Finite Group Databasebeta

← #29714817

Subgroups — #29714817

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×#16839963C211yesmaximal, characteristic
3C2×C4×#11758202C211yesmaximal, characteristic, complemented
4#21451192C211yesmaximal, characteristic, complemented
5#21447057C212yesmaximal, complemented
6#21446914C212yesmaximal, complemented
7#21446754C211yesmaximal, characteristic, complemented
8#21446624C211yesmaximal, characteristic
9#21446154C212yesmaximal, complemented
10#21446067C212yesmaximal, complemented
11C61×#1803C211yesmaximal, characteristic, complemented
12C22×#11758202C2211yescharacteristic
13C22×#11758190C2212yes—
14#16840497—24no—
15#16840497—24no—
16C2×#14519774C2211yescharacteristic
17C2×#14519766C2212yes—
18C2×#14519765—21no—
19C2×#14519762—22no—
20C2×C4×Dic122C2211yescharacteristic
21C2×C4×Dic122C2211yescharacteristic
22C2×C4×D244—21no—
23C2×#14519739C2212yes—
24#16839963—24no—
25C4×#11758202—24no—
26C2×#14519710C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519700C2211yescharacteristic
29C2×#14519699C2212yes—
30C2×#14519697C2212yes—
31C2×#14519697C2212yes—
32C2×C4×#92418C2211yescharacteristic
33#16839692—22no—
34#16839667—21no—
35#16839667C2212yes—
36#16839661—21no—
37#16839661C2212yes—
38#16839516C2212yes—
39#16839516C2212yes—
40C2×C4×Q8×C61C2211yescharacteristic
41C2×C4⋊C8×C61C2211yescharacteristic
42C2×#103×C61C2212yes—
43C61×#384C2211yescharacteristic
44C61×#340C2212yes—
45C22×Dic122D411yescharacteristic
46C22×Dic122C2311yescharacteristic
47C22×D244—21no—
48C22×D244—21no—
49C22×D244C2312yes—
50C22×C4×D61—22no—
51C2×#11758202—22no—
52C2×#11758202—22no—
53C2×#11758202—22no—
54C2×#11758190—24no—
55C2×#11758190—24no—
56C2×#11758190—42no—
57C22×#92418C2311yescharacteristic
58C22×#92418C2311yescharacteristic
59#14519774—22no—
60#14519774D414yes—
61#14519766—24no—
62#14519766—24no—
63#14519765—28no—
64#14519762—216no—
65C4×Dic122—22no—
66C4×Dic122—22no—
67C4×Dic122—22no—
68C4×Dic122—22no—
69C4×D244—28no—
70#14519739—24no—
71#14519739—24no—
72C2×C4⋊Dic61—22no—
73C2×C4⋊Dic61D411yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758196—21no—
76C2×#11758196D412yes—
77C2×#11758198—22no—
78C2×#11758198—22no—
79C2×#11758198—22no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61C2312yes—
82#14519710—22no—
83#14519701—22no—
84#14519700—22no—
85#14519699—24no—
86#14519697—24no—
87#14519697—42no—
88C4×#92418—22no—
89#14519684—22no—
90#14519684—22no—
91#14519684—22no—
92C22×Q8×C61C2311yescharacteristic
93C22×C8×C61C2312yes—
94C4×Q8×C61—22no—
95C4×Q8×C61D414yes—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61C2312yes—
100C2×C4⋊C4×C61C2311yescharacteristic
101C2×C42×C61—21no—
102C2×C42×C61C2311yescharacteristic
103C4⋊C8×C61—22no—
104#103×C61—42no—
105C23×D61—22no—
106C2×Q8×C61—21no—
107C2×Q8×C61—21no—
108C2×Q8×C61—21no—
109C2×Q8×C61—21no—
110C2×Q8×C61—21no—
111C2×Q8×C61C2×D412yes—
112C22×C4×C61—21no—
113C22×C4×C61—21no—
114C22×C4×C61—22no—
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2411yescharacteristic
118C22×Dic61—21no—
119C22×Dic61—21no—
120C22×Dic61C2×D412yes—
121C22×Dic61C2×D412yes—
122C2×D244—22no—
123C2×D244—22no—
124C2×D244—22no—
125C2×D244—22no—
126C2×D244—24no—
127C2×D244—22no—
128C2×D244—24no—
129C2×D244—42no—
130C2×D244—42no—
131C2×D244—81no—
132C2×D244—81no—
133C2×D244C2×D414yes—
134C2×C4×D61—28no—
135C2×C4×D61—82no—
136C2×C4×D61—82no—
137C2×Dic122—21no—
138C2×Dic122—21no—
139C2×Dic122—21no—
140C2×Dic122—21no—
141C2×Dic122—22no—
142C2×Dic122—22no—
143C2×Dic122—22no—
144C2×Dic122—21no—
145C2×Dic122#4112yes—
146C2×C8×C61—22no—
147C2×C8×C61—22no—
148C2×C8×C61—22no—
149C4⋊C4×C61—24no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—42no—
154C4⋊C4×C61—81no—
155C4⋊C4×C61C2×D412yes—
156C42×C61—24no—
157C42×C61C2×D412yes—
158C42×C61C2×D412yes—
159#11758202—28no—
160#11758198—28no—
161#11758198—28no—
162#11758198—28no—
163C4⋊Dic61—22no—
164C4⋊Dic61—21no—
165C4⋊Dic61—21no—
166C4⋊Dic61—82no—
167#11758196—24no—
168#11758196—42no—
169C4×Dic61—22no—
170C4×Dic61—22no—
171C4×Dic61—42no—
172C2×#92418—21no—
173C2×#92418—21no—
174C2×#92418—21no—
175C2×#92418—21no—
176C2×#92418—21no—
177C2×#92418#4112yes—
178#11758190—48no—
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—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—41no—
196C2×C4×C61—41no—
197C2×C4×C61—41no—
198C2×C4×C61—41no—
199C2×C4×C61—42no—
200C2×C4×C61—42no—
201C2×C4×C61—42no—
202C2×C4×C61#12012yes—
203C2×C4×C61#12012yes—
204C2×C4×C61#13512yes—
205C2×C4×C61#13712yes—
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—22no—
214C2×Dic61—22no—
215C2×Dic61—22no—
216C2×Dic61—22no—
217C2×Dic61—22no—
218C2×Dic61—22no—
219C2×Dic61—41no—
220C2×Dic61—41no—
221C2×Dic61—41no—
222C2×Dic61—41no—
223C2×Dic61—41no—
224D244—24no—
225D244—24no—
226D244—42no—
227D244—42no—
228D244—48no—
229D244—84no—
230C4×D61—88no—
231Dic122—22no—
232Dic122—22no—
233Dic122—41no—
234Dic122—41no—
235Dic122—44no—
236C8×C61—42no—
237#92418—22no—
238#92418—41no—
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—82no—
258C4×C61#45214yes—
259C4×C61#52511yescharacteristic
260C4×C61#53711yescharacteristic
261C4×C61#54611yescharacteristic
262C4×C61#54711yescharacteristic
263Dic61—42no—
264Dic61—42no—
265Dic61—42no—
266Dic61—81no—
267C2×#559—611no—
268C2×#460—611no—
269C2×C4×Q16—611no—
270#2582—611no—
271#2042—612no—
272#2015—612no—
273#1984—611no—
274#1957—611no—
275#1877—612no—
276#1862—612no—
277#1803D6111yescharacteristic, complemented
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×Q8—611no—
296C2×C4×Q8C2×D6111yescharacteristic
297C2×C4×D4—612no—
298C2×C4×D4—1221no—
299#460—1224no—
300C4×Q16—1224no—
301C2×#107—611no—
302C2×C4⋊C8C2×D6111yescharacteristic
303C2×#103—611no—
304C2×#103—611no—
305C2×#103C2×D6112yes—
306C2×#102—612no—
307C2×#102—612no—
308C2×#102—612no—
309C2×C4×C8—611no—
310#398—1222no—
311#391—612no—
312#391—1221no—
313#389—612no—
314#389—1221no—
315#384C2×D6111yescharacteristic
316#340C2×D6112yes—
317#336—612no—
318#336—612no—
319C61#2419011yescharacteristic, complemented, Sylow
320C22×Q8—611no—
321C22×Q8—611no—
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—611no—
335C22×C8C22×D6112yes—
336#127—614no—
337#127—1222no—
338#124—1228no—
339#124—1224no—
340#124—1224no—
341#123—12216no—
342C4×Q8—22no—
343C4×Q8—1222no—
344C4×Q8—1222no—
345C4×Q8—1222no—
346C4×Q8—1222no—
347C4×Q8D24414yes—
348C4×D4—1228no—
349C4×D4—1224no—
350C4×D4—1224no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—611no—
355C2×C4⋊C4—612no—
356C2×C4⋊C4—611no—
357C2×C4⋊C4—1222no—
358C2×C4⋊C4—1221no—
359C2×C4⋊C4C22×D6112yes—
360C2×C4⋊C4C22×D6111yescharacteristic
361C2×C22⋊C4—1222no—
362C2×C22⋊C4—1222no—
363C2×C22⋊C4—1222no—
364C2×C42—21no—
365C2×C42—612no—
366C2×C42—1221no—
367C2×C42C22×D6111yescharacteristic
368#107—1222no—
369C4⋊C8—22no—
370#103—42no—
371#103—1222no—
372#103—1222no—
373#102—1224no—
374#102—1224no—
375#102—2442no—
376C4×C8—1222no—
377#95—1222no—
378#95—1222no—
379#95—1222no—
380C24—1222no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—612no—
387C2×Q8—1222no—
388C2×Q8—1221no—
389C2×Q8—1222no—
390C2×Q8—1222no—
391C2×Q8—1221no—
392C2×Q8—1221no—
393C2×Q8—1221no—
394C2×Q8—1221no—
395C2×Q8C2×D24412yes—
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—21no—
410C22×C4—22no—
411C22×C4—612no—
412C22×C4—612no—
413C22×C4—1228no—
414C22×C4—1221no—
415C22×C4—1221no—
416C22×C4—4882no—
417C22×C4—4882no—
418C22×C4D4×D6111yescharacteristic
419C22×C4D4×D6111yescharacteristic
420C22×C4C23×D6111yescharacteristic
421Q16—1228no—
422SD16—2448no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—612no—
427C2×C8—1221no—
428C2×C8—1221no—
429C2×C8—1221no—
430C2×C8—1221no—
431C2×C8—1221no—
432C4⋊C4—21no—
433C4⋊C4—24no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—42no—
437C4⋊C4—81no—
438C4⋊C4—1224no—
439C4⋊C4—1222no—
440C4⋊C4—1221no—
441C4⋊C4—1221no—
442C4⋊C4—4882no—
443C4⋊C4—2442no—
444C4⋊C4C2×D24412yes—
445C22⋊C4—1228no—
446C22⋊C4—1228no—
447C22⋊C4—1228no—
448C42—24no—
449C42—1222no—
450C42—1222no—
451C42—2442no—
452C42C2×D24412yes—
453C42D4×D6112yes—
454C23—1224no—
455C23—2442no—
456C23—2442no—
457C23—2442no—
458C23—2442no—
459C23—2442no—
460C23—2442no—
461C23C2×D4×D6111yescharacteristic
462Q8—22no—
463Q8—22no—
464Q8—41no—
465Q8—41no—
466Q8—1222no—
467Q8—1222no—
468Q8—2441no—
469Q8—2441no—
470Q8—2444no—
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—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—1222no—
489C2×C4—1222no—
490C2×C4—1222no—
491C2×C4—1222no—
492C2×C4—1222no—
493C2×C4—1222no—
494C2×C4—1222no—
495C2×C4—4888no—
496C2×C4—2441no—
497C2×C4—2441no—
498C2×C4—2441no—
499C2×C4—2441no—
500C2×C4—2441no—
501C2×C4#1451974712yes—
502C2×C4#1451974712yes—
503C2×C4#1451979812yes—
504C2×C4#1451980912yes—
505C2×C4C2×D4×D6111yescharacteristic
506C2×C4C22×D24411yescharacteristic
507C2×C4#41×D6111yescharacteristic
508C2×C4#41×D6111yescharacteristic
509C2×C4#1451982211yescharacteristic
510C2×C4#1451982511yescharacteristic
511C8—42no—
512C8—1222no—
513C8—2441no—
514C22—4882no—
515C22—4882no—
516C22—4882no—
517C22—4882no—
518C22—2444no—
519C22—2444no—
520C22—2444no—
521C22C2×#1451974711yescharacteristic
522C22#1684042211yescharacteristic
523C22#1684043011yescharacteristic
524C22C2×#1451979811yescharacteristic
525C22C2×#1451980911yescharacteristic
526C22#1684056611yescharacteristic
527C22#1684057111yescharacteristic
528C4—22no—
529C4—42no—
530C4—42no—
531C4—82no—
532C4—4881no—
533C4—2442no—
534C4—2442no—
535C4—2442no—
536C4#1683992714yes—
537C4#1684027411yescharacteristic
538C4#1684034311yescharacteristic
539C4#1684040911yescharacteristic
540C4#1684041411yescharacteristic
541C2—4884no—
542C2#2144916511yescharacteristic
543C2C2×#1683992711yescharacteristic
544C2#2145512511yescharacteristic
545C2#2145851411yescharacteristic
546C2#2145863211yescharacteristic
547C2#2145883011yescharacteristic
548C2#2145887611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1803C2
socleC23×C61C22×D4
prime radicalC2×#14519774C22
hypercenter#1803D61
hypocenterC61#24190
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
3. C2×C4×#11758202C2semidirect
4. #21451192C2semidirect
5. #21447057C2semidirect
6. #21446914C2semidirect
7. #21446754C2semidirect
9. #21446154C2semidirect
10. #21446067C2semidirect
11. C61×#1803C2semidirect
277. #1803D61semidirect
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×#168399632
C2×C4×#117582022
#214511922
#214470572
#214469142
#214467542
#214466242
#214461542
#214460672
C61×#18032
#2419061