Finite Group Databasebeta

← #29683728

Subgroups — #29683728

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×#16840326C211yesmaximal, characteristic
2C2×#16839763C211yesmaximal, characteristic, complemented
3C2×#16839766C211yesmaximal, characteristic, complemented
4C22×#14519699C211yesmaximal, characteristic, complemented
5C2×#16839717C211yesmaximal, characteristic, complemented
6#21448439C211yesmaximal, characteristic, complemented
7#21446641C212yesmaximal, complemented
8#21446632C212yesmaximal, complemented
9#21446627C212yesmaximal, complemented
10#21446623C212yesmaximal, complemented
11C2×C61×#416C211yesmaximal, characteristic, complemented
12C23×D244C2211yescharacteristic
13C22×#11758194C2211yescharacteristic
14#16840326—24no—
15#16840326C418yescomplemented
16C2×#14519773C2212yes—
17C2×#14519766C2212yes—
18C2×#14519751—22no—
19C2×#14519745—22no—
20C2×#14519736C2211yescharacteristic
21C22×C4⋊Dic61C2211yescharacteristic
22#16839766—24no—
23#16839763—24no—
24C2×#14519701C2212yescomplemented
25C2×#14519699—24no—
26C2×#14519699C2212yes—
27C2×#14519699C2212yes—
28C2×#14519698C2212yescomplemented
29C2×#14519697C2212yes—
30C2×#14519697C2212yes—
31#16839717—24no—
32C2×#14519687C2212yescomplemented
33C2×C8×Dic61C2212yescomplemented
34#16839660—22no—
35#16839634—22no—
36#16839631C2212yescomplemented
37#16839627C2212yescomplemented
38C22×#41×C61C2211yescharacteristic
39C23×C8×C61C2211yescharacteristic
40C2×#117×C61C2211yescharacteristic
41C61×#416—24no—
42C2×#103×C61C2211yescharacteristic, complemented
43C2×#103×C61C2211yescharacteristic, complemented
44C2×#102×C61C2211yescharacteristic, complemented
45C2×#102×C61C2211yescharacteristic, complemented
46C24×D61—21no—
47C22×D244—22no—
48C22×D244—22no—
49C22×D244—22no—
50C22×D244—22no—
51C22×D244—21no—
52C22×D244—22no—
53C22×D244C2×C414yes—
54C22×D244C2312yes—
55C22×C22⋊D61—24no—
56C23×Dic61—21no—
57C2×#11758194—22no—
58C2×#11758194—22no—
59C2×#11758194D414yescomplemented
60C22×#92418C2312yes—
61#14519773—24no—
62#14519773C2×C418yescomplemented
63#14519766—24no—
64#14519766C2×C418yescomplemented
65#14519751—216no—
66#14519745—216no—
67#14519736C2×C414yes—
68#14519736D414yes—
69C2×C4⋊Dic61—22no—
70C2×C4⋊Dic61—22no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61D412yes—
73C2×C4⋊Dic61C2312yes—
74C2×#11758196—24no—
75C2×C22⋊Dic61D412yes—
76C2×#11758198—24no—
77C2×C4×Dic61C2312yes—
78#14519701—24no—
79#14519699—24no—
80#14519699—24no—
81#14519698—24no—
82#14519697—24no—
83#14519697—24no—
84#14519687—24no—
85C8×Dic61—24no—
86#14519684—24no—
87C2×#41×C61—22no—
88C2×#41×C61—22no—
89C2×#41×C61—22no—
90C2×#41×C61—22no—
91C2×#41×C61—22no—
92C2×#41×C61C2×C414yes—
93C22×Q8×C61C2311yescharacteristic
94C22×D4×C61—21no—
95C22×D4×C61C2311yescharacteristic
96C23×C4×C61—21no—
97C23×C4×C61C2311yescharacteristic
98C22×C8×C61—22no—
99C22×C8×C61—22no—
100C22×C8×C61D414yescomplemented
101C22×C8×C61C2311yescharacteristic
102C22×C8×C61C2311yescharacteristic
103#117×C61—22no—
104#117×C61—22no—
105C2×C4⋊C4×C61C2311yescharacteristic
106C2×C4⋊C4×C61C2311yescharacteristic
107C2×C22⋊C4×C61—21no—
108C2×C42×C61—21no—
109#103×C61—22no—
110#103×C61—22no—
111#102×C61—22no—
112#102×C61—22no—
113C24×C61—21no—
114C24×C61C2×D411yescharacteristic
115C23×D61—22no—
116C23×D61—24no—
117C23×D61—42no—
118C23×D61—42no—
119C23×D61—82no—
120C23×D61—82no—
121#41×C61—28no—
122#41×C61—28no—
123#41×C61—44no—
124#41×C61—44no—
125C2×Q8×C61—21no—
126C2×Q8×C61—21no—
127C2×Q8×C61—21no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61C22×C412yes—
131C2×D4×C61—24no—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—42no—
138C2×D4×C61—42no—
139C2×D4×C61C22×C412yes—
140C22×C4×C61—21no—
141C22×C4×C61—24no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—42no—
145C22×C4×C61—42no—
146C22×C4×C61C22×C412yes—
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D412yes—
149C22×C4×C61C2×D412yes—
150C22×C4×C61C2×D412yes—
151C22×C4×C61#4112yes—
152C22×C4×C61#4112yes—
153C22×C4×C61C2411yescharacteristic
154C2×C22⋊D61—216no—
155C2×C22⋊D61—84no—
156C2×C22⋊D61—84no—
157C22×Dic61—22no—
158C22×Dic61—42no—
159C22×Dic61—82no—
160C22×Dic61C2×D414yes—
161C2×D244—28no—
162C2×D244—28no—
163C2×D244—22no—
164C2×D244—22no—
165C2×D244—22no—
166C2×D244—22no—
167C2×D244—24no—
168C2×D244—22no—
169C2×D244—44no—
170C2×D244—44no—
171C2×D244—42no—
172C2×D244—42no—
173C2×D244C22×C414yes—
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—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C22⋊C4×C61—42no—
186C42×C61—42no—
187C22⋊Dic61C22⋊C418yes—
188#11758198—216no—
189C4⋊Dic61—22no—
190C4⋊Dic61—22no—
191C4⋊Dic61—42no—
192C4⋊Dic61C22⋊C418yes—
193#11758196—84no—
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—22no—
203C23×C61—41no—
204C23×C61—41no—
205C23×C61—42no—
206C23×C61—42no—
207C23×C61C2×C22⋊C412yes—
208C23×C61#12412yes—
209C23×C61#12712yes—
210C23×C61C22×D411yescharacteristic
211C22×D61—24no—
212C22×D61—42no—
213C22×D61—42no—
214C22×D61—44no—
215C22×D61—44no—
216C22×D61—48no—
217C22×D61—88no—
218C22×D61—88no—
219C22×D61—84no—
220C22×D61—84no—
221Q8×C61—22no—
222Q8×C61—22no—
223Q8×C61—41no—
224Q8×C61—41no—
225D4×C61—22no—
226D4×C61—22no—
227D4×C61—48no—
228D4×C61—41no—
229D4×C61—41no—
230C2×C4×C61—22no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—48no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—42no—
237C2×C4×C61—42no—
238C2×C4×C61C2×C22⋊C412yes—
239C2×C4×C61C4×D414yes—
240C2×C4×C61C4×D414yes—
241C2×C4×C61#12114yes—
242C2×C4×C61#12114yes—
243C2×C4×C61#12412yes—
244C2×C4×C61#12712yes—
245C2×C4×C61C23×C411yescharacteristic
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C22⋊D61—816no—
252C2×Dic61—24no—
253C2×Dic61—24no—
254C2×Dic61—42no—
255C2×Dic61—44no—
256C2×Dic61—84no—
257C2×Dic61C2×C22⋊C418yes—
258D244—24no—
259D244—24no—
260D244—42no—
261D244—42no—
262D244—48no—
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—44no—
272C22×C61—42no—
273C22×C61—82no—
274C22×C61—81no—
275C22×C61—81no—
276C22×C61#39014yes—
277C22×C61C22×C22⋊C411yescharacteristic
278C22×C61C2×#12411yescharacteristic
279C22×C61C2×#12711yescharacteristic
280C22×C61#57612yes—
281C22×C61#57712yes—
282C2×D61—48no—
283C2×D61—44no—
284C2×D61—816no—
285C2×D61—84no—
286C2×D61—82no—
287C2×D61—82no—
288C4×C61—42no—
289C4×C61—81no—
290C4×C61—81no—
291C4×C61#39014yes—
292C4×C61C2×C4×D411yescharacteristic
293C4×C61C2×C4×D411yescharacteristic
294C4×C61C2×#12111yescharacteristic
295C4×C61C2×#12111yescharacteristic
296Dic61—28no—
297Dic61—82no—
298C2×#535—611no—
299C2×#417—611no—
300C2×#418—611no—
301C2×#416D6111yescharacteristic, complemented
302C22×#102—611no—
303C2×#405—611no—
304#2317—611no—
305#1961—612no—
306#1959—612no—
307#1958—612no—
308#1957—612no—
309C2×C61—24no—
310C2×C61—82no—
311C2×C61C2×#39011yescharacteristic
312C2×C61#289712yes—
313C2×C61#313612yes—
314C2×C61#315411yescharacteristic
315C2×C61#315511yescharacteristic
316D61—84no—
317C22×#41C2×D6111yescharacteristic
318C23×D4—611no—
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×#117C2×D6111yescharacteristic
329C22×C4⋊C4—611no—
330#418—1224no—
331#417—1224no—
332#416—24no—
333C2×#103—612no—
334C2×#103C2×D6111yescharacteristic, complemented
335C2×#103C2×D6111yescharacteristic, complemented
336C2×#102—612no—
337C2×#102—612no—
338C2×#102—612no—
339C2×#102—612no—
340C2×#102—612no—
341C2×#102—1224no—
342C2×#102C2×D6111yescharacteristic, complemented
343C2×#102C2×D6111yescharacteristic, complemented
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—1221no—
353C2×#41—22no—
354C2×#41—22no—
355C2×#41—22no—
356C2×#41—22no—
357C2×#41—22no—
358C2×#41C4×D6114yes—
359C22×Q8C22×D6111yescharacteristic
360C22×D4—21no—
361C22×D4—614no—
362C22×D4—612no—
363C22×D4—1224no—
364C22×D4—1222no—
365C22×D4—1222no—
366C22×D4—1221no—
367C22×D4—1222no—
368C22×D4—1222no—
369C22×D4—1222no—
370C22×D4C22×D6111yescharacteristic
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—22no—
390#117—22no—
391#117—614no—
392#117—614no—
393C2×C4⋊C4—612no—
394C2×C4⋊C4—612no—
395C2×C4⋊C4—1222no—
396C2×C4⋊C4—1222no—
397C2×C4⋊C4—1224no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4C22×D6111yescharacteristic
400C2×C4⋊C4C22×D6111yescharacteristic
401C2×C22⋊C4—21no—
402C2×C22⋊C4—612no—
403C2×C22⋊C4—1224no—
404C2×C42—21no—
405C2×C42—612no—
406#103—22no—
407#103—22no—
408#103—1224no—
409#102—22no—
410#102—22no—
411#102—1224no—
412#102—1224no—
413#102—1224no—
414#102—1224no—
415#102—1224no—
416#97—1224no—
417C4×C8—1224no—
418#95—1224no—
419C24—21no—
420C24—1224no—
421C24—1222no—
422C24—4882no—
423C24—4882no—
424C24—2442no—
425C24—2442no—
426C24D4×D6111yescharacteristic
427#41—28no—
428#41—28no—
429#41—44no—
430#41—44no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8C2×C4×D6112yes—
437C2×D4—24no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—42no—
444C2×D4—42no—
445C2×D4—614no—
446C2×D4—12216no—
447C2×D4—1222no—
448C2×D4—1222no—
449C2×D4—1222no—
450C2×D4—1222no—
451C2×D4—1222no—
452C2×D4—1224no—
453C2×D4—1228no—
454C2×D4—1228no—
455C2×D4—4884no—
456C2×D4—4884no—
457C2×D4—2444no—
458C2×D4—2444no—
459C2×D4—2442no—
460C2×D4—2442no—
461C2×D4C2×C4×D6112yes—
462C22×C4—21no—
463C22×C4—24no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—614no—
469C22×C4—1222no—
470C22×C4—4882no—
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—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—618no—
496C4⋊C4—1222no—
497C4⋊C4—1222no—
498C4⋊C4—4884no—
499C4⋊C4—2442no—
500C22⋊C4—42no—
501C22⋊C4—618no—
502C22⋊C4—12216no—
503C42—42no—
504C42—614no—
505C42—614no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—41no—
510C23—41no—
511C23—42no—
512C23—42no—
513C23—1224no—
514C23—4888no—
515C23—4884no—
516C23—4884no—
517C23—4888no—
518C23—2442no—
519C23—2442no—
520C23—2444no—
521C23—2444no—
522C23—2448no—
523C23C22⋊C4×D6112yes—
524C23#1451976312yes—
525C23#1451977312yes—
526C23C2×D4×D6111yescharacteristic
527Q8—22no—
528Q8—22no—
529Q8—41no—
530Q8—41no—
531D4—22no—
532D4—22no—
533D4—48no—
534D4—41no—
535D4—41no—
536D4—1224no—
537D4—1224no—
538D4—48816no—
539D4—2442no—
540D4—2442no—
541D4—2448no—
542C2×C4—22no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—48no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—618no—
551C2×C4—1224no—
552C2×C4—1224no—
553C2×C4—4884no—
554C2×C4—2442no—
555C2×C4—2444no—
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—44no—
577C22—42no—
578C22—82no—
579C22—81no—
580C22—81no—
581C22—48816no—
582C22—4884no—
583C22—4882no—
584C22—4882no—
585C22—2448no—
586C22—2444no—
587C22#1683966214yes—
588C22C2×C22⋊C4×D6111yescharacteristic
589C22C2×#1451976311yescharacteristic
590C22C2×#1451977311yescharacteristic
591C22#1684056412yes—
592C22#1684056912yes—
593C4—42no—
594C4—81no—
595C4—81no—
596C4—1228no—
597C4—4882no—
598C4#1683966214yes—
599C4C2×#1451973711yescharacteristic
600C4C2×C4×C22⋊D6111yescharacteristic
601C4C2×#1451974911yescharacteristic
602C4C2×#1451975011yescharacteristic
603C2—24no—
604C2—82no—
605C2—4884no—
606C2C2×#1683966211yescharacteristic
607C2#2145434112yes—
608C2#2145617712yes—
609C2#2145629611yescharacteristic
610C2#2145630611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#416C2
socleC23×C61C22×D4
prime radicalC2×#16840326C2
cosocleC22×C4×C61C24
hypercenterC2×#416D61
hypocenterC61#21206
LCSG ▷ C2×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. C2×#16839717C2semidirect
6. #21448439C2semidirect
7. #21446641C2semidirect
8. #21446632C2semidirect
9. #21446627C2semidirect
10. #21446623C2semidirect
11. C2×C61×#416C2semidirect
15. #16840326C4semidirect
24. C2×#14519701C22semidirect
28. C2×#14519698C22semidirect
32. C2×#14519687C22semidirect
33. C2×C8×Dic61C22semidirect
36. #16839631C22semidirect
37. #16839627C22semidirect
42. C2×#103×C61C22semidirect
43. C2×#103×C61C22semidirect
44. C2×#102×C61C22semidirect
45. C2×#102×C61C22semidirect
59. C2×#11758194D4semidirect
62. #14519773C2×C4semidirect
64. #14519766C2×C4semidirect
100. C22×C8×C61D4semidirect
301. C2×#416D61semidirect
334. C2×#103C2×D61semidirect
335. C2×#103C2×D61semidirect
342. C2×#102C2×D61semidirect
343. C2×#102C2×D61semidirect
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×#168403262
C2×#168397632
C2×#168397662
C22×#145196992
C2×#168397172
#214484392
#214466412
#214466322
#214466272
#214466232
C2×C61×#4162
#2120661