Finite Group Databasebeta

← #29711977

Subgroups — #29711977

Order 15616.

1158 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840372C211yesmaximal, characteristic
2C2×#16839918C211yesmaximal, characteristic
3#21448405C211yesmaximal, characteristic
4#21446968C212yesmaximal, complemented
5#21446917C212yesmaximal, complemented
6#21446162C212yesmaximal, complemented
7#21446157C212yesmaximal, complemented
8#21446132C211yesmaximal, characteristic
9#21446129C211yesmaximal, characteristic
10#21445888C211yesmaximal, characteristic
11C2×C61×#451C211yesmaximal, characteristic
12C22×#11758202C2212yes—
13#16840372—24no—
14#16840372—24no—
15C2×#14519775C2211yescharacteristic
16C2×#14519770—21no—
17C2×#14519770—21no—
18C2×#14519768—21no—
19C2×#14519768C2211yescharacteristic
20C2×#14519767—21no—
21C2×#14519756C2211yescharacteristic
22C2×Q8×Dic61C2211yescharacteristic
23C2×#14519742C2211yescharacteristic
24C2×#14519736C2211yescharacteristic
25C2×#14519736C2211yescharacteristic
26#16839918—24no—
27C2×#14519702C2212yes—
28C2×#14519701C2212yes—
29C2×#14519701C2212yes—
30C2×C22⋊#92418C2211yescharacteristic
31#16839669—21no—
32#16839669—21no—
33#16839669C2212yes—
34#16839669C2212yes—
35#16839625—21no—
36#16839625—21no—
37#16839525C2212yes—
38#16839524C2212yes—
39#16839516C2212yes—
40#16839516C2212yes—
41C23×Q8×C61C2211yescharacteristic
42C22×Q16×C61C2212yes—
43C2×C22⋊Q8×C61C2211yescharacteristic
44C61×#451—24no—
45C2×#103×C61C2212yes—
46C2×C22⋊C8×C61C2211yescharacteristic
47C22×Dic122C2311yescharacteristic
48C2×#11758202—24no—
49C2×#11758202—24no—
50C2×#11758202—42no—
51C22×#92418C2312yes—
52#14519775—22no—
53#14519775—22no—
54#14519770—28no—
55#14519770—28no—
56#14519768—28no—
57#14519768—22no—
58#14519768—22no—
59#14519767—28no—
60#14519756—22no—
61#14519756D414yes—
62Q8×Dic61—22no—
63Q8×Dic61—22no—
64#14519742—22no—
65#14519742—22no—
66#14519736—22no—
67#14519736—22no—
68#14519736—42no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61D411yescharacteristic
72C2×C4⋊Dic61D411yescharacteristic
73C2×C4⋊Dic61C2312yes—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61—21no—
82C2×C22⋊Dic61D411yescharacteristic
83C2×C22⋊Dic61D411yescharacteristic
84C2×C4×Dic61—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61C2311yescharacteristic
88C2×C4×Dic61C2311yescharacteristic
89#14519702—42no—
90#14519701—24no—
91#14519701—42no—
92C22⋊#92418—22no—
93#14519684—22no—
94#14519684—22no—
95#14519684—22no—
96#14519684—22no—
97C22×Q8×C61—24no—
98C22×Q8×C61—21no—
99C22×Q8×C61—22no—
100C22×Q8×C61—22no—
101C22×Q8×C61—42no—
102C22×Q8×C61C2311yescharacteristic
103C22×Q8×C61C2312yes—
104C23×C4×C61—21no—
105C23×C4×C61C2311yescharacteristic
106C2×Q16×C61—24no—
107C2×Q16×C61—24no—
108C2×Q16×C61—42no—
109C22×C8×C61C2312yes—
110C22⋊Q8×C61—22no—
111C22⋊Q8×C61D414yes—
112C2×C4⋊C4×C61—21no—
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C22⋊C4×C61—21no—
115#103×C61—24no—
116C22⋊C8×C61—22no—
117C24×C61C2×D411yescharacteristic
118C2×Q8×C61—24no—
119C2×Q8×C61—21no—
120C2×Q8×C61—22no—
121C2×Q8×C61—22no—
122C2×Q8×C61—22no—
123C2×Q8×C61—22no—
124C2×Q8×C61—22no—
125C2×Q8×C61—22no—
126C2×Q8×C61—22no—
127C2×Q8×C61—48no—
128C2×Q8×C61—48no—
129C2×Q8×C61—42no—
130C2×Q8×C61—42no—
131C2×Q8×C61C2×D412yes—
132C2×Q8×C61#4114yes—
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—24no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—22no—
139C22×C4×C61—42no—
140C22×C4×C61—42no—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D412yes—
143C22×C4×C61C2×D412yes—
144C22×C4×C61C2411yescharacteristic
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—22no—
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152C2×Dic122—22no—
153C2×Dic122—22no—
154C2×Dic122—21no—
155C2×Dic122C2×D412yes—
156Q16×C61—48no—
157C2×C8×C61—22no—
158C2×C8×C61—22no—
159C2×C8×C61—22no—
160C4⋊C4×C61—24no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61C2×D412yes—
163C22⋊C4×C61—24no—
164C22⋊Dic61—24no—
165C22⋊Dic61—22no—
166C22⋊Dic61—22no—
167C22⋊Dic61—42no—
168#11758202—48no—
169C4⋊Dic61—24no—
170C4⋊Dic61—24no—
171C4⋊Dic61—22no—
172C4⋊Dic61—22no—
173C4⋊Dic61—21no—
174C4⋊Dic61—22no—
175C4⋊Dic61—22no—
176C4⋊Dic61C2×D412yes—
177#11758196—24no—
178#11758196—24no—
179#11758196—24no—
180#11758196—24no—
181#11758196—24no—
182C4×Dic61—24no—
183C4×Dic61—24no—
184C4×Dic61—21no—
185C4×Dic61—21no—
186C4×Dic61—21no—
187C4×Dic61—21no—
188C4×Dic61—42no—
189C2×#92418—22no—
190C2×#92418—22no—
191C2×#92418—22no—
192C23×C61—21no—
193C23×C61—21no—
194C23×C61—22no—
195C23×C61—22no—
196C23×C61#12012yes—
197C23×C61C22×D411yescharacteristic
198Q8×C61—24no—
199Q8×C61—24no—
200Q8×C61—24no—
201Q8×C61—48no—
202Q8×C61—42no—
203Q8×C61—42no—
204Q8×C61—42no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—24no—
208C2×C4×C61—24no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—24no—
212C2×C4×C61—48no—
213C2×C4×C61—42no—
214C2×C4×C61—41no—
215C2×C4×C61—42no—
216C2×C4×C61—41no—
217C2×C4×C61—42no—
218C2×C4×C61#12012yes—
219C2×C4×C61C22×D411yescharacteristic
220C2×C4×C61C22×D411yescharacteristic
221C2×C4×C61C2×#4112yes—
222C2×C4×C61H(5,𝔽2)12yes—
223C2×Dic61—21no—
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—22no—
227C2×Dic61—21no—
228C2×Dic61—22no—
229C2×Dic61—22no—
230C2×Dic61—22no—
231C2×Dic61—22no—
232C2×Dic61—22no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—42no—
236C2×Dic61—42no—
237C2×Dic61—42no—
238C2×Dic61—42no—
239Dic122—24no—
240Dic122—42no—
241C8×C61—42no—
242#92418—42no—
243#23943—611nomaximal, Sylow
244C22×C61—22no—
245C22×C61—22no—
246C22×C61—22no—
247C22×C61—42no—
248C22×C61—42no—
249C22×C61C2×#12011yescharacteristic
250C22×C61#52511yescharacteristic
251C22×C61#53511yescharacteristic
252C22×C61#57812yes—
253C22×C61#57812yes—
254C4×C61—22no—
255C4×C61—44no—
256C4×C61—42no—
257C4×C61—42no—
258C4×C61#54612yes—
259C4×C61#54612yes—
260Dic61—42no—
261Dic61—41no—
262Dic61—41no—
263Dic61—41no—
264Dic61—41no—
265Dic61—42no—
266Dic61—82no—
267C2×#541—611no—
268C2×#451—611no—
269C2×#451D6111yescharacteristic
270#2310—611no—
271#2026—612no—
272#2016—612no—
273#1878—612no—
274#1878—612no—
275#1873—611no—
276#1873—611no—
277#1818—611no—
278C2×C61—42no—
279C2×C61#240411yescharacteristic
280C2×C61#301111yescharacteristic
281C2×C61#301111yescharacteristic
282C2×C61#312712yes—
283C2×C61#322312yes—
284C23×Q8C2×D6111yescharacteristic
285C22×Q16—612no—
286C22×Q16C2×D6112yes—
287#541—1224no—
288#541—1224no—
289C2×C4⋊Q8—611no—
290C2×#126—1221no—
291C2×#126—1221no—
292C2×#125—611no—
293C2×#125—1221no—
294C2×#125—1221no—
295C2×C22⋊Q8—611no—
296C2×C22⋊Q8C2×D6111yescharacteristic
297C2×C4×Q8—611no—
298C2×C4×Q8—611no—
299C2×#117—611no—
300C2×#117—611no—
301#451—24no—
302#451—1224no—
303C2×#103—612no—
304C2×#103—612no—
305C2×#103—612no—
306C2×#103C2×D6112yes—
307C2×C22⋊C8—611no—
308C2×C22⋊C8C2×D6111yescharacteristic
309#391—612no—
310#391—612no—
311#391—1221no—
312#391—1221no—
313#381—1221no—
314#381—1221no—
315#340—612no—
316#340—612no—
317#336—612no—
318#336—612no—
319C61#2394311yescharacteristic, complemented, Sylow
320C22×Q8—24no—
321C22×Q8—21no—
322C22×Q8—22no—
323C22×Q8—22no—
324C22×Q8—42no—
325C22×Q8—611no—
326C22×Q8C22×D6112yes—
327C22×Q8C22×D6111yescharacteristic
328C23×C4—21no—
329C23×C4C22×D6111yescharacteristic
330C2×Q16—24no—
331C2×Q16—24no—
332C2×Q16—42no—
333C2×Q16—1224no—
334C2×Q16—1224no—
335C2×Q16—2442no—
336C22×C8—612no—
337C22×C8C22×D6112yes—
338C4⋊Q8—1222no—
339C4⋊Q8—1222no—
340#126—1228no—
341#126—1228no—
342#125—1222no—
343#125—1228no—
344#125—1228no—
345#125—1222no—
346C22⋊Q8—22no—
347C22⋊Q8—614no—
348C22⋊Q8—1222no—
349C22⋊Q8C22⋊D6114yes—
350C4×Q8—1222no—
351C4×Q8—1222no—
352C4×Q8—1222no—
353C4×Q8—1222no—
354#117—1222no—
355#117—1222no—
356#117—2442no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—611no—
359C2×C4⋊C4—611no—
360C2×C4⋊C4—611no—
361C2×C4⋊C4—612no—
362C2×C4⋊C4—1221no—
363C2×C4⋊C4—1221no—
364C2×C4⋊C4—1221no—
365C2×C4⋊C4—1221no—
366C2×C4⋊C4—1221no—
367C2×C4⋊C4—1221no—
368C2×C4⋊C4—1221no—
369C2×C4⋊C4C22×D6111yescharacteristic
370C2×C22⋊C4—21no—
371C2×C22⋊C4—611no—
372C2×C22⋊C4—611no—
373C2×C22⋊C4—1221no—
374C2×C22⋊C4—1221no—
375C2×C42—611no—
376C2×C42—611no—
377C2×C42—1221no—
378C2×C42—1221no—
379C2×C42—1221no—
380#103—24no—
381#103—1224no—
382#103—2442no—
383#103—2442no—
384C22⋊C8—22no—
385C22⋊C8—1222no—
386#95—1222no—
387#95—1222no—
388#95—1222no—
389#95—1222no—
390C24D4×D6111yescharacteristic
391C2×Q8—24no—
392C2×Q8—21no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—48no—
401C2×Q8—48no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—612no—
405C2×Q8—1221no—
406C2×Q8—1222no—
407C2×Q8—1222no—
408C2×Q8#1175822414yes—
409C2×Q8C2×C22⋊D6112yes—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—611no—
419C22×C4—611no—
420C22×C4—611no—
421C22×C4—611no—
422C22×C4—1221no—
423C22×C4—1221no—
424C22×C4—1222no—
425C22×C4D4×D6112yes—
426C22×C4D4×D6111yescharacteristic
427C22×C4C2×C22⋊D6112yes—
428C22×C4C23×D6111yescharacteristic
429Q16—48no—
430Q16—2448no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—1222no—
435C2×C8—1222no—
436C2×C8—1222no—
437C4⋊C4—21no—
438C4⋊C4—24no—
439C4⋊C4—612no—
440C4⋊C4—1224no—
441C4⋊C4—1224no—
442C4⋊C4—1224no—
443C4⋊C4—1224no—
444C4⋊C4—1224no—
445C4⋊C4—1224no—
446C4⋊C4—1221no—
447C4⋊C4—1222no—
448C4⋊C4—1222no—
449C4⋊C4—1224no—
450C4⋊C4—1222no—
451C4⋊C4—1222no—
452C4⋊C4C2×C22⋊D6112yes—
453C22⋊C4—24no—
454C22⋊C4—1222no—
455C22⋊C4—1222no—
456C22⋊C4—1224no—
457C22⋊C4—2442no—
458C42—1224no—
459C42—1224no—
460C42—1221no—
461C42—1221no—
462C42—1221no—
463C42—1221no—
464C42—2442no—
465C23—21no—
466C23—22no—
467C23—22no—
468C23—21no—
469C23#1451974512yes—
470C23C2×D4×D6111yescharacteristic
471Q8—24no—
472Q8—24no—
473Q8—24no—
474Q8—48no—
475Q8—42no—
476Q8—42no—
477Q8—42no—
478Q8—1224no—
479Q8—2442no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—48no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—1221no—
494C2×C4—1221no—
495C2×C4—1222no—
496C2×C4—1221no—
497C2×C4—1221no—
498C2×C4—1222no—
499C2×C4—1222no—
500C2×C4—1222no—
501C2×C4—1222no—
502C2×C4—1222no—
503C2×C4—2442no—
504C2×C4—2442no—
505C2×C4—2442no—
506C2×C4—2441no—
507C2×C4—2442no—
508C2×C4—2441no—
509C2×C4#1451974512yes—
510C2×C4C2×D4×D6111yescharacteristic
511C2×C4C22×C22⋊D6111yescharacteristic
512C2×C4C2×#1175822412yes—
513C2×C4#1451982212yes—
514C8—42no—
515C8—2442no—
516C22—22no—
517C22—22no—
518C22—22no—
519C22—42no—
520C22—42no—
521C22C2×#1451974511yescharacteristic
522C22#1684026811yescharacteristic
523C22#1684032711yescharacteristic
524C22#1684057512yes—
525C22#1684057812yes—
526C4—22no—
527C4—42no—
528C4—44no—
529C4—42no—
530C4—4882no—
531C4—2441no—
532C4—2441no—
533C4—2442no—
534C4—2441no—
535C4—2441no—
536C4—2442no—
537C4#1684039912yes—
538C4#1684040612yes—
539C2—42no—
540C2#2144913511yescharacteristic
541C2#2145511011yescharacteristic
542C2#2145511311yescharacteristic
543C2#2145611312yes—
544C2#2145689812yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#451C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenterC2×#451D61
hypocenterC61#23943
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#451

Complements

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

NH ≅ G/Ntype
4. #21446968C2semidirect
5. #21446917C2semidirect
6. #21446162C2semidirect
7. #21446157C2semidirect
319. C61#23943semidirect

Maximal subgroups

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

subgroupindex
C2×#168403722
C2×#168399182
#214484052
#214469682
#214469172
#214461622
#214461572
#214461322
#214461292
#214458882
C2×C61×#4512
#2394361