Finite Group Databasebeta

← #29714757

Subgroups — #29714757

Order 15616.

1150 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840486C211yesmaximal, characteristic, complemented
2C2×C4×#11758199C211yesmaximal, characteristic
3#21450741C211yesmaximal, characteristic, complemented
4#21447056C212yesmaximal, complemented
5#21446893C212yesmaximal
6#21446754C211yesmaximal, characteristic, complemented
7#21446622C211yesmaximal, characteristic, complemented
8#21446176C212yesmaximal, complemented
9#21446071C212yesmaximal, complemented
10#21445820C211yesmaximal, characteristic, complemented
11C2×C61×#461C211yesmaximal, characteristic, complemented
12C22×#11758199C2211yescharacteristic
13#16840486—24no—
14#16840486—24no—
15C2×#14519772—22no—
16C2×#14519766C2212yes—
17C2×#14519757—21no—
18C2×#14519756—21no—
19C2×Q8×Dic61C2212yes—
20C2×C4×Dic122C2211yescharacteristic
21C2×C4×D244C2211yescharacteristic
22C2×#14519735—21no—
23C4×#11758199—24no—
24C2×#14519710C2211yescharacteristic
25C2×#14519701C2212yes—
26C2×#14519701C2212yes—
27C2×#14519699C2212yes—
28C2×#14519698C2211yescharacteristic
29C2×#14519697C2211yescharacteristic
30C2×C4×#92418C2211yescharacteristic
31#16839685—22no—
32#16839665C2212yes—
33#16839658C2212yes—
34#16839632C2211yescharacteristic
35#16839622—21no—
36#16839525C2212yes—
37#16839516C2212yes—
38#16839516C2212yes—
39C22×SD16×C61C2212yes—
40C2×C4⋊Q8×C61C2211yescharacteristic
41C2×C4×D4×C61C2211yescharacteristic
42C61×#461—24no—
43C2×C4⋊C8×C61C2211yescharacteristic
44C2×#103×C61C2212yes—
45C22×Dic122D411yescharacteristic
46C22×D244C2311yescharacteristic
47C22×C4×D61—21no—
48C2×#11758199—22no—
49C2×#11758199—22no—
50C2×#11758199—22no—
51C22×#92418C2311yescharacteristic
52C22×#92418C2311yescharacteristic
53#14519772—216no—
54#14519766—24no—
55#14519766—24no—
56#14519757—28no—
57#14519756—44no—
58Q8×Dic61—24no—
59Q8×Dic61—24no—
60C4×Dic122—22no—
61C4×Dic122—22no—
62C4×D244—22no—
63C4×D244D414yes—
64#14519735—28no—
65C2×C4⋊Dic61—22no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61D411yescharacteristic
69C2×C4⋊Dic61C2312yes—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×#11758196—22no—
72C2×#11758196D412yes—
73C2×C22⋊Dic61—22no—
74C2×C22⋊Dic61—22no—
75C2×#11758198—21no—
76C2×#11758198—22no—
77C2×#11758198—22no—
78C2×C4×Dic61—22no—
79C2×C4×Dic61C2312yes—
80#14519710—22no—
81#14519701—24no—
82#14519701—42no—
83#14519699—42no—
84#14519698—22no—
85#14519697—22no—
86C4×#92418—22no—
87#14519684—22no—
88C22×Q8×C61C2312yes—
89C22×D4×C61C2311yescharacteristic
90C23×C4×C61—21no—
91C2×SD16×C61—24no—
92C2×SD16×C61—24no—
93C2×SD16×C61—42no—
94C22×C8×C61C2312yes—
95C4⋊Q8×C61—22no—
96C4⋊Q8×C61D414yes—
97C4×D4×C61—22no—
98C4×D4×C61—22no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61C2311yescharacteristic
102C2×C22⋊C4×C61—21no—
103C2×C42×C61C2311yescharacteristic
104C4⋊C8×C61—22no—
105#103×C61—24no—
106C24×C61—21no—
107C23×D61—21no—
108C2×Q8×C61—22no—
109C2×Q8×C61—22no—
110C2×Q8×C61—22no—
111C2×Q8×C61—22no—
112C2×Q8×C61—22no—
113C2×Q8×C61C2×D414yes—
114C2×D4×C61—21no—
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61—21no—
119C2×D4×C61#4112yes—
120C22×C4×C61—22no—
121C22×C4×C61—21no—
122C22×C4×C61—42no—
123C22×C4×C61—42no—
124C22×C4×C61—42no—
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2411yescharacteristic
128C22×Dic61—21no—
129C22×Dic61—22no—
130C22×Dic61C2×D412yes—
131C22×Dic61C2×D412yes—
132C2×D244—21no—
133C2×D244—21no—
134C2×D244—21no—
135C2×D244—21no—
136C2×D244—21no—
137C2×D244C2×D412yes—
138C2×C4×D61—24no—
139C2×C4×D61—42no—
140C2×C4×D61—42no—
141C2×Dic122—22no—
142C2×Dic122—22no—
143C2×Dic122—22no—
144SD16×C61—48no—
145C2×C8×C61—22no—
146C2×C8×C61—22no—
147C2×C8×C61—22no—
148C4⋊C4×C61—24no—
149C4⋊C4×C61—24no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—21no—
152C22⋊C4×C61—42no—
153C42×C61C2×D412yes—
154C42×C61C2×D412yes—
155C22⋊Dic61—44no—
156C22⋊Dic61—82no—
157#11758199—28no—
158#11758198—28no—
159#11758198—28no—
160#11758198—24no—
161C4⋊Dic61—28no—
162C4⋊Dic61—22no—
163C4⋊Dic61—21no—
164C4⋊Dic61—22no—
165C4⋊Dic61—22no—
166C4⋊Dic61—42no—
167C4⋊Dic61—81no—
168C4⋊Dic61C2×D412yes—
169#11758196—24no—
170#11758196—44no—
171C4×Dic61—28no—
172C4×Dic61—22no—
173C4×Dic61—22no—
174C2×#92418—21no—
175C2×#92418—21no—
176C2×#92418—21no—
177C2×#92418—21no—
178C2×#92418—21no—
179C2×#92418#4112yes—
180C23×C61—22no—
181C23×C61—41no—
182C23×C61—41no—
183C23×C61—41no—
184C23×C61—41no—
185C23×C61—41no—
186C23×C61—41no—
187C23×C61C22×D411yescharacteristic
188C22×D61—22no—
189C22×D61—41no—
190C22×D61—41no—
191C22×D61—41no—
192C22×D61—41no—
193C22×D61—41no—
194C22×D61—41no—
195Q8×C61—24no—
196Q8×C61—24no—
197Q8×C61—42no—
198Q8×C61—42no—
199D4×C61—22no—
200D4×C61—22no—
201D4×C61—41no—
202D4×C61—41no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—24no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61—41no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61—84no—
212C2×C4×C61#12012yes—
213C2×C4×C61#12012yes—
214C2×C4×C61#13512yes—
215C2×C4×C61AGL(1,ℤ/8)12yes—
216C2×C4×C61C22×D411yescharacteristic
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C2×#4111yescharacteristic
219C2×C4×C61C2×#4111yescharacteristic
220C2×C4×C61H(5,𝔽2)11yescharacteristic
221C2×C4×C612−1+411yescharacteristic
222C2×Dic61—22no—
223C2×Dic61—22no—
224C2×Dic61—22no—
225C2×Dic61—22no—
226C2×Dic61—22no—
227C2×Dic61—22no—
228C2×Dic61—22no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—42no—
232C2×Dic61—42no—
233C2×Dic61—42no—
234D244—22no—
235D244—22no—
236D244—41no—
237D244—41no—
238C4×D61—48no—
239Dic122—44no—
240C8×C61—42no—
241#92418—22no—
242#92418—41no—
243#24185—611nomaximal, Sylow
244C22×C61—42no—
245C22×C61—42no—
246C22×C61—42no—
247C22×C61—81no—
248C22×C61—81no—
249C22×C61—81no—
250C22×C61—81no—
251C22×C61C2×#12011yescharacteristic
252C22×C61#54811yescharacteristic
253C22×C61#55011yescharacteristic
254C22×C61C2×#13511yescharacteristic
255C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
256C22×C61#57711yescharacteristic
257C22×C61#57811yescharacteristic
258C2×D61—42no—
259C2×D61—42no—
260C2×D61—42no—
261C2×D61—81no—
262C2×D61—81no—
263C2×D61—81no—
264C2×D61—81no—
265C4×C61—22no—
266C4×C61—44no—
267C4×C61—81no—
268C4×C61#44914yes—
269C4×C61#52511yescharacteristic
270C4×C61#53711yescharacteristic
271C4×C61#54611yescharacteristic
272C4×C61#54711yescharacteristic
273Dic61—42no—
274Dic61—42no—
275Dic61—42no—
276Dic61—82no—
277C2×#556—611no—
278C2×#461D6111yescharacteristic, complemented
279C2×C4×D8—611no—
280#2538—611no—
281#2042—612no—
282#2012—612no—
283#1984—611no—
284#1956—611no—
285#1880—612no—
286#1863—612no—
287#1802—611no—
288C2×C61—82no—
289C2×C61#240711yescharacteristic
290C2×C61C2×#44911yescharacteristic
291C2×C61#301311yescharacteristic
292C2×C61#336011yescharacteristic
293C2×C61#336711yescharacteristic
294C2×C61#339311yescharacteristic
295C2×C61#339711yescharacteristic
296D61—82no—
297C22×SD16C2×D6112yes—
298C22×D8—611no—
299#556—1224no—
300#556—1224no—
301C2×C4⋊Q8C2×D6111yescharacteristic
302C2×#126—1222no—
303C2×#124—612no—
304C2×C22⋊Q8—1221no—
305C2×C22⋊Q8—1221no—
306C2×C4×Q8—611no—
307C2×C4×Q8—612no—
308C2×C4×D4—611no—
309C2×C4×D4C2×D6111yescharacteristic
310C2×#117—1221no—
311#461—24no—
312C4×D8—1224no—
313C2×#107—611no—
314C2×C4⋊C8C2×D6111yescharacteristic
315C2×#103—612no—
316C2×#103—612no—
317C2×#103C2×D6112yes—
318C2×#102—611no—
319C2×#102—611no—
320C2×#102—612no—
321C2×C4×C8—611no—
322#396—1222no—
323#390—612no—
324#389—612no—
325#384—611no—
326#380—1221no—
327#340—612no—
328#336—612no—
329#336—612no—
330C61#2418511yescharacteristic, complemented, Sylow
331C22×Q8—611no—
332C22×Q8C22×D6112yes—
333C22×D4—611no—
334C22×D4C22×D6111yescharacteristic
335C23×C4—21no—
336C23×C4—1221no—
337C2×SD16—24no—
338C2×SD16—24no—
339C2×SD16—42no—
340C2×D8—1222no—
341C2×D8—1222no—
342C2×D8—1222no—
343C22×C8—611no—
344C22×C8—611no—
345C22×C8C22×D6112yes—
346C4⋊Q8—22no—
347C4⋊Q8C22⋊D6114yes—
348#126—12216no—
349#124—1224no—
350#124—1224no—
351C22⋊Q8—1228no—
352C22⋊Q8—2444no—
353C4×Q8—1224no—
354C4×Q8—1224no—
355C4×Q8—1222no—
356C4×Q8—1222no—
357C4×D4—22no—
358C4×D4—22no—
359C4×D4—614no—
360C4×D4—1222no—
361#117—1228no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—611no—
365C2×C4⋊C4—611no—
366C2×C4⋊C4—612no—
367C2×C4⋊C4—612no—
368C2×C4⋊C4—1222no—
369C2×C4⋊C4—1221no—
370C2×C4⋊C4—1222no—
371C2×C4⋊C4—1221no—
372C2×C4⋊C4C22×D6111yescharacteristic
373C2×C22⋊C4—21no—
374C2×C22⋊C4—1222no—
375C2×C22⋊C4—1222no—
376C2×C22⋊C4—1221no—
377C2×C22⋊C4—1222no—
378C2×C22⋊C4—1222no—
379C2×C42—612no—
380C2×C42—1222no—
381C2×C42C22×D6111yescharacteristic
382#107—1222no—
383C4⋊C8—22no—
384#103—24no—
385#103—1224no—
386#103—2442no—
387#102—1222no—
388#102—1222no—
389#102—2442no—
390C4×C8—1222no—
391#95—1222no—
392C24—21no—
393C24—1221no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—1222no—
400C2×Q8—1222no—
401C2×Q8—1222no—
402C2×Q8C2×C22⋊D6114yes—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—612no—
409C2×D4—1221no—
410C2×D4—1221no—
411C2×D4—1221no—
412C2×D4—1221no—
413C2×D4—1221no—
414C2×D4#1175822412yes—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—612no—
421C22×C4—612no—
422C22×C4—1221no—
423C22×C4—1224no—
424C22×C4—1222no—
425C22×C4—2442no—
426C22×C4—2442no—
427C22×C4D4×D6111yescharacteristic
428C22×C4D4×D6111yescharacteristic
429C22×C4C23×D6111yescharacteristic
430SD16—48no—
431D8—1228no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—612no—
436C2×C8—1221no—
437C2×C8—1221no—
438C2×C8—1221no—
439C2×C8—1221no—
440C2×C8—1221no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—612no—
446C4⋊C4—1228no—
447C4⋊C4—1221no—
448C4⋊C4—1224no—
449C4⋊C4—1222no—
450C4⋊C4—1222no—
451C4⋊C4—1222no—
452C4⋊C4—4881no—
453C4⋊C4—2444no—
454C4⋊C4—2442no—
455C22⋊C4—42no—
456C22⋊C4—1228no—
457C22⋊C4—1228no—
458C22⋊C4—1224no—
459C22⋊C4—4882no—
460C22⋊C4—2444no—
461C42—1228no—
462C42—1222no—
463C42—1222no—
464C42D4×D6112yes—
465C42C2×C22⋊D6112yes—
466C23—22no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—1222no—
474C23—2441no—
475C23—2441no—
476C23—2441no—
477C23—2441no—
478C23—2441no—
479C23—2441no—
480C23C2×D4×D6111yescharacteristic
481Q8—24no—
482Q8—24no—
483Q8—42no—
484Q8—42no—
485Q8—2444no—
486D4—22no—
487D4—22no—
488D4—41no—
489D4—41no—
490D4—1222no—
491D4—1222no—
492D4—2441no—
493D4—2441no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—24no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—84no—
503C2×C4—1222no—
504C2×C4—1222no—
505C2×C4—1222no—
506C2×C4—1222no—
507C2×C4—1222no—
508C2×C4—1222no—
509C2×C4—1222no—
510C2×C4—2442no—
511C2×C4—2442no—
512C2×C4—2442no—
513C2×C4—2441no—
514C2×C4—2441no—
515C2×C4—2448no—
516C2×C4#1451974512yes—
517C2×C4#1451974512yes—
518C2×C4#1451979812yes—
519C2×C4#1451980412yes—
520C2×C4C2×D4×D6111yescharacteristic
521C2×C4C22×C22⋊D6111yescharacteristic
522C2×C4#41×D6111yescharacteristic
523C2×C4C2×#1175822411yescharacteristic
524C2×C4#1451982211yescharacteristic
525C2×C4#1451982411yescharacteristic
526C8—42no—
527C8—1222no—
528C8—2441no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—81no—
533C22—81no—
534C22—81no—
535C22—81no—
536C22—4881no—
537C22—4881no—
538C22—4881no—
539C22—4881no—
540C22—2442no—
541C22—2442no—
542C22—2442no—
543C22C2×#1451974511yescharacteristic
544C22#1684041811yescharacteristic
545C22#1684043011yescharacteristic
546C22C2×#1451979811yescharacteristic
547C22C2×#1451980411yescharacteristic
548C22#1684057111yescharacteristic
549C22#1684057811yescharacteristic
550C4—22no—
551C4—44no—
552C4—81no—
553C4—4882no—
554C4—2442no—
555C4—2442no—
556C4—2442no—
557C4#1683990614yes—
558C4#1684027211yescharacteristic
559C4#1684034111yescharacteristic
560C4#1684040611yescharacteristic
561C4#1684041111yescharacteristic
562C2—82no—
563C2—4882no—
564C2#2144916011yescharacteristic
565C2C2×#1683990611yescharacteristic
566C2#2145513311yescharacteristic
567C2#2145847211yescharacteristic
568C2#2145854111yescharacteristic
569C2#2145880611yescharacteristic
570C2#2145883811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#461C2
socleC23×C61C22×D4
prime radicalC22×#11758199C22
hypercenterC2×#461D61
hypocenterC61#24185
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#461

Complements

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

NH ≅ G/Ntype
1. C2×#16840486C2semidirect
3. #21450741C2semidirect
4. #21447056C2semidirect
6. #21446754C2semidirect
7. #21446622C2semidirect
8. #21446176C2semidirect
9. #21446071C2semidirect
10. #21445820C2semidirect
11. C2×C61×#461C2semidirect
278. C2×#461D61semidirect
330. C61#24185semidirect

Maximal subgroups

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

subgroupindex
C2×#168404862
C2×C4×#117581992
#214507412
#214470562
#214468932
#214467542
#214466222
#214461762
#214460712
#214458202
C2×C61×#4612
#2418561