Finite Group Databasebeta

← #29712646

Subgroups — #29712646

Order 15616.

1844 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519751C211yesmaximal, characteristic
2C2×#16840075C212yesmaximal, complemented
3#21447001C214yesmaximal, complemented
4#21446136C212yesmaximal, complemented
5#21446006C212yesmaximal, complemented
6#21445749C212yesmaximal, complemented
7#21445729C211yesmaximal, characteristic, complemented
8C2×C61×#380C211yesmaximal, characteristic, complemented
9C23×D244C2211yescharacteristic
10C23×C22⋊D61—21no—
11C2×#14519751—28no—
12C2×#14519751—21no—
13C2×#14519751—21no—
14C2×#14519751C2212yes—
15C2×#14519751C2212yes—
16C22×C4⋊Dic61C2211yescharacteristic
17C22×#11758198—21no—
18#16840075—28no—
19C2×#14519710C2214yescomplemented
20C2×#14519697C2214yescomplemented
21C2×#14519697C2214yescomplemented
22C2×C22⋊#92418C2212yes—
23C2×C22⋊#92418C2212yescomplemented
24C2×C22⋊#92418C2212yescomplemented
25#16839524C2214yescomplemented
26C24×C4×C61C2211yescharacteristic
27C22×C4⋊C4×C61C2211yescharacteristic, complemented
28C22×C4⋊C4×C61C2211yescharacteristic, complemented
29C22×C22⋊C4×C61—21no—
30C61×#380—21no—
31C61×#380—21no—
32C61×#380C2212yes—
33C61×#380C2212yescomplemented
34C2×#95×C61—21no—
35C24×D61—21no—
36C22×D244—21no—
37C22×D244—24no—
38C22×D244—24no—
39C22×D244—42no—
40C22×D244C2312yes—
41C22×C22⋊D61—28no—
42C22×C22⋊D61—22no—
43C22×C22⋊D61—22no—
44C22×C22⋊D61—44no—
45C22×C22⋊D61—44no—
46C23×Dic61—21no—
47C22×#92418C2314yes—
48#14519751—28no—
49#14519751—28no—
50#14519751—24no—
51#14519751—24no—
52#14519751—24no—
53#14519751D418yes—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61—24no—
56C2×C4⋊Dic61C2312yes—
57C2×#11758198—28no—
58C2×#11758198—22no—
59C2×#11758198—22no—
60#14519710—28no—
61#14519697—28no—
62#14519697—44no—
63C22⋊#92418—24no—
64C22⋊#92418—24no—
65C22⋊#92418—42no—
66C25×C61D411yescharacteristic
67C23×C4×C61—24no—
68C23×C4×C61—24no—
69C23×C4×C61—21no—
70C23×C4×C61—21no—
71C23×C4×C61—42no—
72C23×C4×C61D411yescharacteristic
73C23×C4×C61D411yescharacteristic
74C23×C4×C61D411yescharacteristic
75C23×C4×C61C2311yescharacteristic
76C23×C4×C61C2311yescharacteristic
77C23×C4×C61C2311yescharacteristic
78C2×C4⋊C4×C61—24no—
79C2×C4⋊C4×C61—21no—
80C2×C4⋊C4×C61—21no—
81C2×C4⋊C4×C61—42no—
82C2×C4⋊C4×C61C2312yes—
83C2×C4⋊C4×C61C2312yes—
84C2×C22⋊C4×C61—22no—
85C2×C22⋊C4×C61—22no—
86C2×C22⋊C4×C61—44no—
87#95×C61—22no—
88#95×C61—22no—
89C24×C61—22no—
90C24×C61—22no—
91C24×C61—22no—
92C24×C61—22no—
93C24×C61—41no—
94C24×C61—41no—
95C24×C61D814yes—
96C24×C61C2×D411yescharacteristic
97C24×C61C2×D411yescharacteristic
98C24×C61C2×D411yescharacteristic
99C23×D61—24no—
100C23×D61—22no—
101C23×D61—42no—
102C23×D61—42no—
103C23×D61—42no—
104C23×D61—42no—
105C23×D61—42no—
106C23×D61—42no—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—41no—
123C22×C4×C61—41no—
124C22×C4×C61—41no—
125C22×C4×C61—416no—
126C22×C4×C61—42no—
127C22×C4×C61—42no—
128C22×C4×C61—42no—
129C22×C4×C61—42no—
130C22×C4×C61—42no—
131C22×C4×C61—42no—
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D414yes—
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61#4114yes—
137C22×C4×C61C2411yescharacteristic
138C2×C22⋊D61—28no—
139C2×C22⋊D61—28no—
140C2×C22⋊D61—432no—
141C2×C22⋊D61—44no—
142C2×C22⋊D61—44no—
143C2×C22⋊D61—44no—
144C2×C22⋊D61—44no—
145C22×Dic61—24no—
146C22×Dic61—22no—
147C22×Dic61—42no—
148C22×Dic61—42no—
149C2×D244—24no—
150C2×D244—24no—
151C2×D244—24no—
152C2×D244—22no—
153C2×D244—416no—
154C2×D244—42no—
155C2×D244—42no—
156C2×D244C2×D414yes—
157C4⋊C4×C61—24no—
158C4⋊C4×C61—24no—
159C4⋊C4×C61—42no—
160C4⋊C4×C61—42no—
161C22⋊C4×C61—44no—
162C22⋊C4×C61—44no—
163#11758198—28no—
164#11758198—28no—
165C4⋊Dic61—24no—
166C4⋊Dic61—22no—
167C4⋊Dic61C2×D414yes—
168C2×#92418—24no—
169C2×#92418—24no—
170C2×#92418—24no—
171C23×C61—21no—
172C23×C61—21no—
173C23×C61—21no—
174C23×C61—21no—
175C23×C61—21no—
176C23×C61—21no—
177C23×C61—21no—
178C23×C61—21no—
179C23×C61—21no—
180C23×C61—21no—
181C23×C61—21no—
182C23×C61—21no—
183C23×C61—21no—
184C23×C61—21no—
185C23×C61—28no—
186C23×C61—28no—
187C23×C61—44no—
188C23×C61—44no—
189C23×C61—44no—
190C23×C61—44no—
191C23×C61—44no—
192C23×C61#12012yes—
193C23×C61#12312yes—
194C23×C61C2×D812yes—
195C23×C61C2×D812yes—
196C23×C61C2×D812yes—
197C23×C61AGL(1,ℤ/8)12yes—
198C23×C61#13712yes—
199C23×C61C22×D411yescharacteristic
200C22×D61—24no—
201C22×D61—48no—
202C22×D61—48no—
203C22×D61—48no—
204C22×D61—42no—
205C22×D61—42no—
206C22×D61—42no—
207C22×D61—42no—
208C22×D61—42no—
209C22×D61—42no—
210C22×D61—84no—
211C22×D61—84no—
212C22×D61—84no—
213C22×D61—84no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—28no—
219C2×C4×C61—28no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—44no—
224C2×C4×C61—44no—
225C2×C4×C61—44no—
226C2×C4×C61—44no—
227C2×C4×C61—42no—
228C2×C4×C61—42no—
229C2×C4×C61—42no—
230C2×C4×C61—42no—
231C2×C4×C61—42no—
232C2×C4×C61—42no—
233C2×C4×C61—84no—
234C2×C4×C61—84no—
235C2×C4×C61#12012yes—
236C2×C4×C61#12312yes—
237C2×C4×C61C22×D412yes—
238C2×C4×C61C2×#4112yes—
239C2×C4×C61H(5,𝔽2)12yes—
240C22⋊D61—416no—
241C22⋊D61—416no—
242C2×Dic61—24no—
243C2×Dic61—48no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246D244—28no—
247D244—48no—
248D244—44no—
249#92418—44no—
250#24002—611nomaximal, Sylow
251C22×C61—22no—
252C22×C61—22no—
253C22×C61—22no—
254C22×C61—22no—
255C22×C61—22no—
256C22×C61—22no—
257C22×C61—22no—
258C22×C61—22no—
259C22×C61—22no—
260C22×C61—22no—
261C22×C61—22no—
262C22×C61—22no—
263C22×C61—22no—
264C22×C61—22no—
265C22×C61—22no—
266C22×C61—416no—
267C22×C61—41no—
268C22×C61—41no—
269C22×C61—41no—
270C22×C61—41no—
271C22×C61#44714yes—
272C22×C61#48014yescomplemented
273C22×C61C2×#12011yescharacteristic
274C22×C61C2×#12311yescharacteristic
275C22×C61#53411yescharacteristic
276C22×C61C22×D811yescharacteristic
277C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
278C22×C61C2×#13711yescharacteristic
279C22×C61#57511yescharacteristic
280C2×D61—44no—
281C2×D61—44no—
282C2×D61—44no—
283C2×D61—816no—
284C2×D61—82no—
285C2×D61—82no—
286C2×D61—82no—
287C2×D61—82no—
288C4×C61—22no—
289C4×C61—22no—
290C4×C61—41no—
291C4×C61—82no—
292C4×C61—82no—
293C4×C61#54614yes—
294Dic61—44no—
295C22×#121—611no—
296C2×#480—612no—
297C2×#380D6111yescharacteristic, complemented
298#2033—614no—
299#1874—612no—
300#1852—612no—
301#1786—612no—
302#1779—611no—
303C2×C61—24no—
304C2×C61—24no—
305C2×C61—42no—
306C2×C61#240511yescharacteristic
307C2×C61C2×#44711yescharacteristic
308C2×C61#300711yescharacteristic
309C2×C61C2×#48011yescharacteristic
310C2×C61#309411yescharacteristic
311C2×C61#318611yescharacteristic
312C2×C61#319311yescharacteristic
313D61—84no—
314C23×D4—611no—
315C23×D4—1221no—
316C24×C4C2×D6111yescharacteristic
317C2×#121—612no—
318C2×#121—612no—
319C2×#121—1228no—
320C2×#121—1221no—
321C2×#121—1221no—
322C22×C4⋊C4—611no—
323C22×C4⋊C4C2×D6111yescharacteristic, complemented
324C22×C4⋊C4C2×D6111yescharacteristic, complemented
325C22×C22⋊C4—21no—
326C22×C22⋊C4—1221no—
327#480—1228no—
328C2×#107—614no—
329C2×#102—614no—
330C2×#102—614no—
331C2×C22⋊C8—612no—
332C2×C22⋊C8—612no—
333C2×C22⋊C8—612no—
334#380—21no—
335#380—21no—
336#380C2×D6112yes—
337#380C2×D6112yescomplemented
338C2×#95—21no—
339#340—614no—
340C61#2400211yescharacteristic, complemented, Sylow
341C25—1221no—
342C25C22⋊D6111yescharacteristic
343C22×D4—612no—
344C22×D4—1224no—
345C22×D4—1228no—
346C22×D4—1222no—
347C22×D4—1222no—
348C22×D4—1224no—
349C22×D4—1221no—
350C22×D4—2442no—
351C22×D4—2444no—
352C22×D4—2444no—
353C23×C4—24no—
354C23×C4—24no—
355C23×C4—21no—
356C23×C4—21no—
357C23×C4—42no—
358C23×C4—1221no—
359C23×C4C22⋊D6111yescharacteristic
360C23×C4C22⋊D6111yescharacteristic
361C23×C4C22⋊D6111yescharacteristic
362C23×C4C22×D6111yescharacteristic
363C23×C4C22×D6111yescharacteristic
364C23×C4C22×D6111yescharacteristic
365C22×C8—614no—
366#121—618no—
367#121—1224no—
368#121—1224no—
369#121—1228no—
370#121—1228no—
371#121—1224no—
372C2×C4⋊C4—24no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—42no—
376C2×C4⋊C4—612no—
377C2×C4⋊C4—1224no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4C22×D6112yes—
380C2×C4⋊C4C22×D6112yes—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—44no—
384C2×C22⋊C4—1228no—
385C2×C22⋊C4—1222no—
386C2×C22⋊C4—1222no—
387#107—1228no—
388#102—1228no—
389#102—2444no—
390C22⋊C8—1224no—
391C22⋊C8—1224no—
392C22⋊C8—2442no—
393#95—22no—
394#95—22no—
395C24—22no—
396C24—22no—
397C24—22no—
398C24—22no—
399C24—41no—
400C24—41no—
401C24—1224no—
402C24—1222no—
403C24—2442no—
404C24—2442no—
405C24—2442no—
406C24—2442no—
407C24—2442no—
408C24—2442no—
409C24#1175819914yes—
410C24C2×C22⋊D6111yescharacteristic
411C24C2×C22⋊D6111yescharacteristic
412C24C2×C22⋊D6111yescharacteristic
413C2×D4—614no—
414C2×D4—1224no—
415C2×D4—1228no—
416C2×D4—1228no—
417C2×D4—1222no—
418C2×D4—1224no—
419C2×D4—1224no—
420C2×D4—24416no—
421C2×D4—2442no—
422C2×D4—24432no—
423C2×D4—2444no—
424C2×D4—2444no—
425C2×D4—2444no—
426C2×D4—2444no—
427C2×D4—2442no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—416no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—1224no—
454C22×C4—1222no—
455C22×C4—2442no—
456C22×C4—2442no—
457C22×C4#1175822214yes—
458C22×C4D4×D6114yes—
459C22×C4C2×C22⋊D6111yescharacteristic
460C22×C4C2×C22⋊D6111yescharacteristic
461C22×C4C2×C22⋊D6111yescharacteristic
462C22×C4C23×D6111yescharacteristic
463C2×C8—1224no—
464C2×C8—1224no—
465C2×C8—1224no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—614no—
471C4⋊C4—1224no—
472C4⋊C4—1222no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—1228no—
476C22⋊C4—1228no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—28no—
492C23—28no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—1224no—
499C23—4884no—
500C23—4884no—
501C23—4884no—
502C23—4884no—
503C23—2448no—
504C23—2448no—
505C23—2448no—
506C23—2442no—
507C23—2442no—
508C23—2442no—
509C23—2442no—
510C23—2442no—
511C23—2442no—
512C23#1451974512yes—
513C23#1451976012yes—
514C23C2×#1175819912yes—
515C23C2×#1175819912yes—
516C23C2×#1175819912yes—
517C23#1451980212yes—
518C23#1451980812yes—
519C23C22×C22⋊D6111yescharacteristic
520D4—1228no—
521D4—2448no—
522D4—24416no—
523D4—24416no—
524D4—2444no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—84no—
545C2×C4—84no—
546C2×C4—1224no—
547C2×C4—2448no—
548C2×C4—2442no—
549C2×C4—2442no—
550C2×C4#1451974512yes—
551C2×C4#1451976012yes—
552C2×C4C2×D4×D6112yes—
553C2×C4C2×#1175822212yes—
554C2×C4#1451982212yes—
555C8—2444no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—416no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—48816no—
577C22—4882no—
578C22—4882no—
579C22—4882no—
580C22—4882no—
581C22—2444no—
582C22—2444no—
583C22—2444no—
584C22#1683989514yes—
585C22#1684007514yescomplemented
586C22C2×#1451974511yescharacteristic
587C22C2×#1451976011yescharacteristic
588C22#1684032111yescharacteristic
589C22C22×#1175819911yescharacteristic
590C22C2×#1451980211yescharacteristic
591C22C2×#1451980811yescharacteristic
592C22#1684056011yescharacteristic
593C4—22no—
594C4—22no—
595C4—41no—
596C4—82no—
597C4—82no—
598C4—2444no—
599C4#1684040214yes—
600C2—24no—
601C2—24no—
602C2—42no—
603C2—4884no—
604C2#2144914611yescharacteristic
605C2C2×#1683989511yescharacteristic
606C2#2145507711yescharacteristic
607C2C2×#1684007511yescharacteristic
608C2#2145575911yescharacteristic
609C2#2145656711yescharacteristic
610C2#2145662111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#380C2
socleC23×C61C22×D4
prime radicalC22×#14519751C2
hypercenterC2×#380D61
hypocenterC61#24002
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. C2×#16840075C2semidirect
3. #21447001C2semidirect
4. #21446136C2semidirect
5. #21446006C2semidirect
6. #21445749C2semidirect
7. #21445729C2semidirect
8. C2×C61×#380C2semidirect
19. C2×#14519710C22semidirect
20. C2×#14519697C22semidirect
21. C2×#14519697C22semidirect
23. C2×C22⋊#92418C22semidirect
24. C2×C22⋊#92418C22semidirect
25. #16839524C22semidirect
27. C22×C4⋊C4×C61C22semidirect
28. C22×C4⋊C4×C61C22semidirect
33. C61×#380C22semidirect
272. C22×C61#480semidirect
297. C2×#380D61semidirect
323. C22×C4⋊C4C2×D61semidirect
324. C22×C4⋊C4C2×D61semidirect
337. #380C2×D61semidirect
340. C61#24002semidirect
585. C22#16840075semidirect

Maximal subgroups

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

subgroupindex
C22×#145197512
C2×#168400752
#214470012
#214461362
#214460062
#214457492
#214457292
C2×C61×#3802
#2400261