Finite Group Databasebeta

← #29713957

Subgroups — #29713957

Order 15616.

1154 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×#16840477C211yesmaximal, characteristic
2C2×#16840057C211yesmaximal, characteristic, complemented
3C2×C4×#11758201C211yesmaximal, characteristic, complemented
4#21447001C212yesmaximal, complemented
5#21446820C212yesmaximal, complemented
6#21446779C211yesmaximal, characteristic, complemented
7#21446614C211yesmaximal, characteristic, complemented
8#21446132C212yesmaximal, complemented
9#21446016C212yesmaximal, complemented
10#21445816C211yesmaximal, characteristic, complemented
11C61×#2538C211yesmaximal, characteristic, complemented
12C22×#11758201C2211yescharacteristic
13#16840477—24no—
14#16840477—24no—
15C2×#14519777C2211yescharacteristic
16C2×#14519763—22no—
17C2×#14519751C2212yes—
18C2×#14519748—21no—
19C2×C4×D244C2211yescharacteristic
20C2×#14519736C2212yes—
21C2×#14519735—21no—
22#16840057—24no—
23C4×#11758201—24no—
24C2×#14519710C2212yes—
25C2×#14519708C2211yescharacteristic
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519701C2212yes—
28C2×#14519701C2211yescharacteristic
29C2×#14519697C2211yescharacteristic
30C2×#14519697C2212yes—
31C2×C22⋊#92418C2212yes—
32C2×C22⋊#92418C2212yes—
33C2×C4×#92418C2211yescharacteristic
34#16839524C2212yes—
35#16839524C2212yes—
36#16839524C2212yes—
37C2×C22⋊Q8×C61C2212yes—
38C2×C4×Q8×C61C2211yescharacteristic
39C2×C4×D4×C61C2211yescharacteristic
40C61×#396—22no—
41C61×#390—21no—
42C61×#389—21no—
43C61×#384C2211yescharacteristic
44C61×#380C2212yes—
45C22×Dic122—21no—
46C22×D244C2311yescharacteristic
47C22×C22⋊D61—22no—
48C22×C4×D61—21no—
49C2×#11758201—22no—
50C2×#11758201—22no—
51C2×#11758201—22no—
52C22×#92418C2311yescharacteristic
53C22×#92418C2312yes—
54C22×#92418C2311yescharacteristic
55#14519777—22no—
56#14519777D414yes—
57#14519763—216no—
58#14519751—24no—
59#14519751—24no—
60#14519748—28no—
61C4×D244—22no—
62C4×D244D414yes—
63#14519736—24no—
64#14519736—24no—
65#14519735—28no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2312yes—
69C2×C22⋊Dic61—22no—
70C2×#11758198—22no—
71C2×#11758198—21no—
72C2×C4×Dic61—22no—
73#14519710—24no—
74#14519708—22no—
75C4⋊#92418—22no—
76#14519701—24no—
77#14519701—22no—
78#14519697—22no—
79#14519697—42no—
80C22⋊#92418—24no—
81C22⋊#92418—42no—
82C4×#92418—22no—
83C22×Q8×C61C2311yescharacteristic
84C22×D4×C61D411yescharacteristic
85C23×C4×C61C2312yes—
86C22⋊Q8×C61—24no—
87C22⋊Q8×C61—42no—
88C4×Q8×C61—22no—
89C4×Q8×C61—22no—
90C4×D4×C61—22no—
91C4×D4×C61—22no—
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61—22no—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61D411yescharacteristic
97C2×C4⋊C4×C61C2312yes—
98C2×C4⋊C4×C61C2311yescharacteristic
99C2×C22⋊C4×C61—22no—
100C2×C22⋊C4×C61—22no—
101C2×C22⋊C4×C61D412yes—
102C2×C42×C61—21no—
103C2×C42×C61C2311yescharacteristic
104#95×C61—22no—
105C24×C61C2×D412yes—
106C23×D61—21no—
107C2×Q8×C61—21no—
108C2×Q8×C61—21no—
109C2×Q8×C61—21no—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61#4112yes—
113C2×D4×C61—22no—
114C2×D4×C61—22no—
115C2×D4×C61—22no—
116C22×C4×C61—21no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—21no—
124C22×C4×C61—22no—
125C22×C4×C61C2×D412yes—
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2411yescharacteristic
129C2×C22⋊D61—28no—
130C2×C22⋊D61—82no—
131C2×C22⋊D61—82no—
132C22×Dic61—21no—
133C22×Dic61—22no—
134C2×D244—21no—
135C2×D244—21no—
136C2×D244—21no—
137C2×D244—21no—
138C2×D244—21no—
139C2×D244C2×D412yes—
140C2×C4×D61—24no—
141C2×C4×D61—42no—
142C2×C4×D61—42no—
143C2×Dic122—24no—
144C2×Dic122—81no—
145C2×Dic122—81no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—44no—
152C4⋊C4×C61—42no—
153C4⋊C4×C61—42no—
154C4⋊C4×C61—81no—
155C22⋊C4×C61—24no—
156C22⋊C4×C61—44no—
157C22⋊C4×C61—82no—
158C42×C61—42no—
159C42×C61C2×D412yes—
160C42×C61C2×D412yes—
161C22⋊Dic61—28no—
162#11758201—28no—
163#11758198—28no—
164#11758198—24no—
165C4⋊Dic61—24no—
166C4⋊Dic61—21no—
167C4⋊Dic61—22no—
168C4⋊Dic61C2×D412yes—
169C4⋊Dic61C2×D414yes—
170C4×Dic61—28no—
171C2×#92418—21no—
172C2×#92418—21no—
173C2×#92418—21no—
174C2×#92418—22no—
175C2×#92418—22no—
176C2×#92418—21no—
177C2×#92418—22no—
178C2×#92418—21no—
179C2×#92418#4112yes—
180C23×C61—22no—
181C23×C61—22no—
182C23×C61—22no—
183C23×C61—22no—
184C23×C61—22no—
185C23×C61—22no—
186C23×C61—22no—
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—22no—
196Q8×C61—22no—
197Q8×C61—41no—
198Q8×C61—41no—
199D4×C61—44no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61—41no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—41no—
216C2×C4×C61—42no—
217C2×C4×C61—41no—
218C2×C4×C61#12012yes—
219C2×C4×C61#12012yes—
220C2×C4×C61#13512yes—
221C2×C4×C61#13712yes—
222C2×C4×C61C22×D411yescharacteristic
223C2×C4×C61C22×D411yescharacteristic
224C2×C4×C61C2×#4111yescharacteristic
225C2×C4×C61C2×#4111yescharacteristic
226C2×C4×C61H(5,𝔽2)11yescharacteristic
227C2×C4×C612−1+411yescharacteristic
228C22⋊D61—88no—
229C2×Dic61—22no—
230C2×Dic61—24no—
231C2×Dic61—41no—
232C2×Dic61—41no—
233C2×Dic61—42no—
234C2×Dic61—42no—
235D244—22no—
236D244—22no—
237D244—41no—
238D244—41no—
239C4×D61—48no—
240Dic122—84no—
241#92418—22no—
242#92418—41no—
243#92418—42no—
244#24121—611nomaximal, Sylow
245C22×C61—42no—
246C22×C61—42no—
247C22×C61—42no—
248C22×C61—42no—
249C22×C61—42no—
250C22×C61—42no—
251C22×C61—42no—
252C22×C61C2×#12011yescharacteristic
253C22×C61#54611yescharacteristic
254C22×C61#54711yescharacteristic
255C22×C61C2×#13511yescharacteristic
256C22×C61C2×#13711yescharacteristic
257C22×C61#57611yescharacteristic
258C22×C61#57711yescharacteristic
259C2×D61—42no—
260C2×D61—42no—
261C2×D61—42no—
262C2×D61—81no—
263C2×D61—81no—
264C2×D61—81no—
265C2×D61—81no—
266C4×C61—22no—
267C4×C61—42no—
268C4×C61—42no—
269C4×C61—81no—
270C4×C61—82no—
271C4×C61#45214yes—
272C4×C61#52511yescharacteristic
273C4×C61#53711yescharacteristic
274C4×C61#54811yescharacteristic
275C4×C61#55011yescharacteristic
276Dic61—42no—
277Dic61—44no—
278C2×#555—611no—
279C2×#477—611no—
280C2×C4×SD16—611no—
281#2538D6111yescharacteristic, complemented
282#2033—612no—
283#2000—612no—
284#1990—611no—
285#1954—611no—
286#1873—612no—
287#1854—612no—
288#1802—611no—
289C2×C61—82no—
290C2×C61#240711yescharacteristic
291C2×C61C2×#45211yescharacteristic
292C2×C61#301211yescharacteristic
293C2×C61#330511yescharacteristic
294C2×C61#332911yescharacteristic
295C2×C61#334011yescharacteristic
296C2×C61#335711yescharacteristic
297D61—82no—
298C22×SD16—611no—
299#555—1224no—
300#555—1224no—
301C2×C4⋊Q8—611no—
302C2×#124—1222no—
303C2×C22⋊Q8C2×D6112yes—
304C2×#121—612no—
305C2×#121—1221no—
306C2×C4×Q8C2×D6111yescharacteristic
307C2×C4×D4—611no—
308C2×C4×D4C2×D6111yescharacteristic
309C2×#117—612no—
310C2×#117—1221no—
311#477—1224no—
312C4×SD16—1224no—
313C2×#107—612no—
314C2×#106—611no—
315C2×C4⋊C8—611no—
316C2×#103—612no—
317C2×#103—611no—
318C2×#102—611no—
319C2×#102—612no—
320C2×C22⋊C8—612no—
321C2×C22⋊C8—612no—
322C2×C4×C8—611no—
323#396—22no—
324#390—21no—
325#389—21no—
326#384C2×D6111yescharacteristic
327#380C2×D6112yes—
328#340—612no—
329#340—612no—
330#340—612no—
331C61#2412111yescharacteristic, complemented, Sylow
332C22×Q8—1221no—
333C22×Q8C22×D6111yescharacteristic
334C22×D4—611no—
335C22×D4—1222no—
336C22×D4C22⋊D6111yescharacteristic
337C23×C4—1221no—
338C23×C4C22×D6112yes—
339C2×SD16—1222no—
340C2×SD16—1222no—
341C2×SD16—1222no—
342C22×C8—611no—
343C22×C8—611no—
344C22×C8—612no—
345C4⋊Q8—614no—
346C4⋊Q8—1222no—
347#124—12216no—
348C22⋊Q8—24no—
349C22⋊Q8—42no—
350#121—1224no—
351#121—1224no—
352#121—1228no—
353C4×Q8—22no—
354C4×Q8—22no—
355C4×D4—22no—
356C4×D4—22no—
357C4×D4—614no—
358C4×D4—1222no—
359#117—1228no—
360#117—1224no—
361#117—1224no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—611no—
367C2×C4⋊C4—612no—
368C2×C4⋊C4—1221no—
369C2×C4⋊C4C22⋊D6111yescharacteristic
370C2×C4⋊C4C22×D6111yescharacteristic
371C2×C4⋊C4C22×D6112yes—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—1222no—
375C2×C22⋊C4—1222no—
376C2×C22⋊C4—1221no—
377C2×C22⋊C4C22⋊D6112yes—
378C2×C42—21no—
379C2×C42—1222no—
380C2×C42C22×D6111yescharacteristic
381#107—1224no—
382#106—1222no—
383C4⋊C8—1222no—
384#103—1222no—
385#103—1224no—
386#102—1222no—
387#102—2442no—
388C22⋊C8—1224no—
389C22⋊C8—2442no—
390C4×C8—1222no—
391#95—22no—
392C24—1221no—
393C24C2×C22⋊D6112yes—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—1224no—
400C2×Q8—4881no—
401C2×Q8—4881no—
402C2×Q8#1175822412yes—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—612no—
407C2×D4—1228no—
408C2×D4—1221no—
409C2×D4—1221no—
410C2×D4—1221no—
411C2×D4—1221no—
412C2×D4—1221no—
413C2×D4—4882no—
414C2×D4—4882no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—1221no—
425C22×C4—1222no—
426C22×C4—1224no—
427C22×C4—2442no—
428C22×C4—2442no—
429C22×C4C2×C22⋊D6112yes—
430C22×C4C2×C22⋊D6111yescharacteristic
431C22×C4C2×C22⋊D6111yescharacteristic
432C22×C4C23×D6111yescharacteristic
433SD16—1228no—
434C2×C8—612no—
435C2×C8—1221no—
436C2×C8—1221no—
437C2×C8—1221no—
438C2×C8—1222no—
439C2×C8—1222no—
440C2×C8—1221no—
441C2×C8—1222no—
442C2×C8—1221no—
443C4⋊C4—22no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—44no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—81no—
452C4⋊C4—612no—
453C4⋊C4—614no—
454C4⋊C4—1221no—
455C4⋊C4—1222no—
456C4⋊C4—1224no—
457C22⋊C4—24no—
458C22⋊C4—44no—
459C22⋊C4—82no—
460C22⋊C4—1228no—
461C22⋊C4—1228no—
462C22⋊C4—1224no—
463C42—42no—
464C42—1228no—
465C42D4×D6112yes—
466C42D4×D6112yes—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—1222no—
475C23—2441no—
476C23—2441no—
477C23—2441no—
478C23—2441no—
479C23—2441no—
480C23—2441no—
481C23C22×C22⋊D6111yescharacteristic
482Q8—22no—
483Q8—22no—
484Q8—41no—
485Q8—41no—
486Q8—4884no—
487D4—44no—
488D4—1222no—
489D4—1222no—
490D4—4888no—
491D4—2441no—
492D4—2441no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—1222no—
512C2×C4—1224no—
513C2×C4—2441no—
514C2×C4—2441no—
515C2×C4—2442no—
516C2×C4—2442no—
517C2×C4—2448no—
518C2×C4#1451974512yes—
519C2×C4#1451974512yes—
520C2×C4#1451979712yes—
521C2×C4#1451980812yes—
522C2×C4C2×D4×D6111yescharacteristic
523C2×C4C2×D4×D6111yescharacteristic
524C2×C4#41×D6111yescharacteristic
525C2×C4C2×#1175822411yescharacteristic
526C2×C4#1451982211yescharacteristic
527C2×C4#1451982411yescharacteristic
528C8—1222no—
529C8—2441no—
530C8—2442no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—4881no—
539C22—4881no—
540C22—4881no—
541C22—4881no—
542C22—2442no—
543C22—2442no—
544C22—2442no—
545C22C2×#1451974511yescharacteristic
546C22#1684040611yescharacteristic
547C22#1684041111yescharacteristic
548C22C2×#1451979711yescharacteristic
549C22C2×#1451980811yescharacteristic
550C22#1684056511yescharacteristic
551C22#1684057011yescharacteristic
552C4—22no—
553C4—42no—
554C4—42no—
555C4—81no—
556C4—82no—
557C4—2444no—
558C4—2442no—
559C4#1683992214yes—
560C4#1684027211yescharacteristic
561C4#1684034111yescharacteristic
562C4#1684041811yescharacteristic
563C4#1684043011yescharacteristic
564C2—82no—
565C2—4882no—
566C2#2144916011yescharacteristic
567C2C2×#1683992211yescharacteristic
568C2#2145512011yescharacteristic
569C2#2145776911yescharacteristic
570C2#2145811311yescharacteristic
571C2#2145822611yescharacteristic
572C2#2145844511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2538C2
socleC23×C61C22×D4
prime radicalC2×#16840477C2
hypercenter#2538D61
hypocenterC61#24121
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2538

Complements

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

NH ≅ G/Ntype
2. C2×#16840057C2semidirect
3. C2×C4×#11758201C2semidirect
4. #21447001C2semidirect
5. #21446820C2semidirect
6. #21446779C2semidirect
7. #21446614C2semidirect
8. #21446132C2semidirect
9. #21446016C2semidirect
10. #21445816C2semidirect
11. C61×#2538C2semidirect
281. #2538D61semidirect
331. C61#24121semidirect

Maximal subgroups

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

subgroupindex
C2×#168404772
C2×#168400572
C2×C4×#117582012
#214470012
#214468202
#214467792
#214466142
#214461322
#214460162
#214458162
C61×#25382
#2412161