
41C2×C4×C8×C61C2211yescharacteristic
42C61×#384C2211yescharacteristic
43C61×#340C2212yes—
44C61×#340C2212yes—
45C22×Dic122C2311yescharacteristic
46C22×Dic122C2311yescharacteristic
47C22×D244D411yescharacteristic
48C22×C4×D61C2312yes—
49C2×Dic244—22no—
50C2×Dic244—22no—
51C2×Dic244—22no—
52C22×#92418C2312yes—
53#14519772—216no—
54#14519766—28no—
55#14519757—24no—
56#14519757—24no—
57#14519757—24no—
58#14519757—42no—
59Q8×Dic61—28no—
60C4×Dic122—22no—
61C4×Dic122—22no—
62C4×Dic122—22no—
63C4×Dic122D414yes—
64C4×D244—22no—
65C4×D244—22no—
66#14519735—24no—
67#14519735—24no—
68C2×C4⋊Dic61—22no—
69C2×C4⋊Dic61—22no—
70C2×C4⋊Dic61D411yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758198—22no—
76C2×#11758198—22no—
77C2×#11758198—22no—
78C2×#11758198—22no—
79C2×#11758198D412yes—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—22no—
82C2×C4×Dic61—21no—
83#14519711—22no—
84#14519708—24no—
85C4⋊#92418—22no—
86#14519702—22no—
87#14519702—22no—
88#14519700—24no—
89#14519700—42no—
90#14519689—24no—
91#14519689—42no—
92#14519684—22no—
93C22×Q8×C61—21no—
94C22×C8×C61C2311yescharacteristic
95C22×C8×C61C2311yescharacteristic
96C4⋊Q8×C61—22no—
97C4⋊Q8×C61D414yes—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61C2312yes—
102C2×C4⋊C4×C61C2312yes—
103C2×C42×C61C2311yescharacteristic
104C4×C8×C61—22no—
105C23×D61C2×D412yes—
106C2×Q8×C61—24no—
107C2×Q8×C61—42no—
108C2×Q8×C61—42no—
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2411yescharacteristic
114C22×Dic61—21no—
115C22×Dic61—21no—
116C22×Dic61—21no—
117C22×Dic61—21no—
118C22×Dic61C2×D412yes—
119C2×D244—22no—
120C2×D244—22no—
121C2×D244—22no—
122C2×C4×D61—22no—
123C2×C4×D61—22no—
124C2×C4×D61—22no—
125C2×C4×D61—22no—
126C2×C4×D61—22no—
127C2×C4×D61—22no—
128C2×Dic122—21no—
129C2×Dic122—21no—
130C2×Dic122—21no—
131C2×Dic122—21no—
132C2×Dic122—21no—
133C2×Dic122—21no—
134C2×Dic122—21no—
135C2×Dic122—21no—
136C2×Dic122—21no—
137C2×Dic122—21no—
138C2×Dic122C2×D412yes—
139C2×Dic122#4112yes—
140C2×C8×C61—21no—
141C2×C8×C61—21no—
142C2×C8×C61—21no—
143C2×C8×C61—21no—
144C2×C8×C61—21no—
145C2×C8×C61#4112yes—
146C4⋊C4×C61—24no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—42no—
151C4⋊C4×C61—81no—
152C4⋊C4×C61C2×D414yes—
153C42×C61C2×D412yes—
154C42×C61C2×D412yes—
155#11758198—28no—
156#11758198—28no—
157#11758198—24no—
158#11758198—44no—
159#11758198—82no—
160C4⋊Dic61—28no—
161C4⋊Dic61—22no—
162C4⋊Dic61—21no—
163C4⋊Dic61—21no—
164C4⋊Dic61—21no—
165C4⋊Dic61—44no—
166C4⋊Dic61C2×D412yes—
167#11758196—24no—
168#11758196—42no—
169C4×Dic61—28no—
170C4×Dic61—24no—
171C4×Dic61—42no—
172C2×#92418—22no—
173C2×#92418—22no—
174C2×#92418—22no—
175Dic244—28no—
176C23×C61C22×D411yescharacteristic
177C22×D61—22no—
178C22×D61—22no—
179C22×D61—22no—
180C22×D61—22no—
181C22×D61—22no—
182C22×D61—22no—
183C22×D61—22no—
184Q8×C61—48no—
185C2×C4×C61—21no—
186C2×C4×C61—21no—
187C2×C4×C61—24no—
188C2×C4×C61—42no—
189C2×C4×C61—42no—
190C2×C4×C61—42no—
191C2×C4×C61—42no—
192C2×C4×C61—42no—
193C2×C4×C61#12012yes—
194C2×C4×C61#12012yes—
195C2×C4×C61#13512yes—
196C2×C4×C61#13712yes—
197C2×C4×C61C22×D411yescharacteristic
198C2×C4×C61C22×D411yescharacteristic
199C2×C4×C61C2×#4111yescharacteristic
200C2×C4×C61C2×#4111yescharacteristic
201C2×C4×C61H(5,𝔽2)11yescharacteristic
202C2×C4×C612−1+411yescharacteristic
203C2×Dic61—22no—
204C2×Dic61—22no—
205C2×Dic61—22no—
206C2×Dic61—22no—
207C2×Dic61—22no—
208C2×Dic61—22no—
209C2×Dic61—41no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61—41no—
213C2×Dic61—41no—
214C2×Dic61—41no—
215C2×Dic61—41no—
216C2×Dic61—41no—
217C2×Dic61—41no—
218D244—44no—
219C4×D61—42no—
220C4×D61—42no—
221C4×D61—42no—
222C4×D61—42no—
223Dic122—22no—
224Dic122—22no—
225Dic122—22no—
226Dic122—22no—
227Dic122—41no—
228Dic122—41no—
229Dic122—41no—
230Dic122—41no—
231C8×C61—22no—
232C8×C61—41no—
233#92418—42no—
234#24123—611nomaximal, Sylow
235C22×C61C2×#12011yescharacteristic
236C22×C61#54611yescharacteristic
237C22×C61#54711yescharacteristic
238C22×C61C2×#13511yescharacteristic
239C22×C61C2×#13711yescharacteristic
240C22×C61#57711yescharacteristic
241C22×C61#57811yescharacteristic
242C2×D61—42no—
243C2×D61—42no—
244C2×D61—42no—
245C2×D61—42no—
246C2×D61—42no—
247C2×D61—42no—
248C2×D61—42no—
249C4×C61—22no—
250C4×C61—44no—
251C4×C61—82no—
252C4×C61#45214yes—
253C4×C61#52511yescharacteristic
254C4×C61#53711yescharacteristic
255C4×C61#54811yescharacteristic
256C4×C61#55011yescharacteristic
257Dic61—42no—
258Dic61—42no—
259Dic61—42no—
260Dic61—42no—
261Dic61—81no—
262C2×#556—611no—
263C2×#475—611no—
264C2×C4×Q16—611no—
265#2538—611no—
266#2035—612no—
267#2001—612no—
268#1990D6111yescharacteristic, complemented
269#1954—611no—
270#1873—612no—
271#1854—612no—
272#1803—611no—
273C2×C61#240711yescharacteristic
274C2×C61C2×#45211yescharacteristic
275C2×C61#301311yescharacteristic
276C2×C61#330411yescharacteristic
277C2×C61#332811yescharacteristic
278C2×C61#334211yescharacteristic
279C2×C61#335611yescharacteristic
280D61—82no—
281C22×Q16—611no—
282#556—1224no—
283#556—1224no—
284C2×C4⋊Q8C2×D6111yescharacteristic
285C2×#126—1222no—
286C2×#124—1221no—
287C2×C22⋊Q8—612no—
288C2×C22⋊Q8—612no—
289C2×C4×Q8—611no—
290C2×C4×Q8—611no—
291C2×C4×Q8—1221no—
292C2×C4×D4—611no—
293C2×#117—612no—
294#475—1224no—
295C4×Q16—1224no—
296C2×#107—611no—
297C2×#106—612no—
298C2×C4⋊C8—611no—
299C2×#103—611no—
300C2×#103—611no—
301C2×#103—612no—
302C2×#103—612no—
303C2×C22⋊C8—612no—
304C2×C22⋊C8—612no—
305C2×C4×C8C2×D6111yescharacteristic
306#396—1222no—
307#390—1221no—
308#389—1221no—
309#384C2×D6111yescharacteristic
310#380—612no—
311#340—612no—
312#340C2×D6112yes—
313#340C2×D6112yes—
314C61#2412311yescharacteristic, complemented, Sylow
315C22×Q8—21no—
316C22×Q8—611no—
317C22×Q8—611no—
318C22×D4—611no—
319C23×C4—612no—
320C2×Q16—1222no—
321C2×Q16—1222no—
322C2×Q16—1222no—
323C22×C8—612no—
324C22×C8C22×D6111yescharacteristic
325C22×C8C22×D6111yescharacteristic
326C4⋊Q8—22no—
327C4⋊Q8D24414yes—
328#126—12216no—
329#124—1228no—
330C22⋊Q8—1224no—
331C22⋊Q8—1224no—
332C22⋊Q8—1224no—
333C22⋊Q8—2442no—
334C4×Q8—614no—
335C4×Q8—1228no—
336C4×Q8—1222no—
337C4×Q8—1222no—
338C4×Q8—1222no—
339C4×D4—1222no—
340C4×D4—1222no—
341#117—1224no—
342#117—1224no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—611no—
347C2×C4⋊C4—611no—
348C2×C4⋊C4—611no—
349C2×C4⋊C4—1222no—
350C2×C4⋊C4—1221no—
351C2×C4⋊C4—1222no—
352C2×C4⋊C4—1221no—
353C2×C4⋊C4C22×D6112yes—
354C2×C4⋊C4C22×D6112yes—
355C2×C22⋊C4—612no—
356C2×C22⋊C4—1222no—
357C2×C22⋊C4—1222no—
358C2×C22⋊C4—1222no—
359C2×C22⋊C4—1222no—
360C2×C42—1222no—
361C2×C42—1221no—
362C2×C42—1221no—
363C2×C42C22×D6111yescharacteristic
364#107—1222no—
365#106—1224no—
366C4⋊C8—1222no—
367#103—1222no—
368#103—1222no—
369#103—1224no—
370#103—2442no—
371C22⋊C8—1224no—
372C22⋊C8—2442no—
373C4×C8—22no—
374#95—1222no—
375C24—612no—
376C2×Q8—24no—
377C2×Q8—42no—
378C2×Q8—42no—
379C2×Q8—612no—
380C2×Q8—612no—
381C2×Q8—1221no—
382C2×Q8—1221no—
383C2×Q8—1221no—
384C2×Q8—1221no—
385C2×Q8—1221no—
386C2×Q8—1221no—
387C2×Q8—1221no—
388C2×Q8—1221no—
389C2×Q8—1221no—
390C2×Q8—1221no—
391C2×D4—1222no—
392C2×D4—1222no—
393C2×D4—1222no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—612no—
397C22×C4—1221no—
398C22×C4—1221no—
399C22×C4—1221no—
400C22×C4—1222no—
401C22×C4—1222no—
402C22×C4—1222no—
403C22×C4—1222no—
404C22×C4—1222no—
405C22×C4—1222no—
406C22×C4—1221no—
407C22×C4D4×D6111yescharacteristic
408C22×C4D4×D6111yescharacteristic
409C22×C4C23×D6111yescharacteristic
410Q16—1228no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—1222no—
417C2×C8—1222no—
418C2×C8—1222no—
419C2×C8#1175822612yes—
420C4⋊C4—22no—
421C4⋊C4—24no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—42no—
425C4⋊C4—81no—
426C4⋊C4—612no—
427C4⋊C4—1228no—
428C4⋊C4—1221no—
429C4⋊C4—1222no—
430C4⋊C4—1224no—
431C4⋊C4—1221no—
432C4⋊C4—1221no—
433C4⋊C4—2444no—
434C4⋊C4—2442no—
435C4⋊C4C2×D24414yes—
436C22⋊C4—1228no—
437C22⋊C4—1228no—
438C22⋊C4—1224no—
439C22⋊C4—4882no—
440C22⋊C4—2444no—
441C42—1228no—
442C42—1224no—
443C42—2442no—
444C42C2×D24412yes—
445C42D4×D6112yes—
446C23—1222no—
447C23—1222no—
448C23—1222no—
449C23—1222no—
450C23—1222no—
451C23—1222no—
452C23—1222no—
453C23C2×D4×D6111yescharacteristic
454Q8—48no—
455Q8—1222no—
456Q8—1222no—
457Q8—1222no—
458Q8—1222no—
459Q8—2441no—
460Q8—2441no—
461Q8—2441no—
462Q8—2441no—
463D4—2444no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—24no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—1222no—
473C2×C4—1222no—
474C2×C4—1222no—
475C2×C4—1222no—
476C2×C4—1222no—
477C2×C4—1222no—
478C2×C4—2441no—
479C2×C4—2441no—
480C2×C4—2441no—
481C2×C4—2441no—
482C2×C4—2441no—
483C2×C4—2441no—
484C2×C4—2441no—
485C2×C4—2441no—
486C2×C4—2441no—
487C2×C4—2442no—
488C2×C4—2442no—
489C2×C4—2442no—
490C2×C4—2442no—
491C2×C4#1451974712yes—
492C2×C4#1451974712yes—
493C2×C4#1451979612yes—
494C2×C4#1451980612yes—
495C2×C4C2×D4×D6111yescharacteristic
496C2×C4C22×D24411yescharacteristic
497C2×C4#41×D6111yescharacteristic
498C2×C4C2×#1175822611yescharacteristic
499C2×C4#1451982311yescharacteristic
500C2×C4#1451982511yescharacteristic
501C8—22no—
502C8—41no—
503C8—2442no—
504C22—2442no—
505C22—2442no—
506C22—2442no—
507C22—2442no—
508C22—2442no—
509C22—2442no—
510C22—2442no—
511C22C2×#1451974711yescharacteristic
512C22#1684040511yescharacteristic
513C22#1684041511yescharacteristic
514C22C2×#1451979611yescharacteristic
515C22C2×#1451980611yescharacteristic
516C22#1684056811yescharacteristic
517C22#1684057511yescharacteristic
518C4—22no—
519C4—44no—
520C4—82no—
521C4—4881no—
522C4—2442no—
523C4—2442no—
524C4—2442no—
525C4—2442no—
526C4#1683992514yes—
527C4#1684027511yescharacteristic
528C4#1684034211yescharacteristic
529C4#1684042411yescharacteristic
530C4#1684043411yescharacteristic
531C2—4882no—
532C2#2144916711yescharacteristic
533C2C2×#1683992511yescharacteristic
534C2#2145513911yescharacteristic
535C2#2145775711yescharacteristic
536C2#2145810111yescharacteristic
537C2#2145824911yescharacteristic
538C2#2145843311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1990C2
socleC23×C61C22×D4
prime radicalC22×D244D4
hypercenter#1990D61
hypocenterC61#24123
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1990

Complements

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

NH ≅ G/Ntype
2. C2×#16840044C2semidirect
3. C2×C4×Dic244C2semidirect
5. #21447018C2semidirect
6. #21446830C2semidirect
8. #21446130C2semidirect
9. #21446021C2semidirect
10. #21445821C2semidirect
11. C61×#1990C2semidirect
268. #1990D61semidirect
314. C61#24123semidirect

Maximal subgroups

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

subgroupindex
C2×#168404862
C2×#168400442
C2×C4×Dic2442
#214507422
#214470182
#214468302
#214466132
#214461302
#214460212
#214458212
C61×#19902
#2412361