Finite Group Databasebeta

← #29734161

Subgroups — #29734161

Order 15616.

1092 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840497C211yesmaximal, characteristic
2C2×#16840124C211yesmaximal, characteristic, complemented
3C2×#16840039C211yesmaximal, characteristic, complemented
4#21447060C212yesmaximal, complemented
5#21447053C212yesmaximal
6#21446760C211yesmaximal, characteristic, complemented
7#21446751C211yesmaximal, characteristic, complemented
8#21446072C212yesmaximal, complemented
9#21446065C212yesmaximal, complemented
10#21445647C211yesmaximal, characteristic, complemented
11C61×#2567C211yesmaximal, characteristic, complemented
12#16840497—24no—
13#16840497—24no—
14C2×#14519774C2211yescharacteristic
15C2×#14519766—21no—
16C2×#14519765C2211yescharacteristic
17C2×#14519765C2211yescharacteristic
18C2×#14519762—21no—
19C2×#14519762—21no—
20C2×C4×Dic122C2211yescharacteristic
21C2×C4×D244C2211yescharacteristic
22C2×C4×D244C2211yescharacteristic
23C2×#14519739—21no—
24#16840124—24no—
25#16840039—24no—
26C2×#14519710C2212yes—
27C2×C4⋊#92418C2211yescharacteristic
28C2×#14519700C2212yes—
29C2×#14519700C2212yes—
30C2×#14519697C2212yes—
31C2×#14519697C2212yes—
32C2×#14519697C2212yes—
33C2×#14519697C2212yes—
34C2×#14519688C2211yescharacteristic
35#16839524C2212yes—
36#16839517C2212yes—
37#16839517C2212yes—
38C2×C4⋊Q8×C61C2211yescharacteristic
39C2×C4×Q8×C61C2211yescharacteristic
40C61×#395—21no—
41C61×#395—21no—
42C61×#391—21no—
43C61×#389C2212yes—
44C61×#389C2212yes—
45C61×#384—21no—
46C4×C4⋊C4×C61C2211yescharacteristic
47C22×Dic122C2311yescharacteristic
48C22×D244—21no—
49C22×D244—21no—
50C22×D244C2311yescharacteristic
51C22×D244C2311yescharacteristic
52C22×C4×D61—21no—
53C22×C4×D61—21no—
54C22×#92418C2312yes—
55C22×#92418C2312yes—
56#14519774—22no—
57#14519774—22no—
58#14519766—28no—
59#14519765—22no—
60#14519765—22no—
61#14519765—22no—
62#14519765D414yes—
63#14519762—28no—
64#14519762—28no—
65C4×Dic122—22no—
66C4×Dic122—22no—
67C4×D244—22no—
68C4×D244—22no—
69C4×D244—22no—
70C4×D244D414yes—
71#14519739—28no—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×#11758196—21no—
74C2×#11758198—21no—
75C2×#11758198—21no—
76C2×#11758198—21no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×C4×Dic61—21no—
81#14519710—24no—
82C4⋊#92418—22no—
83#14519700—24no—
84#14519700—24no—
85#14519697—24no—
86#14519697—24no—
87#14519697—24no—
88#14519697—24no—
89#14519688—22no—
90C22×Q8×C61—21no—
91C22×Q8×C61D411yescharacteristic
92C4⋊Q8×C61—22no—
93C4⋊Q8×C61—22no—
94C4×Q8×C61—22no—
95C4×Q8×C61—22no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61—22no—
98C2×C4⋊C4×C61—22no—
99C2×C4⋊C4×C61D411yescharacteristic
100C2×C4⋊C4×C61D411yescharacteristic
101C2×C4⋊C4×C61D411yescharacteristic
102C2×C4⋊C4×C61C2312yes—
103C2×C4⋊C4×C61C2312yes—
104C2×C42×C61—21no—
105C2×C42×C61C2311yescharacteristic
106C2×C42×C61C2311yescharacteristic
107C2×C42×C61C2311yescharacteristic
108#95×C61—21no—
109#95×C61—22no—
110#95×C61—22no—
111C23×D61—21no—
112C23×D61—21no—
113C2×Q8×C61—22no—
114C2×Q8×C61—22no—
115C2×Q8×C61—41no—
116C2×Q8×C61—42no—
117C2×Q8×C61—82no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2411yescharacteristic
127C22×Dic61—21no—
128C22×Dic61—21no—
129C2×D244—22no—
130C2×D244—22no—
131C2×D244—22no—
132C2×D244—22no—
133C2×D244—24no—
134C2×D244—21no—
135C2×D244—21no—
136C2×D244—24no—
137C2×D244—42no—
138C2×D244—42no—
139C2×D244—82no—
140C2×D244C2×D412yes—
141C2×D244C2×D412yes—
142C2×C4×D61—24no—
143C2×C4×D61—24no—
144C2×C4×D61—82no—
145C2×C4×D61—82no—
146C2×Dic122—22no—
147C2×Dic122—22no—
148C2×Dic122—21no—
149C2×Dic122C2×D412yes—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—42no—
157C4⋊C4×C61—82no—
158C4⋊C4×C61—82no—
159C4⋊C4×C61#4114yes—
160C42×C61—21no—
161C42×C61—21no—
162C42×C61—21no—
163C42×C61—21no—
164C42×C61—42no—
165C42×C61C2×D412yes—
166C42×C61C2×D412yes—
167#11758198—24no—
168#11758198—24no—
169#11758198—24no—
170#11758198—24no—
171#11758198—24no—
172#11758198—24no—
173C4⋊Dic61—21no—
174C4⋊Dic61C2×D412yes—
175#11758196—24no—
176C4×Dic61—24no—
177C2×#92418—22no—
178C2×#92418—22no—
179C2×#92418—22no—
180C2×#92418—22no—
181C2×#92418—22no—
182C2×#92418—22no—
183C23×C61C22×D411yescharacteristic
184C22×D61—22no—
185C22×D61—22no—
186C22×D61—42no—
187C22×D61—42no—
188C22×D61—41no—
189C22×D61—41no—
190C22×D61—41no—
191C22×D61—41no—
192C22×D61—42no—
193C22×D61—42no—
194Q8×C61—44no—
195Q8×C61—84no—
196C2×C4×C61—21no—
197C2×C4×C61—21no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—22no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—24no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—42no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61#12012yes—
215C2×C4×C61#12012yes—
216C2×C4×C61#13712yes—
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C22×D411yescharacteristic
219C2×C4×C61C2×#4112yes—
220C2×C4×C612−1+412yes—
221C2×Dic61—22no—
222C2×Dic61—22no—
223C2×Dic61—42no—
224C2×Dic61—42no—
225D244—22no—
226D244—22no—
227D244—24no—
228D244—48no—
229D244—41no—
230D244—41no—
231D244—42no—
232D244—84no—
233C4×D61—84no—
234C4×D61—84no—
235Dic122—24no—
236Dic122—41no—
237Dic122—41no—
238#92418—42no—
239#92418—42no—
240#25660—611nomaximal, Sylow
241C22×C61C2×#12011yescharacteristic
242C22×C61#54812yes—
243C22×C61C2×#13711yescharacteristic
244C22×C61#57511yescharacteristic
245C22×C61#57612yes—
246C2×D61—42no—
247C2×D61—42no—
248C2×D61—42no—
249C2×D61—42no—
250C2×D61—42no—
251C2×D61—42no—
252C2×D61—82no—
253C2×D61—82no—
254C2×D61—82no—
255C2×D61—82no—
256C4×C61—22no—
257C4×C61—22no—
258C4×C61—22no—
259C4×C61—44no—
260C4×C61—41no—
261C4×C61—41no—
262C4×C61—41no—
263C4×C61—82no—
264C4×C61#53211yescharacteristic
265C4×C61#53611yescharacteristic
266C4×C61#54712yes—
267Dic61—42no—
268Dic61—42no—
269C2×#559—611no—
270C2×#489—611no—
271C2×#474—611no—
272#2567D6111yescharacteristic, complemented
273#2042—612no—
274#2041—612no—
275#1985—611no—
276#1983—611no—
277#1863—612no—
278#1862—612no—
279#1762—611no—
280C2×C61#243211yescharacteristic
281C2×C61#300711yescharacteristic
282C2×C61#300811yescharacteristic
283C2×C61#337012yes—
284C2×C61#337512yes—
285D61—82no—
286D61—82no—
287#559—1224no—
288#559—1224no—
289C2×C4⋊Q8C2×D6111yescharacteristic
290C2×#127—611no—
291C2×#124—611no—
292C2×#124—611no—
293C2×#124—1221no—
294C2×#123—1221no—
295C2×#123—1221no—
296C2×C4×Q8—611no—
297C2×C4×Q8C2×D6111yescharacteristic
298C2×C4×D4—611no—
299C2×C4×D4—611no—
300C2×C4×D4—1221no—
301#489—1224no—
302#474—1224no—
303C2×#107—612no—
304C2×C4⋊C8—611no—
305C2×#103—612no—
306C2×#103—612no—
307C2×#102—612no—
308C2×#102—612no—
309C2×#102—612no—
310C2×#102—612no—
311C2×#97—611no—
312#395—21no—
313#395—21no—
314#391—21no—
315#389C2×D6112yes—
316#389C2×D6112yes—
317#384—21no—
318C4×C4⋊C4C2×D6111yescharacteristic
319#340—612no—
320#336—612no—
321#336—612no—
322C61#2566011yescharacteristic, complemented, Sylow
323C22×Q8—21no—
324C22×Q8—611no—
325C22×Q8C22⋊D6111yescharacteristic
326C22×D4—611no—
327C22×D4—611no—
328C22×D4—1221no—
329C22×D4—1221no—
330C23×C4—1221no—
331C23×C4—1221no—
332C22×C8—612no—
333C22×C8—612no—
334C4⋊Q8—22no—
335C4⋊Q8—22no—
336#127—1222no—
337#127—1222no—
338#124—614no—
339#124—1222no—
340#124—1222no—
341#124—1228no—
342#124—1222no—
343#123—1228no—
344#123—1228no—
345C4×Q8—22no—
346C4×Q8—22no—
347C4×Q8—1222no—
348C4×Q8—1222no—
349C4×D4—614no—
350C4×D4—1222no—
351C4×D4—1222no—
352C4×D4—1228no—
353C4×D4—1222no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—611no—
358C2×C4⋊C4—1221no—
359C2×C4⋊C4C22⋊D6111yescharacteristic
360C2×C4⋊C4C22⋊D6111yescharacteristic
361C2×C4⋊C4C22⋊D6111yescharacteristic
362C2×C4⋊C4C22×D6112yes—
363C2×C4⋊C4C22×D6112yes—
364C2×C22⋊C4—1221no—
365C2×C22⋊C4—1221no—
366C2×C22⋊C4—1221no—
367C2×C22⋊C4—1221no—
368C2×C22⋊C4—1221no—
369C2×C22⋊C4—1221no—
370C2×C42—21no—
371C2×C42—1221no—
372C2×C42C22×D6111yescharacteristic
373C2×C42C22×D6111yescharacteristic
374C2×C42C22×D6111yescharacteristic
375#107—1224no—
376C4⋊C8—1222no—
377#103—1224no—
378#103—1224no—
379#102—1224no—
380#102—1224no—
381#102—1224no—
382#102—1224no—
383#97—1222no—
384#95—21no—
385#95—22no—
386#95—22no—
387C24—1221no—
388C24—1221no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—41no—
392C2×Q8—42no—
393C2×Q8—82no—
394C2×Q8—612no—
395C2×Q8—1221no—
396C2×Q8—1222no—
397C2×Q8—1222no—
398C2×D4—612no—
399C2×D4—612no—
400C2×D4—1224no—
401C2×D4—1224no—
402C2×D4—1221no—
403C2×D4—1221no—
404C2×D4—1222no—
405C2×D4—1222no—
406C2×D4—1222no—
407C2×D4—1222no—
408C2×D4—4882no—
409C2×D4—2442no—
410C2×D4—2442no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—1224no—
414C22×C4—1221no—
415C22×C4—1221no—
416C22×C4—1224no—
417C22×C4—4882no—
418C22×C4—4882no—
419C22×C4C2×C22⋊D6111yescharacteristic
420C22×C4C2×C22⋊D6111yescharacteristic
421C22×C4C2×C22⋊D6111yescharacteristic
422C22×C4C2×C22⋊D6111yescharacteristic
423C22×C4C2×C22⋊D6111yescharacteristic
424C22×C4C2×C22⋊D6111yescharacteristic
425C22×C4C23×D6111yescharacteristic
426C2×C8—1222no—
427C2×C8—1222no—
428C2×C8—1222no—
429C2×C8—1222no—
430C2×C8—1222no—
431C2×C8—1222no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—42no—
439C4⋊C4—82no—
440C4⋊C4—82no—
441C4⋊C4—612no—
442C4⋊C4—1224no—
443C4⋊C4—1221no—
444C4⋊C4#1175822614yes—
445C22⋊C4—1224no—
446C22⋊C4—1224no—
447C22⋊C4—1224no—
448C22⋊C4—1224no—
449C22⋊C4—1224no—
450C22⋊C4—1224no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—21no—
455C42—42no—
456C42—1224no—
457C42D4×D6112yes—
458C42D4×D6112yes—
459C23—1222no—
460C23—1222no—
461C23—2441no—
462C23—2441no—
463C23—2441no—
464C23—2441no—
465C23—2442no—
466C23—2442no—
467C23—2442no—
468C23—2442no—
469C23C22×C22⋊D6111yescharacteristic
470Q8—44no—
471Q8—84no—
472Q8—1224no—
473Q8—2441no—
474Q8—2441no—
475D4—1222no—
476D4—1222no—
477D4—1224no—
478D4—4884no—
479D4—2448no—
480D4—2441no—
481D4—2441no—
482D4—2442no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—1222no—
502C2×C4—1222no—
503C2×C4—4884no—
504C2×C4—4884no—
505C2×C4—2442no—
506C2×C4—2442no—
507C2×C4#1451974512yes—
508C2×C4#1451974512yes—
509C2×C4#1451980812yes—
510C2×C4C2×D4×D6111yescharacteristic
511C2×C4C2×D4×D6111yescharacteristic
512C2×C4C2×#1175822612yes—
513C2×C4#1451982512yes—
514C8—2442no—
515C8—2442no—
516C22—4882no—
517C22—4882no—
518C22—4882no—
519C22—4882no—
520C22—2442no—
521C22—2442no—
522C22—2442no—
523C22—2442no—
524C22—2442no—
525C22—2442no—
526C22C2×#1451974511yescharacteristic
527C22#1684042312yes—
528C22C2×#1451980811yescharacteristic
529C22#1684056011yescharacteristic
530C22#1684056512yes—
531C4—22no—
532C4—22no—
533C4—22no—
534C4—44no—
535C4—41no—
536C4—41no—
537C4—41no—
538C4—82no—
539C4—2442no—
540C4—2442no—
541C4#1684031311yescharacteristic
542C4#1684033711yescharacteristic
543C4#1684041512yes—
544C2—4882no—
545C2—4882no—
546C2#2144944411yescharacteristic
547C2#2145507711yescharacteristic
548C2#2145508511yescharacteristic
549C2#2145857212yes—
550C2#2145863312yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2567C2
socleC23×C61C22×D4
prime radicalC2×#14519774C22
hypercenter#2567D61
hypocenterC61#25660
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2567

Complements

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

NH ≅ G/Ntype
2. C2×#16840124C2semidirect
3. C2×#16840039C2semidirect
4. #21447060C2semidirect
6. #21446760C2semidirect
7. #21446751C2semidirect
8. #21446072C2semidirect
9. #21446065C2semidirect
10. #21445647C2semidirect
11. C61×#2567C2semidirect
272. #2567D61semidirect
322. C61#25660semidirect

Maximal subgroups

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

subgroupindex
C2×#168404972
C2×#168401242
C2×#168400392
#214470602
#214470532
#214467602
#214467512
#214460722
#214460652
#214456472
C61×#25672
#2566061