Finite Group Databasebeta

← #29734160

Subgroups — #29734160

Order 15616.

1092 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840121C211yesmaximal, characteristic, complemented
2C2×#16840042C211yesmaximal, characteristic, complemented
3#21451024C211yesmaximal, characteristic, complemented
4#21447055C212yesmaximal, complemented
5#21447050C212yesmaximal
6#21446760C211yesmaximal, characteristic, complemented
7#21446752C211yesmaximal, characteristic, complemented
8#21446073C212yesmaximal, complemented
9#21446066C212yesmaximal, complemented
10#21445644C211yesmaximal, characteristic, complemented
11C2×C61×#559C211yesmaximal, characteristic
12C2×#14519777C2211yescharacteristic
13#16840121—24no—
14#16840042—24no—
15C2×#14519710C2212yes—
16C2×C4⋊#92418C2211yescharacteristic
17C2×#14519701C2212yes—
18C2×#14519701C2212yes—
19C2×#14519698C2212yes—
20C2×#14519698C2212yes—
21C2×#14519698C2212yes—
22C2×#14519698C2212yes—
23C2×#14519688C2211yescharacteristic
24#16839682—21no—
25#16839682—21no—
26#16839669—21no—
27#16839660C2212yes—
28#16839660C2212yes—
29#16839631—21no—
30C4×C4⋊Dic61C2211yescharacteristic
31#16839524C2212yes—
32#16839517C2212yes—
33#16839517C2212yes—
34C61×#559—24no—
35C61×#559—24no—
36C2×#127×C61C2211yescharacteristic
37C2×#124×C61—21no—
38C2×#124×C61C2211yescharacteristic
39C2×#124×C61C2211yescharacteristic
40C2×#123×C61—21no—
41C2×#123×C61—21no—
42C2×C4×Q8×C61C2211yescharacteristic
43C2×C4×Q8×C61C2211yescharacteristic
44C2×C4×D4×C61—21no—
45C2×C4×D4×C61C2211yescharacteristic
46C2×C4×D4×C61C2211yescharacteristic
47C22×Dic122—21no—
48C22×#92418C2312yes—
49C22×#92418C2312yes—
50#14519777—22no—
51#14519777—22no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61C2312yes—
54C2×C4⋊Dic61C2312yes—
55C2×#11758196—22no—
56C2×#11758196—22no—
57C2×C4×Dic61—21no—
58#14519710—24no—
59C4⋊#92418—22no—
60#14519701—24no—
61#14519701—24no—
62#14519698—24no—
63#14519698—24no—
64#14519698—24no—
65#14519698—24no—
66#14519688—22no—
67#14519684—21no—
68#14519684—22no—
69#14519684—22no—
70C22×Q8×C61D411yescharacteristic
71C22×Q8×C61C2311yescharacteristic
72C22×D4×C61—21no—
73C22×D4×C61—21no—
74C22×D4×C61C2311yescharacteristic
75C22×D4×C61C2311yescharacteristic
76C23×C4×C61—21no—
77C23×C4×C61—21no—
78#127×C61—22no—
79#127×C61—22no—
80#124×C61—28no—
81#124×C61—22no—
82#124×C61—22no—
83#124×C61—22no—
84#124×C61D414yes—
85#123×C61—28no—
86#123×C61—28no—
87C4×Q8×C61—22no—
88C4×Q8×C61—22no—
89C4×Q8×C61—22no—
90C4×Q8×C61—22no—
91C4×D4×C61—28no—
92C4×D4×C61—22no—
93C4×D4×C61—22no—
94C4×D4×C61—22no—
95C4×D4×C61D414yes—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61D411yescharacteristic
98C2×C4⋊C4×C61D411yescharacteristic
99C2×C4⋊C4×C61D411yescharacteristic
100C2×C4⋊C4×C61C2311yescharacteristic
101C2×C22⋊C4×C61—21no—
102C2×C22⋊C4×C61—21no—
103C2×C22⋊C4×C61—21no—
104C2×C22⋊C4×C61—21no—
105C2×C22⋊C4×C61—21no—
106C2×C22⋊C4×C61—21no—
107C2×C42×C61—21no—
108C2×C42×C61C2311yescharacteristic
109C2×C42×C61C2311yescharacteristic
110C2×C42×C61C2311yescharacteristic
111C24×C61—21no—
112C24×C61—21no—
113C2×Q8×C61—22no—
114C2×Q8×C61—22no—
115C2×Q8×C61—21no—
116C2×Q8×C61—22no—
117C2×Q8×C61—22no—
118C2×Q8×C61—41no—
119C2×Q8×C61C2×D412yes—
120C2×D4×C61—24no—
121C2×D4×C61—24no—
122C2×D4×C61—21no—
123C2×D4×C61—21no—
124C2×D4×C61—22no—
125C2×D4×C61—22no—
126C2×D4×C61—22no—
127C2×D4×C61—22no—
128C2×D4×C61—42no—
129C2×D4×C61—42no—
130C2×D4×C61—82no—
131C2×D4×C61C2×D412yes—
132C2×D4×C61C2×D412yes—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—24no—
136C22×C4×C61—24no—
137C22×C4×C61—82no—
138C22×C4×C61—82no—
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2411yescharacteristic
146C22×Dic61—22no—
147C22×Dic61—22no—
148C2×Dic122—42no—
149C2×Dic122—82no—
150C4⋊C4×C61—24no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61C2×D412yes—
156C22⋊C4×C61—24no—
157C22⋊C4×C61—24no—
158C22⋊C4×C61—24no—
159C22⋊C4×C61—24no—
160C22⋊C4×C61—24no—
161C22⋊C4×C61—24no—
162C42×C61—24no—
163C42×C61—21no—
164C42×C61—21no—
165C42×C61—21no—
166C42×C61—21no—
167C42×C61C2×D412yes—
168C42×C61C2×D412yes—
169C4⋊Dic61—22no—
170C4⋊Dic61—22no—
171C4⋊Dic61—22no—
172C4⋊Dic61—42no—
173C4⋊Dic61#4114yes—
174#11758196—82no—
175#11758196—82no—
176C4×Dic61—42no—
177C2×#92418—22no—
178C2×#92418—22no—
179C2×#92418—22no—
180C2×#92418—22no—
181C2×#92418—22no—
182C2×#92418—22no—
183C23×C61—22no—
184C23×C61—22no—
185C23×C61—42no—
186C23×C61—42no—
187C23×C61—41no—
188C23×C61—41no—
189C23×C61—41no—
190C23×C61—41no—
191C23×C61—42no—
192C23×C61—42no—
193C23×C61C22×D411yescharacteristic
194Q8×C61—24no—
195Q8×C61—44no—
196Q8×C61—41no—
197Q8×C61—41no—
198D4×C61—22no—
199D4×C61—22no—
200D4×C61—24no—
201D4×C61—48no—
202D4×C61—41no—
203D4×C61—41no—
204D4×C61—42no—
205D4×C61—84no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—22no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61—84no—
223C2×C4×C61—84no—
224C2×C4×C61#12012yes—
225C2×C4×C61#12012yes—
226C2×C4×C61#13712yes—
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C2×#4112yes—
230C2×C4×C612−1+412yes—
231C2×Dic61—24no—
232C2×Dic61—42no—
233C2×Dic61—42no—
234C2×Dic61—42no—
235C2×Dic61—42no—
236C2×Dic61—42no—
237Dic122—84no—
238#92418—42no—
239#92418—42no—
240#25660—611nomaximal, Sylow
241C22×C61—42no—
242C22×C61—42no—
243C22×C61—42no—
244C22×C61—42no—
245C22×C61—42no—
246C22×C61—42no—
247C22×C61—82no—
248C22×C61—82no—
249C22×C61—82no—
250C22×C61—82no—
251C22×C61C2×#12011yescharacteristic
252C22×C61#54812yes—
253C22×C61C2×#13711yescharacteristic
254C22×C61#57511yescharacteristic
255C22×C61#57612yes—
256C4×C61—22no—
257C4×C61—22no—
258C4×C61—22no—
259C4×C61—42no—
260C4×C61—41no—
261C4×C61—41no—
262C4×C61—41no—
263C4×C61—42no—
264C4×C61#53211yescharacteristic
265C4×C61#53611yescharacteristic
266C4×C61#54712yes—
267Dic61—44no—
268Dic61—82no—
269C2×#559D6111yescharacteristic
270C2×#489—611no—
271C2×#474—611no—
272#2567—611no—
273#2042—612no—
274#2041—612no—
275#1985—611no—
276#1983—611no—
277#1863—612no—
278#1862—612no—
279#1762—611no—
280C2×C61—82no—
281C2×C61—82no—
282C2×C61#243211yescharacteristic
283C2×C61#300711yescharacteristic
284C2×C61#300811yescharacteristic
285C2×C61#337012yes—
286C2×C61#337512yes—
287#559—24no—
288#559—24no—
289C2×C4⋊Q8—611no—
290C2×#127C2×D6111yescharacteristic
291C2×#124—21no—
292C2×#124C2×D6111yescharacteristic
293C2×#124C2×D6111yescharacteristic
294C2×#123—21no—
295C2×#123—21no—
296C2×C4×Q8C2×D6111yescharacteristic
297C2×C4×Q8C2×D6111yescharacteristic
298C2×C4×D4—21no—
299C2×C4×D4C2×D6111yescharacteristic
300C2×C4×D4C2×D6111yescharacteristic
301#489—1224no—
302#474—1224no—
303C2×#107—612no—
304C2×C4⋊C8—611no—
305C2×#103—612no—
306C2×#103—612no—
307C2×#102—612no—
308C2×#102—612no—
309C2×#102—612no—
310C2×#102—612no—
311C2×#97—611no—
312#395—1221no—
313#395—1221no—
314#391—1221no—
315#389—612no—
316#389—612no—
317#384—1221no—
318C4×C4⋊C4—611no—
319#340—612no—
320#336—612no—
321#336—612no—
322C61#2566011yescharacteristic, complemented, Sylow
323C22×Q8—1221no—
324C22×Q8C22⋊D6111yescharacteristic
325C22×Q8C22×D6111yescharacteristic
326C22×D4—21no—
327C22×D4—21no—
328C22×D4C22×D6111yescharacteristic
329C22×D4C22×D6111yescharacteristic
330C23×C4—21no—
331C23×C4—21no—
332C22×C8—612no—
333C22×C8—612no—
334C4⋊Q8—1222no—
335C4⋊Q8—1222no—
336#127—22no—
337#127—22no—
338#124—28no—
339#124—22no—
340#124—22no—
341#124—22no—
342#124C22⋊D6114yes—
343#123—28no—
344#123—28no—
345C4×Q8—22no—
346C4×Q8—22no—
347C4×Q8—22no—
348C4×Q8—22no—
349C4×D4—28no—
350C4×D4—22no—
351C4×D4—22no—
352C4×D4—22no—
353C4×D4C22⋊D6114yes—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—612no—
356C2×C4⋊C4—612no—
357C2×C4⋊C4—1222no—
358C2×C4⋊C4—1222no—
359C2×C4⋊C4—1221no—
360C2×C4⋊C4C22⋊D6111yescharacteristic
361C2×C4⋊C4C22⋊D6111yescharacteristic
362C2×C4⋊C4C22⋊D6111yescharacteristic
363C2×C4⋊C4C22×D6111yescharacteristic
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—21no—
370C2×C42—21no—
371C2×C42—1221no—
372C2×C42C22×D6111yescharacteristic
373C2×C42C22×D6111yescharacteristic
374C2×C42C22×D6111yescharacteristic
375#107—1224no—
376C4⋊C8—1222no—
377#103—1224no—
378#103—1224no—
379#102—1224no—
380#102—1224no—
381#102—1224no—
382#102—1224no—
383#97—1222no—
384#95—1221no—
385#95—1222no—
386#95—1222no—
387C24—21no—
388C24—21no—
389C2×Q8—21no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—41no—
395C2×Q8—4882no—
396C2×Q8—2442no—
397C2×Q8C2×C22⋊D6112yes—
398C2×D4—24no—
399C2×D4—24no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—82no—
409C2×D4C2×C22⋊D6112yes—
410C2×D4C2×C22⋊D6112yes—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—82no—
416C22×C4—82no—
417C22×C4—1222no—
418C22×C4—1222no—
419C22×C4C2×C22⋊D6111yescharacteristic
420C22×C4C2×C22⋊D6111yescharacteristic
421C22×C4C2×C22⋊D6111yescharacteristic
422C22×C4C2×C22⋊D6111yescharacteristic
423C22×C4C2×C22⋊D6111yescharacteristic
424C22×C4C2×C22⋊D6111yescharacteristic
425C22×C4C23×D6111yescharacteristic
426C2×C8—1222no—
427C2×C8—1222no—
428C2×C8—1222no—
429C2×C8—1222no—
430C2×C8—1222no—
431C2×C8—1222no—
432C4⋊C4—24no—
433C4⋊C4—21no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—614no—
438C4⋊C4—1222no—
439C4⋊C4—1222no—
440C4⋊C4—1222no—
441C4⋊C4—4882no—
442C4⋊C4—4882no—
443C4⋊C4—2442no—
444C4⋊C4C2×C22⋊D6112yes—
445C22⋊C4—24no—
446C22⋊C4—24no—
447C22⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C42—24no—
452C42—21no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42—2442no—
457C42C2×C22⋊D6112yes—
458C42C2×C22⋊D6112yes—
459C23—22no—
460C23—22no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23C22×C22⋊D6111yescharacteristic
470Q8—24no—
471Q8—44no—
472Q8—41no—
473Q8—41no—
474Q8—4884no—
475D4—22no—
476D4—22no—
477D4—24no—
478D4—48no—
479D4—41no—
480D4—41no—
481D4—42no—
482D4—84no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—84no—
500C2×C4—84no—
501C2×C4—1224no—
502C2×C4—2442no—
503C2×C4—2442no—
504C2×C4—2442no—
505C2×C4—2442no—
506C2×C4—2442no—
507C2×C4#1451974612yes—
508C2×C4#1451974612yes—
509C2×C4#1451980712yes—
510C2×C4C22×C22⋊D6111yescharacteristic
511C2×C4C22×C22⋊D6111yescharacteristic
512C2×C4#41×D6112yes—
513C2×C4#1451982412yes—
514C8—2442no—
515C8—2442no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—82no—
523C22—82no—
524C22—82no—
525C22—82no—
526C22C2×#1451974611yescharacteristic
527C22#1684041912yes—
528C22C2×#1451980711yescharacteristic
529C22#1684056011yescharacteristic
530C22#1684056512yes—
531C4—22no—
532C4—22no—
533C4—22no—
534C4—42no—
535C4—41no—
536C4—41no—
537C4—41no—
538C4—42no—
539C4—4882no—
540C4—2444no—
541C4#1684030811yescharacteristic
542C4#1684033311yescharacteristic
543C4#1684041112yes—
544C2—82no—
545C2—82no—
546C2#2144944311yescharacteristic
547C2#2145507911yescharacteristic
548C2#2145508711yescharacteristic
549C2#2145856612yes—
550C2#2145862912yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#559C2
socleC23×C61C22×D4
prime radicalC2×#127×C61C22
hypercenterC2×#559D61
hypocenterC61#25660
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#559

Complements

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

NH ≅ G/Ntype
1. C2×#16840121C2semidirect
2. C2×#16840042C2semidirect
3. #21451024C2semidirect
4. #21447055C2semidirect
6. #21446760C2semidirect
7. #21446752C2semidirect
8. #21446073C2semidirect
9. #21446066C2semidirect
10. #21445644C2semidirect
322. C61#25660semidirect

Maximal subgroups

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

subgroupindex
C2×#168401212
C2×#168400422
#214510242
#214470552
#214470502
#214467602
#214467522
#214460732
#214460662
#214456442
C2×C61×#5592
#2566061