Finite Group Databasebeta

← #29708013

Subgroups — #29708013

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
1C2×#16839785C211yesmaximal, characteristic, complemented
2C2×#16839763C211yesmaximal, characteristic, complemented
3C2×#16839767C211yesmaximal, characteristic, complemented
4#21448310C211yesmaximal, characteristic, complemented
5C2×#16839665C211yesmaximal, characteristic, complemented
6#21447132C214yesmaximal, complemented
7#21447127C214yesmaximal, complemented
8#21445539C211yesmaximal, characteristic, complemented
9C22×AGL(1,ℤ/8)×C61C211yesmaximal, characteristic
10C22×#11758194C2211yescharacteristic
11C2×#14519736C2211yescharacteristic
12C22×C4⋊Dic61C2211yescharacteristic
13C22×C22⋊Dic61—22no—
14C22×C4×Dic61C2211yescharacteristic
15#16839785—24no—
16C2×#14519706C2214yescomplemented
17#16839767—24no—
18#16839763—24no—
19C2×#14519701C2212yescomplemented
20C2×#14519698C2214yes—
21C2×#14519698C2212yescomplemented
22#16839669C2212yescomplemented
23#16839665—21no—
24#16839665—21no—
25#16839665C2214yes—
26#16839665C2212yescomplemented
27#16839642—22no—
28C4×C22⋊Dic61—21no—
29C4×C22⋊Dic61—21no—
30#16839525C2214yescomplemented
31C22×#41×C61C2211yescharacteristic
32C23×D4×C61C2211yescharacteristic
33C2×AGL(1,ℤ/8)×C61—24no—
34C2×AGL(1,ℤ/8)×C61—24no—
35C2×AGL(1,ℤ/8)×C61—24no—
36C22×SD16×C61C2212yes—
37C22×D8×C61C2212yes—
38C22×M16×C61C2211yescharacteristic
39C23×Dic61—21no—
40C23×Dic61D412yes—
41C2×#11758194—22no—
42C2×#11758194—22no—
43C2×#11758194—22no—
44C22×#92418C2312yes—
45#14519736—22no—
46#14519736—22no—
47C2×C4⋊Dic61—22no—
48C2×C4⋊Dic61—22no—
49C2×C4⋊Dic61—21no—
50C2×C4⋊Dic61C2312yes—
51C2×C4⋊Dic61C2312yes—
52C2×C22⋊Dic61—21no—
53C2×C22⋊Dic61—24no—
54C2×C22⋊Dic61—24no—
55C2×C22⋊Dic61—22no—
56C2×C22⋊Dic61—22no—
57C2×C22⋊Dic61—48no—
58C2×C4×Dic61—22no—
59C2×C4×Dic61—22no—
60C2×C4×Dic61—21no—
61C2×C4×Dic61D411yescharacteristic
62C2×C4×Dic61D411yescharacteristic
63C2×C4×Dic61C2312yes—
64#14519706—28no—
65#14519701—24no—
66#14519698—28no—
67#14519698—24no—
68#14519684—22no—
69#14519684—22no—
70C25×C61—21no—
71C2×#41×C61—22no—
72C2×#41×C61—22no—
73C2×#41×C61—22no—
74C2×#41×C61—22no—
75C2×#41×C61—22no—
76C2×#41×C61D414yes—
77C22×Q8×C61C2311yescharacteristic
78C22×D4×C61—21no—
79C22×D4×C61—22no—
80C22×D4×C61—22no—
81C22×D4×C61—21no—
82C22×D4×C61—22no—
83C22×D4×C61—22no—
84C22×D4×C61—22no—
85C22×D4×C61D414yes—
86C22×D4×C61C2311yescharacteristic
87C22×D4×C61C2312yes—
88C23×C4×C61—21no—
89C23×C4×C61C2311yescharacteristic
90AGL(1,ℤ/8)×C61—232no—
91C2×SD16×C61—24no—
92C2×SD16×C61—24no—
93C2×SD16×C61—24no—
94C2×D8×C61—24no—
95C2×D8×C61—24no—
96C2×D8×C61—24no—
97C2×M16×C61—22no—
98C2×M16×C61—22no—
99C2×M16×C61D414yes—
100C22×C8×C61C2312yes—
101C24×C61—24no—
102C24×C61—21no—
103C24×C61—22no—
104C24×C61—44no—
105C24×C61—44no—
106C24×C61—42no—
107C24×C61—42no—
108C24×C61C2×D411yescharacteristic
109#41×C61—28no—
110#41×C61—28no—
111#41×C61—44no—
112#41×C61—44no—
113C2×Q8×C61—21no—
114C2×Q8×C61—21no—
115C2×Q8×C61—21no—
116C2×Q8×C61—21no—
117C2×Q8×C61—21no—
118C2×Q8×C61C2×D412yes—
119C2×D4×C61—24no—
120C2×D4×C61—28no—
121C2×D4×C61—24no—
122C2×D4×C61—28no—
123C2×D4×C61—21no—
124C2×D4×C61—21no—
125C2×D4×C61—22no—
126C2×D4×C61—22no—
127C2×D4×C61—22no—
128C2×D4×C61—22no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—22no—
132C2×D4×C61—21no—
133C2×D4×C61—42no—
134C2×D4×C61—42no—
135C2×D4×C61—42no—
136C2×D4×C61—42no—
137C2×D4×C61—44no—
138C2×D4×C61—44no—
139C2×D4×C61C2×D412yes—
140C2×D4×C61C2×D414yes—
141C22×C4×C61—21no—
142C22×C4×C61—24no—
143C22×C4×C61—42no—
144C22×C4×C61—42no—
145C22×C4×C61C2×D412yes—
146C22×C4×C61C2×D412yes—
147C22×C4×C61C2×D412yes—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61#4112yes—
150C22×C4×C61#4112yes—
151C22×C4×C61#4112yes—
152C22×C4×C61C2411yescharacteristic
153C22×Dic61—24no—
154C22×Dic61—24no—
155C22×Dic61—22no—
156C22×Dic61—22no—
157C22×Dic61—42no—
158C22×Dic61—44no—
159C22×Dic61—42no—
160C22×Dic61C2×D412yes—
161C22×Dic61C2×D412yes—
162SD16×C61—216no—
163D8×C61—216no—
164M16×C61—28no—
165C2×C8×C61—22no—
166C2×C8×C61—22no—
167C2×C8×C61C2×D414yes—
168C22⋊Dic61—44no—
169C22⋊Dic61—48no—
170C22⋊Dic61—42no—
171C22⋊Dic61—44no—
172C22⋊Dic61—48no—
173C4⋊Dic61—22no—
174C4⋊Dic61—22no—
175C4⋊Dic61—22no—
176C4⋊Dic61—22no—
177C4⋊Dic61—42no—
178C4×Dic61—22no—
179C4×Dic61—22no—
180C4×Dic61—22no—
181C4×Dic61—22no—
182C4×Dic61—42no—
183#11758194—44no—
184C2×#92418—22no—
185C2×#92418—22no—
186C2×#92418—22no—
187C23×C61—22no—
188C23×C61—22no—
189C23×C61—22no—
190C23×C61—24no—
191C23×C61—416no—
192C23×C61—41no—
193C23×C61—41no—
194C23×C61—42no—
195C23×C61—42no—
196C23×C61—44no—
197C23×C61—42no—
198C23×C61—42no—
199C23×C61—44no—
200C23×C61—48no—
201C23×C61—84no—
202C23×C61—84no—
203C23×C61—88no—
204C23×C61#12112yes—
205C23×C61#12112yes—
206C23×C61#12112yes—
207C23×C61C22×D411yescharacteristic
208Q8×C61—22no—
209Q8×C61—22no—
210Q8×C61—41no—
211Q8×C61—41no—
212D4×C61—22no—
213D4×C61—24no—
214D4×C61—24no—
215D4×C61—22no—
216D4×C61—48no—
217D4×C61—42no—
218D4×C61—41no—
219D4×C61—42no—
220D4×C61—41no—
221D4×C61—48no—
222C2×C4×C61—22no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—48no—
226C2×C4×C61#12014yes—
227C2×C4×C61#12112yes—
228C2×C4×C61#12112yes—
229C2×C4×C61#12112yes—
230C2×C4×C61#12314yes—
231C2×C4×C61#12314yes—
232C2×C4×C61#12414yes—
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C2×Dic61—22no—
240C2×Dic61—22no—
241C2×Dic61—22no—
242C2×Dic61—22no—
243C2×Dic61—22no—
244C2×Dic61—48no—
245C2×Dic61—42no—
246C2×Dic61—42no—
247C2×Dic61—44no—
248C2×Dic61—42no—
249C2×Dic61—42no—
250C2×Dic61—84no—
251C2×Dic61AGL(1,ℤ/8)14yescomplemented
252C8×C61—24no—
253#92418—42no—
254#23546—611nomaximal, Sylow
255C22×C61—24no—
256C22×C61—22no—
257C22×C61—22no—
258C22×C61—24no—
259C22×C61—42no—
260C22×C61—44no—
261C22×C61—48no—
262C22×C61—42no—
263C22×C61—42no—
264C22×C61—816no—
265C22×C61—82no—
266C22×C61—82no—
267C22×C61—82no—
268C22×C61—82no—
269C22×C61—84no—
270C22×C61#39414yes—
271C22×C61C2×#12111yescharacteristic
272C22×C61C2×#12111yescharacteristic
273C22×C61C2×#12111yescharacteristic
274C22×C61C2×AGL(1,ℤ/8)12yes—
275C22×C61#57512yes—
276C4×C61—42no—
277C4×C61#39414yes—
278C4×C61C2×#12011yescharacteristic
279C4×C61C2×#12311yescharacteristic
280C4×C61C2×#12311yescharacteristic
281C4×C61C2×#12411yescharacteristic
282Dic61—24no—
283Dic61—42no—
284Dic61—82no—
285Dic61—82no—
286C22×AGL(1,ℤ/8)D6111yescharacteristic
287C2×#423—611no—
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×C61—82no—
297C2×C61—84no—
298C2×C61C2×#39411yescharacteristic
299C2×C61#303612yes—
300C2×C61#303811yescharacteristic
301C2×C61#304011yescharacteristic
302C2×C61#305512yes—
303C22×#41C2×D6111yescharacteristic
304C23×D4C2×D6111yescharacteristic
305C2×AGL(1,ℤ/8)—24no—
306C2×AGL(1,ℤ/8)—24no—
307C2×AGL(1,ℤ/8)—24no—
308C22×SD16C2×D6112yes—
309C22×D8C2×D6112yes—
310C22×M16—611no—
311C22×M16C2×D6111yescharacteristic
312C2×#117—611no—
313C22×C4⋊C4—611no—
314C22×C22⋊C4—1222no—
315C22×C42—611no—
316#423—1224no—
317C2×C4⋊C8—614no—
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—21no—
334C2×#41—22no—
335C2×#41—22no—
336C2×#41—22no—
337C2×#41—22no—
338C2×#41—22no—
339C2×#41C22⋊D6114yes—
340C22×Q8C22×D6111yescharacteristic
341C22×D4—21no—
342C22×D4—21no—
343C22×D4—22no—
344C22×D4—22no—
345C22×D4—22no—
346C22×D4—22no—
347C22×D4—22no—
348C22×D4C22⋊D6114yes—
349C22×D4C22×D6111yescharacteristic
350C22×D4C22×D6112yes—
351C23×C4—21no—
352C23×C4—612no—
353C23×C4—1221no—
354C23×C4C22×D6111yescharacteristic
355AGL(1,ℤ/8)—232no—
356C2×SD16—24no—
357C2×SD16—24no—
358C2×SD16—24no—
359C2×D8—24no—
360C2×D8—24no—
361C2×D8—24no—
362C2×M16—22no—
363C2×M16—22no—
364C2×M16—1222no—
365C2×M16—1222no—
366C2×M16—1222no—
367C2×M16C22⋊D6114yes—
368C22×C8—612no—
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—1224no—
378C2×C22⋊C4—1224no—
379C2×C22⋊C4—1222no—
380C2×C22⋊C4—1222no—
381C2×C22⋊C4—1221no—
382C2×C22⋊C4—2448no—
383C2×C42—611no—
384C2×C42—611no—
385C2×C42—612no—
386C2×C42—1222no—
387C2×C42—1222no—
388C2×C42—1221no—
389C4⋊C8—1228no—
390#103—1224no—
391#102—1224no—
392#102—1228no—
393#95—1222no—
394#95—1222no—
395C24—21no—
396C24—22no—
397C24—24no—
398C24—44no—
399C24—44no—
400C24—42no—
401C24—42no—
402C24D4×D6111yescharacteristic
403#41—28no—
404#41—28no—
405#41—44no—
406#41—44no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8C2×C22⋊D6112yes—
413C2×D4—24no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—22no—
423C2×D4—21no—
424C2×D4—28no—
425C2×D4—24no—
426C2×D4—28no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4C2×C22⋊D6112yes—
434C2×D4C2×C22⋊D6114yes—
435C22×C4—21no—
436C22×C4—24no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—612no—
440C22×C4—612no—
441C22×C4—1224no—
442C22×C4—1224no—
443C22×C4—1222no—
444C22×C4—1222no—
445C22×C4—2442no—
446C22×C4—2444no—
447C22×C4—2442no—
448C22×C4D4×D6111yescharacteristic
449C22×C4#1175822412yes—
450C22×C4#1175822412yes—
451C22×C4#1175822412yes—
452C22×C4C2×C22⋊D6112yes—
453C22×C4C2×C22⋊D6112yes—
454C22×C4C2×C22⋊D6112yes—
455C22×C4C23×D6111yescharacteristic
456SD16—216no—
457D8—216no—
458M16—28no—
459M16—2444no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—1222no—
463C2×C8—1222no—
464C2×C8—1222no—
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—2448no—
474C22⋊C4—2444no—
475C22⋊C4—2442no—
476C42—1222no—
477C42—1222no—
478C42—1222no—
479C42—1222no—
480C42—2442no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—24no—
485C23—416no—
486C23—41no—
487C23—42no—
488C23—41no—
489C23—42no—
490C23—44no—
491C23—42no—
492C23—42no—
493C23—44no—
494C23—48no—
495C23—88no—
496C23—84no—
497C23—84no—
498C23#1451974812yes—
499C23#1451974912yes—
500C23#1451974912yes—
501C23C2×D4×D6111yescharacteristic
502Q8—22no—
503Q8—22no—
504Q8—41no—
505Q8—41no—
506D4—22no—
507D4—24no—
508D4—24no—
509D4—22no—
510D4—48no—
511D4—42no—
512D4—41no—
513D4—42no—
514D4—41no—
515D4—48no—
516C2×C4—22no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—48no—
520C2×C4—614no—
521C2×C4—1222no—
522C2×C4—1222no—
523C2×C4—1222no—
524C2×C4—1222no—
525C2×C4—1222no—
526C2×C4—4884no—
527C2×C4—2448no—
528C2×C4—2442no—
529C2×C4—2442no—
530C2×C4—2444no—
531C2×C4—2442no—
532C2×C4—2442no—
533C2×C4#1451974614yes—
534C2×C4#1451974812yes—
535C2×C4#1451974912yes—
536C2×C4#1451974912yes—
537C2×C4#1451975814yes—
538C2×C4#1451975814yes—
539C2×C4#1451976414yes—
540C2×C4C22×C22⋊D6111yescharacteristic
541C2×C4C22×C22⋊D6111yescharacteristic
542C2×C4C22×C22⋊D6111yescharacteristic
543C2×C4C2×#1175822411yescharacteristic
544C2×C4C2×#1175822411yescharacteristic
545C2×C4C2×#1175822411yescharacteristic
546C8—24no—
547C8—2442no—
548C22—24no—
549C22—24no—
550C22—22no—
551C22—22no—
552C22—42no—
553C22—44no—
554C22—48no—
555C22—42no—
556C22—42no—
557C22—816no—
558C22—82no—
559C22—84no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22#1683967614yes—
564C22C2×#1451974811yescharacteristic
565C22C2×#1451974911yescharacteristic
566C22C2×#1451974911yescharacteristic
567C22AGL(1,ℤ/8)×D6112yes—
568C22#1684055812yes—
569C4—42no—
570C4—1224no—
571C4—4882no—
572C4—4882no—
573C4—2442no—
574C4#1683967614yes—
575C4C2×#1451974611yescharacteristic
576C4C2×#1451975811yescharacteristic
577C4C2×#1451975811yescharacteristic
578C4C2×#1451976411yescharacteristic
579C2—24no—
580C2—84no—
581C2—82no—
582C2C2×#1683967611yescharacteristic
583C2#2145528012yes—
584C2#2145529711yescharacteristic
585C2#2145531111yescharacteristic
586C2#2145543212yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC22×AGL(1,ℤ/8)×C61C2
socleC23×C61C22×D4
hypercenterC22×AGL(1,ℤ/8)D61
hypocenterC61#23546
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#16839785C2semidirect
2. C2×#16839763C2semidirect
3. C2×#16839767C2semidirect
4. #21448310C2semidirect
5. C2×#16839665C2semidirect
6. #21447132C2semidirect
7. #21447127C2semidirect
8. #21445539C2semidirect
16. C2×#14519706C22semidirect
19. C2×#14519701C22semidirect
21. C2×#14519698C22semidirect
22. #16839669C22semidirect
26. #16839665C22semidirect
30. #16839525C22semidirect
251. C2×Dic61AGL(1,ℤ/8)semidirect
332. C61#23546semidirect

Maximal subgroups

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

subgroupindex
C2×#168397852
C2×#168397632
C2×#168397672
#214483102
C2×#168396652
#214471322
#214471272
#214455392
C22×AGL(1,ℤ/8)×C612
#2354661