Finite Group Databasebeta

← #29732193

Subgroups — #29732193

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×#16839989C212yesmaximal, complemented
2#21454158C211yesmaximal, characteristic, complemented
3#21447149C211yesmaximal, characteristic, complemented
4#21447050C211yesmaximal, characteristic, complemented
5#21446957C211yesmaximal, characteristic, complemented
6#21446907C211yesmaximal, characteristic, complemented
7#21446248C212yesmaximal, complemented
8#21446153C212yesmaximal, complemented
9#21446080C212yesmaximal, complemented
10C2×#16839517C211yesmaximal, characteristic, complemented
11C2×C61×#532C211yesmaximal, characteristic
12C22×#11758200C2212yes—
13C23×#92418C2212yes—
14C2×#14519756C2211yescharacteristic
15#16839989—28no—
16C2×#14519708C2212yes—
17C2×#14519700C2212yes—
18C2×#14519698C2212yes—
19C2×#14519698C2212yes—
20C2×#14519698C2212yes—
21#16839668C2211yescharacteristic
22#16839668C2211yescharacteristic
23#16839660C2211yescharacteristic
24#16839660C2211yescharacteristic
25#16839646—21no—
26#16839646—21no—
27#16839635—21no—
28#16839625—21no—
29#16839622C2211yescharacteristic
30#16839524C2212yes—
31#16839524C2212yes—
32#16839517C2212yes—
33#16839517C2211yescharacteristic
34#16839517C2211yescharacteristic
35C22×#41×C61—21no—
36C61×#532—28no—
37C2×C4⋊Q8×C61C2211yescharacteristic
38C2×#127×C61C2211yescharacteristic
39C2×#124×C61—21no—
40C2×#121×C61—21no—
41C2×#121×C61C2212yes—
42C2×C4×D4×C61—21no—
43C2×C4×D4×C61C2212yes—
44C22×C42×C61C2211yescharacteristic
45C22×Dic122C2311yescharacteristic
46C2×#11758200—24no—
47C2×#11758200—24no—
48C2×#11758200—42no—
49C22×#92418—24no—
50C22×#92418—24no—
51C22×#92418—42no—
52C22×#92418C2312yes—
53C22×#92418C2312yes—
54#14519756—24no—
55C2×C4⋊Dic61C2311yescharacteristic
56C2×C4⋊Dic61C2311yescharacteristic
57C2×C4⋊Dic61C2311yescharacteristic
58C2×#11758196—21no—
59C2×#11758196—21no—
60C2×#11758196—21no—
61C2×C22⋊Dic61—21no—
62C2×C22⋊Dic61—21no—
63#14519708—24no—
64#14519700—24no—
65#14519698—24no—
66#14519698—24no—
67#14519698—42no—
68#14519684—21no—
69#14519684—21no—
70#14519684—21no—
71#14519684—21no—
72#14519684—21no—
73#14519684—21no—
74#14519684—21no—
75C2×#41×C61—28no—
76C2×#41×C61—44no—
77C2×#41×C61—44no—
78C22×Q8×C61—21no—
79C22×D4×C61—22no—
80C22×D4×C61—21no—
81C22×D4×C61—21no—
82C22×D4×C61C2312yes—
83C23×C4×C61—22no—
84C23×C4×C61D411yescharacteristic
85C23×C4×C61D411yescharacteristic
86C23×C4×C61C2311yescharacteristic
87C4⋊Q8×C61—24no—
88#127×C61—24no—
89#124×C61—28no—
90#121×C61—28no—
91#121×C61—24no—
92#121×C61D418yes—
93C4×D4×C61—28no—
94C4×D4×C61—24no—
95C4×D4×C61—24no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61C2312yes—
98C2×C22⋊C4×C61—22no—
99C2×C22⋊C4×C61—22no—
100C2×C42×C61—24no—
101C2×C42×C61D411yescharacteristic
102C2×C42×C61D411yescharacteristic
103C2×C42×C61C2311yescharacteristic
104C2×C42×C61C2311yescharacteristic
105C24×C61—22no—
106C24×C61C2×D411yescharacteristic
107#41×C61—432no—
108C2×Q8×C61—24no—
109C2×Q8×C61—42no—
110C2×Q8×C61—42no—
111C2×D4×C61—24no—
112C2×D4×C61—24no—
113C2×D4×C61—28no—
114C2×D4×C61—22no—
115C2×D4×C61—22no—
116C2×D4×C61—22no—
117C2×D4×C61—22no—
118C2×D4×C61—22no—
119C2×D4×C61—42no—
120C2×D4×C61—42no—
121C2×D4×C61—44no—
122C2×D4×C61—44no—
123C2×D4×C61—42no—
124C2×D4×C61—42no—
125C2×D4×C61C2×D414yes—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—28no—
132C22×C4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—41no—
136C22×C4×C61—44no—
137C22×C4×C61—44no—
138C22×C4×C61SD1614yescomplemented
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D414yes—
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2411yescharacteristic
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C2×Dic122—22no—
151C2×Dic122—22no—
152C2×Dic122—21no—
153C2×Dic122—21no—
154C4⋊C4×C61—24no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61C2×D414yes—
157C22⋊C4×C61—28no—
158C22⋊C4×C61—28no—
159C42×C61—22no—
160C42×C61—22no—
161C42×C61—22no—
162C42×C61—22no—
163C22⋊Dic61—42no—
164C22⋊Dic61—42no—
165#11758200—48no—
166C4⋊Dic61—22no—
167C4⋊Dic61—22no—
168C4⋊Dic61—21no—
169C4⋊Dic61—21no—
170#11758196—42no—
171#11758196—42no—
172#11758196—42no—
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—24no—
185C23×C61—42no—
186C23×C61—42no—
187C23×C61—42no—
188C23×C61—42no—
189C23×C61—42no—
190C23×C61—42no—
191C23×C61#12012yes—
192C23×C61C2×SD1612yes—
193C23×C61AGL(1,ℤ/8)12yes—
194C23×C61C22×D411yescharacteristic
195Q8×C61—48no—
196D4×C61—24no—
197D4×C61—24no—
198D4×C61—48no—
199D4×C61—48no—
200D4×C61—416no—
201D4×C61—42no—
202D4×C61—42no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—28no—
212C2×C4×C61—24no—
213C2×C4×C61—24no—
214C2×C4×C61—21no—
215C2×C4×C61—24no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—44no—
219C2×C4×C61—416no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61#12012yes—
223C2×C4×C61C2×SD1612yes—
224C2×C4×C61C2×SD1612yes—
225C2×C4×C61#13512yes—
226C2×C4×C61C22×D412yes—
227C2×C4×C61C2×#4112yes—
228C2×C4×C61H(5,𝔽2)11yescharacteristic
229C2×C4×C612−1+411yescharacteristic
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61—41no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242Dic122—42no—
243Dic122—42no—
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—44no—
254C22×C61—44no—
255C22×C61—44no—
256C22×C61—82no—
257C22×C61—82no—
258C22×C61—82no—
259C22×C61—82no—
260C22×C61#45014yes—
261C22×C61C2×#12011yescharacteristic
262C22×C61#53211yescharacteristic
263C22×C61#53711yescharacteristic
264C22×C61C22×SD1611yescharacteristic
265C22×C61C2×#13511yescharacteristic
266C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
267C22×C61#57511yescharacteristic
268C4×C61—22no—
269C4×C61—22no—
270C4×C61—22no—
271C4×C61—22no—
272C4×C61—44no—
273C4×C61—41no—
274C4×C61#54612yes—
275C4×C61#54712yes—
276Dic61—81no—
277Dic61—81no—
278Dic61—81no—
279Dic61—81no—
280C2×#532D6111yescharacteristic
281C2×#465—612no—
282#2855—611no—
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—84no—
293C2×C61#243011yescharacteristic
294C2×C61C2×#45011yescharacteristic
295C2×C61#300211yescharacteristic
296C2×C61#316711yescharacteristic
297C2×C61#318011yescharacteristic
298C2×C61#324611yescharacteristic
299C2×C61#324711yescharacteristic
300C22×#41—21no—
301C22×SD16—612no—
302C23×C8—612no—
303#532—28no—
304C2×C4⋊Q8C2×D6111yescharacteristic
305C2×#127C2×D6111yescharacteristic
306C2×#124—21no—
307C2×C22⋊Q8—611no—
308C2×#121—21no—
309C2×#121C2×D6112yes—
310C2×C4×D4—21no—
311C2×C4×D4C2×D6112yes—
312C22×C42C2×D6111yescharacteristic
313#465—1228no—
314C2×#106—612no—
315C2×#103—612no—
316C2×#102—612no—
317C2×#102—612no—
318C2×#102—612no—
319#391—611no—
320#391—611no—
321#389—611no—
322#389—611no—
323#387—1221no—
324#387—1221no—
325#385—1221no—
326#381—1221no—
327#380—611no—
328#340—612no—
329#340—612no—
330#336—611no—
331#336—612no—
332#336—611no—
333C61#2549511yescharacteristic, complemented, Sylow
334C2×#41—28no—
335C2×#41—44no—
336C2×#41—44no—
337C22×Q8—21no—
338C22×Q8—611no—
339C22×D4—22no—
340C22×D4—21no—
341C22×D4—21no—
342C22×D4C22×D6112yes—
343C23×C4—22no—
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—24no—
356#127—24no—
357#124—28no—
358C22⋊Q8—1224no—
359#121—28no—
360#121—24no—
361#121C22⋊D6118yes—
362C4×D4—28no—
363C4×D4—24no—
364C4×D4—24no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—611no—
367C2×C4⋊C4—611no—
368C2×C4⋊C4—611no—
369C2×C4⋊C4—1221no—
370C2×C4⋊C4—1221no—
371C2×C4⋊C4—1221no—
372C2×C4⋊C4C22×D6112yes—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—1221no—
376C2×C22⋊C4—1221no—
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—1221no—
388#95—1221no—
389#95—1221no—
390#95—1221no—
391#95—1221no—
392#95—1221no—
393#95—1221no—
394C24—22no—
395C24C2×C22⋊D6111yescharacteristic
396#41—432no—
397C2×Q8—24no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8—1222no—
401C2×Q8—1222no—
402C2×Q8—1221no—
403C2×Q8—1221no—
404C2×D4—24no—
405C2×D4—24no—
406C2×D4—28no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4C2×C22⋊D6114yes—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—28no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—41no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—1221no—
432C22×C4—1221no—
433C22×C4—1221no—
434C22×C4—1221no—
435C22×C4#1175820014yescomplemented
436C22×C4C2×C22⋊D6111yescharacteristic
437C22×C4C2×C22⋊D6114yes—
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—24no—
452C4⋊C4—22no—
453C4⋊C4—1222no—
454C4⋊C4—1222no—
455C4⋊C4—1221no—
456C4⋊C4—1221no—
457C4⋊C4—2442no—
458C4⋊C4—2442no—
459C4⋊C4—2442no—
460C4⋊C4C2×C22⋊D6114yes—
461C22⋊C4—28no—
462C22⋊C4—28no—
463C22⋊C4—2442no—
464C22⋊C4—2442no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—22no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—24no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23#1451974612yes—
481C23C2×#1175820012yes—
482C23#1451980312yes—
483C23C22×C22⋊D6111yescharacteristic
484Q8—48no—
485Q8—2442no—
486Q8—2442no—
487D4—24no—
488D4—24no—
489D4—48no—
490D4—48no—
491D4—416no—
492D4—42no—
493D4—42no—
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—24no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—44no—
510C2×C4—416no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—2441no—
514C2×C4—2441no—
515C2×C4—2441no—
516C2×C4—2441no—
517C2×C4—2441no—
518C2×C4—2441no—
519C2×C4—2441no—
520C2×C4—2441no—
521C2×C4—2441no—
522C2×C4—2441no—
523C2×C4—2441no—
524C2×C4—2441no—
525C2×C4#1451974612yes—
526C2×C4C2×#1175820012yes—
527C2×C4C2×#1175820012yes—
528C2×C4#1451979712yes—
529C2×C4C22×C22⋊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—44no—
542C22—44no—
543C22—44no—
544C22—82no—
545C22—82no—
546C22—82no—
547C22—82no—
548C22#1683991314yes—
549C22C2×#1451974611yescharacteristic
550C22#1684030811yescharacteristic
551C22#1684034011yescharacteristic
552C22C22×#1175820011yescharacteristic
553C22C2×#1451979711yescharacteristic
554C22C2×#1451980311yescharacteristic
555C22#1684056011yescharacteristic
556C4—22no—
557C4—22no—
558C4—22no—
559C4—22no—
560C4—44no—
561C4—41no—
562C4—4881no—
563C4—4881no—
564C4—4881no—
565C4—4881no—
566C4#1684040012yes—
567C4#1684041112yes—
568C2—24no—
569C2—84no—
570C2#2144942311yescharacteristic
571C2C2×#1683991311yescharacteristic
572C2#2145503711yescharacteristic
573C2#2145637411yescharacteristic
574C2#2145651111yescharacteristic
575C2#2145713511yescharacteristic
576C2#2145713911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC2×C61×#532C2
socleC23×C61C22×D4
hypercenterC2×#532D61
hypocenterC61#25495
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#532

Complements

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

NH ≅ G/Ntype
1. C2×#16839989C2semidirect
2. #21454158C2semidirect
3. #21447149C2semidirect
4. #21447050C2semidirect
5. #21446957C2semidirect
6. #21446907C2semidirect
7. #21446248C2semidirect
8. #21446153C2semidirect
9. #21446080C2semidirect
10. C2×#16839517C2semidirect
138. C22×C4×C61SD16semidirect
179. C2×#92418#41semidirect
333. C61#25495semidirect
435. C22×C4#11758200semidirect

Maximal subgroups

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

subgroupindex
C2×#168399892
#214541582
#214471492
#214470502
#214469572
#214469072
#214462482
#214461532
#214460802
C2×#168395172
C2×C61×#5322
#2549561