Finite Group Databasebeta

← #29685683

Subgroups — #29685683

Order 15616.

1520 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
1C22×#14519742C211yesmaximal, characteristic, complemented
2C2×#16839900C211yesmaximal, characteristic
3C2×#16839859C212yesmaximal, complemented
4#121×Dic61C211yesmaximal, characteristic
5#21446171C212yesmaximal, complemented
6#21446074C212yesmaximal, complemented
7#21446007C211yesmaximal, characteristic, complemented
8#21445755C211yesmaximal, characteristic, complemented
9#21445648C212yesmaximal, complemented
10#21445583C211yesmaximal, characteristic, complemented
11C61×#1874C211yesmaximal, characteristic
12C23×Dic122C2211yescharacteristic
13C22×#11758200C2212yes—
14C2×#14519742—28no—
15C2×#14519742—21no—
16C2×#14519742—21no—
17C2×#14519742C2212yes—
18C2×#14519742C2212yes—
19C2×D4×Dic61—21no—
20C2×D4×Dic61C2211yescharacteristic
21C22×#11758196—21no—
22C22×C4×Dic61—21no—
23C22×C4×Dic61C2211yescharacteristic
24#16839900C418yescomplemented
25#16839859—28no—
26C2×#14519710C2212yes—
27C2×#14519702C2212yes—
28C2×#14519702C2212yes—
29C2×#14519700C2212yes—
30C2×C22⋊#92418C2211yescharacteristic
31C2×#14519687C2212yes—
32#16839665C2211yescharacteristic
33#16839661C2211yescharacteristic
34#16839655—21no—
35#16839642—21no—
36#16839622C2211yescharacteristic
37C4⋊C4×Dic61C2211yescharacteristic
38C22⋊C4×Dic61—21no—
39#16839524C2212yes—
40#16839516C2212yes—
41C2×#121×C61C2211yescharacteristic
42C22×C4⋊C4×C61C2211yescharacteristic
43C2×#102×C61C2212yes—
44C2×C22⋊C8×C61C2211yescharacteristic
45C61×#340C2212yes—
46C22×Dic122—21no—
47C22×Dic122—22no—
48C22×Dic122—24no—
49C22×Dic122—24no—
50C22×Dic122C2×C414yescomplemented
51C22×Dic122C2312yes—
52C23×Dic61—21no—
53C23×Dic61—21no—
54C23×Dic61D411yescharacteristic
55C23×Dic61D411yescharacteristic
56C2×#11758200—24no—
57C2×#11758200—24no—
58C2×#11758200C2×C418yescomplemented
59C22×#92418C2312yes—
60#14519742—28no—
61#14519742—28no—
62#14519742—24no—
63#14519742—24no—
64#14519742—24no—
65#14519742D418yescomplemented
66D4×Dic61—24no—
67D4×Dic61—44no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×#11758196—28no—
72C2×#11758196—22no—
73C2×#11758196—22no—
74C2×C22⋊Dic61—21no—
75C2×C22⋊Dic61—21no—
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61—21no—
78C2×C4×Dic61—28no—
79C2×C4×Dic61—22no—
80C2×C4×Dic61—22no—
81C2×C4×Dic61—24no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61D411yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86C2×C4×Dic61C2311yescharacteristic
87#14519710—24no—
88#14519702—24no—
89#14519702—24no—
90#14519700C2×C418yescomplemented
91C22⋊#92418C2×C414yescomplemented
92#14519687—24no—
93#14519684—21no—
94#14519684—21no—
95#14519684—21no—
96C22×D4×C61—21no—
97C22×D4×C61C2311yescharacteristic
98C23×C4×C61—21no—
99C23×C4×C61C2311yescharacteristic
100C22×C8×C61C2312yes—
101#121×C61—22no—
102#121×C61C2×C414yes—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61—24no—
105C2×C4⋊C4×C61C2311yescharacteristic
106C2×C4⋊C4×C61C2312yes—
107C2×C22⋊C4×C61—21no—
108#102×C61—24no—
109C22⋊C8×C61—22no—
110C24×C61—21no—
111C24×C61C2×D411yescharacteristic
112C2×D4×C61—24no—
113C2×D4×C61—22no—
114C2×D4×C61—22no—
115C2×D4×C61—21no—
116C2×D4×C61—42no—
117C2×D4×C61—42no—
118C2×D4×C61C22×C412yes—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—24no—
122C22×C4×C61—21no—
123C22×C4×C61—22no—
124C22×C4×C61—42no—
125C22×C4×C61—42no—
126C22×C4×C61C22×C412yes—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61#4112yes—
129C22×C4×C61C2411yescharacteristic
130C22×Dic61—24no—
131C22×Dic61—24no—
132C22×Dic61—24no—
133C22×Dic61—22no—
134C22×Dic61—22no—
135C22×Dic61—22no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—22no—
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—41no—
142C22×Dic61—42no—
143C22×Dic61—42no—
144C22×Dic61—42no—
145C22×Dic61C2×D411yescharacteristic
146C22×Dic61C2×D411yescharacteristic
147C22×Dic61C2×D411yescharacteristic
148C22×Dic61C2×D411yescharacteristic
149C2×Dic122—24no—
150C2×Dic122—216no—
151C2×Dic122—24no—
152C2×Dic122—24no—
153C2×Dic122—22no—
154C2×Dic122—22no—
155C2×Dic122—22no—
156C2×Dic122—216no—
157C2×Dic122C22×C414yes—
158C2×Dic122C2×D414yes—
159C2×Dic122#4114yes—
160C2×C8×C61—22no—
161C2×C8×C61—22no—
162C2×C8×C61—22no—
163C4⋊C4×C61—24no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61C22×C412yes—
167C4⋊C4×C61C2×D414yes—
168C22⋊C4×C61—24no—
169C22⋊Dic61—42no—
170C22⋊Dic61—42no—
171C22⋊Dic61—42no—
172C22⋊Dic61—42no—
173#11758200—216no—
174C4⋊Dic61—22no—
175C4⋊Dic61—22no—
176C4⋊Dic61—42no—
177#11758196—28no—
178#11758196—28no—
179C4×Dic61—28no—
180C4×Dic61—22no—
181C4×Dic61—28no—
182C4×Dic61—22no—
183C4×Dic61—22no—
184C4×Dic61—42no—
185C4×Dic61C2×D414yes—
186C2×#92418—22no—
187C2×#92418—22no—
188C2×#92418C22×C414yes—
189C23×C61—21no—
190C23×C61—21no—
191C23×C61—21no—
192C23×C61—21no—
193C23×C61—22no—
194C23×C61—41no—
195C23×C61—41no—
196C23×C61—41no—
197C23×C61—41no—
198C23×C61—41no—
199C23×C61—41no—
200C23×C61C4×D412yes—
201C23×C61AGL(1,ℤ/8)12yes—
202C23×C61#13712yes—
203C23×C61C22×D411yescharacteristic
204D4×C61—24no—
205D4×C61—48no—
206D4×C61—42no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—24no—
210C2×C4×C61—24no—
211C2×C4×C61—22no—
212C2×C4×C61—24no—
213C2×C4×C61—48no—
214C2×C4×C61—41no—
215C2×C4×C61—42no—
216C2×C4×C61—41no—
217C2×C4×C61—42no—
218C2×C4×C61C4×D412yes—
219C2×C4×C61C23×C411yescharacteristic
220C2×C4×C61C22×D412yes—
221C2×C4×C61C2×#4112yes—
222C2×C4×C61C2×#4111yescharacteristic
223C2×Dic61—216no—
224C2×Dic61—28no—
225C2×Dic61—24no—
226C2×Dic61—21no—
227C2×Dic61—24no—
228C2×Dic61—24no—
229C2×Dic61—21no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—44no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—84no—
245C2×Dic61#12012yes—
246C2×Dic61#12012yes—
247C2×Dic61#13712yes—
248Dic122—216no—
249Dic122—24no—
250Dic122—24no—
251Dic122—24no—
252Dic122C4×D418yescomplemented
253C8×C61—42no—
254#92418—24no—
255#21454—611nomaximal, Sylow
256C22×C61—22no—
257C22×C61—22no—
258C22×C61—22no—
259C22×C61—22no—
260C22×C61—22no—
261C22×C61—22no—
262C22×C61—42no—
263C22×C61—42no—
264C22×C61—42no—
265C22×C61—81no—
266C22×C61—81no—
267C22×C61—81no—
268C22×C61—81no—
269C22×C61#44014yescomplemented
270C22×C61C2×C4×D411yescharacteristic
271C22×C61C2×#12011yescharacteristic
272C22×C61#52511yescharacteristic
273C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
274C22×C61C2×#13711yescharacteristic
275C22×C61C2×#13711yescharacteristic
276C22×C61#57511yescharacteristic
277C4×C61—22no—
278C4×C61—42no—
279C4×C61—44no—
280C4×C61C2×C4×D412yes—
281C4×C61#52512yes—
282Dic61—22no—
283Dic61—22no—
284Dic61—22no—
285Dic61—28no—
286Dic61—41no—
287Dic61—81no—
288Dic61—81no—
289C22×C4×Q8—611no—
290C2×#448—611no—
291C2×#440—612no—
292C4×#121—611no—
293#1880—612no—
294#1874D6111yescharacteristic
295#1863—612no—
296#1852—611no—
297#1787—611no—
298#1763—612no—
299#1749—611no—
300C2×C61—24no—
301C2×C61—82no—
302C2×C61C4×#12011yescharacteristic
303C2×C61C2×#44011yescharacteristic
304C2×C61C4×#13711yescharacteristic
305C2×C61#300411yescharacteristic
306C2×C61#300611yescharacteristic
307C2×C61#311211yescharacteristic
308C2×C61#311711yescharacteristic
309C23×Q8—611no—
310C22×SD16—612no—
311C2×#121C2×D6111yescharacteristic
312C2×C4×Q8—612no—
313C2×C4×Q8—612no—
314C2×C4×Q8—1221no—
315C2×C4×Q8—1221no—
316C2×C4×Q8—1228no—
317C2×C4×D4—611no—
318C2×C4×D4—1221no—
319C22×C4⋊C4—1221no—
320C22×C4⋊C4C2×D6111yescharacteristic
321C22×C42—611no—
322C22×C42—1221no—
323#448—618no—
324#440—1228no—
325C2×#107—612no—
326C2×#103—612no—
327C2×#103—612no—
328C2×#103—612no—
329C2×#102C2×D6112yes—
330C2×C22⋊C8—611no—
331C2×C22⋊C8C2×D6111yescharacteristic
332C2×#97—612no—
333#390—611no—
334#389—611no—
335#388—1221no—
336#386—1221no—
337#380—611no—
338C4×C4⋊C4—611no—
339C4×C22⋊C4—1221no—
340#340—612no—
341#340C2×D6112yes—
342#336—612no—
343C61#2145411yescharacteristic, complemented, Sylow
344C22×Q8—614no—
345C22×Q8—612no—
346C22×Q8—1224no—
347C22×Q8—1224no—
348C22×Q8—1221no—
349C22×Q8—1222no—
350C22×D4—21no—
351C22×D4C22×D6111yescharacteristic
352C23×C4—21no—
353C23×C4—611no—
354C23×C4—611no—
355C23×C4—1221no—
356C23×C4—1221no—
357C23×C4C22×D6111yescharacteristic
358C2×SD16—618no—
359C2×SD16—1224no—
360C2×SD16—1224no—
361C22×C8—612no—
362C22×C8C22×D6112yes—
363#121—22no—
364#121C4×D6114yes—
365C4×Q8—618no—
366C4×Q8—1224no—
367C4×Q8—1224no—
368C4×Q8—1228no—
369C4×Q8—1228no—
370C4×Q8—1224no—
371C4×D4—1224no—
372C4×D4—2444no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—24no—
375C2×C4⋊C4—611no—
376C2×C4⋊C4—611no—
377C2×C4⋊C4—1222no—
378C2×C4⋊C4—1222no—
379C2×C4⋊C4—1228no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4C22×D6111yescharacteristic
382C2×C4⋊C4C22×D6112yes—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—1221no—
385C2×C22⋊C4—1221no—
386C2×C22⋊C4—1221no—
387C2×C22⋊C4—1221no—
388C2×C42—611no—
389C2×C42—611no—
390C2×C42—611no—
391C2×C42—611no—
392C2×C42—1224no—
393C2×C42—1222no—
394C2×C42—1222no—
395C2×C42—1228no—
396C2×C42—1221no—
397#107—1224no—
398#103—618no—
399#103—1224no—
400#103—1224no—
401#102—24no—
402C22⋊C8—22no—
403C22⋊C8—614no—
404#97—1224no—
405#95—1221no—
406#95—1221no—
407#95—1221no—
408C24—21no—
409C24D4×D6111yescharacteristic
410C2×Q8—614no—
411C2×Q8—614no—
412C2×Q8—614no—
413C2×Q8—12216no—
414C2×Q8—1224no—
415C2×Q8—1224no—
416C2×Q8—12216no—
417C2×Q8—1222no—
418C2×Q8—1222no—
419C2×Q8—1222no—
420C2×Q8—1224no—
421C2×D4—24no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—21no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4C2×C4×D6112yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—611no—
436C22×C4—611no—
437C22×C4—611no—
438C22×C4—611no—
439C22×C4—1222no—
440C22×C4—1222no—
441C22×C4—1222no—
442C22×C4—1224no—
443C22×C4—1224no—
444C22×C4—1222no—
445C22×C4—1224no—
446C22×C4—1222no—
447C22×C4—1222no—
448C22×C4—1221no—
449C22×C4—1221no—
450C22×C4—2442no—
451C22×C4—2441no—
452C22×C4—2442no—
453C22×C4—2442no—
454C22×C4C2×C4×D6112yes—
455C22×C4D4×D6111yescharacteristic
456C22×C4#1175822412yes—
457C22×C4C23×D6111yescharacteristic
458SD16—12216no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—614no—
463C2×C8—1222no—
464C2×C8—1222no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—1228no—
469C4⋊C4—1228no—
470C4⋊C4—1222no—
471C4⋊C4—1222no—
472C4⋊C4—2442no—
473C4⋊C4C2×C4×D6112yes—
474C4⋊C4D4×D6114yes—
475C22⋊C4—24no—
476C22⋊C4—2442no—
477C22⋊C4—2442no—
478C22⋊C4—2442no—
479C22⋊C4—2442no—
480C42—614no—
481C42—1222no—
482C42—1222no—
483C42—1228no—
484C42—1228no—
485C42—1222no—
486C42—2442no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—22no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23#1451973712yes—
499C23#1451980112yes—
500C23#1451980612yes—
501C23C2×D4×D6111yescharacteristic
502Q8—618no—
503Q8—12216no—
504Q8—1224no—
505Q8—1224no—
506Q8—1224no—
507D4—24no—
508D4—48no—
509D4—42no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—48no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—612no—
522C2×C4—612no—
523C2×C4—612no—
524C2×C4—1228no—
525C2×C4—12216no—
526C2×C4—1224no—
527C2×C4—1221no—
528C2×C4—1224no—
529C2×C4—1224no—
530C2×C4—1221no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—1221no—
534C2×C4—1221no—
535C2×C4—1221no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—4884no—
539C2×C4—2444no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4—2441no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4—2441no—
546C2×C4#1451973712yes—
547C2×C4C22×C4×D6111yescharacteristic
548C2×C4C2×D4×D6112yes—
549C2×C4#41×D6112yes—
550C2×C4C2×#1175822411yescharacteristic
551C8—42no—
552C8—1224no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22#1683985914yescomplemented
567C22C2×#1451973711yescharacteristic
568C22#120×D6111yescharacteristic
569C22#1684027011yescharacteristic
570C22C2×#1451980111yescharacteristic
571C22#137×D6111yescharacteristic
572C22C2×#1451980611yescharacteristic
573C22#1684055811yescharacteristic
574C4—22no—
575C4—42no—
576C4—44no—
577C4—1228no—
578C4—1222no—
579C4—1222no—
580C4—1222no—
581C4—4881no—
582C4—4881no—
583C4—2441no—
584C4C4×D4×D6112yes—
585C4#1684027212yes—
586C2—24no—
587C2—82no—
588C2#2144832411yescharacteristic
589C2C2×#1683985911yescharacteristic
590C2#2145464311yescharacteristic
591C2#2145505011yescharacteristic
592C2#2145506611yescharacteristic
593C2#2145593811yescharacteristic
594C2#2145598711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1874C2
socleC23×C61C22×D4
prime radicalC2×#121×C61C22
cosocleC22×C4×C61C24
hypercenter#1874D61
hypocenterC61#21454
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1874

Complements

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

NH ≅ G/Ntype
1. C22×#14519742C2semidirect
3. C2×#16839859C2semidirect
5. #21446171C2semidirect
6. #21446074C2semidirect
7. #21446007C2semidirect
8. #21445755C2semidirect
9. #21445648C2semidirect
10. #21445583C2semidirect
24. #16839900C4semidirect
50. C22×Dic122C2×C4semidirect
58. C2×#11758200C2×C4semidirect
65. #14519742D4semidirect
90. #14519700C2×C4semidirect
91. C22⋊#92418C2×C4semidirect
252. Dic122C4×D4semidirect
269. C22×C61#440semidirect
343. C61#21454semidirect
566. C22#16839859semidirect

Maximal subgroups

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

subgroupindex
C22×#145197422
C2×#168399002
C2×#168398592
#121×Dic612
#214461712
#214460742
#214460072
#214457552
#214456482
#214455832
C61×#18742
#2145461