Finite Group Databasebeta

← #29716420

Subgroups — #29716420

Order 15616.

1102 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×Dic122C211yesmaximal, characteristic, complemented
2C2×#16840053C211yesmaximal, characteristic
3C2×C4×#11758190C211yesmaximal, characteristic
4#21450737C211yesmaximal, characteristic
5#21447153C212yesmaximal, complemented
6#21446956C212yesmaximal
7#21446780C211yesmaximal, characteristic, complemented
8#21446173C212yesmaximal
9#21446090C212yesmaximal, complemented
10#21445825C211yesmaximal, characteristic, complemented
11C61×#1984C211yesmaximal, characteristic, complemented
12C22×#11758190C2211yescharacteristic
13Q8×Dic122—24no—
14Q8×Dic122—24no—
15C2×#14519777C2212yes—
16C2×#14519777C2211yescharacteristic
17C2×#14519776—22no—
18C2×#14519776—21no—
19C2×#14519753—21no—
20C2×Q8×Dic61—21no—
21C2×C4×Dic122C2212yes—
22C2×C4×Dic122C2211yescharacteristic
23C2×C4×D244C2211yescharacteristic
24#16840053—24no—
25C4×#11758190—24no—
26C2×#14519709C2212yes—
27C2×#14519709C2211yescharacteristic
28C2×#14519702C2212yes—
29C2×#14519702C2211yescharacteristic
30C2×#14519699C2212yes—
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—
47C22×Dic122C2311yescharacteristic
48C22×D244C2311yescharacteristic
49C22×C4×D61—21no—
50C2×#11758190—22no—
51C2×#11758190—22no—
52C2×#11758190Q814yescomplemented
53#14519777—22no—
54#14519777—24no—
55#14519777—24no—
56#14519777D414yes—
57#14519776—216no—
58#14519776—28no—
59#14519753—44no—
60Q8×Dic61—28no—
61C4×Dic122—24no—
62C4×Dic122—22no—
63C4×Dic122—24no—
64C4×Dic122D414yes—
65C4×D244—22no—
66C4×D244—22no—
67C2×C4⋊Dic61—22no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2312yes—
73C2×C4⋊Dic61C2312yes—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758196—22no—
76C2×#11758196—21no—
77C2×#11758196—22no—
78C2×#11758198—21no—
79C2×#11758198—22no—
80C2×#11758198—22no—
81C2×C4×Dic61—22no—
82C2×C4×Dic61—21no—
83#14519709—24no—
84#14519709—22no—
85#14519702—22no—
86#14519702—24no—
87#14519699—24no—
88#14519699—22no—
89#14519684—22no—
90C22×Q8×C61D411yescharacteristic
91C22×C8×C61C2311yescharacteristic
92C22×C8×C61C2312yes—
93C22×C8×C61C2311yescharacteristic
94C4×Q8×C61—22no—
95C4×Q8×C61—22no—
96C2×C4⋊C4×C61D412yes—
97C2×C4⋊C4×C61D411yescharacteristic
98C2×C42×C61C2312yes—
99C2×C42×C61C2311yescharacteristic
100C4⋊C8×C61—22no—
101C4×C8×C61—22no—
102C23×D61—21no—
103C2×Q8×C61—22no—
104C2×Q8×C61—22no—
105C2×Q8×C61SD1614yescomplemented
106C22×C4×C61C2×D412yes—
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D412yes—
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2411yescharacteristic
111C22×Dic61—21no—
112C22×Dic61—22no—
113C22×Dic61—21no—
114C22×Dic61—21no—
115C22×Dic61—22no—
116C2×D244—21no—
117C2×D244—21no—
118C2×D244—21no—
119C2×D244—21no—
120C2×D244—21no—
121C2×D244C2×Q812yes—
122C2×C4×D61—42no—
123C2×C4×D61—42no—
124C2×C4×D61—42no—
125C2×Dic122—21no—
126C2×Dic122—21no—
127C2×Dic122—21no—
128C2×Dic122—28no—
129C2×Dic122—24no—
130C2×Dic122—81no—
131C2×Dic122—81no—
132C2×Dic122—82no—
133C2×Dic122—82no—
134C2×Dic122C2×D412yes—
135C2×Dic122C2×Q812yes—
136C2×Dic122#4112yes—
137C2×C8×C61—21no—
138C2×C8×C61—21no—
139C2×C8×C61—21no—
140C2×C8×C61—22no—
141C2×C8×C61—22no—
142C2×C8×C61—21no—
143C2×C8×C61—21no—
144C2×C8×C61—22no—
145C2×C8×C61C2×Q812yes—
146C4⋊C4×C61—24no—
147C4⋊C4×C61—22no—
148C42×C61—22no—
149C42×C61—22no—
150C42×C61C2×D412yes—
151C42×C61C2×D412yes—
152#11758198—42no—
153#11758198—44no—
154#11758198—82no—
155C4⋊Dic61—28no—
156C4⋊Dic61—24no—
157C4⋊Dic61—22no—
158C4⋊Dic61—21no—
159C4⋊Dic61—21no—
160C4⋊Dic61—21no—
161C4⋊Dic61—22no—
162C4⋊Dic61—22no—
163C4⋊Dic61—42no—
164C4⋊Dic61—81no—
165C4⋊Dic61C2×D414yes—
166C4⋊Dic61C2×D412yes—
167#11758196—28no—
168#11758196—24no—
169#11758196—44no—
170C4×Dic61—28no—
171C4×Dic61—24no—
172#11758190—28no—
173C23×C61C22×D411yescharacteristic
174C22×D61—22no—
175C22×D61—41no—
176C22×D61—41no—
177C22×D61—41no—
178C22×D61—41no—
179C22×D61—41no—
180C22×D61—41no—
181Q8×C61—28no—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—21no—
185C2×C4×C61—21no—
186C2×C4×C61—22no—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61#12012yes—
190C2×C4×C61#12012yes—
191C2×C4×C61C2×SD1614yes—
192C2×C4×C61C2×SD1612yes—
193C2×C4×C61#13712yes—
194C2×C4×C61C22×D411yescharacteristic
195C2×C4×C61C22×D411yescharacteristic
196C2×C4×C61C22×Q811yescharacteristic
197C2×C4×C61C2×#4111yescharacteristic
198C2×C4×C612−1+411yescharacteristic
199C2×C4×C612−1+411yescharacteristic
200C2×Dic61—22no—
201C2×Dic61—22no—
202C2×Dic61—22no—
203C2×Dic61—24no—
204C2×Dic61—22no—
205C2×Dic61—42no—
206C2×Dic61—41no—
207C2×Dic61—42no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210C2×Dic61—42no—
211C2×Dic61—41no—
212C2×Dic61—42no—
213C2×Dic61—41no—
214C2×Dic61—42no—
215D244—22no—
216D244—22no—
217D244—41no—
218D244—41no—
219C4×D61—84no—
220Dic122—22no—
221Dic122—22no—
222Dic122—22no—
223Dic122—84no—
224Dic122—88no—
225Dic122C22⋊Q814yescomplemented
226C8×C61—22no—
227C8×C61—41no—
228C8×C61—42no—
229#24329—611nomaximal, Sylow
230C22×C61C2×#12011yescharacteristic
231C22×C61#54811yescharacteristic
232C22×C61D4×Q811yescharacteristic
233C22×C61C22×SD1611yescharacteristic
234C22×C61C2×#13711yescharacteristic
235C22×C61#57611yescharacteristic
236C22×C61#57711yescharacteristic
237C2×D61—42no—
238C2×D61—42no—
239C2×D61—42no—
240C2×D61—81no—
241C2×D61—81no—
242C2×D61—81no—
243C2×D61—81no—
244C4×C61—24no—
245C4×C61—22no—
246C4×C61—42no—
247C4×C61#44814yes—
248C4×C61C2×C22⋊Q811yescharacteristic
249C4×C61#53611yescharacteristic
250C4×C61#54811yescharacteristic
251C4×C61D4×Q811yescharacteristic
252Dic61—24no—
253Dic61—44no—
254Dic61—42no—
255Dic61—82no—
256Dic61—81no—
257C2×Q82—611no—
258C2×#476—611no—
259C2×C4×SD16—611no—
260#2538—611no—
261#2058—612no—
262#2023—612no—
263#1990—611no—
264#1984D6111yescharacteristic, complemented
265#1880—612no—
266#1865—612no—
267#1803—611no—
268C2×C61#240911yescharacteristic
269C2×C61C2×#44811yescharacteristic
270C2×C61#301211yescharacteristic
271C2×C61#336211yescharacteristic
272C2×C61#336911yescharacteristic
273C2×C61Q8×SD1611yescharacteristic
274C2×C61#338511yescharacteristic
275D61—82no—
276C22×SD16—611no—
277Q82—1224no—
278Q82—1224no—
279C2×C4⋊Q8—611no—
280C2×C4⋊Q8—612no—
281C2×C4⋊Q8—1221no—
282C2×C4⋊Q8—1222no—
283C2×C22⋊Q8—1221no—
284C2×C4×Q8—611no—
285C2×C4×Q8—612no—
286C2×C4×Q8—1221no—
287C2×C4×Q8C2×D6111yescharacteristic
288C2×C4×D4—611no—
289#476—1224no—
290C4×SD16—1224no—
291C2×#106—611no—
292C2×#106—612no—
293C2×C4⋊C8C2×D6111yescharacteristic
294C2×#103—612no—
295C2×#103—611no—
296C2×#102—612no—
297C2×#102—611no—
298C2×C4×C8C2×D6111yescharacteristic
299#396—1222no—
300#390—612no—
301#389—612no—
302#384—611no—
303#380—1221no—
304#340—612no—
305#340—612no—
306#340—612no—
307#336C2×D6112yes—
308#336C2×D6112yes—
309C61#2432911yescharacteristic, complemented, Sylow
310C22×Q8—611no—
311C22×Q8—1222no—
312C22×Q8—1221no—
313C22×Q8D24411yescharacteristic
314C22×D4—611no—
315C23×C4—1221no—
316C2×SD16—614no—
317C2×SD16—1222no—
318C2×SD16—1222no—
319C22×C8C22×D6111yescharacteristic
320C22×C8C22×D6112yes—
321C22×C8C22×D6111yescharacteristic
322C4⋊Q8—614no—
323C4⋊Q8—1224no—
324C4⋊Q8—1224no—
325C4⋊Q8—12216no—
326C4⋊Q8—1228no—
327C4⋊Q8—1222no—
328C22⋊Q8—2444no—
329C4×Q8—22no—
330C4×Q8—22no—
331C4×Q8—614no—
332C4×Q8—1224no—
333C4×Q8—1224no—
334C4×Q8—1228no—
335C4×Q8—1222no—
336C4×D4—1222no—
337C4×D4—1222no—
338C2×C4⋊C4—611no—
339C2×C4⋊C4—612no—
340C2×C4⋊C4—611no—
341C2×C4⋊C4—612no—
342C2×C4⋊C4—1222no—
343C2×C4⋊C4—1221no—
344C2×C4⋊C4—1222no—
345C2×C4⋊C4—1221no—
346C2×C4⋊C4—1221no—
347C2×C4⋊C4—1221no—
348C2×C4⋊C4—1222no—
349C2×C4⋊C4D24411yescharacteristic
350C2×C4⋊C4D24412yes—
351C2×C22⋊C4—1222no—
352C2×C22⋊C4—1221no—
353C2×C22⋊C4—1222no—
354C2×C42—1222no—
355C2×C42—1221no—
356C2×C42C22×D6112yes—
357C2×C42C22×D6111yescharacteristic
358#106—1222no—
359#106—1224no—
360C4⋊C8—22no—
361#103—1224no—
362#103—1222no—
363#102—1222no—
364#102—1224no—
365C4×C8—22no—
366#95—1222no—
367C24—1221no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—612no—
371C2×Q8—612no—
372C2×Q8—612no—
373C2×Q8—1228no—
374C2×Q8—1221no—
375C2×Q8—1221no—
376C2×Q8—1221no—
377C2×Q8—1224no—
378C2×Q8—4881no—
379C2×Q8—4881no—
380C2×Q8—4882no—
381C2×Q8—4882no—
382C2×Q8#1175819014yescomplemented
383C2×D4—612no—
384C2×D4—1221no—
385C2×D4—1221no—
386C2×D4—1221no—
387C2×D4—1221no—
388C2×D4—1221no—
389C22×C4—1221no—
390C22×C4—1222no—
391C22×C4—1221no—
392C22×C4—1221no—
393C22×C4—1222no—
394C22×C4—2442no—
395C22×C4—2442no—
396C22×C4—2442no—
397C22×C4C2×D24412yes—
398C22×C4C2×D24411yescharacteristic
399C22×C4C2×D24412yes—
400C22×C4C2×D24411yescharacteristic
401C22×C4C23×D6111yescharacteristic
402SD16—1228no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—21no—
409C2×C8—22no—
410C2×C8—21no—
411C2×C8Q8×D6112yes—
412C4⋊C4—24no—
413C4⋊C4—22no—
414C4⋊C4—612no—
415C4⋊C4—614no—
416C4⋊C4—1228no—
417C4⋊C4—1228no—
418C4⋊C4—1222no—
419C4⋊C4—1221no—
420C4⋊C4—1224no—
421C4⋊C4—1224no—
422C4⋊C4—1221no—
423C4⋊C4—1221no—
424C4⋊C4—1222no—
425C4⋊C4—1222no—
426C4⋊C4—4881no—
427C4⋊C4—2444no—
428C4⋊C4—2442no—
429C22⋊C4—4882no—
430C22⋊C4—2442no—
431C22⋊C4—2444no—
432C42—22no—
433C42—22no—
434C42—1228no—
435C42—1224no—
436C42D4×D6112yes—
437C42D4×D6112yes—
438C23—1222no—
439C23—2441no—
440C23—2441no—
441C23—2441no—
442C23—2441no—
443C23—2441no—
444C23—2441no—
445C23C22×D24411yescharacteristic
446Q8—28no—
447Q8—614no—
448Q8—1222no—
449Q8—1222no—
450Q8—1222no—
451Q8—4884no—
452Q8—4888no—
453D4—1222no—
454D4—1222no—
455D4—2441no—
456D4—2441no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—1222no—
465C2×C4—1222no—
466C2×C4—1222no—
467C2×C4—1224no—
468C2×C4—1222no—
469C2×C4—4884no—
470C2×C4—2441no—
471C2×C4—2442no—
472C2×C4—2441no—
473C2×C4—2441no—
474C2×C4—2442no—
475C2×C4—2442no—
476C2×C4—2442no—
477C2×C4—2441no—
478C2×C4—2442no—
479C2×C4—2441no—
480C2×C4#1451974712yes—
481C2×C4#1451974712yes—
482C2×C4C2×#1175819014yes—
483C2×C4C2×#1175819012yes—
484C2×C4#1451981012yes—
485C2×C4C2×D4×D6111yescharacteristic
486C2×C4C2×D4×D6111yescharacteristic
487C2×C4C2×Q8×D6111yescharacteristic
488C2×C4#41×D6111yescharacteristic
489C2×C4#1451982411yescharacteristic
490C2×C4#1451982511yescharacteristic
491C8—22no—
492C8—41no—
493C8—42no—
494C22—4881no—
495C22—4881no—
496C22—4881no—
497C22—4881no—
498C22—2442no—
499C22—2442no—
500C22—2442no—
501C22C2×#1451974711yescharacteristic
502C22#1684042411yescharacteristic
503C22Q8×D24411yescharacteristic
504C22C22×#1175819011yescharacteristic
505C22C2×#1451981011yescharacteristic
506C22#1684056711yescharacteristic
507C22#1684057411yescharacteristic
508C4—22no—
509C4—24no—
510C4—42no—
511C4—1224no—
512C4—4882no—
513C4—4881no—
514C4—2444no—
515C4—2442no—
516C4#1683990314yes—
517C4C22⋊Q8×D6111yescharacteristic
518C4#1684033511yescharacteristic
519C4#1684042011yescharacteristic
520C4#1684042711yescharacteristic
521C2—4882no—
522C2#2144918211yescharacteristic
523C2C2×#1683990311yescharacteristic
524C2#2145512611yescharacteristic
525C2#2145849111yescharacteristic
526C2#2145856111yescharacteristic
527C2Q8×#1175819011yescharacteristic
528C2#2145873111yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1984C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1984D61
hypocenterC61#24329
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×Dic122C2semidirect
5. #21447153C2semidirect
7. #21446780C2semidirect
9. #21446090C2semidirect
10. #21445825C2semidirect
11. C61×#1984C2semidirect
52. C2×#11758190Q8semidirect
105. C2×Q8×C61SD16semidirect
225. Dic122C22⋊Q8semidirect
264. #1984D61semidirect
309. C61#24329semidirect
382. C2×Q8#11758190semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic1222
C2×#168400532
C2×C4×#117581902
#214507372
#214471532
#214469562
#214467802
#214461732
#214460902
#214458252
C61×#19842
#2432961