Finite Group Databasebeta

← #29685266

Subgroups — #29685266

Order 15616.

1174 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840298C211yesmaximal, characteristic
2C2×M16×Dic61C211yesmaximal, characteristic, complemented
3#21448214C211yesmaximal, characteristic, complemented
4#21446782C212yesmaximal, complemented
5#21446627C212yesmaximal, complemented
6#21446257C211yesmaximal, characteristic
7#21446255C211yesmaximal, characteristic
8#21446253C211yesmaximal, characteristic, complemented
9#21445651C212yesmaximal, complemented
10#21445643C212yesmaximal, complemented
11C61×#1894C211yesmaximal, characteristic, complemented
12C22×#11758194C2211yescharacteristic
13#16840298—24no—
14#16840298C418yescomplemented
15C2×#14519772—21no—
16C2×#14519768C2211yescharacteristic
17C2×#14519766C2211yescharacteristic
18C2×#14519761—21no—
19C2×#14519751C2211yescharacteristic
20C2×Q8×Dic61C2211yescharacteristic
21C2×#14519739C2211yescharacteristic
22C2×#14519737—21no—
23C2×#14519735—21no—
24C22×C4×Dic61C2211yescharacteristic
25C2×#14519701C2212yes—
26C2×#14519699C2212yes—
27C2×#14519697C2212yes—
28M16×Dic61—24no—
29C2×#14519687C2212yes—
30C2×C8×Dic61C2212yes—
31#16839629—21no—
32#16839627C2212yes—
33#16839623—21no—
34#16839622C2211yescharacteristic
35C4⋊C4×Dic61C2212yes—
36C22⋊C4×Dic61—21no—
37#16839607—21no—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C22×M16×C61C2211yescharacteristic
42C2×C22⋊Q8×C61C2211yescharacteristic
43C2×#103×C61C2212yes—
44C61×#380C2211yescharacteristic
45C61×#340C2212yes—
46C22×D244C2311yescharacteristic
47C22×C22⋊D61—21no—
48C23×Dic61D411yescharacteristic
49C23×Dic61D411yescharacteristic
50C22×C4×D61—21no—
51C2×#11758194—22no—
52C2×#11758194—24no—
53C22×#92418C2312yes—
54#14519772—28no—
55#14519768—22no—
56#14519768C2×C414yes—
57#14519766—22no—
58#14519766C2×C414yes—
59#14519761—28no—
60#14519751—22no—
61#14519751C2×C414yes—
62Q8×Dic61—22no—
63Q8×Dic61C2×C414yes—
64#14519739—22no—
65#14519739C2×C414yes—
66#14519737—28no—
67#14519735—28no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61C2312yes—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×#11758196—21no—
73C2×C22⋊Dic61—21no—
74C2×#11758198—21no—
75C2×#11758198—21no—
76C2×#11758198—21no—
77C2×#11758198—21no—
78C2×C4×Dic61—21no—
79C2×C4×Dic61—22no—
80C2×C4×Dic61C2×C414yes—
81C2×C4×Dic61D411yescharacteristic
82C2×C4×Dic61D411yescharacteristic
83C2×C4×Dic61D414yes—
84C2×C4×Dic61C2311yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86#14519701—24no—
87#14519699—24no—
88#14519697—24no—
89#14519687—24no—
90C8×Dic61—24no—
91#14519684—21no—
92#14519684—22no—
93#14519684—22no—
94#14519684—22no—
95C22×Q8×C61C2311yescharacteristic
96C23×C4×C61C2311yescharacteristic
97C2×M16×C61—22no—
98C2×M16×C61—24no—
99C22×C8×C61C2312yes—
100C22⋊Q8×C61—22no—
101C22⋊Q8×C61C2×C414yes—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61C2312yes—
104C2×C4⋊C4×C61C2311yescharacteristic
105C2×C22⋊C4×C61—21no—
106C2×C22⋊C4×C61—21no—
107#103×C61—24no—
108#95×C61—21no—
109C24×C61C2×D411yescharacteristic
110C23×D61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—22no—
113C2×Q8×C61—22no—
114C2×Q8×C61C22×C412yes—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61—22no—
120C22×C4×C61C22×C412yes—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D412yes—
123C22×C4×C61C2411yescharacteristic
124C2×C22⋊D61—24no—
125C2×C22⋊D61—44no—
126C22×Dic61—21no—
127C22×Dic61—22no—
128C22×Dic61—42no—
129C22×Dic61—41no—
130C22×Dic61—41no—
131C22×Dic61C22⋊C414yes—
132C22×Dic61C22⋊C414yes—
133C22×Dic61C2×D411yescharacteristic
134C22×Dic61C2×D411yescharacteristic
135C22×Dic61C2×D411yescharacteristic
136C22×Dic61C2×D411yescharacteristic
137C2×D244—22no—
138C2×D244—22no—
139C2×D244—21no—
140C2×D244C22×C412yes—
141C2×C4×D61—24no—
142C2×C4×D61—42no—
143C2×C4×D61—42no—
144M16×C61—28no—
145C2×C8×C61—22no—
146C2×C8×C61—22no—
147C2×C8×C61#4114yes—
148C4⋊C4×C61—24no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61C22×C412yes—
153C22⋊C4×C61—24no—
154C22⋊C4×C61—42no—
155C22⋊Dic61—42no—
156#11758198—24no—
157#11758198—24no—
158#11758198—24no—
159#11758198—24no—
160C4⋊Dic61—21no—
161C4⋊Dic61—24no—
162C4⋊Dic61—24no—
163C4⋊Dic61—22no—
164C4⋊Dic61—22no—
165C4⋊Dic61C22×C412yes—
166#11758196—24no—
167C4×Dic61—24no—
168C4×Dic61—44no—
169C4×Dic61C22⋊C414yes—
170C4×Dic61C22⋊C414yes—
171C4×Dic61C22×C412yes—
172C4×Dic61C22×C412yes—
173C4×Dic61C2×D412yes—
174C4×Dic61C2×D412yes—
175#11758194—28no—
176C2×#92418—22no—
177C2×#92418—22no—
178C2×#92418#4114yes—
179C23×C61—22no—
180C23×C61—21no—
181C23×C61—21no—
182C23×C61—22no—
183C23×C61C2×C22⋊C412yes—
184C23×C61C22×D411yescharacteristic
185C22×D61—22no—
186C22×D61—42no—
187C22×D61—41no—
188C22×D61—41no—
189C22×D61—42no—
190Q8×C61—24no—
191Q8×C61—41no—
192Q8×C61—41no—
193C2×C4×C61—22no—
194C2×C4×C61—24no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—42no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61C2×C22⋊C412yes—
206C2×C4×C61C23×C411yescharacteristic
207C2×C4×C61C22×D411yescharacteristic
208C2×C4×C61C2×#4112yes—
209C2×C4×C61C2×#4112yes—
210C22⋊D61—48no—
211C2×Dic61—22no—
212C2×Dic61—21no—
213C2×Dic61—21no—
214C2×Dic61—21no—
215C2×Dic61—21no—
216C2×Dic61—22no—
217C2×Dic61—22no—
218C2×Dic61—44no—
219C2×Dic61—44no—
220C2×Dic61—42no—
221C2×Dic61—42no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224C2×Dic61C2×C22⋊C412yes—
225C2×Dic61C2×C22⋊C412yes—
226C2×Dic61C2×C22⋊C412yes—
227C2×Dic61C2×C22⋊C412yes—
228D244—24no—
229D244—41no—
230D244—41no—
231C4×D61—48no—
232C8×C61—24no—
233#92418—24no—
234#21411—611nomaximal, Sylow
235C22×C61—22no—
236C22×C61—22no—
237C22×C61—22no—
238C22×C61—42no—
239C22×C61—42no—
240C22×C61C22×C22⋊C411yescharacteristic
241C22×C61#53212yes—
242C22×C61#57511yescharacteristic
243C22×C61#57512yes—
244C22×C61#57511yescharacteristic
245C2×D61—42no—
246C2×D61—42no—
247C2×D61—42no—
248C2×D61—82no—
249C2×D61—82no—
250C4×C61—22no—
251C4×C61—42no—
252C4×C61—42no—
253C4×C61—82no—
254C4×C61C4×#4112yes—
255C4×C61#52511yescharacteristic
256C4×C61#52511yescharacteristic
257Dic61—22no—
258Dic61—22no—
259Dic61—22no—
260Dic61—22no—
261Dic61—42no—
262Dic61—82no—
263C2×#529—611no—
264C2×C4×M16—611no—
265#2283—611no—
266#1991—612no—
267#1958—612no—
268#1895—611no—
269#1895—611no—
270#1894—611no—
271#1894D6111yescharacteristic, complemented
272#1763—612no—
273#1762—612no—
274C2×C61—42no—
275C2×C61#229711yescharacteristic
276C2×C61#289611yescharacteristic
277C2×C61#289611yescharacteristic
278C2×C61#318312yes—
279C2×C61#318312yes—
280D61—82no—
281C22×M16—611no—
282C22×M16C2×D6111yescharacteristic
283#529—618no—
284#529—1224no—
285C2×#126—1221no—
286C2×#125—611no—
287C2×#124—611no—
288C2×#123—1221no—
289C2×C22⋊Q8C2×D6111yescharacteristic
290C2×#121—611no—
291C2×C4×Q8—611no—
292C2×C4×D4—611no—
293C2×C4×D4—1221no—
294C2×#117—1221no—
295C22×C42—611no—
296C2×#103—612no—
297C2×#103C2×D6112yes—
298C2×#102—612no—
299C2×#102—612no—
300C4×M16—1224no—
301C2×#97—612no—
302C2×C4×C8—612no—
303#383—1221no—
304#382—612no—
305#381—1221no—
306#380—611no—
307#380C2×D6111yescharacteristic
308C4×C4⋊C4—612no—
309C4×C22⋊C4—1221no—
310#376—1221no—
311#340—612no—
312#340—612no—
313#340—612no—
314#340C2×D6112yes—
315C61#2141111yescharacteristic, complemented, Sylow
316C22×Q8C22×D6111yescharacteristic
317C22×D4—611no—
318C22×D4—1221no—
319C23×C4—611no—
320C23×C4—611no—
321C23×C4—1221no—
322C23×C4C22×D6111yescharacteristic
323C2×M16—22no—
324C2×M16—24no—
325C2×M16—1224no—
326C2×M16—1222no—
327C22×C8—612no—
328C22×C8C22×D6112yes—
329#126—1228no—
330#125—614no—
331#125—1222no—
332#124—614no—
333#124—1222no—
334#123—1228no—
335C22⋊Q8—22no—
336C22⋊Q8C4×D6114yes—
337#121—614no—
338#121—1222no—
339C4×Q8—614no—
340C4×Q8—1222no—
341C4×D4—614no—
342C4×D4—1222no—
343C4×D4—1228no—
344#117—1228no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—611no—
347C2×C4⋊C4—612no—
348C2×C4⋊C4—1221no—
349C2×C4⋊C4—1221no—
350C2×C4⋊C4—1221no—
351C2×C4⋊C4C22×D6112yes—
352C2×C4⋊C4C22×D6111yescharacteristic
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—1221no—
356C2×C22⋊C4—1221no—
357C2×C22⋊C4—1221no—
358C2×C22⋊C4—1221no—
359C2×C22⋊C4—1221no—
360C2×C42—614no—
361C2×C42—611no—
362C2×C42—611no—
363C2×C42—614no—
364C2×C42—611no—
365C2×C42—611no—
366C2×C42—1221no—
367C2×C42—1222no—
368#103—24no—
369#103—1224no—
370#102—1224no—
371#102—1224no—
372#97—1224no—
373C4×C8—1224no—
374#95—21no—
375#95—1222no—
376#95—1222no—
377#95—1222no—
378#95—1221no—
379C24—1221no—
380C24D4×D6111yescharacteristic
381C2×Q8—21no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8C2×C4×D6112yes—
385C2×D4—612no—
386C2×D4—1224no—
387C2×D4—1221no—
388C2×D4—1222no—
389C2×D4—1222no—
390C2×D4—2444no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—611no—
397C22×C4—611no—
398C22×C4—611no—
399C22×C4—611no—
400C22×C4—614no—
401C22×C4—614no—
402C22×C4—1224no—
403C22×C4—1221no—
404C22×C4—1222no—
405C22×C4—2442no—
406C22×C4—2442no—
407C22×C4—2441no—
408C22×C4—2442no—
409C22×C4—2441no—
410C22×C4C2×C4×D6112yes—
411C22×C4D4×D6112yes—
412C22×C4D4×D6111yescharacteristic
413C22×C4C23×D6111yescharacteristic
414M16—28no—
415M16—1228no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—614no—
419C2×C8—1222no—
420C2×C8—1222no—
421C2×C8#1175822414yes—
422C4⋊C4—21no—
423C4⋊C4—24no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—612no—
427C4⋊C4—1224no—
428C4⋊C4—1224no—
429C4⋊C4—1224no—
430C4⋊C4—1221no—
431C4⋊C4—1222no—
432C4⋊C4—1222no—
433C4⋊C4C2×C4×D6112yes—
434C22⋊C4—24no—
435C22⋊C4—42no—
436C22⋊C4—1224no—
437C22⋊C4—1224no—
438C22⋊C4—1224no—
439C22⋊C4—1224no—
440C22⋊C4—2442no—
441C42—612no—
442C42—612no—
443C42—612no—
444C42—612no—
445C42—614no—
446C42—614no—
447C42—1224no—
448C42—2444no—
449C23—21no—
450C23—21no—
451C23—22no—
452C23—22no—
453C23—1222no—
454C23—2441no—
455C23—2441no—
456C23—2442no—
457C23—2442no—
458C23C22⋊C4×D6112yes—
459C23C2×D4×D6111yescharacteristic
460Q8—24no—
461Q8—41no—
462Q8—41no—
463D4—1224no—
464D4—2448no—
465D4—2441no—
466D4—2441no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—612no—
480C2×C4—612no—
481C2×C4—612no—
482C2×C4—612no—
483C2×C4—1221no—
484C2×C4—1221no—
485C2×C4—1222no—
486C2×C4—1222no—
487C2×C4—1221no—
488C2×C4—1221no—
489C2×C4—1222no—
490C2×C4—2448no—
491C2×C4—2444no—
492C2×C4—2444no—
493C2×C4—2442no—
494C2×C4—2442no—
495C2×C4—2442no—
496C2×C4—2442no—
497C2×C4C22⋊C4×D6112yes—
498C2×C4C22×C4×D6111yescharacteristic
499C2×C4C2×D4×D6111yescharacteristic
500C2×C4#41×D6112yes—
501C2×C4C2×#1175822412yes—
502C8—24no—
503C8—1224no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—42no—
508C22—42no—
509C22—4882no—
510C22—4882no—
511C22—2442no—
512C22—2442no—
513C22—2442no—
514C22C2×C22⋊C4×D6111yescharacteristic
515C22#1684030712yes—
516C22#1684055811yescharacteristic
517C22#1684056112yes—
518C22#1684056111yescharacteristic
519C4—22no—
520C4—42no—
521C4—42no—
522C4—82no—
523C4—1222no—
524C4—1222no—
525C4—1222no—
526C4—1222no—
527C4—4882no—
528C4—2442no—
529C4C4×#1175822412yes—
530C4#1684026811yescharacteristic
531C4#1684027011yescharacteristic
532C2—42no—
533C2—4882no—
534C2#2144830411yescharacteristic
535C2#2145433511yescharacteristic
536C2#2145433711yescharacteristic
537C2#2145654112yes—
538C2#2145654912yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1894C2
socleC23×C61C22×D4
prime radicalC2×#14519751C22
cosocleC22×C4×C61C24
hypercenter#1894D61
hypocenterC61#21411
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1894

Complements

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

NH ≅ G/Ntype
2. C2×M16×Dic61C2semidirect
3. #21448214C2semidirect
4. #21446782C2semidirect
5. #21446627C2semidirect
8. #21446253C2semidirect
9. #21445651C2semidirect
10. #21445643C2semidirect
11. C61×#1894C2semidirect
14. #16840298C4semidirect
271. #1894D61semidirect
315. C61#21411semidirect

Maximal subgroups

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

subgroupindex
C2×#168402982
C2×M16×Dic612
#214482142
#214467822
#214466272
#214462572
#214462552
#214462532
#214456512
#214456432
C61×#18942
#2141161