Finite Group Databasebeta

← #29686258

Subgroups — #29686258

Order 15616.

1260 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839958C211yesmaximal, characteristic
2C2×#16839873C212yesmaximal, complemented
3#21448381C211yesmaximal, characteristic, complemented
4C4×#14519739C211yesmaximal, characteristic, complemented
5#21446746C211yesmaximal, characteristic, complemented
6#21446445C212yesmaximal, complemented
7#21446168C212yesmaximal, complemented
8#21445816C211yesmaximal, characteristic, complemented
9#21445674C211yesmaximal, characteristic, complemented
10#21445643C212yesmaximal, complemented
11C61×#1948C211yesmaximal, characteristic, complemented
12C22×D488C2212yes—
13C2×#14519774C2211yescharacteristic
14C2×C4×D244C2211yescharacteristic
15C2×#14519739—21no—
16C2×#14519739—21no—
17C2×#14519739C2212yes—
18C2×#14519739C2211yescharacteristic
19C2×#14519739C2211yescharacteristic
20C2×C42×D61—21no—
21#16839958C418yescomplemented
22#16839873—28no—
23C2×C4⋊#92418C2211yescharacteristic
24C2×#14519699C2212yes—
25C2×#14519697C2212yes—
26C2×#14519697C2212yes—
27C2×#14519697C2212yes—
28C2×#14519687C2212yes—
29#16839663—21no—
30#16839663—21no—
31#16839663C2212yes—
32C4⋊C4×Dic61C2211yescharacteristic
33C4⋊C4×Dic61C2211yescharacteristic
34C4×#11758198—21no—
35#16839609—21no—
36#16839609—21no—
37C42×Dic61C2211yescharacteristic
38#16839524C2212yes—
39#16839516C2212yes—
40C2×#106×C61C2212yes—
41C2×C4⋊C8×C61C2211yescharacteristic
42C61×#384C2211yescharacteristic
43C4×C4⋊C4×C61C2211yescharacteristic
44C61×#340C2212yes—
45C22×D244—21no—
46C22×D244—21no—
47C22×D244C2312yes—
48C22×D244C2311yescharacteristic
49C22×C4×D61—21no—
50C22×C4×D61—21no—
51C22×C4×D61—21no—
52C22×C4×D61—22no—
53C2×D488—24no—
54C2×D488—24no—
55C2×D488C2×C418yescomplemented
56C22×#92418C2312yes—
57#14519774—22no—
58#14519774C2×C414yescomplemented
59C4×D244—22no—
60C4×D244C2×C414yescomplemented
61#14519739—22no—
62#14519739—22no—
63#14519739—24no—
64#14519739—22no—
65#14519739—24no—
66#14519739—44no—
67#14519739—44no—
68#14519739D414yescomplemented
69C2×C4⋊Dic61C2311yescharacteristic
70C2×#11758198—21no—
71C2×#11758198—21no—
72C2×#11758198—21no—
73C2×#11758198—22no—
74C2×#11758198—22no—
75C2×#11758198—22no—
76C2×C4×Dic61—21no—
77C2×C4×Dic61—21no—
78C2×C4×Dic61D411yescharacteristic
79C2×C4×Dic61D411yescharacteristic
80C2×C4×Dic61D411yescharacteristic
81C2×C4×Dic61D411yescharacteristic
82C2×C4×Dic61C2311yescharacteristic
83C2×C4×Dic61C2311yescharacteristic
84C42×D61—44no—
85C4⋊#92418—22no—
86#14519699C2×C418yescomplemented
87#14519697—24no—
88#14519697—24no—
89#14519697—24no—
90#14519687—24no—
91#14519684—21no—
92#14519684—21no—
93C22×C8×C61C2312yes—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61C2312yes—
97C2×C4⋊C4×C61C2311yescharacteristic
98C2×C4⋊C4×C61C2311yescharacteristic
99C2×C42×C61—21no—
100C2×C42×C61C2311yescharacteristic
101#106×C61—24no—
102C4⋊C8×C61C2×C414yescomplemented
103#95×C61—21no—
104C23×D61—21no—
105C23×D61—22no—
106C22×C4×C61—21no—
107C22×C4×C61—21no—
108C22×C4×C61—22no—
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2411yescharacteristic
112C22×Dic61—21no—
113C22×Dic61C2×D411yescharacteristic
114C22×Dic61C2×D411yescharacteristic
115C22×Dic61C2×D411yescharacteristic
116C22×Dic61C2×D411yescharacteristic
117C2×D244—21no—
118C2×D244—21no—
119C2×D244—21no—
120C2×D244—22no—
121C2×D244—22no—
122C2×D244—22no—
123C2×D244—22no—
124C2×D244—24no—
125C2×D244—24no—
126C2×D244—22no—
127C2×D244—42no—
128C2×D244—42no—
129C2×D244—42no—
130C2×D244—42no—
131C2×D244C22×C414yes—
132C2×D244C22×C412yes—
133C2×D244C2×D412yes—
134C2×D244#4112yes—
135C2×C4×D61—24no—
136C2×C4×D61—24no—
137C2×C4×D61—24no—
138C2×C4×D61—42no—
139C2×C4×D61—42no—
140C2×C4×D61—42no—
141C2×C4×D61—42no—
142C2×C4×D61—42no—
143C2×C4×D61—42no—
144C2×C4×D61—44no—
145C2×C4×D61—44no—
146C2×C4×D61—44no—
147C2×C8×C61—22no—
148C2×C8×C61—22no—
149C2×C8×C61C22×C414yes—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—42no—
155C4⋊C4×C61—42no—
156C4⋊C4×C61C2×D412yes—
157C4⋊C4×C61#4112yes—
158C42×C61—42no—
159C42×C61C22×C412yes—
160C42×C61#4112yes—
161#11758198—24no—
162#11758198—24no—
163#11758198—24no—
164#11758198—44no—
165#11758198—44no—
166#11758198—44no—
167C4⋊Dic61—21no—
168C4⋊Dic61C22×C412yes—
169C4×Dic61—22no—
170C4×Dic61—22no—
171C4×Dic61—21no—
172C4×Dic61—21no—
173C4×Dic61—22no—
174C4×Dic61—41no—
175C4×Dic61—42no—
176C4×Dic61—42no—
177C4×Dic61C2×D412yes—
178C4×Dic61#4112yes—
179C2×#92418—22no—
180C2×#92418—22no—
181C2×#92418—22no—
182D488—216no—
183C23×C61C22×D411yescharacteristic
184C22×D61—22no—
185C22×D61—22no—
186C22×D61—22no—
187C22×D61—24no—
188C22×D61—41no—
189C22×D61—41no—
190C22×D61—41no—
191C22×D61—41no—
192C22×D61—42no—
193C22×D61—42no—
194C22×D61—42no—
195C22×D61—42no—
196C22×D61—42no—
197C22×D61—42no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61C4×D412yes—
210C2×C4×C61C4×D412yes—
211C2×C4×C61AGL(1,ℤ/8)12yes—
212C2×C4×C61AGL(1,ℤ/8)12yes—
213C2×C4×C61C23×C411yescharacteristic
214C2×C4×C61C22×D411yescharacteristic
215C2×C4×C61C2×#4111yescharacteristic
216C2×C4×C61C2×#4111yescharacteristic
217C2×C4×C61C2×#4111yescharacteristic
218C2×C4×C61C2×#4111yescharacteristic
219C2×Dic61—21no—
220C2×Dic61—21no—
221C2×Dic61—21no—
222C2×Dic61—21no—
223C2×Dic61—21no—
224C2×Dic61—21no—
225C2×Dic61—22no—
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—21no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61#12112yes—
232C2×Dic61#12112yes—
233C2×Dic61AGL(1,ℤ/8)12yes—
234D244—24no—
235D244—22no—
236D244—24no—
237D244—22no—
238D244—22no—
239D244—48no—
240D244—42no—
241D244—42no—
242D244—48no—
243D244C4×D414yescomplemented
244C4×D61—48no—
245C4×D61—48no—
246C4×D61—48no—
247C4×D61—88no—
248C8×C61—24no—
249#92418—42no—
250#21503—611nomaximal, Sylow
251C22×C61C2×C4×D411yescharacteristic
252C22×C61C2×#12111yescharacteristic
253C22×C61#52511yescharacteristic
254C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
255C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
256C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
257C22×C61#57511yescharacteristic
258C2×D61—24no—
259C2×D61—42no—
260C2×D61—42no—
261C2×D61—42no—
262C2×D61—42no—
263C2×D61—42no—
264C2×D61—42no—
265C2×D61—44no—
266C2×D61—44no—
267C2×D61—44no—
268C2×D61—82no—
269C2×D61—82no—
270C2×D61—82no—
271C2×D61—82no—
272C4×C61—22no—
273C4×C61—42no—
274C4×C61—42no—
275C4×C61—82no—
276C4×C61#44214yescomplemented
277C4×C61C2×C4×D411yescharacteristic
278C4×C61C4×#4111yescharacteristic
279C4×C61#52911yescharacteristic
280C4×C61#53211yescharacteristic
281Dic61—22no—
282Dic61—22no—
283Dic61—22no—
284Dic61—41no—
285Dic61—42no—
286C2×#459—611no—
287C2×#442—612no—
288C4×#127—611no—
289C42×D4—611no—
290#1982—611no—
291#1948D6111yescharacteristic, complemented
292#1926—612no—
293#1879—612no—
294#1802—611no—
295#1767—611no—
296#1762—612no—
297C2×C61C4×#12111yescharacteristic
298C2×C61C2×#44211yescharacteristic
299C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
300C2×C61#303611yescharacteristic
301C2×C61#303811yescharacteristic
302C2×C61#310911yescharacteristic
303C2×C61#311511yescharacteristic
304D61—44no—
305D61—84no—
306C22×D8—612no—
307C2×#127—611no—
308C2×C4×D4—611no—
309C2×C4×D4—612no—
310C2×C4×D4—611no—
311C2×C4×D4—611no—
312C2×C4×D4—1221no—
313C2×C4×D4—1221no—
314C22×C42—1221no—
315#459—618no—
316#442—1228no—
317C2×#106C2×D6112yes—
318C2×C4⋊C8—611no—
319C2×C4⋊C8C2×D6111yescharacteristic
320C2×#102—612no—
321C2×#102—612no—
322C2×#102—612no—
323C2×#102—612no—
324C2×#97—612no—
325#390—612no—
326#390—1221no—
327#390—1221no—
328#384C2×D6111yescharacteristic
329C4×C4⋊C4—611no—
330C4×C4⋊C4—611no—
331C4×C4⋊C4C2×D6111yescharacteristic
332C4×C22⋊C4—1221no—
333C4×C22⋊C4—1221no—
334C4×C22⋊C4—1221no—
335C43—611no—
336#340—612no—
337#340C2×D6112yes—
338#336—612no—
339C61#2150311yescharacteristic, complemented, Sylow
340C22×D4—611no—
341C22×D4—612no—
342C22×D4—1221no—
343C22×D4—1221no—
344C23×C4—1221no—
345C23×C4—1221no—
346C23×C4—1222no—
347C23×C4—1221no—
348C2×D8—618no—
349C2×D8—1224no—
350C2×D8—1224no—
351C22×C8—612no—
352C22×C8C22×D6112yes—
353#127—614no—
354#127—1222no—
355C4×D4—614no—
356C4×D4—614no—
357C4×D4—1222no—
358C4×D4—1224no—
359C4×D4—1224no—
360C4×D4—1222no—
361C4×D4—1222no—
362C4×D4—1222no—
363C4×D4—2444no—
364C4×D4—2444no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—611no—
368C2×C4⋊C4C22×D6112yes—
369C2×C4⋊C4C22×D6111yescharacteristic
370C2×C4⋊C4C22×D6111yescharacteristic
371C2×C22⋊C4—1221no—
372C2×C22⋊C4—1221no—
373C2×C22⋊C4—1221no—
374C2×C22⋊C4—1222no—
375C2×C22⋊C4—1222no—
376C2×C22⋊C4—1222no—
377C2×C42—21no—
378C2×C42—611no—
379C2×C42—611no—
380C2×C42—611no—
381C2×C42—611no—
382C2×C42—611no—
383C2×C42—611no—
384C2×C42—1221no—
385C2×C42—1221no—
386C2×C42—2444no—
387C2×C42C22×D6111yescharacteristic
388#106—24no—
389C4⋊C8—1222no—
390C4⋊C8C4×D6114yescomplemented
391#102—618no—
392#102—1224no—
393#102—1224no—
394#102—1224no—
395#97—1224no—
396#95—21no—
397#95—1221no—
398#95—1221no—
399C24—1221no—
400C24—1222no—
401C2×D4—612no—
402C2×D4—614no—
403C2×D4—612no—
404C2×D4—612no—
405C2×D4—1222no—
406C2×D4—1221no—
407C2×D4—1222no—
408C2×D4—1222no—
409C2×D4—1221no—
410C2×D4—1221no—
411C2×D4—1222no—
412C2×D4—1222no—
413C2×D4—1224no—
414C2×D4—1224no—
415C2×D4—2442no—
416C2×D4—2442no—
417C2×D4—2442no—
418C2×D4—2442no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—611no—
423C22×C4—611no—
424C22×C4—611no—
425C22×C4—611no—
426C22×C4—1224no—
427C22×C4—1224no—
428C22×C4—1221no—
429C22×C4—1224no—
430C22×C4—2444no—
431C22×C4—2442no—
432C22×C4—2444no—
433C22×C4—2444no—
434C22×C4—2442no—
435C22×C4—2442no—
436C22×C4—2442no—
437C22×C4—2442no—
438C22×C4—2442no—
439C22×C4D4×D6111yescharacteristic
440C22×C4D4×D6111yescharacteristic
441C22×C4C23×D6111yescharacteristic
442D8—12216no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—1222no—
446C2×C8—1222no—
447C2×C8—1222no—
448C2×C8C2×C4×D6114yes—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—612no—
456C4⋊C4—1221no—
457C4⋊C4D4×D6112yes—
458C4⋊C4#1175822612yes—
459C22⋊C4—1224no—
460C22⋊C4—1224no—
461C22⋊C4—1224no—
462C22⋊C4—2444no—
463C22⋊C4—2444no—
464C22⋊C4—2444no—
465C42—42no—
466C42—612no—
467C42—612no—
468C42—1222no—
469C42—1221no—
470C42—1221no—
471C42—1222no—
472C42—1222no—
473C42—2441no—
474C42—2442no—
475C42—2442no—
476C42C2×C4×D6112yes—
477C42#1175822612yes—
478C23—1222no—
479C23—1222no—
480C23—1222no—
481C23—1224no—
482C23—2442no—
483C23—2442no—
484C23—2442no—
485C23—2442no—
486C23—2442no—
487C23—2442no—
488C23—2441no—
489C23—2441no—
490C23—2441no—
491C23—2441no—
492C23C2×D4×D6111yescharacteristic
493D4—614no—
494D4—1224no—
495D4—1222no—
496D4—1224no—
497D4—1222no—
498D4—1222no—
499D4—2448no—
500D4—2442no—
501D4—2442no—
502D4—2448no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—612no—
515C2×C4—612no—
516C2×C4—612no—
517C2×C4—1221no—
518C2×C4—1221no—
519C2×C4—1221no—
520C2×C4—1221no—
521C2×C4—1221no—
522C2×C4—1221no—
523C2×C4—1222no—
524C2×C4—1221no—
525C2×C4—1221no—
526C2×C4—1221no—
527C2×C4—4888no—
528C2×C4—2448no—
529C2×C4—2448no—
530C2×C4—2441no—
531C2×C4—2441no—
532C2×C4—2448no—
533C2×C4#1451973912yes—
534C2×C4#1451973912yes—
535C2×C4#1451980412yes—
536C2×C4#1451980412yes—
537C2×C4C22×C4×D6111yescharacteristic
538C2×C4C2×D4×D6111yescharacteristic
539C2×C4#41×D6111yescharacteristic
540C2×C4#41×D6111yescharacteristic
541C2×C4C2×#1175822611yescharacteristic
542C2×C4C2×#1175822611yescharacteristic
543C8—24no—
544C8—2442no—
545C22—1224no—
546C22—4882no—
547C22—4882no—
548C22—4882no—
549C22—4882no—
550C22—2444no—
551C22—2444no—
552C22—2444no—
553C22—2442no—
554C22—2442no—
555C22—2442no—
556C22—2442no—
557C22—2442no—
558C22—2442no—
559C22C2×#1451973911yescharacteristic
560C22#121×D6111yescharacteristic
561C22#1684027511yescharacteristic
562C22AGL(1,ℤ/8)×D6111yescharacteristic
563C22C2×#1451980411yescharacteristic
564C22C2×#1451980411yescharacteristic
565C22#1684056111yescharacteristic
566C4—22no—
567C4—42no—
568C4—42no—
569C4—82no—
570C4—1222no—
571C4—1222no—
572C4—1222no—
573C4—2442no—
574C4—2441no—
575C4#1683987314yescomplemented
576C4C4×D4×D6111yescharacteristic
577C4C4×#1175822611yescharacteristic
578C4#1684029711yescharacteristic
579C4#1684031311yescharacteristic
580C2—4884no—
581C2—2444no—
582C2#2144832711yescharacteristic
583C2C2×#1683987311yescharacteristic
584C2#2145462611yescharacteristic
585C2#2145528411yescharacteristic
586C2#2145529611yescharacteristic
587C2#2145591211yescharacteristic
588C2#2145597311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1948C2
socleC23×C61C22×D4
prime radicalC2×#16839958C2
cosocleC22×C4×C61C24
hypercenter#1948D61
hypocenterC61#21503
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1948

Complements

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

NH ≅ G/Ntype
2. C2×#16839873C2semidirect
3. #21448381C2semidirect
4. C4×#14519739C2semidirect
5. #21446746C2semidirect
6. #21446445C2semidirect
7. #21446168C2semidirect
8. #21445816C2semidirect
9. #21445674C2semidirect
10. #21445643C2semidirect
11. C61×#1948C2semidirect
21. #16839958C4semidirect
55. C2×D488C2×C4semidirect
58. #14519774C2×C4semidirect
60. C4×D244C2×C4semidirect
68. #14519739D4semidirect
86. #14519699C2×C4semidirect
102. C4⋊C8×C61C2×C4semidirect
243. D244C4×D4semidirect
276. C4×C61#442semidirect
291. #1948D61semidirect
339. C61#21503semidirect
390. C4⋊C8C4×D61semidirect
575. C4#16839873semidirect

Maximal subgroups

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

subgroupindex
C2×#168399582
C2×#168398732
#214483812
C4×#145197392
#214467462
#214464452
#214461682
#214458162
#214456742
#214456432
C61×#19482
#2150361