Finite Group Databasebeta

← #29714366

Subgroups — #29714366

Order 15616.

1394 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
1C2×#16840313C211yesmaximal, characteristic
2C2×#16839962C211yesmaximal, characteristic, complemented
3C4×#14519753C211yesmaximal, characteristic, complemented
4#21447010C212yesmaximal, complemented
5#21446821C212yesmaximal, complemented
6#21446772C211yesmaximal, characteristic, complemented
7#21446745C211yesmaximal, characteristic, complemented
8#21446150C212yesmaximal, complemented
9#21446077C212yesmaximal, complemented
10#21445796C211yesmaximal, characteristic, complemented
11C2×C61×#475C211yesmaximal, characteristic, complemented
12C22×#11758226—21no—
13C22×#11758201C2212yes—
14#16840313—24no—
15#16840313—24no—
16C2×#14519774C2211yescharacteristic
17C2×#14519766—21no—
18C2×#14519753C2211yescharacteristic
19C2×#14519752—21no—
20C2×#14519752C2212yes—
21C2×#14519739—21no—
22C2×#14519739C2212yes—
23C2×C42×D61C2211yescharacteristic
24#16839962—24no—
25C2×C4⋊#92418C2211yescharacteristic
26C2×#14519699C2212yes—
27C2×#14519699C2212yes—
28C2×#14519697C2212yes—
29C2×#14519689C2212yes—
30#16839669C2211yescharacteristic
31#16839658C2211yescharacteristic
32#16839643—21no—
33#16839633—21no—
34#16839619C2211yescharacteristic
35C4×#11758196—21no—
36C4×C4⋊Dic61C2211yescharacteristic
37C4×#11758198—21no—
38#16839524C2212yes—
39#16839524C2212yes—
40#16839516C2212yes—
41C2×C4⋊Q8×C61C2211yescharacteristic
42C2×C4×Q8×C61C2211yescharacteristic
43C61×#475—24no—
44C2×#106×C61C2212yes—
45C2×C4⋊C8×C61C2211yescharacteristic
46C2×#103×C61C2212yes—
47C2×#11758226—28no—
48C2×#11758226—44no—
49C2×#11758226—44no—
50C22×D244—22no—
51C22×D244—21no—
52C22×D244—21no—
53C22×D244C2312yes—
54C22×C4×D61—22no—
55C22×C4×D61D411yescharacteristic
56C22×C4×D61D411yescharacteristic
57C22×C4×D61C2311yescharacteristic
58C2×#11758201—24no—
59C2×#11758201—24no—
60C2×#11758201—42no—
61C22×#92418C2312yes—
62#14519774—22no—
63#14519774D414yes—
64#14519766—28no—
65#14519753—22no—
66#14519753—22no—
67#14519752—28no—
68#14519752—24no—
69#14519752—24no—
70#14519739—28no—
71#14519739—24no—
72#14519739—24no—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758198—21no—
78C2×#11758198—22no—
79C2×#11758198—22no—
80C2×#11758198—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61D411yescharacteristic
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61C2311yescharacteristic
85C42×D61—22no—
86C42×D61—22no—
87C4⋊#92418—22no—
88#14519699—24no—
89#14519699—42no—
90#14519697—24no—
91#14519689—42no—
92#14519684—21no—
93#14519684—21no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97C22×Q8×C61—21no—
98C22×Q8×C61C2311yescharacteristic
99C22×C8×C61C2312yes—
100C4⋊Q8×C61—22no—
101C4⋊Q8×C61D414yes—
102C4×Q8×C61—22no—
103C4×Q8×C61—22no—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61—21no—
106C2×C4⋊C4×C61C2311yescharacteristic
107C2×C4⋊C4×C61C2312yes—
108C2×C42×C61—21no—
109C2×C42×C61C2311yescharacteristic
110#106×C61—24no—
111C4⋊C8×C61—22no—
112#103×C61—24no—
113C23×D61—22no—
114C23×D61C2×D411yescharacteristic
115C2×Q8×C61—24no—
116C2×Q8×C61—21no—
117C2×Q8×C61—21no—
118C2×Q8×C61—21no—
119C2×Q8×C61—21no—
120C2×Q8×C61—21no—
121C2×Q8×C61—42no—
122C2×Q8×C61—42no—
123C2×Q8×C61#4112yescomplemented
124C22×C4×C61—22no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2411yescharacteristic
130C22×Dic61—21no—
131C22×Dic61—21no—
132C22×Dic61C2×D411yescharacteristic
133C22×Dic61C2×D411yescharacteristic
134C22×Dic61C2×D411yescharacteristic
135#11758226—432no—
136C2×D244—22no—
137C2×D244—22no—
138C2×D244—22no—
139C2×D244—22no—
140C2×D244—28no—
141C2×D244—24no—
142C2×D244—24no—
143C2×D244—22no—
144C2×D244—42no—
145C2×D244—42no—
146C2×D244—44no—
147C2×D244—44no—
148C2×D244—42no—
149C2×D244—42no—
150C2×D244C2×D414yes—
151C2×C4×D61—22no—
152C2×C4×D61—22no—
153C2×C4×D61—28no—
154C2×C4×D61—22no—
155C2×C4×D61—21no—
156C2×C4×D61—21no—
157C2×C4×D61—21no—
158C2×C4×D61—21no—
159C2×C4×D61—21no—
160C2×C4×D61—21no—
161C2×C4×D61—41no—
162C2×C4×D61—41no—
163C2×C4×D61—44no—
164C2×C4×D61—44no—
165C2×C4×D61SD1614yescomplemented
166C2×C8×C61—22no—
167C2×C8×C61—22no—
168C2×C8×C61—22no—
169C4⋊C4×C61—24no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—42no—
174C4⋊C4×C61C2×D414yes—
175C42×C61—42no—
176C42×C61C2×D412yes—
177C42×C61C2×D412yes—
178#11758201—48no—
179#11758198—28no—
180#11758198—28no—
181#11758198—24no—
182#11758198—42no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61#4112yescomplemented
187#11758196—24no—
188#11758196—42no—
189C4×Dic61—22no—
190C4×Dic61—22no—
191C4×Dic61—21no—
192C4×Dic61—21no—
193C4×Dic61—42no—
194C2×#92418—22no—
195C2×#92418—22no—
196C2×#92418—22no—
197C23×C61C22×D411yescharacteristic
198C22×D61—21no—
199C22×D61—21no—
200C22×D61—21no—
201C22×D61—21no—
202C22×D61—21no—
203C22×D61—21no—
204C22×D61—24no—
205C22×D61—42no—
206C22×D61—42no—
207C22×D61—42no—
208C22×D61—42no—
209C22×D61—42no—
210C22×D61—42no—
211C22×D61C2×SD1612yes—
212Q8×C61—22no—
213Q8×C61—22no—
214Q8×C61—41no—
215Q8×C61—41no—
216Q8×C61—48no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—24no—
220C2×C4×C61—22no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61#12012yes—
229C2×C4×C61#12012yes—
230C2×C4×C61C2×SD1612yes—
231C2×C4×C61AGL(1,ℤ/8)12yes—
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C61H(5,𝔽2)11yescharacteristic
237C2×C4×C612−1+411yescharacteristic
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—22no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61C2×SD1612yes—
253D244—24no—
254D244—24no—
255D244—48no—
256D244—416no—
257D244—42no—
258D244—42no—
259D244—48no—
260C4×D61—28no—
261C4×D61—416no—
262C4×D61—44no—
263C4×D61—41no—
264C4×D61—41no—
265C4×D61—41no—
266C4×D61—41no—
267C8×C61—42no—
268#92418—42no—
269#24154—611nomaximal, Sylow
270C22×C61C2×#12011yescharacteristic
271C22×C61#52511yescharacteristic
272C22×C61#53711yescharacteristic
273C22×C61C22×SD1611yescharacteristic
274C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
275C22×C61#57511yescharacteristic
276C22×C61#57511yescharacteristic
277C2×D61—22no—
278C2×D61—22no—
279C2×D61—22no—
280C2×D61—41no—
281C2×D61—41no—
282C2×D61—41no—
283C2×D61—41no—
284C2×D61—44no—
285C2×D61—44no—
286C2×D61—44no—
287C2×D61—82no—
288C2×D61—82no—
289C2×D61—82no—
290C2×D61—82no—
291C4×C61—22no—
292C4×C61—42no—
293C4×C61—44no—
294C4×C61—81no—
295C4×C61#45014yes—
296C4×C61#54611yescharacteristic
297C4×C61#54711yescharacteristic
298C4×C61#54811yescharacteristic
299C4×C61#55011yescharacteristic
300Dic61—22no—
301Dic61—42no—
302Dic61—41no—
303Dic61—41no—
304Dic61—81no—
305C2×#532—611no—
306C2×#475D6111yescharacteristic, complemented
307C2×#460—611no—
308C4×C22⋊Q8—611no—
309#2034—612no—
310#2000—612no—
311#1988—611no—
312#1982—611no—
313#1877—612no—
314#1864—612no—
315#1797—611no—
316C2×C61#240711yescharacteristic
317C2×C61C2×#45011yescharacteristic
318C2×C61#300511yescharacteristic
319C2×C61#310311yescharacteristic
320C2×C61#311511yescharacteristic
321C2×C61#324611yescharacteristic
322C2×C61#325111yescharacteristic
323D61—42no—
324D61—84no—
325C22×#41—1221no—
326C22×SD16—612no—
327#532—1224no—
328#532—1224no—
329C2×C4⋊Q8C2×D6111yescharacteristic
330C2×#127—611no—
331C2×#124—1221no—
332C2×C22⋊Q8—611no—
333C2×#121—612no—
334C2×#121—1221no—
335C2×C4×Q8C2×D6111yescharacteristic
336C2×C4×D4—612no—
337C2×C4×D4—1221no—
338C22×C42—611no—
339#475—24no—
340#460—1224no—
341C2×#106C2×D6112yes—
342C2×C4⋊C8—611no—
343C2×C4⋊C8C2×D6111yescharacteristic
344C2×#103C2×D6112yes—
345C2×#102—612no—
346C2×#102—612no—
347C2×#102—612no—
348C2×C22⋊C8—612no—
349#391—611no—
350#389—611no—
351#387—1221no—
352#385—1221no—
353#380—611no—
354C4×C4⋊C4—611no—
355C4×C4⋊C4—1221no—
356C4×C22⋊C4—1221no—
357#340—612no—
358#340—612no—
359#336—612no—
360C61#2415411yescharacteristic, complemented, Sylow
361C2×#41—1228no—
362C2×#41—2444no—
363C2×#41—2444no—
364C22×Q8—21no—
365C22×Q8C22×D6111yescharacteristic
366C22×D4—612no—
367C22×D4—1222no—
368C22×D4—1221no—
369C22×D4—1221no—
370C23×C4—611no—
371C23×C4—611no—
372C23×C4—611no—
373C23×C4—1222no—
374C2×SD16—1224no—
375C2×SD16—1224no—
376C2×SD16—2442no—
377C22×C8—612no—
378C22×C8C22×D6112yes—
379C4⋊Q8—22no—
380C4⋊Q8C22⋊D6114yes—
381#127—614no—
382#127—1222no—
383#124—1228no—
384C22⋊Q8—1222no—
385C22⋊Q8—1222no—
386#121—1224no—
387#121—1228no—
388#121—1224no—
389C4×Q8—22no—
390C4×Q8—22no—
391C4×D4—1224no—
392C4×D4—1228no—
393C4×D4—1224no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4C22×D6111yescharacteristic
401C2×C4⋊C4C22×D6112yes—
402C2×C22⋊C4—1222no—
403C2×C22⋊C4—1222no—
404C2×C22⋊C4—1221no—
405C2×C22⋊C4—1221no—
406C2×C42—21no—
407C2×C42—611no—
408C2×C42—611no—
409C2×C42—611no—
410C2×C42—1222no—
411C2×C42—1222no—
412C2×C42—1221no—
413C2×C42C22×D6111yescharacteristic
414#106—24no—
415C4⋊C8—22no—
416C4⋊C8—1222no—
417#103—24no—
418#102—1224no—
419#102—1224no—
420#102—2442no—
421C22⋊C8—2442no—
422#95—1221no—
423#95—1221no—
424#95—1221no—
425#95—1221no—
426#95—1221no—
427C24—611no—
428C24—1222no—
429#41—24432no—
430C2×Q8—21no—
431C2×Q8—24no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—42no—
437C2×Q8—42no—
438C2×Q8#1175822612yescomplemented
439C2×D4—614no—
440C2×D4—1228no—
441C2×D4—1222no—
442C2×D4—1222no—
443C2×D4—1222no—
444C2×D4—1222no—
445C2×D4—1222no—
446C2×D4—1224no—
447C2×D4—1224no—
448C2×D4—2442no—
449C2×D4—2442no—
450C2×D4—2444no—
451C2×D4—2444no—
452C2×D4—2442no—
453C2×D4—2442no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—611no—
458C22×C4—614no—
459C22×C4—611no—
460C22×C4—611no—
461C22×C4—1228no—
462C22×C4—1222no—
463C22×C4—1222no—
464C22×C4—1221no—
465C22×C4—1221no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—1221no—
471C22×C4—1222no—
472C22×C4—1221no—
473C22×C4—2441no—
474C22×C4—2441no—
475C22×C4—2444no—
476C22×C4—2444no—
477C22×C4D4×D6111yescharacteristic
478C22×C4D4×D6111yescharacteristic
479C22×C4C23×D6111yescharacteristic
480SD16—2448no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—1222no—
485C2×C8—1222no—
486C2×C8—1222no—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—42no—
492C4⋊C4—612no—
493C4⋊C4—1224no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—2442no—
498C4⋊C4C2×C22⋊D6114yes—
499C22⋊C4—1228no—
500C22⋊C4—1228no—
501C22⋊C4—1224no—
502C22⋊C4—2442no—
503C42—42no—
504C42—1221no—
505C42—1221no—
506C42—1222no—
507C42—1222no—
508C42—2442no—
509C42D4×D6112yes—
510C42C2×C22⋊D6112yes—
511C23—612no—
512C23—1221no—
513C23—1221no—
514C23—1221no—
515C23—1221no—
516C23—1221no—
517C23—1221no—
518C23—1224no—
519C23—2442no—
520C23—2442no—
521C23—2442no—
522C23—2442no—
523C23—2442no—
524C23—2442no—
525C23C2×D4×D6111yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—41no—
529Q8—48no—
530Q8—41no—
531D4—1224no—
532D4—1224no—
533D4—2448no—
534D4—24416no—
535D4—2448no—
536D4—2442no—
537D4—2442no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—612no—
550C2×C4—1221no—
551C2×C4—1221no—
552C2×C4—1228no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1222no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—2444no—
561C2×C4—24416no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4—2441no—
569C2×C4—2441no—
570C2×C4—2441no—
571C2×C4#1451974512yes—
572C2×C4#1451974512yes—
573C2×C4SD16×D6112yes—
574C2×C4#1451980412yes—
575C2×C4C2×D4×D6111yescharacteristic
576C2×C4C22×C22⋊D6111yescharacteristic
577C2×C4#41×D6111yescharacteristic
578C2×C4C2×#1175822611yescharacteristic
579C2×C4#1451982311yescharacteristic
580C2×C4#1451982511yescharacteristic
581C8—42no—
582C8—2442no—
583C22—1222no—
584C22—1222no—
585C22—1222no—
586C22—4882no—
587C22—4882no—
588C22—4882no—
589C22—4882no—
590C22—2444no—
591C22—2444no—
592C22—2444no—
593C22—2441no—
594C22—2441no—
595C22—2441no—
596C22—2441no—
597C22C2×#1451974511yescharacteristic
598C22#1684027511yescharacteristic
599C22#1684034211yescharacteristic
600C22C2×SD16×D6111yescharacteristic
601C22C2×#1451980411yescharacteristic
602C22#1684056111yescharacteristic
603C22#1684056111yescharacteristic
604C4—22no—
605C4—44no—
606C4—42no—
607C4—81no—
608C4—1222no—
609C4—4881no—
610C4—2442no—
611C4—2441no—
612C4—2441no—
613C4#1683991114yes—
614C4#1684040311yescharacteristic
615C4#1684041411yescharacteristic
616C4#1684042311yescharacteristic
617C4#1684043111yescharacteristic
618C2—4884no—
619C2—2442no—
620C2#2144916611yescharacteristic
621C2C2×#1683991111yescharacteristic
622C2#2145506011yescharacteristic
623C2#2145585211yescharacteristic
624C2#2145597311yescharacteristic
625C2#2145713411yescharacteristic
626C2#2145718411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#475C2
socleC23×C61C22×D4
prime radicalC2×#16840313C2
hypercenterC2×#475D61
hypocenterC61#24154
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#475

Complements

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

NH ≅ G/Ntype
2. C2×#16839962C2semidirect
3. C4×#14519753C2semidirect
4. #21447010C2semidirect
5. #21446821C2semidirect
6. #21446772C2semidirect
7. #21446745C2semidirect
8. #21446150C2semidirect
9. #21446077C2semidirect
10. #21445796C2semidirect
11. C2×C61×#475C2semidirect
123. C2×Q8×C61#41semidirect
165. C2×C4×D61SD16semidirect
186. C4⋊Dic61#41semidirect
306. C2×#475D61semidirect
360. C61#24154semidirect
438. C2×Q8#11758226semidirect

Maximal subgroups

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

subgroupindex
C2×#168403132
C2×#168399622
C4×#145197532
#214470102
#214468212
#214467722
#214467452
#214461502
#214460772
#214457962
C2×C61×#4752
#2415461