Finite Group Databasebeta

← #29695472

Subgroups — #29695472

Order 15616.

1490 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840417C211yesmaximal, characteristic
2C2×#16839759C212yesmaximal, complemented
3#21448768C211yesmaximal, characteristic, complemented
4#21446741C211yesmaximal, characteristic, complemented
5#21446621C211yesmaximal, characteristic, complemented
6#21446242C212yesmaximal, complemented
7#21446129C212yesmaximal, complemented
8#21445820C211yesmaximal, characteristic, complemented
9#21445760C212yesmaximal, complemented
10C4×#14519699C211yesmaximal, characteristic, complemented
11C2×C8×D4×C61C211yesmaximal, characteristic, complemented
12C22×#11758222C2212yes—
13#16840417—24no—
14#16840417C418yescomplemented
15C2×#14519777C2211yescharacteristic
16C2×#14519759—22no—
17C2×#14519756C2212yes—
18C2×#14519748—21no—
19C2×C4×D244C2211yescharacteristic
20C2×#14519736C2212yes—
21C2×C4⋊C4×D61—21no—
22#16839759—28no—
23C2×#14519702C2212yes—
24C2×#14519702C2212yes—
25C2×#14519702C2212yes—
26C2×#14519702C2212yes—
27C2×#14519699C2211yescharacteristic
28C2×#14519699C2212yes—
29C2×#14519699C2211yescharacteristic
30#16839665—21no—
31#16839655—22no—
32#16839632C2211yescharacteristic
33#16839622C2212yes—
34C4⋊C4×Dic61—21no—
35C4×C4⋊Dic61C2211yescharacteristic
36#16839525C2211yescharacteristic
37#16839525C2211yescharacteristic
38#16839525C2212yes—
39C23×C8×C61C2212yes—
40C2×C4×D4×C61C2211yescharacteristic
41C8×D4×C61—24no—
42C2×C4⋊C8×C61C2211yescharacteristic
43C2×C22⋊C8×C61C2212yes—
44C2×C4×C8×C61C2211yescharacteristic
45C2×#11758222—24no—
46C2×#11758222—24no—
47C2×#11758222—24no—
48C2×#11758222—42no—
49C2×#11758222—42no—
50C2×#11758222C2×C418yes—
51C22×Dic122C2312yes—
52C22×D244C2311yescharacteristic
53C22×C22⋊D61—22no—
54C22×C4×D61—21no—
55C22×C4×D61—22no—
56#14519777—22no—
57#14519777C2×C414yes—
58#14519759—216no—
59#14519756—24no—
60#14519756C2×C418yes—
61#14519748—28no—
62C4×D244—22no—
63C4×D244C2×C414yes—
64#14519736—24no—
65#14519736—42no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61C2312yes—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758196—22no—
75C4⋊C4×D61—28no—
76C2×C22⋊Dic61—22no—
77C2×C22⋊Dic61—22no—
78C2×C22⋊Dic61—22no—
79C2×#11758198—21no—
80C2×C4×Dic61—22no—
81C2×C4×Dic61—21no—
82#14519702—24no—
83#14519702—24no—
84#14519702—42no—
85#14519702—42no—
86#14519699—24no—
87#14519699—22no—
88#14519699D414yescomplemented
89#14519684—21no—
90#14519684—22no—
91C22×D4×C61D411yescharacteristic
92C23×C4×C61C2312yes—
93C22×C8×C61—24no—
94C22×C8×C61—24no—
95C22×C8×C61—42no—
96C22×C8×C61C2312yes—
97C22×C8×C61C2311yescharacteristic
98C22×C8×C61C2311yescharacteristic
99C4×D4×C61C2×C414yes—
100C4×D4×C61D414yes—
101C2×C4⋊C4×C61D411yescharacteristic
102C2×C22⋊C4×C61D412yes—
103C2×C42×C61C2311yescharacteristic
104C4⋊C8×C61—22no—
105C22⋊C8×C61—24no—
106C4×C8×C61—22no—
107C24×C61C2×D412yes—
108C23×D61—21no—
109C2×D4×C61—22no—
110C2×D4×C61—22no—
111C2×D4×C61C22⋊C414yes—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61C22×C414yes—
117C22×C4×C61C2×D412yes—
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D414yes—
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2411yescharacteristic
122C2×C22⋊D61—28no—
123C2×C22⋊D61—44no—
124C2×C22⋊D61—44no—
125C22×Dic61—22no—
126C22×Dic61—21no—
127C22×Dic61—21no—
128C22×Dic61—21no—
129C22×Dic61—22no—
130#11758222—216no—
131#11758222—48no—
132#11758222—48no—
133#11758222—48no—
134C2×D244—21no—
135C2×D244—21no—
136C2×D244—21no—
137C2×D244C22×C412yes—
138C2×D244C2×D412yes—
139C2×D244#4112yes—
140C2×C4×D61—28no—
141C2×C4×D61—24no—
142C2×C4×D61—44no—
143C2×C4×D61—44no—
144C2×C4×D61—42no—
145C2×C4×D61—42no—
146C2×Dic122—22no—
147C2×Dic122—22no—
148C2×Dic122—22no—
149C2×Dic122—22no—
150C2×Dic122—22no—
151C2×Dic122C22×C414yes—
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C2×C8×C61—21no—
155C2×C8×C61—22no—
156C2×C8×C61—22no—
157C2×C8×C61—21no—
158C2×C8×C61—22no—
159C2×C8×C61—21no—
160C2×C8×C61—48no—
161C2×C8×C61C2×D412yes—
162C4⋊C4×C61C22⋊C414yes—
163C22⋊C4×C61C22⋊C418yes—
164C42×C61C22×C412yes—
165C42×C61C2×D412yes—
166C22⋊Dic61—28no—
167C22⋊Dic61—44no—
168C22⋊Dic61—82no—
169#11758198—24no—
170C4⋊Dic61—24no—
171C4⋊Dic61—24no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—22no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—22no—
178C4⋊Dic61—42no—
179C4⋊Dic61—81no—
180C4⋊Dic61C22×C412yes—
181C4⋊Dic61C2×D412yes—
182#11758196—28no—
183C4×Dic61—44no—
184C4×Dic61—42no—
185C23×C61—22no—
186C23×C61—22no—
187C23×C61—22no—
188C23×C61—22no—
189C23×C61—22no—
190C23×C61—22no—
191C23×C61C2×C22⋊C414yes—
192C23×C61C22×D411yescharacteristic
193C22×D61—22no—
194C22×D61—22no—
195C22×D61—22no—
196C22×D61—41no—
197C22×D61—41no—
198C22×D61—41no—
199C22×D61—41no—
200D4×C61—28no—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—24no—
206C2×C4×C61—24no—
207C2×C4×C61—24no—
208C2×C4×C61—24no—
209C2×C4×C61C2×C22⋊C414yes—
210C2×C4×C61C2×C22⋊C412yes—
211C2×C4×C61C2×C22⋊C412yes—
212C2×C4×C61#13512yes—
213C2×C4×C61#13512yes—
214C2×C4×C61C23×C411yescharacteristic
215C2×C4×C61C22×D411yescharacteristic
216C2×C4×C61C22×D411yescharacteristic
217C2×C4×C61C2×#4111yescharacteristic
218C2×C4×C61H(5,𝔽2)11yescharacteristic
219C2×C4×C61H(5,𝔽2)11yescharacteristic
220C22⋊D61—416no—
221C2×Dic61—22no—
222C2×Dic61—24no—
223C2×Dic61—22no—
224C2×Dic61—42no—
225C2×Dic61—42no—
226C2×Dic61—41no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—42no—
231C2×Dic61—41no—
232C2×Dic61—41no—
233C2×Dic61—42no—
234C2×Dic61—42no—
235C2×Dic61—41no—
236D244—22no—
237D244—22no—
238D244—22no—
239D244C4×D414yes—
240C4×D61—416no—
241C4×D61—48no—
242Dic122—24no—
243Dic122—24no—
244Dic122—42no—
245Dic122—42no—
246C8×C61—22no—
247C8×C61—42no—
248C8×C61—41no—
249#22417—611nomaximal, Sylow
250C22×C61—24no—
251C22×C61—24no—
252C22×C61—24no—
253C22×C61—42no—
254C22×C61—42no—
255C22×C61—42no—
256C22×C61—42no—
257C22×C61C22×C22⋊C411yescharacteristic
258C22×C61D4211yescharacteristic
259C22×C61#54611yescharacteristic
260C22×C61C2×#13511yescharacteristic
261C22×C61C2×#13511yescharacteristic
262C22×C61#57711yescharacteristic
263C22×C61#57811yescharacteristic
264C2×D61—24no—
265C2×D61—42no—
266C2×D61—42no—
267C2×D61—42no—
268C2×D61—42no—
269C2×D61—42no—
270C2×D61—42no—
271C4×C61—24no—
272C4×C61—22no—
273C4×C61#41614yescomplemented
274C4×C61C2×C4×D411yescharacteristic
275C4×C61#51811yescharacteristic
276C4×C61D4211yescharacteristic
277C4×C61#54611yescharacteristic
278Dic61—44no—
279Dic61—42no—
280Dic61—42no—
281Dic61—82no—
282Dic61—81no—
283C2×#547—611no—
284C2×C8×D4D6111yescharacteristic, complemented
285C2×#416—612no—
286#2354—611no—
287#1981—611no—
288#1956—611no—
289#1892—612no—
290#1873—612no—
291#1802—611no—
292#1788—612no—
293C4×#102—611no—
294C2×C61—44no—
295C2×C61D4×C22⋊C411yescharacteristic
296C2×C61C2×#41611yescharacteristic
297C2×C61#289811yescharacteristic
298C2×C61#328311yescharacteristic
299C2×C61#328411yescharacteristic
300C2×C61#330211yescharacteristic
301C2×C61#330311yescharacteristic
302D61—44no—
303C22×#41—612no—
304C23×C8C2×D6112yes—
305#547—618no—
306#547—1224no—
307C2×C4⋊Q8—611no—
308C2×#123—1222no—
309C2×C22⋊Q8—612no—
310C2×#121—1221no—
311C2×C4×D4—611no—
312C2×C4×D4C2×D6111yescharacteristic
313C2×#117—612no—
314C22×C4⋊C4—1221no—
315C8×D4—24no—
316C2×C4⋊C8C2×D6111yescharacteristic
317#416—1228no—
318C2×#103—612no—
319C2×#103—612no—
320C2×#103—612no—
321C2×#103—612no—
322C2×#102—611no—
323C2×#102—612no—
324C2×#102—611no—
325C2×C22⋊C8C2×D6112yes—
326C2×C4×C8C2×D6111yescharacteristic
327#390—1221no—
328#388—1222no—
329#384—611no—
330#380—612no—
331C4×C4⋊C4—611no—
332C4×C4⋊C4—1221no—
333#340—612no—
334#340—611no—
335#340—611no—
336C61#2241711yescharacteristic, complemented, Sylow
337C2×#41—618no—
338C2×#41—1224no—
339C2×#41—1224no—
340C2×#41—1224no—
341C2×#41—2442no—
342C2×#41—2442no—
343C22×Q8—612no—
344C22×D4—611no—
345C22×D4—1222no—
346C22×D4D24411yescharacteristic
347C23×C4—1221no—
348C23×C4—1222no—
349C23×C4C22×D6112yes—
350C22×C8—24no—
351C22×C8—24no—
352C22×C8—42no—
353C22×C8C22×D6112yes—
354C22×C8C22×D6111yescharacteristic
355C22×C8C22×D6111yescharacteristic
356C4⋊Q8—614no—
357C4⋊Q8—1222no—
358#123—12216no—
359C22⋊Q8—618no—
360C22⋊Q8—1224no—
361#121—1228no—
362C4×D4—614no—
363C4×D4—1222no—
364C4×D4C4×D6114yes—
365C4×D4C22⋊D6114yes—
366#117—1224no—
367#117—2442no—
368C2×C4⋊C4—611no—
369C2×C4⋊C4—611no—
370C2×C4⋊C4—612no—
371C2×C4⋊C4—611no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4—1222no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4—1228no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4—1221no—
378C2×C4⋊C4D24411yescharacteristic
379C2×C22⋊C4—1222no—
380C2×C22⋊C4—1221no—
381C2×C22⋊C4—1222no—
382C2×C22⋊C4—1222no—
383C2×C22⋊C4D24412yes—
384C2×C42—1222no—
385C2×C42—1221no—
386C2×C42C22×D6111yescharacteristic
387C4⋊C8—22no—
388#103—1224no—
389#103—1224no—
390#103—2442no—
391#103—2442no—
392#102—614no—
393#102—1222no—
394#102—1224no—
395C22⋊C8—24no—
396C4×C8—22no—
397#95—1221no—
398#95—1222no—
399C24—1221no—
400C24C2×D24412yes—
401#41—12216no—
402#41—2448no—
403#41—2448no—
404#41—2448no—
405C2×Q8—614no—
406C2×Q8—1222no—
407C2×Q8—1222no—
408C2×Q8—1222no—
409C2×Q8—1222no—
410C2×Q8—1222no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—612no—
414C2×D4—612no—
415C2×D4—612no—
416C2×D4—1228no—
417C2×D4—1221no—
418C2×D4—1221no—
419C2×D4—1221no—
420C2×D4—2444no—
421C2×D4—2444no—
422C2×D4#1175819814yes—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—1221no—
428C22×C4—1221no—
429C22×C4—1222no—
430C22×C4—1224no—
431C22×C4—1228no—
432C22×C4—1221no—
433C22×C4—1222no—
434C22×C4—2444no—
435C22×C4—2444no—
436C22×C4—2442no—
437C22×C4—2442no—
438C22×C4C2×C4×D6114yes—
439C22×C4C2×D24412yes—
440C22×C4C2×D24411yescharacteristic
441C22×C4C2×D24411yescharacteristic
442C22×C4C2×C22⋊D6114yes—
443C22×C4C23×D6111yescharacteristic
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—22no—
447C2×C8—21no—
448C2×C8—22no—
449C2×C8—21no—
450C2×C8—22no—
451C2×C8—21no—
452C2×C8—48no—
453C2×C8D4×D6112yes—
454C4⋊C4—612no—
455C4⋊C4—612no—
456C4⋊C4—1221no—
457C4⋊C4—1221no—
458C4⋊C4—1228no—
459C4⋊C4—1224no—
460C4⋊C4—1224no—
461C4⋊C4—1222no—
462C4⋊C4—1221no—
463C4⋊C4—1221no—
464C4⋊C4—1222no—
465C4⋊C4—4881no—
466C4⋊C4—2442no—
467C4⋊C4#1175819814yes—
468C22⋊C4—1228no—
469C22⋊C4—1224no—
470C22⋊C4—4882no—
471C22⋊C4—2444no—
472C22⋊C4#1175819818yes—
473C42—2444no—
474C42—2442no—
475C42C2×C4×D6112yes—
476C42C2×C22⋊D6112yes—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—1222no—
484C23—1222no—
485C23—1222no—
486C23—2441no—
487C23—2441no—
488C23—2441no—
489C23—2441no—
490C23C2×#1175819814yes—
491C23C22×D24411yescharacteristic
492Q8—1224no—
493Q8—1224no—
494Q8—2442no—
495Q8—2442no—
496D4—28no—
497D4—614no—
498D4—1222no—
499D4—1222no—
500D4—1222no—
501D4—24416no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—1222no—
511C2×C4—1224no—
512C2×C4—1222no—
513C2×C4—24416no—
514C2×C4—2441no—
515C2×C4—2442no—
516C2×C4—2441no—
517C2×C4—2442no—
518C2×C4—2442no—
519C2×C4—2441no—
520C2×C4—2441no—
521C2×C4—2441no—
522C2×C4—2441no—
523C2×C4—2441no—
524C2×C4—2442no—
525C2×C4—2442no—
526C2×C4—2448no—
527C2×C4C2×#1175819812yes—
528C2×C4C2×#1175819814yes—
529C2×C4C2×#1175819812yes—
530C2×C4#1451979912yes—
531C2×C4#1451979912yes—
532C2×C4C22×C4×D6111yescharacteristic
533C2×C4C2×D4×D6111yescharacteristic
534C2×C4C22×C22⋊D6111yescharacteristic
535C2×C4#41×D6111yescharacteristic
536C2×C4#1451982211yescharacteristic
537C2×C4#1451982311yescharacteristic
538C8—22no—
539C8—42no—
540C8—41no—
541C22—24no—
542C22—24no—
543C22—24no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—1224no—
549C22—2442no—
550C22—2442no—
551C22—2442no—
552C22—2442no—
553C22—2442no—
554C22—2442no—
555C22C22×#1175819811yescharacteristic
556C22D4×D24411yescharacteristic
557C22#1684040711yescharacteristic
558C22C2×#1451979911yescharacteristic
559C22C2×#1451979911yescharacteristic
560C22#1684057411yescharacteristic
561C22#1684057911yescharacteristic
562C4—22no—
563C4—24no—
564C4—4881no—
565C4—4882no—
566C4—2442no—
567C4—2444no—
568C4—2442no—
569C4#1683975914yescomplemented
570C4C4×D4×D6111yescharacteristic
571C4#1684023411yescharacteristic
572C4D4×C22⋊D6111yescharacteristic
573C4#1684040011yescharacteristic
574C2—44no—
575C2—2444no—
576C2D4×#1175819811yescharacteristic
577C2C2×#1683975911yescharacteristic
578C2#2145435311yescharacteristic
579C2#2145754111yescharacteristic
580C2#2145755111yescharacteristic
581C2#2145773711yescharacteristic
582C2#2145774811yescharacteristic
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C8×D4×C61C2
socleC23×C61C22×D4
prime radicalC2×#16840417C2
cosocleC22×C4×C61C24
hypercenterC2×C8×D4D61
hypocenterC61#22417
LCSG ▷ C2×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
2. C2×#16839759C2semidirect
3. #21448768C2semidirect
4. #21446741C2semidirect
5. #21446621C2semidirect
6. #21446242C2semidirect
7. #21446129C2semidirect
8. #21445820C2semidirect
9. #21445760C2semidirect
10. C4×#14519699C2semidirect
11. C2×C8×D4×C61C2semidirect
14. #16840417C4semidirect
88. #14519699D4semidirect
273. C4×C61#416semidirect
284. C2×C8×D4D61semidirect
336. C61#22417semidirect
569. C4#16839759semidirect

Maximal subgroups

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

subgroupindex
C2×#168404172
C2×#168397592
#214487682
#214467412
#214466212
#214462422
#214461292
#214458202
#214457602
C4×#145196992
C2×C8×D4×C612
#2241761