Finite Group Databasebeta

← #29714790

Subgroups — #29714790

Order 15616.

1394 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840432C211yesmaximal, characteristic
2C2×C4×#11758201C211yesmaximal, characteristic, complemented
3#21451189C211yesmaximal, characteristic, complemented
4#21447051C212yesmaximal, complemented
5#21446905C212yesmaximal, complemented
6#21446754C211yesmaximal, characteristic, complemented
7#21446626C211yesmaximal, characteristic, complemented
8#21446155C212yesmaximal, complemented
9#21446078C212yesmaximal, complemented
10#21445820C211yesmaximal, characteristic, complemented
11C2×C61×#460C211yesmaximal, characteristic, complemented
12C22×#11758224—21no—
13C22×#11758201C2211yescharacteristic
14#16840432—24no—
15#16840432—24no—
16C2×#14519773C2212yes—
17C2×#14519749—22no—
18C2×#14519748—21no—
19C2×C4×Dic122C2211yescharacteristic
20C2×C4×D244C2211yescharacteristic
21C2×D4×Dic61C2212yes—
22C4×#11758201—24no—
23C2×#14519708C2211yescharacteristic
24C2×#14519701C2211yescharacteristic
25C2×#14519699C2212yes—
26C2×#14519698C2212yes—
27C2×#14519698C2212yes—
28C2×#14519697C2211yescharacteristic
29C2×C4×#92418C2211yescharacteristic
30#16839695—22no—
31#16839669C2212yes—
32#16839668—21no—
33#16839660—21no—
34#16839658C2212yes—
35#16839632C2211yescharacteristic
36#16839525C2212yes—
37#16839516C2212yes—
38#16839516C2212yes—
39C22×SD16×C61C2212yes—
40C2×#127×C61C2211yescharacteristic
41C2×C4×Q8×C61C2211yescharacteristic
42C61×#460—24no—
43C2×C4⋊C8×C61C2211yescharacteristic
44C2×#102×C61C2212yes—
45C2×#11758224—28no—
46C2×#11758224—44no—
47C2×#11758224—44no—
48C22×Dic122D411yescharacteristic
49C22×D244C2311yescharacteristic
50C22×C22⋊D61—22no—
51C22×C22⋊D61—22no—
52C23×Dic61—22no—
53C22×C4×D61—21no—
54C2×#11758201—22no—
55C2×#11758201—22no—
56C2×#11758201—22no—
57C22×#92418C2311yescharacteristic
58C22×#92418C2311yescharacteristic
59#14519773—24no—
60#14519773—24no—
61#14519749—216no—
62#14519748—28no—
63C4×Dic122—22no—
64C4×Dic122—22no—
65C4×D244—22no—
66C4×D244D414yes—
67D4×Dic61—24no—
68D4×Dic61—24no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61D411yescharacteristic
72C2×C4⋊Dic61C2312yes—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758196—22no—
75C2×#11758196D412yes—
76C2×C22⋊Dic61—22no—
77C2×#11758198—21no—
78C2×C4×Dic61C2312yes—
79#14519708—22no—
80#14519701—22no—
81#14519699—42no—
82#14519698—24no—
83#14519698—42no—
84#14519697—22no—
85C4×#92418—22no—
86#14519684—22no—
87#14519684—22no—
88#14519684—22no—
89C22×Q8×C61C2311yescharacteristic
90C22×D4×C61—21no—
91C22×D4×C61—21no—
92C22×D4×C61C2312yes—
93C2×SD16×C61—24no—
94C2×SD16×C61—24no—
95C2×SD16×C61—42no—
96C22×C8×C61C2312yes—
97#127×C61—22no—
98#127×C61D414yes—
99C4×Q8×C61—22no—
100C4×Q8×C61—22no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61C2311yescharacteristic
103C2×C42×C61—21no—
104C2×C42×C61C2311yescharacteristic
105C4⋊C8×C61—22no—
106#102×C61—24no—
107C24×C61—22no—
108C23×D61—21no—
109C2×Q8×C61—21no—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C2×Q8×C61#4112yescomplemented
115C2×D4×C61—24no—
116C2×D4×C61—24no—
117C2×D4×C61—22no—
118C2×D4×C61—22no—
119C2×D4×C61—22no—
120C2×D4×C61—22no—
121C2×D4×C61—22no—
122C2×D4×C61—42no—
123C2×D4×C61—42no—
124C2×D4×C61—81no—
125C2×D4×C61—81no—
126C2×D4×C61C2×D414yes—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2411yescharacteristic
132C2×C22⋊D61—28no—
133C2×C22⋊D61—28no—
134C2×C22⋊D61—44no—
135C2×C22⋊D61—44no—
136C2×C22⋊D61—82no—
137C2×C22⋊D61—82no—
138C22×Dic61—28no—
139C22×Dic61—21no—
140C22×Dic61—22no—
141C22×Dic61—44no—
142C22×Dic61—44no—
143C22×Dic61C2×D412yes—
144C22×Dic61C2×D412yes—
145#11758224—432no—
146C2×D244—21no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244C2×D412yes—
152C2×C4×D61—24no—
153C2×C4×D61—42no—
154C2×C4×D61—42no—
155C2×Dic122—22no—
156C2×Dic122—22no—
157C2×Dic122SD1614yescomplemented
158SD16×C61—48no—
159C2×C8×C61—22no—
160C2×C8×C61—22no—
161C2×C8×C61—22no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—42no—
165C42×C61—42no—
166C42×C61C2×D412yes—
167C42×C61C2×D412yes—
168C22⋊Dic61—28no—
169#11758201—28no—
170#11758198—24no—
171C4⋊Dic61—22no—
172C4⋊Dic61—21no—
173C4⋊Dic61—22no—
174C4⋊Dic61—22no—
175C4⋊Dic61—42no—
176C4⋊Dic61—81no—
177C4⋊Dic61C2×D412yes—
178#11758196—24no—
179#11758196—82no—
180C4×Dic61—22no—
181C4×Dic61—22no—
182C2×#92418—21no—
183C2×#92418—21no—
184C2×#92418—21no—
185C2×#92418—21no—
186C2×#92418—21no—
187C2×#92418#4112yescomplemented
188C23×C61—24no—
189C23×C61—42no—
190C23×C61—42no—
191C23×C61—42no—
192C23×C61—42no—
193C23×C61—42no—
194C23×C61—42no—
195C23×C61C22×D411yescharacteristic
196C22×D61—22no—
197C22×D61—41no—
198C22×D61—41no—
199C22×D61—41no—
200C22×D61—41no—
201C22×D61—41no—
202C22×D61—41no—
203Q8×C61—22no—
204Q8×C61—22no—
205Q8×C61—41no—
206Q8×C61—41no—
207D4×C61—24no—
208D4×C61—24no—
209D4×C61—42no—
210D4×C61—42no—
211D4×C61—48no—
212D4×C61—84no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—22no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61#12012yes—
222C2×C4×C61#12012yes—
223C2×C4×C61C2×SD1612yes—
224C2×C4×C61AGL(1,ℤ/8)12yes—
225C2×C4×C61C22×D411yescharacteristic
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C2×#4111yescharacteristic
228C2×C4×C61C2×#4111yescharacteristic
229C2×C4×C61H(5,𝔽2)11yescharacteristic
230C2×C4×C612−1+411yescharacteristic
231C22⋊D61—416no—
232C22⋊D61—88no—
233C2×Dic61—22no—
234C2×Dic61—22no—
235C2×Dic61—22no—
236C2×Dic61—22no—
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—416no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61C2×SD1614yes—
246D244—22no—
247D244—22no—
248D244—41no—
249D244—41no—
250C4×D61—48no—
251Dic122—28no—
252C8×C61—42no—
253#92418—22no—
254#92418—41no—
255#24188—611nomaximal, Sylow
256C22×C61—44no—
257C22×C61—44no—
258C22×C61—44no—
259C22×C61—82no—
260C22×C61—82no—
261C22×C61—82no—
262C22×C61—82no—
263C22×C61C2×#12011yescharacteristic
264C22×C61#54811yescharacteristic
265C22×C61#55011yescharacteristic
266C22×C61C22×SD1611yescharacteristic
267C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
268C22×C61#57611yescharacteristic
269C22×C61#57711yescharacteristic
270C2×D61—42no—
271C2×D61—42no—
272C2×D61—42no—
273C2×D61—81no—
274C2×D61—81no—
275C2×D61—81no—
276C2×D61—81no—
277C4×C61—22no—
278C4×C61—42no—
279C4×C61—81no—
280C4×C61#45014yes—
281C4×C61#52511yescharacteristic
282C4×C61#53711yescharacteristic
283C4×C61#54611yescharacteristic
284C4×C61#54711yescharacteristic
285Dic61—24no—
286Dic61—42no—
287Dic61—42no—
288Dic61—82no—
289C2×#550—611no—
290C2×#460D6111yescharacteristic, complemented
291C2×C4×SD16—611no—
292#2582—611no—
293#2041—612no—
294#2014—612no—
295#1984—611no—
296#1957—611no—
297#1877—612no—
298#1864—612no—
299#1802—611no—
300C2×C61—84no—
301C2×C61#240711yescharacteristic
302C2×C61C2×#45011yescharacteristic
303C2×C61#301211yescharacteristic
304C2×C61#336211yescharacteristic
305C2×C61#336811yescharacteristic
306C2×C61#339211yescharacteristic
307C2×C61#339911yescharacteristic
308D61—82no—
309C22×#41—1221no—
310C22×SD16—611no—
311C22×SD16C2×D6112yes—
312#550—1224no—
313#550—1224no—
314C2×#127—612no—
315C2×#127C2×D6111yescharacteristic
316C2×#121—1221no—
317C2×#121—1222no—
318C2×C4×Q8—611no—
319C2×C4×Q8C2×D6111yescharacteristic
320C2×C4×D4—612no—
321C2×C4×D4—611no—
322#460—24no—
323C4×SD16—1224no—
324C2×#106—611no—
325C2×C4⋊C8C2×D6111yescharacteristic
326C2×#103—611no—
327C2×#102—612no—
328C2×#102—611no—
329C2×#102—612no—
330C2×#102—612no—
331C2×#102C2×D6112yes—
332C2×C4×C8—611no—
333#398—1222no—
334#391—612no—
335#391—1221no—
336#389—612no—
337#389—1221no—
338#384—611no—
339#340—612no—
340#336—612no—
341#336—612no—
342C61#2418811yescharacteristic, complemented, Sylow
343C2×#41—1228no—
344C2×#41—2444no—
345C2×#41—2444no—
346C22×Q8—611no—
347C22×Q8C22×D6111yescharacteristic
348C22×D4—21no—
349C22×D4—21no—
350C22×D4—611no—
351C22×D4—1222no—
352C22×D4—1222no—
353C22×D4C22×D6112yes—
354C23×C4—1222no—
355C23×C4—1221no—
356C2×SD16—24no—
357C2×SD16—24no—
358C2×SD16—42no—
359C2×SD16—1222no—
360C2×SD16—1222no—
361C2×SD16—1222no—
362C22×C8—611no—
363C22×C8—611no—
364C22×C8C22×D6112yes—
365#127—22no—
366#127—1224no—
367#127—1224no—
368#127C22⋊D6114yes—
369#121—1228no—
370#121—12216no—
371C4×Q8—22no—
372C4×Q8—22no—
373C4×Q8—1222no—
374C4×Q8—1222no—
375C4×D4—614no—
376C4×D4—1224no—
377C4×D4—1224no—
378C4×D4—1222no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—611no—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—612no—
383C2×C4⋊C4—612no—
384C2×C4⋊C4—1222no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4C22×D6111yescharacteristic
388C2×C22⋊C4—1222no—
389C2×C22⋊C4—1221no—
390C2×C42—21no—
391C2×C42—612no—
392C2×C42C22×D6111yescharacteristic
393#106—1222no—
394C4⋊C8—22no—
395#103—1222no—
396#102—24no—
397#102—1222no—
398#102—1224no—
399#102—2442no—
400#102—2442no—
401C4×C8—1222no—
402#95—1222no—
403#95—1222no—
404#95—1222no—
405C24—22no—
406C24—1221no—
407#41—24432no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—614no—
414C2×Q8—1222no—
415C2×Q8—1222no—
416C2×Q8#1175822412yescomplemented
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—81no—
427C2×D4—81no—
428C2×D4—612no—
429C2×D4—1228no—
430C2×D4—1228no—
431C2×D4—1221no—
432C2×D4—1221no—
433C2×D4—1221no—
434C2×D4—1221no—
435C2×D4—1221no—
436C2×D4—4882no—
437C2×D4—4882no—
438C2×D4—2444no—
439C2×D4—2444no—
440C2×D4C2×C22⋊D6114yes—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—612no—
444C22×C4—612no—
445C22×C4—1228no—
446C22×C4—1221no—
447C22×C4—1224no—
448C22×C4—1222no—
449C22×C4—2444no—
450C22×C4—2444no—
451C22×C4—2442no—
452C22×C4—2442no—
453C22×C4D4×D6111yescharacteristic
454C22×C4D4×D6111yescharacteristic
455C22×C4C23×D6111yescharacteristic
456SD16—48no—
457SD16—1228no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—612no—
462C2×C8—1221no—
463C2×C8—1221no—
464C2×C8—1221no—
465C2×C8—1221no—
466C2×C8—1221no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—42no—
470C4⋊C4—612no—
471C4⋊C4—1221no—
472C4⋊C4—1224no—
473C4⋊C4—1222no—
474C4⋊C4—1222no—
475C4⋊C4—1222no—
476C4⋊C4—4881no—
477C4⋊C4—4882no—
478C4⋊C4—2442no—
479C22⋊C4—1228no—
480C22⋊C4—1224no—
481C42—42no—
482C42—1222no—
483C42—1222no—
484C42D4×D6112yes—
485C42C2×C22⋊D6112yes—
486C23—24no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—1222no—
494C23—2441no—
495C23—2441no—
496C23—2441no—
497C23—2441no—
498C23—2441no—
499C23—2441no—
500C23C2×D4×D6111yescharacteristic
501Q8—22no—
502Q8—22no—
503Q8—41no—
504Q8—41no—
505Q8—1228no—
506D4—24no—
507D4—24no—
508D4—42no—
509D4—42no—
510D4—48no—
511D4—84no—
512D4—1222no—
513D4—1222no—
514D4—4888no—
515D4—24416no—
516D4—2441no—
517D4—2441no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—614no—
527C2×C4—1222no—
528C2×C4—1222no—
529C2×C4—1222no—
530C2×C4—1222no—
531C2×C4—1222no—
532C2×C4—1222no—
533C2×C4—24416no—
534C2×C4—2442no—
535C2×C4—2442no—
536C2×C4—2442no—
537C2×C4—2441no—
538C2×C4—2441no—
539C2×C4—2448no—
540C2×C4#1451974512yes—
541C2×C4#1451974512yes—
542C2×C4SD16×D6112yes—
543C2×C4#1451980412yes—
544C2×C4C2×D4×D6111yescharacteristic
545C2×C4C22×C22⋊D6111yescharacteristic
546C2×C4#41×D6111yescharacteristic
547C2×C4C2×#1175822411yescharacteristic
548C2×C4#1451982211yescharacteristic
549C2×C4#1451982411yescharacteristic
550C8—42no—
551C8—1222no—
552C8—2441no—
553C22—44no—
554C22—44no—
555C22—44no—
556C22—82no—
557C22—82no—
558C22—82no—
559C22—82no—
560C22—4881no—
561C22—4881no—
562C22—4881no—
563C22—4881no—
564C22—2442no—
565C22—2442no—
566C22—2442no—
567C22C2×#1451974511yescharacteristic
568C22#1684041811yescharacteristic
569C22#1684043011yescharacteristic
570C22C2×SD16×D6111yescharacteristic
571C22C2×#1451980411yescharacteristic
572C22#1684056611yescharacteristic
573C22#1684057211yescharacteristic
574C4—22no—
575C4—42no—
576C4—81no—
577C4—1224no—
578C4—4882no—
579C4—2442no—
580C4—2442no—
581C4#1683991114yes—
582C4#1684027211yescharacteristic
583C4#1684034111yescharacteristic
584C4#1684040611yescharacteristic
585C4#1684041111yescharacteristic
586C2—84no—
587C2—4882no—
588C2#2144916011yescharacteristic
589C2C2×#1683991111yescharacteristic
590C2#2145512211yescharacteristic
591C2#2145848511yescharacteristic
592C2#2145855211yescharacteristic
593C2#2145879911yescharacteristic
594C2#2145886011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#460C2
socleC23×C61C22×D4
prime radicalC2×#16840432C2
hypercenterC2×#460D61
hypocenterC61#24188
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#460

Complements

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

NH ≅ G/Ntype
2. C2×C4×#11758201C2semidirect
3. #21451189C2semidirect
4. #21447051C2semidirect
5. #21446905C2semidirect
6. #21446754C2semidirect
7. #21446626C2semidirect
8. #21446155C2semidirect
9. #21446078C2semidirect
10. #21445820C2semidirect
11. C2×C61×#460C2semidirect
114. C2×Q8×C61#41semidirect
157. C2×Dic122SD16semidirect
187. C2×#92418#41semidirect
290. C2×#460D61semidirect
342. C61#24188semidirect
416. C2×Q8#11758224semidirect

Maximal subgroups

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

subgroupindex
C2×#168404322
C2×C4×#117582012
#214511892
#214470512
#214469052
#214467542
#214466262
#214461552
#214460782
#214458202
C2×C61×#4602
#2418861