Finite Group Databasebeta

← #29716396

Subgroups — #29716396

Order 15616.

1146 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840469C211yesmaximal, characteristic, complemented
2C2×#16840053C211yesmaximal, characteristic, complemented
3C2×C4×#11758199C211yesmaximal, characteristic
4#21450742C211yesmaximal, characteristic, complemented
5#21447156C212yesmaximal, complemented
6#21446964C212yesmaximal, complemented
7#21446779C211yesmaximal, characteristic, complemented
8#21446754C211yesmaximal, characteristic, complemented
9#21446165C212yesmaximal
10#21446082C212yesmaximal, complemented
11C61×#1802C211yesmaximal, characteristic, complemented
12C22×#11758199C2211yescharacteristic
13#16840469—24no—
14#16840469—24no—
15C2×#14519777C2211yescharacteristic
16C2×#14519768C2212yes—
17C2×#14519757—21no—
18C2×#14519756—21no—
19C2×#14519754—22no—
20C2×C4×Dic122C2211yescharacteristic
21C2×C4×D244C2211yescharacteristic
22C2×D4×Dic61C2212yes—
23C22×C4⋊Dic61—21no—
24#16840053—24no—
25C4×#11758199—24no—
26C2×#14519710C2211yescharacteristic
27C2×#14519709C2212yes—
28C2×#14519699C2212yes—
29C2×#14519698C2211yescharacteristic
30C2×#14519697C2211yescharacteristic
31C2×C4×#92418C2211yescharacteristic
32#16839684—22no—
33#16839663C2212yes—
34#16839661C2212yes—
35#16839619—21no—
36#16839524C2212yes—
37#16839524C2212yes—
38#16839516C2212yes—
39#16839516C2212yes—
40C2×C4×D4×C61C2211yescharacteristic
41C2×C4⋊C8×C61C2211yescharacteristic
42C2×#102×C61C2212yes—
43C61×#384C2211yescharacteristic
44C61×#340C2212yes—
45C22×Dic122—21no—
46C22×Dic122D411yescharacteristic
47C22×D244C2311yescharacteristic
48C23×Dic61—22no—
49C22×C4×D61—21no—
50C2×#11758199—22no—
51C2×#11758199—22no—
52C2×#11758199Q814yescomplemented
53C22×#92418C2311yescharacteristic
54C22×#92418C2311yescharacteristic
55#14519777—22no—
56#14519777D414yes—
57#14519768—24no—
58#14519768—24no—
59#14519757—44no—
60#14519756—28no—
61#14519754—216no—
62C4×Dic122—22no—
63C4×Dic122—22no—
64C4×D244—22no—
65C4×D244—22no—
66D4×Dic61—24no—
67D4×Dic61—24no—
68C2×C4⋊Dic61—28no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—22no—
71C2×C4⋊Dic61—22no—
72C2×C4⋊Dic61D411yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2312yes—
75C2×#11758196—22no—
76C2×#11758196D412yes—
77C2×C22⋊Dic61—22no—
78C2×#11758198—21no—
79C2×#11758198—22no—
80C2×#11758198—22no—
81C2×C4×Dic61C2312yes—
82#14519710—22no—
83#14519709—24no—
84#14519699—24no—
85#14519698—22no—
86#14519697—22no—
87C4×#92418—22no—
88#14519684—22no—
89C22×D4×C61C2311yescharacteristic
90C23×C4×C61—21no—
91C22×C8×C61C2312yes—
92C4×D4×C61—22no—
93C4×D4×C61D414yes—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61C2312yes—
97C2×C4⋊C4×C61C2311yescharacteristic
98C2×C22⋊C4×C61—21no—
99C2×C42×C61C2311yescharacteristic
100C4⋊C8×C61—22no—
101#102×C61—24no—
102C24×C61—21no—
103C23×D61—21no—
104C2×D4×C61—21no—
105C2×D4×C61—21no—
106C2×D4×C61—21no—
107C2×D4×C61C2×D412yes—
108C2×D4×C61C2×Q812yes—
109C2×D4×C61#4112yes—
110C22×C4×C61—21no—
111C22×C4×C61—24no—
112C22×C4×C61—22no—
113C22×C4×C61—42no—
114C22×C4×C61—42no—
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2411yescharacteristic
118C22×Dic61—22no—
119C22×Dic61—28no—
120C22×Dic61—21no—
121C22×Dic61—82no—
122C22×Dic61—82no—
123C22×Dic61C2×D412yes—
124C22×Dic61C2×D412yes—
125C2×D244—21no—
126C2×D244—21no—
127C2×D244—21no—
128C2×D244—21no—
129C2×D244—21no—
130C2×D244C2×Q812yes—
131C2×C4×D61—42no—
132C2×C4×D61—42no—
133C2×C4×D61—42no—
134C2×Dic122—22no—
135C2×Dic122—22no—
136C2×Dic122—22no—
137C2×Dic122—24no—
138C2×Dic122—81no—
139C2×Dic122—81no—
140C2×C8×C61—22no—
141C2×C8×C61—22no—
142C2×C8×C61—22no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—42no—
147C4⋊C4×C61—81no—
148C4⋊C4×C61C2×D412yes—
149C22⋊C4×C61—24no—
150C42×C61C2×D412yes—
151C42×C61C2×D412yes—
152C22⋊Dic61—28no—
153#11758199—28no—
154#11758198—44no—
155#11758198—42no—
156#11758198—82no—
157C4⋊Dic61—28no—
158C4⋊Dic61—22no—
159C4⋊Dic61—24no—
160C4⋊Dic61—22no—
161C4⋊Dic61—21no—
162C4⋊Dic61—21no—
163C4⋊Dic61—44no—
164C4⋊Dic61C2×D414yes—
165#11758196—28no—
166#11758196—24no—
167C4×Dic61—22no—
168C4×Dic61—22no—
169C2×#92418—21no—
170C2×#92418—21no—
171C2×#92418—21no—
172C2×#92418—21no—
173C2×#92418—21no—
174C2×#92418C2×Q812yes—
175C23×C61—22no—
176C23×C61—22no—
177C23×C61—22no—
178C23×C61—41no—
179C23×C61—41no—
180C23×C61—41no—
181C23×C61—41no—
182C23×C61C22×D411yescharacteristic
183C22×D61—22no—
184C22×D61—41no—
185C22×D61—41no—
186C22×D61—41no—
187C22×D61—41no—
188C22×D61—41no—
189C22×D61—41no—
190D4×C61—22no—
191D4×C61—22no—
192D4×C61—22no—
193D4×C61C22⋊Q814yescomplemented
194C2×C4×C61—21no—
195C2×C4×C61—21no—
196C2×C4×C61—22no—
197C2×C4×C61—41no—
198C2×C4×C61—41no—
199C2×C4×C61—48no—
200C2×C4×C61—42no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61#12012yes—
204C2×C4×C61#12012yes—
205C2×C4×C61#13512yes—
206C2×C4×C61AGL(1,ℤ/8)12yes—
207C2×C4×C61C22×D411yescharacteristic
208C2×C4×C61C22×D411yescharacteristic
209C2×C4×C61C22×Q811yescharacteristic
210C2×C4×C61C2×#4111yescharacteristic
211C2×C4×C612−1+411yescharacteristic
212C2×C4×C612−1+411yescharacteristic
213C2×Dic61—22no—
214C2×Dic61—22no—
215C2×Dic61—24no—
216C2×Dic61—22no—
217C2×Dic61—22no—
218C2×Dic61—22no—
219C2×Dic61—22no—
220C2×Dic61—42no—
221C2×Dic61—42no—
222C2×Dic61—41no—
223C2×Dic61—41no—
224C2×Dic61—41no—
225C2×Dic61—88no—
226D244—22no—
227D244—22no—
228D244—41no—
229D244—41no—
230C4×D61—84no—
231Dic122—44no—
232Dic122—84no—
233C8×C61—42no—
234#92418—22no—
235#92418—41no—
236#24327—611nomaximal, Sylow
237C22×C61—24no—
238C22×C61—42no—
239C22×C61—42no—
240C22×C61—42no—
241C22×C61—42no—
242C22×C61—42no—
243C22×C61—42no—
244C22×C61C2×#12011yescharacteristic
245C22×C61#54811yescharacteristic
246C22×C61D4×Q811yescharacteristic
247C22×C61C2×#13511yescharacteristic
248C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
249C22×C61#57611yescharacteristic
250C22×C61#57711yescharacteristic
251C2×D61—42no—
252C2×D61—42no—
253C2×D61—42no—
254C2×D61—81no—
255C2×D61—81no—
256C2×D61—81no—
257C2×D61—81no—
258C4×C61—22no—
259C4×C61—42no—
260C4×C61—82no—
261C4×C61#44914yes—
262C4×C61C2×C22⋊Q811yescharacteristic
263C4×C61#53611yescharacteristic
264C4×C61#54811yescharacteristic
265C4×C61D4×Q811yescharacteristic
266Dic61—42no—
267Dic61—44no—
268Dic61—42no—
269Dic61—81no—
270C2×#554—611no—
271C2×#476—611no—
272C2×C4×D8—611no—
273#2538—611no—
274#2059—612no—
275#2025—612no—
276#1990—611no—
277#1984—611no—
278#1879—612no—
279#1864—612no—
280#1802D6111yescharacteristic, complemented
281C2×C61—44no—
282C2×C61#240911yescharacteristic
283C2×C61C2×#44911yescharacteristic
284C2×C61#301211yescharacteristic
285C2×C61#336011yescharacteristic
286C2×C61#336611yescharacteristic
287C2×C61#338011yescharacteristic
288C2×C61#338411yescharacteristic
289D61—82no—
290C22×D8—611no—
291#554—1224no—
292#554—1224no—
293C2×C4⋊Q8—611no—
294C2×#125—612no—
295C2×C22⋊Q8—1222no—
296C2×C22⋊Q8—1221no—
297C2×C22⋊Q8—1221no—
298C2×C4×Q8—611no—
299C2×C4×D4—612no—
300C2×C4×D4—611no—
301C2×C4×D4C2×D6111yescharacteristic
302C22×C4⋊C4—1221no—
303#476—1224no—
304C4×D8—1224no—
305C2×#107—611no—
306C2×#106—612no—
307C2×C4⋊C8C2×D6111yescharacteristic
308C2×#102—611no—
309C2×#102—612no—
310C2×#102—611no—
311C2×#102C2×D6112yes—
312C2×C4×C8—611no—
313#396—1222no—
314#390—612no—
315#389—612no—
316#384C2×D6111yescharacteristic
317#380—1221no—
318#340—612no—
319#340—612no—
320#340C2×D6112yes—
321#336—612no—
322#336—612no—
323C61#2432711yescharacteristic, complemented, Sylow
324C22×Q8—611no—
325C22×Q8—1221no—
326C22×D4—611no—
327C22×D4C22×D6111yescharacteristic
328C23×C4—21no—
329C23×C4—1222no—
330C23×C4—1221no—
331C2×D8—614no—
332C2×D8—1222no—
333C2×D8—1222no—
334C22×C8—611no—
335C22×C8—611no—
336C22×C8C22×D6112yes—
337C4⋊Q8—614no—
338C4⋊Q8—1222no—
339#125—1224no—
340#125—1224no—
341C22⋊Q8—12216no—
342C22⋊Q8—1228no—
343C22⋊Q8—2444no—
344C4×Q8—1222no—
345C4×Q8—1222no—
346C4×D4—22no—
347C4×D4—1224no—
348C4×D4—1224no—
349C4×D4—1222no—
350C4×D4—1222no—
351C4×D4D24414yes—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—612no—
355C2×C4⋊C4—611no—
356C2×C4⋊C4—612no—
357C2×C4⋊C4—611no—
358C2×C4⋊C4—1222no—
359C2×C4⋊C4—1222no—
360C2×C4⋊C4—1228no—
361C2×C4⋊C4—1221no—
362C2×C4⋊C4—1222no—
363C2×C4⋊C4C22×D6112yes—
364C2×C4⋊C4C22×D6111yescharacteristic
365C2×C22⋊C4—21no—
366C2×C22⋊C4—1222no—
367C2×C22⋊C4—1222no—
368C2×C22⋊C4—1222no—
369C2×C22⋊C4—1221no—
370C2×C42—612no—
371C2×C42C22×D6111yescharacteristic
372#107—1222no—
373#106—1224no—
374C4⋊C8—22no—
375#102—24no—
376#102—1224no—
377#102—1222no—
378#102—1222no—
379C4×C8—1222no—
380#95—1222no—
381C24—21no—
382C24—1221no—
383C2×Q8—1224no—
384C2×Q8—1222no—
385C2×Q8—1222no—
386C2×Q8—1222no—
387C2×Q8—4881no—
388C2×Q8—4881no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—612no—
393C2×D4—1221no—
394C2×D4—1221no—
395C2×D4—1221no—
396C2×D4—1221no—
397C2×D4—1221no—
398C2×D4C2×D24412yes—
399C2×D4#1175822412yes—
400C2×D4Q8×D6112yes—
401C22×C4—21no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—612no—
407C22×C4—612no—
408C22×C4—1228no—
409C22×C4—1222no—
410C22×C4—1221no—
411C22×C4—4882no—
412C22×C4—4882no—
413C22×C4—2442no—
414C22×C4—2442no—
415C22×C4—2442no—
416C22×C4D4×D6111yescharacteristic
417C22×C4D4×D6111yescharacteristic
418C22×C4C23×D6111yescharacteristic
419D8—1228no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—612no—
424C2×C8—1221no—
425C2×C8—1221no—
426C2×C8—1221no—
427C2×C8—1221no—
428C2×C8—1221no—
429C4⋊C4—21no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—42no—
433C4⋊C4—81no—
434C4⋊C4—614no—
435C4⋊C4—1228no—
436C4⋊C4—1228no—
437C4⋊C4—1222no—
438C4⋊C4—1222no—
439C4⋊C4—1224no—
440C4⋊C4—1224no—
441C4⋊C4—1221no—
442C4⋊C4—1221no—
443C4⋊C4—2444no—
444C4⋊C4C2×D24412yes—
445C22⋊C4—24no—
446C22⋊C4—1228no—
447C22⋊C4—4882no—
448C22⋊C4—2444no—
449C22⋊C4—2442no—
450C42—1222no—
451C42—1222no—
452C42C2×D24412yes—
453C42D4×D6112yes—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—1222no—
462C23—2441no—
463C23—2441no—
464C23—2441no—
465C23—2441no—
466C23—2441no—
467C23—2441no—
468C23C2×D4×D6111yescharacteristic
469Q8—4884no—
470Q8—2444no—
471D4—22no—
472D4—22no—
473D4—22no—
474D4—1222no—
475D4—1222no—
476D4—2441no—
477D4—2441no—
478D4#1451975714yescomplemented
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—48no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—1222no—
489C2×C4—1222no—
490C2×C4—1224no—
491C2×C4—1222no—
492C2×C4—1222no—
493C2×C4—1222no—
494C2×C4—1222no—
495C2×C4—4884no—
496C2×C4—4888no—
497C2×C4—2441no—
498C2×C4—2441no—
499C2×C4—2441no—
500C2×C4—2442no—
501C2×C4—2442no—
502C2×C4#1451974712yes—
503C2×C4#1451974712yes—
504C2×C4#1451979812yes—
505C2×C4#1451980112yes—
506C2×C4C2×D4×D6111yescharacteristic
507C2×C4C22×D24411yescharacteristic
508C2×C4C2×Q8×D6111yescharacteristic
509C2×C4C2×#1175822411yescharacteristic
510C2×C4#1451982411yescharacteristic
511C2×C4#1451982411yescharacteristic
512C8—42no—
513C8—1222no—
514C8—2441no—
515C22—24no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—4881no—
523C22—4881no—
524C22—4881no—
525C22—4881no—
526C22—2442no—
527C22—2442no—
528C22—2442no—
529C22C2×#1451974711yescharacteristic
530C22#1684041811yescharacteristic
531C22#1684042511yescharacteristic
532C22C2×#1451979811yescharacteristic
533C22C2×#1451980111yescharacteristic
534C22#1684056611yescharacteristic
535C22#1684057111yescharacteristic
536C4—22no—
537C4—42no—
538C4—82no—
539C4—4881no—
540C4—2442no—
541C4—2444no—
542C4—2442no—
543C4#1683990814yes—
544C4C2×#1451975711yescharacteristic
545C4#1684033811yescharacteristic
546C4#1684041811yescharacteristic
547C4#1684042511yescharacteristic
548C2—44no—
549C2—4882no—
550C2#2144918511yescharacteristic
551C2C2×#1683990811yescharacteristic
552C2#2145512511yescharacteristic
553C2#2145847211yescharacteristic
554C2#2145853011yescharacteristic
555C2#2145868511yescharacteristic
556C2#2145872211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1802C2
socleC23×C61C22×D4
prime radicalC22×#11758199C22
hypercenter#1802D61
hypocenterC61#24327
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
1. C2×#16840469C2semidirect
2. C2×#16840053C2semidirect
4. #21450742C2semidirect
5. #21447156C2semidirect
6. #21446964C2semidirect
7. #21446779C2semidirect
8. #21446754C2semidirect
10. #21446082C2semidirect
11. C61×#1802C2semidirect
52. C2×#11758199Q8semidirect
193. D4×C61C22⋊Q8semidirect
280. #1802D61semidirect
323. C61#24327semidirect
478. D4#14519757semidirect

Maximal subgroups

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

subgroupindex
C2×#168404692
C2×#168400532
C2×C4×#117581992
#214507422
#214471562
#214469642
#214467792
#214467542
#214461652
#214460822
C61×#18022
#2432761