Finite Group Databasebeta

← #29685259

Subgroups — #29685259

Order 15616.

1466 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×#16840307C211yesmaximal, characteristic
2C2×M16×Dic61C211yesmaximal, characteristic, complemented
3#21448221C211yesmaximal, characteristic, complemented
4#21446778C212yesmaximal, complemented
5#21446623C212yesmaximal, complemented
6#21446258C211yesmaximal, characteristic, complemented
7#21446257C211yesmaximal, characteristic, complemented
8#21446255C211yesmaximal, characteristic, complemented
9#21445646C212yesmaximal, complemented
10#21445643C212yesmaximal, complemented
11C61×#1895C211yesmaximal, characteristic, complemented
12C22×#11758224—21no—
13C22×#11758194C2211yescharacteristic
14#16840307—24no—
15#16840307C418yescomplemented
16C2×#14519776C2211yescharacteristic
17C2×#14519773C2211yescharacteristic
18C2×#14519763—21no—
19C2×#14519751C2211yescharacteristic
20C2×#14519749—21no—
21C2×D4×Dic61C2211yescharacteristic
22C2×#14519739C2211yescharacteristic
23C2×#14519737—21no—
24C2×#14519733—21no—
25C22×C4×Dic61C2211yescharacteristic
26C2×#14519699C2212yes—
27C2×#14519698C2212yes—
28C2×#14519697C2212yes—
29M16×Dic61—24no—
30C2×#14519687C2212yes—
31C2×C8×Dic61C2212yes—
32#16839631C2212yes—
33#16839626—21no—
34#16839622C2211yescharacteristic
35#16839620—21no—
36C4⋊C4×Dic61C2212yes—
37#16839613—21no—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C22×M16×C61C2211yescharacteristic
42C2×#121×C61C2211yescharacteristic
43C2×#102×C61C2212yes—
44C61×#380C2211yescharacteristic
45C61×#340C2212yes—
46C2×#11758224—28no—
47C2×#11758224—48no—
48C22×Dic122—21no—
49C22×D244C2311yescharacteristic
50C22×C22⋊D61—21no—
51C22×C22⋊D61—21no—
52C22×C22⋊D61—21no—
53C23×Dic61—21no—
54C23×Dic61D411yescharacteristic
55C23×Dic61D411yescharacteristic
56C22×C4×D61—21no—
57C2×#11758194—22no—
58C2×#11758194—24no—
59C22×#92418C2312yes—
60#14519776—22no—
61#14519776C2×C414yes—
62#14519773—22no—
63#14519773C2×C414yes—
64#14519763—28no—
65#14519751—22no—
66#14519751C2×C414yes—
67#14519749—28no—
68D4×Dic61—22no—
69D4×Dic61C2×C414yes—
70#14519739—22no—
71#14519739C2×C414yes—
72#14519737—28no—
73#14519733—44no—
74C2×C4⋊Dic61C2312yes—
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—21no—
77C2×#11758196—22no—
78C2×#11758196—22no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×C4×Dic61—22no—
84C2×C4×Dic61C2×C414yes—
85C2×C4×Dic61D411yescharacteristic
86C2×C4×Dic61D411yescharacteristic
87C2×C4×Dic61D414yes—
88C2×C4×Dic61C2311yescharacteristic
89C2×C4×Dic61C2311yescharacteristic
90#14519699—24no—
91#14519698—24no—
92#14519697—24no—
93#14519687—24no—
94C8×Dic61—24no—
95#14519684—21no—
96#14519684—22no—
97C22×D4×C61—21no—
98C22×D4×C61C2311yescharacteristic
99C23×C4×C61C2311yescharacteristic
100C2×M16×C61—22no—
101C2×M16×C61—24no—
102C22×C8×C61C2312yes—
103#121×C61—22no—
104#121×C61C2×C414yes—
105C2×C4⋊C4×C61C2312yes—
106C2×C4⋊C4×C61C2311yescharacteristic
107C2×C22⋊C4×C61—21no—
108C2×C22⋊C4×C61—21no—
109#102×C61—24no—
110#95×C61—21no—
111C24×C61—21no—
112C24×C61C2×D411yescharacteristic
113C23×D61—21no—
114C2×D4×C61—24no—
115C2×D4×C61—21no—
116C2×D4×C61—22no—
117C2×D4×C61—22no—
118C2×D4×C61—44no—
119C2×D4×C61C22×C412yes—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—21no—
123C22×C4×C61—22no—
124C22×C4×C61C22×C412yes—
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D412yes—
127C22×C4×C61C2411yescharacteristic
128C2×C22⋊D61—24no—
129C2×C22⋊D61—24no—
130C2×C22⋊D61—24no—
131C2×C22⋊D61—44no—
132C2×C22⋊D61—42no—
133C2×C22⋊D61—44no—
134C2×C22⋊D61—42no—
135C22×Dic61—24no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—21no—
139C22×Dic61—24no—
140C22×Dic61—22no—
141C22×Dic61—44no—
142C22×Dic61C22⋊C414yes—
143C22×Dic61C22⋊C414yes—
144C22×Dic61C2×D411yescharacteristic
145C22×Dic61C2×D411yescharacteristic
146C22×Dic61C2×D411yescharacteristic
147C22×Dic61C2×D411yescharacteristic
148#11758224—432no—
149C2×D244—22no—
150C2×D244—22no—
151C2×D244—21no—
152C2×D244C22×C412yes—
153C2×C4×D61—24no—
154C2×C4×D61—44no—
155C2×Dic122—24no—
156C2×Dic122—44no—
157M16×C61—28no—
158C2×C8×C61—22no—
159C2×C8×C61—22no—
160C2×C8×C61#4114yescomplemented
161C4⋊C4×C61—21no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61C22×C412yes—
165C22⋊C4×C61—24no—
166C22⋊C4×C61—42no—
167C22⋊Dic61—24no—
168C22⋊Dic61—42no—
169#11758198—24no—
170#11758198—24no—
171C4⋊Dic61—21no—
172C4⋊Dic61—22no—
173C4⋊Dic61—22no—
174C4⋊Dic61C22×C412yes—
175#11758196—24no—
176#11758196—44no—
177#11758196—44no—
178C4×Dic61—44no—
179C4×Dic61C22⋊C414yes—
180C4×Dic61C22⋊C414yes—
181C4×Dic61C22×C412yes—
182C4×Dic61C22×C412yes—
183C4×Dic61C2×D412yes—
184C4×Dic61C2×D412yes—
185#11758194—28no—
186C2×#92418—22no—
187C2×#92418—22no—
188C2×#92418#4114yescomplemented
189C23×C61—22no—
190C23×C61—21no—
191C23×C61—22no—
192C23×C61—21no—
193C23×C61—22no—
194C23×C61—42no—
195C23×C61—42no—
196C23×C61—41no—
197C23×C61—41no—
198C23×C61C2×C22⋊C412yes—
199C23×C61C22×D411yescharacteristic
200C22×D61—22no—
201C22×D61—42no—
202C22×D61—42no—
203C22×D61—41no—
204C22×D61—41no—
205D4×C61—24no—
206D4×C61—48no—
207D4×C61—41no—
208D4×C61—41no—
209C2×C4×C61—22no—
210C2×C4×C61—24no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61C2×C22⋊C412yes—
219C2×C4×C61C23×C411yescharacteristic
220C2×C4×C61C22×D411yescharacteristic
221C2×C4×C61C2×#4112yes—
222C2×C4×C61C2×#4112yes—
223C22⋊D61—48no—
224C22⋊D61—48no—
225C22⋊D61—48no—
226C2×Dic61—21no—
227C2×Dic61—22no—
228C2×Dic61—21no—
229C2×Dic61—28no—
230C2×Dic61—28no—
231C2×Dic61—22no—
232C2×Dic61—48no—
233C2×Dic61—42no—
234C2×Dic61—42no—
235C2×Dic61—42no—
236C2×Dic61—42no—
237C2×Dic61C2×C22⋊C412yes—
238C2×Dic61C2×C22⋊C412yes—
239C2×Dic61C2×C22⋊C412yes—
240C2×Dic61C2×C22⋊C412yes—
241C2×Dic61AGL(1,ℤ/8)12yes—
242C2×Dic61AGL(1,ℤ/8)14yes—
243C2×Dic61#13712yes—
244D244—24no—
245D244—41no—
246D244—41no—
247C4×D61—48no—
248Dic122—48no—
249C8×C61—24no—
250#92418—24no—
251#21410—611nomaximal, Sylow
252C22×C61—22no—
253C22×C61—22no—
254C22×C61—22no—
255C22×C61—42no—
256C22×C61—42no—
257C22×C61—42no—
258C22×C61—42no—
259C22×C61—42no—
260C22×C61—82no—
261C22×C61—82no—
262C22×C61C22×C22⋊C411yescharacteristic
263C22×C61#53212yes—
264C22×C61C2×AGL(1,ℤ/8)12yes—
265C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
266C22×C61C2×#13711yescharacteristic
267C2×D61—42no—
268C2×D61—42no—
269C2×D61—42no—
270C2×D61—82no—
271C2×D61—82no—
272C4×C61—22no—
273C4×C61—42no—
274C4×C61—82no—
275C4×C61C4×#4112yes—
276C4×C61#52511yescharacteristic
277C4×C61#52511yescharacteristic
278Dic61—22no—
279Dic61—22no—
280Dic61—42no—
281Dic61—82no—
282Dic61#41714yes—
283Dic61#41814yes—
284C2×#532—611no—
285C2×C4×M16—611no—
286#2285—611no—
287#1990—612no—
288#1957—612no—
289#1895—611no—
290#1895—611no—
291#1895—611no—
292#1895D6111yescharacteristic, complemented
293#1762—612no—
294#1762—612no—
295C2×C61—42no—
296C2×C61—82no—
297C2×C61#229711yescharacteristic
298C2×C61C2×#41811yescharacteristic
299C2×C61C2×#41711yescharacteristic
300C2×C61#318212yes—
301C2×C61#318412yes—
302D61—82no—
303C22×#41—1221no—
304C22×M16—611no—
305C22×M16C2×D6111yescharacteristic
306#532—618no—
307#532—1224no—
308C2×C4⋊Q8—611no—
309C2×#127—611no—
310C2×#124—1221no—
311C2×#121—611no—
312C2×#121—1221no—
313C2×#121C2×D6111yescharacteristic
314C2×C4×D4—611no—
315C2×C4×D4—611no—
316C2×C4×D4—1221no—
317C2×#117—1221no—
318C22×C42—611no—
319C2×#102—612no—
320C2×#102—612no—
321C2×#102—612no—
322C2×#102C2×D6112yes—
323C4×M16—1224no—
324C2×#97—612no—
325C2×C4×C8—612no—
326#384—612no—
327#382—1221no—
328#380—611no—
329#380—1221no—
330#380C2×D6111yescharacteristic
331C4×C4⋊C4—612no—
332C4×C4⋊C4—1221no—
333#340—612no—
334#340—612no—
335#340—612no—
336#340C2×D6112yes—
337C61#2141011yescharacteristic, complemented, Sylow
338C2×#41—1228no—
339C2×#41—2448no—
340C22×Q8—1221no—
341C22×D4—21no—
342C22×D4—611no—
343C22×D4—1221no—
344C22×D4—1221no—
345C22×D4—1221no—
346C22×D4C22×D6111yescharacteristic
347C23×C4—611no—
348C23×C4—611no—
349C23×C4—1221no—
350C23×C4—1221no—
351C23×C4C22×D6111yescharacteristic
352C2×M16—22no—
353C2×M16—24no—
354C2×M16—1224no—
355C2×M16—1222no—
356C22×C8—612no—
357C22×C8C22×D6112yes—
358C4⋊Q8—614no—
359C4⋊Q8—1222no—
360#127—614no—
361#127—1222no—
362#124—1228no—
363#121—22no—
364#121—614no—
365#121—1228no—
366#121—1222no—
367#121C4×D6114yes—
368C4×D4—614no—
369C4×D4—614no—
370C4×D4—1228no—
371C4×D4—1222no—
372C4×D4—1222no—
373#117—2444no—
374C2×C4⋊C4—611no—
375C2×C4⋊C4—612no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4—1222no—
378C2×C4⋊C4—1222no—
379C2×C4⋊C4C22×D6112yes—
380C2×C4⋊C4C22×D6111yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—1221no—
384C2×C22⋊C4—1221no—
385C2×C22⋊C4—1221no—
386C2×C22⋊C4—1221no—
387C2×C42—614no—
388C2×C42—611no—
389C2×C42—611no—
390C2×C42—614no—
391C2×C42—611no—
392C2×C42—611no—
393C2×C42—1222no—
394#102—24no—
395#102—1224no—
396#102—1224no—
397#102—1224no—
398#97—1224no—
399C4×C8—1224no—
400#95—21no—
401#95—1222no—
402#95—1221no—
403C24—21no—
404C24—1221no—
405C24D4×D6111yescharacteristic
406#41—24432no—
407C2×Q8—1224no—
408C2×Q8—2444no—
409C2×D4—24no—
410C2×D4—21no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—44no—
414C2×D4—612no—
415C2×D4—1224no—
416C2×D4—1224no—
417C2×D4—1224no—
418C2×D4—1221no—
419C2×D4—1222no—
420C2×D4—1222no—
421C2×D4—2444no—
422C2×D4—2442no—
423C2×D4—2442no—
424C2×D4—2444no—
425C2×D4C2×C4×D6112yes—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—611no—
431C22×C4—611no—
432C22×C4—611no—
433C22×C4—611no—
434C22×C4—614no—
435C22×C4—614no—
436C22×C4—1224no—
437C22×C4—1222no—
438C22×C4—1224no—
439C22×C4—1222no—
440C22×C4—1224no—
441C22×C4—1221no—
442C22×C4—1222no—
443C22×C4—2444no—
444C22×C4—2444no—
445C22×C4C2×C4×D6112yes—
446C22×C4D4×D6112yes—
447C22×C4D4×D6111yescharacteristic
448C22×C4C23×D6111yescharacteristic
449M16—28no—
450M16—1228no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—614no—
454C2×C8—1222no—
455C2×C8—1222no—
456C2×C8#1175822414yescomplemented
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—612no—
461C4⋊C4—1224no—
462C4⋊C4—1221no—
463C4⋊C4—1222no—
464C4⋊C4—1222no—
465C4⋊C4—2444no—
466C4⋊C4—2444no—
467C4⋊C4C2×C4×D6112yes—
468C22⋊C4—24no—
469C22⋊C4—42no—
470C22⋊C4—1224no—
471C22⋊C4—1224no—
472C22⋊C4—1224no—
473C22⋊C4—2442no—
474C42—612no—
475C42—612no—
476C42—612no—
477C42—612no—
478C42—614no—
479C42—614no—
480C42—2444no—
481C23—21no—
482C23—22no—
483C23—21no—
484C23—22no—
485C23—22no—
486C23—42no—
487C23—41no—
488C23—41no—
489C23—42no—
490C23—1222no—
491C23—2442no—
492C23—2441no—
493C23—2441no—
494C23—2442no—
495C23C22⋊C4×D6112yes—
496C23C2×D4×D6111yescharacteristic
497Q8—2448no—
498D4—24no—
499D4—48no—
500D4—41no—
501D4—41no—
502D4—1224no—
503D4—2448no—
504D4—2448no—
505D4—2448no—
506D4—2441no—
507D4—2441no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—612no—
518C2×C4—612no—
519C2×C4—612no—
520C2×C4—614no—
521C2×C4—612no—
522C2×C4—612no—
523C2×C4—612no—
524C2×C4—1228no—
525C2×C4—1228no—
526C2×C4—1221no—
527C2×C4—1222no—
528C2×C4—1221no—
529C2×C4—1222no—
530C2×C4—2448no—
531C2×C4—2448no—
532C2×C4—2442no—
533C2×C4—2442no—
534C2×C4—2442no—
535C2×C4—2442no—
536C2×C4C22⋊C4×D6112yes—
537C2×C4C22×C4×D6111yescharacteristic
538C2×C4C2×D4×D6111yescharacteristic
539C2×C4#41×D6112yes—
540C2×C4C2×#1175822412yes—
541C8—24no—
542C8—1224no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—82no—
552C22—82no—
553C22—4882no—
554C22—4882no—
555C22—2442no—
556C22—2442no—
557C22—2442no—
558C22C2×C22⋊C4×D6111yescharacteristic
559C22#1684030712yes—
560C22AGL(1,ℤ/8)×D6112yes—
561C22AGL(1,ℤ/8)×D6111yescharacteristic
562C22#137×D6111yescharacteristic
563C4—22no—
564C4—42no—
565C4—82no—
566C4—614no—
567C4—614no—
568C4—1222no—
569C4—1222no—
570C4—4882no—
571C4—2442no—
572C4C4×#1175822412yes—
573C4#1684026811yescharacteristic
574C4#1684027011yescharacteristic
575C2—42no—
576C2—82no—
577C2—4882no—
578C2#2144830411yescharacteristic
579C2#418×D6111yescharacteristic
580C2#417×D6111yescharacteristic
581C2#2145653212yes—
582C2#2145655212yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1895C2
socleC23×C61C22×D4
prime radicalC2×#16840307C2
cosocleC22×C4×C61C24
hypercenter#1895D61
hypocenterC61#21410
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1895

Complements

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

NH ≅ G/Ntype
2. C2×M16×Dic61C2semidirect
3. #21448221C2semidirect
4. #21446778C2semidirect
5. #21446623C2semidirect
6. #21446258C2semidirect
7. #21446257C2semidirect
8. #21446255C2semidirect
9. #21445646C2semidirect
10. #21445643C2semidirect
11. C61×#1895C2semidirect
15. #16840307C4semidirect
160. C2×C8×C61#41semidirect
188. C2×#92418#41semidirect
292. #1895D61semidirect
337. C61#21410semidirect
456. C2×C8#11758224semidirect

Maximal subgroups

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

subgroupindex
C2×#168403072
C2×M16×Dic612
#214482212
#214467782
#214466232
#214462582
#214462572
#214462552
#214456462
#214456432
C61×#18952
#2141061