Finite Group Databasebeta

← #29685199

Subgroups — #29685199

Order 15616.

1226 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840291C211yesmaximal, characteristic
2C2×C4×C8×D61C211yesmaximal, characteristic, complemented
3#21448212C211yesmaximal, characteristic, complemented
4#21446784C211yesmaximal, characteristic, complemented
5#21446783C211yesmaximal, characteristic, complemented
6#21446631C211yesmaximal, characteristic, complemented
7#21446244C212yesmaximal
8#21446239C212yesmaximal, complemented
9#103×Dic61C212yesmaximal, complemented
10#102×Dic61C212yesmaximal, complemented
11C61×#1958C211yesmaximal, characteristic, complemented
12C22×C8×D61C2212yes—
13#16840291—24no—
14#16840291C418yescomplemented
15C2×#14519770—21no—
16C2×#14519769C2211yescharacteristic
17C2×#14519759—21no—
18C2×#14519753C2211yescharacteristic
19C2×#14519748C2211yescharacteristic
20C2×Q8×Dic61C2211yescharacteristic
21C2×D4×Dic61C2211yescharacteristic
22C2×#14519737—21no—
23C2×#14519733—21no—
24C2×C42×D61C2211yescharacteristic
25C2×#14519701C2212yes—
26C2×#14519698C2212yes—
27C2×C4×#92418C2211yescharacteristic
28C2×C8×Dic61C2212yes—
29C4×C8×D61—24no—
30#16839629—21no—
31#16839628C2211yescharacteristic
32#16839624—21no—
33#16839619C2212yes—
34C4⋊C4×Dic61C2212yes—
35#16839609—21no—
36#16839607—21no—
37#16839525C2212yes—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×#124×C61C2211yescharacteristic
42C2×#103×C61C2212yes—
43C2×#102×C61C2212yes—
44C2×C4×C8×C61C2211yescharacteristic
45C61×#382C2211yescharacteristic
46C22×C22⋊D61—21no—
47C23×Dic61—21no—
48C22×C4×D61D411yescharacteristic
49C22×C4×D61D411yescharacteristic
50C22×C4×D61C2311yescharacteristic
51C22×#92418C2312yes—
52C2×C8×D61—24no—
53C2×C8×D61—24no—
54C2×C8×D61—24no—
55#14519770—28no—
56#14519769—22no—
57#14519769C2×C414yes—
58#14519759—28no—
59#14519753—22no—
60#14519753C2×C414yes—
61#14519748—22no—
62#14519748C2×C414yes—
63Q8×Dic61—22no—
64Q8×Dic61C2×C414yes—
65D4×Dic61—22no—
66D4×Dic61C2×C414yes—
67#14519737—28no—
68#14519733—28no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61C2312yes—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61—21no—
78C2×#11758198—21no—
79C2×#11758198—22no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—22no—
82C2×C4×Dic61D411yescharacteristic
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61C2311yescharacteristic
85C42×D61C2×C414yes—
86C42×D61D414yes—
87#14519701—24no—
88#14519698—24no—
89C4×#92418—22no—
90C8×Dic61—24no—
91#14519684—21no—
92#14519684—21no—
93#14519684—22no—
94#14519684—22no—
95C22×Q8×C61C2311yescharacteristic
96C22×D4×C61C2311yescharacteristic
97C22×C8×C61C2312yes—
98#124×C61—22no—
99#124×C61C2×C414yes—
100C2×C4⋊C4×C61C2312yes—
101C2×C22⋊C4×C61—21no—
102C2×C22⋊C4×C61—21no—
103C2×C42×C61C2311yescharacteristic
104#103×C61—24no—
105#102×C61—24no—
106C4×C8×C61—22no—
107#95×C61—21no—
108#95×C61—21no—
109C24×C61—21no—
110C23×D61C2×D411yescharacteristic
111C2×Q8×C61—21no—
112C2×Q8×C61—22no—
113C2×Q8×C61—22no—
114C2×Q8×C61C22×C412yes—
115C2×D4×C61—21no—
116C2×D4×C61—22no—
117C2×D4×C61—22no—
118C2×D4×C61C22×C412yes—
119C22×C4×C61—21no—
120C22×C4×C61—22no—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2411yescharacteristic
124C2×C22⋊D61—24no—
125C2×C22⋊D61—44no—
126C22×Dic61—21no—
127C22×Dic61—21no—
128C22×Dic61—24no—
129C22×Dic61—22no—
130C22×Dic61—44no—
131C22×Dic61C2×D411yescharacteristic
132C22×Dic61C2×D411yescharacteristic
133C22×Dic61C2×D411yescharacteristic
134C2×C4×D61—24no—
135C2×C4×D61—22no—
136C2×C4×D61—22no—
137C2×C4×D61—24no—
138C2×C4×D61C22⋊C414yes—
139C2×C4×D61C22⋊C414yes—
140C2×C4×D61C22×C412yes—
141C2×C4×D61C2×D412yes—
142C2×C8×C61—22no—
143C2×C8×C61—22no—
144C2×C8×C61#4114yes—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—22no—
147C22⋊C4×C61—24no—
148C22⋊C4×C61—24no—
149C42×C61C22×C412yes—
150C42×C61C2×D412yes—
151C22⋊Dic61—24no—
152C22⋊Dic61—24no—
153#11758198—24no—
154#11758198—44no—
155C4⋊Dic61—21no—
156C4⋊Dic61—21no—
157C4⋊Dic61—24no—
158C4⋊Dic61—22no—
159C4⋊Dic61—22no—
160C4⋊Dic61C22×C412yes—
161C4⋊Dic61C22×C412yes—
162#11758196—24no—
163#11758196—24no—
164#11758196—24no—
165C4×Dic61—24no—
166C4×Dic61—44no—
167C4×Dic61C22⋊C414yes—
168C4×Dic61C22⋊C414yes—
169C4×Dic61C22×C412yes—
170C4×Dic61C2×D412yes—
171C2×#92418—22no—
172C2×#92418—22no—
173C2×#92418#4114yes—
174C8×D61—216no—
175C23×C61—22no—
176C23×C61—41no—
177C23×C61—41no—
178C23×C61—42no—
179C23×C61—42no—
180C23×C61C22×D411yescharacteristic
181C22×D61—21no—
182C22×D61—21no—
183C22×D61—22no—
184C22×D61C2×C22⋊C412yes—
185C22×D61#13514yes—
186Q8×C61—24no—
187Q8×C61—42no—
188D4×C61—24no—
189D4×C61—42no—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—41no—
194C2×C4×C61—41no—
195C2×C4×C61—42no—
196C2×C4×C61—42no—
197C2×C4×C61—42no—
198C2×C4×C61C2×C22⋊C412yes—
199C2×C4×C61C2×C22⋊C412yes—
200C2×C4×C61C23×C411yescharacteristic
201C2×C4×C61C22×D411yescharacteristic
202C2×C4×C61C2×#4112yes—
203C2×C4×C61C2×#4112yes—
204C22⋊D61—48no—
205C2×Dic61—21no—
206C2×Dic61—21no—
207C2×Dic61—22no—
208C2×Dic61—22no—
209C2×Dic61—22no—
210C2×Dic61—42no—
211C2×Dic61—42no—
212C2×Dic61—48no—
213C2×Dic61—42no—
214C2×Dic61—42no—
215C2×Dic61—42no—
216C2×Dic61C2×C22⋊C412yes—
217C2×Dic61C2×C22⋊C412yes—
218C2×Dic61C2×C22⋊C412yes—
219C2×Dic61#13514yes—
220C4×D61—28no—
221C4×D61—24no—
222C4×D61—24no—
223C4×D61—28no—
224C8×C61—24no—
225#92418—24no—
226#21402—611nomaximal, Sylow
227C22×C61—42no—
228C22×C61—42no—
229C22×C61—42no—
230C22×C61—82no—
231C22×C61—82no—
232C22×C61C22×C22⋊C411yescharacteristic
233C22×C61#53211yescharacteristic
234C22×C61#53211yescharacteristic
235C22×C61C2×#13512yes—
236C22×C61C2×#13512yes—
237C2×D61—22no—
238C2×D61—24no—
239C2×D61—24no—
240C2×D61—22no—
241C2×D61#41614yes—
242C4×C61—22no—
243C4×C61—22no—
244C4×C61—42no—
245C4×C61—82no—
246C4×C61C4×#4112yes—
247C4×C61#52512yes—
248Dic61—22no—
249Dic61—22no—
250Dic61—42no—
251Dic61—42no—
252Dic61—82no—
253Dic61#41614yes—
254C2×#529—611no—
255C22×C4×C8—611no—
256#2283—611no—
257#1991—611no—
258#1991—611no—
259#1958—611no—
260#1958D6111yescharacteristic, complemented
261#1893—612no—
262#1892—612no—
263C4×#103—612no—
264C4×#102—612no—
265C2×C61—82no—
266C2×C61#229711yescharacteristic
267C2×C61C2×#41611yescharacteristic
268C2×C61C2×#41611yescharacteristic
269C2×C61#316812yes—
270C2×C61#316812yes—
271D61—24no—
272C23×C8—612no—
273#529—618no—
274#529—1224no—
275C2×#126—1221no—
276C2×#125—611no—
277C2×#124C2×D6111yescharacteristic
278C2×#123—1221no—
279C2×C22⋊Q8—611no—
280C2×#121—611no—
281C2×C4×Q8—611no—
282C2×C4×D4—611no—
283C2×C4×D4—1221no—
284C2×#117—1221no—
285C22×C42—611no—
286C2×#103—612no—
287C2×#103C2×D6112yes—
288C2×#102—612no—
289C2×#102C2×D6112yes—
290C2×C4×C8—612no—
291C2×C4×C8—611no—
292C2×C4×C8—1224no—
293C2×C4×C8C2×D6111yescharacteristic
294#383—1221no—
295#382—611no—
296#382C2×D6111yescharacteristic
297#381—1221no—
298#380—612no—
299C4×C4⋊C4—612no—
300C4×C22⋊C4—1221no—
301#376—1221no—
302#340—612no—
303#340—612no—
304#340—612no—
305#340—612no—
306C61#2140211yescharacteristic, complemented, Sylow
307C22×Q8C22×D6111yescharacteristic
308C22×D4—1221no—
309C22×D4C22×D6111yescharacteristic
310C23×C4—611no—
311C23×C4—611no—
312C23×C4—611no—
313C23×C4—1221no—
314C22×C8—612no—
315C22×C8—1224no—
316C22×C8—1224no—
317C22×C8—1224no—
318C22×C8C22×D6112yes—
319#126—1228no—
320#125—614no—
321#125—1222no—
322#124—22no—
323#124C4×D6114yes—
324#123—1228no—
325C22⋊Q8—614no—
326C22⋊Q8—1222no—
327#121—614no—
328#121—1222no—
329C4×Q8—614no—
330C4×Q8—1222no—
331C4×D4—614no—
332C4×D4—1222no—
333C4×D4—1228no—
334#117—1228no—
335C2×C4⋊C4—611no—
336C2×C4⋊C4—611no—
337C2×C4⋊C4—612no—
338C2×C4⋊C4—1221no—
339C2×C4⋊C4—1221no—
340C2×C4⋊C4—1221no—
341C2×C4⋊C4—1221no—
342C2×C4⋊C4C22×D6112yes—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—1221no—
346C2×C22⋊C4—1221no—
347C2×C22⋊C4—1221no—
348C2×C22⋊C4—1222no—
349C2×C42—611no—
350C2×C42—611no—
351C2×C42—611no—
352C2×C42—614no—
353C2×C42—614no—
354C2×C42—1221no—
355C2×C42—1222no—
356C2×C42C22×D6111yescharacteristic
357#103—24no—
358#103—1224no—
359#102—24no—
360#102—1224no—
361C4×C8—22no—
362C4×C8—1224no—
363C4×C8—1222no—
364#95—21no—
365#95—21no—
366#95—1222no—
367#95—1222no—
368#95—1221no—
369#95—1221no—
370C24—21no—
371C24—611no—
372C2×Q8—21no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8C2×C4×D6112yes—
376C2×D4—21no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—1224no—
380C2×D4—2444no—
381C2×D4C2×C4×D6112yes—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—611no—
385C22×C4—611no—
386C22×C4—611no—
387C22×C4—612no—
388C22×C4—612no—
389C22×C4—614no—
390C22×C4—614no—
391C22×C4—1224no—
392C22×C4—1224no—
393C22×C4—1224no—
394C22×C4—1222no—
395C22×C4—1222no—
396C22×C4—1221no—
397C22×C4—1221no—
398C22×C4—1222no—
399C22×C4—2444no—
400C22×C4D4×D6111yescharacteristic
401C22×C4D4×D6111yescharacteristic
402C22×C4C23×D6111yescharacteristic
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—614no—
406C2×C8—12216no—
407C2×C8—1222no—
408C2×C8—1222no—
409C2×C8#1175822414yes—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—612no—
413C4⋊C4—612no—
414C4⋊C4—1224no—
415C4⋊C4—1224no—
416C4⋊C4—1221no—
417C4⋊C4—1221no—
418C4⋊C4—1224no—
419C4⋊C4—1224no—
420C4⋊C4—1222no—
421C4⋊C4—1222no—
422C22⋊C4—24no—
423C22⋊C4—24no—
424C22⋊C4—1224no—
425C22⋊C4—1224no—
426C22⋊C4—1224no—
427C22⋊C4—2444no—
428C42—612no—
429C42—612no—
430C42—614no—
431C42—614no—
432C42—1224no—
433C42—2444no—
434C42C2×C4×D6112yes—
435C42D4×D6112yes—
436C23—22no—
437C23—42no—
438C23—42no—
439C23—41no—
440C23—41no—
441C23—612no—
442C23—614no—
443C23—1222no—
444C23—1221no—
445C23—1221no—
446C23C2×D4×D6111yescharacteristic
447Q8—24no—
448Q8—42no—
449D4—24no—
450D4—42no—
451D4—2448no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—612no—
461C2×C4—614no—
462C2×C4—612no—
463C2×C4—612no—
464C2×C4—1228no—
465C2×C4—1228no—
466C2×C4—1222no—
467C2×C4—1221no—
468C2×C4—1221no—
469C2×C4—1224no—
470C2×C4—1224no—
471C2×C4—1222no—
472C2×C4—1222no—
473C2×C4—2448no—
474C2×C4—2442no—
475C2×C4—2442no—
476C2×C4—2442no—
477C2×C4—2442no—
478C2×C4—2442no—
479C2×C4C22⋊C4×D6112yes—
480C2×C4C22⋊C4×D6112yes—
481C2×C4C22×C4×D6111yescharacteristic
482C2×C4C2×D4×D6111yescharacteristic
483C2×C4#41×D6112yes—
484C2×C4C2×#1175822412yes—
485C8—24no—
486C8—1224no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22—82no—
491C22—82no—
492C22—614no—
493C22—1224no—
494C22—1224no—
495C22—1222no—
496C22—1222no—
497C22C2×C22⋊C4×D6111yescharacteristic
498C22#1684031011yescharacteristic
499C22#1684031111yescharacteristic
500C22#135×D6112yes—
501C22#135×D6112yes—
502C4—22no—
503C4—22no—
504C4—42no—
505C4—82no—
506C4—614no—
507C4—1222no—
508C4—1222no—
509C4—4882no—
510C4—2442no—
511C4—2442no—
512C4C4×#1175822412yes—
513C4#1684027212yes—
514C2—82no—
515C2—1224no—
516C2#2144830611yescharacteristic
517C2#416×D6111yescharacteristic
518C2#416×D6111yescharacteristic
519C2#2145638612yes—
520C2#2145638812yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1958C2
socleC23×C61C22×D4
prime radicalC2×#14519748C22
cosocleC22×C4×C61C24
hypercenter#1958D61
hypocenterC61#21402
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1958

Complements

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

NH ≅ G/Ntype
2. C2×C4×C8×D61C2semidirect
3. #21448212C2semidirect
4. #21446784C2semidirect
5. #21446783C2semidirect
6. #21446631C2semidirect
8. #21446239C2semidirect
9. #103×Dic61C2semidirect
10. #102×Dic61C2semidirect
11. C61×#1958C2semidirect
14. #16840291C4semidirect
260. #1958D61semidirect
306. C61#21402semidirect

Maximal subgroups

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

subgroupindex
C2×#168402912
C2×C4×C8×D612
#214482122
#214467842
#214467832
#214466312
#214462442
#214462392
#103×Dic612
#102×Dic612
C61×#19582
#2140261