Finite Group Databasebeta

← #29714818

Subgroups — #29714818

Order 15616.

1146 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840497C211yesmaximal, characteristic
2C2×#16839962C211yesmaximal, characteristic
3C2×C4×#11758202C211yesmaximal, characteristic, complemented
4#21447060C212yesmaximal, complemented
5#21446912C212yesmaximal, complemented
6#21446756C211yesmaximal, characteristic, complemented
7#21446624C211yesmaximal, characteristic
8#21446156C212yesmaximal, complemented
9#21446065C212yesmaximal, complemented
10#21445821C211yesmaximal, characteristic, complemented
11C61×#2582C211yesmaximal, characteristic, complemented
12C22×#11758202C2211yescharacteristic
13C22×#11758201C2212yes—
14#16840497—24no—
15#16840497—24no—
16C2×#14519774C2211yescharacteristic
17C2×#14519766—21no—
18C2×#14519765C2212yes—
19C2×#14519762—22no—
20C2×C4×Dic122C2211yescharacteristic
21C2×C4×D244C2212yes—
22C2×#14519739—21no—
23#16839962—24no—
24C4×#11758202—24no—
25C2×#14519710C2211yescharacteristic
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519700C2211yescharacteristic
29C2×#14519700C2212yes—
30C2×#14519697C2212yes—
31C2×#14519697C2212yes—
32C2×#14519697C2212yes—
33C2×C4×#92418C2211yescharacteristic
34#16839524C2212yes—
35#16839517C2212yes—
36#16839517C2212yes—
37C2×C4×Q8×C61C2211yescharacteristic
38C2×C4×Q8×C61C2211yescharacteristic
39C61×#398—22no—
40C61×#391—21no—
41C61×#391C2212yes—
42C61×#389—21no—
43C61×#389C2212yes—
44C61×#384C2211yescharacteristic
45C22×Dic122C2311yescharacteristic
46C22×D244—21no—
47C22×D244—21no—
48C22×D244C2312yes—
49C22×C4×D61—22no—
50C2×#11758202—22no—
51C2×#11758202—22no—
52C2×#11758202—22no—
53C2×#11758201—24no—
54C2×#11758201—24no—
55C2×#11758201—42no—
56C22×#92418C2311yescharacteristic
57C22×#92418C2312yes—
58C22×#92418C2311yescharacteristic
59#14519774—22no—
60#14519774D414yes—
61#14519766—28no—
62#14519765—24no—
63#14519765—24no—
64#14519762—216no—
65C4×Dic122—22no—
66C4×Dic122D414yes—
67C4×D244—24no—
68C4×D244—24no—
69#14519739—28no—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×#11758196—21no—
72C2×#11758198—22no—
73C2×#11758198—22no—
74C2×#11758198—22no—
75C2×C4×Dic61—21no—
76#14519710—22no—
77C4⋊#92418—22no—
78#14519701—22no—
79#14519700—22no—
80#14519700—42no—
81#14519697—24no—
82#14519697—24no—
83#14519697—42no—
84C4×#92418—22no—
85C22×Q8×C61D411yescharacteristic
86C22×Q8×C61C2311yescharacteristic
87C4×Q8×C61—22no—
88C4×Q8×C61—22no—
89C4×Q8×C61—22no—
90C4×Q8×C61—22no—
91C2×C4⋊C4×C61—21no—
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61—22no—
95C2×C4⋊C4×C61D411yescharacteristic
96C2×C4⋊C4×C61D412yes—
97C2×C4⋊C4×C61C2312yes—
98C2×C4⋊C4×C61C2311yescharacteristic
99C2×C42×C61—21no—
100C2×C42×C61C2312yes—
101C2×C42×C61C2311yescharacteristic
102#95×C61—22no—
103#95×C61—22no—
104#95×C61—22no—
105C23×D61—22no—
106C2×Q8×C61—22no—
107C2×Q8×C61—22no—
108C2×Q8×C61—22no—
109C2×Q8×C61—21no—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C2×Q8×C61#4112yes—
115C22×C4×C61—21no—
116C22×C4×C61—22no—
117C22×C4×C61—21no—
118C22×C4×C61C2×D412yes—
119C22×C4×C61C2×D412yes—
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2411yescharacteristic
123C22×Dic61—21no—
124C22×Dic61—21no—
125C2×D244—22no—
126C2×D244—22no—
127C2×D244—22no—
128C2×D244—22no—
129C2×D244—24no—
130C2×D244—22no—
131C2×D244—24no—
132C2×D244—42no—
133C2×D244—42no—
134C2×D244—81no—
135C2×D244—81no—
136C2×D244C2×D414yes—
137C2×C4×D61—28no—
138C2×C4×D61—82no—
139C2×C4×D61—82no—
140C2×Dic122—21no—
141C2×Dic122—21no—
142C2×Dic122—21no—
143C2×Dic122—21no—
144C2×Dic122—21no—
145C2×Dic122C2×D412yes—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—24no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—42no—
153C4⋊C4×C61—42no—
154C4⋊C4×C61—81no—
155C4⋊C4×C61—82no—
156C42×C61—22no—
157C42×C61—22no—
158C42×C61—42no—
159C42×C61C2×D412yes—
160C42×C61C2×D412yes—
161#11758202—28no—
162#11758201—48no—
163#11758198—28no—
164#11758198—28no—
165#11758198—28no—
166C4⋊Dic61—21no—
167C4⋊Dic61C2×D412yes—
168#11758196—24no—
169C4×Dic61—24no—
170C2×#92418—21no—
171C2×#92418—21no—
172C2×#92418—21no—
173C2×#92418—22no—
174C2×#92418—22no—
175C2×#92418—21no—
176C2×#92418—22no—
177C2×#92418—21no—
178C2×#92418#4112yes—
179C23×C61C22×D411yescharacteristic
180C22×D61—24no—
181C22×D61—42no—
182C22×D61—42no—
183C22×D61—42no—
184C22×D61—42no—
185C22×D61—42no—
186C22×D61—42no—
187Q8×C61—22no—
188Q8×C61—22no—
189Q8×C61—41no—
190Q8×C61—41no—
191Q8×C61—44no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61#12012yes—
210C2×C4×C61#12012yes—
211C2×C4×C61#13512yes—
212C2×C4×C61#13712yes—
213C2×C4×C61C22×D411yescharacteristic
214C2×C4×C61C22×D411yescharacteristic
215C2×C4×C61C2×#4111yescharacteristic
216C2×C4×C61C2×#4111yescharacteristic
217C2×C4×C61H(5,𝔽2)11yescharacteristic
218C2×C4×C612−1+411yescharacteristic
219C2×Dic61—22no—
220C2×Dic61—22no—
221C2×Dic61—41no—
222C2×Dic61—41no—
223C2×Dic61—41no—
224C2×Dic61—41no—
225D244—24no—
226D244—24no—
227D244—42no—
228D244—42no—
229D244—48no—
230D244—84no—
231C4×D61—88no—
232Dic122—22no—
233Dic122—22no—
234Dic122—41no—
235Dic122—41no—
236#92418—22no—
237#92418—41no—
238#92418—42no—
239#24190—611nomaximal, Sylow
240C22×C61C2×#12011yescharacteristic
241C22×C61#54811yescharacteristic
242C22×C61#55011yescharacteristic
243C22×C61C2×#13511yescharacteristic
244C22×C61C2×#13711yescharacteristic
245C22×C61#57611yescharacteristic
246C22×C61#57711yescharacteristic
247C2×D61—44no—
248C2×D61—44no—
249C2×D61—44no—
250C2×D61—82no—
251C2×D61—82no—
252C2×D61—82no—
253C2×D61—82no—
254C4×C61—22no—
255C4×C61—42no—
256C4×C61—42no—
257C4×C61—42no—
258C4×C61—82no—
259C4×C61—81no—
260C4×C61#45214yes—
261C4×C61#52511yescharacteristic
262C4×C61#53711yescharacteristic
263C4×C61#54611yescharacteristic
264C4×C61#54711yescharacteristic
265Dic61—42no—
266Dic61—42no—
267C2×#559—611no—
268C2×#460—611no—
269C2×C4×Q16—611no—
270#2582D6111yescharacteristic, complemented
271#2042—612no—
272#2015—612no—
273#1984—611no—
274#1957—611no—
275#1877—612no—
276#1862—612no—
277#1803—611no—
278C2×C61#240711yescharacteristic
279C2×C61C2×#45211yescharacteristic
280C2×C61#301211yescharacteristic
281C2×C61#336511yescharacteristic
282C2×C61#337511yescharacteristic
283C2×C61#339611yescharacteristic
284C2×C61#340011yescharacteristic
285D61—84no—
286C22×Q16—611no—
287C22×SD16—612no—
288#559—1224no—
289#559—1224no—
290C2×#127—611no—
291C2×#124—612no—
292C2×#124—1221no—
293C2×#123—1222no—
294C2×C4×Q8—611no—
295C2×C4×Q8C2×D6111yescharacteristic
296C2×C4×Q8C2×D6111yescharacteristic
297C2×C4×D4—612no—
298C2×C4×D4—1221no—
299#460—1224no—
300C4×Q16—1224no—
301C2×#107—611no—
302C2×C4⋊C8—611no—
303C2×#103—611no—
304C2×#103—611no—
305C2×#103—612no—
306C2×#102—612no—
307C2×#102—612no—
308C2×#102—612no—
309C2×C4×C8—611no—
310#398—22no—
311#391—21no—
312#391C2×D6112yes—
313#389—21no—
314#389C2×D6112yes—
315#384C2×D6111yescharacteristic
316#340—612no—
317#336—612no—
318#336—612no—
319C61#2419011yescharacteristic, complemented, Sylow
320C22×Q8—611no—
321C22×Q8C22⋊D6111yescharacteristic
322C22×Q8C22×D6111yescharacteristic
323C22×D4—612no—
324C22×D4—1221no—
325C22×D4—1221no—
326C23×C4—1222no—
327C2×Q16—1222no—
328C2×Q16—1222no—
329C2×Q16—1222no—
330C2×SD16—1224no—
331C2×SD16—1224no—
332C2×SD16—2442no—
333C22×C8—611no—
334C22×C8—612no—
335C22×C8—611no—
336#127—614no—
337#127—1222no—
338#124—1224no—
339#124—1224no—
340#124—1228no—
341#123—12216no—
342C4×Q8—22no—
343C4×Q8—22no—
344C4×Q8—22no—
345C4×Q8—22no—
346C4×Q8—614no—
347C4×Q8—1222no—
348C4×D4—1224no—
349C4×D4—1224no—
350C4×D4—1228no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—611no—
356C2×C4⋊C4—1221no—
357C2×C4⋊C4C22⋊D6111yescharacteristic
358C2×C4⋊C4C22⋊D6112yes—
359C2×C4⋊C4C22×D6111yescharacteristic
360C2×C4⋊C4C22×D6112yes—
361C2×C22⋊C4—1222no—
362C2×C22⋊C4—1222no—
363C2×C22⋊C4—1222no—
364C2×C42—21no—
365C2×C42—1221no—
366C2×C42C22×D6112yes—
367C2×C42C22×D6111yescharacteristic
368#107—1222no—
369C4⋊C8—1222no—
370#103—1222no—
371#103—1222no—
372#103—2442no—
373#102—1224no—
374#102—1224no—
375#102—2442no—
376C4×C8—1222no—
377#95—22no—
378#95—22no—
379#95—22no—
380C24—1222no—
381C2×Q8—22no—
382C2×Q8—21no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—612no—
390C2×Q8—1221no—
391C2×Q8—1221no—
392C2×Q8—1221no—
393C2×Q8—1221no—
394C2×Q8—1221no—
395C2×Q8#1175822612yes—
396C2×D4—614no—
397C2×D4—1222no—
398C2×D4—1222no—
399C2×D4—1222no—
400C2×D4—1222no—
401C2×D4—1222no—
402C2×D4—1224no—
403C2×D4—1224no—
404C2×D4—4881no—
405C2×D4—4881no—
406C2×D4—2442no—
407C2×D4—2442no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—1228no—
412C22×C4—1221no—
413C22×C4—1221no—
414C22×C4—4882no—
415C22×C4—4882no—
416C22×C4C2×C22⋊D6112yes—
417C22×C4C2×C22⋊D6112yes—
418C22×C4C2×C22⋊D6111yescharacteristic
419C22×C4C2×C22⋊D6111yescharacteristic
420C22×C4C23×D6111yescharacteristic
421Q16—1228no—
422SD16—2448no—
423C2×C8—612no—
424C2×C8—1221no—
425C2×C8—1221no—
426C2×C8—1221no—
427C2×C8—1222no—
428C2×C8—1222no—
429C2×C8—1221no—
430C2×C8—1222no—
431C2×C8—1221no—
432C4⋊C4—22no—
433C4⋊C4—24no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—81no—
441C4⋊C4—82no—
442C4⋊C4—612no—
443C4⋊C4—1221no—
444C4⋊C4—1224no—
445C22⋊C4—1228no—
446C22⋊C4—1228no—
447C22⋊C4—1228no—
448C42—22no—
449C42—22no—
450C42—42no—
451C42—1224no—
452C42D4×D6112yes—
453C42D4×D6112yes—
454C23—1224no—
455C23—2442no—
456C23—2442no—
457C23—2442no—
458C23—2442no—
459C23—2442no—
460C23—2442no—
461C23C22×C22⋊D6111yescharacteristic
462Q8—22no—
463Q8—22no—
464Q8—41no—
465Q8—41no—
466Q8—44no—
467Q8—1222no—
468Q8—1222no—
469Q8—2441no—
470Q8—2441no—
471D4—1224no—
472D4—1224no—
473D4—4884no—
474D4—2442no—
475D4—2442no—
476D4—2448no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—1222no—
495C2×C4—1222no—
496C2×C4—4888no—
497C2×C4—2441no—
498C2×C4—2441no—
499C2×C4—2441no—
500C2×C4—2441no—
501C2×C4#1451974512yes—
502C2×C4#1451974512yes—
503C2×C4#1451979712yes—
504C2×C4#1451980812yes—
505C2×C4C2×D4×D6111yescharacteristic
506C2×C4C2×D4×D6111yescharacteristic
507C2×C4#41×D6111yescharacteristic
508C2×C4C2×#1175822611yescharacteristic
509C2×C4#1451982311yescharacteristic
510C2×C4#1451982511yescharacteristic
511C8—1222no—
512C8—2441no—
513C8—2442no—
514C22—4882no—
515C22—4882no—
516C22—4882no—
517C22—4882no—
518C22—2444no—
519C22—2444no—
520C22—2444no—
521C22C2×#1451974511yescharacteristic
522C22#1684042311yescharacteristic
523C22#1684043111yescharacteristic
524C22C2×#1451979711yescharacteristic
525C22C2×#1451980811yescharacteristic
526C22#1684056511yescharacteristic
527C22#1684057011yescharacteristic
528C4—22no—
529C4—42no—
530C4—42no—
531C4—42no—
532C4—81no—
533C4—82no—
534C4—2442no—
535C4—2442no—
536C4#1683992214yes—
537C4#1684027511yescharacteristic
538C4#1684034211yescharacteristic
539C4#1684040311yescharacteristic
540C4#1684041411yescharacteristic
541C2—4884no—
542C2#2144916611yescharacteristic
543C2C2×#1683992211yescharacteristic
544C2#2145512011yescharacteristic
545C2#2145851511yescharacteristic
546C2#2145863311yescharacteristic
547C2#2145882711yescharacteristic
548C2#2145886811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2582C2
socleC23×C61C22×D4
prime radicalC2×#14519774C22
hypercenter#2582D61
hypocenterC61#24190
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2582

Complements

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

NH ≅ G/Ntype
3. C2×C4×#11758202C2semidirect
4. #21447060C2semidirect
5. #21446912C2semidirect
6. #21446756C2semidirect
8. #21446156C2semidirect
9. #21446065C2semidirect
10. #21445821C2semidirect
11. C61×#2582C2semidirect
270. #2582D61semidirect
319. C61#24190semidirect

Maximal subgroups

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

subgroupindex
C2×#168404972
C2×#168399622
C2×C4×#117582022
#214470602
#214469122
#214467562
#214466242
#214461562
#214460652
#214458212
C61×#25822
#2419061