Finite Group Databasebeta

← #29734350

Subgroups — #29734350

Order 15616.

1114 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840296C211yesmaximal, characteristic
2C2×#16840171C211yesmaximal, characteristic, complemented
3C2×#16840122C211yesmaximal, characteristic, complemented
4#21454171C211yesmaximal, characteristic, complemented
5#21447023C212yesmaximal, complemented
6#21447009C212yesmaximal
7#21446788C211yesmaximal, characteristic, complemented
8#21446635C211yesmaximal, characteristic, complemented
9#21446090C212yesmaximal, complemented
10#21446069C212yesmaximal, complemented
11C61×#1749C211yesmaximal, characteristic, complemented
12C2×#14519777C2211yescharacteristic
13#16840296—24no—
14#16840296—24no—
15C2×#14519771—21no—
16C2×#14519769C2211yescharacteristic
17C2×#14519765C2211yescharacteristic
18C2×#14519760—21no—
19C2×#14519756—21no—
20C2×#14519756C2211yescharacteristic
21C2×#14519751C2211yescharacteristic
22C2×C4×Dic122C2211yescharacteristic
23C2×C4×D244C2211yescharacteristic
24C2×C4×C22⋊D61—21no—
25C2×#14519734—21no—
26#16840171—24no—
27#16840122—24no—
28C2×#14519711C2212yes—
29C2×#14519709C2212yes—
30C2×#14519702C2212yes—
31C2×#14519702C2212yes—
32C2×#14519699C2212yes—
33C2×#14519699C2212yes—
34#16839668—21no—
35#16839660—21no—
36#16839660C2212yes—
37#16839632C2211yescharacteristic
38#16839622—21no—
39#16839622C2212yes—
40#16839525C2212yes—
41#16839525C2212yes—
42C22×C42×C61C2211yescharacteristic
43C2×C22⋊C8×C61C2212yes—
44C2×#97×C61C2211yescharacteristic
45C2×#97×C61C2211yescharacteristic
46C61×#336C2212yes—
47C22×Dic122—21no—
48C22×Dic122C2311yescharacteristic
49C22×D244C2311yescharacteristic
50C22×C22⋊D61—21no—
51C22×C4×D61—21no—
52#14519777—22no—
53#14519777—22no—
54#14519771—28no—
55#14519769—22no—
56#14519769D414yes—
57#14519765—22no—
58#14519765D414yes—
59#14519760—28no—
60#14519756—22no—
61#14519756—22no—
62#14519756—44no—
63#14519751—22no—
64#14519751—22no—
65C4×Dic122—22no—
66C4×Dic122—22no—
67C4×D244—22no—
68C4×D244—22no—
69C4×C22⋊D61—28no—
70#14519734—28no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61C2312yes—
75C2×C4⋊Dic61C2312yes—
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—22no—
83C2×C22⋊Dic61—21no—
84C2×C22⋊Dic61—22no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×C4×Dic61—21no—
90#14519711—24no—
91#14519709—24no—
92#14519702—24no—
93#14519702—24no—
94#14519699—24no—
95#14519699—24no—
96#14519684—22no—
97#14519684—22no—
98C23×C4×C61D411yescharacteristic
99C23×C4×C61D411yescharacteristic
100C23×C4×C61C2311yescharacteristic
101C22×C8×C61C2312yes—
102C22×C8×C61C2312yes—
103C2×C42×C61—22no—
104C2×C42×C61—22no—
105C2×C42×C61D411yescharacteristic
106C2×C42×C61D411yescharacteristic
107C2×C42×C61C2311yescharacteristic
108C2×C42×C61C2311yescharacteristic
109C22⋊C8×C61—24no—
110#97×C61—22no—
111#97×C61—22no—
112C24×C61C2×D411yescharacteristic
113C23×D61—21no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—41no—
123C22×C4×C61—41no—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2411yescharacteristic
130C2×C22⋊D61—24no—
131C2×C22⋊D61—42no—
132C2×C22⋊D61—42no—
133C22×Dic61—21no—
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C2×D244—22no—
139C2×D244—22no—
140C2×D244—21no—
141C2×D244C2×D412yes—
142C2×C4×D61—24no—
143C2×C4×D61—42no—
144C2×C4×D61—42no—
145C2×Dic122—22no—
146C2×Dic122—22no—
147C2×Dic122—21no—
148C2×Dic122—42no—
149C2×Dic122—42no—
150C2×Dic122—42no—
151C2×Dic122C2×D412yes—
152C2×C8×C61—22no—
153C2×C8×C61—22no—
154C2×C8×C61—22no—
155C2×C8×C61—22no—
156C2×C8×C61—22no—
157C2×C8×C61#4114yes—
158C42×C61—22no—
159C42×C61—22no—
160C42×C61—21no—
161C42×C61—21no—
162C42×C61C2×D412yes—
163C42×C61C2×D412yes—
164C22⋊Dic61—24no—
165C22⋊Dic61—44no—
166#11758198—24no—
167#11758198—24no—
168#11758198—24no—
169#11758198—24no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—22no—
173C4⋊Dic61—22no—
174C4⋊Dic61—22no—
175C4⋊Dic61—22no—
176C4⋊Dic61—42no—
177C4⋊Dic61—42no—
178C4⋊Dic61—42no—
179C4⋊Dic61C2×D412yes—
180C4⋊Dic61C2×D412yes—
181#11758196—24no—
182#11758196—24no—
183#11758196—24no—
184#11758196—24no—
185#11758196—44no—
186C4×Dic61—24no—
187C23×C61—21no—
188C23×C61—21no—
189C23×C61—21no—
190C23×C61—21no—
191C23×C61—21no—
192C23×C61—21no—
193C23×C61#13712yes—
194C23×C61C22×D411yescharacteristic
195C22×D61—22no—
196C22×D61—41no—
197C22×D61—41no—
198C22×D61—41no—
199C22×D61—41no—
200C22×D61—41no—
201C22×D61—41no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—44no—
215C2×C4×C61—44no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61#12012yes—
219C2×C4×C61#12012yes—
220C2×C4×C61AGL(1,ℤ/8)12yes—
221C2×C4×C61C22×D411yescharacteristic
222C2×C4×C61C22×D411yescharacteristic
223C2×C4×C61C2×#4112yes—
224C2×C4×C612−1+412yes—
225C22⋊D61—48no—
226C2×Dic61—22no—
227C2×Dic61—22no—
228C2×Dic61—22no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61—41no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—42no—
236C2×Dic61—42no—
237C2×Dic61—42no—
238C2×Dic61—42no—
239C2×Dic61—42no—
240C2×Dic61—42no—
241D244—24no—
242D244—42no—
243C4×D61—48no—
244Dic122—24no—
245Dic122—42no—
246Dic122—84no—
247C8×C61—24no—
248C8×C61—42no—
249#25678—611nomaximal, Sylow
250C22×C61—22no—
251C22×C61—22no—
252C22×C61—22no—
253C22×C61—41no—
254C22×C61—41no—
255C22×C61—41no—
256C22×C61—41no—
257C22×C61C2×#12011yescharacteristic
258C22×C61#53211yescharacteristic
259C22×C61#53611yescharacteristic
260C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
261C22×C61C2×#13711yescharacteristic
262C22×C61#57511yescharacteristic
263C22×C61#57511yescharacteristic
264C2×D61—42no—
265C2×D61—42no—
266C2×D61—42no—
267C2×D61—81no—
268C2×D61—81no—
269C2×D61—81no—
270C2×D61—81no—
271C4×C61—22no—
272C4×C61—22no—
273C4×C61—22no—
274C4×C61—41no—
275C4×C61—41no—
276C4×C61#54712yes—
277C4×C61#54812yes—
278Dic61—42no—
279Dic61—42no—
280Dic61—42no—
281Dic61—82no—
282Dic61—82no—
283C2×#529—611no—
284C2×#501—611no—
285C2×#489—611no—
286#2857—611no—
287#2036—612no—
288#2034—612no—
289#1992—611no—
290#1959—611no—
291#1865—612no—
292#1862—612no—
293#1749D6111yescharacteristic, complemented
294C2×C61—42no—
295C2×C61#243211yescharacteristic
296C2×C61#300611yescharacteristic
297C2×C61#300711yescharacteristic
298C2×C61#317111yescharacteristic
299C2×C61#318111yescharacteristic
300C2×C61#323911yescharacteristic
301C2×C61#324011yescharacteristic
302D61—82no—
303C2×C4⋊Q8—611no—
304#529—1224no—
305#529—1224no—
306C2×#126—1221no—
307C2×#125—611no—
308C2×#124—611no—
309C2×#123—1221no—
310C2×C22⋊Q8—611no—
311C2×C22⋊Q8—1221no—
312C2×#121—611no—
313C2×C4×Q8—611no—
314C2×C4×D4—611no—
315C2×C4×D4—1221no—
316C2×#117—1221no—
317C22×C42C2×D6111yescharacteristic
318#501—1224no—
319#489—1224no—
320C2×#107—612no—
321C2×#106—612no—
322C2×#103—612no—
323C2×#103—612no—
324C2×#102—612no—
325C2×#102—612no—
326C2×C22⋊C8C2×D6112yes—
327C2×#97C2×D6111yescharacteristic
328C2×#97C2×D6111yescharacteristic
329#391—1221no—
330#389—612no—
331#389—1221no—
332#384—611no—
333#380—612no—
334#380—1221no—
335#340—612no—
336#340—612no—
337#336C2×D6112yes—
338C61#2567811yescharacteristic, complemented, Sylow
339C22×Q8—611no—
340C22×Q8—1221no—
341C22×D4—611no—
342C22×D4—1221no—
343C23×C4—1221no—
344C23×C4D24411yescharacteristic
345C23×C4D24411yescharacteristic
346C23×C4C22×D6111yescharacteristic
347C22×C8C22×D6112yes—
348C22×C8C22×D6112yes—
349C4⋊Q8—1222no—
350C4⋊Q8—1222no—
351#126—1228no—
352#125—614no—
353#125—1222no—
354#124—614no—
355#124—1222no—
356#123—1228no—
357C22⋊Q8—1222no—
358C22⋊Q8—1222no—
359C22⋊Q8—2444no—
360#121—1222no—
361#121—1222no—
362C4×Q8—1222no—
363C4×Q8—1222no—
364C4×D4—1222no—
365C4×D4—1222no—
366C4×D4—1228no—
367#117—1228no—
368C2×C4⋊C4—611no—
369C2×C4⋊C4—611no—
370C2×C4⋊C4—612no—
371C2×C4⋊C4—612no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4—1221no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4—1222no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4—1221no—
380C2×C22⋊C4—1221no—
381C2×C22⋊C4—1221no—
382C2×C22⋊C4—1221no—
383C2×C22⋊C4—1221no—
384C2×C22⋊C4—1221no—
385C2×C22⋊C4—1222no—
386C2×C42—22no—
387C2×C42—22no—
388C2×C42—1221no—
389C2×C42D24411yescharacteristic
390C2×C42D24411yescharacteristic
391C2×C42C22×D6111yescharacteristic
392C2×C42C22×D6111yescharacteristic
393#107—1224no—
394#106—1224no—
395#103—1224no—
396#103—1224no—
397#102—1224no—
398#102—1224no—
399C22⋊C8—24no—
400#97—22no—
401#97—22no—
402#95—1222no—
403#95—1222no—
404C24—1221no—
405C24C2×D24411yescharacteristic
406C2×Q8—612no—
407C2×Q8—1221no—
408C2×Q8—1222no—
409C2×Q8—1222no—
410C2×Q8—2442no—
411C2×Q8—2442no—
412C2×Q8—2442no—
413C2×D4—612no—
414C2×D4—1224no—
415C2×D4—1221no—
416C2×D4—1222no—
417C2×D4—1222no—
418C2×D4—2442no—
419C2×D4—2442no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—1224no—
431C22×C4—1221no—
432C22×C4—1221no—
433C22×C4—1221no—
434C22×C4—1222no—
435C22×C4—1222no—
436C22×C4—2442no—
437C22×C4—2442no—
438C22×C4C2×D24411yescharacteristic
439C22×C4C2×D24411yescharacteristic
440C22×C4C2×D24411yescharacteristic
441C22×C4C2×D24411yescharacteristic
442C22×C4C2×D24411yescharacteristic
443C22×C4C23×D6111yescharacteristic
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8#1175822214yes—
450C4⋊C4—612no—
451C4⋊C4—612no—
452C4⋊C4—1224no—
453C4⋊C4—1224no—
454C4⋊C4—1221no—
455C4⋊C4—1221no—
456C4⋊C4—1224no—
457C4⋊C4—1224no—
458C4⋊C4—1222no—
459C4⋊C4—1222no—
460C4⋊C4—1222no—
461C4⋊C4—1222no—
462C4⋊C4—2442no—
463C4⋊C4—2442no—
464C4⋊C4—2442no—
465C4⋊C4—2444no—
466C22⋊C4—1224no—
467C22⋊C4—1224no—
468C22⋊C4—1224no—
469C22⋊C4—1224no—
470C22⋊C4—1224no—
471C22⋊C4—2444no—
472C42—21no—
473C42—21no—
474C42—22no—
475C42—22no—
476C42—1224no—
477C42D4×D6112yes—
478C42D4×D6112yes—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—1222no—
486C23—2441no—
487C23—2441no—
488C23—2441no—
489C23—2441no—
490C23—2441no—
491C23—2441no—
492C23#1451981012yes—
493C23C22×D24411yescharacteristic
494Q8—1224no—
495Q8—4884no—
496Q8—2442no—
497D4—1224no—
498D4—2448no—
499D4—2442no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—1222no—
517C2×C4—1222no—
518C2×C4—1222no—
519C2×C4—2448no—
520C2×C4—2442no—
521C2×C4—2442no—
522C2×C4—2442no—
523C2×C4—2442no—
524C2×C4—2442no—
525C2×C4—2442no—
526C2×C4—2441no—
527C2×C4—2441no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4—2441no—
531C2×C4—2441no—
532C2×C4#1451974712yes—
533C2×C4#1451974712yes—
534C2×C4#1451980512yes—
535C2×C4C2×D4×D6111yescharacteristic
536C2×C4C2×D4×D6111yescharacteristic
537C2×C4C2×#1175822212yes—
538C2×C4#1451982412yes—
539C8—24no—
540C8—42no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—4881no—
549C22—4881no—
550C22—4881no—
551C22—4881no—
552C22—2442no—
553C22—2442no—
554C22—2442no—
555C22C2×#1451974711yescharacteristic
556C22#1684031211yescharacteristic
557C22#1684033811yescharacteristic
558C22C2×#1451980511yescharacteristic
559C22C2×#1451981011yescharacteristic
560C22#1684056211yescharacteristic
561C22#1684056211yescharacteristic
562C4—22no—
563C4—22no—
564C4—22no—
565C4—41no—
566C4—41no—
567C4—4882no—
568C4—4882no—
569C4—2442no—
570C4—2442no—
571C4—2442no—
572C4#1684041712yes—
573C4#1684042112yes—
574C2—42no—
575C2—4882no—
576C2#2144944711yescharacteristic
577C2#2145507311yescharacteristic
578C2#2145508111yescharacteristic
579C2#2145641411yescharacteristic
580C2#2145652811yescharacteristic
581C2#2145705111yescharacteristic
582C2#2145706111yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1749C2
socleC23×C61C22×D4
prime radicalC2×#14519751C22
hypercenter#1749D61
hypocenterC61#25678
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
2. C2×#16840171C2semidirect
3. C2×#16840122C2semidirect
4. #21454171C2semidirect
5. #21447023C2semidirect
7. #21446788C2semidirect
8. #21446635C2semidirect
9. #21446090C2semidirect
10. #21446069C2semidirect
11. C61×#1749C2semidirect
293. #1749D61semidirect
338. C61#25678semidirect

Maximal subgroups

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

subgroupindex
C2×#168402962
C2×#168401712
C2×#168401222
#214541712
#214470232
#214470092
#214467882
#214466352
#214460902
#214460692
C61×#17492
#2567861