Finite Group Databasebeta

← #29736627

Subgroups — #29736627

Order 15616.

1446 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519776C211yesmaximal, characteristic, complemented
2C2×#16840107C212yesmaximal, complemented
3#21454162C211yesmaximal, characteristic
4#21446954C212yesmaximal, complemented
5#21446953C212yesmaximal, complemented
6#21446778C212yesmaximal, complemented
7#21446137C211yesmaximal, characteristic, complemented
8#21446136C211yesmaximal, characteristic, complemented
9#21446134C211yesmaximal, characteristic, complemented
10C22⋊C8×Dic61C211yesmaximal, characteristic, complemented
11C61×#1874C211yesmaximal, characteristic, complemented
12C23×Dic122—21no—
13C2×#14519776—28no—
14C2×#14519776—21no—
15C2×#14519776—21no—
16C2×#14519776C2212yes—
17C2×#14519776C2212yes—
18C2×#14519773C2211yescharacteristic
19C2×#14519751C2211yescharacteristic
20C2×#14519749—21no—
21C2×#14519749—21no—
22C22×C4⋊Dic61C2211yescharacteristic
23C22×#11758196—21no—
24C22×C4×Dic61C2211yescharacteristic
25#16840107—28no—
26C2×#14519699C2212yes—
27C2×#14519698C2212yes—
28C2×#14519697C2212yes—
29C2×C22⋊#92418C2211yescharacteristic
30C2×C8×Dic61C2212yes—
31#16839665C2211yescharacteristic
32#16839663C2211yescharacteristic
33#16839662—21no—
34#16839662—21no—
35#16839631C2211yescharacteristic
36#16839525C2212yes—
37#16839524C2212yes—
38#16839524C2212yes—
39#16839516C2212yes—
40C2×#121×C61C2211yescharacteristic
41C22×C4⋊C4×C61C2211yescharacteristic
42C2×#102×C61C2212yes—
43C2×C22⋊C8×C61C2211yescharacteristic
44C61×#340C2212yes—
45C22×Dic122—28no—
46C22×Dic122—22no—
47C22×Dic122—22no—
48C22×Dic122—44no—
49C22×Dic122—44no—
50C22×D244C2311yescharacteristic
51C22×C22⋊D61—21no—
52C22×C22⋊D61—21no—
53C22×C22⋊D61—21no—
54C23×Dic61—21no—
55C23×Dic61D411yescharacteristic
56C23×Dic61D411yescharacteristic
57C22×#92418C2312yes—
58#14519776—28no—
59#14519776—28no—
60#14519776—24no—
61#14519776—24no—
62#14519776—24no—
63#14519776D418yescomplemented
64#14519773—24no—
65#14519751—22no—
66#14519751—22no—
67#14519749—44no—
68#14519749—44no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—24no—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2312yes—
73C2×#11758196—28no—
74C2×#11758196—22no—
75C2×#11758196—22no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×C4×Dic61—24no—
84C2×C4×Dic61D411yescharacteristic
85C2×C4×Dic61D411yescharacteristic
86C2×C4×Dic61C2311yescharacteristic
87C2×C4×Dic61C2311yescharacteristic
88#14519699—24no—
89#14519698—24no—
90#14519697—24no—
91C22⋊#92418—22no—
92C8×Dic61—24no—
93C22×D4×C61—21no—
94C22×D4×C61C2311yescharacteristic
95C23×C4×C61—21no—
96C23×C4×C61C2311yescharacteristic
97C22×C8×C61C2312yes—
98#121×C61—22no—
99#121×C61—22no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—24no—
102C2×C4⋊C4×C61C2311yescharacteristic
103C2×C4⋊C4×C61C2312yes—
104C2×C22⋊C4×C61—21no—
105#102×C61—24no—
106C22⋊C8×C61—22no—
107C24×C61—21no—
108C24×C61C2×D411yescharacteristic
109C23×D61—21no—
110C2×D4×C61—22no—
111C2×D4×C61—21no—
112C2×D4×C61—21no—
113C2×D4×C61—22no—
114C2×D4×C61—42no—
115C2×D4×C61—42no—
116C2×D4×C61—42no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—24no—
120C22×C4×C61—22no—
121C22×C4×C61—21no—
122C22×C4×C61—42no—
123C22×C4×C61—42no—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61#4112yes—
126C22×C4×C61#4112yes—
127C22×C4×C61C2411yescharacteristic
128C2×C22⋊D61—42no—
129C2×C22⋊D61—42no—
130C2×C22⋊D61—42no—
131C2×C22⋊D61—42no—
132C2×C22⋊D61—42no—
133C2×C22⋊D61—42no—
134C2×C22⋊D61—42no—
135C2×C22⋊D61—42no—
136C2×C22⋊D61—42no—
137C22×Dic61—22no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—22no—
141C22×Dic61—24no—
142C22×Dic61—22no—
143C22×Dic61—21no—
144C22×Dic61—42no—
145C22×Dic61—42no—
146C22×Dic61D814yes—
147C22×Dic61SD1614yes—
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152C2×D244—22no—
153C2×D244—21no—
154C2×D244—21no—
155C2×D244—22no—
156C2×Dic122—28no—
157C2×Dic122—28no—
158C2×Dic122—44no—
159C2×Dic122—432no—
160C2×Dic122—44no—
161C2×Dic122—44no—
162C2×Dic122—44no—
163C2×C8×C61—22no—
164C2×C8×C61—22no—
165C2×C8×C61—22no—
166C4⋊C4×C61—24no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61C2×D414yes—
171C22⋊C4×C61—42no—
172C22⋊Dic61—42no—
173C22⋊Dic61—42no—
174#11758198—42no—
175#11758198—42no—
176#11758198—42no—
177C4⋊Dic61—24no—
178C4⋊Dic61—22no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61C2×D414yes—
182#11758196—28no—
183#11758196—28no—
184#11758196—42no—
185#11758196—42no—
186C4×Dic61—22no—
187C4×Dic61—22no—
188C4×Dic61—22no—
189C4×Dic61C2×D414yes—
190C2×#92418—22no—
191C2×#92418—22no—
192C2×#92418—22no—
193C23×C61—21no—
194C23×C61—21no—
195C23×C61—21no—
196C23×C61—21no—
197C23×C61—41no—
198C23×C61—41no—
199C23×C61—41no—
200C23×C61—41no—
201C23×C61—41no—
202C23×C61—41no—
203C23×C61—41no—
204C23×C61#12412yes—
205C23×C61C2×D812yes—
206C23×C61C2×SD1612yes—
207C23×C61C22×D411yescharacteristic
208C22×D61—41no—
209C22×D61—41no—
210C22×D61—41no—
211C22×D61—41no—
212C22×D61—41no—
213C22×D61—41no—
214C22×D61—41no—
215D4×C61—42no—
216D4×C61—42no—
217D4×C61—84no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—24no—
221C2×C4×C61—24no—
222C2×C4×C61—24no—
223C2×C4×C61—48no—
224C2×C4×C61—42no—
225C2×C4×C61—42no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61#12412yes—
230C2×C4×C61C22×D412yes—
231C2×C4×C61C2×#4111yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×C4×C612−1+412yes—
234C22⋊D61—84no—
235C22⋊D61—84no—
236C22⋊D61—84no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—28no—
242C2×Dic61—28no—
243C2×Dic61—24no—
244C2×Dic61—48no—
245C2×Dic61—42no—
246C2×Dic61—42no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61#12012yes—
251C2×Dic61#12012yes—
252C2×Dic61C2×D812yes—
253C2×Dic61C2×D812yes—
254C2×Dic61C2×SD1612yes—
255C2×Dic61C2×SD1612yes—
256C2×Dic61AGL(1,ℤ/8)12yes—
257C2×Dic61#13712yes—
258D244—42no—
259D244—42no—
260Dic122—416no—
261Dic122—416no—
262C8×C61—42no—
263#92418—42no—
264#25903—611nomaximal, Sylow
265C22×C61—22no—
266C22×C61—22no—
267C22×C61—22no—
268C22×C61—22no—
269C22×C61—22no—
270C22×C61—22no—
271C22×C61—81no—
272C22×C61—81no—
273C22×C61—81no—
274C22×C61—81no—
275C22×C61—81no—
276C22×C61—81no—
277C22×C61—81no—
278C22×C61#48614yescomplemented
279C22×C61C2×#12011yescharacteristic
280C22×C61C2×#12411yescharacteristic
281C22×C61#53611yescharacteristic
282C22×C61C22×D811yescharacteristic
283C22×C61C22×SD1611yescharacteristic
284C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
285C22×C61C2×#13711yescharacteristic
286C2×D61—81no—
287C2×D61—81no—
288C2×D61—81no—
289C2×D61—81no—
290C2×D61—81no—
291C2×D61—81no—
292C2×D61—81no—
293C4×C61—22no—
294C4×C61—44no—
295C4×C61—81no—
296C4×C61#54812yes—
297C4×C61#54812yes—
298Dic61—22no—
299Dic61—22no—
300Dic61—44no—
301Dic61—81no—
302Dic61#44714yes—
303Dic61#44814yes—
304C22×C4⋊Q8—611no—
305C2×#486—612no—
306#2856—611no—
307#2023—612no—
308#2023—612no—
309#1990—612no—
310#1874—611no—
311#1874—611no—
312#1874—611no—
313#1874D6111yescharacteristic, complemented
314C4×C22⋊C8—611no—
315C2×C61—24no—
316C2×C61—161no—
317C2×C61#243611yescharacteristic
318C2×C61C2×#44711yescharacteristic
319C2×C61C2×#44811yescharacteristic
320C2×C61C2×#48611yescharacteristic
321C2×C61#314111yescharacteristic
322C2×C61#322611yescharacteristic
323C2×C61#322711yescharacteristic
324D61—161no—
325C23×Q8—1221no—
326C2×C4⋊Q8—612no—
327C2×C4⋊Q8—612no—
328C2×C4⋊Q8—1221no—
329C2×C4⋊Q8—1221no—
330C2×C4⋊Q8—1228no—
331C2×#127—611no—
332C2×#121—611no—
333C2×#121—1221no—
334C2×#121—1221no—
335C2×#121C2×D6111yescharacteristic
336C22×C4⋊C4—611no—
337C22×C4⋊C4—1221no—
338C22×C4⋊C4C2×D6111yescharacteristic
339C22×C42—611no—
340#486—1228no—
341C2×#102—612no—
342C2×#102—612no—
343C2×#102—612no—
344C2×#102C2×D6112yes—
345C2×C22⋊C8—611no—
346C2×C22⋊C8C2×D6111yescharacteristic
347C2×C4×C8—612no—
348#390—611no—
349#390—611no—
350#390—1221no—
351#390—1221no—
352#384—611no—
353#340—612no—
354#340—612no—
355#340—612no—
356#340C2×D6112yes—
357#336—612no—
358C61#2590311yescharacteristic, complemented, Sylow
359C22×Q8—1222no—
360C22×Q8—1222no—
361C22×Q8—1228no—
362C22×Q8—2444no—
363C22×Q8—2444no—
364C22×D4—21no—
365C22×D4—611no—
366C22×D4—1221no—
367C22×D4—1221no—
368C22×D4—1221no—
369C22×D4C22×D6111yescharacteristic
370C23×C4—21no—
371C23×C4—611no—
372C23×C4—611no—
373C23×C4—1221no—
374C23×C4C22×D6111yescharacteristic
375C22×C8—612no—
376C22×C8C22×D6112yes—
377C4⋊Q8—618no—
378C4⋊Q8—1224no—
379C4⋊Q8—1224no—
380C4⋊Q8—1228no—
381C4⋊Q8—1228no—
382C4⋊Q8—1224no—
383#127—1224no—
384#121—22no—
385#121—22no—
386#121—1222no—
387#121—1222no—
388#121—2444no—
389#121—2444no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—24no—
392C2×C4⋊C4—612no—
393C2×C4⋊C4—611no—
394C2×C4⋊C4—1224no—
395C2×C4⋊C4—1222no—
396C2×C4⋊C4—1222no—
397C2×C4⋊C4—1228no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4C22×D6111yescharacteristic
402C2×C4⋊C4C22×D6112yes—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—1221no—
405C2×C22⋊C4—1221no—
406C2×C22⋊C4—1221no—
407C2×C22⋊C4—1221no—
408C2×C22⋊C4—1221no—
409C2×C42—611no—
410C2×C42—611no—
411C2×C42—611no—
412C2×C42—611no—
413C2×C42—1224no—
414#102—24no—
415#102—1224no—
416#102—1224no—
417#102—1224no—
418C22⋊C8—22no—
419C22⋊C8—1222no—
420C4×C8—1224no—
421C24—21no—
422C24—1221no—
423C24D4×D6111yescharacteristic
424C2×Q8—1228no—
425C2×Q8—1228no—
426C2×Q8—2444no—
427C2×Q8—2444no—
428C2×Q8—2444no—
429C2×Q8—24432no—
430C2×Q8—2444no—
431C2×D4—22no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—22no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—1222no—
439C2×D4—1221no—
440C2×D4—1221no—
441C2×D4—1222no—
442C2×D4—2442no—
443C2×D4—2442no—
444C2×D4—2442no—
445C2×D4—2442no—
446C2×D4—2442no—
447C2×D4—2442no—
448C2×D4—2442no—
449C2×D4—2442no—
450C2×D4—2442no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—611no—
459C22×C4—611no—
460C22×C4—611no—
461C22×C4—611no—
462C22×C4—614no—
463C22×C4—614no—
464C22×C4—1222no—
465C22×C4—1222no—
466C22×C4—1222no—
467C22×C4—1222no—
468C22×C4—1224no—
469C22×C4—1222no—
470C22×C4—1221no—
471C22×C4—2442no—
472C22×C4—2442no—
473C22×C4#1175822212yes—
474C22×C4D4×D6111yescharacteristic
475C22×C4#1175822412yes—
476C22×C4C23×D6111yescharacteristic
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—1222no—
481C2×C8—1222no—
482C2×C8—1222no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—614no—
488C4⋊C4—1228no—
489C4⋊C4—1228no—
490C4⋊C4—1224no—
491C4⋊C4—1222no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—2442no—
495C4⋊C4—2442no—
496C4⋊C4D4×D6114yes—
497C22⋊C4—42no—
498C22⋊C4—2442no—
499C22⋊C4—2442no—
500C22⋊C4—2442no—
501C22⋊C4—2442no—
502C22⋊C4—2442no—
503C42—614no—
504C42—1222no—
505C42—1222no—
506C42—1222no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—2441no—
519C23—2441no—
520C23—2441no—
521C23—2441no—
522C23—2441no—
523C23—2441no—
524C23—2441no—
525C23#1451976312yes—
526C23D8×D6112yes—
527C23SD16×D6112yes—
528C23C2×D4×D6111yescharacteristic
529Q8—24416no—
530Q8—24416no—
531D4—42no—
532D4—42no—
533D4—84no—
534D4—4884no—
535D4—4884no—
536D4—4884no—
537D4—2442no—
538D4—2442no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—48no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—612no—
551C2×C4—612no—
552C2×C4—612no—
553C2×C4—612no—
554C2×C4—612no—
555C2×C4—612no—
556C2×C4—612no—
557C2×C4—612no—
558C2×C4—1228no—
559C2×C4—1228no—
560C2×C4—1221no—
561C2×C4—1221no—
562C2×C4—1221no—
563C2×C4—1221no—
564C2×C4—1224no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4—2448no—
569C2×C4—2442no—
570C2×C4—2442no—
571C2×C4#1451976312yes—
572C2×C4C2×D4×D6112yes—
573C2×C4C2×#1175822411yescharacteristic
574C2×C4C2×#1175822211yescharacteristic
575C2×C4#1451982412yes—
576C8—42no—
577C8—2442no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22—4881no—
592C22—4881no—
593C22—4881no—
594C22—4881no—
595C22—4881no—
596C22—4881no—
597C22—4881no—
598C22#1684010714yescomplemented
599C22#120×D6111yescharacteristic
600C22C2×#1451976311yescharacteristic
601C22#1684033411yescharacteristic
602C22C2×D8×D6111yescharacteristic
603C22C2×SD16×D6111yescharacteristic
604C22AGL(1,ℤ/8)×D6111yescharacteristic
605C22#137×D6111yescharacteristic
606C4—22no—
607C4—44no—
608C4—81no—
609C4—614no—
610C4—614no—
611C4—1222no—
612C4—1222no—
613C4—4881no—
614C4—2444no—
615C4#1684041812yes—
616C4#1684042112yes—
617C2—24no—
618C2—161no—
619C2—9761no—
620C2#2144947511yescharacteristic
621C2#447×D6111yescharacteristic
622C2#448×D6111yescharacteristic
623C2C2×#1684010711yescharacteristic
624C2#2145621011yescharacteristic
625C2#2145693811yescharacteristic
626C2#2145694511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1874C2
socleC23×C61C22×D4
prime radical#21454162C2
hypercenter#1874D61
hypocenterC61#25903
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1874

Complements

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

NH ≅ G/Ntype
1. C22×#14519776C2semidirect
2. C2×#16840107C2semidirect
4. #21446954C2semidirect
5. #21446953C2semidirect
6. #21446778C2semidirect
7. #21446137C2semidirect
8. #21446136C2semidirect
9. #21446134C2semidirect
10. C22⋊C8×Dic61C2semidirect
11. C61×#1874C2semidirect
63. #14519776D4semidirect
278. C22×C61#486semidirect
313. #1874D61semidirect
358. C61#25903semidirect
598. C22#16840107semidirect

Maximal subgroups

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

subgroupindex
C22×#145197762
C2×#168401072
#214541622
#214469542
#214469532
#214467782
#214461372
#214461362
#214461342
C22⋊C8×Dic612
C61×#18742
#2590361