Finite Group Databasebeta

← #29689692

Subgroups — #29689692

Order 15616.

1222 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840314C211yesmaximal, characteristic
2C2×#16839812C211yesmaximal, characteristic
3C2×#16839631C211yesmaximal, characteristic
4#21446778C212yesmaximal, complemented
5#21446619C212yesmaximal, complemented
6#21446127C211yesmaximal, characteristic
7#21446126C211yesmaximal, characteristic
8#21446124C211yesmaximal, characteristic
9#21445823C212yesmaximal, complemented
10#21445822C212yesmaximal, complemented
11C61×#1872C211yesmaximal, characteristic
12#16840314—24no—
13#16840314C418yescomplemented
14C2×#14519776C2211yescharacteristic
15C2×#14519775C2211yescharacteristic
16C2×#14519767—21no—
17C2×#14519756C2211yescharacteristic
18C2×#14519754—21no—
19C2×Q8×Dic61C2211yescharacteristic
20C2×#14519742—21no—
21C2×#14519742C2211yescharacteristic
22C2×#14519733—21no—
23C22×C4⋊Dic61C2211yescharacteristic
24C22×#11758196—21no—
25C22×#11758196—21no—
26C22×C4×Dic61C2211yescharacteristic
27#16839812—24no—
28C2×#14519706C2212yes—
29C2×#14519702C2212yes—
30C2×#14519701C2212yes—
31C2×#14519700C2212yes—
32C2×C22⋊#92418C2211yescharacteristic
33C2×C8×Dic61C2212yes—
34#16839631—21no—
35#16839631—21no—
36#16839631C2212yes—
37#16839631C2212yes—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×C22⋊Q8×C61C2211yescharacteristic
42C22×C4⋊C4×C61C2211yescharacteristic
43C2×#103×C61C2212yes—
44C2×C22⋊C8×C61C2211yescharacteristic
45C61×#340C2212yes—
46C22×Dic122—21no—
47C22×Dic122C2311yescharacteristic
48C23×Dic61—21no—
49C23×Dic61D411yescharacteristic
50C23×Dic61D411yescharacteristic
51C22×#92418C2312yes—
52#14519776—22no—
53#14519776C2×C414yescomplemented
54#14519775—22no—
55#14519775C2×C414yescomplemented
56#14519767—28no—
57#14519756—22no—
58#14519756C2×C414yes—
59#14519754—28no—
60Q8×Dic61—22no—
61Q8×Dic61C2×C414yescomplemented
62#14519742—28no—
63#14519742—22no—
64#14519742C2×C414yescomplemented
65#14519733—28no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—42no—
69C2×C4⋊Dic61C2312yes—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×#11758196—21no—
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—22no—
77C2×#11758196—22no—
78C2×#11758196—22no—
79C2×#11758196—22no—
80C2×#11758196—44no—
81C2×#11758196—44no—
82C2×C22⋊Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61C2×C414yes—
86C2×C4×Dic61D411yescharacteristic
87C2×C4×Dic61D411yescharacteristic
88C2×C4×Dic61D414yes—
89C2×C4×Dic61C2311yescharacteristic
90C2×C4×Dic61C2311yescharacteristic
91#14519706—24no—
92#14519702—42no—
93#14519701—42no—
94#14519700—42no—
95C22⋊#92418—22no—
96C8×Dic61—24no—
97C22×Q8×C61C2311yescharacteristic
98C23×C4×C61—21no—
99C23×C4×C61C2311yescharacteristic
100C22×C8×C61C2312yes—
101C22⋊Q8×C61—22no—
102C22⋊Q8×C61C2×C414yes—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61—42no—
106C2×C4⋊C4×C61C2312yes—
107C2×C4⋊C4×C61C2311yescharacteristic
108C2×C22⋊C4×C61—21no—
109#103×C61—42no—
110C22⋊C8×C61—22no—
111C24×C61C2×D411yescharacteristic
112C2×Q8×C61—21no—
113C2×Q8×C61—22no—
114C2×Q8×C61—22no—
115C2×Q8×C61C22×C412yes—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61—22no—
121C22×C4×C61—42no—
122C22×C4×C61—42no—
123C22×C4×C61—42no—
124C22×C4×C61C22×C412yes—
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D412yes—
127C22×C4×C61C2411yescharacteristic
128C22×Dic61—22no—
129C22×Dic61—22no—
130C22×Dic61—21no—
131C22×Dic61—21no—
132C22×Dic61—22no—
133C22×Dic61—42no—
134C22×Dic61—42no—
135C22×Dic61—42no—
136C22×Dic61C22⋊C414yes—
137C22×Dic61C22⋊C414yes—
138C22×Dic61SD1614yes—
139C22×Dic61Q1614yes—
140C22×Dic61C2×D411yescharacteristic
141C22×Dic61C2×D411yescharacteristic
142C22×Dic61C2×D411yescharacteristic
143C22×Dic61C2×D411yescharacteristic
144C2×Dic122—22no—
145C2×Dic122—22no—
146C2×Dic122—21no—
147C2×Dic122—24no—
148C2×Dic122—42no—
149C2×Dic122—42no—
150C2×Dic122C22×C412yes—
151C2×C8×C61—22no—
152C2×C8×C61—22no—
153C2×C8×C61—22no—
154C4⋊C4×C61—24no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—42no—
159C4⋊C4×C61C22×C412yes—
160C22⋊C4×C61—24no—
161C22⋊Dic61—24no—
162C4⋊Dic61—21no—
163C4⋊Dic61—24no—
164C4⋊Dic61—22no—
165C4⋊Dic61—22no—
166C4⋊Dic61—42no—
167C4⋊Dic61C22×C412yes—
168#11758196—24no—
169#11758196—24no—
170#11758196—24no—
171#11758196—24no—
172#11758196—24no—
173#11758196—44no—
174#11758196—44no—
175#11758196—44no—
176#11758196—44no—
177C4×Dic61—24no—
178C4×Dic61—24no—
179C4×Dic61C22⋊C414yes—
180C4×Dic61C22⋊C414yes—
181C4×Dic61C22×C412yes—
182C4×Dic61C22×C412yes—
183C4×Dic61C2×D412yes—
184C4×Dic61C2×D412yes—
185C2×#92418—22no—
186C2×#92418—22no—
187C2×#92418—22no—
188C23×C61—21no—
189C23×C61—21no—
190C23×C61—21no—
191C23×C61—21no—
192C23×C61C2×C22⋊C412yes—
193C23×C61C2×SD1612yes—
194C23×C61C2×Q1612yes—
195C23×C61C22×D411yescharacteristic
196Q8×C61—24no—
197Q8×C61—42no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—24no—
201C2×C4×C61—24no—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61—42no—
211C2×C4×C61—84no—
212C2×C4×C61C2×C22⋊C412yes—
213C2×C4×C61C23×C411yescharacteristic
214C2×C4×C61C22×D411yescharacteristic
215C2×C4×C61H(5,𝔽2)12yes—
216C2×C4×C612−1+412yes—
217C2×Dic61—21no—
218C2×Dic61—21no—
219C2×Dic61—28no—
220C2×Dic61—22no—
221C2×Dic61—22no—
222C2×Dic61—41no—
223C2×Dic61—41no—
224C2×Dic61—41no—
225C2×Dic61—41no—
226C2×Dic61—42no—
227C2×Dic61—42no—
228C2×Dic61—42no—
229C2×Dic61—84no—
230C2×Dic61#103116yescomplemented
231C2×Dic61C2×C22⋊C412yes—
232C2×Dic61C2×C22⋊C412yes—
233C2×Dic61C2×C22⋊C412yes—
234C2×Dic61C2×C22⋊C412yes—
235C2×Dic61C2×SD1612yes—
236C2×Dic61C2×SD1612yes—
237C2×Dic61C2×Q1612yes—
238C2×Dic61C2×Q1612yes—
239C2×Dic61AGL(1,ℤ/8)12yes—
240C2×Dic61#13712yes—
241Dic122—24no—
242Dic122—42no—
243Dic122—48no—
244C8×C61—42no—
245#92418—42no—
246#21860—611nomaximal, Sylow
247C22×C61—22no—
248C22×C61—22no—
249C22×C61—22no—
250C22×C61—22no—
251C22×C61—22no—
252C22×C61—22no—
253C22×C61C2×#10314yes—
254C22×C61C22×C22⋊C411yescharacteristic
255C22×C61#53511yescharacteristic
256C22×C61#53611yescharacteristic
257C22×C61C22×SD1611yescharacteristic
258C22×C61C22×Q1611yescharacteristic
259C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
260C22×C61C2×#13711yescharacteristic
261C4×C61—22no—
262C4×C61—42no—
263C4×C61—42no—
264C4×C61—82no—
265C4×C61#52012yes—
266C4×C61#53712yes—
267Dic61—22no—
268Dic61—42no—
269Dic61—42no—
270Dic61—82no—
271Dic61C2×#10314yes—
272Dic61C2×#10314yes—
273Dic61#41714yes—
274C2×#533—611no—
275C2×#431—611no—
276C2×#384—611no—
277#1990—612no—
278#1956—612no—
279#1872—611no—
280#1872—611no—
281#1872—611no—
282#1872D6111yescharacteristic
283#1803—612no—
284#1803—612no—
285C2×C61—24no—
286C2×C61#231111yescharacteristic
287C2×C61C22×#10311yescharacteristic
288C2×C61C2×#41711yescharacteristic
289C2×C61#320811yescharacteristic
290C2×C61#320911yescharacteristic
291C2×C61#322911yescharacteristic
292C2×C61#323011yescharacteristic
293#533—618no—
294#533—1224no—
295C2×C4⋊Q8—611no—
296C2×C4⋊Q8—611no—
297C2×#125—1221no—
298C2×C22⋊Q8—611no—
299C2×C22⋊Q8—1221no—
300C2×C22⋊Q8C2×D6111yescharacteristic
301C2×C4×Q8—611no—
302C2×C4×Q8—611no—
303C2×C4×Q8—1221no—
304C2×#117—1221no—
305C22×C4⋊C4—611no—
306C22×C4⋊C4—1221no—
307C22×C4⋊C4—1221no—
308C22×C4⋊C4C2×D6111yescharacteristic
309C22×C42—611no—
310#431—1224no—
311C2×C4⋊C8—612no—
312C2×#103—612no—
313C2×#103—612no—
314C2×#103—612no—
315C2×#103C2×D6112yes—
316C2×C22⋊C8—611no—
317C2×C22⋊C8C2×D6111yescharacteristic
318C2×C4×C8—612no—
319#384—612no—
320#384—612no—
321#384—1221no—
322#384—1221no—
323#340—612no—
324#340—612no—
325#340—612no—
326#340C2×D6112yes—
327C61#2186011yescharacteristic, complemented, Sylow
328C22×Q8—611no—
329C22×Q8—1221no—
330C22×Q8C22×D6111yescharacteristic
331C23×C4—21no—
332C23×C4—611no—
333C23×C4—611no—
334C23×C4—1221no—
335C23×C4C22×D6111yescharacteristic
336C22×C8—612no—
337C22×C8C22×D6112yes—
338C4⋊Q8—614no—
339C4⋊Q8—614no—
340C4⋊Q8—1222no—
341C4⋊Q8—1222no—
342#125—1228no—
343C22⋊Q8—22no—
344C22⋊Q8—614no—
345C22⋊Q8—1228no—
346C22⋊Q8—1222no—
347C22⋊Q8C4×D6114yes—
348C4×Q8—614no—
349C4×Q8—614no—
350C4×Q8—1222no—
351C4×Q8—1228no—
352C4×Q8—1222no—
353#117—1228no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—42no—
357C2×C4⋊C4—611no—
358C2×C4⋊C4—612no—
359C2×C4⋊C4—1221no—
360C2×C4⋊C4—1221no—
361C2×C4⋊C4—1221no—
362C2×C4⋊C4—1221no—
363C2×C4⋊C4—1221no—
364C2×C4⋊C4—1221no—
365C2×C4⋊C4—1222no—
366C2×C4⋊C4—1222no—
367C2×C4⋊C4—1222no—
368C2×C4⋊C4—1222no—
369C2×C4⋊C4—1221no—
370C2×C4⋊C4—2442no—
371C2×C4⋊C4—2444no—
372C2×C4⋊C4—2444no—
373C2×C4⋊C4C22×D6112yes—
374C2×C4⋊C4C22×D6111yescharacteristic
375C2×C22⋊C4—21no—
376C2×C22⋊C4—1221no—
377C2×C42—614no—
378C2×C42—611no—
379C2×C42—611no—
380C2×C42—614no—
381C2×C42—611no—
382C2×C42—611no—
383C2×C42—1221no—
384C2×C42—1221no—
385C4⋊C8—1224no—
386#103—42no—
387#103—2442no—
388#103—2442no—
389#103—2442no—
390C22⋊C8—22no—
391C22⋊C8—1222no—
392C4×C8—1224no—
393C24D4×D6111yescharacteristic
394C2×Q8—21no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—612no—
398C2×Q8—1224no—
399C2×Q8—1221no—
400C2×Q8—1222no—
401C2×Q8—1222no—
402C2×Q8—2442no—
403C2×Q8—2442no—
404C2×Q8C2×C4×D6112yes—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—611no—
414C22×C4—611no—
415C22×C4—611no—
416C22×C4—614no—
417C22×C4—614no—
418C22×C4—614no—
419C22×C4—611no—
420C22×C4—614no—
421C22×C4—1222no—
422C22×C4—1222no—
423C22×C4—1221no—
424C22×C4—1221no—
425C22×C4—1222no—
426C22×C4—2442no—
427C22×C4—2442no—
428C22×C4—2442no—
429C22×C4C2×C4×D6112yes—
430C22×C4D4×D6112yes—
431C22×C4D4×D6111yescharacteristic
432C22×C4C23×D6111yescharacteristic
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—1222no—
437C2×C8—1222no—
438C2×C8—1222no—
439C4⋊C4—21no—
440C4⋊C4—24no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—42no—
444C4⋊C4—612no—
445C4⋊C4—1224no—
446C4⋊C4—1224no—
447C4⋊C4—1224no—
448C4⋊C4—1224no—
449C4⋊C4—1224no—
450C4⋊C4—1221no—
451C4⋊C4—1224no—
452C4⋊C4—1222no—
453C4⋊C4—1222no—
454C4⋊C4—2442no—
455C4⋊C4—2444no—
456C4⋊C4—2444no—
457C4⋊C4—2444no—
458C4⋊C4—2444no—
459C4⋊C4C2×C4×D6112yes—
460C22⋊C4—24no—
461C22⋊C4—1224no—
462C42—612no—
463C42—612no—
464C42—612no—
465C42—612no—
466C42—614no—
467C42—614no—
468C42—1224no—
469C42—1224no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23C22⋊C4×D6112yes—
475C23SD16×D6112yes—
476C23Q16×D6112yes—
477C23C2×D4×D6111yescharacteristic
478Q8—24no—
479Q8—42no—
480Q8—1224no—
481Q8—2448no—
482Q8—2442no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—84no—
497C2×C4—612no—
498C2×C4—612no—
499C2×C4—612no—
500C2×C4—6116no—
501C2×C4—612no—
502C2×C4—612no—
503C2×C4—612no—
504C2×C4—612no—
505C2×C4—612no—
506C2×C4—612no—
507C2×C4—612no—
508C2×C4—1228no—
509C2×C4—1221no—
510C2×C4—1221no—
511C2×C4—1222no—
512C2×C4—1222no—
513C2×C4—4884no—
514C2×C4—2442no—
515C2×C4—2442no—
516C2×C4—2442no—
517C2×C4—2441no—
518C2×C4—2441no—
519C2×C4—2441no—
520C2×C4—2441no—
521C2×C4C22⋊C4×D6112yes—
522C2×C4C22×C4×D6111yescharacteristic
523C2×C4C2×D4×D6111yescharacteristic
524C2×C4#1451982212yes—
525C2×C4#1451982412yes—
526C8—42no—
527C8—2442no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22#103×D6114yes—
535C22C2×C22⋊C4×D6111yescharacteristic
536C22#1684032711yescharacteristic
537C22#1684033411yescharacteristic
538C22C2×SD16×D6111yescharacteristic
539C22C2×Q16×D6111yescharacteristic
540C22AGL(1,ℤ/8)×D6111yescharacteristic
541C22#137×D6111yescharacteristic
542C4—22no—
543C4—42no—
544C4—42no—
545C4—82no—
546C4—614no—
547C4—614no—
548C4—614no—
549C4—1222no—
550C4—4882no—
551C4—2442no—
552C4—2442no—
553C4#1684024112yes—
554C4#1684033912yes—
555C2—24no—
556C2#2144840611yescharacteristic
557C2C2×#103×D6111yescharacteristic
558C2#417×D6111yescharacteristic
559C2#2145675811yescharacteristic
560C2#2145676511yescharacteristic
561C2#2145696111yescharacteristic
562C2#2145696811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1872C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
cosocleC22×C4×C61C24
hypercenter#1872D61
hypocenterC61#21860
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
4. #21446778C2semidirect
5. #21446619C2semidirect
9. #21445823C2semidirect
10. #21445822C2semidirect
13. #16840314C4semidirect
53. #14519776C2×C4semidirect
55. #14519775C2×C4semidirect
61. Q8×Dic61C2×C4semidirect
64. #14519742C2×C4semidirect
230. C2×Dic61#103semidirect
327. C61#21860semidirect

Maximal subgroups

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

subgroupindex
C2×#168403142
C2×#168398122
C2×#168396312
#214467782
#214466192
#214461272
#214461262
#214461242
#214458232
#214458222
C61×#18722
#2186061