Finite Group Databasebeta

← #29695831

Subgroups — #29695831

Order 15616.

1830 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11758199C211yesmaximal, characteristic
2C2×#16839842C212yesmaximal, complemented
3D4×C22⋊Dic61C211yesmaximal, characteristic, complemented
4#21448639C211yesmaximal, characteristic, complemented
5#21446682C212yesmaximal, complemented
6#21446165C212yesmaximal, complemented
7#21446164C212yesmaximal, complemented
8#21445846C211yesmaximal, characteristic, complemented
9#21445751C211yesmaximal, characteristic, complemented
10#21445593C211yesmaximal, characteristic, complemented
11C61×#1786C211yesmaximal, characteristic, complemented
12C23×D244C2211yescharacteristic
13C22×#11758199—28no—
14C22×#11758199C418yescomplemented
15C22×#11758199C2211yescharacteristic
16C22×#11758199C2211yescharacteristic
17C22×#11758199C2211yescharacteristic
18C22×#11758199C2211yescharacteristic
19C23×#92418C2211yescharacteristic
20C2×D4×Dic61C2212yescomplemented
21C2×#14519739C2212yescomplemented
22C22×C22⋊Dic61—21no—
23C2×C22⋊C4×D61—21no—
24#16839842—28no—
25C2×#14519710C2212yescomplemented
26C2×#14519699C2212yescomplemented
27C2×#14519699C2212yescomplemented
28C2×C8×Dic61C2212yescomplemented
29#16839665C2212yescomplemented
30#16839663C2212yescomplemented
31#16839642—22no—
32#16839640—22no—
33#16839622C2211yescharacteristic
34#16839618—21no—
35#16839617—21no—
36C4×C22⋊Dic61C2211yescharacteristic
37#16839524C2212yescomplemented
38#16839516C2212yescomplemented
39C23×D4×C61C2211yescharacteristic
40C2×#102×C61C2212yescomplemented
41C2×#102×C61C2212yescomplemented
42C2×C22⋊C8×C61C2211yescharacteristic
43C61×#380C2211yescharacteristic
44C24×D61—21no—
45C22×D244—24no—
46C22×D244—21no—
47C22×D244—22no—
48C22×D244—42no—
49C22×D244C2×C414yescomplemented
50C22×D244C2311yescharacteristic
51C22×D244C2311yescharacteristic
52C23×Dic61—22no—
53C22×C4×D61—22no—
54C2×#11758199—24no—
55C2×#11758199—24no—
56C2×#11758199—232no—
57C2×#11758199—24no—
58C2×#11758199C2×C414yes—
59C2×#11758199C2×C414yescomplemented
60C2×#11758199C2×C414yescomplemented
61C2×#11758199C2×C414yescomplemented
62C2×#11758199D418yes—
63C22×#92418—24no—
64C22×#92418C2×C414yescomplemented
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67D4×Dic61—24no—
68D4×Dic61—42no—
69#14519739—24no—
70#14519739—42no—
71C2×C4⋊Dic61C2312yes—
72C2×C22⋊Dic61—21no—
73C2×C22⋊Dic61—22no—
74C2×C22⋊Dic61—22no—
75C2×C22⋊Dic61—22no—
76C2×C22⋊Dic61—44no—
77C2×C22⋊Dic61D411yescharacteristic
78C2×C22⋊Dic61D411yescharacteristic
79C2×#11758198—22no—
80C2×#11758198—22no—
81C2×#11758198—22no—
82C22⋊C4×D61—44no—
83C2×C4×Dic61C2312yes—
84#14519710—24no—
85#14519699—24no—
86#14519699—42no—
87C8×Dic61—24no—
88#14519684—21no—
89#14519684D412yes—
90C25×C61—21no—
91C22×D4×C61—24no—
92C22×D4×C61—21no—
93C22×D4×C61—22no—
94C22×D4×C61—42no—
95C22×D4×C61C2×C414yescomplemented
96C22×D4×C61C2311yescharacteristic
97C22×D4×C61C2311yescharacteristic
98C23×C4×C61C2311yescharacteristic
99C22×C8×C61C2312yes—
100C2×C4⋊C4×C61C2312yes—
101C2×C22⋊C4×C61—21no—
102#102×C61—24no—
103#102×C61—42no—
104C22⋊C8×C61—22no—
105#95×C61—21no—
106C24×C61—21no—
107C24×C61—21no—
108C24×C61—24no—
109C24×C61—44no—
110C24×C61—42no—
111C24×C61—42no—
112C24×C61—82no—
113C24×C61C2×D411yescharacteristic
114C23×D61—21no—
115C23×D61—21no—
116C23×D61—24no—
117C23×D61—44no—
118C23×D61—42no—
119C23×D61—42no—
120C23×D61—82no—
121C2×D4×C61—28no—
122C2×D4×C61—24no—
123C2×D4×C61—22no—
124C2×D4×C61—21no—
125C2×D4×C61—21no—
126C2×D4×C61—22no—
127C2×D4×C61—22no—
128C2×D4×C61—44no—
129C2×D4×C61—48no—
130C2×D4×C61—44no—
131C2×D4×C61C22×C412yes—
132C2×D4×C61C22×C412yes—
133C2×D4×C61C2×D414yes—
134C22×C4×C61—22no—
135C22×C4×C61—22no—
136C22×C4×C61—22no—
137C22×C4×C61C22×C412yes—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61#4112yes—
140C22×C4×C61C2411yescharacteristic
141C22×Dic61—22no—
142C22×Dic61—44no—
143C22×Dic61—44no—
144C22×Dic61—82no—
145C22×Dic61C2×D412yes—
146C22×Dic61C2×D412yes—
147C2×D244—21no—
148C2×D244—22no—
149C2×D244—22no—
150C2×D244—22no—
151C2×D244—28no—
152C2×D244—24no—
153C2×D244—21no—
154C2×D244—44no—
155C2×D244—48no—
156C2×D244—44no—
157C2×D244C22×C412yes—
158C2×D244C22×C412yes—
159C2×D244C2×D414yes—
160C2×C4×D61—44no—
161C2×C4×D61—44no—
162C2×C4×D61—82no—
163C2×C8×C61—22no—
164C2×C8×C61—22no—
165C2×C8×C61—22no—
166C4⋊C4×C61—22no—
167C4⋊C4×C61—22no—
168C22⋊C4×C61—42no—
169C22⋊Dic61—22no—
170C22⋊Dic61—44no—
171C22⋊Dic61—42no—
172C22⋊Dic61—44no—
173C22⋊Dic61—82no—
174C22⋊Dic61D814yescomplemented
175#11758199—28no—
176#11758199—28no—
177#11758199—28no—
178#11758199C22⋊C4116yescomplemented
179#11758198—44no—
180#11758198—44no—
181#11758198—82no—
182C4⋊Dic61—22no—
183C4⋊Dic61—22no—
184C4×Dic61—22no—
185C4×Dic61—22no—
186C2×#92418—22no—
187C2×#92418—28no—
188C2×#92418C22×C412yes—
189C2×#92418C22×C412yes—
190C2×#92418C2×D414yes—
191C23×C61—22no—
192C23×C61—22no—
193C23×C61—22no—
194C23×C61—24no—
195C23×C61—22no—
196C23×C61—48no—
197C23×C61—41no—
198C23×C61—41no—
199C23×C61—42no—
200C23×C61—42no—
201C23×C61—41no—
202C23×C61—41no—
203C23×C61—42no—
204C23×C61—48no—
205C23×C61—84no—
206C23×C61—88no—
207C23×C61—88no—
208C23×C61C4×D412yes—
209C23×C61C2×D812yes—
210C23×C61#13512yes—
211C23×C61C22×D411yescharacteristic
212C22×D61—22no—
213C22×D61—24no—
214C22×D61—22no—
215C22×D61—48no—
216C22×D61—41no—
217C22×D61—41no—
218C22×D61—42no—
219C22×D61—42no—
220C22×D61—41no—
221C22×D61—41no—
222C22×D61—42no—
223C22×D61—48no—
224C22×D61—84no—
225C22×D61—88no—
226C22×D61—88no—
227D4×C61—22no—
228D4×C61—22no—
229D4×C61—22no—
230D4×C61—24no—
231D4×C61—48no—
232D4×C61—42no—
233D4×C61C2×C22⋊C414yescomplemented
234C2×C4×C61—22no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—24no—
238C2×C4×C61—42no—
239C2×C4×C61—42no—
240C2×C4×C61—42no—
241C2×C4×C61C4×D412yes—
242C2×C4×C61C23×C411yescharacteristic
243C2×C4×C61C22×D412yes—
244C2×C4×C61C2×#4111yescharacteristic
245C2×C4×C61H(5,𝔽2)12yes—
246C2×Dic61—22no—
247C2×Dic61—22no—
248C2×Dic61—22no—
249C2×Dic61—22no—
250C2×Dic61—22no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254C2×Dic61—88no—
255C2×Dic61C2×D814yescomplemented
256D244—22no—
257D244—22no—
258D244—22no—
259D244—24no—
260D244—48no—
261D244—42no—
262D244C2×C22⋊C414yescomplemented
263C4×D61—88no—
264C8×C61—42no—
265#92418—22no—
266#92418C2×C22⋊C414yescomplemented
267C22⋊C4×D8—611nomaximal, Sylow
268C22×C61—22no—
269C22×C61—22no—
270C22×C61—24no—
271C22×C61—24no—
272C22×C61—24no—
273C22×C61—42no—
274C22×C61—42no—
275C22×C61—44no—
276C22×C61—44no—
277C22×C61—42no—
278C22×C61—42no—
279C22×C61—42no—
280C22×C61—816no—
281C22×C61—82no—
282C22×C61—82no—
283C22×C61C4×D814yescomplemented
284C22×C61C2×C4×D411yescharacteristic
285C22×C61D4212yes—
286C22×C61C22×D811yescharacteristic
287C22×C61C2×#13511yescharacteristic
288C22×C61#57612yes—
289C2×D61—24no—
290C2×D61—42no—
291C2×D61—42no—
292C2×D61—44no—
293C2×D61—44no—
294C2×D61—42no—
295C2×D61—42no—
296C2×D61—42no—
297C2×D61—816no—
298C2×D61—82no—
299C2×D61—82no—
300C4×C61—22no—
301C4×C61—82no—
302C4×C61C22×C22⋊C411yescharacteristic
303C4×C61#51811yescharacteristic
304C4×C61#54612yes—
305Dic61—24no—
306Dic61—42no—
307Dic61—82no—
308C23×D8—611no—
309C2×C4×D8—612no—
310D4×C22⋊C4—611no—
311D4×C22⋊C4—611no—
312#1970—612no—
313#1879—612no—
314#1879—612no—
315#1808—611no—
316#1786—611no—
317#1786D6111yescharacteristic, complemented
318C8×C22⋊C4—611no—
319C2×C61—24no—
320C2×C61—44no—
321C2×C61—82no—
322C2×C61D4×C22⋊C411yescharacteristic
323C2×C61C2×C4×D811yescharacteristic
324C2×C61#294811yescharacteristic
325C2×C61D4×D812yes—
326C2×C61#328012yes—
327D61—44no—
328D61—82no—
329C23×D4—611no—
330C23×D4C2×D6111yescharacteristic
331C22×D8—611no—
332C22×D8—618no—
333C22×D8—611no—
334C22×D8—611no—
335C22×D8—611no—
336C22×D8—1228no—
337C23×C8—611no—
338C2×C4×D4—612no—
339C2×C4×D4—612no—
340C22×C22⋊C4—1221no—
341C22×C22⋊C4—1221no—
342C4×D8—1228no—
343C2×#107—612no—
344C2×#102—612no—
345C2×#102—612no—
346C2×#102C2×D6112yescomplemented
347C2×#102C2×D6112yescomplemented
348C2×C22⋊C8C2×D6111yescharacteristic
349C2×C4×C8—612no—
350#390—612no—
351#390—612no—
352#386—1222no—
353#386—1222no—
354#380—611no—
355#380C2×D6111yescharacteristic
356#379—1221no—
357#379—1221no—
358C4×C22⋊C4—611no—
359#340—612no—
360#336—612no—
361C61C22⋊C4×D811yescharacteristic, complemented, Sylow
362C25—21no—
363C25—1221no—
364C22×D4—24no—
365C22×D4—21no—
366C22×D4—22no—
367C22×D4—42no—
368C22×D4—614no—
369C22×D4—611no—
370C22×D4—611no—
371C22×D4—1224no—
372C22×D4—1221no—
373C22×D4—1222no—
374C22×D4—2442no—
375C22×D4C4×D6114yescomplemented
376C22×D4C22×D6111yescharacteristic
377C22×D4C22×D6111yescharacteristic
378C23×C4—1222no—
379C23×C4—1222no—
380C23×C4C22×D6111yescharacteristic
381C2×D8—614no—
382C2×D8—618no—
383C2×D8—614no—
384C2×D8—614no—
385C2×D8—614no—
386C2×D8—1224no—
387C2×D8—12232no—
388C2×D8—1224no—
389C2×D8—1224no—
390C22×C8—611no—
391C22×C8—614no—
392C22×C8—611no—
393C22×C8—1224no—
394C22×C8C22×D6112yes—
395C4×D4—1224no—
396C4×D4—1224no—
397C4×D4—2442no—
398C4×D4—2442no—
399C2×C4⋊C4—612no—
400C2×C4⋊C4C22×D6112yes—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—611no—
403C2×C22⋊C4—611no—
404C2×C22⋊C4—1222no—
405C2×C22⋊C4—1222no—
406C2×C22⋊C4—1222no—
407C2×C22⋊C4—1222no—
408C2×C22⋊C4—1222no—
409C2×C22⋊C4—1222no—
410C2×C22⋊C4—1221no—
411C2×C22⋊C4—2444no—
412C2×C22⋊C4—2444no—
413C2×C42—612no—
414#107—1224no—
415#102—24no—
416#102—42no—
417#102—1224no—
418#102—2442no—
419C22⋊C8—22no—
420C4×C8—1224no—
421#95—21no—
422#95—612no—
423#95—1221no—
424C24—21no—
425C24—21no—
426C24—24no—
427C24—44no—
428C24—42no—
429C24—42no—
430C24—82no—
431C24—1221no—
432C24—1224no—
433C24—1221no—
434C24—4882no—
435C24—2444no—
436C24—2442no—
437C24—2442no—
438C24D4×D6111yescharacteristic
439C2×D4—28no—
440C2×D4—22no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—24no—
446C2×D4—44no—
447C2×D4—48no—
448C2×D4—44no—
449C2×D4—612no—
450C2×D4—612no—
451C2×D4—614no—
452C2×D4—1228no—
453C2×D4—1222no—
454C2×D4—1221no—
455C2×D4—1221no—
456C2×D4—1222no—
457C2×D4—1222no—
458C2×D4—1224no—
459C2×D4—2444no—
460C2×D4—2448no—
461C2×D4—2444no—
462C2×D4C2×C4×D6112yes—
463C2×D4C2×C4×D6112yes—
464C2×D4D4×D6114yes—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—612no—
469C22×C4—612no—
470C22×C4—1222no—
471C22×C4—4882no—
472C22×C4—4882no—
473C22×C4—2444no—
474C22×C4—2444no—
475C22×C4—2444no—
476C22×C4—2444no—
477C22×C4C2×C4×D6112yes—
478C22×C4D4×D6111yescharacteristic
479C22×C4#1175822412yes—
480C22×C4C23×D6111yescharacteristic
481D8—6116no—
482D8—1228no—
483D8—1228no—
484D8—1228no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—612no—
489C2×C8—614no—
490C2×C8—612no—
491C2×C8—1228no—
492C2×C8—1222no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—1222no—
496C4⋊C4—1222no—
497C22⋊C4—42no—
498C22⋊C4—614no—
499C22⋊C4—1222no—
500C22⋊C4—4882no—
501C22⋊C4—4882no—
502C22⋊C4—2444no—
503C22⋊C4—2444no—
504C22⋊C4—2444no—
505C22⋊C4—2444no—
506C22⋊C4—2442no—
507C42—1222no—
508C42—1222no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—24no—
513C23—22no—
514C23—48no—
515C23—41no—
516C23—41no—
517C23—42no—
518C23—41no—
519C23—41no—
520C23—42no—
521C23—48no—
522C23—42no—
523C23—84no—
524C23—88no—
525C23—88no—
526C23—1222no—
527C23—1224no—
528C23—1222no—
529C23—4888no—
530C23—4888no—
531C23—4884no—
532C23—2448no—
533C23—2441no—
534C23—2441no—
535C23—2442no—
536C23—2448no—
537C23—2442no—
538C23—2441no—
539C23—2441no—
540C23—2442no—
541C23#1451973712yes—
542C23D8×D6112yes—
543C23#1451979812yes—
544C23C2×D4×D6111yescharacteristic
545D4—22no—
546D4—22no—
547D4—22no—
548D4—24no—
549D4—48no—
550D4—42no—
551D4—614no—
552D4—1222no—
553D4—1222no—
554D4—1222no—
555D4—1224no—
556D4—2448no—
557D4—2442no—
558D4C22⋊C4×D6114yescomplemented
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—614no—
567C2×C4—1222no—
568C2×C4—1222no—
569C2×C4—1222no—
570C2×C4—1222no—
571C2×C4—1222no—
572C2×C4—4888no—
573C2×C4—4888no—
574C2×C4—2442no—
575C2×C4—2442no—
576C2×C4—2442no—
577C2×C4#1451973712yes—
578C2×C4C22×C4×D6111yescharacteristic
579C2×C4C2×D4×D6112yes—
580C2×C4C2×#1175822411yescharacteristic
581C2×C4#1451982212yes—
582C8—42no—
583C8—614no—
584C8—1222no—
585C22—22no—
586C22—24no—
587C22—24no—
588C22—22no—
589C22—24no—
590C22—42no—
591C22—44no—
592C22—44no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—816no—
598C22—82no—
599C22—82no—
600C22—1224no—
601C22—48816no—
602C22—4882no—
603C22—4882no—
604C22—2442no—
605C22—2442no—
606C22—2442no—
607C22—2442no—
608C22—2444no—
609C22—2444no—
610C22—2442no—
611C22#1683984214yescomplemented
612C22C2×#1451973711yescharacteristic
613C22#1684039412yes—
614C22C2×D8×D6111yescharacteristic
615C22C2×#1451979811yescharacteristic
616C22#1684056612yes—
617C4—22no—
618C4—82no—
619C4—1224no—
620C4—4882no—
621C4—2442no—
622C4C2×C22⋊C4×D6111yescharacteristic
623C4#1684023511yescharacteristic
624C4#1684039912yes—
625C2—24no—
626C2—44no—
627C2—82no—
628C2—4882no—
629C2—2444no—
630C2#2144863511yescharacteristic
631C2C2×#1683984211yescharacteristic
632C2#2145466711yescharacteristic
633C2#2145748912yes—
634C2#2145749712yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1786C2
socleC23×C61C22×D4
prime radicalC23×#11758199C2
cosocleC22×C4×C61C24
hypercenter#1786D61
hypocenterC61C22⋊C4×D8
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1786

Complements

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

NH ≅ G/Ntype
2. C2×#16839842C2semidirect
3. D4×C22⋊Dic61C2semidirect
4. #21448639C2semidirect
5. #21446682C2semidirect
6. #21446165C2semidirect
7. #21446164C2semidirect
8. #21445846C2semidirect
9. #21445751C2semidirect
10. #21445593C2semidirect
11. C61×#1786C2semidirect
14. C22×#11758199C4semidirect
20. C2×D4×Dic61C22semidirect
21. C2×#14519739C22semidirect
25. C2×#14519710C22semidirect
26. C2×#14519699C22semidirect
27. C2×#14519699C22semidirect
28. C2×C8×Dic61C22semidirect
29. #16839665C22semidirect
30. #16839663C22semidirect
37. #16839524C22semidirect
38. #16839516C22semidirect
40. C2×#102×C61C22semidirect
41. C2×#102×C61C22semidirect
49. C22×D244C2×C4semidirect
59. C2×#11758199C2×C4semidirect
60. C2×#11758199C2×C4semidirect
61. C2×#11758199C2×C4semidirect
64. C22×#92418C2×C4semidirect
95. C22×D4×C61C2×C4semidirect
174. C22⋊Dic61D8semidirect
178. #11758199C22⋊C4semidirect
233. D4×C61C2×C22⋊C4semidirect
255. C2×Dic61C2×D8semidirect
262. D244C2×C22⋊C4semidirect
266. #92418C2×C22⋊C4semidirect
283. C22×C61C4×D8semidirect
317. #1786D61semidirect
346. C2×#102C2×D61semidirect
347. C2×#102C2×D61semidirect
361. C61C22⋊C4×D8semidirect
375. C22×D4C4×D61semidirect
558. D4C22⋊C4×D61semidirect
611. C22#16839842semidirect

Maximal subgroups

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

subgroupindex
C23×#117581992
C2×#168398422
D4×C22⋊Dic612
#214486392
#214466822
#214461652
#214461642
#214458462
#214457512
#214455932
C61×#17862
C22⋊C4×D861