Finite Group Databasebeta

← #29714092

Subgroups — #29714092

Order 15616.

1336 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840054C211yesmaximal, characteristic, complemented
2C2×#16839862C211yesmaximal, characteristic, complemented
3#21450728C211yesmaximal, characteristic, complemented
4#21446998C212yesmaximal, complemented
5#21446822C212yesmaximal, complemented
6#21446793C211yesmaximal, characteristic, complemented
7#21446618C211yesmaximal, characteristic, complemented
8#21446126C212yesmaximal, complemented
9#21446007C212yesmaximal, complemented
10#21445829C211yesmaximal, characteristic, complemented
11C2×C61×#547C211yesmaximal, characteristic
12C22×#11758200C2211yescharacteristic
13C2×#14519756C2212yes—
14C2×C4×Dic122C2211yescharacteristic
15#16840054—24no—
16#16839862—24no—
17C2×#14519710C2211yescharacteristic
18C2×#14519710C2212yes—
19C2×C4⋊#92418C2211yescharacteristic
20C2×#14519700C2211yescharacteristic
21C2×#14519700C2212yes—
22C2×#14519698C2211yescharacteristic
23C2×#14519698C2212yes—
24C2×C22⋊#92418C2212yes—
25C2×C22⋊#92418C2212yes—
26C2×#14519688C2211yescharacteristic
27#16839693—21no—
28#16839676—21no—
29#16839650—22no—
30#16839628C2211yescharacteristic
31#16839622C2212yes—
32#16839524C2212yes—
33#16839524C2212yes—
34#16839524C2212yes—
35C22×#41×C61—21no—
36C61×#547—24no—
37C61×#547—24no—
38C2×C4⋊Q8×C61C2211yescharacteristic
39C2×#123×C61—22no—
40C2×C22⋊Q8×C61—21no—
41C2×#121×C61C2212yes—
42C2×C4×D4×C61C2211yescharacteristic
43C2×C4×D4×C61C2211yescharacteristic
44C22×C4⋊C4×C61C2212yes—
45C22×Dic122C2311yescharacteristic
46C2×#11758200—22no—
47C2×#11758200—22no—
48C2×#11758200—22no—
49C22×#92418C2311yescharacteristic
50C22×#92418C2312yes—
51C22×#92418C2311yescharacteristic
52#14519756—24no—
53#14519756—42no—
54C4×Dic122—22no—
55C4×Dic122—22no—
56C2×C4⋊Dic61C2311yescharacteristic
57C2×C4⋊Dic61C2312yes—
58C2×#11758196—21no—
59C2×#11758196—22no—
60C2×C22⋊Dic61—22no—
61C2×C22⋊Dic61—22no—
62C2×C4×Dic61—21no—
63#14519710—24no—
64#14519710—22no—
65C4⋊#92418—22no—
66#14519700—24no—
67#14519700—22no—
68#14519698—22no—
69#14519698—42no—
70C22⋊#92418—24no—
71C22⋊#92418—42no—
72#14519688—22no—
73#14519684—22no—
74#14519684—21no—
75#14519684—21no—
76C2×#41×C61—28no—
77C2×#41×C61—48no—
78C22×Q8×C61—21no—
79C22×D4×C61—22no—
80C22×D4×C61D411yescharacteristic
81C22×D4×C61C2311yescharacteristic
82C23×C4×C61—22no—
83C23×C4×C61—21no—
84C23×C4×C61C2312yes—
85C4⋊Q8×C61—22no—
86C4⋊Q8×C61D414yes—
87#123×C61—216no—
88C22⋊Q8×C61—28no—
89#121×C61—24no—
90#121×C61—24no—
91C4×D4×C61—22no—
92C4×D4×C61—22no—
93C4×D4×C61—22no—
94C4×D4×C61D414yes—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61—22no—
97C2×C4⋊C4×C61—24no—
98C2×C4⋊C4×C61—24no—
99C2×C4⋊C4×C61D411yescharacteristic
100C2×C4⋊C4×C61C2312yes—
101C2×C4⋊C4×C61C2311yescharacteristic
102C2×C22⋊C4×C61—21no—
103C2×C22⋊C4×C61—22no—
104C2×C22⋊C4×C61D412yes—
105C2×C42×C61C2311yescharacteristic
106C24×C61—21no—
107C24×C61C2×D412yes—
108#41×C61—432no—
109C2×Q8×C61—24no—
110C2×Q8×C61—44no—
111C2×D4×C61—24no—
112C2×D4×C61—28no—
113C2×D4×C61—22no—
114C2×D4×C61—21no—
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61—21no—
119C2×D4×C61—44no—
120C2×D4×C61—44no—
121C2×D4×C61C2×D412yes—
122C22×C4×C61—28no—
123C22×C4×C61—21no—
124C22×C4×C61—22no—
125C22×C4×C61—24no—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61—44no—
133C22×C4×C61—44no—
134C22×C4×C61—44no—
135C22×C4×C61C2×D412yes—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2411yescharacteristic
139C22×Dic61—21no—
140C22×Dic61—22no—
141C22×Dic61—21no—
142C2×Dic122—21no—
143C2×Dic122—21no—
144C2×Dic122—21no—
145C2×Dic122—21no—
146C2×Dic122—21no—
147C2×Dic122#4112yescomplemented
148C4⋊C4×C61—28no—
149C4⋊C4×C61—24no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61C2×D412yes—
154C4⋊C4×C61C2×D414yes—
155C22⋊C4×C61—28no—
156C22⋊C4×C61—24no—
157C22⋊C4×C61—24no—
158C42×C61C2×D412yes—
159C42×C61C2×D412yes—
160C22⋊Dic61—44no—
161C22⋊Dic61—44no—
162#11758200—28no—
163C4⋊Dic61—21no—
164C4⋊Dic61—21no—
165C4⋊Dic61—22no—
166C4⋊Dic61—22no—
167#11758196—44no—
168#11758196—42no—
169C4×Dic61—42no—
170C2×#92418—21no—
171C2×#92418—21no—
172C2×#92418—21no—
173C2×#92418—22no—
174C2×#92418—22no—
175C2×#92418—21no—
176C2×#92418—22no—
177C2×#92418—21no—
178C2×#92418#4112yescomplemented
179C23×C61—22no—
180C23×C61—22no—
181C23×C61—22no—
182C23×C61—22no—
183C23×C61—22no—
184C23×C61—22no—
185C23×C61—22no—
186C23×C61—41no—
187C23×C61—41no—
188C23×C61—42no—
189C23×C61—42no—
190C23×C61AGL(1,ℤ/8)14yes—
191C23×C61C22×D411yescharacteristic
192Q8×C61—48no—
193D4×C61—28no—
194D4×C61—22no—
195D4×C61—22no—
196D4×C61—416no—
197D4×C61—41no—
198D4×C61—41no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61—24no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—48no—
207C2×C4×C61—416no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61#12012yes—
216C2×C4×C61#12012yes—
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C22×D411yescharacteristic
219C2×C4×C61C2×#4111yescharacteristic
220C2×C4×C61C2×#4111yescharacteristic
221C2×C4×C61H(5,𝔽2)11yescharacteristic
222C2×C4×C612−1+411yescharacteristic
223C2×Dic61—22no—
224C2×Dic61—41no—
225C2×Dic61—41no—
226C2×Dic61—41no—
227C2×Dic61—42no—
228C2×Dic61—42no—
229C2×Dic61—42no—
230C2×Dic61—42no—
231Dic122—22no—
232Dic122—22no—
233Dic122—41no—
234Dic122—41no—
235#92418—22no—
236#92418—41no—
237#92418—42no—
238#24132—611nomaximal, Sylow
239C22×C61—24no—
240C22×C61—24no—
241C22×C61—24no—
242C22×C61—42no—
243C22×C61—42no—
244C22×C61—42no—
245C22×C61—42no—
246C22×C61—42no—
247C22×C61—42no—
248C22×C61—42no—
249C22×C61—82no—
250C22×C61—82no—
251C22×C61C2×#12011yescharacteristic
252C22×C61#54611yescharacteristic
253C22×C61#54711yescharacteristic
254C22×C61C2×AGL(1,ℤ/8)12yes—
255C22×C61#57812yes—
256C4×C61—22no—
257C4×C61—22no—
258C4×C61—42no—
259C4×C61—42no—
260C4×C61—44no—
261C4×C61#52511yescharacteristic
262C4×C61#53711yescharacteristic
263C4×C61#54811yescharacteristic
264C4×C61#55011yescharacteristic
265Dic61—42no—
266Dic61—81no—
267Dic61—82no—
268C2×#547D6111yescharacteristic
269C2×#477—611no—
270C2×#440—611no—
271#2536—611no—
272#2033—612no—
273#2000—612no—
274#1993—611no—
275#1955—611no—
276#1872—612no—
277#1852—612no—
278#1804—611no—
279C2×C61—44no—
280C2×C61—82no—
281C2×C61#240711yescharacteristic
282C2×C61#300311yescharacteristic
283C2×C61#301111yescharacteristic
284C2×C61#330812yes—
285C2×C61#334312yes—
286C22×#41—21no—
287C22×SD16—611no—
288#547—24no—
289#547—24no—
290C2×C4⋊Q8C2×D6111yescharacteristic
291C2×#123—22no—
292C2×C22⋊Q8—21no—
293C2×C22⋊Q8—612no—
294C2×#121C2×D6112yes—
295C2×C4×Q8—611no—
296C2×C4×D4C2×D6111yescharacteristic
297C2×C4×D4C2×D6111yescharacteristic
298C22×C4⋊C4C2×D6112yes—
299#477—1224no—
300#440—1224no—
301C2×#107—612no—
302C2×#107—611no—
303C2×C4⋊C8—611no—
304C2×#103—612no—
305C2×#103—611no—
306C2×#102—611no—
307C2×#102—612no—
308C2×C22⋊C8—612no—
309C2×C22⋊C8—612no—
310C2×#97—611no—
311#398—1221no—
312#394—1221no—
313#388—1222no—
314#382—611no—
315#380—612no—
316#340—612no—
317#340—612no—
318#340—612no—
319C61#2413211yescharacteristic, complemented, Sylow
320C2×#41—28no—
321C2×#41—48no—
322C22×Q8—21no—
323C22×Q8—611no—
324C22×D4—22no—
325C22×D4C22⋊D6111yescharacteristic
326C22×D4C22×D6111yescharacteristic
327C23×C4—22no—
328C23×C4—21no—
329C23×C4C22×D6112yes—
330C2×SD16—1222no—
331C2×SD16—1222no—
332C2×SD16—1222no—
333C22×C8—611no—
334C22×C8—611no—
335C22×C8—612no—
336C4⋊Q8—22no—
337C4⋊Q8C22⋊D6114yes—
338#123—216no—
339C22⋊Q8—28no—
340C22⋊Q8—1224no—
341C22⋊Q8—2442no—
342#121—24no—
343#121—24no—
344C4×Q8—1222no—
345C4×Q8—1222no—
346C4×D4—22no—
347C4×D4—22no—
348C4×D4—22no—
349C4×D4C22⋊D6114yes—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—611no—
355C2×C4⋊C4—612no—
356C2×C4⋊C4—1221no—
357C2×C4⋊C4—1222no—
358C2×C4⋊C4C22⋊D6111yescharacteristic
359C2×C4⋊C4C22×D6111yescharacteristic
360C2×C4⋊C4C22×D6112yes—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—1222no—
364C2×C22⋊C4—1222no—
365C2×C22⋊C4C22⋊D6112yes—
366C2×C42—1221no—
367C2×C42C22×D6111yescharacteristic
368#107—1222no—
369#107—1224no—
370C4⋊C8—1222no—
371#103—1222no—
372#103—1224no—
373#102—1222no—
374#102—2442no—
375C22⋊C8—1224no—
376C22⋊C8—2442no—
377#97—1222no—
378#95—1222no—
379#95—1221no—
380#95—1221no—
381C24—21no—
382C24C2×C22⋊D6112yes—
383#41—432no—
384C2×Q8—24no—
385C2×Q8—44no—
386C2×Q8—612no—
387C2×Q8—1221no—
388C2×Q8—1221no—
389C2×Q8—1221no—
390C2×Q8—1221no—
391C2×Q8—1221no—
392C2×D4—24no—
393C2×D4—28no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—22no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4C2×C22⋊D6112yes—
403C22×C4—28no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—1221no—
417C22×C4—1222no—
418C22×C4—1221no—
419C22×C4C2×C22⋊D6112yes—
420C22×C4C2×C22⋊D6111yescharacteristic
421C22×C4C2×C22⋊D6111yescharacteristic
422C22×C4C23×D6111yescharacteristic
423SD16—1228no—
424C2×C8—612no—
425C2×C8—1221no—
426C2×C8—1221no—
427C2×C8—1221no—
428C2×C8—1222no—
429C2×C8—1222no—
430C2×C8—1221no—
431C2×C8—1222no—
432C2×C8—1221no—
433C4⋊C4—28no—
434C4⋊C4—21no—
435C4⋊C4—22no—
436C4⋊C4—24no—
437C4⋊C4—22no—
438C4⋊C4—1221no—
439C4⋊C4—1221no—
440C4⋊C4—1222no—
441C4⋊C4—1222no—
442C4⋊C4—2444no—
443C4⋊C4—2442no—
444C4⋊C4C2×C22⋊D6112yes—
445C4⋊C4C2×C22⋊D6114yes—
446C22⋊C4—28no—
447C22⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—2444no—
450C22⋊C4—2444no—
451C42—2442no—
452C42C2×C22⋊D6112yes—
453C42C2×C22⋊D6112yes—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—42no—
462C23—42no—
463C23—41no—
464C23—41no—
465C23#1451980314yes—
466C23C22×C22⋊D6111yescharacteristic
467Q8—48no—
468Q8—1222no—
469Q8—1222no—
470Q8—2441no—
471Q8—2441no—
472D4—28no—
473D4—22no—
474D4—22no—
475D4—416no—
476D4—41no—
477D4—41no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—48no—
486C2×C4—416no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—1222no—
495C2×C4—2441no—
496C2×C4—2442no—
497C2×C4—2442no—
498C2×C4—2442no—
499C2×C4—2442no—
500C2×C4—2441no—
501C2×C4—2441no—
502C2×C4#1451974612yes—
503C2×C4#1451974612yes—
504C2×C4C22×C22⋊D6111yescharacteristic
505C2×C4C22×C22⋊D6111yescharacteristic
506C2×C4#41×D6111yescharacteristic
507C2×C4#41×D6111yescharacteristic
508C2×C4#1451982311yescharacteristic
509C2×C4#1451982411yescharacteristic
510C8—1222no—
511C8—2441no—
512C8—2442no—
513C22—24no—
514C22—24no—
515C22—24no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—82no—
524C22—82no—
525C22C2×#1451974611yescharacteristic
526C22#1684040011yescharacteristic
527C22#1684041111yescharacteristic
528C22C2×#1451980312yes—
529C22#1684057712yes—
530C4—22no—
531C4—22no—
532C4—42no—
533C4—42no—
534C4—44no—
535C4—4882no—
536C4—4881no—
537C4—2442no—
538C4#1684026911yescharacteristic
539C4#1684034011yescharacteristic
540C4#1684041911yescharacteristic
541C4#1684043111yescharacteristic
542C2—44no—
543C2—82no—
544C2#2144915811yescharacteristic
545C2#2145504611yescharacteristic
546C2#2145511411yescharacteristic
547C2#2145780412yes—
548C2#2145825312yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC2×C61×#547C2
socleC23×C61C22×D4
hypercenterC2×#547D61
hypocenterC61#24132
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. C2×#16840054C2semidirect
2. C2×#16839862C2semidirect
3. #21450728C2semidirect
4. #21446998C2semidirect
5. #21446822C2semidirect
6. #21446793C2semidirect
7. #21446618C2semidirect
8. #21446126C2semidirect
9. #21446007C2semidirect
10. #21445829C2semidirect
147. C2×Dic122#41semidirect
178. C2×#92418#41semidirect
319. C61#24132semidirect

Maximal subgroups

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

subgroupindex
C2×#168400542
C2×#168398622
#214507282
#214469982
#214468222
#214467932
#214466182
#214461262
#214460072
#214458292
C2×C61×#5472
#2413261