Finite Group Databasebeta

← #29721568

Subgroups — #29721568

Order 15616.

1118 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839963C211yesmaximal, characteristic, complemented
2C2×#16839961C212yesmaximal, complemented
3C2×C4×#11758200C211yesmaximal, characteristic, complemented
4#21450765C211yesmaximal, characteristic
5#21448731C211yesmaximal, characteristic, complemented
6#21447194C212yesmaximal, complemented
7#21447051C212yesmaximal, complemented
8#21446624C211yesmaximal, characteristic, complemented
9#21446535C211yesmaximal, characteristic, complemented
10#21446175C212yesmaximal, complemented
11C61×#1802C211yesmaximal, characteristic, complemented
12C22×#11758190C2212yes—
13C22×#11758200C2211yescharacteristic
14C2×#14519774C2211yescharacteristic
15C2×#14519773C2212yes—
16C2×#14519751—21no—
17C2×#14519751—21no—
18C2×C4×Dic122C2211yescharacteristic
19C2×#14519742C2212yes—
20#16839963—24no—
21#16839961—28no—
22C4×#11758200—24no—
23C2×#14519708C2211yescharacteristic
24C2×#14519706C2212yes—
25C2×#14519700C2211yescharacteristic
26C2×#14519699C2212yes—
27C2×#14519698C2211yescharacteristic
28C2×#14519697C2212yes—
29C2×#14519697C2212yes—
30C2×C4×#92418C2211yescharacteristic
31#16839671—22no—
32#16839663C2212yes—
33#16839658C2212yes—
34#16839657—22no—
35#16839631—21no—
36C4⋊C4×Dic61—21no—
37C4×C4⋊Dic61C2211yescharacteristic
38#16839516C2212yes—
39C2×C4×D4×C61C2211yescharacteristic
40C2×C4⋊C8×C61C2211yescharacteristic
41C2×#102×C61C2212yes—
42C61×#384C2211yescharacteristic
43C61×#340C2212yes—
44C22×Dic122C2311yescharacteristic
45C22×D244—21no—
46C22×D244—21no—
47C22×D244C2312yes—
48C22×C22⋊D61—22no—
49C22×C22⋊D61—22no—
50C2×#11758190—24no—
51C2×#11758190—24no—
52C2×#11758190—24no—
53C2×#11758200—22no—
54C2×#11758200—22no—
55C2×#11758200D414yescomplemented
56C22×#92418C2311yescharacteristic
57C22×#92418C2311yescharacteristic
58#14519774—22no—
59#14519774—22no—
60#14519773—24no—
61#14519773—42no—
62#14519751—44no—
63#14519751—44no—
64C4×Dic122—22no—
65C4×Dic122D414yescomplemented
66#14519742—24no—
67#14519742—42no—
68C2×C4⋊Dic61D411yescharacteristic
69C2×C4⋊Dic61D411yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×#11758196—21no—
72C2×#11758196—22no—
73C2×#11758196—22no—
74C2×#11758198—22no—
75C2×#11758198—22no—
76C2×C4×Dic61—21no—
77C2×C4×Dic61—22no—
78C2×C4×Dic61C2312yes—
79#14519708—22no—
80#14519706—24no—
81#14519700—22no—
82#14519699—24no—
83#14519698—22no—
84#14519697—24no—
85#14519697—42no—
86C4×#92418—22no—
87#14519684—22no—
88#14519684D412yes—
89C22×D4×C61C2311yescharacteristic
90C23×C4×C61—21no—
91C22×C8×C61C2312yes—
92C4×D4×C61—22no—
93C4×D4×C61—22no—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61C2312yes—
97C2×C4⋊C4×C61C2311yescharacteristic
98C2×C22⋊C4×C61—21no—
99C2×C42×C61C2311yescharacteristic
100C4⋊C8×C61—22no—
101#102×C61—24no—
102C24×C61—21no—
103C23×D61—22no—
104C2×D4×C61—21no—
105C2×D4×C61—21no—
106C2×D4×C61—21no—
107C2×D4×C61—21no—
108C2×D4×C61—21no—
109C2×D4×C61C2×D412yes—
110C22×C4×C61—21no—
111C22×C4×C61—22no—
112C22×C4×C61—42no—
113C22×C4×C61—42no—
114C22×C4×C61—42no—
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2411yescharacteristic
118C2×C22⋊D61—44no—
119C2×C22⋊D61—44no—
120C2×C22⋊D61—44no—
121C2×C22⋊D61—44no—
122C2×C22⋊D61—82no—
123C2×C22⋊D61—82no—
124C22×Dic61—21no—
125C22×Dic61—21no—
126C22×Dic61C2×D412yes—
127C22×Dic61C2×D412yes—
128C2×D244—22no—
129C2×D244—22no—
130C2×D244—22no—
131C2×D244—22no—
132C2×D244—22no—
133C2×D244—22no—
134C2×D244—42no—
135C2×D244—42no—
136C2×D244—42no—
137C2×D244—42no—
138C2×D244—81no—
139C2×D244—81no—
140C2×Dic122—21no—
141C2×Dic122—21no—
142C2×Dic122—21no—
143C2×Dic122C2×D412yes—
144C2×Dic122C2×D412yes—
145C2×Dic122#4112yes—
146C2×C8×C61—22no—
147C2×C8×C61—22no—
148C2×C8×C61—22no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—42no—
154C4⋊C4×C61—81no—
155C22⋊C4×C61—42no—
156C42×C61C2×D412yes—
157C42×C61#4112yes—
158#11758200—28no—
159#11758198—44no—
160#11758198—44no—
161C4⋊Dic61—22no—
162C4⋊Dic61—21no—
163C4⋊Dic61SD1614yescomplemented
164C4⋊Dic61C2×D412yes—
165#11758196—24no—
166#11758196—44no—
167#11758196—82no—
168C4×Dic61—24no—
169C4×Dic61—22no—
170C4×Dic61—22no—
171C4×Dic61—44no—
172C2×#92418—21no—
173C2×#92418—21no—
174C2×#92418—21no—
175C2×#92418—21no—
176C2×#92418—21no—
177C2×#92418C2×D412yes—
178#11758190—48no—
179C23×C61—22no—
180C23×C61—41no—
181C23×C61—41no—
182C23×C61—41no—
183C23×C61—41no—
184C23×C61—41no—
185C23×C61—41no—
186C23×C61C22×D411yescharacteristic
187C22×D61—42no—
188C22×D61—42no—
189C22×D61—42no—
190C22×D61—42no—
191C22×D61—42no—
192C22×D61—42no—
193C22×D61—42no—
194D4×C61—22no—
195D4×C61—22no—
196D4×C61—41no—
197D4×C61—41no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—41no—
201C2×C4×C61—42no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61—84no—
207C2×C4×C61#12112yes—
208C2×C4×C61#12112yes—
209C2×C4×C61C2×SD1612yes—
210C2×C4×C61AGL(1,ℤ/8)12yes—
211C2×C4×C61C22×D411yescharacteristic
212C2×C4×C61C22×D411yescharacteristic
213C2×C4×C61C2×#4111yescharacteristic
214C2×C4×C61C2×#4111yescharacteristic
215C2×C4×C61H(5,𝔽2)11yescharacteristic
216C2×C4×C612−1+411yescharacteristic
217C22⋊D61—88no—
218C22⋊D61—88no—
219C2×Dic61—22no—
220C2×Dic61—22no—
221C2×Dic61—22no—
222C2×Dic61—22no—
223C2×Dic61—22no—
224C2×Dic61—22no—
225C2×Dic61—22no—
226C2×Dic61—22no—
227C2×Dic61—22no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61C2×SD1614yes—
231D244—42no—
232D244—42no—
233D244—42no—
234D244—42no—
235D244—84no—
236D244—84no—
237Dic122—22no—
238Dic122—22no—
239Dic122—22no—
240Dic122#12114yescomplemented
241C8×C61—42no—
242#92418—22no—
243#92418—41no—
244#24728—611nomaximal, Sylow
245C22×C61—42no—
246C22×C61—42no—
247C22×C61—42no—
248C22×C61—81no—
249C22×C61—81no—
250C22×C61—81no—
251C22×C61—81no—
252C22×C61C2×#12111yescharacteristic
253C22×C61D4211yescharacteristic
254C22×C61#54811yescharacteristic
255C22×C61C22×SD1611yescharacteristic
256C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
257C22×C61#57611yescharacteristic
258C22×C61#57711yescharacteristic
259C2×D61—82no—
260C2×D61—82no—
261C2×D61—82no—
262C2×D61—82no—
263C2×D61—82no—
264C2×D61—82no—
265C2×D61—82no—
266C4×C61—22no—
267C4×C61—82no—
268C4×C61—81no—
269C4×C61#46014yescomplemented
270C4×C61C2×#12111yescharacteristic
271C4×C61#53711yescharacteristic
272C4×C61#54711yescharacteristic
273C4×C61#55311yescharacteristic
274Dic61—24no—
275Dic61—24no—
276Dic61—42no—
277Dic61—42no—
278C2×#460—611no—
279C2×#460—612no—
280C2×C4×SD16—611no—
281#2541—611no—
282Q8×C4⋊C4—611no—
283#2065—612no—
284#2041—612no—
285#1957—611no—
286#1939—611no—
287#1880—612no—
288#1802D6111yescharacteristic, complemented
289C2×C61—82no—
290C2×C61#241611yescharacteristic
291C2×C61C2×#46011yescharacteristic
292C2×C61#304511yescharacteristic
293C2×C61#327411yescharacteristic
294C2×C61D4×SD1611yescharacteristic
295C2×C61#336211yescharacteristic
296C2×C61#336811yescharacteristic
297D61—162no—
298C22×SD16—611no—
299C22×SD16—612no—
300C2×#127—612no—
301C2×#127—611no—
302C2×#121—1221no—
303C2×#121—1221no—
304C2×C4×Q8—612no—
305C2×C4×Q8—611no—
306C2×C4×D4C2×D6111yescharacteristic
307#460—1228no—
308#460—1224no—
309C4×SD16—1224no—
310C2×#106—611no—
311C2×C4⋊C8—612no—
312C2×C4⋊C8C2×D6111yescharacteristic
313C2×#103—611no—
314C2×#102—611no—
315C2×#102—612no—
316C2×#102—612no—
317C2×#102—612no—
318C2×#102C2×D6112yes—
319C2×C4×C8—611no—
320#392—1222no—
321#390—612no—
322#389—612no—
323#389—1222no—
324#384—1221no—
325#384C2×D6111yescharacteristic
326C4×C4⋊C4—611no—
327C4×C4⋊C4—1221no—
328#340C2×D6112yes—
329#336—612no—
330C61#2472811yescharacteristic, complemented, Sylow
331C22×Q8—611no—
332C22×D4—612no—
333C22×D4—1222no—
334C22×D4—1222no—
335C22×D4—1221no—
336C22×D4—1221no—
337C22×D4C22×D6111yescharacteristic
338C23×C4—21no—
339C2×SD16—614no—
340C2×SD16—1224no—
341C2×SD16—1224no—
342C2×SD16—1224no—
343C2×SD16—1222no—
344C2×SD16—1222no—
345C22×C8—611no—
346C22×C8—611no—
347C22×C8C22×D6112yes—
348#127—1224no—
349#127—1222no—
350#127—1222no—
351#127—2442no—
352#121—2444no—
353#121—2444no—
354C4×Q8—614no—
355C4×Q8—1222no—
356C4×Q8—1224no—
357C4×Q8—2442no—
358C4×D4—22no—
359C4×D4—22no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—611no—
363C2×C4⋊C4—611no—
364C2×C4⋊C4—611no—
365C2×C4⋊C4—1221no—
366C2×C4⋊C4—1222no—
367C2×C4⋊C4—1222no—
368C2×C4⋊C4C22×D6111yescharacteristic
369C2×C4⋊C4C22×D6112yes—
370C2×C22⋊C4—21no—
371C2×C22⋊C4—1222no—
372C2×C22⋊C4—1222no—
373C2×C42—612no—
374C2×C42—1221no—
375C2×C42—1222no—
376C2×C42C22×D6111yescharacteristic
377#106—1222no—
378C4⋊C8—22no—
379C4⋊C8—1224no—
380#103—1222no—
381#102—24no—
382#102—1224no—
383#102—1224no—
384#102—1222no—
385#102—2442no—
386C4×C8—1222no—
387#95—612no—
388#95—1222no—
389C24—21no—
390C24—1222no—
391C2×Q8—612no—
392C2×Q8—612no—
393C2×Q8—612no—
394C2×Q8—1221no—
395C2×Q8—1221no—
396C2×Q8—1221no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—1222no—
403C2×D4—1222no—
404C2×D4—1222no—
405C2×D4—1222no—
406C2×D4—1222no—
407C2×D4—1222no—
408C2×D4—4881no—
409C2×D4—4881no—
410C2×D4—4882no—
411C2×D4—4882no—
412C2×D4—2444no—
413C2×D4—2444no—
414C2×D4—2444no—
415C2×D4—2444no—
416C2×D4—2442no—
417C2×D4—2442no—
418C2×D4—2442no—
419C2×D4—2442no—
420C2×D4D4×D6112yes—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—612no—
427C22×C4—612no—
428C22×C4—1221no—
429C22×C4—1221no—
430C22×C4D4×D6111yescharacteristic
431C22×C4D4×D6111yescharacteristic
432C22×C4C23×D6111yescharacteristic
433SD16—1228no—
434SD16—2448no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—612no—
439C2×C8—1221no—
440C2×C8—1221no—
441C2×C8—1221no—
442C2×C8—1221no—
443C2×C8—1221no—
444C4⋊C4—22no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—42no—
449C4⋊C4—81no—
450C4⋊C4—614no—
451C4⋊C4—612no—
452C4⋊C4—1221no—
453C4⋊C4—1224no—
454C4⋊C4—1222no—
455C4⋊C4—4882no—
456C4⋊C4—2444no—
457C22⋊C4—42no—
458C22⋊C4—2444no—
459C22⋊C4—2444no—
460C42—1222no—
461C42—1222no—
462C42—1224no—
463C42—2444no—
464C42#1175822212yes—
465C42D4×D6112yes—
466C23—22no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—2442no—
474C23—2442no—
475C23—2442no—
476C23—2442no—
477C23—2442no—
478C23—2442no—
479C23—2442no—
480C23C2×D4×D6111yescharacteristic
481Q8—614no—
482Q8—1222no—
483Q8—1222no—
484Q8—1222no—
485D4—22no—
486D4—22no—
487D4—41no—
488D4—41no—
489D4—4884no—
490D4—4888no—
491D4—4888no—
492D4—4884no—
493D4—2442no—
494D4—2442no—
495D4—2442no—
496D4—2442no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—84no—
506C2×C4—614no—
507C2×C4—1222no—
508C2×C4—1222no—
509C2×C4—1222no—
510C2×C4—1222no—
511C2×C4—1222no—
512C2×C4—1222no—
513C2×C4—1222no—
514C2×C4—1222no—
515C2×C4—1222no—
516C2×C4—2441no—
517C2×C4—2441no—
518C2×C4#1451975012yes—
519C2×C4#1451975012yes—
520C2×C4SD16×D6112yes—
521C2×C4#1451980112yes—
522C2×C4C2×D4×D6111yescharacteristic
523C2×C4C2×D4×D6111yescharacteristic
524C2×C4#41×D6111yescharacteristic
525C2×C4C2×#1175822211yescharacteristic
526C2×C4#1451982311yescharacteristic
527C2×C4#1451982511yescharacteristic
528C8—42no—
529C8—1222no—
530C8—2441no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22—4882no—
539C22—4882no—
540C22—4882no—
541C22—4882no—
542C22—4882no—
543C22—4882no—
544C22—4882no—
545C22C2×#1451975011yescharacteristic
546C22#1684039611yescharacteristic
547C22#1684042211yescharacteristic
548C22C2×SD16×D6111yescharacteristic
549C22C2×#1451980111yescharacteristic
550C22#1684056611yescharacteristic
551C22#1684057111yescharacteristic
552C4—22no—
553C4—82no—
554C4—81no—
555C4—1224no—
556C4—1224no—
557C4—2442no—
558C4—2442no—
559C4#1683996114yescomplemented
560C4#121×D6111yescharacteristic
561C4#1684034211yescharacteristic
562C4#1684041311yescharacteristic
563C4#1684045611yescharacteristic
564C2—82no—
565C2—9762no—
566C2#2144926111yescharacteristic
567C2C2×#1683996111yescharacteristic
568C2#2145535711yescharacteristic
569C2#2145744211yescharacteristic
570C2#2145751811yescharacteristic
571C2#2145848911yescharacteristic
572C2#2145854711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1802C2
socleC23×C61C22×D4
prime radical#21450765C2
hypercenter#1802D61
hypocenterC61#24728
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
1. C2×#16839963C2semidirect
2. C2×#16839961C2semidirect
3. C2×C4×#11758200C2semidirect
5. #21448731C2semidirect
6. #21447194C2semidirect
7. #21447051C2semidirect
8. #21446624C2semidirect
9. #21446535C2semidirect
10. #21446175C2semidirect
11. C61×#1802C2semidirect
55. C2×#11758200D4semidirect
65. C4×Dic122D4semidirect
163. C4⋊Dic61SD16semidirect
240. Dic122#121semidirect
269. C4×C61#460semidirect
288. #1802D61semidirect
330. C61#24728semidirect
559. C4#16839961semidirect

Maximal subgroups

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

subgroupindex
C2×#168399632
C2×#168399612
C2×C4×#117582002
#214507652
#214487312
#214471942
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