Finite Group Databasebeta

← #29711099

Subgroups — #29711099

Order 15616.

1930 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11758200C211yesmaximal, characteristic, complemented
2C2×#16840425C211yesmaximal, characteristic, complemented
3C2×#16839899C212yesmaximal, complemented
4#21449169C211yesmaximal, characteristic
5#21446900C212yesmaximal, complemented
6#21446823C212yesmaximal, complemented
7#21446243C211yesmaximal, characteristic, complemented
8#21446171C212yesmaximal, complemented
9#21446109C211yesmaximal, characteristic, complemented
10#21445755C211yesmaximal, characteristic, complemented
11C2×C61×#450C211yesmaximal, characteristic, complemented
12C23×Dic122C2211yescharacteristic
13C22×Q8×D61C2211yescharacteristic, complemented
14C22×#11758200—28no—
15C22×#11758200—24no—
16C22×#11758200C2212yes—
17C22×#11758200C2212yes—
18C23×#92418C2211yescharacteristic
19#16840425—24no—
20#16840425—24no—
21C2×#14519776—21no—
22C2×#14519775C2211yescharacteristic, complemented
23C2×#14519757C2211yescharacteristic, complemented
24C2×#14519755—21no—
25C2×#14519754—21no—
26C2×#14519748C2212yes—
27C2×#14519746—21no—
28C2×#14519742C2211yescharacteristic, complemented
29C2×C4×C22⋊D61C2211yescharacteristic
30C2×#14519737—21no—
31#16839899—28no—
32C2×#14519702C2212yescomplemented
33C2×#14519702C2212yescomplemented
34C2×#14519701C2212yescomplemented
35C2×#14519689C2212yescomplemented
36#16839676—21no—
37#16839670—21no—
38#16839665C2212yes—
39#16839622C2211yescharacteristic
40#16839618—21no—
41#16839524C2212yescomplemented
42#16839516C2212yescomplemented
43C23×D4×C61C2211yescharacteristic
44C22×SD16×C61C2212yes—
45C2×C22⋊Q8×C61C2211yescharacteristic
46C61×#450—24no—
47C2×#102×C61C2212yes—
48C2×C22⋊C8×C61C2211yescharacteristic
49C22×Dic122—24no—
50C22×Dic122—21no—
51C22×Dic122—22no—
52C22×Dic122—21no—
53C22×Dic122—42no—
54C22×Dic122D414yescomplemented
55C22×Dic122C2311yescharacteristic
56C22×Dic122C2311yescharacteristic
57C2×Q8×D61—22no—
58C2×Q8×D61—22no—
59C2×Q8×D61—24no—
60C2×Q8×D61—42no—
61C22×C22⋊D61—22no—
62C22×C22⋊D61D411yescharacteristic
63C23×Dic61—21no—
64C22×C4×D61—21no—
65C22×C4×D61C2311yescharacteristic
66C2×#11758200—232no—
67C2×#11758200—24no—
68C2×#11758200—24no—
69C2×#11758200—24no—
70C2×#11758200—24no—
71C2×#11758200—24no—
72C2×#11758200D418yescomplemented
73C22×#92418—22no—
74C22×#92418—24no—
75C22×#92418C2312yes—
76#14519776—28no—
77#14519775—22no—
78#14519775—22no—
79#14519757—22no—
80#14519757—22no—
81#14519755—28no—
82#14519754—28no—
83#14519748—24no—
84#14519748—42no—
85#14519746—44no—
86#14519742—22no—
87#14519742—22no—
88C4×C22⋊D61—22no—
89C4×C22⋊D61—22no—
90#14519737—28no—
91C2×C4⋊Dic61—21no—
92C2×C4⋊Dic61C2312yes—
93C2×#11758196—21no—
94C2×#11758196—21no—
95C2×#11758196—21no—
96C2×#11758196D411yescharacteristic
97C2×C22⋊Dic61—21no—
98C2×C22⋊Dic61—22no—
99C2×C22⋊Dic61—22no—
100C2×C22⋊Dic61—22no—
101C2×C22⋊Dic61D411yescharacteristic
102C2×#11758198—21no—
103C2×#11758198D411yescharacteristic
104C2×C4×Dic61—21no—
105C2×C4×Dic61C2311yescharacteristic
106#14519702—24no—
107#14519702—42no—
108#14519701—24no—
109#14519689—24no—
110#14519684—21no—
111C25×C61—21no—
112C22×Q8×C61C2311yescharacteristic
113C22×D4×C61—24no—
114C22×D4×C61—22no—
115C22×D4×C61—22no—
116C22×D4×C61—21no—
117C22×D4×C61—42no—
118C22×D4×C61C2312yes—
119C23×C4×C61C2311yescharacteristic
120C2×SD16×C61—24no—
121C2×SD16×C61—24no—
122C2×SD16×C61—42no—
123C22×C8×C61C2312yes—
124C22⋊Q8×C61—22no—
125C22⋊Q8×C61D414yescomplemented
126C2×C4⋊C4×C61—21no—
127C2×C4⋊C4×C61C2311yescharacteristic
128C2×C22⋊C4×C61—21no—
129#102×C61—24no—
130C22⋊C8×C61—22no—
131C24×C61—22no—
132C24×C61—44no—
133C24×C61—42no—
134C24×C61—42no—
135C24×C61—42no—
136C24×C61—82no—
137C24×C61C2×D411yescharacteristic
138C23×D61C2×D411yescharacteristic
139C2×Q8×C61—21no—
140C2×Q8×C61—22no—
141C2×Q8×C61—22no—
142C2×Q8×C61C2×D412yes—
143C2×D4×C61—28no—
144C2×D4×C61—22no—
145C2×D4×C61—22no—
146C2×D4×C61—22no—
147C2×D4×C61—22no—
148C2×D4×C61—22no—
149C2×D4×C61—48no—
150C2×D4×C61—44no—
151C2×D4×C61—44no—
152C2×D4×C61—42no—
153C2×D4×C61C2×D414yes—
154C22×C4×C61—22no—
155C22×C4×C61—22no—
156C22×C4×C61—21no—
157C22×C4×C61—21no—
158C22×C4×C61C2×D411yescharacteristic
159C22×C4×C61C2×D412yes—
160C22×C4×C61C2×D412yes—
161C22×C4×C61C2411yescharacteristic
162C2×C22⋊D61—22no—
163C2×C22⋊D61—22no—
164C2×C22⋊D61—44no—
165C2×C22⋊D61—44no—
166C2×C22⋊D61—82no—
167C2×C22⋊D61SD1614yescomplemented
168C22×Dic61—21no—
169C22×Dic61—24no—
170C22×Dic61—21no—
171C22×Dic61—22no—
172C22×Dic61—44no—
173C22×Dic61C2×D411yescharacteristic
174C22×Dic61C2×D411yescharacteristic
175C22×Dic61C2×D411yescharacteristic
176Q8×D61—48no—
177Q8×D61—44no—
178Q8×D61—44no—
179C2×C4×D61—24no—
180C2×C4×D61—22no—
181C2×C4×D61—21no—
182C2×C4×D61—21no—
183C2×C4×D61—22no—
184C2×C4×D61—44no—
185C2×Dic122—21no—
186C2×Dic122—22no—
187C2×Dic122—22no—
188C2×Dic122—22no—
189C2×Dic122—28no—
190C2×Dic122—24no—
191C2×Dic122—24no—
192C2×Dic122—21no—
193C2×Dic122—44no—
194C2×Dic122—48no—
195C2×Dic122—44no—
196C2×Dic122—44no—
197C2×Dic122C2×D412yes—
198C2×Dic122C2×D414yes—
199C2×Dic122C2×D412yes—
200SD16×C61—48no—
201C2×C8×C61—22no—
202C2×C8×C61—22no—
203C2×C8×C61—22no—
204C4⋊C4×C61—24no—
205C4⋊C4×C61—21no—
206C4⋊C4×C61C2×D412yes—
207C22⋊C4×C61—24no—
208C22⋊Dic61—22no—
209C22⋊Dic61—44no—
210C22⋊Dic61—42no—
211C22⋊Dic61—44no—
212C22⋊Dic61—82no—
213#11758200—216no—
214#11758200—216no—
215#11758198—24no—
216#11758198—22no—
217C4⋊Dic61—24no—
218C4⋊Dic61—22no—
219C4⋊Dic61—22no—
220#11758196—24no—
221#11758196—24no—
222#11758196—24no—
223#11758196—22no—
224C4×Dic61—24no—
225C4×Dic61—21no—
226C4×Dic61—21no—
227C2×#92418—28no—
228C2×#92418—22no—
229C2×#92418—22no—
230C2×#92418C2×D414yes—
231C23×C61—22no—
232C23×C61—22no—
233C23×C61—24no—
234C23×C61—48no—
235C23×C61—42no—
236C23×C61—42no—
237C23×C61—42no—
238C23×C61—42no—
239C23×C61—42no—
240C23×C61—42no—
241C23×C61—88no—
242C23×C61—88no—
243C23×C61—84no—
244C23×C61—84no—
245C23×C61#12012yes—
246C23×C61C2×SD1612yes—
247C23×C61#13712yes—
248C23×C61C22×D411yescharacteristic
249C22×D61—21no—
250C22×D61—22no—
251C22×D61—21no—
252C22×D61—22no—
253C22×D61C2×SD1612yescomplemented
254Q8×C61—24no—
255Q8×C61—41no—
256Q8×C61—41no—
257D4×C61—24no—
258D4×C61—24no—
259D4×C61—48no—
260D4×C61—42no—
261D4×C61—42no—
262C2×C4×C61—22no—
263C2×C4×C61—22no—
264C2×C4×C61—22no—
265C2×C4×C61—24no—
266C2×C4×C61—22no—
267C2×C4×C61—22no—
268C2×C4×C61—42no—
269C2×C4×C61—42no—
270C2×C4×C61#12012yes—
271C2×C4×C61C22×D411yescharacteristic
272C2×C4×C61C22×D412yes—
273C2×C4×C61C22×D411yescharacteristic
274C2×C4×C61H(5,𝔽2)12yes—
275C22⋊D61—28no—
276C22⋊D61—88no—
277C2×Dic61—21no—
278C2×Dic61—22no—
279C2×Dic61—24no—
280C2×Dic61—21no—
281C2×Dic61—21no—
282C2×Dic61—22no—
283C2×Dic61—21no—
284C2×Dic61—22no—
285C2×Dic61—22no—
286C2×Dic61—48no—
287C2×Dic61—42no—
288C2×Dic61—42no—
289C2×Dic61—42no—
290C2×Dic61—42no—
291C2×Dic61C2×SD1612yescomplemented
292C4×D61—48no—
293C4×D61—41no—
294C4×D61—41no—
295C4×D61—42no—
296Dic122—22no—
297Dic122—24no—
298Dic122—22no—
299Dic122—22no—
300Dic122—48no—
301Dic122—48no—
302Dic122—42no—
303Dic122#12014yescomplemented
304C8×C61—42no—
305#92418—24no—
306#23855—611nomaximal, Sylow
307C22×C61—22no—
308C22×C61—22no—
309C22×C61—24no—
310C22×C61—24no—
311C22×C61—44no—
312C22×C61—44no—
313C22×C61—44no—
314C22×C61—816no—
315C22×C61—82no—
316C22×C61—82no—
317C22×C61—82no—
318C22×C61—82no—
319C22×C61#44814yescomplemented
320C22×C61C2×#12011yescharacteristic
321C22×C61D4212yes—
322C22×C61C22×SD1611yescharacteristic
323C22×C61C2×#13711yescharacteristic
324C22×C61#57712yes—
325C2×D61—22no—
326C2×D61—24no—
327C2×D61—41no—
328C2×D61—41no—
329C2×D61—42no—
330C4×C61—22no—
331C4×C61—42no—
332C4×C61—42no—
333C4×C61C2×#12011yescharacteristic
334C4×C61#53411yescharacteristic
335C4×C61D4212yes—
336Dic61—24no—
337Dic61—22no—
338Dic61—42no—
339Dic61—41no—
340Dic61—41no—
341Dic61—82no—
342C23×SD16—611no—
343C2×D4×Q8—611no—
344C2×#448—612no—
345C2×#450D6111yescharacteristic, complemented
346#2408—611no—
347#2013—612no—
348#2000—612no—
349#1892—611no—
350#1880—612no—
351#1869—611no—
352#1787—611no—
353C2×C61—24no—
354C2×C61—84no—
355C2×C61#240311yescharacteristic
356C2×C61C2×#44811yescharacteristic
357C2×C61#301011yescharacteristic
358C2×C61#326612yes—
359C2×C61D4×SD1612yes—
360D61—42no—
361C23×Q8—611no—
362C23×Q8—611no—
363C23×D4C2×D6111yescharacteristic
364C22×SD16—612no—
365C22×SD16—612no—
366C22×SD16—1228no—
367C22×SD16—1224no—
368C22×SD16C2×D6112yes—
369C23×C8—611no—
370D4×Q8—1224no—
371D4×Q8—1224no—
372C2×C4⋊Q8—611no—
373C2×C4⋊Q8—1221no—
374C2×C22⋊Q8—611no—
375C2×C22⋊Q8—1221no—
376C2×C22⋊Q8—1221no—
377C2×C22⋊Q8C2×D6111yescharacteristic
378C2×#121—612no—
379C2×#120—1221no—
380C2×C4×Q8—611no—
381C2×C4×D4—611no—
382C2×C4×D4—1221no—
383#450—24no—
384#448—1228no—
385C2×#103—612no—
386C2×#103—612no—
387C2×#103—612no—
388C2×#102C2×D6112yes—
389C2×C22⋊C8—612no—
390C2×C22⋊C8C2×D6111yescharacteristic
391#394—1221no—
392#392—1221no—
393#390—612no—
394#380—611no—
395#379—1221no—
396#340—612no—
397#336—612no—
398C61#2385511yescharacteristic, complemented, Sylow
399C25—21no—
400C22×Q8—614no—
401C22×Q8—611no—
402C22×Q8—611no—
403C22×Q8—1221no—
404C22×Q8—1224no—
405C22×Q8—1222no—
406C22×Q8—1222no—
407C22×Q8—1221no—
408C22×Q8—1222no—
409C22×Q8—1224no—
410C22×Q8—2442no—
411C22×Q8—2442no—
412C22×Q8C22×D6111yescharacteristic
413C22×D4—24no—
414C22×D4—22no—
415C22×D4—21no—
416C22×D4—22no—
417C22×D4—42no—
418C22×D4—611no—
419C22×D4—1222no—
420C22×D4C22×D6112yes—
421C23×C4—611no—
422C23×C4—1221no—
423C23×C4—1221no—
424C23×C4C22×D6111yescharacteristic
425C2×SD16—24no—
426C2×SD16—24no—
427C2×SD16—42no—
428C2×SD16—618no—
429C2×SD16—1224no—
430C2×SD16—1224no—
431C2×SD16—1224no—
432C2×SD16—1224no—
433C2×SD16—12232no—
434C2×SD16—1224no—
435C22×C8—612no—
436C22×C8—1224no—
437C22×C8—1222no—
438C22×C8C22×D6112yes—
439C4⋊Q8—1228no—
440C4⋊Q8—1222no—
441C4⋊Q8—1222no—
442C22⋊Q8—22no—
443C22⋊Q8—1228no—
444C22⋊Q8—1222no—
445C22⋊Q8—1228no—
446C22⋊Q8—1222no—
447C22⋊Q8C22⋊D6114yescomplemented
448#121—1224no—
449#121—2442no—
450#120—2444no—
451C4×Q8—1222no—
452C4×Q8—1222no—
453C4×D4—1228no—
454C4×D4—1222no—
455C4×D4—1222no—
456C2×C4⋊C4—21no—
457C2×C4⋊C4—611no—
458C2×C4⋊C4—612no—
459C2×C4⋊C4—1221no—
460C2×C4⋊C4—1221no—
461C2×C4⋊C4—1221no—
462C2×C4⋊C4—1221no—
463C2×C4⋊C4C22×D6111yescharacteristic
464C2×C22⋊C4—21no—
465C2×C22⋊C4—611no—
466C2×C22⋊C4—611no—
467C2×C22⋊C4—1221no—
468C2×C22⋊C4—1222no—
469C2×C22⋊C4—1222no—
470C2×C22⋊C4—1222no—
471C2×C22⋊C4—1221no—
472C2×C42—611no—
473C2×C42—1221no—
474#103—1224no—
475#103—1224no—
476#103—2442no—
477#102—24no—
478C22⋊C8—22no—
479C22⋊C8—1224no—
480#95—1221no—
481C24—22no—
482C24—44no—
483C24—42no—
484C24—42no—
485C24—42no—
486C24—82no—
487C24—611no—
488C24D4×D6111yescharacteristic
489C2×Q8—21no—
490C2×Q8—22no—
491C2×Q8—22no—
492C2×Q8—614no—
493C2×Q8—612no—
494C2×Q8—612no—
495C2×Q8—1224no—
496C2×Q8—1228no—
497C2×Q8—1221no—
498C2×Q8—1221no—
499C2×Q8—1222no—
500C2×Q8—1224no—
501C2×Q8—1222no—
502C2×Q8—1222no—
503C2×Q8—2444no—
504C2×Q8—2444no—
505C2×Q8—2448no—
506C2×Q8—2444no—
507C2×Q8—2448no—
508C2×Q8—2444no—
509C2×Q8—2444no—
510C2×Q8C2×C22⋊D6112yes—
511C2×D4—28no—
512C2×D4—22no—
513C2×D4—22no—
514C2×D4—22no—
515C2×D4—22no—
516C2×D4—22no—
517C2×D4—48no—
518C2×D4—44no—
519C2×D4—44no—
520C2×D4—42no—
521C2×D4—614no—
522C2×D4—1222no—
523C2×D4—1222no—
524C2×D4—4882no—
525C2×D4—2444no—
526C2×D4—2444no—
527C2×D4D4×D6114yes—
528C22×C4—22no—
529C22×C4—22no—
530C22×C4—21no—
531C22×C4—21no—
532C22×C4—611no—
533C22×C4—611no—
534C22×C4—611no—
535C22×C4—1224no—
536C22×C4—1221no—
537C22×C4—1221no—
538C22×C4—1224no—
539C22×C4—1222no—
540C22×C4—1221no—
541C22×C4—1221no—
542C22×C4—1222no—
543C22×C4—1222no—
544C22×C4—2444no—
545C22×C4—2444no—
546C22×C4D4×D6112yes—
547C22×C4D4×D6111yescharacteristic
548C22×C4C2×C22⋊D6112yes—
549C22×C4C23×D6111yescharacteristic
550SD16—48no—
551SD16—12216no—
552SD16—12216no—
553C2×C8—22no—
554C2×C8—22no—
555C2×C8—22no—
556C2×C8—614no—
557C2×C8—1228no—
558C2×C8—1222no—
559C2×C8—1222no—
560C4⋊C4—21no—
561C4⋊C4—24no—
562C4⋊C4—1224no—
563C4⋊C4—1224no—
564C4⋊C4—1224no—
565C4⋊C4—1224no—
566C4⋊C4—1222no—
567C4⋊C4—1222no—
568C4⋊C4—1222no—
569C4⋊C4C2×C22⋊D6112yes—
570C22⋊C4—24no—
571C22⋊C4—1224no—
572C22⋊C4—1222no—
573C22⋊C4—1222no—
574C22⋊C4—4882no—
575C22⋊C4—2444no—
576C22⋊C4—2444no—
577C22⋊C4—2442no—
578C42—1224no—
579C42—1221no—
580C42—1221no—
581C23—22no—
582C23—22no—
583C23—24no—
584C23—48no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—88no—
592C23—88no—
593C23—84no—
594C23—84no—
595C23—612no—
596C23—1221no—
597C23—1221no—
598C23—1222no—
599C23—1222no—
600C23#1451974512yes—
601C23SD16×D6112yes—
602C23#1451980612yes—
603C23C2×D4×D6111yescharacteristic
604Q8—24no—
605Q8—41no—
606Q8—41no—
607Q8—614no—
608Q8—1222no—
609Q8—1224no—
610Q8—1222no—
611Q8—1222no—
612Q8—2448no—
613Q8—2448no—
614Q8—2442no—
615D4—24no—
616D4—24no—
617D4—48no—
618D4—42no—
619D4—42no—
620D4—1228no—
621D4—4888no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—24no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—612no—
631C2×C4—1221no—
632C2×C4—1222no—
633C2×C4—1221no—
634C2×C4—1224no—
635C2×C4—1221no—
636C2×C4—1221no—
637C2×C4—1222no—
638C2×C4—1222no—
639C2×C4—1222no—
640C2×C4—2448no—
641C2×C4—2448no—
642C2×C4—2442no—
643C2×C4—2442no—
644C2×C4—2442no—
645C2×C4—2442no—
646C2×C4—2441no—
647C2×C4—2441no—
648C2×C4—2442no—
649C2×C4#1451974512yes—
650C2×C4C2×D4×D6111yescharacteristic
651C2×C4C2×D4×D6112yes—
652C2×C4C22×C22⋊D6111yescharacteristic
653C2×C4#1451982212yes—
654C8—42no—
655C8—1224no—
656C22—22no—
657C22—24no—
658C22—22no—
659C22—24no—
660C22—44no—
661C22—44no—
662C22—44no—
663C22—816no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—1224no—
669C22—1222no—
670C22—2441no—
671C22—2441no—
672C22—2442no—
673C22#1683989914yescomplemented
674C22C2×#1451974511yescharacteristic
675C22#1684039412yes—
676C22C2×SD16×D6111yescharacteristic
677C22C2×#1451980611yescharacteristic
678C22#1684057212yes—
679C4—22no—
680C4—42no—
681C4—42no—
682C4—1222no—
683C4—1224no—
684C4—4882no—
685C4—2441no—
686C4—2441no—
687C4—2442no—
688C4#120×D6111yescharacteristic
689C4#1684032211yescharacteristic
690C4D4×C22⋊D6112yes—
691C2—24no—
692C2—84no—
693C2—2442no—
694C2#2144913011yescharacteristic
695C2C2×#1683989911yescharacteristic
696C2#2145510411yescharacteristic
697C2#2145733312yes—
698C2#2145751312yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#450C2
socleC23×C61C22×D4
prime radical#21449169C2
hypercenterC2×#450D61
hypocenterC61#23855
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#450

Complements

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

NH ≅ G/Ntype
1. C23×#11758200C2semidirect
2. C2×#16840425C2semidirect
3. C2×#16839899C2semidirect
5. #21446900C2semidirect
6. #21446823C2semidirect
7. #21446243C2semidirect
8. #21446171C2semidirect
9. #21446109C2semidirect
10. #21445755C2semidirect
11. C2×C61×#450C2semidirect
13. C22×Q8×D61C22semidirect
22. C2×#14519775C22semidirect
23. C2×#14519757C22semidirect
28. C2×#14519742C22semidirect
32. C2×#14519702C22semidirect
33. C2×#14519702C22semidirect
34. C2×#14519701C22semidirect
35. C2×#14519689C22semidirect
41. #16839524C22semidirect
42. #16839516C22semidirect
54. C22×Dic122D4semidirect
72. C2×#11758200D4semidirect
125. C22⋊Q8×C61D4semidirect
167. C2×C22⋊D61SD16semidirect
253. C22×D61C2×SD16semidirect
291. C2×Dic61C2×SD16semidirect
303. Dic122#120semidirect
319. C22×C61#448semidirect
345. C2×#450D61semidirect
398. C61#23855semidirect
447. C22⋊Q8C22⋊D61semidirect
673. C22#16839899semidirect

Maximal subgroups

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

subgroupindex
C23×#117582002
C2×#168404252
C2×#168398992
#214491692
#214469002
#214468232
#214462432
#214461712
#214461092
#214457552
C2×C61×#4502
#2385561