Finite Group Databasebeta

← #29718355

Subgroups — #29718355

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
1C22×#14519697C211yesmaximal, characteristic, complemented
2#21449457C211yesmaximal, characteristic
3#21448688C211yesmaximal, characteristic, complemented
4#21447202C212yesmaximal, complemented
5#21447052C212yesmaximal, complemented
6#21446744C212yesmaximal, complemented
7#21446438C212yesmaximal, complemented
8#21446248C211yesmaximal, characteristic, complemented
9#21445996C211yesmaximal, characteristic, complemented
10#21445751C211yesmaximal, characteristic, complemented
11C2×C61×#480C211yesmaximal, characteristic, complemented
12C23×D244C2211yescharacteristic
13C23×#92418C2211yescharacteristic
14C2×#14519773C2212yes—
15C2×#14519745—21no—
16C2×#14519745—21no—
17C2×#14519736—21no—
18C2×#14519706C2212yescomplemented
19C2×#14519699C2212yes—
20C2×#14519699C2212yes—
21C2×#14519698C2212yescomplemented
22C2×#14519697—24no—
23C2×#14519697C2212yes—
24C2×#14519697C2212yes—
25#16839671—22no—
26#16839663C2212yes—
27#16839661C2212yescomplemented
28#16839648—22no—
29#16839625—21no—
30#16839622C2211yescharacteristic
31C4⋊C4×Dic61C2212yescomplemented
32C4×C22⋊Dic61C2211yescharacteristic
33#16839524C2212yescomplemented
34#16839516C2212yescomplemented
35C2×#121×C61C2211yescharacteristic
36C22×C4⋊C4×C61C2211yescharacteristic
37C61×#480—24no—
38C2×#107×C61C2211yescharacteristic, complemented
39C2×#107×C61C2211yescharacteristic, complemented
40C2×#102×C61C2211yescharacteristic, complemented
41C2×#102×C61C2211yescharacteristic, complemented
42C2×C22⋊C8×C61C2211yescharacteristic
43C24×D61—21no—
44C22×D244—22no—
45C22×D244—24no—
46C22×D244—22no—
47C22×D244—21no—
48C22×D244—42no—
49C22×D244C2312yes—
50C22×C22⋊D61—22no—
51C22×C22⋊D61—22no—
52C22×#92418—22no—
53C22×#92418—42no—
54C22×#92418C2312yes—
55#14519773—24no—
56#14519773—42no—
57#14519745—44no—
58#14519745—44no—
59#14519736—44no—
60C2×C4⋊Dic61—22no—
61C2×C4⋊Dic61C2312yes—
62C2×C22⋊Dic61—21no—
63C2×C22⋊Dic61D411yescharacteristic
64C2×C22⋊Dic61D411yescharacteristic
65C2×#11758198—22no—
66C2×#11758198—22no—
67C2×C4×Dic61—22no—
68C2×C4×Dic61C2312yes—
69#14519706—42no—
70#14519699—24no—
71#14519699—42no—
72#14519698—42no—
73#14519697—24no—
74#14519697—24no—
75#14519684—21no—
76#14519684—22no—
77#14519684—22no—
78#14519684D412yes—
79C22×D4×C61—21no—
80C22×D4×C61C2311yescharacteristic
81C23×C4×C61—21no—
82C23×C4×C61C2311yescharacteristic
83C22×C8×C61C2311yescharacteristic
84C22×C8×C61C2311yescharacteristic
85#121×C61—22no—
86#121×C61—22no—
87C2×C4⋊C4×C61—21no—
88C2×C4⋊C4×C61—22no—
89C2×C4⋊C4×C61D414yescomplemented
90C2×C4⋊C4×C61C2311yescharacteristic
91C2×C4⋊C4×C61C2311yescharacteristic
92C2×C4⋊C4×C61C2311yescharacteristic
93C2×C22⋊C4×C61—21no—
94#107×C61—22no—
95#107×C61—22no—
96#102×C61—22no—
97#102×C61—22no—
98C22⋊C8×C61—22no—
99C24×C61—21no—
100C24×C61C2×D411yescharacteristic
101C23×D61—22no—
102C23×D61—44no—
103C23×D61—42no—
104C23×D61—42no—
105C23×D61—42no—
106C23×D61—82no—
107C2×D4×C61—21no—
108C2×D4×C61—21no—
109C2×D4×C61—22no—
110C2×D4×C61—22no—
111C2×D4×C61—42no—
112C2×D4×C61—42no—
113C2×D4×C61—42no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—24no—
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61—42no—
121C22×C4×C61—42no—
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D412yes—
124C22×C4×C61#4112yes—
125C22×C4×C61C2411yescharacteristic
126C2×C22⋊D61—44no—
127C2×C22⋊D61—44no—
128C2×C22⋊D61—44no—
129C2×C22⋊D61—44no—
130C2×C22⋊D61—82no—
131C2×C22⋊D61—82no—
132C22×Dic61—22no—
133C22×Dic61C2×D412yes—
134C22×Dic61C2×D412yes—
135C2×D244—28no—
136C2×D244—22no—
137C2×D244—22no—
138C2×D244—22no—
139C2×D244—22no—
140C2×D244—22no—
141C2×D244—48no—
142C2×D244—44no—
143C2×D244—44no—
144C2×D244—42no—
145C2×D244#4114yes—
146C2×C8×C61—21no—
147C2×C8×C61—21no—
148C2×C8×C61—22no—
149C2×C8×C61—22no—
150C4⋊C4×C61—24no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61C2×D412yes—
156C4⋊C4×C61C2×D412yes—
157C22⋊C4×C61—42no—
158C22⋊Dic61—22no—
159C22⋊Dic61—42no—
160C22⋊Dic61D814yes—
161#11758198—44no—
162#11758198—44no—
163C4⋊Dic61—22no—
164C4⋊Dic61—22no—
165C4⋊Dic61—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—41no—
176C23×C61—42no—
177C23×C61—41no—
178C23×C61—41no—
179C23×C61—42no—
180C23×C61#12112yes—
181C23×C61C2×D812yes—
182C23×C61AGL(1,ℤ/8)12yes—
183C23×C61C22×D411yescharacteristic
184C22×D61—24no—
185C22×D61—48no—
186C22×D61—42no—
187C22×D61—42no—
188C22×D61—42no—
189C22×D61—42no—
190C22×D61—42no—
191C22×D61—42no—
192C22×D61—88no—
193C22×D61—88no—
194C22×D61—84no—
195C22×D61—84no—
196D4×C61—41no—
197D4×C61—41no—
198D4×C61—42no—
199D4×C61—84no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61—24no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61—41no—
210C2×C4×C61—48no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61#12112yes—
215C2×C4×C61C22×D411yescharacteristic
216C2×C4×C61C2×#4112yes—
217C2×C4×C61C2×#4111yescharacteristic
218C2×C4×C61H(5,𝔽2)12yes—
219C22⋊D61—88no—
220C22⋊D61—88no—
221C2×Dic61—22no—
222C2×Dic61—22no—
223C2×Dic61—22no—
224C2×Dic61—22no—
225C2×Dic61—22no—
226C2×Dic61—42no—
227C2×Dic61—42no—
228C2×Dic61—42no—
229C2×Dic61C2×D814yes—
230D244—24no—
231D244—24no—
232D244—48no—
233D244—42no—
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—81no—
244C22×C61—81no—
245C22×C61—82no—
246C22×C61—81no—
247C22×C61—82no—
248C22×C61#45914yes—
249C22×C61C2×#12111yescharacteristic
250C22×C61#54612yes—
251C22×C61C22×D811yescharacteristic
252C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
253C22×C61#57712yes—
254C2×D61—44no—
255C2×D61—44no—
256C2×D61—44no—
257C2×D61—816no—
258C2×D61—82no—
259C2×D61—82no—
260C2×D61—82no—
261C2×D61—82no—
262C4×C61—22no—
263C4×C61—42no—
264C4×C61—42no—
265C4×C61—81no—
266C4×C61#52511yescharacteristic
267C4×C61#53511yescharacteristic
268C4×C61#55112yes—
269Dic61—24no—
270Dic61—42no—
271Dic61—82no—
272C2×#480D6111yescharacteristic, complemented
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#1786—611no—
283C2×C61—24no—
284C2×C61—161no—
285C2×C61#241111yescharacteristic
286C2×C61C2×#45911yescharacteristic
287C2×C61#304311yescharacteristic
288C2×C61#329412yes—
289C2×C61#331012yes—
290D61—84no—
291C23×D4—611no—
292C23×C8—611no—
293C2×#127—612no—
294C2×#121C2×D6111yescharacteristic
295C2×#120—1221no—
296C2×#120—1221no—
297C2×#117—1221no—
298C22×C4⋊C4C2×D6111yescharacteristic
299#480—24no—
300C2×#107C2×D6111yescharacteristic, complemented
301C2×#107C2×D6111yescharacteristic, complemented
302C2×C4⋊C8—612no—
303C2×#102—612no—
304C2×#102—612no—
305C2×#102—612no—
306C2×#102—612no—
307C2×#102—612no—
308C2×#102—1224no—
309C2×#102C2×D6111yescharacteristic, complemented
310C2×#102C2×D6111yescharacteristic, complemented
311C2×C22⋊C8C2×D6111yescharacteristic
312#392—1222no—
313#390—612no—
314#389—612no—
315#387—1222no—
316#381—1221no—
317#380—611no—
318C4×C4⋊C4—612no—
319C4×C22⋊C4—611no—
320#340—612no—
321#336—612no—
322C61#2447611yescharacteristic, complemented, Sylow
323C25—1221no—
324C22×D4—21no—
325C22×D4—612no—
326C22×D4—1224no—
327C22×D4—1222no—
328C22×D4—1222no—
329C22×D4—1221no—
330C22×D4—1222no—
331C22×D4—1222no—
332C22×D4—2442no—
333C22×D4C22×D6111yescharacteristic
334C23×C4—21no—
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—22no—
344#121—22no—
345#120—2444no—
346#120—2444no—
347#117—2444no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—612no—
351C2×C4⋊C4—1222no—
352C2×C4⋊C4C22⋊D6114yescomplemented
353C2×C4⋊C4C22×D6111yescharacteristic
354C2×C4⋊C4C22×D6111yescharacteristic
355C2×C4⋊C4C22×D6111yescharacteristic
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—22no—
365#107—22no—
366C4⋊C8—2442no—
367#102—22no—
368#102—22no—
369#102—1224no—
370#102—1224no—
371#102—1224no—
372#102—2442no—
373#102—2442no—
374C22⋊C8—22no—
375#95—612no—
376#95—1222no—
377#95—1222no—
378#95—1221no—
379C24—21no—
380C24—1222no—
381C24—4882no—
382C24—2444no—
383C24—2442no—
384C24—2442no—
385C24—2442no—
386C24D4×D6111yescharacteristic
387C2×D4—21no—
388C2×D4—22no—
389C2×D4—21no—
390C2×D4—22no—
391C2×D4—42no—
392C2×D4—42no—
393C2×D4—42no—
394C2×D4—614no—
395C2×D4—1228no—
396C2×D4—1222no—
397C2×D4—1222no—
398C2×D4—1222no—
399C2×D4—1222no—
400C2×D4—1222no—
401C2×D4—4882no—
402C2×D4—4882no—
403C2×D4—2448no—
404C2×D4—2444no—
405C2×D4—2444no—
406C2×D4—2444no—
407C2×D4—2444no—
408C2×D4—2444no—
409C2×D4—2444no—
410C2×D4—2442no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—24no—
416C22×C4—21no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—612no—
420C22×C4—612no—
421C22×C4—1222no—
422C22×C4D4×D6111yescharacteristic
423C22×C4#1175822412yes—
424C22×C4C2×C22⋊D6112yes—
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—21no—
435C4⋊C4—21no—
436C4⋊C4—24no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—1222no—
440C4⋊C4—1222no—
441C4⋊C4—2444no—
442C4⋊C4C2×C22⋊D6112yes—
443C4⋊C4C2×C22⋊D6112yes—
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—41no—
456C23—41no—
457C23—41no—
458C23—42no—
459C23—42no—
460C23—1224no—
461C23—4884no—
462C23—4888no—
463C23—4888no—
464C23—4884no—
465C23—2448no—
466C23—2442no—
467C23—2442no—
468C23—2442no—
469C23—2442no—
470C23—2442no—
471C23—2442no—
472C23#1451974912yes—
473C23D8×D6112yes—
474C23#1451980112yes—
475C23C2×D4×D6111yescharacteristic
476D4—41no—
477D4—41no—
478D4—42no—
479D4—84no—
480D4—1224no—
481D4—1224no—
482D4—4888no—
483D4—4888no—
484D4—2448no—
485D4—2442no—
486D4—2442no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—48no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—614no—
502C2×C4—1222no—
503C2×C4—1222no—
504C2×C4—1222no—
505C2×C4—1222no—
506C2×C4—1222no—
507C2×C4—2442no—
508C2×C4—2442no—
509C2×C4—2442no—
510C2×C4#1451974912yes—
511C2×C4C22×C22⋊D6111yescharacteristic
512C2×C4#41×D6112yes—
513C2×C4C2×#1175822411yescharacteristic
514C2×C4#1451982312yes—
515C8—41no—
516C8—41no—
517C8—2442no—
518C22—24no—
519C22—24no—
520C22—22no—
521C22—22no—
522C22—81no—
523C22—82no—
524C22—82no—
525C22—81no—
526C22—81no—
527C22—48816no—
528C22—4882no—
529C22—4882no—
530C22—4882no—
531C22—4882no—
532C22—2444no—
533C22—2444no—
534C22—2444no—
535C22#1683995514yes—
536C22C2×#1451974911yescharacteristic
537C22#1684040412yes—
538C22C2×D8×D6111yescharacteristic
539C22C2×#1451980111yescharacteristic
540C22#1684057112yes—
541C4—22no—
542C4—42no—
543C4—42no—
544C4—81no—
545C4—1224no—
546C4—4882no—
547C4—2442no—
548C4#1684026911yescharacteristic
549C4#1684032611yescharacteristic
550C4#1684043512yes—
551C2—24no—
552C2—161no—
553C2—4884no—
554C2#2144920511yescharacteristic
555C2C2×#1683995511yescharacteristic
556C2#2145534211yescharacteristic
557C2#2145763012yes—
558C2#2145783812yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#480C2
socleC23×C61C22×D4
prime radical#21449457C2
hypercenterC2×#480D61
hypocenterC61#24476
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#480

Complements

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

NH ≅ G/Ntype
1. C22×#14519697C2semidirect
3. #21448688C2semidirect
4. #21447202C2semidirect
5. #21447052C2semidirect
6. #21446744C2semidirect
7. #21446438C2semidirect
8. #21446248C2semidirect
9. #21445996C2semidirect
10. #21445751C2semidirect
11. C2×C61×#480C2semidirect
18. C2×#14519706C22semidirect
21. C2×#14519698C22semidirect
27. #16839661C22semidirect
31. C4⋊C4×Dic61C22semidirect
33. #16839524C22semidirect
34. #16839516C22semidirect
38. C2×#107×C61C22semidirect
39. C2×#107×C61C22semidirect
40. C2×#102×C61C22semidirect
41. C2×#102×C61C22semidirect
89. C2×C4⋊C4×C61D4semidirect
272. C2×#480D61semidirect
300. C2×#107C2×D61semidirect
301. C2×#107C2×D61semidirect
309. C2×#102C2×D61semidirect
310. C2×#102C2×D61semidirect
322. C61#24476semidirect
352. C2×C4⋊C4C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C22×#145196972
#214494572
#214486882
#214472022
#214470522
#214467442
#214464382
#214462482
#214459962
#214457512
C2×C61×#4802
#2447661