Finite Group Databasebeta

← #29685246

Subgroups — #29685246

Order 15616.

1434 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840309C211yesmaximal, characteristic
2C2×#97×D61C211yesmaximal, characteristic, complemented
3#21448215C211yesmaximal, characteristic, complemented
4#21446790C212yesmaximal, complemented
5#21446641C212yesmaximal, complemented
6#21446246C212yesmaximal, complemented
7#21446244C212yesmaximal, complemented
8#21445647C211yesmaximal, characteristic, complemented
9#21445645C211yesmaximal, characteristic, complemented
10#21445644C211yesmaximal, characteristic, complemented
11C61×#1762C211yesmaximal, characteristic, complemented
12C22×#11758222—21no—
13C22×C8×D61C2212yes—
14#16840309—24no—
15#16840309C418yescomplemented
16C2×#14519776C2211yescharacteristic
17C2×#14519773C2211yescharacteristic
18C2×#14519763—21no—
19C2×#14519752C2211yescharacteristic
20C2×#14519750—21no—
21C2×#14519748C2211yescharacteristic
22C2×C4×D244C2211yescharacteristic
23C2×C4×C22⋊D61—21no—
24C2×C42×D61C2211yescharacteristic
25C2×#14519699C2212yes—
26C2×#14519698C2212yes—
27C2×#14519697C2212yes—
28C2×#14519688C2211yescharacteristic
29C2×#14519687C2212yes—
30#97×D61—24no—
31#16839629—21no—
32#16839627C2212yes—
33#16839624—21no—
34#16839619C2212yes—
35C4×C4⋊Dic61C2211yescharacteristic
36C4×#11758198—21no—
37#16839608—21no—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×C4×D4×C61C2211yescharacteristic
42C2×#102×C61C2212yes—
43C2×#97×C61C2211yescharacteristic
44C4×C4⋊C4×C61C2211yescharacteristic
45C61×#340C2212yes—
46C2×#11758222—28no—
47C2×#11758222—48no—
48C22×Dic122—21no—
49C22×D244—21no—
50C22×D244C2311yescharacteristic
51C22×C22⋊D61—21no—
52C22×C22⋊D61—21no—
53C22×C22⋊D61—21no—
54C22×C4×D61—21no—
55C22×C4×D61D411yescharacteristic
56C22×C4×D61D411yescharacteristic
57C22×C4×D61C2311yescharacteristic
58C22×#92418C2312yes—
59C2×C8×D61—24no—
60C2×C8×D61—24no—
61C2×C8×D61—24no—
62#14519776—22no—
63#14519776C2×C414yes—
64#14519773—22no—
65#14519773C2×C414yes—
66#14519763—28no—
67#14519752—22no—
68#14519752C2×C414yes—
69#14519750—28no—
70#14519748—22no—
71#14519748C2×C414yes—
72C4×D244—22no—
73C4×D244C2×C414yes—
74C4×C22⋊D61—28no—
75C2×C4⋊Dic61C2312yes—
76C2×C4⋊Dic61C2311yescharacteristic
77C2×#11758196—21no—
78C2×C22⋊Dic61—21no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×#11758198—22no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61D411yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86C42×D61C2×C414yes—
87C42×D61D414yes—
88#14519699—24no—
89#14519698—24no—
90#14519697—24no—
91#14519688—22no—
92#14519687—24no—
93#14519684—21no—
94#14519684—22no—
95#14519684—22no—
96#14519684—22no—
97C22×D4×C61C2311yescharacteristic
98C23×C4×C61—21no—
99C22×C8×C61C2312yes—
100C4×D4×C61—22no—
101C4×D4×C61C2×C414yes—
102C2×C4⋊C4×C61C2312yes—
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C22⋊C4×C61—21no—
105C2×C42×C61—21no—
106C2×C42×C61C2311yescharacteristic
107#102×C61—24no—
108#97×C61—22no—
109#95×C61—21no—
110C24×C61—21no—
111C23×D61—21no—
112C23×D61C2×D411yescharacteristic
113C2×D4×C61—21no—
114C2×D4×C61—22no—
115C2×D4×C61—22no—
116C2×D4×C61C22×C412yes—
117C22×C4×C61—21no—
118C22×C4×C61—24no—
119C22×C4×C61—22no—
120C22×C4×C61—44no—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2411yescharacteristic
124C2×C22⋊D61—24no—
125C2×C22⋊D61—24no—
126C2×C22⋊D61—24no—
127C2×C22⋊D61—42no—
128C2×C22⋊D61—42no—
129C2×C22⋊D61—44no—
130C2×C22⋊D61—44no—
131C22×Dic61—21no—
132C22×Dic61—22no—
133C22×Dic61C2×D411yescharacteristic
134C22×Dic61C2×D411yescharacteristic
135C22×Dic61C2×D411yescharacteristic
136#11758222—432no—
137C2×D244—22no—
138C2×D244—22no—
139C2×D244—24no—
140C2×D244—21no—
141C2×D244—44no—
142C2×D244C22×C412yes—
143C2×C4×D61—24no—
144C2×C4×D61—22no—
145C2×C4×D61—22no—
146C2×C4×D61—24no—
147C2×C4×D61—41no—
148C2×C4×D61—41no—
149C2×C4×D61—44no—
150C2×C4×D61C22⋊C414yes—
151C2×C4×D61C22⋊C414yes—
152C2×C4×D61C22×C412yes—
153C2×C4×D61C2×D412yes—
154C2×Dic122—24no—
155C2×Dic122—44no—
156C2×C8×C61—22no—
157C2×C8×C61—22no—
158C2×C8×C61#4114yescomplemented
159C4⋊C4×C61—21no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61C22×C412yes—
163C22⋊C4×C61—24no—
164C42×C61—42no—
165C42×C61C22×C412yes—
166C42×C61C2×D412yes—
167C22⋊Dic61—24no—
168#11758198—24no—
169#11758198—24no—
170#11758198—44no—
171C4⋊Dic61—21no—
172C4⋊Dic61—22no—
173C4⋊Dic61—22no—
174C4⋊Dic61C22×C412yes—
175#11758196—24no—
176C4×Dic61—42no—
177C4×Dic61C22⋊C414yes—
178C4×Dic61C22⋊C414yes—
179C4×Dic61C22×C412yes—
180C4×Dic61C2×D412yes—
181C2×#92418—22no—
182C2×#92418—22no—
183C2×#92418#4114yescomplemented
184C8×D61—216no—
185C23×C61—22no—
186C23×C61—42no—
187C23×C61—42no—
188C23×C61—41no—
189C23×C61—41no—
190C23×C61C22×D411yescharacteristic
191C22×D61—22no—
192C22×D61—21no—
193C22×D61—21no—
194C22×D61—22no—
195C22×D61—42no—
196C22×D61—42no—
197C22×D61—41no—
198C22×D61—41no—
199C22×D61C2×C22⋊C412yes—
200C22×D61AGL(1,ℤ/8)14yes—
201D4×C61—22no—
202D4×C61—22no—
203D4×C61—42no—
204C2×C4×C61—22no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—22no—
208C2×C4×C61—48no—
209C2×C4×C61—42no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61C2×C22⋊C412yes—
214C2×C4×C61C2×C22⋊C412yes—
215C2×C4×C61C23×C411yescharacteristic
216C2×C4×C61C22×D411yescharacteristic
217C2×C4×C61C2×#4112yes—
218C2×C4×C61C2×#4112yes—
219C22⋊D61—48no—
220C22⋊D61—48no—
221C22⋊D61—48no—
222C2×Dic61—22no—
223C2×Dic61—22no—
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—22no—
227C2×Dic61—42no—
228C2×Dic61—42no—
229C2×Dic61—42no—
230C2×Dic61—42no—
231C2×Dic61C2×C22⋊C412yes—
232C2×Dic61C2×C22⋊C412yes—
233C2×Dic61C2×C22⋊C412yes—
234D244—22no—
235D244—22no—
236D244—48no—
237D244—42no—
238C4×D61—28no—
239C4×D61—22no—
240C4×D61—22no—
241C4×D61—24no—
242C4×D61—44no—
243C4×D61—48no—
244Dic122—48no—
245C8×C61—24no—
246#92418—24no—
247#21408—611nomaximal, Sylow
248C22×C61—42no—
249C22×C61—42no—
250C22×C61—42no—
251C22×C61—82no—
252C22×C61—82no—
253C22×C61C22×C22⋊C411yescharacteristic
254C22×C61#53212yes—
255C22×C61C2×AGL(1,ℤ/8)12yes—
256C22×C61#57511yescharacteristic
257C22×C61#57511yescharacteristic
258C2×D61—22no—
259C2×D61—24no—
260C2×D61—24no—
261C2×D61—22no—
262C2×D61—42no—
263C2×D61—42no—
264C2×D61—42no—
265C2×D61—82no—
266C2×D61—82no—
267C2×D61#41814yes—
268C4×C61—22no—
269C4×C61—22no—
270C4×C61—42no—
271C4×C61—82no—
272C4×C61C4×#4111yescharacteristic
273C4×C61C4×#4111yescharacteristic
274C4×C61#52512yes—
275Dic61—22no—
276Dic61—22no—
277Dic61—22no—
278Dic61—42no—
279Dic61—82no—
280C2×#532—611no—
281C22×#97—611no—
282#2283—611no—
283#1993—612no—
284#1961—612no—
285#1893—612no—
286#1893—612no—
287#1762—611no—
288#1762—611no—
289#1762—611no—
290#1762D6111yescharacteristic, complemented
291C2×C61—82no—
292C2×C61#229711yescharacteristic
293C2×C61C2×#41811yescharacteristic
294C2×C61#289611yescharacteristic
295C2×C61#318012yes—
296C2×C61#318012yes—
297D61—24no—
298D61—82no—
299C22×#41—1221no—
300C23×C8—612no—
301#532—618no—
302#532—1224no—
303C2×C4⋊Q8—611no—
304C2×#127—611no—
305C2×#124—1221no—
306C2×#121—611no—
307C2×#121—611no—
308C2×#121—1221no—
309C2×C4×D4—611no—
310C2×C4×D4—1221no—
311C2×C4×D4C2×D6111yescharacteristic
312C22×C42—611no—
313C2×#102—612no—
314C2×#102—612no—
315C2×#102—612no—
316C2×#102C2×D6112yes—
317C2×#97—612no—
318C2×#97—611no—
319C2×#97—1224no—
320C2×#97C2×D6111yescharacteristic
321#383—1221no—
322#382—612no—
323#381—1221no—
324#380—612no—
325C4×C4⋊C4—611no—
326C4×C4⋊C4C2×D6111yescharacteristic
327C4×C22⋊C4—1221no—
328#376—1221no—
329#340—612no—
330#340—612no—
331#340—612no—
332#340C2×D6112yes—
333C61#2140811yescharacteristic, complemented, Sylow
334C2×#41—1228no—
335C2×#41—2448no—
336C22×Q8—1221no—
337C22×D4—611no—
338C22×D4—1221no—
339C22×D4—1221no—
340C22×D4—1221no—
341C22×D4—1221no—
342C22×D4C22×D6111yescharacteristic
343C23×C4—21no—
344C23×C4—611no—
345C23×C4—611no—
346C23×C4—611no—
347C23×C4—1221no—
348C22×C8—612no—
349C22×C8—1224no—
350C22×C8—1224no—
351C22×C8—1224no—
352C22×C8C22×D6112yes—
353C4⋊Q8—614no—
354C4⋊Q8—1222no—
355#127—614no—
356#127—1222no—
357#124—1228no—
358#121—614no—
359#121—614no—
360#121—1228no—
361#121—1222no—
362#121—1222no—
363C4×D4—22no—
364C4×D4—614no—
365C4×D4—1228no—
366C4×D4—1222no—
367C4×D4C4×D6114yes—
368C2×C4⋊C4—611no—
369C2×C4⋊C4—612no—
370C2×C4⋊C4—1221no—
371C2×C4⋊C4C22×D6112yes—
372C2×C4⋊C4C22×D6111yescharacteristic
373C2×C22⋊C4—21no—
374C2×C22⋊C4—1221no—
375C2×C22⋊C4—1221no—
376C2×C22⋊C4—1221no—
377C2×C22⋊C4—1222no—
378C2×C42—21no—
379C2×C42—611no—
380C2×C42—611no—
381C2×C42—611no—
382C2×C42—614no—
383C2×C42—614no—
384C2×C42—1221no—
385C2×C42C22×D6111yescharacteristic
386#102—24no—
387#102—1224no—
388#102—1224no—
389#102—1224no—
390#97—22no—
391#97—1224no—
392#97—1222no—
393#95—21no—
394#95—1222no—
395#95—1222no—
396#95—1222no—
397#95—1221no—
398C24—21no—
399C24—611no—
400C24—1221no—
401#41—24432no—
402C2×Q8—1224no—
403C2×Q8—2444no—
404C2×D4—21no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—612no—
408C2×D4—1224no—
409C2×D4—1224no—
410C2×D4—1224no—
411C2×D4—1224no—
412C2×D4—1221no—
413C2×D4—1222no—
414C2×D4—1222no—
415C2×D4—2444no—
416C2×D4—2444no—
417C2×D4—2442no—
418C2×D4—2442no—
419C2×D4—2444no—
420C2×D4C2×C4×D6112yes—
421C22×C4—21no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—44no—
425C22×C4—611no—
426C22×C4—611no—
427C22×C4—611no—
428C22×C4—612no—
429C22×C4—612no—
430C22×C4—614no—
431C22×C4—614no—
432C22×C4—1224no—
433C22×C4—1222no—
434C22×C4—1222no—
435C22×C4—1221no—
436C22×C4—1224no—
437C22×C4—1222no—
438C22×C4—2441no—
439C22×C4—2441no—
440C22×C4—2444no—
441C22×C4D4×D6111yescharacteristic
442C22×C4D4×D6111yescharacteristic
443C22×C4C23×D6111yescharacteristic
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—614no—
447C2×C8—12216no—
448C2×C8—1222no—
449C2×C8—1222no—
450C2×C8#1175822214yescomplemented
451C4⋊C4—21no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—612no—
455C4⋊C4—1224no—
456C4⋊C4—1221no—
457C4⋊C4—1222no—
458C4⋊C4—1222no—
459C4⋊C4C2×C4×D6112yes—
460C22⋊C4—24no—
461C22⋊C4—1224no—
462C22⋊C4—1224no—
463C22⋊C4—1224no—
464C22⋊C4—2444no—
465C42—42no—
466C42—612no—
467C42—612no—
468C42—614no—
469C42—614no—
470C42—2442no—
471C42C2×C4×D6112yes—
472C42D4×D6112yes—
473C23—22no—
474C23—42no—
475C23—41no—
476C23—41no—
477C23—42no—
478C23—612no—
479C23—614no—
480C23—1221no—
481C23—1221no—
482C23—1222no—
483C23—1222no—
484C23—2442no—
485C23—2441no—
486C23—2441no—
487C23—2442no—
488C23C2×D4×D6111yescharacteristic
489Q8—2448no—
490D4—22no—
491D4—22no—
492D4—42no—
493D4—1222no—
494D4—1222no—
495D4—2448no—
496D4—2448no—
497D4—2448no—
498D4—2448no—
499D4—2442no—
500C2×C4—22no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—48no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—612no—
510C2×C4—612no—
511C2×C4—612no—
512C2×C4—1228no—
513C2×C4—1222no—
514C2×C4—1221no—
515C2×C4—1221no—
516C2×C4—1222no—
517C2×C4—1222no—
518C2×C4—1222no—
519C2×C4—1224no—
520C2×C4—1222no—
521C2×C4—2444no—
522C2×C4—2448no—
523C2×C4—2442no—
524C2×C4—2442no—
525C2×C4—2442no—
526C2×C4—2442no—
527C2×C4C22⋊C4×D6112yes—
528C2×C4C22⋊C4×D6112yes—
529C2×C4C22×C4×D6111yescharacteristic
530C2×C4C2×D4×D6111yescharacteristic
531C2×C4#41×D6112yes—
532C2×C4C2×#1175822212yes—
533C8—24no—
534C8—1224no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—82no—
539C22—82no—
540C22—614no—
541C22—1222no—
542C22—1224no—
543C22—1224no—
544C22—1222no—
545C22—4882no—
546C22—4882no—
547C22—2442no—
548C22—2442no—
549C22—2442no—
550C22C2×C22⋊C4×D6111yescharacteristic
551C22#1684030912yes—
552C22AGL(1,ℤ/8)×D6112yes—
553C22#1684055811yescharacteristic
554C22#1684056111yescharacteristic
555C4—22no—
556C4—22no—
557C4—42no—
558C4—82no—
559C4—1222no—
560C4—1222no—
561C4—1222no—
562C4—4882no—
563C4—2442no—
564C4#1684023011yescharacteristic
565C4C4×#1175822211yescharacteristic
566C4#1684027112yes—
567C2—82no—
568C2—1224no—
569C2—4882no—
570C2#2144830511yescharacteristic
571C2#418×D6111yescharacteristic
572C2#2145433511yescharacteristic
573C2#2145651312yes—
574C2#2145651512yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1762C2
socleC23×C61C22×D4
prime radicalC2×#16840309C2
cosocleC22×C4×C61C24
hypercenter#1762D61
hypocenterC61#21408
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
2. C2×#97×D61C2semidirect
3. #21448215C2semidirect
4. #21446790C2semidirect
5. #21446641C2semidirect
6. #21446246C2semidirect
7. #21446244C2semidirect
8. #21445647C2semidirect
9. #21445645C2semidirect
10. #21445644C2semidirect
11. C61×#1762C2semidirect
15. #16840309C4semidirect
158. C2×C8×C61#41semidirect
183. C2×#92418#41semidirect
290. #1762D61semidirect
333. C61#21408semidirect
450. C2×C8#11758222semidirect

Maximal subgroups

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

subgroupindex
C2×#168403092
C2×#97×D612
#214482152
#214467902
#214466412
#214462462
#214462442
#214456472
#214456452
#214456442
C61×#17622
#2140861