Finite Group Databasebeta

← #29732201

Subgroups — #29732201

Order 15616.

1412 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840312C211yesmaximal, characteristic
2C2×#16839993C212yesmaximal, complemented
3#21447152C211yesmaximal, characteristic, complemented
4#21447053C211yesmaximal, characteristic, complemented
5#21446960C211yesmaximal, characteristic, complemented
6#21446906C211yesmaximal, characteristic, complemented
7#21446245C212yesmaximal, complemented
8#21446156C212yesmaximal, complemented
9#21446079C212yesmaximal, complemented
10C2×#16839517C211yesmaximal, characteristic, complemented
11C61×#2855C211yesmaximal, characteristic, complemented
12C22×#11758222—21no—
13C22×#11758201C2212yes—
14C23×#92418C2212yes—
15#16840312—28no—
16C2×#14519777C2211yescharacteristic
17C2×#14519774C2211yescharacteristic
18C2×#14519765—21no—
19C2×#14519751—21no—
20C2×#14519751C2212yes—
21C2×C4×D244—21no—
22C2×C4×D244C2212yes—
23#16839993—28no—
24C2×#14519708C2212yes—
25C2×#14519701C2212yes—
26C2×#14519697C2212yes—
27C2×#14519697C2212yes—
28C2×#14519697C2212yes—
29#16839524C2212yes—
30#16839524C2212yes—
31#16839517C2212yes—
32#16839517C2211yescharacteristic
33#16839517C2211yescharacteristic
34C2×C22⋊Q8×C61C2211yescharacteristic
35C22×C42×C61C2211yescharacteristic
36C61×#391C2211yescharacteristic
37C61×#391C2211yescharacteristic
38C61×#389C2211yescharacteristic
39C61×#389C2211yescharacteristic
40C61×#387—21no—
41C61×#387—21no—
42C61×#385—21no—
43C61×#381—21no—
44C61×#380C2211yescharacteristic
45C2×#11758222—28no—
46C2×#11758222—44no—
47C2×#11758222—44no—
48C22×Dic122—21no—
49C22×D244—21no—
50C22×D244—21no—
51C22×D244C2312yes—
52C22×C22⋊D61—22no—
53C22×C4×D61—22no—
54C2×#11758201—24no—
55C2×#11758201—24no—
56C2×#11758201—42no—
57C22×#92418—24no—
58C22×#92418—24no—
59C22×#92418—42no—
60C22×#92418C2312yes—
61C22×#92418C2312yes—
62#14519777—24no—
63#14519774—24no—
64#14519765—28no—
65#14519751—28no—
66#14519751—24no—
67#14519751D418yes—
68C4×D244—28no—
69C4×D244—24no—
70C4×D244—24no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61C2312yes—
73C2×#11758198—22no—
74C2×#11758198—22no—
75#14519708—24no—
76#14519701—24no—
77#14519697—24no—
78#14519697—24no—
79#14519697—42no—
80C22×Q8×C61C2311yescharacteristic
81C23×C4×C61D411yescharacteristic
82C23×C4×C61D411yescharacteristic
83C23×C4×C61C2311yescharacteristic
84C22⋊Q8×C61—24no—
85C2×C4⋊C4×C61—21no—
86C2×C4⋊C4×C61—21no—
87C2×C4⋊C4×C61—21no—
88C2×C4⋊C4×C61C2311yescharacteristic
89C2×C4⋊C4×C61C2311yescharacteristic
90C2×C4⋊C4×C61C2311yescharacteristic
91C2×C22⋊C4×C61—21no—
92C2×C22⋊C4×C61—21no—
93C2×C42×C61—24no—
94C2×C42×C61D411yescharacteristic
95C2×C42×C61D411yescharacteristic
96C2×C42×C61C2311yescharacteristic
97C2×C42×C61C2311yescharacteristic
98#95×C61—21no—
99#95×C61—21no—
100#95×C61—21no—
101#95×C61—21no—
102#95×C61—21no—
103#95×C61—21no—
104#95×C61—21no—
105C24×C61C2×D411yescharacteristic
106C23×D61—22no—
107C2×Q8×C61—22no—
108C2×Q8×C61—22no—
109C2×Q8×C61—21no—
110C2×Q8×C61—21no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—21no—
115C22×C4×C61—21no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—41no—
123C22×C4×C61SD1614yescomplemented
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D414yes—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2411yescharacteristic
131C2×C22⋊D61—28no—
132C2×C22⋊D61—44no—
133C2×C22⋊D61—44no—
134C22×Dic61—22no—
135#11758222—432no—
136C2×D244—22no—
137C2×D244—22no—
138C2×D244—22no—
139C2×D244—22no—
140C2×D244—24no—
141C2×D244—24no—
142C2×D244—22no—
143C2×D244—42no—
144C2×D244—42no—
145C2×D244—42no—
146C2×D244—42no—
147C2×D244C2×D414yes—
148C2×C4×D61—28no—
149C2×C4×D61—44no—
150C2×C4×D61—44no—
151C2×Dic122—24no—
152C2×Dic122—42no—
153C2×Dic122—42no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—42no—
159C4⋊C4×C61—42no—
160C4⋊C4×C61—42no—
161C22⋊C4×C61—42no—
162C22⋊C4×C61—42no—
163C42×C61—22no—
164C42×C61—22no—
165C42×C61—22no—
166C42×C61—22no—
167#11758201—48no—
168#11758198—28no—
169#11758198—28no—
170C4⋊Dic61—24no—
171C4⋊Dic61—22no—
172C4⋊Dic61C2×D414yes—
173C2×#92418—22no—
174C2×#92418—22no—
175C2×#92418—22no—
176C2×#92418—22no—
177C2×#92418—22no—
178C2×#92418—48no—
179C2×#92418#4114yescomplemented
180C23×C61—21no—
181C23×C61—21no—
182C23×C61—21no—
183C23×C61—21no—
184C23×C61#12012yes—
185C23×C61C2×SD1612yes—
186C23×C61AGL(1,ℤ/8)12yes—
187C23×C61C22×D411yescharacteristic
188C22×D61—24no—
189C22×D61—42no—
190C22×D61—42no—
191C22×D61—42no—
192C22×D61—42no—
193C22×D61—42no—
194C22×D61—42no—
195Q8×C61—42no—
196Q8×C61—42no—
197C2×C4×C61—21no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—28no—
206C2×C4×C61—24no—
207C2×C4×C61—24no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—44no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61#12012yes—
225C2×C4×C61C2×SD1612yes—
226C2×C4×C61C2×SD1612yes—
227C2×C4×C61#13512yes—
228C2×C4×C61C22×D412yes—
229C2×C4×C61C2×#4112yes—
230C2×C4×C61H(5,𝔽2)11yescharacteristic
231C2×C4×C612−1+411yescharacteristic
232C22⋊D61—416no—
233C2×Dic61—24no—
234C2×Dic61—42no—
235C2×Dic61—42no—
236D244—24no—
237D244—24no—
238D244—48no—
239D244—48no—
240D244—42no—
241D244—42no—
242C4×D61—416no—
243Dic122—48no—
244#92418—24no—
245#92418—42no—
246#25495—611nomaximal, Sylow
247C22×C61—22no—
248C22×C61—22no—
249C22×C61—22no—
250C22×C61—22no—
251C22×C61—22no—
252C22×C61—22no—
253C22×C61#45014yes—
254C22×C61C2×#12011yescharacteristic
255C22×C61#53211yescharacteristic
256C22×C61#53711yescharacteristic
257C22×C61C22×SD1611yescharacteristic
258C22×C61C2×#13511yescharacteristic
259C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
260C22×C61#57511yescharacteristic
261C2×D61—44no—
262C2×D61—44no—
263C2×D61—44no—
264C2×D61—82no—
265C2×D61—82no—
266C2×D61—82no—
267C2×D61—82no—
268C4×C61—22no—
269C4×C61—22no—
270C4×C61—22no—
271C4×C61—22no—
272C4×C61—41no—
273C4×C61—81no—
274C4×C61—81no—
275C4×C61—81no—
276C4×C61—81no—
277C4×C61#54612yes—
278C4×C61#54712yes—
279Dic61—44no—
280C2×#532—611no—
281C2×#465—612no—
282#2855D6111yescharacteristic, complemented
283#2058—611no—
284#2041—611no—
285#2024—611no—
286#2014—611no—
287#1893—612no—
288#1877—612no—
289#1864—612no—
290C2×#336—611no—
291C2×C61—24no—
292C2×C61#243011yescharacteristic
293C2×C61C2×#45011yescharacteristic
294C2×C61#300211yescharacteristic
295C2×C61#316711yescharacteristic
296C2×C61#318011yescharacteristic
297C2×C61#324611yescharacteristic
298C2×C61#324711yescharacteristic
299D61—84no—
300C22×#41—1221no—
301C22×SD16—612no—
302C23×C8—612no—
303#532—1228no—
304C2×C4⋊Q8—611no—
305C2×#127—611no—
306C2×#124—1221no—
307C2×C22⋊Q8C2×D6111yescharacteristic
308C2×#121—612no—
309C2×#121—1221no—
310C2×C4×D4—612no—
311C2×C4×D4—1221no—
312C22×C42C2×D6111yescharacteristic
313#465—1228no—
314C2×#106—612no—
315C2×#103—612no—
316C2×#102—612no—
317C2×#102—612no—
318C2×#102—612no—
319#391C2×D6111yescharacteristic
320#391C2×D6111yescharacteristic
321#389C2×D6111yescharacteristic
322#389C2×D6111yescharacteristic
323#387—21no—
324#387—21no—
325#385—21no—
326#381—21no—
327#380C2×D6111yescharacteristic
328#340—612no—
329#340—612no—
330#336—611no—
331#336—612no—
332#336—611no—
333C61#2549511yescharacteristic, complemented, Sylow
334C2×#41—1228no—
335C2×#41—2444no—
336C2×#41—2444no—
337C22×Q8—1221no—
338C22×Q8C22×D6111yescharacteristic
339C22×D4—612no—
340C22×D4—1222no—
341C22×D4—1221no—
342C22×D4—1221no—
343C23×C4—1222no—
344C23×C4C22⋊D6111yescharacteristic
345C23×C4C22⋊D6111yescharacteristic
346C23×C4C22×D6111yescharacteristic
347C2×SD16—1224no—
348C2×SD16—1224no—
349C2×SD16—2442no—
350C22×C8—612no—
351C22×C8—612no—
352C22×C8—1224no—
353C22×C8—1224no—
354C22×C8—2442no—
355C4⋊Q8—1224no—
356#127—1224no—
357#124—1228no—
358C22⋊Q8—24no—
359#121—618no—
360#121—1228no—
361#121—1224no—
362C4×D4—1224no—
363C4×D4—1224no—
364C4×D4—1228no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—612no—
369C2×C4⋊C4—1221no—
370C2×C4⋊C4C22×D6111yescharacteristic
371C2×C4⋊C4C22×D6111yescharacteristic
372C2×C4⋊C4C22×D6111yescharacteristic
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—1222no—
376C2×C22⋊C4—1222no—
377C2×C42—24no—
378C2×C42C22⋊D6111yescharacteristic
379C2×C42C22⋊D6111yescharacteristic
380C2×C42C22×D6111yescharacteristic
381C2×C42C22×D6111yescharacteristic
382#106—1224no—
383#103—1224no—
384#102—1224no—
385#102—1224no—
386#102—2442no—
387#95—21no—
388#95—21no—
389#95—21no—
390#95—21no—
391#95—21no—
392#95—21no—
393#95—21no—
394C24—1222no—
395C24C2×C22⋊D6111yescharacteristic
396#41—24432no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—1224no—
402C2×Q8—2442no—
403C2×Q8—2442no—
404C2×D4—614no—
405C2×D4—1224no—
406C2×D4—1224no—
407C2×D4—1228no—
408C2×D4—1222no—
409C2×D4—1222no—
410C2×D4—1222no—
411C2×D4—1222no—
412C2×D4—1222no—
413C2×D4—2442no—
414C2×D4—2442no—
415C2×D4—2444no—
416C2×D4—2444no—
417C2×D4—2442no—
418C2×D4—2442no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—41no—
431C22×C4—1228no—
432C22×C4—1222no—
433C22×C4—2444no—
434C22×C4—2444no—
435C22×C4#1175820114yescomplemented
436C22×C4D4×D6114yes—
437C22×C4C2×C22⋊D6111yescharacteristic
438C22×C4C2×C22⋊D6111yescharacteristic
439C22×C4C2×C22⋊D6111yescharacteristic
440C22×C4C2×C22⋊D6111yescharacteristic
441C22×C4C2×C22⋊D6111yescharacteristic
442C22×C4C23×D6111yescharacteristic
443SD16—2448no—
444C2×C8—614no—
445C2×C8—1222no—
446C2×C8—1222no—
447C2×C8—1222no—
448C2×C8—1222no—
449C2×C8—1222no—
450C2×C8—2448no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—614no—
459C4⋊C4—1224no—
460C4⋊C4—1222no—
461C22⋊C4—42no—
462C22⋊C4—42no—
463C22⋊C4—1228no—
464C22⋊C4—1228no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—22no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—1224no—
474C23—2442no—
475C23—2442no—
476C23—2442no—
477C23—2442no—
478C23—2442no—
479C23—2442no—
480C23#1451974512yes—
481C23C2×#1175820112yes—
482C23#1451980212yes—
483C23C22×C22⋊D6111yescharacteristic
484Q8—42no—
485Q8—42no—
486Q8—2448no—
487D4—1224no—
488D4—1224no—
489D4—2448no—
490D4—2448no—
491D4—24416no—
492D4—2442no—
493D4—2442no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—28no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—44no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—1224no—
522C2×C4—24416no—
523C2×C4—2442no—
524C2×C4—2442no—
525C2×C4#1451974512yes—
526C2×C4C2×#1175820112yes—
527C2×C4C2×#1175820112yes—
528C2×C4#1451979712yes—
529C2×C4C2×D4×D6112yes—
530C2×C4#41×D6112yes—
531C2×C4#1451982311yescharacteristic
532C2×C4#1451982411yescharacteristic
533C8—1224no—
534C8—2442no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—4882no—
542C22—4882no—
543C22—4882no—
544C22—4882no—
545C22—2444no—
546C22—2444no—
547C22—2444no—
548C22#1683991214yes—
549C22C2×#1451974511yescharacteristic
550C22#1684030811yescharacteristic
551C22#1684034011yescharacteristic
552C22C22×#1175820111yescharacteristic
553C22C2×#1451979711yescharacteristic
554C22C2×#1451980211yescharacteristic
555C22#1684056011yescharacteristic
556C4—22no—
557C4—22no—
558C4—22no—
559C4—22no—
560C4—41no—
561C4—81no—
562C4—81no—
563C4—81no—
564C4—81no—
565C4—2444no—
566C4#1684040412yes—
567C4#1684041012yes—
568C2—24no—
569C2—4884no—
570C2#2144943111yescharacteristic
571C2C2×#1683991211yescharacteristic
572C2#2145503511yescharacteristic
573C2#2145637911yescharacteristic
574C2#2145651811yescharacteristic
575C2#2145713111yescharacteristic
576C2#2145714511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2855C2
socleC23×C61C22×D4
prime radicalC2×#16840312C2
hypercenter#2855D61
hypocenterC61#25495
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2855

Complements

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

NH ≅ G/Ntype
2. C2×#16839993C2semidirect
3. #21447152C2semidirect
4. #21447053C2semidirect
5. #21446960C2semidirect
6. #21446906C2semidirect
7. #21446245C2semidirect
8. #21446156C2semidirect
9. #21446079C2semidirect
10. C2×#16839517C2semidirect
11. C61×#2855C2semidirect
123. C22×C4×C61SD16semidirect
179. C2×#92418#41semidirect
282. #2855D61semidirect
333. C61#25495semidirect
435. C22×C4#11758201semidirect

Maximal subgroups

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

subgroupindex
C2×#168403122
C2×#168399932
#214471522
#214470532
#214469602
#214469062
#214462452
#214461562
#214460792
C2×#168395172
C61×#28552
#2549561