Finite Group Databasebeta

← #29707739

Subgroups — #29707739

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×#16840329C211yesmaximal, characteristic
2C2×#16839764C211yesmaximal, characteristic, complemented
3C2×#16839756C211yesmaximal, characteristic, complemented
4C22×#14519698C211yesmaximal, characteristic, complemented
5#21448442C211yesmaximal, characteristic, complemented
6#21447059C214yesmaximal, complemented
7#21447052C214yesmaximal, complemented
8#21445517C211yesmaximal, characteristic, complemented
9C2×C61×#418C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C23×#92418C2211yescharacteristic
12#16840329—24no—
13#16840329—24no—
14C2×#14519773C2212yes—
15C2×#14519764C2212yes—
16C2×#14519748—21no—
17C2×#14519748—21no—
18C2×#14519746—22no—
19C2×#14519736C2211yescharacteristic
20C22×C4⋊Dic61C2211yescharacteristic
21#16839764—24no—
22#16839756—24no—
23C2×#14519702C2212yescomplemented
24C2×#14519700C2212yescomplemented
25C2×#14519699C2212yescomplemented
26C2×#14519698—24no—
27C2×#14519698C2214yes—
28C2×#14519697C2212yescomplemented
29#16839661C2214yescomplemented
30#16839637—22no—
31#16839632—21no—
32#16839628—21no—
33#16839516C2214yescomplemented
34C23×D4×C61C2211yescharacteristic
35C22×M16×C61C2211yescharacteristic
36C2×#117×C61C2211yescharacteristic
37C61×#418—24no—
38C2×#102×C61C2214yes—
39C2×#11758222—22no—
40C2×#11758222—22no—
41C2×#11758222—22no—
42C2×#11758222—22no—
43C2×#11758222—22no—
44C2×#11758222D414yes—
45C22×Dic122C2311yescharacteristic
46C22×D244C2311yescharacteristic
47C22×C22⋊D61—21no—
48C22×C22⋊D61—22no—
49C22×C22⋊D61—22no—
50C23×Dic61—21no—
51C22×C4×D61—21no—
52C22×#92418—22no—
53C22×#92418—22no—
54C22×#92418—22no—
55C22×#92418C2312yes—
56#14519773—24no—
57#14519773—24no—
58#14519764—24no—
59#14519764—24no—
60#14519748—28no—
61#14519748—28no—
62#14519746—216no—
63#14519736—22no—
64#14519736—22no—
65C2×C4⋊Dic61—22no—
66C2×C4⋊Dic61—22no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—24no—
69C2×C4⋊Dic61D411yescharacteristic
70C2×C4⋊Dic61D411yescharacteristic
71C2×C4⋊Dic61C2312yes—
72C2×C22⋊Dic61—22no—
73C2×C22⋊Dic61—22no—
74C2×C22⋊Dic61D412yes—
75C2×C4×Dic61C2312yes—
76#14519702—24no—
77#14519700—24no—
78#14519699—24no—
79#14519698—28no—
80#14519697—24no—
81#14519684—24no—
82C25×C61—21no—
83C22×D4×C61—22no—
84C22×D4×C61—21no—
85C22×D4×C61—22no—
86C22×D4×C61—22no—
87C22×D4×C61—22no—
88C22×D4×C61—22no—
89C22×D4×C61D414yes—
90C22×D4×C61C2312yes—
91C23×C4×C61C2311yescharacteristic
92C2×M16×C61—22no—
93C2×M16×C61—22no—
94C2×M16×C61—22no—
95C22×C8×C61C2312yes—
96#117×C61—22no—
97#117×C61D414yescomplemented
98C2×C4⋊C4×C61C2312yes—
99C2×C22⋊C4×C61—21no—
100C2×C42×C61—21no—
101#102×C61—28no—
102C24×C61—22no—
103C24×C61—24no—
104C24×C61—42no—
105C24×C61—42no—
106C24×C61—82no—
107C24×C61—82no—
108C24×C61C2×D411yescharacteristic
109C23×D61—21no—
110C2×D4×C61—24no—
111C2×D4×C61—28no—
112C2×D4×C61—28no—
113C2×D4×C61—22no—
114C2×D4×C61—22no—
115C2×D4×C61—22no—
116C2×D4×C61—22no—
117C2×D4×C61—22no—
118C2×D4×C61—44no—
119C2×D4×C61—42no—
120C2×D4×C61—44no—
121C2×D4×C61—42no—
122C2×D4×C61C2×D414yes—
123C22×C4×C61—22no—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D412yes—
126C22×C4×C61C2×D412yes—
127C22×C4×C61C2×D412yes—
128C22×C4×C61#4112yes—
129C22×C4×C61#4112yes—
130C22×C4×C61#4112yes—
131C22×C4×C61C2411yescharacteristic
132C2×C22⋊D61—24no—
133C2×C22⋊D61—28no—
134C2×C22⋊D61—28no—
135C2×C22⋊D61—42no—
136C2×C22⋊D61—42no—
137C2×C22⋊D61—84no—
138C2×C22⋊D61—84no—
139C22×Dic61—21no—
140C22×Dic61—22no—
141C22×Dic61—42no—
142C22×Dic61—82no—
143C22×Dic61C2×D412yes—
144C22×Dic61C2×D412yes—
145#11758222—28no—
146#11758222—28no—
147#11758222—44no—
148#11758222—44no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244C2×D412yes—
155C2×C4×D61—24no—
156C2×C4×D61—42no—
157C2×C4×D61—42no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122C2×D412yes—
164M16×C61—44no—
165C2×C8×C61—22no—
166C2×C8×C61—22no—
167C2×C8×C61—22no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61C2×D414yes—
170C22⋊C4×C61—24no—
171C42×C61—24no—
172C22⋊Dic61—28no—
173C22⋊Dic61—28no—
174C22⋊Dic61—24no—
175C4⋊Dic61—22no—
176C4⋊Dic61—22no—
177C4⋊Dic61—22no—
178C4⋊Dic61—22no—
179C4⋊Dic61—42no—
180C4⋊Dic61—84no—
181C4×Dic61—22no—
182C4×Dic61—22no—
183C2×#92418—22no—
184C2×#92418—22no—
185C2×#92418—22no—
186C2×#92418—44no—
187C23×C61—22no—
188C23×C61—22no—
189C23×C61—24no—
190C23×C61—42no—
191C23×C61—42no—
192C23×C61—44no—
193C23×C61—44no—
194C23×C61—48no—
195C23×C61—84no—
196C23×C61—84no—
197C23×C61—88no—
198C23×C61—88no—
199C23×C61#12012yes—
200C23×C61#12112yes—
201C23×C61#12412yes—
202C23×C61C22×D411yescharacteristic
203C22×D61—22no—
204C22×D61—41no—
205C22×D61—41no—
206C22×D61—42no—
207C22×D61—42no—
208D4×C61—24no—
209D4×C61—24no—
210D4×C61—42no—
211D4×C61—42no—
212D4×C61—48no—
213C2×C4×C61—24no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61#12012yes—
217C2×C4×C61#12114yes—
218C2×C4×C61#12114yes—
219C2×C4×C61#12112yes—
220C2×C4×C61#12314yes—
221C2×C4×C61#12314yes—
222C2×C4×C61#12412yes—
223C2×C4×C61C22×D411yescharacteristic
224C2×C4×C61C22×D411yescharacteristic
225C2×C4×C61C22×D411yescharacteristic
226C2×C4×C61C2×#4111yescharacteristic
227C2×C4×C61C2×#4111yescharacteristic
228C2×C4×C61C2×#4111yescharacteristic
229C22⋊D61—48no—
230C22⋊D61—88no—
231C22⋊D61—88no—
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—22no—
235C2×Dic61—24no—
236C2×Dic61—24no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—42no—
240C2×Dic61—44no—
241C2×Dic61—84no—
242D244—22no—
243D244—22no—
244D244—41no—
245D244—41no—
246C4×D61—48no—
247Dic122—22no—
248Dic122—22no—
249Dic122—41no—
250Dic122—41no—
251C8×C61—42no—
252#92418—42no—
253#23509—611nomaximal, Sylow
254C22×C61—24no—
255C22×C61—22no—
256C22×C61—22no—
257C22×C61—24no—
258C22×C61—44no—
259C22×C61—44no—
260C22×C61—44no—
261C22×C61—816no—
262C22×C61—84no—
263C22×C61—84no—
264C22×C61#39414yes—
265C22×C61C2×#12011yescharacteristic
266C22×C61C2×#12111yescharacteristic
267C22×C61C2×#12411yescharacteristic
268C22×C61#57612yes—
269C22×C61#57712yes—
270C2×D61—42no—
271C2×D61—42no—
272C2×D61—42no—
273C2×D61—82no—
274C2×D61—82no—
275C4×C61—44no—
276C4×C61#39414yes—
277C4×C61C2×#12111yescharacteristic
278C4×C61C2×#12111yescharacteristic
279C4×C61C2×#12311yescharacteristic
280C4×C61C2×#12311yescharacteristic
281Dic61—42no—
282Dic61—42no—
283Dic61—42no—
284Dic61—82no—
285C2×#535—611no—
286C2×#417—611no—
287C2×#418D6111yescharacteristic, complemented
288C2×#416—611no—
289C22×#102—611no—
290#2317—611no—
291#2042—614no—
292#2041—614no—
293#1729—611no—
294C2×C61—24no—
295C2×C61—84no—
296C2×C61C2×#39411yescharacteristic
297C2×C61#300811yescharacteristic
298C2×C61#300911yescharacteristic
299C2×C61#304512yes—
300C2×C61#313612yes—
301D61—82no—
302C22×#41—611no—
303C23×D4C2×D6111yescharacteristic
304C22×M16C2×D6111yescharacteristic
305C23×C8—611no—
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—24no—
317#417—1224no—
318#416—1224no—
319C2×#103—612no—
320C2×#103—612no—
321C2×#102—614no—
322C2×#102—612no—
323C2×#102—612no—
324C2×#102—1224no—
325C2×#102C2×D6114yes—
326#389—614no—
327#385—1222no—
328#384—1221no—
329#382—1221no—
330#336—614no—
331C61#2350911yescharacteristic, complemented, Sylow
332C25—21no—
333C2×#41—614no—
334C2×#41—1222no—
335C2×#41—1222no—
336C2×#41—1222no—
337C2×#41—1222no—
338C2×#41—1222no—
339C22×Q8—611no—
340C22×D4—22no—
341C22×D4—21no—
342C22×D4—22no—
343C22×D4—22no—
344C22×D4—22no—
345C22×D4—22no—
346C22×D4—611no—
347C22×D4—1222no—
348C22×D4—1222no—
349C22×D4—1221no—
350C22×D4D24414yes—
351C22×D4C22×D6112yes—
352C23×C4—1221no—
353C23×C4—1221no—
354C23×C4C22×D6111yescharacteristic
355C2×M16—22no—
356C2×M16—22no—
357C2×M16—22no—
358C22×C8—612no—
359C22×C8—1222no—
360C22×C8—1222no—
361C22×C8—1222no—
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—1224no—
389#103—1224no—
390#102—28no—
391#102—1224no—
392#102—1228no—
393#102—1224no—
394#95—1224no—
395C24—22no—
396C24—24no—
397C24—42no—
398C24—42no—
399C24—82no—
400C24—82no—
401C24—1221no—
402C24D4×D6111yescharacteristic
403#41—1228no—
404#41—1228no—
405#41—2444no—
406#41—2444no—
407C2×Q8—612no—
408C2×Q8—1221no—
409C2×Q8—1221no—
410C2×Q8—1221no—
411C2×Q8—1221no—
412C2×Q8—1221no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—24no—
419C2×D4—28no—
420C2×D4—28no—
421C2×D4—44no—
422C2×D4—42no—
423C2×D4—44no—
424C2×D4—42no—
425C2×D4—612no—
426C2×D4—1228no—
427C2×D4—1228no—
428C2×D4—1224no—
429C2×D4—1221no—
430C2×D4—1221no—
431C2×D4—1221no—
432C2×D4—1221no—
433C2×D4—1221no—
434C2×D4—4884no—
435C2×D4—4884no—
436C2×D4—2442no—
437C2×D4—2442no—
438C2×D4C2×D24414yes—
439C22×C4—22no—
440C22×C4—612no—
441C22×C4—612no—
442C22×C4—1221no—
443C22×C4—1224no—
444C22×C4—1222no—
445C22×C4—4882no—
446C22×C4—2442no—
447C22×C4—2442no—
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—44no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—1222no—
462C2×C8—1222no—
463C2×C8—1222no—
464C2×C8—2444no—
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—24no—
483C23—42no—
484C23—42no—
485C23—44no—
486C23—44no—
487C23—48no—
488C23—84no—
489C23—84no—
490C23—88no—
491C23—88no—
492C23—1222no—
493C23—2441no—
494C23—2441no—
495C23—2442no—
496C23—2442no—
497C23#1451974712yes—
498C23#1451974912yes—
499C23#1451976312yes—
500C23C2×D4×D6111yescharacteristic
501Q8—1222no—
502Q8—1222no—
503Q8—2441no—
504Q8—2441no—
505D4—24no—
506D4—24no—
507D4—42no—
508D4—42no—
509D4—48no—
510D4—1222no—
511D4—1222no—
512D4—4888no—
513D4—4888no—
514D4—2448no—
515D4—2441no—
516D4—2441no—
517C2×C4—24no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—1222no—
521C2×C4—1222no—
522C2×C4—1222no—
523C2×C4—1224no—
524C2×C4—1224no—
525C2×C4—4884no—
526C2×C4—2442no—
527C2×C4—2444no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4—2448no—
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—24no—
548C22—22no—
549C22—22no—
550C22—44no—
551C22—44no—
552C22—44no—
553C22—816no—
554C22—84no—
555C22—84no—
556C22—4882no—
557C22—4882no—
558C22—2442no—
559C22—2442no—
560C22—2442no—
561C22#1683967814yes—
562C22C2×#1451974711yescharacteristic
563C22C2×#1451974911yescharacteristic
564C22C2×#1451976311yescharacteristic
565C22#1684056612yes—
566C22#1684057212yes—
567C4—44no—
568C4—4882no—
569C4—2442no—
570C4—2442no—
571C4—2442no—
572C4#1683967814yes—
573C4C2×#1451974811yescharacteristic
574C4C2×#1451975111yescharacteristic
575C4C2×#1451975911yescharacteristic
576C4C2×#1451976111yescharacteristic
577C2—24no—
578C2—84no—
579C2—4882no—
580C2C2×#1683967811yescharacteristic
581C2#2145508811yescharacteristic
582C2#2145509711yescharacteristic
583C2#2145536212yes—
584C2#2145616912yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#418C2
socleC23×C61C22×D4
prime radicalC2×#16840329C2
hypercenterC2×#418D61
hypocenterC61#23509
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#418

Complements

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

NH ≅ G/Ntype
2. C2×#16839764C2semidirect
3. C2×#16839756C2semidirect
4. C22×#14519698C2semidirect
5. #21448442C2semidirect
6. #21447059C2semidirect
7. #21447052C2semidirect
8. #21445517C2semidirect
9. C2×C61×#418C2semidirect
23. C2×#14519702C22semidirect
24. C2×#14519700C22semidirect
25. C2×#14519699C22semidirect
28. C2×#14519697C22semidirect
29. #16839661C22semidirect
33. #16839516C22semidirect
97. #117×C61D4semidirect
287. C2×#418D61semidirect
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×#168403292
C2×#168397642
C2×#168397562
C22×#145196982
#214484422
#214470592
#214470522
#214455172
C2×C61×#4182
#2350961