Finite Group Databasebeta

← #29708019

Subgroups — #29708019

Order 15616.

1674 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519805C211yesmaximal, characteristic
2C2×#16839764C211yesmaximal, characteristic, complemented
3C2×#16839768C211yesmaximal, characteristic, complemented
4#21448309C211yesmaximal, characteristic, complemented
5C2×#16839664C211yesmaximal, characteristic, complemented
6#21447137C214yesmaximal, complemented
7#21447130C214yesmaximal, complemented
8#21445539C211yesmaximal, characteristic, complemented
9C2×C61×#423C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C23×D244C2211yescharacteristic
12C2×#14519805—24no—
13C2×#14519805—24no—
14C2×#14519805—24no—
15C22×#11758190C2212yes—
16C22×D488C2212yes—
17C2×#14519736C2211yescharacteristic
18C22×C4⋊Dic61C2211yescharacteristic
19C22×#11758198—22no—
20#16839768—24no—
21#16839764—24no—
22C2×#14519702C2212yescomplemented
23C2×#14519699C2214yes—
24C2×#14519699C2212yescomplemented
25#16839668C2212yescomplemented
26#16839664—21no—
27#16839664—21no—
28#16839664C2214yes—
29#16839664C2212yescomplemented
30#16839641—22no—
31C4×#11758198—21no—
32C4×#11758198—21no—
33#16839525C2214yescomplemented
34C22×M16×C61C2211yescharacteristic
35C22×M16×C61C2211yescharacteristic
36C22×C42×C61C2211yescharacteristic
37C61×#423—24no—
38C2×C4⋊C8×C61C2214yescomplemented
39C24×D61—21no—
40C2×#11758222—22no—
41C2×#11758222—22no—
42C2×#11758222—22no—
43C2×#11758222—22no—
44C2×#11758222—22no—
45C2×#11758222D414yes—
46C22×Dic122C2311yescharacteristic
47C22×D244—22no—
48C22×D244—22no—
49C22×D244—22no—
50C22×D244—22no—
51C22×D244—21no—
52C22×D244—22no—
53C22×D244D414yes—
54C22×D244C2312yes—
55C22×D244C2311yescharacteristic
56C22×C22⋊D61—21no—
57C23×Dic61—21no—
58C22×C4×D61—21no—
59#14519805—232no—
60C2×#11758190—24no—
61C2×#11758190—24no—
62C2×#11758190—24no—
63C2×D488—24no—
64C2×D488—24no—
65C2×D488—24no—
66#14519736—22no—
67#14519736—22no—
68C2×C4⋊Dic61—22no—
69C2×C4⋊Dic61—22no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2312yes—
72C2×C4⋊Dic61C2312yes—
73C2×C22⋊Dic61—21no—
74C2×#11758198—24no—
75C2×#11758198—24no—
76C2×#11758198—22no—
77C2×#11758198—22no—
78C2×#11758198—48no—
79C2×C4×Dic61—21no—
80#14519702—24no—
81#14519699—28no—
82#14519699—24no—
83#14519684—22no—
84#14519684—22no—
85C23×C4×C61D412yes—
86C23×C4×C61C2311yescharacteristic
87C2×M16×C61—22no—
88C2×M16×C61—22no—
89C2×M16×C61—22no—
90C2×M16×C61—22no—
91C2×M16×C61—22no—
92C2×M16×C61D414yes—
93C22×C8×C61C2312yes—
94C22×C8×C61C2312yes—
95C2×C42×C61—22no—
96C2×C42×C61—22no—
97C2×C42×C61D411yescharacteristic
98C2×C42×C61D411yescharacteristic
99C2×C42×C61C2312yes—
100C4⋊C8×C61—28no—
101C24×C61C2×D411yescharacteristic
102C23×D61—24no—
103C23×D61—21no—
104C23×D61—22no—
105C23×D61—44no—
106C23×D61—44no—
107C23×D61—42no—
108C23×D61—42no—
109C22×C4×C61—24no—
110C22×C4×C61—24no—
111C22×C4×C61—42no—
112C22×C4×C61C2×D412yes—
113C22×C4×C61C2×D412yes—
114C22×C4×C61C2×D412yes—
115C22×C4×C61C2×D412yes—
116C22×C4×C61C2×D412yes—
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61#4112yes—
119C22×C4×C61#4112yes—
120C22×C4×C61#4112yes—
121C22×C4×C61C2411yescharacteristic
122C2×C22⋊D61—24no—
123C2×C22⋊D61—42no—
124C2×C22⋊D61—42no—
125C22×Dic61—21no—
126C22×Dic61—22no—
127C22×Dic61—22no—
128C22×Dic61—44no—
129C22×Dic61—42no—
130#11758222—28no—
131#11758222—28no—
132#11758222—44no—
133#11758222—44no—
134C2×D244—28no—
135C2×D244—28no—
136C2×D244—22no—
137C2×D244—22no—
138C2×D244—21no—
139C2×D244—21no—
140C2×D244—22no—
141C2×D244—22no—
142C2×D244—21no—
143C2×D244—21no—
144C2×D244—24no—
145C2×D244—22no—
146C2×D244—21no—
147C2×D244—42no—
148C2×D244—42no—
149C2×D244—44no—
150C2×D244—44no—
151C2×D244C2×D414yes—
152C2×D244C2×D412yes—
153C2×C4×D61—24no—
154C2×C4×D61—42no—
155C2×C4×D61—42no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122C2×D412yes—
162M16×C61—28no—
163M16×C61—44no—
164C2×C8×C61—22no—
165C2×C8×C61—22no—
166C2×C8×C61—22no—
167C2×C8×C61—22no—
168C2×C8×C61—22no—
169C2×C8×C61C2×D414yes—
170C42×C61—22no—
171C42×C61—22no—
172C42×C61—22no—
173C42×C61—22no—
174C22⋊Dic61—42no—
175#11758198—44no—
176#11758198—48no—
177#11758198—44no—
178#11758198—48no—
179C4⋊Dic61—22no—
180C4⋊Dic61—22no—
181C4⋊Dic61—22no—
182C4⋊Dic61—22no—
183C4⋊Dic61—42no—
184C4×Dic61—42no—
185D488—216no—
186#11758190—216no—
187C23×C61—22no—
188C23×C61—22no—
189C23×C61#12112yes—
190C23×C61#12112yes—
191C23×C61#12112yes—
192C23×C61C22×D411yescharacteristic
193C22×D61—22no—
194C22×D61—24no—
195C22×D61—416no—
196C22×D61—41no—
197C22×D61—41no—
198C22×D61—42no—
199C22×D61—42no—
200C22×D61—44no—
201C22×D61—42no—
202C22×D61—42no—
203C22×D61—44no—
204C22×D61—48no—
205C22×D61—84no—
206C22×D61—84no—
207C22×D61—88no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—48no—
214C2×C4×C61#12014yes—
215C2×C4×C61#12112yes—
216C2×C4×C61#12112yes—
217C2×C4×C61#12112yes—
218C2×C4×C61#12314yes—
219C2×C4×C61#12314yes—
220C2×C4×C61#12414yes—
221C2×C4×C61AGL(1,ℤ/8)14yescomplemented
222C2×C4×C61C22×D411yescharacteristic
223C2×C4×C61C22×D411yescharacteristic
224C2×C4×C61C22×D411yescharacteristic
225C2×C4×C61C2×#4111yescharacteristic
226C2×C4×C61C2×#4111yescharacteristic
227C2×C4×C61C2×#4111yescharacteristic
228C22⋊D61—48no—
229C2×Dic61—22no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61—42no—
233C2×Dic61—42no—
234C2×Dic61—44no—
235C2×Dic61—42no—
236C2×Dic61—42no—
237C2×Dic61—84no—
238D244—24no—
239D244—22no—
240D244—24no—
241D244—22no—
242D244—42no—
243D244—41no—
244D244—42no—
245D244—41no—
246D244—48no—
247C4×D61—48no—
248Dic122—22no—
249Dic122—22no—
250Dic122—41no—
251Dic122—41no—
252C8×C61—24no—
253C8×C61—42no—
254#23546—611nomaximal, Sylow
255C22×C61—24no—
256C22×C61—22no—
257C22×C61—22no—
258C22×C61—24no—
259C22×C61#39414yes—
260C22×C61C2×#12111yescharacteristic
261C22×C61C2×#12111yescharacteristic
262C22×C61C2×#12111yescharacteristic
263C22×C61C2×AGL(1,ℤ/8)12yes—
264C22×C61#57512yes—
265C2×D61—42no—
266C2×D61—44no—
267C2×D61—48no—
268C2×D61—42no—
269C2×D61—42no—
270C2×D61—816no—
271C2×D61—82no—
272C2×D61—82no—
273C2×D61—82no—
274C2×D61—82no—
275C2×D61—84no—
276C4×C61—24no—
277C4×C61—42no—
278C4×C61#39414yes—
279C4×C61C2×#12011yescharacteristic
280C4×C61C2×#12311yescharacteristic
281C4×C61C2×#12311yescharacteristic
282C4×C61C2×#12411yescharacteristic
283Dic61—42no—
284Dic61—82no—
285Dic61—82no—
286C22×AGL(1,ℤ/8)—611no—
287C2×#423D6111yescharacteristic, complemented
288C2×#417—611no—
289C2×#418—611no—
290#2297—611no—
291C2×#390—611no—
292#2055—614no—
293#2054—614no—
294#1736—611no—
295C2×C61—24no—
296C2×C61C2×#39411yescharacteristic
297C2×C61#303612yes—
298C2×C61#303811yescharacteristic
299C2×C61#304011yescharacteristic
300C2×C61#305512yes—
301D61—82no—
302D61—84no—
303C22×#41—611no—
304C23×D4—611no—
305C2×AGL(1,ℤ/8)—1224no—
306C2×AGL(1,ℤ/8)—1224no—
307C2×AGL(1,ℤ/8)—1224no—
308C22×SD16—612no—
309C22×D8—612no—
310C22×M16C2×D6111yescharacteristic
311C22×M16C2×D6111yescharacteristic
312C2×#117—611no—
313C22×C4⋊C4—611no—
314C22×C22⋊C4—1222no—
315C22×C42C2×D6111yescharacteristic
316#423—24no—
317C2×C4⋊C8C2×D6114yescomplemented
318#418—1224no—
319#417—1224no—
320C2×#103—612no—
321C2×#102—612no—
322C2×#102—614no—
323#391—612no—
324#390—612no—
325#390—614no—
326#390—1221no—
327#390—1221no—
328#386—1222no—
329C4×C22⋊C4—1221no—
330C4×C22⋊C4—1221no—
331#340—614no—
332C61#2354611yescharacteristic, complemented, Sylow
333C25—1221no—
334C2×#41—614no—
335C2×#41—1222no—
336C2×#41—1222no—
337C2×#41—1222no—
338C2×#41—1222no—
339C2×#41—1222no—
340C22×Q8—611no—
341C22×D4—614no—
342C22×D4—611no—
343C22×D4—612no—
344C22×D4—1221no—
345C22×D4—1221no—
346C22×D4—1222no—
347C22×D4—1222no—
348C22×D4—1222no—
349C22×D4—1222no—
350C22×D4—1222no—
351C23×C4—1221no—
352C23×C4—1221no—
353C23×C4D24412yes—
354C23×C4C22×D6111yescharacteristic
355AGL(1,ℤ/8)—12232no—
356C2×SD16—1224no—
357C2×SD16—1224no—
358C2×SD16—1224no—
359C2×D8—1224no—
360C2×D8—1224no—
361C2×D8—1224no—
362C2×M16—22no—
363C2×M16—22no—
364C2×M16—22no—
365C2×M16—22no—
366C2×M16—22no—
367C2×M16C22⋊D6114yes—
368C22×C8C22×D6112yes—
369C22×C8C22×D6112yes—
370#117—1222no—
371#117—1222no—
372C2×C4⋊C4—612no—
373C2×C4⋊C4—612no—
374C2×C4⋊C4—1222no—
375C2×C4⋊C4—1222no—
376C2×C4⋊C4—1221no—
377C2×C22⋊C4—1221no—
378C2×C22⋊C4—1224no—
379C2×C22⋊C4—1224no—
380C2×C22⋊C4—1222no—
381C2×C22⋊C4—1222no—
382C2×C22⋊C4—2448no—
383C2×C42—22no—
384C2×C42—22no—
385C2×C42—1221no—
386C2×C42D24411yescharacteristic
387C2×C42D24411yescharacteristic
388C2×C42C22×D6112yes—
389C4⋊C8—28no—
390#103—1224no—
391#102—1224no—
392#102—1228no—
393#95—1222no—
394#95—1222no—
395C24—1221no—
396C24—1222no—
397C24—1224no—
398C24—2444no—
399C24—2444no—
400C24—2442no—
401C24—2442no—
402C24C2×D24411yescharacteristic
403#41—1228no—
404#41—1228no—
405#41—2444no—
406#41—2444no—
407C2×Q8—612no—
408C2×Q8—1221no—
409C2×Q8—1221no—
410C2×Q8—1221no—
411C2×Q8—1221no—
412C2×Q8—1221no—
413C2×D4—612no—
414C2×D4—614no—
415C2×D4—1224no—
416C2×D4—1221no—
417C2×D4—1222no—
418C2×D4—1221no—
419C2×D4—1222no—
420C2×D4—1222no—
421C2×D4—1221no—
422C2×D4—1222no—
423C2×D4—1221no—
424C2×D4—1222no—
425C2×D4—1221no—
426C2×D4—1228no—
427C2×D4—1224no—
428C2×D4—1228no—
429C2×D4—2442no—
430C2×D4—2442no—
431C2×D4—2442no—
432C2×D4—2442no—
433C2×D4—2444no—
434C2×D4—2444no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—42no—
438C22×C4—1221no—
439C22×C4—1224no—
440C22×C4—1222no—
441C22×C4—1222no—
442C22×C4—2444no—
443C22×C4—2442no—
444C22×C4—2442no—
445C22×C4—2442no—
446C22×C4C2×D24412yes—
447C22×C4C2×D24412yes—
448C22×C4C2×D24411yescharacteristic
449C22×C4#1175822212yes—
450C22×C4D4×D6112yes—
451C22×C4D4×D6112yes—
452C22×C4#1175822612yes—
453C22×C4#1175822612yes—
454C22×C4C2×C22⋊D6112yes—
455C22×C4C23×D6111yescharacteristic
456SD16—12216no—
457D8—12216no—
458M16—28no—
459M16—44no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8C2×C22⋊D6114yes—
466C4⋊C4—1222no—
467C4⋊C4—1222no—
468C4⋊C4—1222no—
469C4⋊C4—1222no—
470C4⋊C4—2442no—
471C22⋊C4—2448no—
472C22⋊C4—2444no—
473C22⋊C4—2442no—
474C22⋊C4—2448no—
475C22⋊C4—2444no—
476C42—22no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—2442no—
481C23—22no—
482C23—22no—
483C23—1222no—
484C23—1224no—
485C23—4888no—
486C23—4884no—
487C23—4884no—
488C23—24416no—
489C23—2441no—
490C23—2442no—
491C23—2441no—
492C23—2442no—
493C23—2444no—
494C23—2442no—
495C23—2442no—
496C23—2444no—
497C23—2448no—
498C23#1451975112yes—
499C23#1451975212yes—
500C23#1451975212yes—
501C23C22×D24411yescharacteristic
502Q8—1222no—
503Q8—1222no—
504Q8—2441no—
505Q8—2441no—
506D4—1222no—
507D4—1224no—
508D4—1224no—
509D4—1222no—
510D4—2448no—
511D4—2442no—
512D4—2441no—
513D4—2442no—
514D4—2441no—
515D4—2448no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—48no—
522C2×C4—1222no—
523C2×C4—4884no—
524C2×C4—2442no—
525C2×C4—2442no—
526C2×C4—2444no—
527C2×C4—2442no—
528C2×C4—2442no—
529C2×C4—2441no—
530C2×C4—2441no—
531C2×C4—2448no—
532C2×C4#1451974514yes—
533C2×C4#1451975112yes—
534C2×C4#1451975212yes—
535C2×C4#1451975212yes—
536C2×C4#1451976214yes—
537C2×C4#1451976214yes—
538C2×C4#1451976614yes—
539C2×C4#1451980514yescomplemented
540C2×C4C2×D4×D6111yescharacteristic
541C2×C4C2×D4×D6111yescharacteristic
542C2×C4C22×C22⋊D6111yescharacteristic
543C2×C4C2×#1175822211yescharacteristic
544C2×C4C2×#1175822611yescharacteristic
545C2×C4C2×#1175822611yescharacteristic
546C8—24no—
547C8—42no—
548C22—24no—
549C22—24no—
550C22—22no—
551C22—22no—
552C22—48816no—
553C22—4884no—
554C22—4882no—
555C22—4882no—
556C22—4882no—
557C22—4882no—
558C22—2442no—
559C22—2444no—
560C22—2448no—
561C22—2442no—
562C22—2442no—
563C22#1683967914yes—
564C22C2×#1451975111yescharacteristic
565C22C2×#1451975211yescharacteristic
566C22C2×#1451975211yescharacteristic
567C22C2×#1451980512yes—
568C22#1684056212yes—
569C4—24no—
570C4—42no—
571C4—4882no—
572C4—4882no—
573C4—2442no—
574C4#1683967914yes—
575C4C2×#1451974511yescharacteristic
576C4C2×#1451976211yescharacteristic
577C4C2×#1451976211yescharacteristic
578C4C2×#1451976611yescharacteristic
579C2—24no—
580C2—4884no—
581C2—4882no—
582C2C2×#1683967911yescharacteristic
583C2#2145528512yes—
584C2#2145530011yescharacteristic
585C2#2145531411yescharacteristic
586C2#2145543812yes—
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#423C2
socleC23×C61C22×D4
prime radicalC22×#14519805C2
hypercenterC2×#423D61
hypocenterC61#23546
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#423

Complements

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

NH ≅ G/Ntype
2. C2×#16839764C2semidirect
3. C2×#16839768C2semidirect
4. #21448309C2semidirect
5. C2×#16839664C2semidirect
6. #21447137C2semidirect
7. #21447130C2semidirect
8. #21445539C2semidirect
9. C2×C61×#423C2semidirect
22. C2×#14519702C22semidirect
24. C2×#14519699C22semidirect
25. #16839668C22semidirect
29. #16839664C22semidirect
33. #16839525C22semidirect
38. C2×C4⋊C8×C61C22semidirect
221. C2×C4×C61AGL(1,ℤ/8)semidirect
287. C2×#423D61semidirect
317. C2×C4⋊C8C2×D61semidirect
332. C61#23546semidirect
539. C2×C4#14519805semidirect

Maximal subgroups

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

subgroupindex
C22×#145198052
C2×#168397642
C2×#168397682
#214483092
C2×#168396642
#214471372
#214471302
#214455392
C2×C61×#4232
#2354661