Finite Group Databasebeta

← #29714911

Subgroups — #29714911

Order 15616.

1116 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840485C211yesmaximal, characteristic, complemented
2C2×#16839967C211yesmaximal, characteristic, complemented
3C2×#16839872C211yesmaximal, characteristic
4#21447060C212yesmaximal, complemented
5#21446895C212yesmaximal
6#21446760C211yesmaximal, characteristic, complemented
7#21446638C211yesmaximal, characteristic, complemented
8#21446177C212yesmaximal, complemented
9#21446086C212yesmaximal, complemented
10#21445826C211yesmaximal, characteristic, complemented
11C61×#2580C211yesmaximal, characteristic, complemented
12C22×#11758200C2212yes—
13C22×#11758199C2211yescharacteristic
14#16840485—24no—
15#16840485—24no—
16C2×#14519777C2211yescharacteristic
17C2×#14519772—22no—
18C2×#14519765C2212yes—
19C2×#14519753—21no—
20C2×C4×Dic122C2212yes—
21C2×C4×D244C2211yescharacteristic
22C2×#14519735—21no—
23#16839967—24no—
24#16839872—24no—
25C2×#14519708C2211yescharacteristic
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519700C2212yes—
28C2×#14519700C2212yes—
29C2×#14519700C2212yes—
30C2×#14519698C2211yescharacteristic
31C2×#14519697C2211yescharacteristic
32C2×#14519697C2212yes—
33C2×#14519688C2211yescharacteristic
34#16839524C2212yes—
35#16839517C2212yes—
36#16839517C2212yes—
37C2×C4×Q8×C61C2211yescharacteristic
38C2×C4×D4×C61C2211yescharacteristic
39C61×#399—21no—
40C61×#393—21no—
41C61×#390C2212yes—
42C61×#389C2212yes—
43C61×#385—22no—
44C61×#382C2211yescharacteristic
45C22×Dic122C2312yes—
46C22×D244C2311yescharacteristic
47C22×C4×D61—21no—
48C2×#11758200—24no—
49C2×#11758200—24no—
50C2×#11758200—42no—
51C2×#11758199—22no—
52C2×#11758199—22no—
53C2×#11758199—22no—
54C22×#92418C2311yescharacteristic
55C22×#92418C2312yes—
56C22×#92418C2311yescharacteristic
57#14519777—22no—
58#14519777D414yes—
59#14519772—216no—
60#14519765—24no—
61#14519765—24no—
62#14519753—28no—
63C4×Dic122—24no—
64C4×Dic122—24no—
65C4×D244—22no—
66C4×D244D414yes—
67#14519735—28no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×#11758196—22no—
72C2×#11758198—22no—
73C2×#11758198—21no—
74C2×#11758198—22no—
75C2×C4×Dic61—22no—
76#14519708—22no—
77C4⋊#92418—22no—
78#14519700—24no—
79#14519700—24no—
80#14519700—42no—
81#14519698—22no—
82#14519697—22no—
83#14519697—42no—
84#14519688—22no—
85C22×Q8×C61D411yescharacteristic
86C22×D4×C61C2311yescharacteristic
87C23×C4×C61—21no—
88C4×Q8×C61—22no—
89C4×Q8×C61—22no—
90C4×D4×C61—22no—
91C4×D4×C61—22no—
92C2×C4⋊C4×C61—22no—
93C2×C4⋊C4×C61D411yescharacteristic
94C2×C4⋊C4×C61D412yes—
95C2×C4⋊C4×C61C2312yes—
96C2×C4⋊C4×C61C2311yescharacteristic
97C2×C22⋊C4×C61—22no—
98C2×C22⋊C4×C61—21no—
99C2×C22⋊C4×C61—22no—
100C2×C42×C61C2312yes—
101C2×C42×C61C2311yescharacteristic
102#95×C61—21no—
103#95×C61—21no—
104#95×C61—22no—
105C24×C61—21no—
106C23×D61—21no—
107C2×Q8×C61—22no—
108C2×Q8×C61—42no—
109C2×D4×C61—21no—
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61—21no—
113C2×D4×C61—21no—
114C2×D4×C61#4112yes—
115C22×C4×C61—22no—
116C22×C4×C61—21no—
117C22×C4×C61—42no—
118C22×C4×C61—82no—
119C22×C4×C61C2×D412yes—
120C22×C4×C61C2×D412yes—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2411yescharacteristic
124C22×Dic61—21no—
125C22×Dic61—22no—
126C2×D244—21no—
127C2×D244—21no—
128C2×D244—21no—
129C2×D244—21no—
130C2×D244—21no—
131C2×D244C2×D412yes—
132C2×C4×D61—24no—
133C2×C4×D61—44no—
134C2×Dic122—22no—
135C2×Dic122—22no—
136C2×Dic122—22no—
137C2×Dic122—22no—
138C2×Dic122—22no—
139C2×Dic122C2×D414yes—
140C4⋊C4×C61—22no—
141C4⋊C4×C61—24no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—82no—
147C22⋊C4×C61—44no—
148C22⋊C4×C61—42no—
149C22⋊C4×C61—82no—
150C42×C61—22no—
151C42×C61—22no—
152C42×C61C2×D412yes—
153C42×C61C2×D412yes—
154#11758200—48no—
155#11758199—28no—
156#11758198—28no—
157#11758198—28no—
158#11758198—24no—
159C4⋊Dic61—24no—
160C4⋊Dic61—24no—
161C4⋊Dic61—21no—
162C4⋊Dic61C2×D412yes—
163#11758196—28no—
164C4×Dic61—28no—
165C2×#92418—21no—
166C2×#92418—21no—
167C2×#92418—21no—
168C2×#92418—22no—
169C2×#92418—22no—
170C2×#92418—22no—
171C2×#92418—21no—
172C2×#92418—21no—
173C2×#92418#4112yes—
174C23×C61—22no—
175C23×C61—41no—
176C23×C61—41no—
177C23×C61—42no—
178C23×C61—42no—
179C23×C61C22×D411yescharacteristic
180C22×D61—22no—
181C22×D61—41no—
182C22×D61—41no—
183C22×D61—42no—
184C22×D61—42no—
185Q8×C61—44no—
186D4×C61—22no—
187D4×C61—22no—
188D4×C61—41no—
189D4×C61—41no—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—41no—
199C2×C4×C61—42no—
200C2×C4×C61—42no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61—84no—
204C2×C4×C61#12012yes—
205C2×C4×C61#12012yes—
206C2×C4×C61C22×D411yescharacteristic
207C2×C4×C61C22×D411yescharacteristic
208C2×C4×C61C2×#4111yescharacteristic
209C2×C4×C61C2×#4111yescharacteristic
210C2×C4×C61H(5,𝔽2)11yescharacteristic
211C2×C4×C612−1+411yescharacteristic
212C2×Dic61—22no—
213C2×Dic61—24no—
214C2×Dic61—42no—
215C2×Dic61—42no—
216C2×Dic61—42no—
217D244—22no—
218D244—22no—
219D244—41no—
220D244—41no—
221C4×D61—48no—
222Dic122—24no—
223Dic122—24no—
224Dic122—42no—
225Dic122—42no—
226#92418—22no—
227#92418—41no—
228#92418—42no—
229#24197—611nomaximal, Sylow
230C22×C61—44no—
231C22×C61—42no—
232C22×C61—82no—
233C22×C61—81no—
234C22×C61—81no—
235C22×C61C2×#12011yescharacteristic
236C22×C61#54811yescharacteristic
237C22×C61#55011yescharacteristic
238C22×C61#57512yes—
239C22×C61#57712yes—
240C2×D61—42no—
241C2×D61—42no—
242C2×D61—42no—
243C2×D61—82no—
244C2×D61—82no—
245C4×C61—22no—
246C4×C61—22no—
247C4×C61—42no—
248C4×C61—42no—
249C4×C61—82no—
250C4×C61—81no—
251C4×C61#52511yescharacteristic
252C4×C61#53711yescharacteristic
253C4×C61#54611yescharacteristic
254C4×C61#54711yescharacteristic
255Dic61—44no—
256Dic61—42no—
257C2×#556—611no—
258C2×#461—611no—
259C2×#442—611no—
260#2580D6111yescharacteristic, complemented
261#2042—612no—
262#2012—612no—
263#1985—611no—
264#1960—611no—
265#1880—612no—
266#1865—612no—
267#1804—611no—
268C2×C61—82no—
269C2×C61#240711yescharacteristic
270C2×C61#300511yescharacteristic
271C2×C61#301011yescharacteristic
272C2×C61#337612yes—
273C2×C61#340112yes—
274D61—82no—
275C22×SD16—612no—
276C22×D8—611no—
277#556—1224no—
278#556—1224no—
279C2×C4⋊Q8—611no—
280C2×#126—1222no—
281C2×#124—612no—
282C2×C22⋊Q8—1221no—
283C2×C4×Q8—612no—
284C2×C4×Q8C2×D6111yescharacteristic
285C2×C4×D4—611no—
286C2×C4×D4C2×D6111yescharacteristic
287C2×#117—1221no—
288#461—1224no—
289#442—1224no—
290C2×#106—611no—
291C2×C4⋊C8—611no—
292C2×#103—612no—
293C2×#103—612no—
294C2×#103—612no—
295C2×#102—611no—
296C2×#102—611no—
297C2×#102—612no—
298C2×#97—611no—
299#399—21no—
300#393—21no—
301#390C2×D6112yes—
302#389C2×D6112yes—
303#385—22no—
304#382C2×D6111yescharacteristic
305#340—612no—
306#336—612no—
307#336—612no—
308C61#2419711yescharacteristic, complemented, Sylow
309C22×Q8—612no—
310C22×Q8C22⋊D6111yescharacteristic
311C22×D4—611no—
312C22×D4C22×D6111yescharacteristic
313C23×C4—21no—
314C23×C4—1221no—
315C2×SD16—1224no—
316C2×SD16—1224no—
317C2×SD16—2442no—
318C2×D8—1222no—
319C2×D8—1222no—
320C2×D8—1222no—
321C22×C8—611no—
322C22×C8—612no—
323C22×C8—611no—
324C4⋊Q8—614no—
325C4⋊Q8—1222no—
326#126—12216no—
327#124—1224no—
328#124—1224no—
329C22⋊Q8—1228no—
330C4×Q8—22no—
331C4×Q8—22no—
332C4×Q8—1224no—
333C4×Q8—1224no—
334C4×D4—22no—
335C4×D4—22no—
336C4×D4—614no—
337C4×D4—1222no—
338#117—1228no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—611no—
341C2×C4⋊C4—1221no—
342C2×C4⋊C4—1222no—
343C2×C4⋊C4—1221no—
344C2×C4⋊C4C22⋊D6111yescharacteristic
345C2×C4⋊C4C22⋊D6112yes—
346C2×C4⋊C4C22×D6111yescharacteristic
347C2×C4⋊C4C22×D6112yes—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—1222no—
352C2×C22⋊C4—1221no—
353C2×C22⋊C4—1222no—
354C2×C42—1222no—
355C2×C42C22×D6112yes—
356C2×C42C22×D6111yescharacteristic
357#106—1222no—
358C4⋊C8—1222no—
359#103—1224no—
360#103—1224no—
361#103—2442no—
362#102—1222no—
363#102—1222no—
364#102—2442no—
365#97—1222no—
366#95—21no—
367#95—21no—
368#95—22no—
369C24—21no—
370C24—1221no—
371C2×Q8—22no—
372C2×Q8—42no—
373C2×Q8—614no—
374C2×Q8—1222no—
375C2×Q8—1222no—
376C2×Q8—1222no—
377C2×Q8—1222no—
378C2×Q8—1222no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—612no—
385C2×D4—1221no—
386C2×D4—1221no—
387C2×D4—1221no—
388C2×D4—1221no—
389C2×D4—1221no—
390C2×D4#1175822612yes—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—42no—
394C22×C4—82no—
395C22×C4—1221no—
396C22×C4—1222no—
397C22×C4—1224no—
398C22×C4—2444no—
399C22×C4C2×C22⋊D6112yes—
400C22×C4C2×C22⋊D6112yes—
401C22×C4C2×C22⋊D6111yescharacteristic
402C22×C4C2×C22⋊D6111yescharacteristic
403C22×C4C23×D6111yescharacteristic
404SD16—2448no—
405D8—1228no—
406C2×C8—612no—
407C2×C8—1221no—
408C2×C8—1221no—
409C2×C8—1221no—
410C2×C8—1222no—
411C2×C8—1222no—
412C2×C8—1222no—
413C2×C8—1221no—
414C2×C8—1221no—
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—82no—
422C4⋊C4—612no—
423C4⋊C4—1228no—
424C4⋊C4—1221no—
425C4⋊C4—1224no—
426C4⋊C4—1224no—
427C22⋊C4—44no—
428C22⋊C4—42no—
429C22⋊C4—82no—
430C22⋊C4—1228no—
431C22⋊C4—1228no—
432C22⋊C4—1224no—
433C42—22no—
434C42—22no—
435C42—1228no—
436C42D4×D6112yes—
437C42D4×D6112yes—
438C23—22no—
439C23—41no—
440C23—42no—
441C23—42no—
442C23—41no—
443C23—1222no—
444C23—2442no—
445C23—2442no—
446C23—2441no—
447C23—2441no—
448C23C22×C22⋊D6111yescharacteristic
449Q8—44no—
450Q8—1224no—
451Q8—1224no—
452Q8—2442no—
453Q8—2442no—
454D4—22no—
455D4—22no—
456D4—41no—
457D4—41no—
458D4—1222no—
459D4—1222no—
460D4—2441no—
461D4—2441no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—84no—
476C2×C4—1222no—
477C2×C4—1224no—
478C2×C4—2448no—
479C2×C4—2442no—
480C2×C4—2442no—
481C2×C4—2442no—
482C2×C4#1451974512yes—
483C2×C4#1451974512yes—
484C2×C4C2×D4×D6111yescharacteristic
485C2×C4C2×D4×D6111yescharacteristic
486C2×C4#41×D6111yescharacteristic
487C2×C4C2×#1175822611yescharacteristic
488C2×C4#1451982311yescharacteristic
489C2×C4#1451982511yescharacteristic
490C8—1222no—
491C8—2441no—
492C8—2442no—
493C22—42no—
494C22—44no—
495C22—82no—
496C22—81no—
497C22—81no—
498C22—4882no—
499C22—4882no—
500C22—2442no—
501C22—2442no—
502C22—2442no—
503C22C2×#1451974511yescharacteristic
504C22#1684042311yescharacteristic
505C22#1684043111yescharacteristic
506C22#1684056012yes—
507C22#1684057312yes—
508C4—22no—
509C4—22no—
510C4—42no—
511C4—42no—
512C4—82no—
513C4—81no—
514C4—2444no—
515C4—2442no—
516C4#1684027511yescharacteristic
517C4#1684034211yescharacteristic
518C4#1684040311yescharacteristic
519C4#1684041411yescharacteristic
520C2—82no—
521C2—4882no—
522C2#2144916611yescharacteristic
523C2#2145505911yescharacteristic
524C2#2145510311yescharacteristic
525C2#2145864612yes—
526C2#2145887912yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2580C2
socleC23×C61C22×D4
prime radicalC22×#11758199C22
hypercenter#2580D61
hypocenterC61#24197
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2580

Complements

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

NH ≅ G/Ntype
1. C2×#16840485C2semidirect
2. C2×#16839967C2semidirect
4. #21447060C2semidirect
6. #21446760C2semidirect
7. #21446638C2semidirect
8. #21446177C2semidirect
9. #21446086C2semidirect
10. #21445826C2semidirect
11. C61×#2580C2semidirect
260. #2580D61semidirect
308. C61#24197semidirect

Maximal subgroups

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

subgroupindex
C2×#168404852
C2×#168399672
C2×#168398722
#214470602
#214468952
#214467602
#214466382
#214461772
#214460862
#214458262
C61×#25802
#2419761