Finite Group Databasebeta

← #29730519

Subgroups — #29730519

Order 15616.

1014 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×#16840135C211yesmaximal, characteristic
2C2×#16839985C212yesmaximal, complemented
3C2×#16839973C211yesmaximal, characteristic, complemented
4#21450821C211yesmaximal, characteristic
5#21448875C211yesmaximal, characteristic, complemented
6#21447137C212yesmaximal, complemented
7#21446916C212yesmaximal, complemented
8#21446745C211yesmaximal, characteristic
9#21446173C212yesmaximal, complemented
10C4×#14519702C211yesmaximal, characteristic, complemented
11C61×#1939C211yesmaximal, characteristic, complemented
12C22×Dic244C2212yes—
13C22×#11758190C2212yes—
14C22×#11758190C2212yes—
15C2×#14519777C2211yescharacteristic
16C2×#14519774C2211yescharacteristic
17C2×#14519765C2212yes—
18C2×C4×Dic122C2211yescharacteristic
19C2×C4×Dic122C2212yes—
20#16840135—24no—
21#16839985—28no—
22#16839973—24no—
23C2×#14519702C2211yescharacteristic
24C2×#14519702C2212yes—
25C2×#14519702C2211yescharacteristic
26C2×#14519699C2212yes—
27C2×#14519699C2212yes—
28#16839680—22no—
29#16839668C2212yes—
30#16839664C2212yes—
31#16839663—21no—
32#16839663—21no—
33#16839658—21no—
34#16839645—22no—
35C4⋊C4×Dic61—21no—
36C4×C4⋊Dic61C2211yescharacteristic
37#16839525C2211yescharacteristic
38#16839525C2211yescharacteristic
39C2×C4⋊C8×C61C2211yescharacteristic
40C2×C4⋊C8×C61C2212yes—
41C2×C4×C8×C61C2211yescharacteristic
42C4×C4⋊C4×C61C2211yescharacteristic
43C61×#336C2212yes—
44C22×Dic122C2311yescharacteristic
45C22×Dic122C2312yes—
46C22×D244—21no—
47C22×D244—21no—
48C22×D244C2312yes—
49C2×Dic244—24no—
50C2×Dic244—24no—
51C2×Dic244—42no—
52C2×#11758190—24no—
53C2×#11758190—24no—
54C2×#11758190—24no—
55C2×#11758190—24no—
56C2×#11758190—42no—
57C2×#11758190—42no—
58#14519777—22no—
59#14519777D414yes—
60#14519774—22no—
61#14519774—22no—
62#14519765—24no—
63#14519765—42no—
64C4×Dic122—22no—
65C4×Dic122—22no—
66C4×Dic122—24no—
67C4×Dic122—42no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×#11758196—21no—
74C2×#11758196—22no—
75C2×#11758198—22no—
76C2×#11758198—22no—
77C2×#11758198—22no—
78C2×#11758198—22no—
79C2×C4×Dic61—22no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—21no—
82#14519702—24no—
83#14519702—22no—
84#14519702D414yescomplemented
85#14519699—24no—
86#14519699—42no—
87#14519684—21no—
88#14519684—22no—
89#14519684—22no—
90C22×C8×C61C2312yes—
91C22×C8×C61C2311yescharacteristic
92C22×C8×C61C2311yescharacteristic
93C2×C4⋊C4×C61D411yescharacteristic
94C2×C4⋊C4×C61D411yescharacteristic
95C2×C42×C61C2312yes—
96C2×C42×C61C2311yescharacteristic
97C4⋊C8×C61—24no—
98C4⋊C8×C61—22no—
99C4×C8×C61—22no—
100#95×C61D412yes—
101C23×D61—22no—
102C22×C4×C61C2×D412yes—
103C22×C4×C61C2×D412yes—
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2411yescharacteristic
107C22×Dic61—22no—
108C22×Dic61—21no—
109C22×Dic61—21no—
110C22×Dic61—21no—
111C22×Dic61—21no—
112C2×D244—22no—
113C2×D244—22no—
114C2×D244—22no—
115C2×D244—22no—
116C2×D244—22no—
117C2×D244—22no—
118C2×D244—42no—
119C2×D244—42no—
120C2×D244—42no—
121C2×D244—42no—
122C2×D244—81no—
123C2×D244—81no—
124C2×Dic122—22no—
125C2×Dic122—21no—
126C2×Dic122—21no—
127C2×Dic122—21no—
128C2×Dic122—22no—
129C2×Dic122—22no—
130C2×Dic122—22no—
131C2×Dic122—21no—
132C2×Dic122—22no—
133C2×Dic122—21no—
134C2×Dic122C2×D414yes—
135C2×Dic122C2×D412yes—
136C2×C8×C61—21no—
137C2×C8×C61—21no—
138C2×C8×C61—21no—
139C2×C8×C61—21no—
140C2×C8×C61—22no—
141C2×C8×C61—22no—
142C2×C8×C61—22no—
143C2×C8×C61C2×D412yes—
144C2×C8×C61#4112yes—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—22no—
147C42×C61—22no—
148C42×C61—22no—
149C42×C61C2×D412yes—
150C42×C61#4112yes—
151#11758198—44no—
152#11758198—44no—
153#11758198—82no—
154#11758198—82no—
155C4⋊Dic61—24no—
156C4⋊Dic61—24no—
157C4⋊Dic61—21no—
158C4⋊Dic61—21no—
159C4⋊Dic61—21no—
160C4⋊Dic61—21no—
161C4⋊Dic61—21no—
162C4⋊Dic61C2×D412yes—
163#11758196—44no—
164#11758196—42no—
165C4×Dic61—44no—
166C4×Dic61—42no—
167C4×Dic61—42no—
168Dic244—48no—
169#11758190—48no—
170#11758190—48no—
171C23×C61C22×D411yescharacteristic
172C22×D61—42no—
173C22×D61—42no—
174C22×D61—42no—
175C22×D61—42no—
176C22×D61—42no—
177C22×D61—42no—
178C22×D61—42no—
179C2×C4×C61—21no—
180C2×C4×C61—21no—
181C2×C4×C61—22no—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—22no—
185C2×C4×C61—22no—
186C2×C4×C61—22no—
187C2×C4×C61#12112yes—
188C2×C4×C61#12112yes—
189C2×C4×C61#13512yes—
190C2×C4×C61#13712yes—
191C2×C4×C61C22×D411yescharacteristic
192C2×C4×C61C22×D411yescharacteristic
193C2×C4×C61C2×#4111yescharacteristic
194C2×C4×C61C2×#4111yescharacteristic
195C2×C4×C61H(5,𝔽2)11yescharacteristic
196C2×C4×C61H(5,𝔽2)11yescharacteristic
197C2×Dic61—24no—
198C2×Dic61—22no—
199C2×Dic61—22no—
200C2×Dic61—41no—
201C2×Dic61—42no—
202C2×Dic61—41no—
203C2×Dic61—42no—
204C2×Dic61—41no—
205C2×Dic61—41no—
206C2×Dic61—41no—
207C2×Dic61—41no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212D244—42no—
213D244—42no—
214D244—42no—
215D244—42no—
216D244—84no—
217D244—84no—
218Dic122—22no—
219Dic122—24no—
220Dic122—22no—
221Dic122—24no—
222Dic122—42no—
223Dic122—41no—
224Dic122—42no—
225Dic122—41no—
226C8×C61—22no—
227C8×C61—22no—
228C8×C61—42no—
229#25369—611nomaximal, Sylow
230C22×C61C2×#12111yescharacteristic
231C22×C61D4211yescharacteristic
232C22×C61#55011yescharacteristic
233C22×C61C2×#13511yescharacteristic
234C22×C61C2×#13711yescharacteristic
235C22×C61#57711yescharacteristic
236C22×C61#57711yescharacteristic
237C2×D61—82no—
238C2×D61—82no—
239C2×D61—82no—
240C2×D61—82no—
241C2×D61—82no—
242C2×D61—82no—
243C2×D61—82no—
244C4×C61—22no—
245C4×C61—42no—
246C4×C61—42no—
247C4×C61#46414yescomplemented
248C4×C61#53211yescharacteristic
249C4×C61#54011yescharacteristic
250C4×C61D4211yescharacteristic
251C4×C61#54611yescharacteristic
252Dic61—42no—
253Dic61—44no—
254Dic61—42no—
255Dic61—81no—
256Dic61—81no—
257C2×#492—611no—
258C2×#464—612no—
259C2×#462—611no—
260#2548—611no—
261#2365—611no—
262#2055—612no—
263#2015—612no—
264#1982—611no—
265#1939D6111yescharacteristic, complemented
266#1880—612no—
267C4×#103—611no—
268C2×C61#242811yescharacteristic
269C2×C61C2×#46411yescharacteristic
270C2×C61#304311yescharacteristic
271C2×C61#327811yescharacteristic
272C2×C61#328411yescharacteristic
273C2×C61#339411yescharacteristic
274C2×C61#340011yescharacteristic
275D61—162no—
276C22×Q16—612no—
277C22×SD16—612no—
278C22×SD16—612no—
279C2×C4⋊Q8—611no—
280C2×#127—611no—
281C2×#124—612no—
282C2×C4×Q8—611no—
283C2×C4×Q8—612no—
284#492—1224no—
285#464—1228no—
286#462—1224no—
287C2×C4⋊C8C2×D6112yes—
288C2×C4⋊C8C2×D6111yescharacteristic
289C2×#103—611no—
290C2×#103—612no—
291C2×#103—611no—
292C2×#102—612no—
293C2×#102—612no—
294C2×C4×C8C2×D6111yescharacteristic
295#394—1222no—
296#391—612no—
297#390—612no—
298#390—1221no—
299#390—1221no—
300#389—1221no—
301#387—1222no—
302C4×C4⋊C4—611no—
303C4×C4⋊C4—1221no—
304C4×C4⋊C4C2×D6111yescharacteristic
305#340—611no—
306#340—611no—
307#336C2×D6112yes—
308C61#2536911yescharacteristic, complemented, Sylow
309C22×Q8—611no—
310C22×Q8—612no—
311C22×D4—612no—
312C22×D4—1221no—
313C22×D4—1221no—
314C2×Q16—1224no—
315C2×Q16—1224no—
316C2×Q16—2442no—
317C2×SD16—1224no—
318C2×SD16—1224no—
319C2×SD16—1224no—
320C2×SD16—1224no—
321C2×SD16—2442no—
322C2×SD16—2442no—
323C22×C8C22×D6112yes—
324C22×C8C22×D6111yescharacteristic
325C22×C8C22×D6111yescharacteristic
326C4⋊Q8—614no—
327C4⋊Q8—1222no—
328#127—1222no—
329#127—1222no—
330#124—1224no—
331#124—2442no—
332C4×Q8—1224no—
333C4×Q8—1222no—
334C4×Q8—1222no—
335C4×Q8—2442no—
336C2×C4⋊C4—611no—
337C2×C4⋊C4—611no—
338C2×C4⋊C4—611no—
339C2×C4⋊C4—1221no—
340C2×C4⋊C4—1221no—
341C2×C4⋊C4—1222no—
342C2×C4⋊C4—1221no—
343C2×C4⋊C4D24411yescharacteristic
344C2×C4⋊C4D24411yescharacteristic
345C2×C22⋊C4—1222no—
346C2×C22⋊C4—1222no—
347C2×C22⋊C4—1222no—
348C2×C22⋊C4—1222no—
349C2×C42—1222no—
350C2×C42—1221no—
351C2×C42—1221no—
352C2×C42C22×D6112yes—
353C2×C42C22×D6111yescharacteristic
354C4⋊C8—24no—
355C4⋊C8—22no—
356#103—614no—
357#103—1224no—
358#103—1222no—
359#102—1224no—
360#102—2442no—
361C4×C8—22no—
362#95—1221no—
363#95—1222no—
364#95—1222no—
365#95D24412yes—
366C24—1222no—
367C2×Q8—612no—
368C2×Q8—614no—
369C2×Q8—1222no—
370C2×Q8—1221no—
371C2×Q8—1222no—
372C2×Q8—1221no—
373C2×Q8—1221no—
374C2×Q8—1221no—
375C2×Q8—1222no—
376C2×Q8—1222no—
377C2×Q8—1222no—
378C2×Q8—1221no—
379C2×D4—1222no—
380C2×D4—1222no—
381C2×D4—1222no—
382C2×D4—1222no—
383C2×D4—1222no—
384C2×D4—1222no—
385C2×D4—4881no—
386C2×D4—4881no—
387C2×D4—2442no—
388C2×D4—2442no—
389C2×D4—2442no—
390C2×D4—2442no—
391C22×C4—1222no—
392C22×C4—1221no—
393C22×C4—1221no—
394C22×C4—1221no—
395C22×C4—1221no—
396C22×C4C2×D24412yes—
397C22×C4C2×D24412yes—
398C22×C4C2×D24411yescharacteristic
399C22×C4C2×D24411yescharacteristic
400C22×C4C23×D6111yescharacteristic
401Q16—2448no—
402SD16—2448no—
403SD16—2448no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8D4×D6112yes—
412C2×C8#1175822612yes—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—612no—
416C4⋊C4—1221no—
417C4⋊C4—1224no—
418C4⋊C4—1224no—
419C4⋊C4—1221no—
420C4⋊C4—1221no—
421C4⋊C4—1221no—
422C4⋊C4—1221no—
423C4⋊C4—2444no—
424C4⋊C4—2442no—
425C22⋊C4—4882no—
426C22⋊C4—4882no—
427C22⋊C4—2444no—
428C22⋊C4—2444no—
429C42—22no—
430C42—22no—
431C42—2444no—
432C42—2442no—
433C42—2442no—
434C42D4×D6112yes—
435C42#1175822612yes—
436C23—2442no—
437C23—2442no—
438C23—2442no—
439C23—2442no—
440C23—2442no—
441C23—2442no—
442C23—2442no—
443C23C22×D24411yescharacteristic
444Q8—1222no—
445Q8—1224no—
446Q8—1222no—
447Q8—1224no—
448Q8—2442no—
449Q8—2442no—
450Q8—2441no—
451Q8—2441no—
452D4—4884no—
453D4—4884no—
454D4—2442no—
455D4—2442no—
456D4—2442no—
457D4—2442no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—1224no—
467C2×C4—1222no—
468C2×C4—1222no—
469C2×C4—2441no—
470C2×C4—2441no—
471C2×C4—2441no—
472C2×C4—2441no—
473C2×C4—2441no—
474C2×C4—2441no—
475C2×C4—2441no—
476C2×C4—2441no—
477C2×C4—2441no—
478C2×C4—2442no—
479C2×C4—2442no—
480C2×C4—2441no—
481C2×C4#1451975212yes—
482C2×C4#1451975212yes—
483C2×C4#1451979912yes—
484C2×C4#1451981012yes—
485C2×C4C2×D4×D6111yescharacteristic
486C2×C4C2×D4×D6111yescharacteristic
487C2×C4C2×#1175822611yescharacteristic
488C2×C4C2×#1175822611yescharacteristic
489C2×C4#1451982311yescharacteristic
490C2×C4#1451982311yescharacteristic
491C8—22no—
492C8—22no—
493C8—42no—
494C22—4882no—
495C22—4882no—
496C22—4882no—
497C22—4882no—
498C22—4882no—
499C22—4882no—
500C22—4882no—
501C22C2×#1451975211yescharacteristic
502C22D4×D24411yescharacteristic
503C22#1684043411yescharacteristic
504C22C2×#1451979911yescharacteristic
505C22C2×#1451981011yescharacteristic
506C22#1684057411yescharacteristic
507C22#1684057411yescharacteristic
508C4—22no—
509C4—42no—
510C4—42no—
511C4—4881no—
512C4—4881no—
513C4—2442no—
514C4—2444no—
515C4—2442no—
516C4#1683998514yescomplemented
517C4#1684031311yescharacteristic
518C4#1684036811yescharacteristic
519C4#1684039611yescharacteristic
520C4#1684040311yescharacteristic
521C2—9762no—
522C2#2144940511yescharacteristic
523C2C2×#1683998511yescharacteristic
524C2#2145534111yescharacteristic
525C2#2145748711yescharacteristic
526C2#2145755111yescharacteristic
527C2#2145881711yescharacteristic
528C2#2145887411yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1939C2
socleC23×C61C22×D4
prime radicalC2×#14519774C22
hypercenter#1939D61
hypocenterC61#25369
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1939

Complements

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

NH ≅ G/Ntype
2. C2×#16839985C2semidirect
3. C2×#16839973C2semidirect
5. #21448875C2semidirect
6. #21447137C2semidirect
7. #21446916C2semidirect
9. #21446173C2semidirect
10. C4×#14519702C2semidirect
11. C61×#1939C2semidirect
84. #14519702D4semidirect
247. C4×C61#464semidirect
265. #1939D61semidirect
308. C61#25369semidirect
516. C4#16839985semidirect

Maximal subgroups

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

subgroupindex
C2×#168401352
C2×#168399852
C2×#168399732
#214508212
#214488752
#214471372
#214469162
#214467452
#214461732
C4×#145197022
C61×#19392
#2536961