Finite Group Databasebeta

← #29708014

Subgroups — #29708014

Order 15616.

1674 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519802C211yesmaximal, characteristic
2C2×#16839785C211yesmaximal, characteristic, complemented
3C2×#16839768C211yesmaximal, characteristic, complemented
4#21448310C211yesmaximal, characteristic, complemented
5C2×#16839663C211yesmaximal, characteristic, complemented
6#21447133C214yesmaximal, complemented
7#21447128C214yesmaximal, complemented
8#21445538C211yesmaximal, characteristic, complemented
9C2×C61×#417C211yesmaximal, characteristic, complemented
10C23×D244C2211yescharacteristic
11C2×#14519802—24no—
12C2×#14519802—24no—
13C2×#14519802—24no—
14C22×#11758201C2212yes—
15C22×#11758199C2212yes—
16C22×#11758194C2211yescharacteristic
17C2×#14519736C2211yescharacteristic
18C22×#11758198—22no—
19C22×C4×Dic61C2211yescharacteristic
20#16839785—24no—
21C2×#14519706C2214yescomplemented
22#16839768—24no—
23C2×#14519699C2214yes—
24#16839669C2212yescomplemented
25#16839665C2212yescomplemented
26#16839663—21no—
27#16839663—21no—
28#16839663C2214yes—
29#16839642—22no—
30C4×C22⋊Dic61—21no—
31C4×C22⋊Dic61—21no—
32#16839524C2214yescomplemented
33C22×#41×C61C2211yescharacteristic
34C22×M16×C61C2211yescharacteristic
35C22×C4⋊C4×C61C2211yescharacteristic
36C61×#417—24no—
37C2×#103×C61C2212yescomplemented
38C2×#102×C61C2212yescomplemented
39C24×D61—21no—
40C22×D244—22no—
41C22×D244—22no—
42C22×D244—22no—
43C22×D244—22no—
44C22×D244—21no—
45C22×D244—22no—
46C22×D244D414yes—
47C22×D244C2312yes—
48C23×Dic61D412yes—
49#14519802—232no—
50C2×#11758201—24no—
51C2×#11758201—24no—
52C2×#11758201—24no—
53C2×#11758199—24no—
54C2×#11758199—24no—
55C2×#11758199—24no—
56C2×#11758194—22no—
57C2×#11758194—22no—
58C2×#11758194D414yes—
59C22×#92418C2312yes—
60#14519736—22no—
61#14519736—22no—
62C2×C4⋊Dic61C2312yes—
63C2×C22⋊Dic61—21no—
64C2×C22⋊Dic61—22no—
65C2×C22⋊Dic61—22no—
66C2×#11758198—24no—
67C2×#11758198—24no—
68C2×#11758198—48no—
69C2×C4×Dic61—22no—
70C2×C4×Dic61—22no—
71C2×C4×Dic61—21no—
72C2×C4×Dic61D411yescharacteristic
73C2×C4×Dic61D411yescharacteristic
74C2×C4×Dic61C2312yes—
75#14519706—28no—
76#14519699—28no—
77#14519684—22no—
78#14519684—22no—
79C2×#41×C61—22no—
80C2×#41×C61—22no—
81C2×#41×C61—22no—
82C2×#41×C61—22no—
83C2×#41×C61—22no—
84C2×#41×C61D414yes—
85C22×Q8×C61C2311yescharacteristic
86C22×D4×C61—21no—
87C22×D4×C61C2311yescharacteristic
88C23×C4×C61—21no—
89C23×C4×C61—21no—
90C23×C4×C61C2311yescharacteristic
91C2×M16×C61—22no—
92C2×M16×C61—22no—
93C2×M16×C61—22no—
94C22×C8×C61C2312yes—
95C2×C4⋊C4×C61—22no—
96C2×C4⋊C4×C61—22no—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61C2312yes—
99#103×C61—24no—
100#102×C61—24no—
101C24×C61—21no—
102C24×C61C2×D411yescharacteristic
103C23×D61—24no—
104C23×D61—22no—
105C23×D61—44no—
106C23×D61—44no—
107C23×D61—42no—
108C23×D61—42no—
109#41×C61—28no—
110#41×C61—28no—
111#41×C61—44no—
112#41×C61—44no—
113C2×Q8×C61—21no—
114C2×Q8×C61—21no—
115C2×Q8×C61—21no—
116C2×Q8×C61—21no—
117C2×Q8×C61—21no—
118C2×Q8×C61C2×D412yes—
119C2×D4×C61—24no—
120C2×D4×C61—21no—
121C2×D4×C61—21no—
122C2×D4×C61—21no—
123C2×D4×C61—21no—
124C2×D4×C61—21no—
125C2×D4×C61—42no—
126C2×D4×C61—42no—
127C2×D4×C61C2×D412yes—
128C22×C4×C61—21no—
129C22×C4×C61—24no—
130C22×C4×C61—22no—
131C22×C4×C61—42no—
132C22×C4×C61—42no—
133C22×C4×C61—44no—
134C22×C4×C61—42no—
135C22×C4×C61C2×D412yes—
136C22×C4×C61C2×D412yes—
137C22×C4×C61C2×D412yes—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61#4112yes—
140C22×C4×C61#4112yes—
141C22×C4×C61#4112yes—
142C22×C4×C61C2411yescharacteristic
143C22×Dic61—24no—
144C22×Dic61—24no—
145C22×Dic61—22no—
146C22×Dic61—42no—
147C22×Dic61C2×D412yes—
148C22×Dic61C2×D412yes—
149C2×D244—28no—
150C2×D244—28no—
151C2×D244—22no—
152C2×D244—22no—
153C2×D244—22no—
154C2×D244—22no—
155C2×D244—24no—
156C2×D244—22no—
157C2×D244—42no—
158C2×D244—42no—
159C2×D244—44no—
160C2×D244—44no—
161C2×D244C2×D414yes—
162M16×C61—44no—
163C2×C8×C61—22no—
164C2×C8×C61—22no—
165C2×C8×C61—22no—
166C4⋊C4×C61—22no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—42no—
169C22⋊Dic61—44no—
170C22⋊Dic61—42no—
171C22⋊Dic61—44no—
172#11758201—216no—
173#11758199—216no—
174#11758198—48no—
175#11758198—48no—
176C4⋊Dic61—22no—
177C4⋊Dic61—22no—
178C4×Dic61—22no—
179C4×Dic61—22no—
180C4×Dic61—22no—
181C4×Dic61—22no—
182C4×Dic61—42no—
183#11758194—28no—
184C2×#92418—22no—
185C2×#92418—22no—
186C2×#92418C2×D414yes—
187C23×C61—22no—
188C23×C61—22no—
189C23×C61—22no—
190C23×C61—41no—
191C23×C61—41no—
192C23×C61—42no—
193C23×C61—42no—
194C23×C61#12112yes—
195C23×C61#12112yes—
196C23×C61#12112yes—
197C23×C61C22×D411yescharacteristic
198C22×D61—24no—
199C22×D61—416no—
200C22×D61—42no—
201C22×D61—42no—
202C22×D61—44no—
203C22×D61—44no—
204C22×D61—48no—
205C22×D61—84no—
206C22×D61—84no—
207C22×D61—88no—
208Q8×C61—22no—
209Q8×C61—22no—
210Q8×C61—41no—
211Q8×C61—41no—
212D4×C61—22no—
213D4×C61—22no—
214D4×C61—48no—
215D4×C61—41no—
216D4×C61—41no—
217C2×C4×C61—22no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—48no—
221C2×C4×C61—42no—
222C2×C4×C61—44no—
223C2×C4×C61—84no—
224C2×C4×C61#12014yes—
225C2×C4×C61#12112yes—
226C2×C4×C61#12112yes—
227C2×C4×C61#12112yes—
228C2×C4×C61#12314yes—
229C2×C4×C61#12314yes—
230C2×C4×C61#12414yes—
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—22no—
241C2×Dic61—22no—
242C2×Dic61—48no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246C2×Dic61AGL(1,ℤ/8)14yescomplemented
247D244—24no—
248D244—24no—
249D244—42no—
250D244—42no—
251D244—48no—
252C8×C61—42no—
253#92418—24no—
254#23546—611nomaximal, Sylow
255C22×C61—24no—
256C22×C61—22no—
257C22×C61—22no—
258C22×C61—24no—
259C22×C61—42no—
260C22×C61—42no—
261C22×C61—42no—
262C22×C61—82no—
263C22×C61—82no—
264C22×C61#39414yes—
265C22×C61C2×#12111yescharacteristic
266C22×C61C2×#12111yescharacteristic
267C22×C61C2×#12111yescharacteristic
268C22×C61C2×AGL(1,ℤ/8)12yes—
269C22×C61#57512yes—
270C2×D61—44no—
271C2×D61—48no—
272C2×D61—816no—
273C2×D61—82no—
274C2×D61—82no—
275C2×D61—84no—
276C4×C61—42no—
277C4×C61—82no—
278C4×C61#39414yes—
279C4×C61C2×#12011yescharacteristic
280C4×C61C2×#12311yescharacteristic
281C4×C61C2×#12311yescharacteristic
282C4×C61C2×#12411yescharacteristic
283Dic61—24no—
284Dic61—42no—
285Dic61—82no—
286C22×AGL(1,ℤ/8)—611no—
287C2×#423—611no—
288C2×#417D6111yescharacteristic, complemented
289C2×#418—611no—
290#2297—611no—
291C2×#390—611no—
292#2055—614no—
293#2054—614no—
294#1736—611no—
295C2×C61—24no—
296C2×C61—82no—
297C2×C61C2×#39411yescharacteristic
298C2×C61#303612yes—
299C2×C61#303811yescharacteristic
300C2×C61#304011yescharacteristic
301C2×C61#305512yes—
302D61—84no—
303C22×#41C2×D6111yescharacteristic
304C23×D4—611no—
305C2×AGL(1,ℤ/8)—1224no—
306C2×AGL(1,ℤ/8)—1224no—
307C2×AGL(1,ℤ/8)—1224no—
308C22×SD16—612no—
309C22×D8—612no—
310C22×M16—611no—
311C22×M16C2×D6111yescharacteristic
312C2×#117—611no—
313C22×C4⋊C4C2×D6111yescharacteristic
314C22×C22⋊C4—1222no—
315C22×C42—611no—
316#423—1224no—
317C2×C4⋊C8—614no—
318#418—1224no—
319#417—24no—
320C2×#103C2×D6112yescomplemented
321C2×#102—614no—
322C2×#102C2×D6112yescomplemented
323#391—612no—
324#390—612no—
325#390—614no—
326#390—1221no—
327#390—1221no—
328#386—1222no—
329C4×C22⋊C4—1221no—
330C4×C22⋊C4—1221no—
331#340—614no—
332C61#2354611yescharacteristic, complemented, Sylow
333C25—1221no—
334C2×#41—22no—
335C2×#41—22no—
336C2×#41—22no—
337C2×#41—22no—
338C2×#41—22no—
339C2×#41C22⋊D6114yes—
340C22×Q8C22×D6111yescharacteristic
341C22×D4—21no—
342C22×D4—614no—
343C22×D4—612no—
344C22×D4—1222no—
345C22×D4—1222no—
346C22×D4—1222no—
347C22×D4—1221no—
348C22×D4—1222no—
349C22×D4—1222no—
350C22×D4C22×D6111yescharacteristic
351C23×C4—21no—
352C23×C4—21no—
353C23×C4—612no—
354C23×C4C22×D6111yescharacteristic
355AGL(1,ℤ/8)—12232no—
356C2×SD16—1224no—
357C2×SD16—1224no—
358C2×SD16—1224no—
359C2×D8—1224no—
360C2×D8—1224no—
361C2×D8—1224no—
362C2×M16—22no—
363C2×M16—22no—
364C2×M16—22no—
365C2×M16—614no—
366C2×M16—1222no—
367C2×M16—1222no—
368C22×C8—612no—
369C22×C8C22×D6112yes—
370#117—1222no—
371#117—1222no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—612no—
376C2×C4⋊C4C22×D6112yes—
377C2×C22⋊C4—1224no—
378C2×C22⋊C4—1224no—
379C2×C22⋊C4—1222no—
380C2×C22⋊C4—1222no—
381C2×C22⋊C4—1221no—
382C2×C22⋊C4—2448no—
383C2×C42—611no—
384C2×C42—611no—
385C2×C42—612no—
386C2×C42—1222no—
387C2×C42—1222no—
388C2×C42—1221no—
389C4⋊C8—1228no—
390#103—24no—
391#102—24no—
392#102—1228no—
393#95—1222no—
394#95—1222no—
395C24—21no—
396C24—1222no—
397C24—1224no—
398C24—2444no—
399C24—2444no—
400C24—2442no—
401C24—2442no—
402C24D4×D6111yescharacteristic
403#41—28no—
404#41—28no—
405#41—44no—
406#41—44no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8C2×C22⋊D6112yes—
413C2×D4—24no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—614no—
422C2×D4—1222no—
423C2×D4—1222no—
424C2×D4—1222no—
425C2×D4—1222no—
426C2×D4—1222no—
427C2×D4—1228no—
428C2×D4—1224no—
429C2×D4—1228no—
430C2×D4—2442no—
431C2×D4—2442no—
432C2×D4—2444no—
433C2×D4—2444no—
434C2×D4C2×C22⋊D6112yes—
435C22×C4—21no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—44no—
441C22×C4—42no—
442C22×C4—612no—
443C22×C4—612no—
444C22×C4—1224no—
445C22×C4—1224no—
446C22×C4—1222no—
447C22×C4—2442no—
448C22×C4#1175822212yes—
449C22×C4#1175822212yes—
450C22×C4D4×D6112yes—
451C22×C4D4×D6112yes—
452C22×C4D4×D6111yescharacteristic
453C22×C4#1175822412yes—
454C22×C4C2×C22⋊D6112yes—
455C22×C4C23×D6111yescharacteristic
456SD16—12216no—
457D8—12216no—
458M16—44no—
459M16—1228no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—614no—
464C2×C8—1222no—
465C2×C8—1222no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—42no—
469C4⋊C4—1222no—
470C4⋊C4—1222no—
471C22⋊C4—2448no—
472C22⋊C4—2448no—
473C22⋊C4—2444no—
474C22⋊C4—2444no—
475C22⋊C4—2442no—
476C42—1222no—
477C42—1222no—
478C42—1222no—
479C42—1222no—
480C42—2442no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—41no—
485C23—41no—
486C23—42no—
487C23—42no—
488C23—1224no—
489C23—4888no—
490C23—4884no—
491C23—4884no—
492C23—24416no—
493C23—2442no—
494C23—2442no—
495C23—2444no—
496C23—2444no—
497C23—2448no—
498C23#1451974912yes—
499C23#1451975012yes—
500C23#1451975012yes—
501C23C2×D4×D6111yescharacteristic
502Q8—22no—
503Q8—22no—
504Q8—41no—
505Q8—41no—
506D4—22no—
507D4—22no—
508D4—48no—
509D4—41no—
510D4—41no—
511D4—1224no—
512D4—1224no—
513D4—2442no—
514D4—2442no—
515D4—2448no—
516C2×C4—22no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—48no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—84no—
523C2×C4—614no—
524C2×C4—1222no—
525C2×C4—1222no—
526C2×C4—1222no—
527C2×C4—1222no—
528C2×C4—1222no—
529C2×C4—2448no—
530C2×C4—2442no—
531C2×C4—2442no—
532C2×C4—2442no—
533C2×C4#1451974514yes—
534C2×C4#1451974912yes—
535C2×C4#1451975012yes—
536C2×C4#1451975012yes—
537C2×C4#1451975914yes—
538C2×C4#1451976014yes—
539C2×C4#1451976314yes—
540C2×C4C2×D4×D6111yescharacteristic
541C2×C4C2×D4×D6111yescharacteristic
542C2×C4C22×C22⋊D6111yescharacteristic
543C2×C4C2×#1175822411yescharacteristic
544C2×C4C2×#1175822211yescharacteristic
545C2×C4C2×#1175822211yescharacteristic
546C8—42no—
547C8—1224no—
548C22—24no—
549C22—24no—
550C22—22no—
551C22—22no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—82no—
556C22—82no—
557C22—48816no—
558C22—4884no—
559C22—4882no—
560C22—4882no—
561C22—2444no—
562C22—2448no—
563C22#1683967714yes—
564C22C2×#1451974911yescharacteristic
565C22C2×#1451975011yescharacteristic
566C22C2×#1451975011yescharacteristic
567C22AGL(1,ℤ/8)×D6112yes—
568C22#1684056112yes—
569C4—42no—
570C4—82no—
571C4—1224no—
572C4—4882no—
573C4—2442no—
574C4#1683967714yes—
575C4C2×#1451974511yescharacteristic
576C4C2×#1451975911yescharacteristic
577C4C2×#1451976011yescharacteristic
578C4C2×#1451976311yescharacteristic
579C2—24no—
580C2—82no—
581C2—4884no—
582C2C2×#1683967711yescharacteristic
583C2#2145528212yes—
584C2#2145529711yescharacteristic
585C2#2145531511yescharacteristic
586C2#2145544112yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#417C2
socleC23×C61C22×D4
prime radicalC22×#14519802C2
hypercenterC2×#417D61
hypocenterC61#23546
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
2. C2×#16839785C2semidirect
3. C2×#16839768C2semidirect
4. #21448310C2semidirect
5. C2×#16839663C2semidirect
6. #21447133C2semidirect
7. #21447128C2semidirect
8. #21445538C2semidirect
9. C2×C61×#417C2semidirect
21. C2×#14519706C22semidirect
24. #16839669C22semidirect
25. #16839665C22semidirect
32. #16839524C22semidirect
37. C2×#103×C61C22semidirect
38. C2×#102×C61C22semidirect
246. C2×Dic61AGL(1,ℤ/8)semidirect
288. C2×#417D61semidirect
320. C2×#103C2×D61semidirect
322. C2×#102C2×D61semidirect
332. C61#23546semidirect

Maximal subgroups

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

subgroupindex
C22×#145198022
C2×#168397852
C2×#168397682
#214483102
C2×#168396632
#214471332
#214471282
#214455382
C2×C61×#4172
#2354661