Finite Group Databasebeta

← #29707736

Subgroups — #29707736

Order 15616.

1606 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×#16840326C211yesmaximal, characteristic
2C2×#16839763C211yesmaximal, characteristic, complemented
3C2×#16839766C211yesmaximal, characteristic, complemented
4C22×#14519699C211yesmaximal, characteristic, complemented
5#21448442C211yesmaximal, characteristic, complemented
6#21447057C214yesmaximal, complemented
7#21447052C214yesmaximal, complemented
8#21445515C211yesmaximal, characteristic, complemented
9C2×C61×#416C211yesmaximal, characteristic, complemented
10C23×D244C2211yescharacteristic
11C22×#11758194C2211yescharacteristic
12#16840326—24no—
13#16840326—24no—
14C2×#14519773C2212yes—
15C2×#14519766C2212yes—
16C2×#14519751—21no—
17C2×#14519751—21no—
18C2×#14519745—22no—
19C2×#14519736C2211yescharacteristic
20C22×C4⋊Dic61C2211yescharacteristic
21#16839766—24no—
22#16839763—24no—
23C2×#14519701C2212yescomplemented
24C2×#14519699—24no—
25C2×#14519699C2214yes—
26C2×#14519698C2212yescomplemented
27C2×#14519697C2214yes—
28#16839661C2214yescomplemented
29#16839637—22no—
30#16839632—21no—
31#16839628—21no—
32#16839516C2214yescomplemented
33C22×#41×C61C2211yescharacteristic
34C23×C8×C61C2211yescharacteristic
35C2×#117×C61C2211yescharacteristic
36C61×#416—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—
48C22×C22⋊D61—22no—
49C22×C22⋊D61—22no—
50C23×Dic61—21no—
51C2×#11758194—22no—
52C2×#11758194—22no—
53C2×#11758194—22no—
54C22×#92418C2312yes—
55#14519773—24no—
56#14519773—24no—
57#14519766—24no—
58#14519766—24no—
59#14519751—28no—
60#14519751—28no—
61#14519745—216no—
62#14519736—22no—
63#14519736—22no—
64C2×C4⋊Dic61—22no—
65C2×C4⋊Dic61—22no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61—24no—
68C2×C4⋊Dic61D411yescharacteristic
69C2×C4⋊Dic61D411yescharacteristic
70C2×C4⋊Dic61C2312yes—
71C2×C22⋊Dic61D412yes—
72C2×#11758198—22no—
73C2×#11758198—22no—
74C2×C4×Dic61C2312yes—
75#14519701—24no—
76#14519699—28no—
77#14519698—24no—
78#14519697—28no—
79#14519684—24no—
80C2×#41×C61—22no—
81C2×#41×C61—22no—
82C2×#41×C61—22no—
83C2×#41×C61—22no—
84C2×#41×C61—22no—
85C2×#41×C61D414yes—
86C22×Q8×C61C2311yescharacteristic
87C22×D4×C61—21no—
88C22×D4×C61C2311yescharacteristic
89C23×C4×C61—21no—
90C23×C4×C61C2311yescharacteristic
91C22×C8×C61—22no—
92C22×C8×C61—22no—
93C22×C8×C61—22no—
94C22×C8×C61C2312yes—
95#117×C61—22no—
96#117×C61D414yescomplemented
97C2×C4⋊C4×C61C2312yes—
98C2×C22⋊C4×C61—21no—
99C2×C42×C61—21no—
100#103×C61—24no—
101#102×C61—24no—
102C24×C61—21no—
103C24×C61C2×D411yescharacteristic
104C23×D61—22no—
105C23×D61—24no—
106C23×D61—42no—
107C23×D61—42no—
108C23×D61—82no—
109C23×D61—82no—
110#41×C61—28no—
111#41×C61—28no—
112#41×C61—44no—
113#41×C61—44no—
114C2×Q8×C61—21no—
115C2×Q8×C61—21no—
116C2×Q8×C61—21no—
117C2×Q8×C61—21no—
118C2×Q8×C61—21no—
119C2×Q8×C61C2×D412yes—
120C2×D4×C61—24no—
121C2×D4×C61—21no—
122C2×D4×C61—21no—
123C2×D4×C61—21no—
124C2×D4×C61—21no—
125C2×D4×C61—21no—
126C2×D4×C61—42no—
127C2×D4×C61—42no—
128C2×D4×C61C2×D412yes—
129C22×C4×C61—22no—
130C22×C4×C61—21no—
131C22×C4×C61—24no—
132C22×C4×C61—42no—
133C22×C4×C61—42no—
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D412yes—
136C22×C4×C61C2×D412yes—
137C22×C4×C61C2×D412yes—
138C22×C4×C61#4112yes—
139C22×C4×C61#4112yes—
140C22×C4×C61#4112yes—
141C22×C4×C61C2411yescharacteristic
142C2×C22⋊D61—28no—
143C2×C22⋊D61—28no—
144C2×C22⋊D61—84no—
145C2×C22⋊D61—84no—
146C22×Dic61—22no—
147C22×Dic61—42no—
148C22×Dic61—82no—
149C22×Dic61C2×D412yes—
150C22×Dic61C2×D412yes—
151C2×D244—28no—
152C2×D244—28no—
153C2×D244—22no—
154C2×D244—22no—
155C2×D244—22no—
156C2×D244—22no—
157C2×D244—24no—
158C2×D244—22no—
159C2×D244—44no—
160C2×D244—42no—
161C2×D244—44no—
162C2×D244—42no—
163C2×D244C2×D414yes—
164C2×C8×C61—22no—
165C2×C8×C61—22no—
166C2×C8×C61—22no—
167C2×C8×C61—44no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61C2×D414yes—
170C22⋊C4×C61—24no—
171C42×C61—24no—
172C22⋊Dic61—24no—
173#11758198—28no—
174#11758198—28no—
175C4⋊Dic61—22no—
176C4⋊Dic61—22no—
177C4⋊Dic61—22no—
178C4⋊Dic61—22no—
179C4⋊Dic61—42no—
180C4⋊Dic61—84no—
181C4×Dic61—22no—
182C4×Dic61—22no—
183#11758194—44no—
184C2×#92418—22no—
185C2×#92418—22no—
186C2×#92418—22no—
187C23×C61—22no—
188C23×C61—22no—
189C23×C61—22no—
190C23×C61—41no—
191C23×C61—41no—
192C23×C61—42no—
193C23×C61—42no—
194C23×C61#12012yes—
195C23×C61#12112yes—
196C23×C61#12412yes—
197C23×C61C22×D411yescharacteristic
198C22×D61—24no—
199C22×D61—42no—
200C22×D61—42no—
201C22×D61—44no—
202C22×D61—44no—
203C22×D61—48no—
204C22×D61—84no—
205C22×D61—84no—
206C22×D61—88no—
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—24no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—42no—
222C2×C4×C61—42no—
223C2×C4×C61—48no—
224C2×C4×C61#12012yes—
225C2×C4×C61#12114yes—
226C2×C4×C61#12114yes—
227C2×C4×C61#12112yes—
228C2×C4×C61#12314yes—
229C2×C4×C61#12314yes—
230C2×C4×C61#12412yes—
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C22⋊D61—88no—
238C22⋊D61—88no—
239C2×Dic61—22no—
240C2×Dic61—22no—
241C2×Dic61—24no—
242C2×Dic61—24no—
243C2×Dic61—42no—
244C2×Dic61—44no—
245C2×Dic61—84no—
246D244—24no—
247D244—24no—
248D244—42no—
249D244—42no—
250D244—48no—
251C8×C61—42no—
252#92418—42no—
253#23509—611nomaximal, Sylow
254C22×C61—24no—
255C22×C61—22no—
256C22×C61—22no—
257C22×C61—24no—
258C22×C61—42no—
259C22×C61—42no—
260C22×C61—42no—
261C22×C61—82no—
262C22×C61—82no—
263C22×C61#39414yes—
264C22×C61C2×#12011yescharacteristic
265C22×C61C2×#12111yescharacteristic
266C22×C61C2×#12411yescharacteristic
267C22×C61#57612yes—
268C22×C61#57712yes—
269C2×D61—44no—
270C2×D61—44no—
271C2×D61—44no—
272C2×D61—816no—
273C2×D61—84no—
274C2×D61—84no—
275C4×C61—42no—
276C4×C61—44no—
277C4×C61#39414yes—
278C4×C61C2×#12111yescharacteristic
279C4×C61C2×#12111yescharacteristic
280C4×C61C2×#12311yescharacteristic
281C4×C61C2×#12311yescharacteristic
282Dic61—42no—
283Dic61—42no—
284Dic61—82no—
285C2×#535—611no—
286C2×#417—611no—
287C2×#418—611no—
288C2×#416D6111yescharacteristic, complemented
289C22×#102—611no—
290#2317—611no—
291#2042—614no—
292#2041—614no—
293#1729—611no—
294C2×C61—24no—
295C2×C61—82no—
296C2×C61C2×#39411yescharacteristic
297C2×C61#300811yescharacteristic
298C2×C61#300911yescharacteristic
299C2×C61#304512yes—
300C2×C61#313612yes—
301D61—84no—
302C22×#41C2×D6111yescharacteristic
303C23×D4—611no—
304C22×M16—611no—
305C23×C8C2×D6111yescharacteristic
306#535—1224no—
307#535—1224no—
308C2×#127—612no—
309C2×#124—612no—
310C2×#121—1221no—
311C2×#121—1221no—
312C2×#120—1222no—
313C2×#117—611no—
314C2×#117C2×D6111yescharacteristic
315C22×C4⋊C4—611no—
316#418—1224no—
317#417—1224no—
318#416—24no—
319C2×#103—612no—
320C2×#103C2×D6112yescomplemented
321C2×#102—612no—
322C2×#102—614no—
323C2×#102—614no—
324C2×#102—1224no—
325C2×#102C2×D6112yescomplemented
326#389—614no—
327#385—1222no—
328#384—1221no—
329#382—1221no—
330#336—614no—
331C61#2350911yescharacteristic, complemented, Sylow
332C25—1221no—
333C2×#41—22no—
334C2×#41—22no—
335C2×#41—22no—
336C2×#41—22no—
337C2×#41—22no—
338C2×#41D24414yes—
339C22×Q8C22×D6111yescharacteristic
340C22×D4—21no—
341C22×D4—614no—
342C22×D4—612no—
343C22×D4—1222no—
344C22×D4—1222no—
345C22×D4—1222no—
346C22×D4—1222no—
347C22×D4—1221no—
348C22×D4—1222no—
349C22×D4—1222no—
350C22×D4—1222no—
351C22×D4C22×D6111yescharacteristic
352C23×C4—21no—
353C23×C4—1221no—
354C23×C4C22×D6111yescharacteristic
355C2×M16—1222no—
356C2×M16—1222no—
357C2×M16—1222no—
358C22×C8—22no—
359C22×C8—22no—
360C22×C8—22no—
361C22×C8—612no—
362C22×C8C22×D6112yes—
363#127—1224no—
364#127—1224no—
365#124—1224no—
366#124—1224no—
367#121—1228no—
368#121—1228no—
369#120—12216no—
370#117—22no—
371#117—1222no—
372#117—1222no—
373#117C22⋊D6114yescomplemented
374C2×C4⋊C4—611no—
375C2×C4⋊C4—611no—
376C2×C4⋊C4—612no—
377C2×C4⋊C4—1222no—
378C2×C4⋊C4—1222no—
379C2×C4⋊C4—1224no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4C22×D6112yes—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—612no—
384C2×C22⋊C4—1222no—
385C2×C22⋊C4—1222no—
386C2×C42—21no—
387C2×C42—612no—
388#103—24no—
389#103—1224no—
390#102—24no—
391#102—1228no—
392#102—1224no—
393#102—1228no—
394#95—1224no—
395C24—21no—
396C24—1224no—
397C24—1222no—
398C24—4882no—
399C24—4882no—
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×D24412yes—
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—1228no—
423C2×D4—1228no—
424C2×D4—1222no—
425C2×D4—1222no—
426C2×D4—1222no—
427C2×D4—1222no—
428C2×D4—1222no—
429C2×D4—1224no—
430C2×D4—1228no—
431C2×D4—1228no—
432C2×D4—4884no—
433C2×D4—4884no—
434C2×D4—2442no—
435C2×D4—2444no—
436C2×D4—2444no—
437C2×D4—2442no—
438C2×D4C2×D24412yes—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—24no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—612no—
445C22×C4—612no—
446C22×C4—1222no—
447C22×C4—4882no—
448C22×C4—2442no—
449C22×C4C2×D24412yes—
450C22×C4#1175822212yes—
451C22×C4D4×D6112yes—
452C22×C4D4×D6111yescharacteristic
453C22×C4#1175822412yes—
454C22×C4#1175822412yes—
455C22×C4C2×C22⋊D6112yes—
456C22×C4C23×D6111yescharacteristic
457M16—2444no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—44no—
462C2×C8—1222no—
463C2×C8—1222no—
464C2×C8—1222no—
465C4⋊C4—22no—
466C4⋊C4—1222no—
467C4⋊C4—1222no—
468C4⋊C4—1222no—
469C4⋊C4—1222no—
470C4⋊C4—4884no—
471C4⋊C4—2442no—
472C4⋊C4C2×C22⋊D6114yes—
473C22⋊C4—24no—
474C22⋊C4—1228no—
475C22⋊C4—1228no—
476C22⋊C4—1224no—
477C42—24no—
478C42—1222no—
479C42—1222no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—41no—
484C23—41no—
485C23—42no—
486C23—42no—
487C23—1224no—
488C23—4888no—
489C23—4888no—
490C23—4884no—
491C23—4884no—
492C23—2442no—
493C23—2442no—
494C23—2444no—
495C23—2444no—
496C23—2448no—
497C23#1451974712yes—
498C23#1451974912yes—
499C23#1451976312yes—
500C23C2×D4×D6111yescharacteristic
501Q8—22no—
502Q8—22no—
503Q8—41no—
504Q8—41no—
505D4—22no—
506D4—22no—
507D4—48no—
508D4—41no—
509D4—41no—
510D4—1224no—
511D4—1224no—
512D4—4888no—
513D4—4888no—
514D4—2442no—
515D4—2442no—
516D4—2448no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—48no—
524C2×C4—1222no—
525C2×C4—1222no—
526C2×C4—1224no—
527C2×C4—1224no—
528C2×C4—4884no—
529C2×C4—2442no—
530C2×C4—2444no—
531C2×C4#1451974712yes—
532C2×C4#1451974814yes—
533C2×C4#1451974912yes—
534C2×C4#1451975114yes—
535C2×C4#1451975914yes—
536C2×C4#1451976114yes—
537C2×C4#1451976312yes—
538C2×C4C2×D4×D6111yescharacteristic
539C2×C4C22×C22⋊D6111yescharacteristic
540C2×C4C22×D24411yescharacteristic
541C2×C4C2×#1175822411yescharacteristic
542C2×C4C2×#1175822411yescharacteristic
543C2×C4C2×#1175822211yescharacteristic
544C8—42no—
545C8—2442no—
546C22—24no—
547C22—22no—
548C22—22no—
549C22—24no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—82no—
554C22—82no—
555C22—48816no—
556C22—4884no—
557C22—4884no—
558C22—2444no—
559C22—2444no—
560C22—2444no—
561C22#1683967814yes—
562C22C2×#1451974711yescharacteristic
563C22C2×#1451974911yescharacteristic
564C22C2×#1451976311yescharacteristic
565C22#1684056412yes—
566C22#1684056912yes—
567C4—42no—
568C4—44no—
569C4—4882no—
570C4—2442no—
571C4—2442no—
572C4#1683967814yes—
573C4C2×#1451974811yescharacteristic
574C4C2×#1451975111yescharacteristic
575C4C2×#1451975911yescharacteristic
576C4C2×#1451976111yescharacteristic
577C2—24no—
578C2—82no—
579C2—4884no—
580C2C2×#1683967811yescharacteristic
581C2#2145509011yescharacteristic
582C2#2145509911yescharacteristic
583C2#2145535612yes—
584C2#2145617712yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#416C2
socleC23×C61C22×D4
prime radicalC2×#16840326C2
hypercenterC2×#416D61
hypocenterC61#23509
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#416

Complements

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

NH ≅ G/Ntype
2. C2×#16839763C2semidirect
3. C2×#16839766C2semidirect
4. C22×#14519699C2semidirect
5. #21448442C2semidirect
6. #21447057C2semidirect
7. #21447052C2semidirect
8. #21445515C2semidirect
9. C2×C61×#416C2semidirect
23. C2×#14519701C22semidirect
26. C2×#14519698C22semidirect
28. #16839661C22semidirect
32. #16839516C22semidirect
37. C2×#103×C61C22semidirect
38. C2×#102×C61C22semidirect
96. #117×C61D4semidirect
288. C2×#416D61semidirect
320. C2×#103C2×D61semidirect
325. C2×#102C2×D61semidirect
331. C61#23509semidirect
373. #117C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403262
C2×#168397632
C2×#168397662
C22×#145196992
#214484422
#214470572
#214470522
#214455152
C2×C61×#4162
#2350961