Finite Group Databasebeta

← #29713926

Subgroups — #29713926

Order 15616.

1218 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840469C211yesmaximal, characteristic, complemented
2C2×#16840053C211yesmaximal, characteristic, complemented
3C2×C4×#11758190C211yesmaximal, characteristic, complemented
4#21450732C211yesmaximal, characteristic
5#21447016C212yesmaximal, complemented
6#21446824C212yesmaximal, complemented
7#21446780C211yesmaximal, characteristic, complemented
8#21446134C212yesmaximal, complemented
9#21446014C212yesmaximal, complemented
10#21445825C211yesmaximal, characteristic, complemented
11C61×#1954C211yesmaximal, characteristic, complemented
12C22×#11758190C2211yescharacteristic
13#16840469—24no—
14#16840469—24no—
15C2×#14519777C2211yescharacteristic
16C2×#14519768—21no—
17C2×#14519756C2212yes—
18C2×#14519754—22no—
19C2×#14519751C2212yes—
20C2×#14519748—21no—
21C2×C4×Dic122C2211yescharacteristic
22C2×C4×D244C2211yescharacteristic
23C2×D4×Dic61—21no—
24C22×C4⋊Dic61C2212yes—
25#16840053—24no—
26C4×#11758190—24no—
27C2×#14519709C2212yes—
28C2×#14519709C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519702C2212yes—
31C2×#14519699C2212yes—
32C2×#14519699C2211yescharacteristic
33#16839678—22no—
34#16839632C2211yescharacteristic
35#16839622C2212yes—
36#16839619—21no—
37#16839525C2212yes—
38#16839525C2212yes—
39#16839525C2212yes—
40C2×C4×D4×C61C2211yescharacteristic
41C2×C4⋊C8×C61C2211yescharacteristic
42C2×C22⋊C8×C61C2212yes—
43C2×C22⋊C8×C61C2212yes—
44C2×C4×C8×C61C2211yescharacteristic
45C22×Dic122—21no—
46C22×Dic122C2311yescharacteristic
47C22×D244C2311yescharacteristic
48C22×C22⋊D61—22no—
49C23×Dic61—22no—
50C22×C4×D61—21no—
51C2×#11758190—22no—
52C2×#11758190—22no—
53C2×#11758190—22no—
54#14519777—22no—
55#14519777D414yescomplemented
56#14519768—28no—
57#14519756—24no—
58#14519756—24no—
59#14519754—216no—
60#14519751—24no—
61#14519751—42no—
62#14519748—44no—
63C4×Dic122—22no—
64C4×Dic122D414yescomplemented
65C4×D244—22no—
66C4×D244—22no—
67D4×Dic61—28no—
68C2×C4⋊Dic61—22no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—24no—
71C2×C4⋊Dic61—24no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2312yes—
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2312yes—
78C2×#11758196—21no—
79C2×#11758196—22no—
80C2×C22⋊Dic61—22no—
81C2×C22⋊Dic61—22no—
82C2×#11758198—21no—
83C2×#11758198—22no—
84C2×C4×Dic61—21no—
85#14519709—24no—
86#14519709—22no—
87#14519702—22no—
88#14519702—42no—
89#14519699—24no—
90#14519699—22no—
91#14519684—22no—
92C22×D4×C61D411yescharacteristic
93C23×C4×C61C2312yes—
94C22×C8×C61C2311yescharacteristic
95C22×C8×C61C2312yes—
96C22×C8×C61C2311yescharacteristic
97C4×D4×C61—22no—
98C4×D4×C61—22no—
99C2×C4⋊C4×C61D411yescharacteristic
100C2×C22⋊C4×C61D412yes—
101C2×C42×C61C2311yescharacteristic
102C4⋊C8×C61—22no—
103C22⋊C8×C61—24no—
104C22⋊C8×C61—42no—
105C4×C8×C61—22no—
106C24×C61C2×D412yes—
107C23×D61—21no—
108C2×D4×C61—22no—
109C2×D4×C61—22no—
110C2×D4×C61SD1614yescomplemented
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×C61C2×D412yes—
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2411yescharacteristic
121C2×C22⋊D61—44no—
122C2×C22⋊D61—44no—
123C2×C22⋊D61—82no—
124C22×Dic61—21no—
125C22×Dic61—22no—
126C22×Dic61—21no—
127C22×Dic61—28no—
128C22×Dic61—21no—
129C22×Dic61—22no—
130C22×Dic61—82no—
131C22×Dic61—82no—
132C2×D244—21no—
133C2×D244—21no—
134C2×D244—21no—
135C2×D244—21no—
136C2×D244—21no—
137C2×D244#4112yes—
138C2×C4×D61—42no—
139C2×C4×D61—42no—
140C2×C4×D61—42no—
141C2×Dic122—21no—
142C2×Dic122—21no—
143C2×Dic122—21no—
144C2×Dic122—21no—
145C2×Dic122—21no—
146C2×Dic122—24no—
147C2×Dic122—81no—
148C2×Dic122—81no—
149C2×Dic122C2×D412yes—
150C2×C8×C61—21no—
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61—22no—
154C2×C8×C61—22no—
155C2×C8×C61—21no—
156C2×C8×C61—22no—
157C2×C8×C61—21no—
158C2×C8×C61#4112yes—
159C4⋊C4×C61—22no—
160C22⋊C4×C61—24no—
161C42×C61C2×D412yes—
162C42×C61C2×D412yes—
163C22⋊Dic61—28no—
164C22⋊Dic61—44no—
165#11758198—44no—
166#11758198—42no—
167C4⋊Dic61—28no—
168C4⋊Dic61—24no—
169C4⋊Dic61—21no—
170C4⋊Dic61—22no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—22no—
174C4⋊Dic61—22no—
175C4⋊Dic61—42no—
176C4⋊Dic61—81no—
177C4⋊Dic61C2×D412yes—
178C4⋊Dic61C2×D414yes—
179#11758196—28no—
180#11758196—24no—
181C4×Dic61—24no—
182#11758190—28no—
183C23×C61—22no—
184C23×C61—22no—
185C23×C61—22no—
186C23×C61—22no—
187C23×C61—22no—
188C23×C61—22no—
189C23×C61C2×SD1614yes—
190C23×C61C22×D411yescharacteristic
191C22×D61—22no—
192C22×D61—41no—
193C22×D61—41no—
194C22×D61—41no—
195C22×D61—41no—
196C22×D61—41no—
197C22×D61—41no—
198D4×C61—28no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61#12012yes—
209C2×C4×C61#12012yes—
210C2×C4×C61C2×SD1612yes—
211C2×C4×C61#13712yes—
212C2×C4×C61C22×D411yescharacteristic
213C2×C4×C61C22×D411yescharacteristic
214C2×C4×C61C2×#4111yescharacteristic
215C2×C4×C61C2×#4111yescharacteristic
216C2×C4×C61H(5,𝔽2)11yescharacteristic
217C2×C4×C612−1+411yescharacteristic
218C22⋊D61—88no—
219C2×Dic61—22no—
220C2×Dic61—22no—
221C2×Dic61—24no—
222C2×Dic61—41no—
223C2×Dic61—41no—
224C2×Dic61—42no—
225C2×Dic61—41no—
226C2×Dic61—42no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61—42no—
230C2×Dic61—42no—
231C2×Dic61—41no—
232C2×Dic61—41no—
233C2×Dic61—42no—
234C2×Dic61—88no—
235D244—22no—
236D244—22no—
237D244—41no—
238D244—41no—
239C4×D61—84no—
240Dic122—22no—
241Dic122—22no—
242Dic122—41no—
243Dic122—41no—
244Dic122—84no—
245C8×C61—22no—
246C8×C61—41no—
247C8×C61—42no—
248#24118—611nomaximal, Sylow
249C22×C61—24no—
250C22×C61—24no—
251C22×C61—24no—
252C22×C61—42no—
253C22×C61—42no—
254C22×C61—42no—
255C22×C61—42no—
256C22×C61C2×#12011yescharacteristic
257C22×C61#54611yescharacteristic
258C22×C61#54711yescharacteristic
259C22×C61C22×SD1611yescharacteristic
260C22×C61C2×#13711yescharacteristic
261C22×C61#57611yescharacteristic
262C22×C61#57711yescharacteristic
263C2×D61—42no—
264C2×D61—42no—
265C2×D61—42no—
266C2×D61—81no—
267C2×D61—81no—
268C2×D61—81no—
269C2×D61—81no—
270C4×C61—22no—
271C4×C61—42no—
272C4×C61#44814yes—
273C4×C61#52511yescharacteristic
274C4×C61#53711yescharacteristic
275C4×C61#54811yescharacteristic
276C4×C61#55011yescharacteristic
277Dic61—42no—
278Dic61—44no—
279Dic61—42no—
280Dic61—81no—
281Dic61—82no—
282C2×#554—611no—
283C2×#476—611no—
284C2×C4×SD16—611no—
285#2537—611no—
286#2035—612no—
287#2000—612no—
288#1990—611no—
289#1954D6111yescharacteristic, complemented
290#1874—612no—
291#1853—612no—
292#1803—611no—
293C2×C61—44no—
294C2×C61#240711yescharacteristic
295C2×C61C2×#44811yescharacteristic
296C2×C61#301211yescharacteristic
297C2×C61#329511yescharacteristic
298C2×C61#333111yescharacteristic
299C2×C61#333511yescharacteristic
300C2×C61#335711yescharacteristic
301D61—82no—
302C22×SD16—611no—
303#554—1224no—
304#554—1224no—
305C2×C4⋊Q8—611no—
306C2×#125—1221no—
307C2×C22⋊Q8—612no—
308C2×C22⋊Q8—1222no—
309C2×#121—612no—
310C2×#121—1221no—
311C2×C4×Q8—611no—
312C2×C4×D4—611no—
313C2×C4×D4—1221no—
314C2×C4×D4C2×D6111yescharacteristic
315C22×C4⋊C4—612no—
316#476—1224no—
317C4×SD16—1224no—
318C2×#106—611no—
319C2×#106—612no—
320C2×C4⋊C8C2×D6111yescharacteristic
321C2×#103—611no—
322C2×#103—612no—
323C2×#102—611no—
324C2×#102—612no—
325C2×C22⋊C8C2×D6112yes—
326C2×C22⋊C8C2×D6112yes—
327C2×C4×C8C2×D6111yescharacteristic
328#394—1222no—
329#384—611no—
330#380—612no—
331#380—1221no—
332#340—612no—
333#340—612no—
334#340—612no—
335C61#2411811yescharacteristic, complemented, Sylow
336C22×Q8—611no—
337C22×Q8—1221no—
338C22×D4—611no—
339C22×D4—1222no—
340C22×D4D24411yescharacteristic
341C23×C4—1222no—
342C23×C4—1221no—
343C23×C4C22×D6112yes—
344C2×SD16—1222no—
345C2×SD16—1222no—
346C2×SD16—1222no—
347C22×C8C22×D6111yescharacteristic
348C22×C8C22×D6112yes—
349C22×C8C22×D6111yescharacteristic
350C4⋊Q8—614no—
351C4⋊Q8—1222no—
352#125—1228no—
353C22⋊Q8—12216no—
354C22⋊Q8—1224no—
355C22⋊Q8—1224no—
356#121—1224no—
357#121—2442no—
358#121—2444no—
359C4×Q8—614no—
360C4×Q8—1222no—
361C4×D4—22no—
362C4×D4—22no—
363C4×D4—1228no—
364C4×D4—1222no—
365C4×D4—1222no—
366C2×C4⋊C4—611no—
367C2×C4⋊C4—612no—
368C2×C4⋊C4—611no—
369C2×C4⋊C4—612no—
370C2×C4⋊C4—1224no—
371C2×C4⋊C4—1224no—
372C2×C4⋊C4—1222no—
373C2×C4⋊C4—1221no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4—1222no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4—1221no—
378C2×C4⋊C4D24411yescharacteristic
379C2×C22⋊C4—1222no—
380C2×C22⋊C4—1222no—
381C2×C22⋊C4—1221no—
382C2×C22⋊C4—1222no—
383C2×C22⋊C4D24412yes—
384C2×C42—1221no—
385C2×C42C22×D6111yescharacteristic
386#106—1222no—
387#106—1224no—
388C4⋊C8—22no—
389#103—1222no—
390#103—2442no—
391#102—1222no—
392#102—1224no—
393C22⋊C8—24no—
394C22⋊C8—42no—
395C4×C8—22no—
396#95—1222no—
397C24—1221no—
398C24C2×D24412yes—
399C2×Q8—612no—
400C2×Q8—1221no—
401C2×Q8—1224no—
402C2×Q8—1221no—
403C2×Q8—1221no—
404C2×Q8—1221no—
405C2×Q8—1221no—
406C2×Q8—4881no—
407C2×Q8—4881no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—612no—
411C2×D4—1221no—
412C2×D4—1221no—
413C2×D4—1221no—
414C2×D4—1221no—
415C2×D4—1221no—
416C2×D4—4882no—
417C2×D4—2444no—
418C2×D4—2444no—
419C2×D4#1175819014yescomplemented
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—1228no—
427C22×C4—1221no—
428C22×C4—1221no—
429C22×C4—1222no—
430C22×C4—1221no—
431C22×C4—1222no—
432C22×C4—4882no—
433C22×C4—4882no—
434C22×C4—2442no—
435C22×C4—2442no—
436C22×C4—2442no—
437C22×C4C2×D24412yes—
438C22×C4C2×D24411yescharacteristic
439C22×C4C2×D24411yescharacteristic
440C22×C4C23×D6111yescharacteristic
441SD16—1228no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—21no—
448C2×C8—22no—
449C2×C8—21no—
450C2×C8#1175822412yes—
451C4⋊C4—22no—
452C4⋊C4—612no—
453C4⋊C4—614no—
454C4⋊C4—1228no—
455C4⋊C4—1228no—
456C4⋊C4—1221no—
457C4⋊C4—1222no—
458C4⋊C4—1224no—
459C4⋊C4—1224no—
460C4⋊C4—1221no—
461C4⋊C4—1221no—
462C4⋊C4—1222no—
463C4⋊C4—1222no—
464C4⋊C4—4881no—
465C4⋊C4—2442no—
466C22⋊C4—24no—
467C22⋊C4—1228no—
468C22⋊C4—2444no—
469C22⋊C4—2444no—
470C22⋊C4—2442no—
471C42—1224no—
472C42D4×D6112yes—
473C42D4×D6112yes—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—1222no—
481C23—2441no—
482C23—2441no—
483C23—2441no—
484C23—2441no—
485C23—2441no—
486C23—2441no—
487C23C2×#1175819014yes—
488C23C22×D24411yescharacteristic
489Q8—1222no—
490Q8—1222no—
491Q8—4884no—
492Q8—2441no—
493Q8—2441no—
494D4—28no—
495D4—1222no—
496D4—1222no—
497D4—4888no—
498D4—2441no—
499D4—2441no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—1222no—
510C2×C4—1222no—
511C2×C4—1224no—
512C2×C4—4884no—
513C2×C4—4888no—
514C2×C4—2441no—
515C2×C4—2441no—
516C2×C4—2442no—
517C2×C4—2441no—
518C2×C4—2442no—
519C2×C4—2442no—
520C2×C4—2441no—
521C2×C4—2442no—
522C2×C4—2441no—
523C2×C4—2442no—
524C2×C4—2441no—
525C2×C4—2441no—
526C2×C4#1451974712yes—
527C2×C4#1451974712yes—
528C2×C4C2×#1175819012yes—
529C2×C4#1451981012yes—
530C2×C4C2×D4×D6111yescharacteristic
531C2×C4C2×D4×D6111yescharacteristic
532C2×C4#41×D6111yescharacteristic
533C2×C4C2×#1175822411yescharacteristic
534C2×C4#1451982211yescharacteristic
535C2×C4#1451982411yescharacteristic
536C8—22no—
537C8—41no—
538C8—42no—
539C22—24no—
540C22—24no—
541C22—24no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—4881no—
547C22—4881no—
548C22—4881no—
549C22—4881no—
550C22—2442no—
551C22—2442no—
552C22—2442no—
553C22C2×#1451974711yescharacteristic
554C22#1684040711yescharacteristic
555C22#1684041611yescharacteristic
556C22C22×#1175819011yescharacteristic
557C22C2×#1451981011yescharacteristic
558C22#1684056711yescharacteristic
559C22#1684057411yescharacteristic
560C4—22no—
561C4—42no—
562C4—4881no—
563C4—4882no—
564C4—2442no—
565C4—2444no—
566C4—2442no—
567C4#1683990314yes—
568C4#1684027211yescharacteristic
569C4#1684034111yescharacteristic
570C4#1684042011yescharacteristic
571C4#1684043211yescharacteristic
572C2—44no—
573C2—4882no—
574C2#2144916211yescharacteristic
575C2C2×#1683990311yescharacteristic
576C2#2145512611yescharacteristic
577C2#2145764411yescharacteristic
578C2#2145813611yescharacteristic
579C2#2145817711yescharacteristic
580C2#2145844611yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1954C2
socleC23×C61C22×D4
prime radical#21450732C2
hypercenter#1954D61
hypocenterC61#24118
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #1954

Complements

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

NH ≅ G/Ntype
1. C2×#16840469C2semidirect
2. C2×#16840053C2semidirect
3. C2×C4×#11758190C2semidirect
5. #21447016C2semidirect
6. #21446824C2semidirect
7. #21446780C2semidirect
8. #21446134C2semidirect
9. #21446014C2semidirect
10. #21445825C2semidirect
11. C61×#1954C2semidirect
55. #14519777D4semidirect
64. C4×Dic122D4semidirect
110. C2×D4×C61SD16semidirect
289. #1954D61semidirect
335. C61#24118semidirect
419. C2×D4#11758190semidirect

Maximal subgroups

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

subgroupindex
C2×#168404692
C2×#168400532
C2×C4×#117581902
#214507322
#214470162
#214468242
#214467802
#214461342
#214460142
#214458252
C61×#19542
#2411861