Finite Group Databasebeta

← #29707172

Subgroups — #29707172

Order 15616.

2146 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519802C211yesmaximal, characteristic
2C22×#14519766C211yesmaximal, characteristic, complemented
3C2×#16839771C211yesmaximal, characteristic, complemented
4C2×#16839768C211yesmaximal, characteristic, complemented
5#21448310C211yesmaximal, characteristic, complemented
6#21446910C214yesmaximal, complemented
7#21446893C214yesmaximal, complemented
8#21445512C211yesmaximal, characteristic, complemented
9C22×#137×C61C211yesmaximal, characteristic, complemented
10C23×D244C2211yescharacteristic
11C2×#14519802—24no—
12C2×#14519802—24no—
13C2×#14519802—24no—
14C22×#11758201C2212yes—
15C22×#11758199C2212yes—
16C22×#11758194C2211yescharacteristic
17C2×#14519766—22no—
18C2×#14519766—24no—
19C2×#14519766—24no—
20C2×#14519766C2214yes—
21C2×#14519736C2211yescharacteristic
22C22×#11758198—22no—
23C22×C4×Dic61C2211yescharacteristic
24#16839771—24no—
25#16839768—24no—
26C2×#14519701C2214yescomplemented
27C2×#14519699C2214yes—
28#16839669C2212yescomplemented
29#16839665C2212yescomplemented
30#16839642—22no—
31C4×C22⋊Dic61—22no—
32#16839516C2214yescomplemented
33C22×#41×C61C2211yescharacteristic
34C23×Q8×C61C2211yescharacteristic
35C2×#137×C61—24no—
36C2×#137×C61—24no—
37C2×#137×C61—24no—
38C22×Q16×C61C2212yescomplemented
39C22×SD16×C61C2212yescomplemented
40C22×M16×C61C2211yescharacteristic
41C24×D61—21no—
42C22×D244—22no—
43C22×D244—22no—
44C22×D244—22no—
45C22×D244—22no—
46C22×D244—21no—
47C22×D244—22no—
48C22×D244D414yes—
49C22×D244C2312yes—
50C23×Dic61D412yes—
51#14519802—232no—
52C2×#11758201—24no—
53C2×#11758201—24no—
54C2×#11758201—24no—
55C2×#11758199—24no—
56C2×#11758199—24no—
57C2×#11758199—24no—
58C2×#11758194—22no—
59C2×#11758194—22no—
60C2×#11758194D414yes—
61C22×#92418C2312yes—
62#14519766—216no—
63#14519766—28no—
64#14519766—28no—
65#14519736—22no—
66#14519736—22no—
67C2×C4⋊Dic61C2312yes—
68C2×C22⋊Dic61—21no—
69C2×C22⋊Dic61—24no—
70C2×#11758198—216no—
71C2×#11758198—28no—
72C2×C4×Dic61—22no—
73C2×C4×Dic61—22no—
74C2×C4×Dic61—21no—
75C2×C4×Dic61D412yes—
76C2×C4×Dic61C2312yes—
77#14519701—28no—
78#14519699—28no—
79#14519684—24no—
80C2×#41×C61—22no—
81C2×#41×C61—22no—
82C2×#41×C61—22no—
83C2×#41×C61—22no—
84C2×#41×C61—22no—
85C2×#41×C61D414yes—
86C22×Q8×C61—22no—
87C22×Q8×C61—21no—
88C22×Q8×C61—22no—
89C22×Q8×C61—22no—
90C22×Q8×C61—22no—
91C22×Q8×C61—22no—
92C22×Q8×C61D414yes—
93C22×Q8×C61C2312yes—
94C22×Q8×C61C2311yescharacteristic
95C22×D4×C61—21no—
96C22×D4×C61C2311yescharacteristic
97C23×C4×C61—21no—
98C23×C4×C61—21no—
99C23×C4×C61C2311yescharacteristic
100#137×C61—232no—
101C2×Q16×C61—24no—
102C2×Q16×C61—24no—
103C2×Q16×C61—24no—
104C2×SD16×C61—24no—
105C2×SD16×C61—24no—
106C2×SD16×C61—24no—
107C2×M16×C61—22no—
108C2×M16×C61—22no—
109C2×M16×C61D414yes—
110C22×C8×C61C2312yes—
111C24×C61—21no—
112C24×C61C2×D411yescharacteristic
113C23×D61—24no—
114C23×D61—22no—
115C23×D61—48no—
116C23×D61—42no—
117C23×D61—42no—
118#41×C61—28no—
119#41×C61—28no—
120#41×C61—44no—
121#41×C61—44no—
122C2×Q8×C61—24no—
123C2×Q8×C61—22no—
124C2×Q8×C61—21no—
125C2×Q8×C61—28no—
126C2×Q8×C61—28no—
127C2×Q8×C61—22no—
128C2×Q8×C61—22no—
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61—22no—
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—22no—
135C2×Q8×C61—44no—
136C2×Q8×C61—42no—
137C2×Q8×C61—44no—
138C2×Q8×C61—42no—
139C2×Q8×C61C2×D414yes—
140C2×Q8×C61C2×D412yes—
141C2×D4×C61—24no—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61—21no—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—42no—
148C2×D4×C61—42no—
149C2×D4×C61C2×D412yes—
150C22×C4×C61—24no—
151C22×C4×C61—21no—
152C22×C4×C61—22no—
153C22×C4×C61—24no—
154C22×C4×C61—42no—
155C22×C4×C61—42no—
156C22×C4×C61—42no—
157C22×C4×C61—42no—
158C22×C4×C61C2×D412yes—
159C22×C4×C61C2×D412yes—
160C22×C4×C61C2×D412yes—
161C22×C4×C61C2×D411yescharacteristic
162C22×C4×C61C2×D412yes—
163C22×C4×C61C2×D412yes—
164C22×C4×C61#4112yes—
165C22×C4×C61C2411yescharacteristic
166C22×Dic61—24no—
167C22×Dic61—22no—
168C22×Dic61—44no—
169C22×Dic61C2×D414yes—
170C2×D244—28no—
171C2×D244—28no—
172C2×D244—22no—
173C2×D244—22no—
174C2×D244—22no—
175C2×D244—22no—
176C2×D244—24no—
177C2×D244—22no—
178C2×D244—44no—
179C2×D244—44no—
180C2×D244—42no—
181C2×D244—42no—
182C2×D244C2×D414yes—
183Q16×C61—216no—
184SD16×C61—216no—
185M16×C61—28no—
186C2×C8×C61—22no—
187C2×C8×C61—22no—
188C2×C8×C61C2×D414yes—
189C22⋊Dic61—48no—
190C22⋊Dic61—42no—
191#11758201—216no—
192#11758199—216no—
193#11758198—232no—
194C4⋊Dic61—22no—
195C4⋊Dic61—22no—
196C4×Dic61—24no—
197C4×Dic61—22no—
198C4×Dic61—22no—
199C4×Dic61—42no—
200#11758194—28no—
201C2×#92418—22no—
202C2×#92418—22no—
203C2×#92418C2×D414yes—
204C23×C61—24no—
205C23×C61—22no—
206C23×C61—41no—
207C23×C61—41no—
208C23×C61—44no—
209C23×C61#12012yes—
210C23×C61#12112yes—
211C23×C61#12712yes—
212C23×C61C22×D411yescharacteristic
213C22×D61—24no—
214C22×D61—416no—
215C22×D61—42no—
216C22×D61—42no—
217C22×D61—48no—
218C22×D61—48no—
219C22×D61—88no—
220C22×D61—88no—
221Q8×C61—22no—
222Q8×C61—24no—
223Q8×C61—22no—
224Q8×C61—24no—
225Q8×C61—42no—
226Q8×C61—42no—
227Q8×C61—41no—
228Q8×C61—41no—
229Q8×C61—48no—
230D4×C61—22no—
231D4×C61—22no—
232D4×C61—48no—
233D4×C61—41no—
234D4×C61—41no—
235C2×C4×C61—24no—
236C2×C4×C61—22no—
237C2×C4×C61—41no—
238C2×C4×C61—42no—
239C2×C4×C61—41no—
240C2×C4×C61—42no—
241C2×C4×C61—48no—
242C2×C4×C61—48no—
243C2×C4×C61#12014yes—
244C2×C4×C61#12014yes—
245C2×C4×C61#12012yes—
246C2×C4×C61#12114yes—
247C2×C4×C61#12114yes—
248C2×C4×C61#12112yes—
249C2×C4×C61#12712yes—
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×Dic61—24no—
257C2×Dic61—24no—
258C2×Dic61—24no—
259C2×Dic61—48no—
260C2×Dic61—42no—
261C2×Dic61—44no—
262D244—24no—
263D244—24no—
264D244—42no—
265D244—42no—
266D244—48no—
267C8×C61—24no—
268#92418—24no—
269#23431—611nomaximal, Sylow
270C22×C61—24no—
271C22×C61—24no—
272C22×C61—24no—
273C22×C61—42no—
274C22×C61—44no—
275C22×C61—82no—
276C22×C61—82no—
277C22×C61#39214yes—
278C22×C61C2×#12011yescharacteristic
279C22×C61C2×#12111yescharacteristic
280C22×C61C2×#12711yescharacteristic
281C22×C61#57514yes—
282C2×D61—44no—
283C2×D61—48no—
284C2×D61—816no—
285C2×D61—84no—
286C2×D61—84no—
287C4×C61—42no—
288C4×C61—44no—
289C4×C61#39214yes—
290C4×C61C2×#12011yescharacteristic
291C4×C61C2×#12011yescharacteristic
292C4×C61C2×#12111yescharacteristic
293C4×C61C2×#12111yescharacteristic
294Dic61—44no—
295Dic61—82no—
296C22×#137D6111yescharacteristic, complemented
297C22×AGL(1,ℤ/8)—611no—
298C22×#124—611no—
299C2×#419—611no—
300C2×#418—611no—
301#2297—611no—
302#2015—614no—
303#2012—614no—
304#1728—611no—
305C2×C61—24no—
306C2×C61—82no—
307C2×C61C2×#39211yescharacteristic
308C2×C61#300512yes—
309C2×C61#304012yes—
310C2×C61#315312yes—
311D61—84no—
312C22×#41C2×D6111yescharacteristic
313C23×Q8C2×D6111yescharacteristic
314C23×D4—611no—
315C2×#137—24no—
316C2×#137—24no—
317C2×#137—24no—
318C2×AGL(1,ℤ/8)—1224no—
319C2×AGL(1,ℤ/8)—1224no—
320C2×AGL(1,ℤ/8)—1224no—
321C22×Q16C2×D6112yescomplemented
322C22×SD16—612no—
323C22×SD16C2×D6112yescomplemented
324C22×D8—612no—
325C22×M16—611no—
326C22×M16C2×D6111yescharacteristic
327C2×#124—614no—
328C2×#124—1222no—
329C2×#124—1224no—
330C2×#124—1224no—
331C2×#117—611no—
332C22×C22⋊C4—1222no—
333C22×C42—611no—
334#419—1224no—
335#418—1224no—
336C2×#103—614no—
337C2×#102—614no—
338#391—612no—
339#390—612no—
340#386—1222no—
341C4×C22⋊C4—1222no—
342#336—614no—
343C61#2343111yescharacteristic, complemented, Sylow
344C25—1221no—
345C2×#41—22no—
346C2×#41—22no—
347C2×#41—22no—
348C2×#41—22no—
349C2×#41—22no—
350C2×#41C22⋊D6114yes—
351C22×Q8—21no—
352C22×Q8—22no—
353C22×Q8—22no—
354C22×Q8—22no—
355C22×Q8—22no—
356C22×Q8—22no—
357C22×Q8C22⋊D6114yes—
358C22×Q8C22×D6112yes—
359C22×Q8C22×D6111yescharacteristic
360C22×D4—21no—
361C22×D4—614no—
362C22×D4—612no—
363C22×D4—1222no—
364C22×D4—1221no—
365C22×D4—1222no—
366C22×D4—1222no—
367C22×D4—1222no—
368C22×D4—1222no—
369C22×D4C22×D6111yescharacteristic
370C23×C4—21no—
371C23×C4—21no—
372C23×C4—612no—
373C23×C4C22×D6111yescharacteristic
374#137—232no—
375AGL(1,ℤ/8)—12232no—
376C2×Q16—24no—
377C2×Q16—24no—
378C2×Q16—24no—
379C2×SD16—24no—
380C2×SD16—24no—
381C2×SD16—24no—
382C2×SD16—1224no—
383C2×SD16—1224no—
384C2×SD16—1224no—
385C2×D8—1224no—
386C2×D8—1224no—
387C2×D8—1224no—
388C2×M16—22no—
389C2×M16—22no—
390C2×M16—614no—
391C2×M16—1222no—
392C2×M16—1222no—
393C2×M16C22⋊D6114yes—
394C22×C8—612no—
395C22×C8C22×D6112yes—
396#124—1228no—
397#124—12216no—
398#124—1228no—
399#117—1222no—
400#117—1222no—
401C2×C4⋊C4—612no—
402C2×C22⋊C4—1228no—
403C2×C22⋊C4—12216no—
404C2×C22⋊C4—1224no—
405C2×C22⋊C4—1221no—
406C2×C42—612no—
407C2×C42—612no—
408C2×C42—1222no—
409C2×C42—1222no—
410C2×C42—1221no—
411#103—1228no—
412#102—1228no—
413#95—1224no—
414C24—21no—
415C24—1222no—
416C24—1224no—
417C24—2448no—
418C24—2442no—
419C24—2442no—
420C24D4×D6111yescharacteristic
421#41—28no—
422#41—28no—
423#41—44no—
424#41—44no—
425C2×Q8—22no—
426C2×Q8—21no—
427C2×Q8—24no—
428C2×Q8—22no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—22no—
436C2×Q8—28no—
437C2×Q8—28no—
438C2×Q8—44no—
439C2×Q8—44no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8C2×C22⋊D6114yes—
443C2×Q8C2×C22⋊D6112yes—
444C2×D4—24no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—42no—
451C2×D4—42no—
452C2×D4—614no—
453C2×D4—1222no—
454C2×D4—1222no—
455C2×D4—1222no—
456C2×D4—1222no—
457C2×D4—1222no—
458C2×D4—1228no—
459C2×D4—1224no—
460C2×D4—1228no—
461C2×D4—2444no—
462C2×D4—2444no—
463C2×D4—2442no—
464C2×D4—2442no—
465C2×D4C2×C22⋊D6112yes—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—614no—
475C22×C4—1224no—
476C22×C4—1222no—
477C22×C4—2444no—
478C22×C4D4×D6112yes—
479C22×C4D4×D6112yes—
480C22×C4D4×D6111yescharacteristic
481C22×C4#1175822412yes—
482C22×C4C2×C22⋊D6112yes—
483C22×C4C2×C22⋊D6112yes—
484C22×C4C2×C22⋊D6112yes—
485C22×C4C23×D6111yescharacteristic
486Q16—216no—
487SD16—216no—
488SD16—12216no—
489D8—12216no—
490M16—28no—
491M16—1228no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—614no—
495C2×C8—1222no—
496C2×C8—1222no—
497C2×C8C2×C22⋊D6114yes—
498C4⋊C4—1222no—
499C4⋊C4—1222no—
500C22⋊C4—12232no—
501C22⋊C4—2448no—
502C22⋊C4—2442no—
503C42—1222no—
504C42—1222no—
505C42—1224no—
506C42—2442no—
507C23—24no—
508C23—22no—
509C23—41no—
510C23—41no—
511C23—44no—
512C23—1224no—
513C23—4888no—
514C23—4888no—
515C23—24416no—
516C23—2442no—
517C23—2442no—
518C23—2448no—
519C23—2448no—
520C23#1451974512yes—
521C23#1451974912yes—
522C23#1451977312yes—
523C23C2×D4×D6111yescharacteristic
524Q8—22no—
525Q8—24no—
526Q8—22no—
527Q8—24no—
528Q8—42no—
529Q8—42no—
530Q8—41no—
531Q8—41no—
532Q8—48no—
533D4—22no—
534D4—22no—
535D4—48no—
536D4—41no—
537D4—41no—
538D4—1224no—
539D4—1224no—
540D4—2442no—
541D4—2442no—
542D4—2448no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—48no—
550C2×C4—48no—
551C2×C4—1224no—
552C2×C4—1224no—
553C2×C4—1224no—
554C2×C4—2442no—
555C2×C4—2444no—
556C2×C4—2448no—
557C2×C4#1451974514yes—
558C2×C4#1451974512yes—
559C2×C4#1451974614yes—
560C2×C4#1451974814yes—
561C2×C4#1451974914yes—
562C2×C4#1451974912yes—
563C2×C4#1451977312yes—
564C2×C4C2×D4×D6111yescharacteristic
565C2×C4C2×D4×D6111yescharacteristic
566C2×C4C22×C22⋊D6111yescharacteristic
567C2×C4C22×C22⋊D6111yescharacteristic
568C2×C4C22×C22⋊D6111yescharacteristic
569C2×C4C2×#1175822411yescharacteristic
570C8—24no—
571C8—1224no—
572C22—24no—
573C22—24no—
574C22—24no—
575C22—42no—
576C22—44no—
577C22—82no—
578C22—82no—
579C22—48816no—
580C22—4884no—
581C22—4884no—
582C22—2444no—
583C22—2448no—
584C22#1683967014yes—
585C22C2×#1451974511yescharacteristic
586C22C2×#1451974911yescharacteristic
587C22C2×#1451977311yescharacteristic
588C22#1684056114yes—
589C4—42no—
590C4—44no—
591C4—4882no—
592C4—2444no—
593C4#1683967014yes—
594C4C2×#1451974511yescharacteristic
595C4C2×#1451974611yescharacteristic
596C4C2×#1451974811yescharacteristic
597C4C2×#1451974911yescharacteristic
598C2—24no—
599C2—82no—
600C2—4884no—
601C2C2×#1683967011yescharacteristic
602C2#2145506012yes—
603C2#2145531512yes—
604C2#2145628612yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC22×#137×C61C2
socleC23×C61C22×D4
prime radicalC22×#14519802C2
hypercenterC22×#137D61
hypocenterC61#23431
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#137

Complements

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

NH ≅ G/Ntype
2. C22×#14519766C2semidirect
3. C2×#16839771C2semidirect
4. C2×#16839768C2semidirect
5. #21448310C2semidirect
6. #21446910C2semidirect
7. #21446893C2semidirect
8. #21445512C2semidirect
9. C22×#137×C61C2semidirect
26. C2×#14519701C22semidirect
28. #16839669C22semidirect
29. #16839665C22semidirect
32. #16839516C22semidirect
38. C22×Q16×C61C22semidirect
39. C22×SD16×C61C22semidirect
296. C22×#137D61semidirect
321. C22×Q16C2×D61semidirect
323. C22×SD16C2×D61semidirect
343. C61#23431semidirect

Maximal subgroups

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

subgroupindex
C22×#145198022
C22×#145197662
C2×#168397712
C2×#168397682
#214483102
#214469102
#214468932
#214455122
C22×#137×C612
#2343161