Finite Group Databasebeta

← #29711957

Subgroups — #29711957

Order 15616.

1334 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839905C212yesmaximal, complemented
2#21448407C211yesmaximal, characteristic, complemented
3#21446967C211yesmaximal, characteristic, complemented
4#21446962C211yesmaximal, characteristic, complemented
5#21446907C211yesmaximal, characteristic, complemented
6#21446890C211yesmaximal, characteristic, complemented
7#21446167C212yesmaximal, complemented
8#21446153C212yesmaximal, complemented
9#21446137C212yesmaximal, complemented
10#21445890C211yesmaximal, characteristic, complemented
11C2×C61×#540C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C22×#11758200C2212yes—
14C22×#11758199C2212yes—
15C22×C4⋊Dic61C2211yescharacteristic
16#16839905—28no—
17C2×#14519700C2212yes—
18C2×#14519698C2212yes—
19C2×#14519698C2212yes—
20C2×#14519697C2212yes—
21C2×C22⋊#92418C2212yes—
22#16839668C2211yescharacteristic
23#16839668C2211yescharacteristic
24#16839664C2211yescharacteristic
25#16839664C2211yescharacteristic
26#16839639—22no—
27#16839619—22no—
28#16839524C2212yes—
29#16839524C2212yes—
30#16839517C2212yes—
31#16839517C2211yescharacteristic
32#16839517C2211yescharacteristic
33C61×#540—28no—
34C2×#127×C61C2211yescharacteristic
35C2×#125×C61C2211yescharacteristic
36C2×#123×C61—22no—
37C2×#121×C61—22no—
38C2×#121×C61C2212yes—
39C2×C4×D4×C61C2212yes—
40C2×#117×C61C2211yescharacteristic
41C2×#11758222—22no—
42C2×#11758222—24no—
43C2×#11758222—22no—
44C2×#11758222—42no—
45C22×Dic122C2311yescharacteristic
46C22×D244C2311yescharacteristic
47C22×C22⋊D61—21no—
48C23×Dic61—21no—
49C22×C4×D61—21no—
50C2×#11758200—24no—
51C2×#11758200—24no—
52C2×#11758200—42no—
53C2×#11758199—24no—
54C2×#11758199—24no—
55C2×#11758199—42no—
56C22×#92418C2312yes—
57C22×#92418C2312yes—
58C2×C4⋊Dic61—21no—
59C2×C4⋊Dic61—42no—
60C2×C4⋊Dic61C2311yescharacteristic
61C2×C4⋊Dic61C2311yescharacteristic
62C2×#11758198—22no—
63C2×#11758198—22no—
64#14519700—24no—
65#14519698—42no—
66#14519698—42no—
67#14519697—24no—
68C22⋊#92418—24no—
69#14519684—22no—
70#14519684—22no—
71C22×D4×C61—22no—
72C22×D4×C61—22no—
73C22×D4×C61C2312yes—
74C23×C4×C61—22no—
75C23×C4×C61C2311yescharacteristic
76#127×C61—24no—
77#125×C61—24no—
78#123×C61—216no—
79#121×C61—216no—
80#121×C61—24no—
81#121×C61D418yes—
82C4×D4×C61—24no—
83C4×D4×C61—24no—
84#117×C61—24no—
85C2×C4⋊C4×C61—22no—
86C2×C4⋊C4×C61D412yes—
87C2×C4⋊C4×C61C2312yes—
88C2×C22⋊C4×C61—22no—
89C2×C22⋊C4×C61—22no—
90C2×C22⋊C4×C61D412yes—
91C2×C42×C61C2311yescharacteristic
92C2×C42×C61C2311yescharacteristic
93C24×C61—22no—
94C24×C61C2×D411yescharacteristic
95C23×D61—21no—
96C2×D4×C61—28no—
97C2×D4×C61—28no—
98C2×D4×C61—22no—
99C2×D4×C61—22no—
100C2×D4×C61—22no—
101C2×D4×C61—22no—
102C2×D4×C61—22no—
103C2×D4×C61—82no—
104C2×D4×C61—82no—
105C2×D4×C61—82no—
106C2×D4×C61—82no—
107C2×D4×C61C2×D414yes—
108C22×C4×C61—21no—
109C22×C4×C61—21no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—28no—
113C22×C4×C61—82no—
114C22×C4×C61—82no—
115C22×C4×C61C2×D412yes—
116C22×C4×C61C2×D412yes—
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D414yes—
119C22×C4×C61C2411yescharacteristic
120C2×C22⋊D61—24no—
121C2×C22⋊D61—42no—
122C2×C22⋊D61—42no—
123C22×Dic61—21no—
124C22×Dic61—21no—
125C22×Dic61—21no—
126C22×Dic61—42no—
127C22×Dic61—82no—
128#11758222—48no—
129#11758222—48no—
130C2×D244—22no—
131C2×D244—22no—
132C2×D244—21no—
133C2×D244#4112yes—
134C2×C4×D61—24no—
135C2×C4×D61—42no—
136C2×C4×D61—42no—
137C2×Dic122—22no—
138C2×Dic122—22no—
139C2×Dic122—21no—
140C2×Dic122#4112yes—
141C4⋊C4×C61—28no—
142C4⋊C4×C61—24no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61C2×D414yes—
145C22⋊C4×C61—28no—
146C22⋊C4×C61—24no—
147C22⋊C4×C61—28no—
148C42×C61—22no—
149C42×C61—22no—
150#11758200—48no—
151#11758199—48no—
152#11758198—82no—
153#11758198—82no—
154C4⋊Dic61—22no—
155C4⋊Dic61—22no—
156C4⋊Dic61—42no—
157C2×#92418—22no—
158C2×#92418—22no—
159C2×#92418—22no—
160C2×#92418—22no—
161C2×#92418—22no—
162C2×#92418—22no—
163C23×C61—21no—
164C23×C61—21no—
165C23×C61—22no—
166C23×C61—22no—
167C23×C61—24no—
168C23×C61—42no—
169C23×C61—42no—
170C23×C61—42no—
171C23×C61—42no—
172C23×C61—42no—
173C23×C61—42no—
174C23×C61#12012yes—
175C23×C61C22×D411yescharacteristic
176C22×D61—22no—
177C22×D61—41no—
178C22×D61—41no—
179C22×D61—42no—
180C22×D61—42no—
181D4×C61—24no—
182D4×C61—24no—
183D4×C61—42no—
184D4×C61—42no—
185D4×C61—88no—
186D4×C61—88no—
187C2×C4×C61—21no—
188C2×C4×C61—21no—
189C2×C4×C61—24no—
190C2×C4×C61—24no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—24no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—42no—
199C2×C4×C61—42no—
200C2×C4×C61—88no—
201C2×C4×C61#12012yes—
202C2×C4×C61C22×D412yes—
203C2×C4×C61C2×#4111yescharacteristic
204C2×C4×C61C2×#4111yescharacteristic
205C2×C4×C61H(5,𝔽2)12yes—
206C22⋊D61—48no—
207C2×Dic61—22no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61—42no—
213C2×Dic61—42no—
214C2×Dic61—84no—
215D244—24no—
216D244—42no—
217C4×D61—48no—
218Dic122—24no—
219Dic122—42no—
220#92418—42no—
221#92418—42no—
222#23941—611nomaximal, Sylow
223C22×C61—22no—
224C22×C61—24no—
225C22×C61—42no—
226C22×C61—41no—
227C22×C61—41no—
228C22×C61—44no—
229C22×C61—44no—
230C22×C61—44no—
231C22×C61—82no—
232C22×C61—82no—
233C22×C61—82no—
234C22×C61—82no—
235C22×C61C2×#12011yescharacteristic
236C22×C61#52511yescharacteristic
237C22×C61#53511yescharacteristic
238C22×C61#57612yes—
239C22×C61#57712yes—
240C2×D61—44no—
241C2×D61—42no—
242C2×D61—81no—
243C2×D61—81no—
244C2×D61—82no—
245C4×C61—22no—
246C4×C61—42no—
247C4×C61—42no—
248C4×C61—44no—
249C4×C61#54612yes—
250C4×C61#54612yes—
251Dic61—42no—
252Dic61—81no—
253Dic61—81no—
254C2×#540D6111yescharacteristic, complemented
255C2×#449—612no—
256#2311—611no—
257#2026—611no—
258#2025—611no—
259#2014—611no—
260#2011—611no—
261#1879—612no—
262#1877—612no—
263#1874—612no—
264#1818—611no—
265C2×C61—42no—
266C2×C61—84no—
267C2×C61#240411yescharacteristic
268C2×C61#301212yes—
269C2×C61#311912yes—
270C2×C61#322011yescharacteristic
271C2×C61#322111yescharacteristic
272D61—82no—
273C22×#41—611no—
274C22×SD16—612no—
275C22×D8—612no—
276#540—28no—
277C2×#127C2×D6111yescharacteristic
278C2×#125C2×D6111yescharacteristic
279C2×#123—22no—
280C2×#121—22no—
281C2×#121C2×D6112yes—
282C2×C4×D4C2×D6112yes—
283C2×#117C2×D6111yescharacteristic
284C22×C4⋊C4—611no—
285#449—1228no—
286C2×#103—612no—
287C2×#102—612no—
288C2×#102—612no—
289C2×#102—612no—
290C2×C22⋊C8—612no—
291#391—611no—
292#391—611no—
293#390—611no—
294#390—611no—
295#386—1222no—
296#380—1222no—
297#340—612no—
298#340—612no—
299#336—612no—
300#336—611no—
301#336—611no—
302C61#2394111yescharacteristic, complemented, Sylow
303C2×#41—1222no—
304C2×#41—1222no—
305C2×#41—1224no—
306C2×#41—2442no—
307C22×Q8—611no—
308C22×D4—22no—
309C22×D4—22no—
310C22×D4—611no—
311C22×D4—1221no—
312C22×D4C22×D6112yes—
313C23×C4—22no—
314C23×C4—1221no—
315C23×C4—1221no—
316C23×C4C22×D6111yescharacteristic
317C2×SD16—1224no—
318C2×SD16—1224no—
319C2×SD16—2442no—
320C2×D8—1224no—
321C2×D8—1224no—
322C2×D8—2442no—
323C22×C8—612no—
324C22×C8—612no—
325#127—24no—
326#125—24no—
327#123—216no—
328#121—216no—
329#121—24no—
330#121C22⋊D6118yes—
331C4×D4—24no—
332C4×D4—24no—
333#117—24no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—611no—
336C2×C4⋊C4—611no—
337C2×C4⋊C4—1221no—
338C2×C4⋊C4—2442no—
339C2×C4⋊C4C22⋊D6112yes—
340C2×C4⋊C4C22×D6112yes—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—1222no—
344C2×C22⋊C4—1222no—
345C2×C22⋊C4C22⋊D6112yes—
346C2×C42C22×D6111yescharacteristic
347C2×C42C22×D6111yescharacteristic
348#103—1224no—
349#102—1224no—
350#102—2442no—
351#102—2442no—
352C22⋊C8—1224no—
353#95—1222no—
354#95—1222no—
355C24—22no—
356C24—1221no—
357C24C2×C22⋊D6111yescharacteristic
358#41—2448no—
359#41—2448no—
360C2×Q8—612no—
361C2×Q8—1221no—
362C2×Q8—1222no—
363C2×Q8—1222no—
364C2×D4—28no—
365C2×D4—28no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—82no—
372C2×D4—82no—
373C2×D4—82no—
374C2×D4—82no—
375C2×D4—612no—
376C2×D4—1224no—
377C2×D4—1222no—
378C2×D4—1221no—
379C2×D4—1222no—
380C2×D4—2442no—
381C2×D4—2442no—
382C2×D4C2×C22⋊D6114yes—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—28no—
388C22×C4—82no—
389C22×C4—82no—
390C22×C4—1224no—
391C22×C4—1221no—
392C22×C4—1221no—
393C22×C4—1221no—
394C22×C4—4882no—
395C22×C4—2442no—
396C22×C4—2442no—
397C22×C4—2442no—
398C22×C4C2×C22⋊D6112yes—
399C22×C4C2×C22⋊D6112yes—
400C22×C4C2×C22⋊D6114yes—
401C22×C4C2×C22⋊D6111yescharacteristic
402C22×C4C23×D6111yescharacteristic
403SD16—2448no—
404D8—2448no—
405C2×C8—1222no—
406C2×C8—1222no—
407C2×C8—1222no—
408C2×C8—1222no—
409C2×C8—1222no—
410C2×C8—1222no—
411C4⋊C4—28no—
412C4⋊C4—22no—
413C4⋊C4—24no—
414C4⋊C4—1222no—
415C4⋊C4—1222no—
416C4⋊C4—2442no—
417C4⋊C4C2×C22⋊D6114yes—
418C22⋊C4—28no—
419C22⋊C4—24no—
420C22⋊C4—28no—
421C22⋊C4—4882no—
422C22⋊C4—4882no—
423C42—22no—
424C42—22no—
425C23—21no—
426C23—22no—
427C23—22no—
428C23—21no—
429C23—24no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—1222no—
437C23—2441no—
438C23—2442no—
439C23—2442no—
440C23—2441no—
441C23#1451974612yes—
442C23C22×C22⋊D6111yescharacteristic
443Q8—1224no—
444Q8—2442no—
445D4—24no—
446D4—24no—
447D4—42no—
448D4—42no—
449D4—88no—
450D4—88no—
451D4—1224no—
452D4—2448no—
453D4—2442no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—88no—
468C2×C4—1222no—
469C2×C4—4884no—
470C2×C4—2448no—
471C2×C4—2442no—
472C2×C4—2442no—
473C2×C4—2441no—
474C2×C4—2441no—
475C2×C4—2441no—
476C2×C4—2441no—
477C2×C4#1451974612yes—
478C2×C4C22×C22⋊D6112yes—
479C2×C4#41×D6111yescharacteristic
480C2×C4#41×D6111yescharacteristic
481C2×C4#1451982312yes—
482C8—2442no—
483C8—2442no—
484C22—24no—
485C22—22no—
486C22—42no—
487C22—41no—
488C22—41no—
489C22—44no—
490C22—44no—
491C22—44no—
492C22—82no—
493C22—82no—
494C22—82no—
495C22—82no—
496C22—4882no—
497C22—4881no—
498C22—4881no—
499C22—2444no—
500C22—2442no—
501C22C2×#1451974611yescharacteristic
502C22#1684026911yescharacteristic
503C22#1684032611yescharacteristic
504C22#1684056512yes—
505C22#1684057312yes—
506C4—22no—
507C4—42no—
508C4—42no—
509C4—44no—
510C4—4881no—
511C4—4881no—
512C4—2442no—
513C4#1684040012yes—
514C4#1684040012yes—
515C2—42no—
516C2—84no—
517C2—4882no—
518C2#2144913411yescharacteristic
519C2#2145512312yes—
520C2#2145601212yes—
521C2#2145686311yescharacteristic
522C2#2145687411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#540C2
socleC23×C61C22×D4
hypercenterC2×#540D61
hypocenterC61#23941
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#540

Complements

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

NH ≅ G/Ntype
1. C2×#16839905C2semidirect
2. #21448407C2semidirect
3. #21446967C2semidirect
4. #21446962C2semidirect
5. #21446907C2semidirect
6. #21446890C2semidirect
7. #21446167C2semidirect
8. #21446153C2semidirect
9. #21446137C2semidirect
10. #21445890C2semidirect
11. C2×C61×#540C2semidirect
254. C2×#540D61semidirect
302. C61#23941semidirect

Maximal subgroups

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

subgroupindex
C2×#168399052
#214484072
#214469672
#214469622
#214469072
#214468902
#214461672
#214461532
#214461372
#214458902
C2×C61×#5402
#2394161