Finite Group Databasebeta

← #29718356

Subgroups — #29718356

Order 15616.

1260 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840076C211yesmaximal, characteristic, complemented
2C22×#14519698C211yesmaximal, characteristic, complemented
3#21449452C211yesmaximal, characteristic
4#21448690C211yesmaximal, characteristic, complemented
5#21447205C212yesmaximal, complemented
6#21447051C212yesmaximal, complemented
7#21446744C212yesmaximal, complemented
8#21446441C212yesmaximal, complemented
9#21446245C211yesmaximal, characteristic, complemented
10#21445996C211yesmaximal, characteristic, complemented
11C61×#1786C211yesmaximal, characteristic, complemented
12C23×#92418C2211yescharacteristic
13C2×#14519773C2212yes—
14C2×#14519751C2211yescharacteristic
15C2×#14519746—21no—
16C2×#14519746—21no—
17C2×#14519733—21no—
18C22×C4⋊Dic61C2211yescharacteristic
19#16840076—24no—
20C2×#14519711C2211yescharacteristic, complemented
21C2×#14519711C2211yescharacteristic, complemented
22C2×#14519706C2212yescomplemented
23C2×#14519699C2211yescharacteristic, complemented
24C2×#14519699C2211yescharacteristic, complemented
25C2×#14519698—24no—
26C2×#14519698C2212yes—
27C2×#14519698C2212yes—
28C2×#14519697C2212yescomplemented
29#16839670—22no—
30#16839665C2212yes—
31#16839658C2212yescomplemented
32#16839645—22no—
33#16839623—21no—
34C4⋊C4×Dic61C2212yescomplemented
35C4×C22⋊Dic61C2211yescharacteristic
36#16839524C2212yescomplemented
37#16839516C2212yescomplemented
38C23×D4×C61C2211yescharacteristic
39C2×#102×C61C2212yes—
40C2×#102×C61C2212yes—
41C2×C22⋊C8×C61C2211yescharacteristic
42C61×#380C2211yescharacteristic
43C22×D244C2311yescharacteristic
44C22×C22⋊D61—21no—
45C22×C22⋊D61—22no—
46C22×C22⋊D61—22no—
47C23×Dic61—21no—
48C22×#92418—22no—
49C22×#92418—42no—
50C22×#92418C2312yes—
51#14519773—24no—
52#14519773—42no—
53#14519751—22no—
54#14519751—22no—
55#14519746—44no—
56#14519746—44no—
57#14519733—44no—
58C2×C4⋊Dic61—21no—
59C2×C4⋊Dic61—22no—
60C2×C4⋊Dic61D414yescomplemented
61C2×C4⋊Dic61C2311yescharacteristic
62C2×C4⋊Dic61C2311yescharacteristic
63C2×C4⋊Dic61C2311yescharacteristic
64C2×#11758196—22no—
65C2×C22⋊Dic61—22no—
66C2×C22⋊Dic61—22no—
67C2×C22⋊Dic61D411yescharacteristic
68C2×C22⋊Dic61D411yescharacteristic
69C2×#11758198—21no—
70C2×C4×Dic61—22no—
71C2×C4×Dic61C2312yes—
72#14519711—22no—
73#14519711—22no—
74#14519706—42no—
75#14519699—22no—
76#14519699—22no—
77#14519698—24no—
78#14519698—24no—
79#14519697—42no—
80#14519684—22no—
81#14519684—22no—
82#14519684D412yes—
83C25×C61—21no—
84C22×D4×C61—24no—
85C22×D4×C61—22no—
86C22×D4×C61—21no—
87C22×D4×C61—22no—
88C22×D4×C61—42no—
89C22×D4×C61C2312yes—
90C23×C4×C61C2311yescharacteristic
91C22×C8×C61C2311yescharacteristic
92C22×C8×C61C2311yescharacteristic
93C2×C4⋊C4×C61C2312yes—
94C2×C22⋊C4×C61—21no—
95#102×C61—24no—
96#102×C61—42no—
97C22⋊C8×C61—22no—
98#95×C61—21no—
99C24×C61—22no—
100C24×C61—44no—
101C24×C61—42no—
102C24×C61—42no—
103C24×C61—42no—
104C24×C61—82no—
105C24×C61C2×D411yescharacteristic
106C23×D61—21no—
107C2×D4×C61—28no—
108C2×D4×C61—22no—
109C2×D4×C61—22no—
110C2×D4×C61—22no—
111C2×D4×C61—22no—
112C2×D4×C61—22no—
113C2×D4×C61—48no—
114C2×D4×C61—44no—
115C2×D4×C61—44no—
116C2×D4×C61—42no—
117C2×D4×C61#4114yes—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D412yes—
123C22×C4×C61#4112yes—
124C22×C4×C61C2411yescharacteristic
125C2×C22⋊D61—44no—
126C2×C22⋊D61—44no—
127C2×C22⋊D61—44no—
128C2×C22⋊D61—42no—
129C2×C22⋊D61—42no—
130C2×C22⋊D61—42no—
131C2×C22⋊D61—44no—
132C2×C22⋊D61—82no—
133C2×C22⋊D61—82no—
134C22×Dic61—24no—
135C22×Dic61—21no—
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—42no—
139C22×Dic61—42no—
140C22×Dic61C2×D412yes—
141C22×Dic61C2×D412yes—
142C2×D244—21no—
143C2×D244—21no—
144C2×D244—22no—
145C2×D244—22no—
146C2×C8×C61—21no—
147C2×C8×C61—21no—
148C2×C8×C61—22no—
149C2×C8×C61—22no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—22no—
152C22⋊C4×C61—42no—
153C22⋊Dic61—22no—
154C22⋊Dic61—44no—
155C22⋊Dic61—44no—
156C22⋊Dic61D814yes—
157#11758198—42no—
158C4⋊Dic61—24no—
159C4⋊Dic61—21no—
160C4⋊Dic61—21no—
161C4⋊Dic61—21no—
162C4⋊Dic61—21no—
163C4⋊Dic61C2×D412yes—
164C4⋊Dic61C2×D412yes—
165#11758196—44no—
166C4×Dic61—22no—
167C4×Dic61—22no—
168C4×Dic61—44no—
169C2×#92418—22no—
170C2×#92418—22no—
171C2×#92418—22no—
172C2×#92418—44no—
173C23×C61—22no—
174C23×C61—22no—
175C23×C61—24no—
176C23×C61—48no—
177C23×C61—42no—
178C23×C61—42no—
179C23×C61—42no—
180C23×C61—42no—
181C23×C61—42no—
182C23×C61—42no—
183C23×C61—88no—
184C23×C61—88no—
185C23×C61—84no—
186C23×C61—84no—
187C23×C61#12112yes—
188C23×C61C2×D812yes—
189C23×C61AGL(1,ℤ/8)12yes—
190C23×C61C22×D411yescharacteristic
191C22×D61—41no—
192C22×D61—42no—
193C22×D61—41no—
194C22×D61—41no—
195C22×D61—42no—
196D4×C61—24no—
197D4×C61—24no—
198D4×C61—48no—
199D4×C61—42no—
200D4×C61—42no—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—24no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61#12112yes—
209C2×C4×C61C22×D411yescharacteristic
210C2×C4×C61C2×#4112yes—
211C2×C4×C61C2×#4111yescharacteristic
212C2×C4×C61H(5,𝔽2)12yes—
213C22⋊D61—88no—
214C22⋊D61—88no—
215C22⋊D61—84no—
216C2×Dic61—22no—
217C2×Dic61—22no—
218C2×Dic61—22no—
219C2×Dic61—22no—
220C2×Dic61—22no—
221C2×Dic61—22no—
222C2×Dic61—22no—
223C2×Dic61—41no—
224C2×Dic61—41no—
225C2×Dic61—41no—
226C2×Dic61—41no—
227C2×Dic61—48no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61C2×D814yes—
232D244—41no—
233D244—41no—
234D244—42no—
235C8×C61—41no—
236C8×C61—41no—
237#92418—42no—
238#24476—611nomaximal, Sylow
239C22×C61—22no—
240C22×C61—24no—
241C22×C61—22no—
242C22×C61—24no—
243C22×C61—44no—
244C22×C61—44no—
245C22×C61—44no—
246C22×C61—816no—
247C22×C61—82no—
248C22×C61—82no—
249C22×C61—82no—
250C22×C61—82no—
251C22×C61#45914yes—
252C22×C61C2×#12111yescharacteristic
253C22×C61#54612yes—
254C22×C61C22×D811yescharacteristic
255C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
256C22×C61#57712yes—
257C2×D61—81no—
258C2×D61—81no—
259C2×D61—82no—
260C2×D61—81no—
261C2×D61—82no—
262C4×C61—22no—
263C4×C61—82no—
264C4×C61#52511yescharacteristic
265C4×C61#53511yescharacteristic
266C4×C61#55112yes—
267Dic61—24no—
268Dic61—42no—
269Dic61—42no—
270Dic61—42no—
271Dic61—81no—
272C2×#480—611no—
273C22×#102—611no—
274#2433—611no—
275#2344—611no—
276#2067—612no—
277#2041—612no—
278#1982—612no—
279#1925—612no—
280#1893—611no—
281#1850—611no—
282#1786D6111yescharacteristic, complemented
283C2×C61—24no—
284C2×C61—84no—
285C2×C61#241111yescharacteristic
286C2×C61C2×#45911yescharacteristic
287C2×C61#304311yescharacteristic
288C2×C61#329412yes—
289C2×C61#331012yes—
290D61—161no—
291C23×D4C2×D6111yescharacteristic
292C23×C8—611no—
293C2×#127—612no—
294C2×#121—611no—
295C2×#120—1221no—
296C2×#120—1221no—
297C2×#117—1221no—
298C22×C4⋊C4—611no—
299#480—1224no—
300C2×#107—611no—
301C2×#107—611no—
302C2×C4⋊C8—612no—
303C2×#102—612no—
304C2×#102—612no—
305C2×#102—611no—
306C2×#102—611no—
307C2×#102—612no—
308C2×#102—1224no—
309C2×#102C2×D6112yes—
310C2×#102C2×D6112yes—
311C2×C22⋊C8C2×D6111yescharacteristic
312#392—1222no—
313#390—612no—
314#389—612no—
315#387—1222no—
316#381—1221no—
317#380C2×D6111yescharacteristic
318C4×C4⋊C4—612no—
319C4×C22⋊C4—611no—
320#340—612no—
321#336—612no—
322C61#2447611yescharacteristic, complemented, Sylow
323C25—21no—
324C22×D4—24no—
325C22×D4—22no—
326C22×D4—22no—
327C22×D4—21no—
328C22×D4—42no—
329C22×D4—611no—
330C22×D4—1222no—
331C22×D4—1222no—
332C22×D4—1221no—
333C22×D4C22×D6112yes—
334C23×C4—1221no—
335C23×C4C22×D6111yescharacteristic
336C22×C8—612no—
337C22×C8—1222no—
338C22×C8—2442no—
339C22×C8C22×D6111yescharacteristic
340C22×C8C22×D6111yescharacteristic
341#127—1224no—
342#127—2442no—
343#121—1222no—
344#121—1222no—
345#120—2444no—
346#120—2444no—
347#117—2444no—
348C2×C4⋊C4—614no—
349C2×C4⋊C4—611no—
350C2×C4⋊C4—611no—
351C2×C4⋊C4—611no—
352C2×C4⋊C4—1222no—
353C2×C4⋊C4—1221no—
354C2×C4⋊C4—1222no—
355C2×C4⋊C4C22×D6112yes—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—611no—
358C2×C22⋊C4—611no—
359C2×C22⋊C4—1222no—
360C2×C22⋊C4—1222no—
361C2×C22⋊C4—1221no—
362C2×C42—612no—
363C2×C42—1222no—
364#107—1222no—
365#107—1222no—
366C4⋊C8—2442no—
367#102—24no—
368#102—42no—
369#102—1222no—
370#102—1222no—
371#102—1224no—
372#102—1224no—
373#102—2442no—
374C22⋊C8—22no—
375#95—21no—
376#95—612no—
377#95—1222no—
378#95—1222no—
379C24—22no—
380C24—44no—
381C24—42no—
382C24—42no—
383C24—42no—
384C24—82no—
385C24—1221no—
386C24D4×D6111yescharacteristic
387C2×D4—28no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—48no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—42no—
397C2×D4—1221no—
398C2×D4—1222no—
399C2×D4—1221no—
400C2×D4—1222no—
401C2×D4—4882no—
402C2×D4—4882no—
403C2×D4—2444no—
404C2×D4—2444no—
405C2×D4—2444no—
406C2×D4—2444no—
407C2×D4—2442no—
408C2×D4—2442no—
409C2×D4—2442no—
410C2×D4#1175822414yes—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—612no—
415C22×C4—612no—
416C22×C4—1221no—
417C22×C4—1221no—
418C22×C4—1224no—
419C22×C4—1221no—
420C22×C4—2442no—
421C22×C4—2442no—
422C22×C4#1175822212yes—
423C22×C4D4×D6112yes—
424C22×C4D4×D6111yescharacteristic
425C22×C4C23×D6111yescharacteristic
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—1222no—
431C2×C8—1222no—
432C2×C8—1222no—
433C2×C8—2444no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—612no—
437C4⋊C4—612no—
438C4⋊C4—1221no—
439C4⋊C4—1221no—
440C4⋊C4—1224no—
441C4⋊C4—1221no—
442C4⋊C4—1221no—
443C4⋊C4—2444no—
444C22⋊C4—42no—
445C22⋊C4—614no—
446C22⋊C4—1222no—
447C22⋊C4—2444no—
448C22⋊C4—2444no—
449C22⋊C4—2442no—
450C42—1222no—
451C42—1222no—
452C42—2444no—
453C23—22no—
454C23—22no—
455C23—24no—
456C23—48no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—88no—
464C23—88no—
465C23—84no—
466C23—84no—
467C23—2441no—
468C23—2441no—
469C23—2441no—
470C23—2442no—
471C23—2442no—
472C23#1451975012yes—
473C23D8×D6112yes—
474C23#1451980412yes—
475C23C2×D4×D6111yescharacteristic
476D4—24no—
477D4—24no—
478D4—48no—
479D4—42no—
480D4—42no—
481D4—4888no—
482D4—4884no—
483D4—4888no—
484D4—2441no—
485D4—2441no—
486D4—2442no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—614no—
495C2×C4—1222no—
496C2×C4—1222no—
497C2×C4—1222no—
498C2×C4—1222no—
499C2×C4—1222no—
500C2×C4—1222no—
501C2×C4—1222no—
502C2×C4—2441no—
503C2×C4—2441no—
504C2×C4—2441no—
505C2×C4—2448no—
506C2×C4—2441no—
507C2×C4—2441no—
508C2×C4—2441no—
509C2×C4—2441no—
510C2×C4#1451975012yes—
511C2×C4C2×D4×D6111yescharacteristic
512C2×C4C2×#1175822412yes—
513C2×C4C2×#1175822211yescharacteristic
514C2×C4#1451982212yes—
515C8—41no—
516C8—41no—
517C8—2442no—
518C22—24no—
519C22—24no—
520C22—22no—
521C22—22no—
522C22—44no—
523C22—44no—
524C22—44no—
525C22—816no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22—82no—
530C22—4881no—
531C22—4882no—
532C22—4882no—
533C22—4881no—
534C22—4881no—
535C22#1683995614yes—
536C22C2×#1451975011yescharacteristic
537C22#1684039912yes—
538C22C2×D8×D6111yescharacteristic
539C22C2×#1451980411yescharacteristic
540C22#1684057212yes—
541C4—22no—
542C4—82no—
543C4—1224no—
544C4—4881no—
545C4—2442no—
546C4—2442no—
547C4—2442no—
548C4#1684026811yescharacteristic
549C4#1684032711yescharacteristic
550C4#1684043812yes—
551C2—24no—
552C2—84no—
553C2—9761no—
554C2#2144920611yescharacteristic
555C2C2×#1683995611yescharacteristic
556C2#2145533911yescharacteristic
557C2#2145762512yes—
558C2#2145784412yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1786C2
socleC23×C61C22×D4
prime radical#21449452C2
hypercenter#1786D61
hypocenterC61#24476
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1786

Complements

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

NH ≅ G/Ntype
1. C2×#16840076C2semidirect
2. C22×#14519698C2semidirect
4. #21448690C2semidirect
5. #21447205C2semidirect
6. #21447051C2semidirect
7. #21446744C2semidirect
8. #21446441C2semidirect
9. #21446245C2semidirect
10. #21445996C2semidirect
11. C61×#1786C2semidirect
20. C2×#14519711C22semidirect
21. C2×#14519711C22semidirect
22. C2×#14519706C22semidirect
23. C2×#14519699C22semidirect
24. C2×#14519699C22semidirect
28. C2×#14519697C22semidirect
31. #16839658C22semidirect
34. C4⋊C4×Dic61C22semidirect
36. #16839524C22semidirect
37. #16839516C22semidirect
60. C2×C4⋊Dic61D4semidirect
282. #1786D61semidirect
322. C61#24476semidirect

Maximal subgroups

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

subgroupindex
C2×#168400762
C22×#145196982
#214494522
#214486902
#214472052
#214470512
#214467442
#214464412
#214462452
#214459962
C61×#17862
#2447661