Finite Group Databasebeta

← #29714788

Subgroups — #29714788

Order 15616.

1394 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839959C211yesmaximal, characteristic, complemented
2C2×C4×#11758200C211yesmaximal, characteristic, complemented
3#21451189C211yesmaximal, characteristic, complemented
4#21447050C212yesmaximal, complemented
5#21446907C212yesmaximal, complemented
6#21446756C211yesmaximal, characteristic, complemented
7#21446626C211yesmaximal, characteristic, complemented
8#21446153C212yesmaximal, complemented
9#21446080C212yesmaximal, complemented
10#21445819C211yesmaximal, characteristic, complemented
11C2×C61×#550C211yesmaximal, characteristic
12C22×#11758200C2211yescharacteristic
13C22×#11758200C2212yes—
14C2×C4×Dic122C2211yescharacteristic
15#16839959—24no—
16C4×#11758200—24no—
17C2×#14519708C2211yescharacteristic
18C2×C4⋊#92418C2211yescharacteristic
19C2×#14519700C2211yescharacteristic
20C2×#14519698C2211yescharacteristic
21C2×#14519698C2212yes—
22C2×#14519698C2212yes—
23C2×#14519698C2212yes—
24C2×#14519698C2212yes—
25C2×C4×#92418C2211yescharacteristic
26#16839695—22no—
27#16839669—21no—
28#16839668C2212yes—
29#16839660C2212yes—
30#16839658—21no—
31#16839632C2211yescharacteristic
32#16839524C2212yes—
33#16839517C2212yes—
34#16839517C2212yes—
35C22×#41×C61—21no—
36C61×#550—24no—
37C61×#550—24no—
38C2×#127×C61C2212yes—
39C2×#127×C61C2211yescharacteristic
40C2×#121×C61—22no—
41C2×#121×C61—21no—
42C2×C4×Q8×C61C2211yescharacteristic
43C2×C4×D4×C61C2212yes—
44C2×C4×D4×C61C2211yescharacteristic
45C22×Dic122C2311yescharacteristic
46C2×#11758200—22no—
47C2×#11758200—22no—
48C2×#11758200—24no—
49C2×#11758200—24no—
50C2×#11758200—22no—
51C2×#11758200—42no—
52C22×#92418C2311yescharacteristic
53C22×#92418C2312yes—
54C22×#92418C2311yescharacteristic
55C4×Dic122—22no—
56C4×Dic122—22no—
57C2×C4⋊Dic61—21no—
58C2×C4⋊Dic61—21no—
59C2×C4⋊Dic61C2311yescharacteristic
60C2×C4⋊Dic61C2312yes—
61C2×#11758196—21no—
62C2×#11758196—22no—
63C2×C4×Dic61—21no—
64#14519708—22no—
65C4⋊#92418—22no—
66#14519700—22no—
67#14519698—24no—
68#14519698—22no—
69#14519698—24no—
70#14519698—42no—
71#14519698—42no—
72C4×#92418—22no—
73#14519684—22no—
74#14519684—22no—
75#14519684—22no—
76C2×#41×C61—28no—
77C2×#41×C61—44no—
78C2×#41×C61—44no—
79C22×Q8×C61D411yescharacteristic
80C22×D4×C61—22no—
81C22×D4×C61—21no—
82C22×D4×C61—22no—
83C22×D4×C61—21no—
84C22×D4×C61C2312yes—
85C22×D4×C61C2311yescharacteristic
86C23×C4×C61—22no—
87C23×C4×C61—21no—
88#127×C61—24no—
89#127×C61—24no—
90#127×C61—22no—
91#127×C61D414yes—
92#121×C61—28no—
93#121×C61—216no—
94C4×Q8×C61—22no—
95C4×Q8×C61—22no—
96C4×D4×C61—24no—
97C4×D4×C61—24no—
98C4×D4×C61—22no—
99C4×D4×C61D414yes—
100C2×C4⋊C4×C61D412yes—
101C2×C4⋊C4×C61D411yescharacteristic
102C2×C4⋊C4×C61C2311yescharacteristic
103C2×C22⋊C4×C61—22no—
104C2×C22⋊C4×C61—21no—
105C2×C42×C61C2312yes—
106C2×C42×C61C2311yescharacteristic
107C24×C61—22no—
108C24×C61—21no—
109#41×C61—432no—
110C2×Q8×C61—22no—
111C2×Q8×C61—22no—
112C2×Q8×C61SD1614yescomplemented
113C2×D4×C61—28no—
114C2×D4×C61—28no—
115C2×D4×C61—24no—
116C2×D4×C61—24no—
117C2×D4×C61—22no—
118C2×D4×C61—21no—
119C2×D4×C61—22no—
120C2×D4×C61—21no—
121C2×D4×C61—22no—
122C2×D4×C61—21no—
123C2×D4×C61—21no—
124C2×D4×C61—22no—
125C2×D4×C61—21no—
126C2×D4×C61—22no—
127C2×D4×C61—44no—
128C2×D4×C61—44no—
129C2×D4×C61—42no—
130C2×D4×C61—42no—
131C2×D4×C61—81no—
132C2×D4×C61—81no—
133C2×D4×C61—82no—
134C2×D4×C61—82no—
135C2×D4×C61C2×D414yes—
136C2×D4×C61C2×D412yes—
137C22×C4×C61—28no—
138C22×C4×C61—21no—
139C22×C4×C61—24no—
140C22×C4×C61—44no—
141C22×C4×C61—44no—
142C22×C4×C61—42no—
143C22×C4×C61—42no—
144C22×C4×C61C2×D412yes—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D412yes—
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2411yescharacteristic
149C22×Dic61—21no—
150C22×Dic61—22no—
151C22×Dic61—21no—
152C2×Dic122—21no—
153C2×Dic122—21no—
154C2×Dic122—21no—
155C2×Dic122—21no—
156C2×Dic122—21no—
157C2×Dic122#4112yescomplemented
158C4⋊C4×C61—24no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61C2×D412yes—
162C22⋊C4×C61—28no—
163C22⋊C4×C61—24no—
164C42×C61—22no—
165C42×C61—22no—
166C42×C61C2×D412yes—
167C42×C61C2×D412yes—
168#11758200—28no—
169#11758200—48no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—22no—
173C4⋊Dic61—22no—
174C4⋊Dic61—42no—
175C4⋊Dic61—81no—
176#11758196—42no—
177#11758196—82no—
178C4×Dic61—42no—
179C2×#92418—21no—
180C2×#92418—21no—
181C2×#92418—21no—
182C2×#92418—22no—
183C2×#92418—22no—
184C2×#92418—21no—
185C2×#92418—22no—
186C2×#92418—21no—
187C2×#92418#4112yescomplemented
188C23×C61—24no—
189C23×C61—22no—
190C23×C61—41no—
191C23×C61—42no—
192C23×C61—42no—
193C23×C61—41no—
194C23×C61—42no—
195C23×C61—41no—
196C23×C61—42no—
197C23×C61—41no—
198C23×C61—42no—
199C23×C61—41no—
200C23×C61—42no—
201C23×C61—41no—
202C23×C61C22×D411yescharacteristic
203Q8×C61—28no—
204D4×C61—22no—
205D4×C61—22no—
206D4×C61—24no—
207D4×C61—24no—
208D4×C61—416no—
209D4×C61—42no—
210D4×C61—42no—
211D4×C61—41no—
212D4×C61—41no—
213D4×C61—48no—
214D4×C61—84no—
215D4×C61—88no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—416no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—48no—
228C2×C4×C61#12012yes—
229C2×C4×C61#12012yes—
230C2×C4×C61C2×SD1614yes—
231C2×C4×C61C2×SD1612yes—
232C2×C4×C61AGL(1,ℤ/8)12yes—
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×C4×C61H(5,𝔽2)11yescharacteristic
238C2×C4×C612−1+411yescharacteristic
239C2×Dic61—22no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61—41no—
246C2×Dic61—42no—
247C2×Dic61—41no—
248Dic122—22no—
249Dic122—22no—
250Dic122—41no—
251Dic122—41no—
252#92418—22no—
253#92418—41no—
254#92418—42no—
255#24188—611nomaximal, Sylow
256C22×C61—42no—
257C22×C61—44no—
258C22×C61—44no—
259C22×C61—42no—
260C22×C61—44no—
261C22×C61—42no—
262C22×C61—81no—
263C22×C61—82no—
264C22×C61—82no—
265C22×C61—81no—
266C22×C61—82no—
267C22×C61—81no—
268C22×C61—82no—
269C22×C61—81no—
270C22×C61C2×#12011yescharacteristic
271C22×C61#54811yescharacteristic
272C22×C61#55011yescharacteristic
273C22×C61C22×SD1611yescharacteristic
274C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
275C22×C61#57611yescharacteristic
276C22×C61#57711yescharacteristic
277C4×C61—24no—
278C4×C61—22no—
279C4×C61—42no—
280C4×C61—42no—
281C4×C61#45014yes—
282C4×C61#52511yescharacteristic
283C4×C61#53711yescharacteristic
284C4×C61#54611yescharacteristic
285C4×C61#54711yescharacteristic
286Dic61—42no—
287Dic61—82no—
288Dic61—81no—
289C2×#550D6111yescharacteristic
290C2×#460—611no—
291C2×C4×SD16—611no—
292#2582—611no—
293#2041—612no—
294#2014—612no—
295#1984—611no—
296#1957—611no—
297#1877—612no—
298#1864—612no—
299#1802—611no—
300C2×C61—82no—
301C2×C61—84no—
302C2×C61#240711yescharacteristic
303C2×C61C2×#45011yescharacteristic
304C2×C61#301211yescharacteristic
305C2×C61#336211yescharacteristic
306C2×C61#336811yescharacteristic
307C2×C61#339211yescharacteristic
308C2×C61#339911yescharacteristic
309C22×#41—21no—
310C22×SD16—611no—
311C22×SD16—612no—
312#550—24no—
313#550—24no—
314C2×#127C2×D6112yes—
315C2×#127C2×D6111yescharacteristic
316C2×#121—21no—
317C2×#121—22no—
318C2×C4×Q8—611no—
319C2×C4×Q8C2×D6111yescharacteristic
320C2×C4×D4C2×D6112yes—
321C2×C4×D4C2×D6111yescharacteristic
322#460—1224no—
323C4×SD16—1224no—
324C2×#106—611no—
325C2×C4⋊C8—611no—
326C2×#103—611no—
327C2×#102—612no—
328C2×#102—612no—
329C2×#102—611no—
330C2×#102—612no—
331C2×#102—612no—
332C2×C4×C8—611no—
333#398—1222no—
334#391—612no—
335#391—1221no—
336#389—612no—
337#389—1221no—
338#384—611no—
339#340—612no—
340#336—612no—
341#336—612no—
342C61#2418811yescharacteristic, complemented, Sylow
343C2×#41—28no—
344C2×#41—44no—
345C2×#41—44no—
346C22×Q8—611no—
347C22×Q8C22⋊D6111yescharacteristic
348C22×D4—22no—
349C22×D4—21no—
350C22×D4—22no—
351C22×D4—21no—
352C22×D4C22×D6112yes—
353C22×D4C22×D6111yescharacteristic
354C23×C4—22no—
355C23×C4—21no—
356C2×SD16—1222no—
357C2×SD16—1222no—
358C2×SD16—1222no—
359C2×SD16—1224no—
360C2×SD16—1224no—
361C2×SD16—2442no—
362C22×C8—611no—
363C22×C8—612no—
364C22×C8—611no—
365#127—24no—
366#127—24no—
367#127—22no—
368#127C22⋊D6114yes—
369#121—28no—
370#121—216no—
371C4×Q8—22no—
372C4×Q8—22no—
373C4×Q8—1222no—
374C4×Q8—1222no—
375C4×D4—24no—
376C4×D4—24no—
377C4×D4—22no—
378C4×D4C22⋊D6114yes—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—612no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—1222no—
383C2×C4⋊C4—1221no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4C22⋊D6111yescharacteristic
386C2×C4⋊C4C22⋊D6112yes—
387C2×C4⋊C4C22×D6111yescharacteristic
388C2×C22⋊C4—22no—
389C2×C22⋊C4—21no—
390C2×C42—1221no—
391C2×C42C22×D6112yes—
392C2×C42C22×D6111yescharacteristic
393#106—1222no—
394C4⋊C8—1222no—
395#103—1222no—
396#102—1224no—
397#102—1222no—
398#102—1224no—
399#102—2442no—
400#102—2442no—
401C4×C8—1222no—
402#95—1222no—
403#95—1222no—
404#95—1222no—
405C24—22no—
406C24—21no—
407#41—432no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—612no—
411C2×Q8—1221no—
412C2×Q8—1221no—
413C2×Q8—1221no—
414C2×Q8—1221no—
415C2×Q8—1221no—
416C2×Q8#1175820014yescomplemented
417C2×D4—28no—
418C2×D4—28no—
419C2×D4—22no—
420C2×D4—21no—
421C2×D4—22no—
422C2×D4—21no—
423C2×D4—22no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—22no—
427C2×D4—21no—
428C2×D4—22no—
429C2×D4—24no—
430C2×D4—24no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—81no—
436C2×D4—81no—
437C2×D4—82no—
438C2×D4—82no—
439C2×D4C2×C22⋊D6114yes—
440C2×D4C2×C22⋊D6112yes—
441C22×C4—28no—
442C22×C4—21no—
443C22×C4—24no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—1221no—
449C22×C4—1222no—
450C22×C4—1221no—
451C22×C4C2×C22⋊D6112yes—
452C22×C4C2×C22⋊D6111yescharacteristic
453C22×C4C2×C22⋊D6112yes—
454C22×C4C2×C22⋊D6111yescharacteristic
455C22×C4C23×D6111yescharacteristic
456SD16—1228no—
457SD16—2448no—
458C2×C8—612no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8—1221no—
462C2×C8—1222no—
463C2×C8—1222no—
464C2×C8—1221no—
465C2×C8—1222no—
466C2×C8—1221no—
467C4⋊C4—21no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—1222no—
473C4⋊C4—1222no—
474C4⋊C4—4882no—
475C4⋊C4—4881no—
476C4⋊C4—2442no—
477C4⋊C4—2442no—
478C4⋊C4C2×C22⋊D6112yes—
479C22⋊C4—28no—
480C22⋊C4—24no—
481C42—22no—
482C42—22no—
483C42—2442no—
484C42C2×C22⋊D6112yes—
485C42C2×C22⋊D6112yes—
486C23—24no—
487C23—22no—
488C23—42no—
489C23—41no—
490C23—42no—
491C23—41no—
492C23—42no—
493C23—41no—
494C23—42no—
495C23—41no—
496C23—41no—
497C23—42no—
498C23—42no—
499C23—41no—
500C23C22×C22⋊D6111yescharacteristic
501Q8—28no—
502Q8—1222no—
503Q8—1222no—
504Q8—2441no—
505Q8—2441no—
506D4—22no—
507D4—22no—
508D4—24no—
509D4—24no—
510D4—416no—
511D4—42no—
512D4—42no—
513D4—41no—
514D4—41no—
515D4—48no—
516D4—84no—
517D4—88no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—416no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—48no—
530C2×C4—1222no—
531C2×C4—2441no—
532C2×C4—2442no—
533C2×C4—2441no—
534C2×C4—2442no—
535C2×C4—2441no—
536C2×C4—2442no—
537C2×C4—2441no—
538C2×C4—2441no—
539C2×C4#1451974612yes—
540C2×C4#1451974612yes—
541C2×C4C2×#1175820014yes—
542C2×C4C2×#1175820012yes—
543C2×C4#1451980312yes—
544C2×C4C22×C22⋊D6111yescharacteristic
545C2×C4C22×C22⋊D6111yescharacteristic
546C2×C4#41×D6111yescharacteristic
547C2×C4#41×D6111yescharacteristic
548C2×C4#1451982311yescharacteristic
549C2×C4#1451982411yescharacteristic
550C8—1222no—
551C8—2441no—
552C8—2442no—
553C22—44no—
554C22—42no—
555C22—44no—
556C22—42no—
557C22—42no—
558C22—44no—
559C22—82no—
560C22—81no—
561C22—82no—
562C22—81no—
563C22—82no—
564C22—81no—
565C22—82no—
566C22—81no—
567C22C2×#1451974611yescharacteristic
568C22#1684041911yescharacteristic
569C22#1684043111yescharacteristic
570C22C22×#1175820011yescharacteristic
571C22C2×#1451980311yescharacteristic
572C22#1684056511yescharacteristic
573C22#1684057311yescharacteristic
574C4—22no—
575C4—24no—
576C4—42no—
577C4—42no—
578C4—4881no—
579C4—4882no—
580C4—2442no—
581C4#1683991314yes—
582C4#1684026911yescharacteristic
583C4#1684034011yescharacteristic
584C4#1684040011yescharacteristic
585C4#1684041111yescharacteristic
586C2—84no—
587C2—82no—
588C2#2144915811yescharacteristic
589C2C2×#1683991311yescharacteristic
590C2#2145512311yescharacteristic
591C2#2145848611yescharacteristic
592C2#2145854411yescharacteristic
593C2#2145880011yescharacteristic
594C2#2145886111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC2×C61×#550C2
socleC23×C61C22×D4
hypercenterC2×#550D61
hypocenterC61#24188
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#550

Complements

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

NH ≅ G/Ntype
1. C2×#16839959C2semidirect
2. C2×C4×#11758200C2semidirect
3. #21451189C2semidirect
4. #21447050C2semidirect
5. #21446907C2semidirect
6. #21446756C2semidirect
7. #21446626C2semidirect
8. #21446153C2semidirect
9. #21446080C2semidirect
10. #21445819C2semidirect
112. C2×Q8×C61SD16semidirect
157. C2×Dic122#41semidirect
187. C2×#92418#41semidirect
342. C61#24188semidirect
416. C2×Q8#11758200semidirect

Maximal subgroups

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

subgroupindex
C2×#168399592
C2×C4×#117582002
#214511892
#214470502
#214469072
#214467562
#214466262
#214461532
#214460802
#214458192
C2×C61×#5502
#2418861