Finite Group Databasebeta

← #29715686

Subgroups — #29715686

Order 15616.

1606 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11758190C211yesmaximal, characteristic, complemented
2C2×#16839991C212yesmaximal, complemented
3#21449184C211yesmaximal, characteristic, complemented
4#21449175C211yesmaximal, characteristic
5#21447137C212yesmaximal, complemented
6#21446824C212yesmaximal, complemented
7#21446673C212yesmaximal, complemented
8#21445844C211yesmaximal, characteristic, complemented
9#21445755C211yesmaximal, characteristic, complemented
10#21445751C211yesmaximal, characteristic, complemented
11C2×C8×D4×C61C211yesmaximal, characteristic, complemented
12C23×Dic122C2211yescharacteristic
13C23×D244C2211yescharacteristic
14C22×#11758190—28no—
15C22×#11758190—24no—
16C22×#11758190C2212yes—
17C22×#11758190C2212yes—
18C2×#14519756C2212yescomplemented
19C2×#14519751C2212yes—
20C2×#14519745—21no—
21#16839991—28no—
22C2×#14519709C2212yescomplemented
23C2×#14519702C2212yescomplemented
24C2×#14519702C2212yescomplemented
25C2×#14519699C2212yes—
26C2×#14519699C2212yes—
27#16839682—21no—
28#16839679—21no—
29#16839674—21no—
30#16839671—21no—
31#16839668C2212yescomplemented
32#16839664C2212yes—
33#16839642—21no—
34#16839634—21no—
35#16839622C2211yescharacteristic
36#16839622C2211yescharacteristic
37#16839617—21no—
38#16839525C2212yescomplemented
39C23×C8×C61C2211yescharacteristic
40C23×C8×C61C2211yescharacteristic, complemented
41C2×C4×D4×C61C2211yescharacteristic
42C8×D4×C61—24no—
43C2×C4⋊C8×C61C2211yescharacteristic, complemented
44C2×C22⋊C8×C61C2211yescharacteristic
45C2×C22⋊C8×C61C2211yescharacteristic, complemented
46C2×C4×C8×C61C2211yescharacteristic, complemented
47C24×D61—21no—
48C22×Dic122—22no—
49C22×Dic122—24no—
50C22×Dic122—22no—
51C22×Dic122—21no—
52C22×Dic122—42no—
53C22×Dic122C2312yes—
54C22×D244—22no—
55C22×D244—24no—
56C22×D244—22no—
57C22×D244—21no—
58C22×D244—42no—
59C22×D244C2312yes—
60C22×C22⋊D61—22no—
61C23×Dic61—21no—
62C2×#11758190—232no—
63C2×#11758190—24no—
64C2×#11758190—24no—
65C2×#11758190—24no—
66C2×#11758190—24no—
67C2×#11758190—24no—
68C2×#11758190D418yescomplemented
69#14519756—24no—
70#14519756—42no—
71#14519751—24no—
72#14519751—42no—
73#14519745—44no—
74C2×C4⋊Dic61C2312yes—
75C2×C4⋊Dic61C2312yes—
76C2×#11758196—22no—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61—22no—
79C2×C22⋊Dic61—21no—
80C2×#11758198—22no—
81C2×#11758198—22no—
82C2×#11758198—22no—
83#14519709—24no—
84#14519702—24no—
85#14519702—42no—
86#14519699—24no—
87#14519699—42no—
88#14519684—21no—
89#14519684—22no—
90#14519684—21no—
91#14519684—22no—
92C22×D4×C61D411yescharacteristic
93C23×C4×C61C2311yescharacteristic
94C23×C4×C61C2311yescharacteristic
95C22×C8×C61—24no—
96C22×C8×C61—24no—
97C22×C8×C61—22no—
98C22×C8×C61D414yescomplemented
99C22×C8×C61C2311yescharacteristic
100C22×C8×C61C2311yescharacteristic
101C22×C8×C61C2311yescharacteristic
102C22×C8×C61C2311yescharacteristic
103C4×D4×C61—22no—
104C4×D4×C61—22no—
105C2×C4⋊C4×C61D411yescharacteristic
106C2×C22⋊C4×C61D411yescharacteristic
107C2×C22⋊C4×C61D411yescharacteristic
108C2×C42×C61C2311yescharacteristic
109C4⋊C8×C61—22no—
110C22⋊C8×C61—22no—
111C22⋊C8×C61—22no—
112C4×C8×C61—22no—
113C24×C61C2×D411yescharacteristic
114C24×C61C2×D411yescharacteristic
115C23×D61—22no—
116C23×D61—44no—
117C23×D61—42no—
118C23×D61—42no—
119C23×D61—42no—
120C23×D61—82no—
121C2×D4×C61—22no—
122C2×D4×C61—22no—
123C2×D4×C61SD1614yescomplemented
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—21no—
127C22×C4×C61—22no—
128C22×C4×C61—21no—
129C22×C4×C61—22no—
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D412yes—
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61#4112yes—
136C22×C4×C61C2411yescharacteristic
137C2×C22⋊D61—44no—
138C2×C22⋊D61—44no—
139C2×C22⋊D61—82no—
140C22×Dic61—22no—
141C22×Dic61—22no—
142C22×Dic61—22no—
143C22×Dic61—44no—
144C22×Dic61—42no—
145C2×D244—28no—
146C2×D244—22no—
147C2×D244—22no—
148C2×D244—22no—
149C2×D244—22no—
150C2×D244—22no—
151C2×D244—48no—
152C2×D244—44no—
153C2×D244—44no—
154C2×D244—42no—
155C2×D244C2×D414yes—
156C2×Dic122—28no—
157C2×Dic122—22no—
158C2×Dic122—22no—
159C2×Dic122—22no—
160C2×Dic122—22no—
161C2×Dic122—22no—
162C2×Dic122—48no—
163C2×Dic122—44no—
164C2×Dic122—44no—
165C2×Dic122—42no—
166C2×Dic122C2×D414yes—
167C2×C8×C61—22no—
168C2×C8×C61—28no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—22no—
172C2×C8×C61—22no—
173C2×C8×C61—44no—
174C2×C8×C61C2×D412yes—
175C2×C8×C61C2×D414yes—
176C2×C8×C61C2×D412yes—
177C4⋊C4×C61—22no—
178C22⋊C4×C61—22no—
179C22⋊C4×C61—22no—
180C42×C61—21no—
181C42×C61—21no—
182C22⋊Dic61—44no—
183C22⋊Dic61—42no—
184C22⋊Dic61—42no—
185#11758198—44no—
186#11758198—44no—
187#11758198—82no—
188C4⋊Dic61—22no—
189C4⋊Dic61—22no—
190C4⋊Dic61—22no—
191C4⋊Dic61—22no—
192#11758196—44no—
193#11758190—216no—
194#11758190—216no—
195C23×C61—22no—
196C23×C61—22no—
197C23×C61—21no—
198C23×C61—22no—
199C23×C61—21no—
200C23×C61—22no—
201C23×C61#12112yes—
202C23×C61C2×SD1612yes—
203C23×C61C2×SD1612yescomplemented
204C23×C61#13512yes—
205C23×C61C22×D411yescharacteristic
206C22×D61—24no—
207C22×D61—48no—
208C22×D61—42no—
209C22×D61—42no—
210C22×D61—42no—
211C22×D61—42no—
212C22×D61—42no—
213C22×D61—42no—
214C22×D61—88no—
215C22×D61—88no—
216C22×D61—84no—
217C22×D61—84no—
218D4×C61—28no—
219C2×C4×C61—21no—
220C2×C4×C61—22no—
221C2×C4×C61—21no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—24no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—42no—
232C2×C4×C61#12112yes—
233C2×C4×C61C2×SD1612yescomplemented
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C22×D412yes—
236C2×C4×C61C2×#4111yescharacteristic
237C2×C4×C61H(5,𝔽2)12yes—
238C22⋊D61—88no—
239C2×Dic61—24no—
240C2×Dic61—48no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246C2×Dic61—42no—
247C2×Dic61—42no—
248C2×Dic61—42no—
249D244—24no—
250D244—24no—
251D244—48no—
252D244—42no—
253D244—42no—
254Dic122—24no—
255Dic122—24no—
256Dic122—48no—
257Dic122—42no—
258Dic122—42no—
259C8×C61—22no—
260C8×C61—41no—
261C8×C61—41no—
262C8×C61#12014yescomplemented
263#24264—611nomaximal, Sylow
264C22×C61—22no—
265C22×C61—24no—
266C22×C61—24no—
267C22×C61—22no—
268C22×C61—24no—
269C22×C61—22no—
270C22×C61—41no—
271C22×C61—42no—
272C22×C61—41no—
273C22×C61#46514yescomplemented
274C22×C61C2×#12111yescharacteristic
275C22×C61D4212yes—
276C22×C61C22×SD1611yescharacteristic
277C22×C61C2×#13511yescharacteristic
278C22×C61#57712yes—
279C2×D61—44no—
280C2×D61—44no—
281C2×D61—44no—
282C2×D61—816no—
283C2×D61—82no—
284C2×D61—82no—
285C2×D61—82no—
286C2×D61—82no—
287C4×C61—22no—
288C4×C61—22no—
289C4×C61—41no—
290C4×C61—41no—
291C4×C61C2×#12011yescharacteristic
292C4×C61#53511yescharacteristic
293C4×C61#54612yes—
294Dic61—44no—
295Dic61—82no—
296Dic61—82no—
297C23×SD16—611no—
298C2×#465—612no—
299C2×C8×D4D6111yescharacteristic, complemented
300#2409—611no—
301#2408—611no—
302#2055—612no—
303#2000—612no—
304#1968—612no—
305#1807—611no—
306#1787—611no—
307#1786—611no—
308C2×C61—24no—
309C2×C61—42no—
310C2×C61#240811yescharacteristic
311C2×C61C2×#46511yescharacteristic
312C2×C61#305711yescharacteristic
313C2×C61D4×SD1612yes—
314C2×C61#328212yes—
315D61—84no—
316C23×Q8—611no—
317C23×D4—611no—
318C22×SD16—612no—
319C22×SD16—612no—
320C22×SD16—1228no—
321C22×SD16—1224no—
322C23×C8C2×D6111yescharacteristic
323C23×C8C2×D6111yescharacteristic, complemented
324C2×C22⋊Q8—612no—
325C2×#121—612no—
326C2×#120—1221no—
327C2×C4×D4C2×D6111yescharacteristic
328#465—1228no—
329C8×D4—24no—
330C2×#106—612no—
331C2×C4⋊C8C2×D6111yescharacteristic, complemented
332C2×#103—612no—
333C2×#103—612no—
334C2×#102—612no—
335C2×#102—612no—
336C2×C22⋊C8C2×D6111yescharacteristic
337C2×C22⋊C8C2×D6111yescharacteristic, complemented
338C2×C4×C8C2×D6111yescharacteristic, complemented
339#395—1221no—
340#394—1221no—
341#393—1221no—
342#392—1221no—
343#391—612no—
344#390—612no—
345#386—1221no—
346#385—1221no—
347#380—611no—
348#380—611no—
349#379—1221no—
350#340—612no—
351C61#2426411yescharacteristic, complemented, Sylow
352C25—1221no—
353C22×Q8—612no—
354C22×Q8—1224no—
355C22×Q8—1221no—
356C22×Q8—1222no—
357C22×Q8—1222no—
358C22×Q8—2442no—
359C22×D4—612no—
360C22×D4—1224no—
361C22×D4—1222no—
362C22×D4—1221no—
363C22×D4—1222no—
364C22×D4—1222no—
365C22×D4—2442no—
366C22×D4D24411yescharacteristic
367C23×C4—1221no—
368C23×C4C22×D6111yescharacteristic
369C23×C4C22×D6111yescharacteristic
370C2×SD16—618no—
371C2×SD16—1224no—
372C2×SD16—1224no—
373C2×SD16—12232no—
374C2×SD16—1224no—
375C2×SD16—1224no—
376C2×SD16—1224no—
377C22×C8—24no—
378C22×C8—24no—
379C22×C8—22no—
380C22×C8C22⋊D6114yescomplemented
381C22×C8C22×D6111yescharacteristic
382C22×C8C22×D6111yescharacteristic
383C22×C8C22×D6111yescharacteristic
384C22×C8C22×D6111yescharacteristic
385C22⋊Q8—1224no—
386C22⋊Q8—2442no—
387#121—1224no—
388#121—2442no—
389#120—2444no—
390C4×D4—22no—
391C4×D4—22no—
392C2×C4⋊C4—612no—
393C2×C4⋊C4—612no—
394C2×C4⋊C4—1222no—
395C2×C4⋊C4D24411yescharacteristic
396C2×C22⋊C4—1222no—
397C2×C22⋊C4—1221no—
398C2×C22⋊C4—1222no—
399C2×C22⋊C4—1221no—
400C2×C22⋊C4—1222no—
401C2×C22⋊C4—1222no—
402C2×C22⋊C4D24411yescharacteristic
403C2×C22⋊C4D24411yescharacteristic
404C2×C42C22×D6111yescharacteristic
405#106—1224no—
406C4⋊C8—22no—
407#103—1224no—
408#103—2442no—
409#102—1224no—
410#102—2442no—
411C22⋊C8—22no—
412C22⋊C8—22no—
413C4×C8—22no—
414#95—1221no—
415#95—1221no—
416#95—1222no—
417#95—1222no—
418C24—1222no—
419C24—4882no—
420C24—2444no—
421C24—2442no—
422C24—2442no—
423C24—2442no—
424C24C2×D24411yescharacteristic
425C24C2×D24411yescharacteristic
426C2×Q8—614no—
427C2×Q8—1228no—
428C2×Q8—1222no—
429C2×Q8—1222no—
430C2×Q8—1222no—
431C2×Q8—1222no—
432C2×Q8—1222no—
433C2×Q8—2444no—
434C2×Q8—2448no—
435C2×Q8—2444no—
436C2×Q8—2442no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—614no—
440C2×D4—1228no—
441C2×D4—1222no—
442C2×D4—1222no—
443C2×D4—1222no—
444C2×D4—1222no—
445C2×D4—1222no—
446C2×D4—4882no—
447C2×D4—2448no—
448C2×D4—2444no—
449C2×D4—2444no—
450C2×D4—2444no—
451C2×D4—2444no—
452C2×D4—2442no—
453C2×D4#1175819014yescomplemented
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—1222no—
461C22×C4—1222no—
462C22×C4—1222no—
463C22×C4—2444no—
464C22×C4—2442no—
465C22×C4C2×D24411yescharacteristic
466C22×C4C2×D24411yescharacteristic
467C22×C4C2×D24411yescharacteristic
468C22×C4C2×D24411yescharacteristic
469C22×C4#1175822212yes—
470C22×C4C2×C22⋊D6112yes—
471C22×C4C23×D6111yescharacteristic
472SD16—12216no—
473SD16—12216no—
474C2×C8—22no—
475C2×C8—28no—
476C2×C8—22no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—22no—
480C2×C8—44no—
481C2×C8D4×D6114yes—
482C2×C8C2×C22⋊D6112yes—
483C2×C8C2×C22⋊D6112yes—
484C4⋊C4—22no—
485C4⋊C4—1222no—
486C4⋊C4—1222no—
487C4⋊C4—1222no—
488C4⋊C4—1222no—
489C4⋊C4—2444no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—4882no—
493C22⋊C4—2444no—
494C22⋊C4—2444no—
495C22⋊C4—2442no—
496C22⋊C4—2442no—
497C22⋊C4—2444no—
498C42—21no—
499C42—21no—
500C23—22no—
501C23—22no—
502C23—21no—
503C23—21no—
504C23—22no—
505C23—22no—
506C23—1224no—
507C23—4884no—
508C23—4888no—
509C23—4888no—
510C23—4884no—
511C23—2448no—
512C23—2442no—
513C23—2442no—
514C23—2442no—
515C23—2442no—
516C23—2442no—
517C23—2442no—
518C23#1451975112yes—
519C23C2×#1175819012yes—
520C23C2×#1175819012yescomplemented
521C23#1451979912yes—
522C23C22×D24411yescharacteristic
523Q8—1224no—
524Q8—1224no—
525Q8—2448no—
526Q8—2442no—
527Q8—2442no—
528D4—28no—
529D4—1224no—
530D4—1224no—
531D4—4888no—
532D4—2448no—
533D4—2442no—
534D4—2442no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—1224no—
549C2×C4—2442no—
550C2×C4—2442no—
551C2×C4—2442no—
552C2×C4—2442no—
553C2×C4—2442no—
554C2×C4—2442no—
555C2×C4—2448no—
556C2×C4—2442no—
557C2×C4—2442no—
558C2×C4#1451975112yes—
559C2×C4C2×#1175819012yescomplemented
560C2×C4C2×D4×D6112yes—
561C2×C4C22×C22⋊D6111yescharacteristic
562C2×C4C2×#1175822211yescharacteristic
563C2×C4#1451982312yes—
564C8—22no—
565C8—41no—
566C8—41no—
567C8#1451974514yescomplemented
568C22—22no—
569C22—24no—
570C22—24no—
571C22—22no—
572C22—22no—
573C22—24no—
574C22—41no—
575C22—42no—
576C22—41no—
577C22—48816no—
578C22—4882no—
579C22—4882no—
580C22—4882no—
581C22—4882no—
582C22—2444no—
583C22—2444no—
584C22—2444no—
585C22#1683999114yescomplemented
586C22C2×#1451975111yescharacteristic
587C22D4×D24412yes—
588C22C22×#1175819011yescharacteristic
589C22C2×#1451979911yescharacteristic
590C22#1684057412yes—
591C4—22no—
592C4—22no—
593C4—41no—
594C4—41no—
595C4—4882no—
596C4—4882no—
597C4—2444no—
598C4C2×#1451974511yescharacteristic
599C4#1684032611yescharacteristic
600C4#1684040812yes—
601C2—24no—
602C2—42no—
603C2—4884no—
604C2#2144917511yescharacteristic
605C2C2×#1683999111yescharacteristic
606C2#2145545811yescharacteristic
607C2D4×#1175819012yes—
608C2#2145753112yes—
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C8×D4×C61C2
socleC23×C61C22×D4
prime radical#21449175C2
hypercenterC2×C8×D4D61
hypocenterC61#24264
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4

Complements

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

NH ≅ G/Ntype
1. C23×#11758190C2semidirect
2. C2×#16839991C2semidirect
3. #21449184C2semidirect
5. #21447137C2semidirect
6. #21446824C2semidirect
7. #21446673C2semidirect
8. #21445844C2semidirect
9. #21445755C2semidirect
10. #21445751C2semidirect
11. C2×C8×D4×C61C2semidirect
18. C2×#14519756C22semidirect
22. C2×#14519709C22semidirect
23. C2×#14519702C22semidirect
24. C2×#14519702C22semidirect
31. #16839668C22semidirect
38. #16839525C22semidirect
40. C23×C8×C61C22semidirect
43. C2×C4⋊C8×C61C22semidirect
45. C2×C22⋊C8×C61C22semidirect
46. C2×C4×C8×C61C22semidirect
68. C2×#11758190D4semidirect
98. C22×C8×C61D4semidirect
123. C2×D4×C61SD16semidirect
203. C23×C61C2×SD16semidirect
233. C2×C4×C61C2×SD16semidirect
262. C8×C61#120semidirect
273. C22×C61#465semidirect
299. C2×C8×D4D61semidirect
323. C23×C8C2×D61semidirect
331. C2×C4⋊C8C2×D61semidirect
337. C2×C22⋊C8C2×D61semidirect
338. C2×C4×C8C2×D61semidirect
351. C61#24264semidirect
380. C22×C8C22⋊D61semidirect
453. C2×D4#11758190semidirect
520. C23C2×#11758190semidirect
559. C2×C4C2×#11758190semidirect
567. C8#14519745semidirect
585. C22#16839991semidirect

Maximal subgroups

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

subgroupindex
C23×#117581902
C2×#168399912
#214491842
#214491752
#214471372
#214468242
#214466732
#214458442
#214457552
#214457512
C2×C8×D4×C612
#2426461