Finite Group Databasebeta

← #29685135

Subgroups — #29685135

Order 15616.

1518 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840309C211yesmaximal, characteristic
2C2×#16839817C211yesmaximal, characteristic, complemented
3C4×C4⋊C4×D61C211yesmaximal, characteristic, complemented
4#21446765C212yesmaximal, complemented
5#21446744C212yesmaximal, complemented
6#21446135C212yesmaximal, complemented
7#21446133C212yesmaximal, complemented
8#21445647C211yesmaximal, characteristic, complemented
9#21445645C211yesmaximal, characteristic, complemented
10#21445644C211yesmaximal, characteristic, complemented
11C61×#1762C211yesmaximal, characteristic, complemented
12C22×#11758222—21no—
13#16840309—24no—
14#16840309C418yescomplemented
15C2×#14519776C2211yescharacteristic
16C2×#14519773C2211yescharacteristic
17C2×#14519763—21no—
18C2×#14519752C2211yescharacteristic
19C2×#14519750—21no—
20C2×#14519748C2211yescharacteristic
21C2×C4×D244C2211yescharacteristic
22C2×C4×C22⋊D61—21no—
23C2×C4⋊C4×D61C2212yes—
24C2×C42×D61—21no—
25C2×C42×D61C2211yescharacteristic
26#16839817—24no—
27C2×#14519706C2212yes—
28C2×#14519699C2212yes—
29C2×#14519698C2212yes—
30C2×#14519697C2212yes—
31C2×#14519689C2212yes—
32C2×#14519688C2211yescharacteristic
33C4⋊C4×Dic61C2212yes—
34C4×#11758196—21no—
35C4×C4⋊Dic61C2211yescharacteristic
36#16839613—21no—
37#95×D61—21no—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×C4×D4×C61C2211yescharacteristic
42C2×#102×C61C2212yes—
43C2×#97×C61C2211yescharacteristic
44C4×C4⋊C4×C61C2211yescharacteristic
45C61×#340C2212yes—
46C2×#11758222—28no—
47C2×#11758222—48no—
48C22×Dic122—21no—
49C22×D244—21no—
50C22×D244C2311yescharacteristic
51C22×C22⋊D61—21no—
52C22×C22⋊D61—21no—
53C22×C22⋊D61—21no—
54C22×C4×D61—22no—
55C22×C4×D61—21no—
56C22×C4×D61D411yescharacteristic
57C22×C4×D61D411yescharacteristic
58C22×C4×D61C2311yescharacteristic
59C22×#92418C2312yes—
60#14519776—22no—
61#14519776C2×C414yescomplemented
62#14519773—22no—
63#14519773C2×C414yescomplemented
64#14519763—28no—
65#14519752—22no—
66#14519752C2×C414yes—
67#14519750—28no—
68#14519748—22no—
69#14519748C2×C414yes—
70C4×D244—22no—
71C4×D244C2×C414yescomplemented
72C4×C22⋊D61—28no—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2312yes—
75C2×#11758196—22no—
76C2×#11758196—21no—
77C4⋊C4×D61—24no—
78C4⋊C4×D61—24no—
79C2×C22⋊Dic61—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—22no—
84C2×C4×Dic61D411yescharacteristic
85C2×C4×Dic61D411yescharacteristic
86C2×C4×Dic61C2311yescharacteristic
87C42×D61—44no—
88C42×D61C2×C414yes—
89C42×D61D414yes—
90#14519706—24no—
91#14519699—24no—
92#14519698—24no—
93#14519697—24no—
94#14519689—24no—
95#14519688—22no—
96#14519684—21no—
97#14519684—22no—
98C22×D4×C61C2311yescharacteristic
99C23×C4×C61—21no—
100C22×C8×C61C2312yes—
101C4×D4×C61—22no—
102C4×D4×C61C2×C414yescomplemented
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C4⋊C4×C61C2312yes—
105C2×C22⋊C4×C61—21no—
106C2×C42×C61—21no—
107C2×C42×C61C2311yescharacteristic
108#102×C61—24no—
109#97×C61—22no—
110#95×C61—21no—
111C24×C61—21no—
112C23×D61—21no—
113C23×D61C2×D411yescharacteristic
114C2×D4×C61—21no—
115C2×D4×C61—22no—
116C2×D4×C61—22no—
117C2×D4×C61C22×C412yes—
118C22×C4×C61—22no—
119C22×C4×C61—21no—
120C22×C4×C61—24no—
121C22×C4×C61—44no—
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2411yescharacteristic
125C2×C22⋊D61—24no—
126C2×C22⋊D61—24no—
127C2×C22⋊D61—24no—
128C2×C22⋊D61—44no—
129C2×C22⋊D61—44no—
130C2×C22⋊D61—42no—
131C2×C22⋊D61—42no—
132C22×Dic61—21no—
133C22×Dic61—22no—
134C22×Dic61C2×D411yescharacteristic
135C22×Dic61C2×D411yescharacteristic
136C22×Dic61C2×D411yescharacteristic
137#11758222—432no—
138C2×D244—22no—
139C2×D244—22no—
140C2×D244—24no—
141C2×D244—21no—
142C2×D244—44no—
143C2×D244C22×C412yes—
144C2×C4×D61—24no—
145C2×C4×D61—22no—
146C2×C4×D61—22no—
147C2×C4×D61—22no—
148C2×C4×D61—22no—
149C2×C4×D61—28no—
150C2×C4×D61—24no—
151C2×C4×D61—44no—
152C2×C4×D61—44no—
153C2×C4×D61—44no—
154C2×C4×D61C22⋊C414yes—
155C2×C4×D61C22⋊C414yes—
156C2×C4×D61C22×C412yes—
157C2×C4×D61C2×D412yes—
158C2×Dic122—24no—
159C2×Dic122—44no—
160C2×C8×C61—22no—
161C2×C8×C61—22no—
162C2×C8×C61—22no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61C22×C412yes—
166C4⋊C4×C61#4114yescomplemented
167C22⋊C4×C61—24no—
168C42×C61—42no—
169C42×C61C22×C412yes—
170C42×C61C2×D412yes—
171C22⋊Dic61—24no—
172#11758198—24no—
173#11758198—24no—
174C4⋊Dic61—21no—
175C4⋊Dic61—22no—
176C4⋊Dic61C22×C412yes—
177C4⋊Dic61#4114yescomplemented
178#11758196—28no—
179#11758196—24no—
180C4×Dic61—42no—
181C4×Dic61—44no—
182C4×Dic61C22⋊C414yes—
183C4×Dic61C22⋊C414yes—
184C4×Dic61C22×C412yes—
185C4×Dic61C2×D412yes—
186C2×#92418—22no—
187C2×#92418—22no—
188C2×#92418—22no—
189C23×C61—22no—
190C23×C61—42no—
191C23×C61—42no—
192C23×C61—41no—
193C23×C61—41no—
194C23×C61C22×D411yescharacteristic
195C22×D61—22no—
196C22×D61—21no—
197C22×D61—21no—
198C22×D61—22no—
199C22×D61—42no—
200C22×D61—42no—
201C22×D61—41no—
202C22×D61—41no—
203C22×D61C2×C22⋊C412yes—
204C22×D61AGL(1,ℤ/8)14yes—
205D4×C61—22no—
206D4×C61—22no—
207D4×C61—42no—
208C2×C4×C61—21no—
209C2×C4×C61—22no—
210C2×C4×C61—21no—
211C2×C4×C61—22no—
212C2×C4×C61—24no—
213C2×C4×C61—48no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61C2×C22⋊C412yes—
218C2×C4×C61C2×C22⋊C412yes—
219C2×C4×C61C23×C411yescharacteristic
220C2×C4×C61C22×D411yescharacteristic
221C2×C4×C61C2×#4112yes—
222C2×C4×C61C2×#4112yes—
223C22⋊D61—48no—
224C22⋊D61—48no—
225C22⋊D61—48no—
226C2×Dic61—22no—
227C2×Dic61—22no—
228C2×Dic61—22no—
229C2×Dic61—24no—
230C2×Dic61—42no—
231C2×Dic61—42no—
232C2×Dic61—42no—
233C2×Dic61C2×C22⋊C412yes—
234C2×Dic61C2×C22⋊C412yes—
235C2×Dic61C2×C22⋊C412yes—
236C2×Dic61AGL(1,ℤ/8)12yes—
237C2×Dic61#13712yes—
238D244—22no—
239D244—22no—
240D244—48no—
241D244—42no—
242C4×D61—28no—
243C4×D61—22no—
244C4×D61—22no—
245C4×D61—24no—
246C4×D61—28no—
247C4×D61—48no—
248C4×D61—416no—
249Dic122—48no—
250C8×C61—42no—
251#92418—42no—
252#21395—611nomaximal, Sylow
253C22×C61—42no—
254C22×C61—42no—
255C22×C61—42no—
256C22×C61—82no—
257C22×C61—82no—
258C22×C61C22×C22⋊C411yescharacteristic
259C22×C61#52512yes—
260C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
261C22×C61C2×AGL(1,ℤ/8)12yes—
262C22×C61C2×#13711yescharacteristic
263C2×D61—24no—
264C2×D61—24no—
265C2×D61—22no—
266C2×D61—22no—
267C2×D61—42no—
268C2×D61—42no—
269C2×D61—42no—
270C2×D61—82no—
271C2×D61—82no—
272C2×D61#41814yes—
273C4×C61—22no—
274C4×C61—22no—
275C4×C61—44no—
276C4×C61—42no—
277C4×C61C4×#4111yescharacteristic
278C4×C61C4×#4111yescharacteristic
279C4×C61#53212yes—
280Dic61—22no—
281Dic61—22no—
282Dic61—44no—
283Dic61—42no—
284Dic61#41714yes—
285C2×#532—611no—
286C2×#432—611no—
287C2×C4×C4⋊C4—611no—
288#1987—612no—
289#1982—612no—
290#1874—612no—
291#1874—612no—
292#1762—611no—
293#1762—611no—
294#1762—611no—
295#1762D6111yescharacteristic, complemented
296C2×C61—82no—
297C2×C61#229711yescharacteristic
298C2×C61C2×#41811yescharacteristic
299C2×C61C2×#41711yescharacteristic
300C2×C61#310912yes—
301C2×C61#311212yes—
302D61—24no—
303D61—82no—
304C22×#41—1221no—
305#532—618no—
306#532—1224no—
307C2×C4⋊Q8—611no—
308C2×#127—611no—
309C2×#124—1221no—
310C2×#121—611no—
311C2×#121—611no—
312C2×#121—1221no—
313C2×C4×D4—611no—
314C2×C4×D4—1221no—
315C2×C4×D4C2×D6111yescharacteristic
316C22×C4⋊C4—612no—
317C22×C42—611no—
318C22×C42—1221no—
319#432—1224no—
320C2×C4⋊C8—612no—
321C2×#102—612no—
322C2×#102—612no—
323C2×#102—612no—
324C2×#102C2×D6112yes—
325C2×C22⋊C8—612no—
326C2×#97—611no—
327C2×#97C2×D6111yescharacteristic
328C4×C4⋊C4—612no—
329C4×C4⋊C4—611no—
330C4×C4⋊C4—1221no—
331C4×C4⋊C4—1221no—
332C4×C4⋊C4C2×D6111yescharacteristic
333C2×#95—1221no—
334#340—612no—
335#340—612no—
336#340—612no—
337#340C2×D6112yes—
338C61#2139511yescharacteristic, complemented, Sylow
339C2×#41—1228no—
340C2×#41—2448no—
341C22×Q8—1221no—
342C22×D4—611no—
343C22×D4—1221no—
344C22×D4—1221no—
345C22×D4—1221no—
346C22×D4—1221no—
347C22×D4C22×D6111yescharacteristic
348C23×C4—21no—
349C23×C4—611no—
350C23×C4—611no—
351C23×C4—611no—
352C23×C4—1222no—
353C23×C4—1221no—
354C22×C8—612no—
355C22×C8C22×D6112yes—
356C4⋊Q8—614no—
357C4⋊Q8—1222no—
358#127—614no—
359#127—1222no—
360#124—1228no—
361#121—614no—
362#121—614no—
363#121—1228no—
364#121—1222no—
365#121—1222no—
366C4×D4—22no—
367C4×D4—614no—
368C4×D4—1228no—
369C4×D4—1222no—
370C4×D4C4×D6114yescomplemented
371C2×C4⋊C4—612no—
372C2×C4⋊C4—611no—
373C2×C4⋊C4—1224no—
374C2×C4⋊C4—1222no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4—1224no—
377C2×C4⋊C4C22×D6111yescharacteristic
378C2×C4⋊C4C22×D6112yes—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—1221no—
381C2×C22⋊C4—1221no—
382C2×C22⋊C4—1221no—
383C2×C42—21no—
384C2×C42—611no—
385C2×C42—611no—
386C2×C42—611no—
387C2×C42—614no—
388C2×C42—614no—
389C2×C42—1222no—
390C2×C42—1221no—
391C2×C42—2444no—
392C2×C42C22×D6111yescharacteristic
393C4⋊C8—1224no—
394#102—24no—
395#102—1224no—
396#102—1224no—
397#102—1224no—
398C22⋊C8—1224no—
399#97—22no—
400#97—1222no—
401#95—21no—
402#95—1222no—
403#95—1221no—
404C24—21no—
405C24—611no—
406C24—1221no—
407#41—24432no—
408C2×Q8—1224no—
409C2×Q8—2444no—
410C2×D4—21no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—612no—
414C2×D4—1224no—
415C2×D4—1224no—
416C2×D4—1224no—
417C2×D4—1224no—
418C2×D4—1221no—
419C2×D4—1222no—
420C2×D4—1222no—
421C2×D4—2444no—
422C2×D4—2442no—
423C2×D4—2442no—
424C2×D4—2444no—
425C2×D4—2444no—
426C2×D4C2×C4×D6112yes—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—24no—
430C22×C4—44no—
431C22×C4—611no—
432C22×C4—611no—
433C22×C4—611no—
434C22×C4—612no—
435C22×C4—612no—
436C22×C4—614no—
437C22×C4—614no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4—1224no—
441C22×C4—1228no—
442C22×C4—1222no—
443C22×C4—1222no—
444C22×C4—1222no—
445C22×C4—1221no—
446C22×C4—1224no—
447C22×C4—2444no—
448C22×C4—2444no—
449C22×C4—2444no—
450C22×C4D4×D6111yescharacteristic
451C22×C4D4×D6111yescharacteristic
452C22×C4C23×D6111yescharacteristic
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—1222no—
457C2×C8—1222no—
458C2×C8—1222no—
459C4⋊C4—21no—
460C4⋊C4—22no—
461C4⋊C4—612no—
462C4⋊C4—614no—
463C4⋊C4—1224no—
464C4⋊C4—1228no—
465C4⋊C4—1221no—
466C4⋊C4—1222no—
467C4⋊C4C2×C4×D6112yes—
468C4⋊C4#1175822214yescomplemented
469C22⋊C4—24no—
470C22⋊C4—1224no—
471C22⋊C4—1224no—
472C22⋊C4—1224no—
473C42—42no—
474C42—612no—
475C42—612no—
476C42—614no—
477C42—614no—
478C42—2442no—
479C42—2444no—
480C42C2×C4×D6112yes—
481C42D4×D6112yes—
482C23—22no—
483C23—42no—
484C23—42no—
485C23—41no—
486C23—41no—
487C23—612no—
488C23—614no—
489C23—1222no—
490C23—1221no—
491C23—1221no—
492C23—1222no—
493C23—2442no—
494C23—2442no—
495C23—2441no—
496C23—2441no—
497C23C2×D4×D6111yescharacteristic
498Q8—2448no—
499D4—22no—
500D4—22no—
501D4—42no—
502D4—1222no—
503D4—1222no—
504D4—2448no—
505D4—2448no—
506D4—2448no—
507D4—2448no—
508D4—2442no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—48no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—612no—
519C2×C4—612no—
520C2×C4—612no—
521C2×C4—612no—
522C2×C4—612no—
523C2×C4—1228no—
524C2×C4—1228no—
525C2×C4—1222no—
526C2×C4—1222no—
527C2×C4—1222no—
528C2×C4—1222no—
529C2×C4—1224no—
530C2×C4—1222no—
531C2×C4—1224no—
532C2×C4—2448no—
533C2×C4—24416no—
534C2×C4—2442no—
535C2×C4—2442no—
536C2×C4—2442no—
537C2×C4C22⋊C4×D6112yes—
538C2×C4C22⋊C4×D6112yes—
539C2×C4C22×C4×D6111yescharacteristic
540C2×C4C2×D4×D6111yescharacteristic
541C2×C4#41×D6112yes—
542C2×C4C2×#1175822212yes—
543C8—42no—
544C8—2442no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—82no—
549C22—82no—
550C22—614no—
551C22—1224no—
552C22—1224no—
553C22—1222no—
554C22—1222no—
555C22—4882no—
556C22—4882no—
557C22—2442no—
558C22—2442no—
559C22—2442no—
560C22C2×C22⋊C4×D6111yescharacteristic
561C22#1684027112yes—
562C22AGL(1,ℤ/8)×D6111yescharacteristic
563C22AGL(1,ℤ/8)×D6112yes—
564C22#137×D6111yescharacteristic
565C4—22no—
566C4—22no—
567C4—44no—
568C4—42no—
569C4—614no—
570C4—1222no—
571C4—1222no—
572C4—2444no—
573C4—2442no—
574C4#1684023011yescharacteristic
575C4C4×#1175822211yescharacteristic
576C4#1684030912yes—
577C2—82no—
578C2—1224no—
579C2—4882no—
580C2#2144830511yescharacteristic
581C2#418×D6111yescharacteristic
582C2#417×D6111yescharacteristic
583C2#2145590812yes—
584C2#2145593912yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1762C2
socleC23×C61C22×D4
prime radicalC2×#16840309C2
cosocleC22×C4×C61C24
hypercenter#1762D61
hypocenterC61#21395
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
2. C2×#16839817C2semidirect
3. C4×C4⋊C4×D61C2semidirect
4. #21446765C2semidirect
5. #21446744C2semidirect
6. #21446135C2semidirect
7. #21446133C2semidirect
8. #21445647C2semidirect
9. #21445645C2semidirect
10. #21445644C2semidirect
11. C61×#1762C2semidirect
14. #16840309C4semidirect
61. #14519776C2×C4semidirect
63. #14519773C2×C4semidirect
71. C4×D244C2×C4semidirect
102. C4×D4×C61C2×C4semidirect
166. C4⋊C4×C61#41semidirect
177. C4⋊Dic61#41semidirect
295. #1762D61semidirect
338. C61#21395semidirect
370. C4×D4C4×D61semidirect
468. C4⋊C4#11758222semidirect

Maximal subgroups

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

subgroupindex
C2×#168403092
C2×#168398172
C4×C4⋊C4×D612
#214467652
#214467442
#214461352
#214461332
#214456472
#214456452
#214456442
C61×#17622
#2139561