Finite Group Databasebeta

← #29708165

Subgroups — #29708165

Order 15616.

1520 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840312C211yesmaximal, characteristic, complemented
2C2×C4×D488C212yesmaximal, complemented
3C4×#14519751C211yesmaximal, characteristic, complemented
4#21447148C211yesmaximal, characteristic, complemented
5#21447054C211yesmaximal, characteristic, complemented
6#21447002C212yesmaximal, complemented
7#21446956C211yesmaximal, characteristic, complemented
8#21446892C211yesmaximal, characteristic, complemented
9#21446817C212yesmaximal, complemented
10C4×#14519699C212yesmaximal, complemented
11C4×C22⋊C8×C61C211yesmaximal, characteristic, complemented
12C22×#11758222—21no—
13C22×D488C2212yes—
14#16840312—28no—
15C2×#14519777C2211yescharacteristic
16C2×#14519774C2211yescharacteristic
17C2×#14519765—21no—
18C2×#14519751—21no—
19C2×#14519751C2212yes—
20C2×#14519751C2211yescharacteristic
21C2×C4×D244—21no—
22C2×C4×D244C2211yescharacteristic
23C2×C4×D244C2212yes—
24C2×C4×C22⋊D61—21no—
25C4×D488—28no—
26C2×#14519711C2212yes—
27C2×#14519699C2212yes—
28C2×#14519699C2212yes—
29C2×#14519699C2212yes—
30C2×#14519699C2212yes—
31#16839664C2211yescharacteristic
32#16839660C2211yescharacteristic
33#16839653—21no—
34#16839641—21no—
35#16839622C2211yescharacteristic
36C4×C4⋊Dic61C2211yescharacteristic
37C4×#11758198—21no—
38#16839525C2212yes—
39#16839525C2212yes—
40C22×C42×C61C2211yescharacteristic
41C2×C22⋊C8×C61C2212yes—
42C2×C4×C8×C61C2212yes—
43C61×#336C2211yescharacteristic
44C61×#336C2211yescharacteristic
45C2×#11758222—28no—
46C2×#11758222—44no—
47C2×#11758222—44no—
48C22×Dic122—21no—
49C22×D244—21no—
50C22×D244—21no—
51C22×D244C2312yes—
52C22×D244C2311yescharacteristic
53C22×C22⋊D61—21no—
54C22×C22⋊D61—22no—
55C22×C4×D61—22no—
56C22×C4×D61—21no—
57C2×D488—24no—
58C2×D488—24no—
59C2×D488—24no—
60#14519777—24no—
61#14519774—24no—
62#14519765—28no—
63#14519751—28no—
64#14519751—24no—
65#14519751—22no—
66#14519751—24no—
67#14519751—22no—
68C4×D244—28no—
69C4×D244—24no—
70C4×D244—24no—
71C4×D244D418yescomplemented
72C4×C22⋊D61—44no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2312yes—
77C2×C4⋊Dic61C2311yescharacteristic
78C2×#11758196—21no—
79C2×C22⋊Dic61—21no—
80C2×#11758198—22no—
81C2×#11758198—22no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×C4×Dic61—21no—
87#14519711—24no—
88#14519699—24no—
89#14519699—24no—
90#14519699—24no—
91#14519699—24no—
92#14519684—21no—
93#14519684—21no—
94#14519684—21no—
95C23×C4×C61D411yescharacteristic
96C23×C4×C61D411yescharacteristic
97C23×C4×C61C2311yescharacteristic
98C22×C8×C61C2312yes—
99C22×C8×C61C2312yes—
100C2×C42×C61—24no—
101C2×C42×C61D411yescharacteristic
102C2×C42×C61D411yescharacteristic
103C2×C42×C61C2311yescharacteristic
104C2×C42×C61C2311yescharacteristic
105C22⋊C8×C61—24no—
106C4×C8×C61—24no—
107C24×C61C2×D411yescharacteristic
108C23×D61—22no—
109C23×D61—21no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61—41no—
119C22×C4×C61D814yescomplemented
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2411yescharacteristic
126C2×C22⋊D61—28no—
127C2×C22⋊D61—24no—
128C2×C22⋊D61—42no—
129C2×C22⋊D61—44no—
130C2×C22⋊D61—44no—
131C2×C22⋊D61—42no—
132C22×Dic61—22no—
133C22×Dic61—21no—
134C22×Dic61—21no—
135C22×Dic61—21no—
136#11758222—432no—
137C2×D244—22no—
138C2×D244—22no—
139C2×D244—22no—
140C2×D244—22no—
141C2×D244—22no—
142C2×D244—22no—
143C2×D244—24no—
144C2×D244—24no—
145C2×D244—22no—
146C2×D244—21no—
147C2×D244—42no—
148C2×D244—42no—
149C2×D244—42no—
150C2×D244—42no—
151C2×D244C2×D414yes—
152C2×D244#4112yescomplemented
153C2×C4×D61—28no—
154C2×C4×D61—44no—
155C2×C4×D61—44no—
156C2×C4×D61—42no—
157C2×C4×D61—42no—
158C2×C4×D61—42no—
159C2×Dic122—24no—
160C2×Dic122—42no—
161C2×Dic122—42no—
162C2×C8×C61—22no—
163C2×C8×C61—22no—
164C2×C8×C61—22no—
165C2×C8×C61—22no—
166C2×C8×C61—22no—
167C2×C8×C61#4114yescomplemented
168C42×C61—22no—
169C42×C61—22no—
170C42×C61—22no—
171C42×C61C2×D414yes—
172C22⋊Dic61—42no—
173#11758198—28no—
174#11758198—24no—
175#11758198—28no—
176#11758198—42no—
177#11758198—42no—
178#11758198—42no—
179C4⋊Dic61—24no—
180C4⋊Dic61—22no—
181C4⋊Dic61—21no—
182C4⋊Dic61—22no—
183C4⋊Dic61—22no—
184C4⋊Dic61C2×D414yes—
185C4⋊Dic61#4112yescomplemented
186#11758196—42no—
187C4×Dic61—42no—
188D488—216no—
189C23×C61—21no—
190C23×C61—21no—
191C23×C61—21no—
192C23×C61—21no—
193C23×C61—21no—
194C23×C61—21no—
195C23×C61C2×D812yes—
196C23×C61C22×D411yescharacteristic
197C22×D61—24no—
198C22×D61—22no—
199C22×D61—41no—
200C22×D61—42no—
201C22×D61—42no—
202C22×D61—41no—
203C22×D61—41no—
204C22×D61—42no—
205C22×D61—41no—
206C22×D61—42no—
207C22×D61—42no—
208C22×D61—41no—
209C22×D61—42no—
210C22×D61—41no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—28no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—44no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61#12012yes—
225C2×C4×C61#12012yes—
226C2×C4×C61C2×D812yes—
227C2×C4×C61C2×D812yes—
228C2×C4×C61#13512yes—
229C2×C4×C61AGL(1,ℤ/8)12yes—
230C2×C4×C61C22×D412yes—
231C2×C4×C61C2×#4111yescharacteristic
232C2×C4×C61C2×#4112yes—
233C2×C4×C61C2×#4111yescharacteristic
234C22⋊D61—416no—
235C22⋊D61—48no—
236C2×Dic61—24no—
237C2×Dic61—22no—
238C2×Dic61—41no—
239C2×Dic61—42no—
240C2×Dic61—42no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248D244—24no—
249D244—24no—
250D244—24no—
251D244—48no—
252D244—48no—
253D244—42no—
254D244—42no—
255D244—42no—
256C4×D61—416no—
257C4×D61—84no—
258Dic122—48no—
259C8×C61—24no—
260C8×C61—42no—
261#23567—611nomaximal, Sylow
262C22×C61—22no—
263C22×C61—22no—
264C22×C61—22no—
265C22×C61—41no—
266C22×C61—41no—
267C22×C61—41no—
268C22×C61—41no—
269C22×C61C2×#12011yescharacteristic
270C22×C61#52511yescharacteristic
271C22×C61#53211yescharacteristic
272C22×C61C22×D811yescharacteristic
273C22×C61C2×#13511yescharacteristic
274C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
275C22×C61#57511yescharacteristic
276C2×D61—42no—
277C2×D61—42no—
278C2×D61—44no—
279C2×D61—42no—
280C2×D61—44no—
281C2×D61—44no—
282C2×D61—82no—
283C2×D61—82no—
284C2×D61—81no—
285C2×D61—82no—
286C2×D61—82no—
287C2×D61—81no—
288C2×D61—81no—
289C2×D61—81no—
290C4×C61—22no—
291C4×C61—22no—
292C4×C61—22no—
293C4×C61—41no—
294C4×C61#44714yes—
295C4×C61#52512yes—
296C4×C61#53212yes—
297Dic61—42no—
298Dic61—44no—
299Dic61—81no—
300Dic61—81no—
301C2×#532—611no—
302C2×C4×D8—612no—
303C4×#121—611no—
304#2057—611no—
305#2041—611no—
306#2033—612no—
307#2023—611no—
308#2011—611no—
309#1999—612no—
310C4×#102—612no—
311C4×C22⋊C8D6111yescharacteristic, complemented
312C2×C61—42no—
313C2×C61#238811yescharacteristic
314C2×C61C2×#44711yescharacteristic
315C2×C61#300211yescharacteristic
316C2×C61#310011yescharacteristic
317C2×C61#310611yescharacteristic
318C2×C61#316911yescharacteristic
319C2×C61#317111yescharacteristic
320D61—82no—
321D61—84no—
322C22×#41—1221no—
323C22×D8—612no—
324#532—1228no—
325C2×C4⋊Q8—611no—
326C2×#127—611no—
327C2×#124—1221no—
328C2×#121—612no—
329C2×#121—611no—
330C2×#121—1221no—
331C2×C4×D4—612no—
332C2×C4×D4—611no—
333C2×C4×D4—1221no—
334C2×C4×D4—1221no—
335C22×C42C2×D6111yescharacteristic
336C4×D8—1228no—
337C2×#107—612no—
338C2×#102—612no—
339C2×#102—612no—
340C2×#102—612no—
341C2×#102—612no—
342C2×C22⋊C8C2×D6112yes—
343C2×C4×C8C2×D6112yes—
344#390—611no—
345#389—611no—
346#388—1221no—
347#386—1221no—
348#380—611no—
349C4×C4⋊C4—611no—
350C4×C22⋊C4—1221no—
351#340—612no—
352#340—612no—
353#336C2×D6111yescharacteristic
354#336C2×D6111yescharacteristic
355C61#2356711yescharacteristic, complemented, Sylow
356C2×#41—1228no—
357C2×#41—2444no—
358C2×#41—2444no—
359C22×Q8—1221no—
360C22×D4—612no—
361C22×D4—611no—
362C22×D4—1221no—
363C22×D4—1222no—
364C22×D4—1221no—
365C22×D4—1221no—
366C23×C4—1222no—
367C23×C4—1221no—
368C23×C4D24411yescharacteristic
369C23×C4D24411yescharacteristic
370C23×C4C22×D6111yescharacteristic
371C2×D8—1224no—
372C2×D8—1224no—
373C2×D8—1224no—
374C22×C8C22×D6112yes—
375C22×C8C22×D6112yes—
376C4⋊Q8—1224no—
377#127—1224no—
378#124—1228no—
379#121—1224no—
380#121—1222no—
381#121—1224no—
382#121—1222no—
383#121—1228no—
384C4×D4—618no—
385C4×D4—1228no—
386C4×D4—1224no—
387C4×D4—1224no—
388C4×D4—2444no—
389C2×C4⋊C4—612no—
390C2×C4⋊C4—611no—
391C2×C4⋊C4—611no—
392C2×C4⋊C4—611no—
393C2×C4⋊C4—1221no—
394C2×C4⋊C4—1221no—
395C2×C22⋊C4—1222no—
396C2×C22⋊C4—1222no—
397C2×C22⋊C4—1221no—
398C2×C22⋊C4—1221no—
399C2×C22⋊C4—1221no—
400C2×C22⋊C4—1221no—
401C2×C22⋊C4—1221no—
402C2×C42—24no—
403C2×C42—1221no—
404C2×C42D24411yescharacteristic
405C2×C42D24411yescharacteristic
406C2×C42C22×D6111yescharacteristic
407C2×C42C22×D6111yescharacteristic
408#107—1224no—
409#102—1224no—
410#102—1224no—
411#102—1224no—
412#102—1224no—
413C22⋊C8—24no—
414C4×C8—24no—
415#95—1221no—
416#95—1221no—
417#95—1221no—
418C24—1221no—
419C24—1222no—
420C24C2×D24411yescharacteristic
421#41—24432no—
422C2×Q8—1224no—
423C2×Q8—2442no—
424C2×Q8—2442no—
425C2×D4—614no—
426C2×D4—612no—
427C2×D4—1224no—
428C2×D4—1224no—
429C2×D4—1228no—
430C2×D4—1224no—
431C2×D4—1222no—
432C2×D4—1222no—
433C2×D4—1222no—
434C2×D4—1222no—
435C2×D4—1221no—
436C2×D4—1222no—
437C2×D4—1222no—
438C2×D4—1222no—
439C2×D4—2442no—
440C2×D4—2442no—
441C2×D4—2444no—
442C2×D4—2442no—
443C2×D4—2442no—
444C2×D4—2444no—
445C2×D4—2442no—
446C2×D4—2442no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—41no—
456C22×C4—1221no—
457C22×C4—1222no—
458C22×C4—1228no—
459C22×C4—1221no—
460C22×C4—1221no—
461C22×C4—2442no—
462C22×C4—2442no—
463C22×C4—2442no—
464C22×C4—2444no—
465C22×C4—2444no—
466C22×C4D48814yescomplemented
467C22×C4C2×D24411yescharacteristic
468C22×C4C2×D24411yescharacteristic
469C22×C4C2×D24411yescharacteristic
470C22×C4C2×D24411yescharacteristic
471C22×C4C2×D24411yescharacteristic
472C22×C4C23×D6111yescharacteristic
473D8—12216no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8#1175822214yescomplemented
480C4⋊C4—614no—
481C4⋊C4—612no—
482C4⋊C4—1224no—
483C4⋊C4—1222no—
484C4⋊C4—1221no—
485C4⋊C4—1222no—
486C4⋊C4—1222no—
487C4⋊C4—2442no—
488C22⋊C4—1228no—
489C22⋊C4—1224no—
490C22⋊C4—1228no—
491C22⋊C4—2442no—
492C22⋊C4—2442no—
493C22⋊C4—2442no—
494C22⋊C4—2442no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—2442no—
499C42D4×D6114yes—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—1224no—
507C23—1222no—
508C23—2441no—
509C23—2442no—
510C23—2441no—
511C23—2442no—
512C23—2441no—
513C23—2442no—
514C23—2442no—
515C23—2441no—
516C23—2441no—
517C23—2442no—
518C23—2442no—
519C23—2441no—
520C23C2×D48812yes—
521C23C22×D24411yescharacteristic
522Q8—2448no—
523D4—1224no—
524D4—1224no—
525D4—1224no—
526D4—2448no—
527D4—2448no—
528D4—2448no—
529D4—24416no—
530D4—2442no—
531D4—2442no—
532D4—2442no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—28no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—1224no—
547C2×C4—1222no—
548C2×C4—4884no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4—2442no—
557C2×C4—2442no—
558C2×C4—2441no—
559C2×C4—24416no—
560C2×C4#1451974712yes—
561C2×C4#1451974712yes—
562C2×C4C2×D48812yes—
563C2×C4C2×D48812yes—
564C2×C4#1451979912yes—
565C2×C4#1451980512yes—
566C2×C4C2×D4×D6112yes—
567C2×C4#41×D6111yescharacteristic
568C2×C4#41×D6111yescharacteristic
569C2×C4C2×#1175822212yes—
570C8—24no—
571C8—42no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—4882no—
580C22—4881no—
581C22—4882no—
582C22—4882no—
583C22—4881no—
584C22—4881no—
585C22—4881no—
586C22—4882no—
587C22—2442no—
588C22—2442no—
589C22—2444no—
590C22—2444no—
591C22—2444no—
592C22—2442no—
593C22C2×#1451974711yescharacteristic
594C22#1684027411yescharacteristic
595C22#1684031211yescharacteristic
596C22C22×D48811yescharacteristic
597C22C2×#1451979911yescharacteristic
598C22C2×#1451980511yescharacteristic
599C22#1684056211yescharacteristic
600C4—22no—
601C4—22no—
602C4—22no—
603C4—41no—
604C4—4881no—
605C4—4881no—
606C4—2444no—
607C4—2442no—
608C4#1683989814yes—
609C4#1684027112yes—
610C4#1684030912yes—
611C2—42no—
612C2—4884no—
613C2—4882no—
614C2#2144903511yescharacteristic
615C2C2×#1683989811yescharacteristic
616C2#2145504111yescharacteristic
617C2#2145582011yescharacteristic
618C2#2145588011yescharacteristic
619C2#2145639811yescharacteristic
620C2#2145641411yescharacteristic
subgroupG/H
centerC22×C4C2×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC4×C22⋊C8×C61C2
socleC23×C61C22×D4
hypercenterC4×C22⋊C8D61
hypocenterC61#23567
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C4×C22⋊C8

Complements

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

NH ≅ G/Ntype
1. C2×#16840312C2semidirect
2. C2×C4×D488C2semidirect
3. C4×#14519751C2semidirect
4. #21447148C2semidirect
5. #21447054C2semidirect
6. #21447002C2semidirect
7. #21446956C2semidirect
8. #21446892C2semidirect
9. #21446817C2semidirect
10. C4×#14519699C2semidirect
11. C4×C22⋊C8×C61C2semidirect
71. C4×D244D4semidirect
119. C22×C4×C61D8semidirect
152. C2×D244#41semidirect
167. C2×C8×C61#41semidirect
185. C4⋊Dic61#41semidirect
311. C4×C22⋊C8D61semidirect
355. C61#23567semidirect
466. C22×C4D488semidirect
479. C2×C8#11758222semidirect

Maximal subgroups

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

subgroupindex
C2×#168403122
C2×C4×D4882
C4×#145197512
#214471482
#214470542
#214470022
#214469562
#214468922
#214468172
C4×#145196992
C4×C22⋊C8×C612
#2356761