Finite Group Databasebeta

← #29685223

Subgroups — #29685223

Order 15616.

1422 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840309C211yesmaximal, characteristic
2C2×C4×#11758189C211yesmaximal, characteristic, complemented
3#21448215C211yesmaximal, characteristic, complemented
4#21446790C212yesmaximal, complemented
5#21446641C212yesmaximal, complemented
6#21446256C212yesmaximal, complemented
7#21446254C212yesmaximal, complemented
8C4×#14519697C211yesmaximal, characteristic, complemented
9C4×#14519699C211yesmaximal, characteristic, complemented
10C4×#14519698C211yesmaximal, characteristic, complemented
11C4×#102×C61C211yesmaximal, characteristic, complemented
12C22×#11758222—21no—
13C22×#11758189C2212yes—
14#16840309—24no—
15#16840309C418yescomplemented
16C2×#14519776C2211yescharacteristic
17C2×#14519773C2211yescharacteristic
18C2×#14519763—21no—
19C2×#14519752C2211yescharacteristic
20C2×#14519750—21no—
21C2×#14519748C2211yescharacteristic
22C2×C4×D244C2211yescharacteristic
23C2×C4×C22⋊D61—21no—
24C2×C42×D61C2211yescharacteristic
25C2×#14519699C2212yes—
26C2×#14519698C2212yes—
27C2×#14519697C2212yes—
28C4×#11758189—24no—
29C2×#14519687C2212yes—
30C2×C4×#92418C2211yescharacteristic
31#16839629—21no—
32#16839627C2212yes—
33#16839624—21no—
34#16839619C2212yes—
35C4×C4⋊Dic61C2211yescharacteristic
36C4×#11758198—21no—
37#16839608—21no—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×C4×D4×C61C2211yescharacteristic
42C2×#102×C61C2212yes—
43C2×C4×C8×C61C2211yescharacteristic
44C4×C4⋊C4×C61C2211yescharacteristic
45C61×#340C2212yes—
46C2×#11758222—28no—
47C2×#11758222—44no—
48C2×#11758222—44no—
49C22×Dic122—21no—
50C22×D244—21no—
51C22×D244C2311yescharacteristic
52C22×C22⋊D61—21no—
53C22×C22⋊D61—21no—
54C22×C22⋊D61—21no—
55C22×C4×D61—21no—
56C22×C4×D61D411yescharacteristic
57C22×C4×D61D411yescharacteristic
58C22×C4×D61C2311yescharacteristic
59C2×#11758189—24no—
60C2×#11758189—24no—
61C2×#11758189—24no—
62C22×#92418C2312yes—
63#14519776—22no—
64#14519776C2×C414yes—
65#14519773—22no—
66#14519773C2×C414yes—
67#14519763—28no—
68#14519752—22no—
69#14519752C2×C414yes—
70#14519750—28no—
71#14519748—22no—
72#14519748C2×C414yes—
73C4×D244—22no—
74C4×D244C2×C414yes—
75C4×C22⋊D61—28no—
76C2×C4⋊Dic61C2312yes—
77C2×C4⋊Dic61C2311yescharacteristic
78C2×#11758196—21no—
79C2×C22⋊Dic61—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—22no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61D411yescharacteristic
85C2×C4×Dic61D411yescharacteristic
86C2×C4×Dic61C2311yescharacteristic
87C42×D61C2×C414yes—
88C42×D61D414yes—
89#14519699—24no—
90#14519698—24no—
91#14519697—24no—
92#14519687—24no—
93C4×#92418—22no—
94#14519684—21no—
95#14519684—22no—
96#14519684—22no—
97#14519684—22no—
98C22×D4×C61C2311yescharacteristic
99C23×C4×C61—21no—
100C22×C8×C61C2312yes—
101C4×D4×C61—22no—
102C4×D4×C61C2×C414yes—
103C2×C4⋊C4×C61C2312yes—
104C2×C4⋊C4×C61C2311yescharacteristic
105C2×C22⋊C4×C61—21no—
106C2×C42×C61—21no—
107C2×C42×C61C2311yescharacteristic
108#102×C61—24no—
109C4×C8×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—21no—
119C22×C4×C61—24no—
120C22×C4×C61—22no—
121C22×C4×C61—42no—
122C22×C4×C61—42no—
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2411yescharacteristic
126C2×C22⋊D61—24no—
127C2×C22⋊D61—24no—
128C2×C22⋊D61—24no—
129C2×C22⋊D61—44no—
130C2×C22⋊D61—42no—
131C2×C22⋊D61—42no—
132C2×C22⋊D61—42no—
133C2×C22⋊D61—42no—
134C22×Dic61—21no—
135C22×Dic61—22no—
136C22×Dic61C2×D411yescharacteristic
137C22×Dic61C2×D411yescharacteristic
138C22×Dic61C2×D411yescharacteristic
139#11758222—432no—
140C2×D244—22no—
141C2×D244—22no—
142C2×D244—24no—
143C2×D244—21no—
144C2×D244—42no—
145C2×D244—42no—
146C2×D244C22×C412yes—
147C2×C4×D61—22no—
148C2×C4×D61—22no—
149C2×C4×D61—22no—
150C2×C4×D61—22no—
151C2×C4×D61—24no—
152C2×C4×D61—42no—
153C2×C4×D61—42no—
154C2×C4×D61—42no—
155C2×C4×D61C22⋊C414yes—
156C2×C4×D61C22⋊C414yes—
157C2×C4×D61C22×C412yes—
158C2×C4×D61C2×D412yes—
159C2×Dic122—24no—
160C2×Dic122—42no—
161C2×Dic122—42no—
162C2×C8×C61—22no—
163C2×C8×C61—22no—
164C2×C8×C61#4114yescomplemented
165C4⋊C4×C61—21no—
166C4⋊C4×C61—22no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61C22×C412yes—
169C22⋊C4×C61—24no—
170C42×C61—42no—
171C42×C61C22×C412yes—
172C42×C61C2×D412yes—
173C22⋊Dic61—24no—
174#11758198—24no—
175#11758198—24no—
176#11758198—44no—
177C4⋊Dic61—21no—
178C4⋊Dic61—22no—
179C4⋊Dic61—22no—
180C4⋊Dic61C22×C412yes—
181#11758196—24no—
182C4×Dic61—42no—
183C4×Dic61C22⋊C414yes—
184C4×Dic61C22⋊C414yes—
185C4×Dic61C22×C412yes—
186C4×Dic61C2×D412yes—
187C2×#92418—22no—
188C2×#92418—22no—
189C2×#92418#4114yescomplemented
190#11758189—216no—
191C23×C61—22no—
192C23×C61—42no—
193C23×C61—41no—
194C23×C61—41no—
195C23×C61—42no—
196C23×C61C22×D411yescharacteristic
197C22×D61—22no—
198C22×D61—21no—
199C22×D61—21no—
200C22×D61—22no—
201C22×D61—22no—
202C22×D61—42no—
203C22×D61—41no—
204C22×D61—41no—
205C22×D61—42no—
206C22×D61C2×C22⋊C412yes—
207D4×C61—22no—
208D4×C61—22no—
209D4×C61—42no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—48no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61C2×C22⋊C412yes—
219C2×C4×C61C2×C22⋊C412yes—
220C2×C4×C61AGL(1,ℤ/8)12yes—
221C2×C4×C61AGL(1,ℤ/8)12yes—
222C2×C4×C61C23×C411yescharacteristic
223C2×C4×C61C22×D411yescharacteristic
224C2×C4×C61C2×#4112yes—
225C2×C4×C61C2×#4112yes—
226C22⋊D61—48no—
227C22⋊D61—48no—
228C22⋊D61—48no—
229C2×Dic61—22no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—42no—
237C2×Dic61—42no—
238C2×Dic61—42no—
239C2×Dic61C2×C22⋊C412yes—
240C2×Dic61C2×C22⋊C412yes—
241C2×Dic61C2×C22⋊C412yes—
242D244—22no—
243D244—22no—
244D244—48no—
245D244—42no—
246C4×D61—22no—
247C4×D61—22no—
248C4×D61—24no—
249C4×D61—28no—
250C4×D61—44no—
251C4×D61—48no—
252Dic122—48no—
253C8×C61—24no—
254#92418—24no—
255#21405—611nomaximal, Sylow
256C22×C61—42no—
257C22×C61—42no—
258C22×C61—42no—
259C22×C61—82no—
260C22×C61—82no—
261C22×C61C22×C22⋊C411yescharacteristic
262C22×C61#53212yes—
263C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
264C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
265C22×C61#57512yes—
266C2×D61—22no—
267C2×D61—22no—
268C2×D61—22no—
269C2×D61—42no—
270C2×D61—42no—
271C2×D61—42no—
272C2×D61—42no—
273C2×D61—42no—
274C2×D61—82no—
275C2×D61—82no—
276C4×C61—22no—
277C4×C61—42no—
278C4×C61—82no—
279C4×C61#41814yes—
280C4×C61C4×#4111yescharacteristic
281C4×C61C4×#4111yescharacteristic
282C4×C61#52512yes—
283Dic61—22no—
284Dic61—22no—
285Dic61—22no—
286Dic61—42no—
287Dic61—82no—
288C2×#532—611no—
289C2×C4×M16—611no—
290#2283—611no—
291#1993—612no—
292#1961—612no—
293#1895—612no—
294#1895—612no—
295C4×#102—611no—
296C4×#102—611no—
297C4×#102—611no—
298C4×#102D6111yescharacteristic, complemented
299C2×C61—82no—
300C2×C61#229711yescharacteristic
301C2×C61C2×#41811yescharacteristic
302C2×C61#289611yescharacteristic
303C2×C61#317112yes—
304C2×C61#317112yes—
305D61—42no—
306D61—82no—
307C22×#41—1221no—
308C22×M16—612no—
309#532—618no—
310#532—1224no—
311C2×C4⋊Q8—611no—
312C2×#127—611no—
313C2×#124—1221no—
314C2×#121—611no—
315C2×#121—611no—
316C2×#121—1221no—
317C2×C4×D4—611no—
318C2×C4×D4—1221no—
319C2×C4×D4C2×D6111yescharacteristic
320C22×C42—611no—
321C2×#102—612no—
322C2×#102—612no—
323C2×#102—612no—
324C2×#102C2×D6112yes—
325C4×M16—1224no—
326C2×#97—612no—
327C2×C4×C8—611no—
328C2×C4×C8C2×D6111yescharacteristic
329#383—1221no—
330#382—612no—
331#381—1221no—
332#380—612no—
333C4×C4⋊C4—611no—
334C4×C4⋊C4C2×D6111yescharacteristic
335C4×C22⋊C4—1221no—
336#376—1221no—
337#340—612no—
338#340—612no—
339#340—612no—
340#340C2×D6112yes—
341C61#2140511yescharacteristic, complemented, Sylow
342C2×#41—1228no—
343C2×#41—2444no—
344C2×#41—2444no—
345C22×Q8—1221no—
346C22×D4—611no—
347C22×D4—1221no—
348C22×D4—1221no—
349C22×D4—1221no—
350C22×D4—1221no—
351C22×D4C22×D6111yescharacteristic
352C23×C4—21no—
353C23×C4—611no—
354C23×C4—611no—
355C23×C4—611no—
356C23×C4—1221no—
357C2×M16—1224no—
358C2×M16—1224no—
359C2×M16—1224no—
360C22×C8—612no—
361C22×C8C22×D6112yes—
362C4⋊Q8—614no—
363C4⋊Q8—1222no—
364#127—614no—
365#127—1222no—
366#124—1228no—
367#121—614no—
368#121—614no—
369#121—1228no—
370#121—1222no—
371#121—1222no—
372C4×D4—22no—
373C4×D4—614no—
374C4×D4—1228no—
375C4×D4—1222no—
376C4×D4C4×D6114yes—
377C2×C4⋊C4—611no—
378C2×C4⋊C4—612no—
379C2×C4⋊C4—1221no—
380C2×C4⋊C4C22×D6112yes—
381C2×C4⋊C4C22×D6111yescharacteristic
382C2×C22⋊C4—21no—
383C2×C22⋊C4—1221no—
384C2×C22⋊C4—1221no—
385C2×C22⋊C4—1221no—
386C2×C22⋊C4—1222no—
387C2×C42—21no—
388C2×C42—611no—
389C2×C42—611no—
390C2×C42—611no—
391C2×C42—614no—
392C2×C42—614no—
393C2×C42—1221no—
394C2×C42C22×D6111yescharacteristic
395#102—24no—
396#102—1224no—
397#102—1224no—
398#102—1224no—
399#97—1224no—
400C4×C8—22no—
401C4×C8—1222no—
402#95—21no—
403#95—1222no—
404#95—1222no—
405#95—1222no—
406#95—1221no—
407C24—21no—
408C24—611no—
409C24—1221no—
410#41—24432no—
411C2×Q8—1224no—
412C2×Q8—2442no—
413C2×Q8—2442no—
414C2×D4—21no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—612no—
418C2×D4—1224no—
419C2×D4—1224no—
420C2×D4—1224no—
421C2×D4—1224no—
422C2×D4—1221no—
423C2×D4—1222no—
424C2×D4—1222no—
425C2×D4—2442no—
426C2×D4—2442no—
427C2×D4—2442no—
428C2×D4—2442no—
429C2×D4—2444no—
430C2×D4—2442no—
431C2×D4—2442no—
432C2×D4C2×C4×D6112yes—
433C22×C4—21no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—611no—
439C22×C4—611no—
440C22×C4—611no—
441C22×C4—612no—
442C22×C4—612no—
443C22×C4—614no—
444C22×C4—614no—
445C22×C4—1222no—
446C22×C4—1222no—
447C22×C4—1222no—
448C22×C4—1222no—
449C22×C4—1221no—
450C22×C4—1224no—
451C22×C4—1222no—
452C22×C4—2442no—
453C22×C4—2442no—
454C22×C4—2442no—
455C22×C4D4×D6111yescharacteristic
456C22×C4D4×D6111yescharacteristic
457C22×C4C23×D6111yescharacteristic
458M16—12216no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—614no—
462C2×C8—1222no—
463C2×C8—1222no—
464C2×C8#1175822214yescomplemented
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—612no—
469C4⋊C4—1224no—
470C4⋊C4—1221no—
471C4⋊C4—1222no—
472C4⋊C4—1222no—
473C4⋊C4C2×C4×D6112yes—
474C22⋊C4—24no—
475C22⋊C4—1224no—
476C22⋊C4—1224no—
477C22⋊C4—1224no—
478C22⋊C4—2444no—
479C42—42no—
480C42—612no—
481C42—612no—
482C42—614no—
483C42—614no—
484C42—2442no—
485C42C2×C4×D6112yes—
486C42D4×D6112yes—
487C23—22no—
488C23—41no—
489C23—41no—
490C23—42no—
491C23—42no—
492C23—612no—
493C23—1221no—
494C23—1221no—
495C23—1222no—
496C23—1222no—
497C23—1222no—
498C23—2441no—
499C23—2441no—
500C23—2442no—
501C23—2442no—
502C23C2×D4×D6111yescharacteristic
503Q8—2448no—
504D4—22no—
505D4—22no—
506D4—42no—
507D4—1222no—
508D4—1222no—
509D4—2448no—
510D4—2448no—
511D4—2448no—
512D4—2448no—
513D4—2442no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—48no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—612no—
523C2×C4—612no—
524C2×C4—612no—
525C2×C4—1221no—
526C2×C4—1221no—
527C2×C4—1222no—
528C2×C4—1222no—
529C2×C4—1222no—
530C2×C4—1222no—
531C2×C4—1224no—
532C2×C4—1228no—
533C2×C4—1222no—
534C2×C4—2444no—
535C2×C4—2442no—
536C2×C4—2442no—
537C2×C4—2442no—
538C2×C4—2441no—
539C2×C4—2441no—
540C2×C4—2448no—
541C2×C4C22⋊C4×D6112yes—
542C2×C4C22⋊C4×D6112yes—
543C2×C4#1451980112yes—
544C2×C4#1451980412yes—
545C2×C4C22×C4×D6111yescharacteristic
546C2×C4C2×D4×D6111yescharacteristic
547C2×C4#41×D6112yes—
548C2×C4C2×#1175822212yes—
549C8—24no—
550C8—1224no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—82no—
555C22—82no—
556C22—1222no—
557C22—1222no—
558C22—1222no—
559C22—4882no—
560C22—4882no—
561C22—2442no—
562C22—2442no—
563C22—2442no—
564C22—2442no—
565C22—2442no—
566C22C2×C22⋊C4×D6111yescharacteristic
567C22#1684030912yes—
568C22C2×#1451980111yescharacteristic
569C22C2×#1451980411yescharacteristic
570C22#1684055912yes—
571C4—22no—
572C4—42no—
573C4—82no—
574C4—1222no—
575C4—1222no—
576C4—1222no—
577C4—4882no—
578C4—2442no—
579C4#1683976514yes—
580C4#1684023011yescharacteristic
581C4C4×#1175822211yescharacteristic
582C4#1684027112yes—
583C2—82no—
584C2—4882no—
585C2—2442no—
586C2#2144830511yescharacteristic
587C2C2×#1683976511yescharacteristic
588C2#2145433411yescharacteristic
589C2#2145641012yes—
590C2#2145641212yes—
subgroupG/H
centerC22×C4D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC4×#102×C61C2
socleC23×C61C22×D4
prime radicalC2×#16840309C2
cosocleC22×C4×C61C24
hypercenterC4×#102D61
hypocenterC61#21405
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
2. C2×C4×#11758189C2semidirect
3. #21448215C2semidirect
4. #21446790C2semidirect
5. #21446641C2semidirect
6. #21446256C2semidirect
7. #21446254C2semidirect
8. C4×#14519697C2semidirect
9. C4×#14519699C2semidirect
10. C4×#14519698C2semidirect
11. C4×#102×C61C2semidirect
15. #16840309C4semidirect
164. C2×C8×C61#41semidirect
189. C2×#92418#41semidirect
298. C4×#102D61semidirect
341. C61#21405semidirect
464. C2×C8#11758222semidirect

Maximal subgroups

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

subgroupindex
C2×#168403092
C2×C4×#117581892
#214482152
#214467902
#214466412
#214462562
#214462542
C4×#145196972
C4×#145196992
C4×#145196982
C4×#102×C612
#2140561