Finite Group Databasebeta

← #29683721

Subgroups — #29683721

Order 15616.

1718 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840329C211yesmaximal, characteristic
2C2×#16839761C211yesmaximal, characteristic, complemented
3C2×#16839767C211yesmaximal, characteristic, complemented
4C2×#16839759C211yesmaximal, characteristic, complemented
5C2×#16839717C211yesmaximal, characteristic, complemented
6#21448443C211yesmaximal, characteristic, complemented
7#21446641C212yesmaximal, complemented
8#21446634C212yesmaximal, complemented
9#21446629C212yesmaximal, complemented
10#21446623C212yesmaximal, complemented
11C22×#102×C61C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C22×#11758194C2211yescharacteristic
14#16840329—24no—
15#16840329C418yescomplemented
16C2×#14519773C2212yes—
17C2×#14519764C2212yes—
18C2×#14519748—22no—
19C2×#14519746—22no—
20C2×#14519736C2211yescharacteristic
21C2×#14519736C2211yescharacteristic
22#16839767—24no—
23#16839761—24no—
24#16839759—24no—
25C2×#14519702C2211yescharacteristic, complemented
26C2×#14519702C2211yescharacteristic, complemented
27C2×#14519700C2212yescomplemented
28C2×#14519699C2211yescharacteristic, complemented
29C2×#14519699C2211yescharacteristic, complemented
30C2×#14519698C2212yes—
31C2×#14519698C2212yes—
32C2×#14519697C2212yescomplemented
33#16839717—24no—
34C2×#14519687C2212yescomplemented
35C2×C8×Dic61C2212yescomplemented
36#16839658—22no—
37#16839635—22no—
38#16839631C2212yescomplemented
39#16839627C2212yescomplemented
40C23×D4×C61C2211yescharacteristic
41C23×C8×C61C2211yescharacteristic
42C22×C4⋊C4×C61C2211yescharacteristic
43C2×#102×C61—24no—
44C2×#102×C61C2212yes—
45C2×#102×C61C2212yes—
46C2×#11758222—22no—
47C2×#11758222—22no—
48C2×#11758222—22no—
49C2×#11758222—22no—
50C2×#11758222—22no—
51C2×#11758222C2×C414yes—
52C22×Dic122C2311yescharacteristic
53C22×D244C2311yescharacteristic
54C22×C22⋊D61—21no—
55C22×C22⋊D61—24no—
56C22×C4×D61—21no—
57C2×#11758194—22no—
58C2×#11758194—22no—
59C2×#11758194D414yescomplemented
60C22×#92418C2312yes—
61#14519773—24no—
62#14519773C2×C418yescomplemented
63#14519764—24no—
64#14519764C2×C418yescomplemented
65#14519748—216no—
66#14519746—216no—
67#14519736—22no—
68#14519736—22no—
69#14519736C2×C414yes—
70#14519736D414yes—
71C2×C4⋊Dic61D412yes—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758196—24no—
75C2×C22⋊Dic61—24no—
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61D412yes—
78C2×C4×Dic61—21no—
79C2×C4×Dic61C2312yes—
80#14519702—22no—
81#14519702—22no—
82#14519700—24no—
83#14519699—22no—
84#14519699—22no—
85#14519698—24no—
86#14519698—24no—
87#14519697—24no—
88#14519687—24no—
89C8×Dic61—24no—
90#14519684—24no—
91C25×C61—21no—
92C22×D4×C61—22no—
93C22×D4×C61—22no—
94C22×D4×C61—21no—
95C22×D4×C61—22no—
96C22×D4×C61—22no—
97C22×D4×C61—22no—
98C22×D4×C61C2×C414yes—
99C22×D4×C61C2312yes—
100C23×C4×C61—21no—
101C23×C4×C61C2311yescharacteristic
102C22×C8×C61—22no—
103C22×C8×C61—22no—
104C22×C8×C61D414yescomplemented
105C22×C8×C61C2311yescharacteristic
106C22×C8×C61C2311yescharacteristic
107C2×C4⋊C4×C61—22no—
108C2×C4⋊C4×C61—22no—
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61C2312yes—
111#102×C61—24no—
112#102×C61—24no—
113C24×C61—22no—
114C24×C61—24no—
115C24×C61—42no—
116C24×C61—42no—
117C24×C61—82no—
118C24×C61—82no—
119C24×C61C2×D411yescharacteristic
120C23×D61—21no—
121C2×D4×C61—24no—
122C2×D4×C61—28no—
123C2×D4×C61—28no—
124C2×D4×C61—22no—
125C2×D4×C61—22no—
126C2×D4×C61—22no—
127C2×D4×C61—22no—
128C2×D4×C61—22no—
129C2×D4×C61—44no—
130C2×D4×C61—44no—
131C2×D4×C61—42no—
132C2×D4×C61—42no—
133C2×D4×C61C22×C414yes—
134C22×C4×C61—22no—
135C22×C4×C61—42no—
136C22×C4×C61—82no—
137C22×C4×C61C22×C412yes—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D412yes—
140C22×C4×C61C2×D412yes—
141C22×C4×C61C2×D412yes—
142C22×C4×C61#4112yes—
143C22×C4×C61#4112yes—
144C22×C4×C61C2411yescharacteristic
145C2×C22⋊D61—24no—
146C2×C22⋊D61—216no—
147C2×C22⋊D61—42no—
148C2×C22⋊D61—42no—
149C2×C22⋊D61—84no—
150C2×C22⋊D61—84no—
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61C2×D414yes—
155#11758222—28no—
156#11758222—28no—
157#11758222—44no—
158#11758222—44no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244—21no—
164C2×D244C22×C412yes—
165C2×C4×D61—24no—
166C2×C4×D61—42no—
167C2×C4×D61—42no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122C22×C412yes—
174C2×C8×C61—28no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61C2×D412yes—
180C2×C8×C61C2×D412yes—
181C4⋊C4×C61—22no—
182C4⋊C4×C61—22no—
183C4⋊C4×C61—42no—
184C22⋊Dic61—216no—
185C22⋊Dic61—42no—
186C22⋊Dic61C22⋊C418yes—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61C22⋊C418yes—
192#11758196—84no—
193C4×Dic61—42no—
194C4×Dic61C22×C414yes—
195C4×Dic61C2×D414yes—
196#11758194—28no—
197C2×#92418—22no—
198C2×#92418—22no—
199C2×#92418C2×D414yes—
200C23×C61—22no—
201C23×C61—22no—
202C23×C61—24no—
203C23×C61—42no—
204C23×C61—42no—
205C23×C61—44no—
206C23×C61—44no—
207C23×C61—48no—
208C23×C61—88no—
209C23×C61—88no—
210C23×C61—84no—
211C23×C61—84no—
212C23×C61C2×C22⋊C412yes—
213C23×C61#12412yes—
214C23×C61#12712yes—
215C23×C61C22×D411yescharacteristic
216C22×D61—22no—
217C22×D61—41no—
218C22×D61—41no—
219C22×D61—42no—
220C22×D61—42no—
221D4×C61—24no—
222D4×C61—24no—
223D4×C61—42no—
224D4×C61—42no—
225D4×C61—48no—
226C2×C4×C61—42no—
227C2×C4×C61—44no—
228C2×C4×C61—84no—
229C2×C4×C61C2×C22⋊C412yes—
230C2×C4×C61C4×D414yes—
231C2×C4×C61C4×D414yes—
232C2×C4×C61#12114yes—
233C2×C4×C61#12114yes—
234C2×C4×C61#12412yes—
235C2×C4×C61#12712yes—
236C2×C4×C61C23×C411yescharacteristic
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C2×#4111yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C22⋊D61—48no—
243C22⋊D61—816no—
244C2×Dic61—24no—
245C2×Dic61—24no—
246C2×Dic61—22no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61C2×C22⋊C418yes—
254D244—22no—
255D244—22no—
256D244—41no—
257D244—41no—
258C4×D61—48no—
259Dic122—22no—
260Dic122—22no—
261Dic122—41no—
262Dic122—41no—
263C8×C61—22no—
264C8×C61—22no—
265#92418—24no—
266#21206—611nomaximal, Sylow
267C22×C61—24no—
268C22×C61—24no—
269C22×C61—22no—
270C22×C61—22no—
271C22×C61—48no—
272C22×C61—44no—
273C22×C61—816no—
274C22×C61—84no—
275C22×C61—82no—
276C22×C61—82no—
277C22×C61#39014yes—
278C22×C61C22×C22⋊C411yescharacteristic
279C22×C61C2×#12411yescharacteristic
280C22×C61C2×#12711yescharacteristic
281C22×C61#57612yes—
282C22×C61#57712yes—
283C2×D61—44no—
284C2×D61—42no—
285C2×D61—82no—
286C2×D61—81no—
287C2×D61—81no—
288C4×C61—82no—
289C4×C61#39014yes—
290C4×C61C2×C4×D411yescharacteristic
291C4×C61C2×C4×D411yescharacteristic
292C4×C61C2×#12111yescharacteristic
293C4×C61C2×#12111yescharacteristic
294Dic61—28no—
295Dic61—42no—
296Dic61—81no—
297Dic61—81no—
298C2×#535—611no—
299C2×#417—611no—
300C2×#418—611no—
301C2×#416—611no—
302C22×#102D6111yescharacteristic, complemented
303C2×#405—611no—
304#2317—611no—
305#1961—612no—
306#1959—612no—
307#1958—612no—
308#1957—612no—
309C2×C61—24no—
310C2×C61—84no—
311C2×C61C2×#39011yescharacteristic
312C2×C61#289712yes—
313C2×C61#313612yes—
314C2×C61#315411yescharacteristic
315C2×C61#315511yescharacteristic
316D61—82no—
317C22×#41—611no—
318C23×D4C2×D6111yescharacteristic
319C22×M16—611no—
320C23×C8C2×D6111yescharacteristic
321#535—618no—
322#535—1224no—
323C2×#127—612no—
324C2×#124—612no—
325C2×#121—1222no—
326C2×#120—1222no—
327C2×#117—611no—
328C2×#117—611no—
329C22×C4⋊C4C2×D6111yescharacteristic
330#418—1224no—
331#417—1224no—
332#416—1224no—
333C2×#103—611no—
334C2×#103—611no—
335C2×#103—612no—
336C2×#102—24no—
337C2×#102—612no—
338C2×#102—612no—
339C2×#102—611no—
340C2×#102—611no—
341C2×#102—612no—
342C2×#102C2×D6112yes—
343C2×#102C2×D6112yes—
344#405—1224no—
345C2×#97—612no—
346C2×C4×C8—612no—
347#389—1222no—
348#385—1222no—
349#384—612no—
350#382—612no—
351C61#2120611yescharacteristic, complemented, Sylow
352C25—21no—
353C2×#41—614no—
354C2×#41—1222no—
355C2×#41—1222no—
356C2×#41—1222no—
357C2×#41—1222no—
358C2×#41—1222no—
359C22×Q8—611no—
360C22×D4—22no—
361C22×D4—22no—
362C22×D4—21no—
363C22×D4—22no—
364C22×D4—22no—
365C22×D4—22no—
366C22×D4—611no—
367C22×D4—1224no—
368C22×D4—1221no—
369C22×D4C4×D6114yes—
370C22×D4C22×D6112yes—
371C23×C4—21no—
372C23×C4—1221no—
373C23×C4C22×D6111yescharacteristic
374C2×M16—614no—
375C2×M16—1222no—
376C2×M16—1222no—
377C22×C8—22no—
378C22×C8—22no—
379C22×C8—612no—
380C22×C8C22⋊D6114yescomplemented
381C22×C8C22×D6111yescharacteristic
382C22×C8C22×D6111yescharacteristic
383#127—618no—
384#127—1224no—
385#124—618no—
386#124—1224no—
387#121—12216no—
388#120—12216no—
389#117—614no—
390#117—614no—
391#117—1222no—
392#117—1222no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—22no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—612no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—611no—
399C2×C4⋊C4—1224no—
400C2×C4⋊C4C22×D6112yes—
401C2×C22⋊C4—612no—
402C2×C22⋊C4—1224no—
403C2×C22⋊C4—1221no—
404C2×C42—612no—
405C2×C42—1221no—
406#103—1224no—
407#103—1222no—
408#103—1222no—
409#102—24no—
410#102—24no—
411#102—1224no—
412#102—1224no—
413#102—1224no—
414#102—1222no—
415#102—1222no—
416#97—1224no—
417C4×C8—1224no—
418#95—1224no—
419C24—22no—
420C24—24no—
421C24—42no—
422C24—42no—
423C24—82no—
424C24—82no—
425C24—1221no—
426C24D4×D6111yescharacteristic
427#41—1228no—
428#41—1228no—
429#41—2444no—
430#41—2444no—
431C2×Q8—612no—
432C2×Q8—1221no—
433C2×Q8—1221no—
434C2×Q8—1221no—
435C2×Q8—1221no—
436C2×Q8—1221no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—24no—
443C2×D4—28no—
444C2×D4—28no—
445C2×D4—44no—
446C2×D4—44no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—612no—
450C2×D4—12216no—
451C2×D4—1224no—
452C2×D4—1221no—
453C2×D4—1221no—
454C2×D4—1221no—
455C2×D4—1221no—
456C2×D4—1221no—
457C2×D4—4884no—
458C2×D4—4884no—
459C2×D4—2442no—
460C2×D4—2442no—
461C2×D4C2×C4×D6114yes—
462C22×C4—22no—
463C22×C4—42no—
464C22×C4—82no—
465C22×C4—614no—
466C22×C4—1221no—
467C22×C4—1224no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—2442no—
471C22×C4—2442no—
472C22×C4C2×C4×D6112yes—
473C22×C4#1175822212yes—
474C22×C4D4×D6112yes—
475C22×C4D4×D6112yes—
476C22×C4D4×D6111yescharacteristic
477C22×C4#1175822412yes—
478C22×C4C2×C22⋊D6112yes—
479C22×C4C23×D6111yescharacteristic
480M16—1228no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—28no—
486C2×C8—614no—
487C2×C8—1222no—
488C2×C8—1222no—
489C2×C8C2×C22⋊D6112yes—
490C2×C8C2×C22⋊D6112yes—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—42no—
494C4⋊C4—618no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—4884no—
500C22⋊C4—618no—
501C22⋊C4—12216no—
502C22⋊C4—2442no—
503C42—614no—
504C42—614no—
505C42—2442no—
506C23—22no—
507C23—22no—
508C23—24no—
509C23—42no—
510C23—42no—
511C23—44no—
512C23—44no—
513C23—48no—
514C23—88no—
515C23—88no—
516C23—84no—
517C23—84no—
518C23—1222no—
519C23—2441no—
520C23—2441no—
521C23—2442no—
522C23—2442no—
523C23C22⋊C4×D6112yes—
524C23#1451976312yes—
525C23#1451977312yes—
526C23C2×D4×D6111yescharacteristic
527Q8—1222no—
528Q8—1222no—
529Q8—2441no—
530Q8—2441no—
531D4—24no—
532D4—24no—
533D4—42no—
534D4—42no—
535D4—48no—
536D4—1222no—
537D4—1222no—
538D4—48816no—
539D4—2448no—
540D4—2441no—
541D4—2441no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—84no—
545C2×C4—618no—
546C2×C4—1224no—
547C2×C4—1224no—
548C2×C4—1222no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2442no—
552C2×C4—2442no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2448no—
556C2×C4C22⋊C4×D6112yes—
557C2×C4#1451973714yes—
558C2×C4C4×C22⋊D6114yes—
559C2×C4#1451974914yes—
560C2×C4#1451975014yes—
561C2×C4#1451976312yes—
562C2×C4#1451977312yes—
563C2×C4C22×C4×D6111yescharacteristic
564C2×C4C2×D4×D6111yescharacteristic
565C2×C4C2×D4×D6111yescharacteristic
566C2×C4C22×C22⋊D6111yescharacteristic
567C2×C4C2×#1175822411yescharacteristic
568C2×C4C2×#1175822211yescharacteristic
569C8—22no—
570C8—22no—
571C8—1224no—
572C22—24no—
573C22—24no—
574C22—22no—
575C22—22no—
576C22—48no—
577C22—44no—
578C22—816no—
579C22—84no—
580C22—82no—
581C22—82no—
582C22—4882no—
583C22—4881no—
584C22—4881no—
585C22—2444no—
586C22—2442no—
587C22#1683966214yes—
588C22C2×C22⋊C4×D6111yescharacteristic
589C22C2×#1451976311yescharacteristic
590C22C2×#1451977311yescharacteristic
591C22#1684056312yes—
592C22#1684056812yes—
593C4—82no—
594C4—1228no—
595C4—4881no—
596C4—4881no—
597C4—2442no—
598C4#1683966214yes—
599C4C2×#1451973711yescharacteristic
600C4C2×C4×C22⋊D6111yescharacteristic
601C4C2×#1451974911yescharacteristic
602C4C2×#1451975011yescharacteristic
603C2—24no—
604C2—84no—
605C2—4882no—
606C2C2×#1683966211yescharacteristic
607C2#2145434012yes—
608C2#2145617112yes—
609C2#2145628911yescharacteristic
610C2#2145630011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC22×#102×C61C2
socleC23×C61C22×D4
prime radicalC2×#16840329C2
cosocleC22×C4×C61C24
hypercenterC22×#102D61
hypocenterC61#21206
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#102

Complements

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

NH ≅ G/Ntype
2. C2×#16839761C2semidirect
3. C2×#16839767C2semidirect
4. C2×#16839759C2semidirect
5. C2×#16839717C2semidirect
6. #21448443C2semidirect
7. #21446641C2semidirect
8. #21446634C2semidirect
9. #21446629C2semidirect
10. #21446623C2semidirect
11. C22×#102×C61C2semidirect
15. #16840329C4semidirect
25. C2×#14519702C22semidirect
26. C2×#14519702C22semidirect
27. C2×#14519700C22semidirect
28. C2×#14519699C22semidirect
29. C2×#14519699C22semidirect
32. C2×#14519697C22semidirect
34. C2×#14519687C22semidirect
35. C2×C8×Dic61C22semidirect
38. #16839631C22semidirect
39. #16839627C22semidirect
59. C2×#11758194D4semidirect
62. #14519773C2×C4semidirect
64. #14519764C2×C4semidirect
104. C22×C8×C61D4semidirect
302. C22×#102D61semidirect
351. C61#21206semidirect
380. C22×C8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403292
C2×#168397612
C2×#168397672
C2×#168397592
C2×#168397172
#214484432
#214466412
#214466342
#214466292
#214466232
C22×#102×C612
#2120661