Finite Group Databasebeta

← #29716387

Subgroups — #29716387

Order 15616.

1330 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×D244C211yesmaximal, characteristic, complemented
2C2×#16840043C211yesmaximal, characteristic, complemented
3C2×C4×D488C211yesmaximal, characteristic
4#21450737C211yesmaximal, characteristic, complemented
5#21447148C212yesmaximal, complemented
6#21446956C212yesmaximal, complemented
7#21446780C211yesmaximal, characteristic, complemented
8#21446166C212yesmaximal
9#21446069C212yesmaximal, complemented
10#21445820C211yesmaximal, characteristic, complemented
11C61×#1984C211yesmaximal, characteristic, complemented
12C22×Q8×D61—21no—
13C22×D488C2211yescharacteristic
14Q8×D244—24no—
15Q8×D244—24no—
16C2×#14519777C2212yes—
17C2×#14519777C2211yescharacteristic
18C2×#14519757—22no—
19C2×#14519757—21no—
20C2×#14519753—21no—
21C2×C4×D244C2212yes—
22C2×C4×D244C2211yescharacteristic
23C2×C4×D244C2211yescharacteristic
24#16840043—24no—
25C4×D488—24no—
26C2×#14519711C2212yes—
27C2×#14519711C2211yescharacteristic
28C2×#14519699C2212yes—
29C2×#14519699C2212yes—
30C2×#14519699C2211yescharacteristic
31C2×#14519699C2211yescharacteristic
32#16839684—22no—
33#16839664C2212yes—
34#16839660C2212yes—
35#16839632C2211yescharacteristic
36#16839619—21no—
37#16839525C2212yes—
38#16839525C2212yes—
39#16839525C2212yes—
40C2×C4×Q8×C61C2211yescharacteristic
41C2×C4⋊C8×C61C2211yescharacteristic
42C2×C4×C8×C61C2211yescharacteristic
43C61×#336C2212yes—
44C61×#336C2212yes—
45C22×Dic122—22no—
46C22×Dic122—21no—
47C2×Q8×D61—28no—
48C2×Q8×D61—44no—
49C2×Q8×D61—44no—
50C22×D244C2311yescharacteristic
51C22×D244C2311yescharacteristic
52C22×C4×D61—22no—
53C22×C4×D61—21no—
54C22×C4×D61—21no—
55C2×D488—22no—
56C2×D488—22no—
57C2×D488Q814yescomplemented
58#14519777—22no—
59#14519777—24no—
60#14519777—24no—
61#14519777D414yes—
62#14519757—28no—
63#14519757—216no—
64#14519753—44no—
65C4×D244—24no—
66C4×D244—22no—
67C4×D244—22no—
68C4×D244—22no—
69C4×D244—24no—
70C4×D244D414yes—
71C2×C4⋊Dic61—22no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2312yes—
77C2×C4⋊Dic61C2312yes—
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—22no—
80C2×#11758198—22no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—22no—
84C2×#11758198—22no—
85#14519711—24no—
86#14519711—22no—
87#14519699—22no—
88#14519699—24no—
89#14519699—24no—
90#14519699—22no—
91#14519684—22no—
92C22×Q8×C61D411yescharacteristic
93C22×C8×C61C2311yescharacteristic
94C22×C8×C61C2312yes—
95C22×C8×C61C2311yescharacteristic
96C4×Q8×C61—22no—
97C4×Q8×C61—22no—
98C2×C4⋊C4×C61D412yes—
99C2×C4⋊C4×C61D411yescharacteristic
100C2×C42×C61C2312yes—
101C2×C42×C61C2311yescharacteristic
102C4⋊C8×C61—22no—
103C4×C8×C61—22no—
104C23×D61—21no—
105C23×D61—21no—
106C2×Q8×C61—22no—
107C2×Q8×C61—22no—
108C2×Q8×C61D814yescomplemented
109C22×C4×C61C2×D412yes—
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2×D412yes—
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2411yescharacteristic
114C22×Dic61—21no—
115C22×Dic61—22no—
116C22×Dic61—21no—
117C22×Dic61—22no—
118Q8×D61—432no—
119C2×D244—21no—
120C2×D244—21no—
121C2×D244—21no—
122C2×D244—21no—
123C2×D244—21no—
124C2×D244—21no—
125C2×D244—21no—
126C2×D244—21no—
127C2×D244C2×D412yes—
128C2×D244C2×Q812yes—
129C2×D244C2×Q812yescomplemented
130C2×D244#4112yes—
131C2×C4×D61—28no—
132C2×C4×D61—24no—
133C2×C4×D61—44no—
134C2×C4×D61—44no—
135C2×C4×D61—42no—
136C2×C4×D61—42no—
137C2×C4×D61—42no—
138C2×C4×D61—42no—
139C2×C4×D61—42no—
140C2×Dic122—28no—
141C2×Dic122—24no—
142C2×Dic122—44no—
143C2×Dic122—44no—
144C2×Dic122—42no—
145C2×Dic122—42no—
146C2×C8×C61—21no—
147C2×C8×C61—21no—
148C2×C8×C61—21no—
149C2×C8×C61—22no—
150C2×C8×C61—22no—
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61—22no—
154C2×C8×C61C2×Q812yescomplemented
155C4⋊C4×C61—24no—
156C4⋊C4×C61—22no—
157C42×C61—22no—
158C42×C61—22no—
159C42×C61C2×D412yes—
160C42×C61C2×D412yes—
161#11758198—28no—
162#11758198—24no—
163#11758198—42no—
164#11758198—44no—
165#11758198—82no—
166C4⋊Dic61—28no—
167C4⋊Dic61—24no—
168C4⋊Dic61—22no—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—22no—
173C4⋊Dic61—22no—
174C4⋊Dic61—42no—
175C4⋊Dic61—81no—
176C4⋊Dic61C2×D414yes—
177C4⋊Dic61C2×D412yes—
178#11758196—44no—
179D488—28no—
180C23×C61C22×D411yescharacteristic
181C22×D61—22no—
182C22×D61—22no—
183C22×D61—22no—
184C22×D61—22no—
185C22×D61—41no—
186C22×D61—41no—
187C22×D61—41no—
188C22×D61—41no—
189C22×D61—41no—
190C22×D61—41no—
191C22×D61—41no—
192C22×D61—41no—
193C22×D61—41no—
194C22×D61—41no—
195Q8×C61—28no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61#12012yes—
204C2×C4×C61#12012yes—
205C2×C4×C61C2×D814yes—
206C2×C4×C61C2×D812yes—
207C2×C4×C61AGL(1,ℤ/8)12yes—
208C2×C4×C61C22×D411yescharacteristic
209C2×C4×C61C22×D411yescharacteristic
210C2×C4×C61C22×Q811yescharacteristic
211C2×C4×C61C2×#4111yescharacteristic
212C2×C4×C612−1+411yescharacteristic
213C2×C4×C612−1+411yescharacteristic
214C2×Dic61—22no—
215C2×Dic61—24no—
216C2×Dic61—41no—
217C2×Dic61—42no—
218C2×Dic61—41no—
219C2×Dic61—42no—
220C2×Dic61—41no—
221C2×Dic61—42no—
222C2×Dic61—41no—
223C2×Dic61—42no—
224C2×Dic61—41no—
225C2×Dic61—42no—
226D244—22no—
227D244—22no—
228D244—22no—
229D244—22no—
230D244—22no—
231D244—41no—
232D244—41no—
233D244C22⋊Q814yescomplemented
234C4×D61—416no—
235C4×D61—48no—
236C4×D61—84no—
237Dic122—416no—
238Dic122—48no—
239C8×C61—22no—
240C8×C61—41no—
241C8×C61—42no—
242#24326—611nomaximal, Sylow
243C22×C61C2×#12011yescharacteristic
244C22×C61#54811yescharacteristic
245C22×C61D4×Q811yescharacteristic
246C22×C61C22×D811yescharacteristic
247C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
248C22×C61#57711yescharacteristic
249C22×C61#57811yescharacteristic
250C2×D61—24no—
251C2×D61—42no—
252C2×D61—42no—
253C2×D61—42no—
254C2×D61—42no—
255C2×D61—42no—
256C2×D61—42no—
257C2×D61—42no—
258C2×D61—42no—
259C2×D61—42no—
260C2×D61—81no—
261C2×D61—81no—
262C2×D61—81no—
263C2×D61—81no—
264C4×C61—24no—
265C4×C61—22no—
266C4×C61—42no—
267C4×C61#44714yes—
268C4×C61C2×C22⋊Q811yescharacteristic
269C4×C61#53611yescharacteristic
270C4×C61#54811yescharacteristic
271C4×C61D4×Q811yescharacteristic
272Dic61—42no—
273Dic61—44no—
274Dic61—82no—
275Dic61—81no—
276C2×D4×Q8—611no—
277C2×#474—611no—
278C2×C4×D8—611no—
279#2538—611no—
280#2057—612no—
281#2023—612no—
282#1990—611no—
283#1984D6111yescharacteristic, complemented
284#1879—612no—
285#1862—612no—
286#1802—611no—
287C2×C61#240911yescharacteristic
288C2×C61C2×#44711yescharacteristic
289C2×C61#301311yescharacteristic
290C2×C61#335811yescharacteristic
291C2×C61#336711yescharacteristic
292C2×C61Q8×D811yescharacteristic
293C2×C61#338411yescharacteristic
294D61—44no—
295D61—82no—
296C23×Q8—1221no—
297C22×D8—611no—
298D4×Q8—1224no—
299D4×Q8—1224no—
300C2×C4⋊Q8—611no—
301C2×C4⋊Q8—612no—
302C2×C22⋊Q8—1221no—
303C2×C22⋊Q8—1221no—
304C2×C22⋊Q8—1222no—
305C2×C4×Q8C2×D6111yescharacteristic
306C2×C4×D4—611no—
307C2×C4×D4—612no—
308C2×C4×D4—611no—
309#474—1224no—
310C4×D8—1224no—
311C2×#107—611no—
312C2×#107—612no—
313C2×C4⋊C8C2×D6111yescharacteristic
314C2×#102—612no—
315C2×#102—611no—
316C2×#102—612no—
317C2×#102—611no—
318C2×C4×C8C2×D6111yescharacteristic
319#396—1222no—
320#390—612no—
321#389—612no—
322#384—611no—
323#380—1221no—
324#340—612no—
325#340—612no—
326#340—612no—
327#336C2×D6112yes—
328#336C2×D6112yes—
329C61#2432611yescharacteristic, complemented, Sylow
330C22×Q8—1228no—
331C22×Q8—1222no—
332C22×Q8—1221no—
333C22×Q8—2444no—
334C22×Q8—2444no—
335C22×Q8D24411yescharacteristic
336C22×D4—611no—
337C22×D4—611no—
338C23×C4—1222no—
339C23×C4—1221no—
340C23×C4—1221no—
341C2×D8—614no—
342C2×D8—1222no—
343C2×D8—1222no—
344C22×C8C22×D6111yescharacteristic
345C22×C8C22×D6112yes—
346C22×C8C22×D6111yescharacteristic
347C4⋊Q8—614no—
348C4⋊Q8—1224no—
349C4⋊Q8—1224no—
350C4⋊Q8—1222no—
351C22⋊Q8—1228no—
352C22⋊Q8—12216no—
353C22⋊Q8—2444no—
354C4×Q8—22no—
355C4×Q8—22no—
356C4×D4—614no—
357C4×D4—1224no—
358C4×D4—1224no—
359C4×D4—1222no—
360C4×D4—1222no—
361C4×D4—1222no—
362C2×C4⋊C4—611no—
363C2×C4⋊C4—612no—
364C2×C4⋊C4—611no—
365C2×C4⋊C4—612no—
366C2×C4⋊C4—1222no—
367C2×C4⋊C4—1221no—
368C2×C4⋊C4—1221no—
369C2×C4⋊C4—1221no—
370C2×C4⋊C4—1222no—
371C2×C4⋊C4D24411yescharacteristic
372C2×C4⋊C4D24412yes—
373C2×C22⋊C4—1222no—
374C2×C22⋊C4—1221no—
375C2×C22⋊C4—1222no—
376C2×C22⋊C4—1221no—
377C2×C22⋊C4—1222no—
378C2×C42C22×D6112yes—
379C2×C42C22×D6111yescharacteristic
380#107—1222no—
381#107—1224no—
382C4⋊C8—22no—
383#102—1222no—
384#102—1224no—
385#102—1224no—
386#102—1222no—
387C4×C8—22no—
388#95—1222no—
389C24—1221no—
390C24—1221no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—1228no—
394C2×Q8—1224no—
395C2×Q8—2444no—
396C2×Q8—24432no—
397C2×Q8—2444no—
398C2×Q8—2442no—
399C2×Q8—2442no—
400C2×Q8D48814yescomplemented
401C2×D4—612no—
402C2×D4—612no—
403C2×D4—612no—
404C2×D4—612no—
405C2×D4—1221no—
406C2×D4—1221no—
407C2×D4—1221no—
408C2×D4—1221no—
409C2×D4—1221no—
410C2×D4—1221no—
411C2×D4—1221no—
412C2×D4—1221no—
413C22×C4—1228no—
414C22×C4—1222no—
415C22×C4—1221no—
416C22×C4—1224no—
417C22×C4—1221no—
418C22×C4—1222no—
419C22×C4—2444no—
420C22×C4—2442no—
421C22×C4—2442no—
422C22×C4—2442no—
423C22×C4—2444no—
424C22×C4—2442no—
425C22×C4—2442no—
426C22×C4C2×D24412yes—
427C22×C4C2×D24411yescharacteristic
428C22×C4C2×D24412yes—
429C22×C4C2×D24411yescharacteristic
430C22×C4C23×D6111yescharacteristic
431D8—1228no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—21no—
438C2×C8—22no—
439C2×C8—21no—
440C2×C8Q8×D6112yescomplemented
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—612no—
444C4⋊C4—614no—
445C4⋊C4—1228no—
446C4⋊C4—1222no—
447C4⋊C4—1221no—
448C4⋊C4—1224no—
449C4⋊C4—1221no—
450C4⋊C4—1221no—
451C4⋊C4—1222no—
452C4⋊C4—1222no—
453C4⋊C4—4881no—
454C4⋊C4—2444no—
455C4⋊C4—2442no—
456C22⋊C4—1228no—
457C22⋊C4—1224no—
458C22⋊C4—4882no—
459C22⋊C4—2442no—
460C22⋊C4—2444no—
461C42—22no—
462C42—22no—
463C42D4×D6112yes—
464C42D4×D6112yes—
465C23—1222no—
466C23—1222no—
467C23—1222no—
468C23—1222no—
469C23—2441no—
470C23—2441no—
471C23—2441no—
472C23—2441no—
473C23—2441no—
474C23—2441no—
475C23—2441no—
476C23—2441no—
477C23—2441no—
478C23—2441no—
479C23C22×D24411yescharacteristic
480Q8—28no—
481Q8—24416no—
482Q8—2448no—
483D4—614no—
484D4—1222no—
485D4—1222no—
486D4—1222no—
487D4—1222no—
488D4—1222no—
489D4—2441no—
490D4—2441no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—1224no—
499C2×C4—1222no—
500C2×C4—4884no—
501C2×C4—24416no—
502C2×C4—2441no—
503C2×C4—2442no—
504C2×C4—2441no—
505C2×C4—2441no—
506C2×C4—2442no—
507C2×C4—2442no—
508C2×C4—2442no—
509C2×C4—2441no—
510C2×C4—2442no—
511C2×C4—2441no—
512C2×C4—2448no—
513C2×C4#1451974712yes—
514C2×C4#1451974712yes—
515C2×C4C2×D48814yes—
516C2×C4C2×D48812yes—
517C2×C4#1451980512yes—
518C2×C4C2×D4×D6111yescharacteristic
519C2×C4C2×D4×D6111yescharacteristic
520C2×C4C2×Q8×D6111yescharacteristic
521C2×C4#41×D6111yescharacteristic
522C2×C4#1451982411yescharacteristic
523C2×C4#1451982511yescharacteristic
524C8—22no—
525C8—41no—
526C8—42no—
527C22—1224no—
528C22—4881no—
529C22—4881no—
530C22—4881no—
531C22—4881no—
532C22—2442no—
533C22—2442no—
534C22—2442no—
535C22—2442no—
536C22—2442no—
537C22—2442no—
538C22—2442no—
539C22—2442no—
540C22—2442no—
541C22C2×#1451974711yescharacteristic
542C22#1684042411yescharacteristic
543C22Q8×D24411yescharacteristic
544C22C22×D48811yescharacteristic
545C22C2×#1451980511yescharacteristic
546C22#1684057411yescharacteristic
547C22#1684057911yescharacteristic
548C4—22no—
549C4—24no—
550C4—42no—
551C4—4882no—
552C4—4881no—
553C4—2444no—
554C4—2442no—
555C4#1683989814yes—
556C4C22⋊Q8×D6111yescharacteristic
557C4#1684033511yescharacteristic
558C4#1684042011yescharacteristic
559C4#1684042711yescharacteristic
560C2—4882no—
561C2—2444no—
562C2#2144918211yescharacteristic
563C2C2×#1683989811yescharacteristic
564C2#2145513711yescharacteristic
565C2#2145845511yescharacteristic
566C2#2145853911yescharacteristic
567C2Q8×D48811yescharacteristic
568C2#2145872011yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1984C2
socleC23×C61C22×D4
prime radicalC22×D488C22
hypercenter#1984D61
hypocenterC61#24326
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #1984

Complements

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

NH ≅ G/Ntype
1. C2×Q8×D244C2semidirect
2. C2×#16840043C2semidirect
4. #21450737C2semidirect
5. #21447148C2semidirect
6. #21446956C2semidirect
7. #21446780C2semidirect
9. #21446069C2semidirect
10. #21445820C2semidirect
11. C61×#1984C2semidirect
57. C2×D488Q8semidirect
108. C2×Q8×C61D8semidirect
129. C2×D244C2×Q8semidirect
154. C2×C8×C61C2×Q8semidirect
233. D244C22⋊Q8semidirect
283. #1984D61semidirect
329. C61#24326semidirect
400. C2×Q8D488semidirect
440. C2×C8Q8×D61semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×D2442
C2×#168400432
C2×C4×D4882
#214507372
#214471482
#214469562
#214467802
#214461662
#214460692
#214458202
C61×#19842
#2432661