Finite Group Databasebeta

← #29683725

Subgroups — #29683725

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×#16839768C211yesmaximal, characteristic, complemented
3C2×#16839758C211yesmaximal, characteristic, complemented
4C22×#14519697C211yesmaximal, characteristic, complemented
5C2×#16839720C211yesmaximal, characteristic, complemented
6#21448443C211yesmaximal, characteristic, complemented
7#21446641C212yesmaximal, complemented
8#21446632C212yesmaximal, complemented
9#21446627C212yesmaximal, complemented
10#21446623C212yesmaximal, complemented
11C2×C61×#417C211yesmaximal, characteristic, complemented
12C23×D244C2211yescharacteristic
13C23×#92418C2211yescharacteristic
14#16840326—24no—
15#16840326C418yescomplemented
16C2×#14519773C2212yes—
17C2×#14519766C2212yes—
18C2×#14519751—22no—
19C2×#14519745—22no—
20C2×#14519736C2211yescharacteristic
21C2×#14519736C2211yescharacteristic
22#16839768—24no—
23#16839758—24no—
24C2×#14519701C2211yescharacteristic, complemented
25C2×#14519701C2211yescharacteristic, complemented
26C2×#14519699C2212yes—
27C2×#14519699C2212yes—
28C2×#14519698C2211yescharacteristic, complemented
29C2×#14519698C2211yescharacteristic, complemented
30C2×#14519697—24no—
31C2×#14519697C2212yes—
32C2×#14519697C2212yes—
33#16839720—24no—
34C2×#14519687C2212yescomplemented
35C2×C8×Dic61C2212yescomplemented
36#16839658—22no—
37#16839635—22no—
38#16839631C2212yescomplemented
39#16839627C2212yescomplemented
40C22×#41×C61C2211yescharacteristic
41C22×M16×C61C2211yescharacteristic
42C22×C4⋊C4×C61C2211yescharacteristic
43C61×#417—24no—
44C2×#103×C61C2212yescomplemented
45C2×#102×C61C2212yescomplemented
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—
56C22×#92418—22no—
57C22×#92418—22no—
58C22×#92418D414yescomplemented
59C22×#92418C2311yescharacteristic
60C22×#92418C2311yescharacteristic
61#14519773—24no—
62#14519773C2×C418yescomplemented
63#14519766—24no—
64#14519766C2×C418yescomplemented
65#14519751—216no—
66#14519745—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—21no—
76C2×C22⋊Dic61D412yes—
77C2×#11758198—24no—
78C2×C4×Dic61—21no—
79C2×C4×Dic61C2312yes—
80#14519701—22no—
81#14519701—22no—
82#14519699—24no—
83#14519699—24no—
84#14519698—22no—
85#14519698—22no—
86#14519697—24no—
87#14519697—24no—
88#14519687—24no—
89C8×Dic61—24no—
90#14519684—24no—
91C2×#41×C61—22no—
92C2×#41×C61—22no—
93C2×#41×C61—22no—
94C2×#41×C61—22no—
95C2×#41×C61—22no—
96C2×#41×C61C2×C414yes—
97C22×Q8×C61C2311yescharacteristic
98C22×D4×C61—21no—
99C22×D4×C61C2311yescharacteristic
100C23×C4×C61—21no—
101C23×C4×C61—21no—
102C23×C4×C61C2311yescharacteristic
103C2×M16×C61—22no—
104C2×M16×C61—22no—
105C2×M16×C61D414yescomplemented
106C22×C8×C61C2312yes—
107C2×C4⋊C4×C61—22no—
108C2×C4⋊C4×C61—22no—
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61C2312yes—
111#103×C61—24no—
112#102×C61—24no—
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—22no—
143C22×C4×C61—42no—
144C22×C4×C61—42no—
145C22×C4×C61—42no—
146C22×C4×C61—82no—
147C22×C4×C61C22×C412yes—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D412yes—
150C22×C4×C61C2×D412yes—
151C22×C4×C61C2×D412yes—
152C22×C4×C61#4112yes—
153C22×C4×C61#4112yes—
154C22×C4×C61C2411yescharacteristic
155C2×C22⋊D61—216no—
156C2×C22⋊D61—84no—
157C2×C22⋊D61—84no—
158C22×Dic61—21no—
159C22×Dic61—21no—
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—
174M16×C61—28no—
175C2×C8×C61—22no—
176C2×C8×C61—22no—
177C2×C8×C61C2×D414yes—
178C4⋊C4×C61—22no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61—42no—
181C22⋊Dic61—42no—
182C22⋊Dic61C22⋊C418yes—
183#11758198—216no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61C22⋊C418yes—
189#11758196—84no—
190C4×Dic61—42no—
191C4×Dic61C22×C414yes—
192C4×Dic61C2×D414yes—
193C2×#92418—28no—
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418C2×D412yes—
199C2×#92418C2×D412yes—
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—42no—
235C2×C4×C61—44no—
236C2×C4×C61—84no—
237C2×C4×C61C2×C22⋊C412yes—
238C2×C4×C61C4×D414yes—
239C2×C4×C61C4×D414yes—
240C2×C4×C61#12114yes—
241C2×C4×C61#12114yes—
242C2×C4×C61#12412yes—
243C2×C4×C61#12712yes—
244C2×C4×C61C23×C411yescharacteristic
245C2×C4×C61C22×D411yescharacteristic
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C22⋊D61—816no—
251C2×Dic61—24no—
252C2×Dic61—24no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—42no—
256C2×Dic61—42no—
257C2×Dic61C2×C22⋊C418yes—
258D244—24no—
259D244—24no—
260D244—42no—
261D244—42no—
262D244—48no—
263C8×C61—24no—
264#92418—22no—
265#92418—22no—
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—82no—
290C4×C61#39014yes—
291C4×C61C2×C4×D411yescharacteristic
292C4×C61C2×C4×D411yescharacteristic
293C4×C61C2×#12111yescharacteristic
294C4×C61C2×#12111yescharacteristic
295Dic61—28no—
296Dic61—81no—
297Dic61—81no—
298C2×#535—611no—
299C2×#417D6111yescharacteristic, complemented
300C2×#418—611no—
301C2×#416—611no—
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×M16C2×D6111yescharacteristic
320C23×C8—611no—
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—24no—
332#416—1224no—
333C2×#103—611no—
334C2×#103—611no—
335C2×#103C2×D6112yescomplemented
336C2×#102—611no—
337C2×#102—611no—
338C2×#102—612no—
339C2×#102—612no—
340C2×#102—612no—
341C2×#102—612no—
342C2×#102—1224no—
343C2×#102C2×D6112yescomplemented
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—21no—
373C23×C4C22×D6111yescharacteristic
374C2×M16—22no—
375C2×M16—22no—
376C2×M16C22⋊D6114yescomplemented
377C22×C8—614no—
378C22×C8—611no—
379C22×C8—611no—
380C22×C8—1222no—
381C22×C8—1222no—
382C22×C8C22×D6112yes—
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—24no—
407#103—1222no—
408#103—1222no—
409#102—24no—
410#102—1224no—
411#102—1224no—
412#102—1222no—
413#102—1222no—
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—22no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—82no—
469C22×C4—614no—
470C22×C4—1221no—
471C22×C4—1221no—
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—28no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—612no—
484C2×C8—612no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8—1228no—
490C2×C8C2×C22⋊D6114yes—
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—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—42no—
547C2×C4—44no—
548C2×C4—84no—
549C2×C4—618no—
550C2×C4—1224no—
551C2×C4—1224no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2442no—
555C2×C4—2442no—
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—24no—
570C8—1222no—
571C8—1222no—
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#1684056612yes—
592C22#1684057112yes—
593C4—42no—
594C4—82no—
595C4—1228no—
596C4—4881no—
597C4—4881no—
598C4#1683966214yes—
599C4C2×#1451973711yescharacteristic
600C4C2×C4×C22⋊D6111yescharacteristic
601C4C2×#1451974911yescharacteristic
602C4C2×#1451975011yescharacteristic
603C2—24no—
604C2—82no—
605C2—4884no—
606C2C2×#1683966211yescharacteristic
607C2#2145434312yes—
608C2#2145617612yes—
609C2#2145629811yescharacteristic
610C2#2145630411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#417C2
socleC23×C61C22×D4
prime radicalC2×#16840326C2
cosocleC22×C4×C61C24
hypercenterC2×#417D61
hypocenterC61#21206
LCSG ▷ C2×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×#16839768C2semidirect
3. C2×#16839758C2semidirect
4. C22×#14519697C2semidirect
5. C2×#16839720C2semidirect
6. #21448443C2semidirect
7. #21446641C2semidirect
8. #21446632C2semidirect
9. #21446627C2semidirect
10. #21446623C2semidirect
11. C2×C61×#417C2semidirect
15. #16840326C4semidirect
24. C2×#14519701C22semidirect
25. C2×#14519701C22semidirect
28. C2×#14519698C22semidirect
29. C2×#14519698C22semidirect
34. C2×#14519687C22semidirect
35. C2×C8×Dic61C22semidirect
38. #16839631C22semidirect
39. #16839627C22semidirect
44. C2×#103×C61C22semidirect
45. C2×#102×C61C22semidirect
58. C22×#92418D4semidirect
62. #14519773C2×C4semidirect
64. #14519766C2×C4semidirect
105. C2×M16×C61D4semidirect
299. C2×#417D61semidirect
335. C2×#103C2×D61semidirect
343. C2×#102C2×D61semidirect
351. C61#21206semidirect
376. C2×M16C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403262
C2×#168397682
C2×#168397582
C22×#145196972
C2×#168397202
#214484432
#214466412
#214466322
#214466272
#214466232
C2×C61×#4172
#2120661