Finite Group Databasebeta

← #29713952

Subgroups — #29713952

Order 15616.

1350 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840415C211yesmaximal, characteristic
2C2×C4×#11758201C211yesmaximal, characteristic, complemented
3#21450737C211yesmaximal, characteristic, complemented
4#21447010C212yesmaximal, complemented
5#21446821C212yesmaximal, complemented
6#21446781C211yesmaximal, characteristic, complemented
7#21446612C211yesmaximal, characteristic, complemented
8#21446123C212yesmaximal, complemented
9#21446010C212yesmaximal, complemented
10#21445820C211yesmaximal, characteristic, complemented
11C2×C61×#475C211yesmaximal, characteristic, complemented
12C22×#11758226—21no—
13C22×#11758201C2211yescharacteristic
14#16840415—24no—
15#16840415—24no—
16C2×#14519762—22no—
17C2×#14519753—21no—
18C2×#14519753C2212yes—
19C2×#14519752C2212yes—
20C2×C4×D244C2211yescharacteristic
21C2×C4×D244C2211yescharacteristic
22C2×C4⋊C4×D61C2212yes—
23C4×#11758201—24no—
24C2×#14519708C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519699C2212yes—
27C2×#14519697C2211yescharacteristic
28C2×#14519689C2212yes—
29C2×#14519689C2212yes—
30C2×C4×#92418C2211yescharacteristic
31#16839684—22no—
32#16839664—21no—
33#16839660—21no—
34#16839632C2211yescharacteristic
35#16839619C2212yes—
36#16839525C2212yes—
37#16839524C2212yes—
38#16839524C2212yes—
39C2×C4⋊Q8×C61C2211yescharacteristic
40C2×C4×Q8×C61C2211yescharacteristic
41C61×#475—24no—
42C2×#106×C61C2212yes—
43C2×C4⋊C8×C61C2211yescharacteristic
44C2×#103×C61C2212yes—
45C2×#11758226—28no—
46C2×#11758226—44no—
47C2×#11758226—44no—
48C22×D244—22no—
49C22×D244D411yescharacteristic
50C22×D244C2311yescharacteristic
51C22×C4×D61—22no—
52C22×C4×D61—21no—
53C22×C4×D61C2312yes—
54C2×#11758201—22no—
55C2×#11758201—22no—
56C2×#11758201—22no—
57C22×#92418C2311yescharacteristic
58C22×#92418C2311yescharacteristic
59#14519762—216no—
60#14519753—28no—
61#14519753—24no—
62#14519753—42no—
63#14519752—24no—
64#14519752—24no—
65C4×D244—22no—
66C4×D244—22no—
67C4×D244—22no—
68C4×D244D414yes—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61D411yescharacteristic
72C2×C4⋊Dic61C2312yes—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758196—22no—
75C2×#11758196—22no—
76C4⋊C4×D61—24no—
77C4⋊C4×D61—24no—
78C2×#11758198—21no—
79C2×#11758198—22no—
80C2×#11758198—22no—
81C2×#11758198—22no—
82C2×#11758198D412yes—
83#14519708—22no—
84#14519701—22no—
85#14519699—42no—
86#14519697—22no—
87#14519689—24no—
88#14519689—42no—
89C4×#92418—22no—
90#14519684—22no—
91C22×Q8×C61—21no—
92C22×Q8×C61C2311yescharacteristic
93C22×C8×C61C2312yes—
94C4⋊Q8×C61—22no—
95C4⋊Q8×C61D414yes—
96C4×Q8×C61—22no—
97C4×Q8×C61—22no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61C2311yescharacteristic
101C2×C4⋊C4×C61C2312yes—
102C2×C42×C61—21no—
103C2×C42×C61C2311yescharacteristic
104#106×C61—24no—
105C4⋊C8×C61—22no—
106#103×C61—24no—
107C23×D61—21no—
108C23×D61C2×D412yes—
109C2×Q8×C61—24no—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C2×Q8×C61—21no—
115C2×Q8×C61—42no—
116C2×Q8×C61—42no—
117C2×Q8×C61#4112yescomplemented
118C22×C4×C61—22no—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2411yescharacteristic
124C22×Dic61—21no—
125C22×Dic61—22no—
126C22×Dic61C2×D412yes—
127#11758226—432no—
128C2×D244—22no—
129C2×D244—22no—
130C2×D244—21no—
131C2×D244—21no—
132C2×D244—21no—
133C2×D244—21no—
134C2×D244—28no—
135C2×D244—21no—
136C2×D244—44no—
137C2×D244—44no—
138C2×D244SD1614yescomplemented
139C2×D244C2×D412yes—
140C2×C4×D61—28no—
141C2×C4×D61—24no—
142C2×C4×D61—22no—
143C2×C4×D61—22no—
144C2×C4×D61—22no—
145C2×C4×D61—22no—
146C2×C4×D61—22no—
147C2×C4×D61—22no—
148C2×C4×D61—44no—
149C2×C4×D61—44no—
150C2×C4×D61—42no—
151C2×C4×D61—42no—
152C2×C8×C61—22no—
153C2×C8×C61—22no—
154C2×C8×C61—22no—
155C4⋊C4×C61—24no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—42no—
160C4⋊C4×C61C2×D414yes—
161C42×C61—42no—
162C42×C61C2×D412yes—
163C42×C61C2×D412yes—
164#11758201—28no—
165#11758198—28no—
166#11758198—24no—
167#11758198—24no—
168#11758198—44no—
169#11758198—82no—
170C4⋊Dic61—22no—
171C4⋊Dic61—21no—
172C4⋊Dic61—22no—
173C4⋊Dic61—22no—
174C4⋊Dic61—42no—
175C4⋊Dic61—81no—
176C4⋊Dic61C2×D412yes—
177#11758196—28no—
178#11758196—44no—
179C2×#92418—21no—
180C2×#92418—21no—
181C2×#92418—21no—
182C2×#92418—21no—
183C2×#92418—21no—
184C2×#92418#4112yescomplemented
185C23×C61C22×D411yescharacteristic
186C22×D61—22no—
187C22×D61—22no—
188C22×D61—22no—
189C22×D61—22no—
190C22×D61—22no—
191C22×D61—22no—
192C22×D61—22no—
193C22×D61—41no—
194C22×D61—41no—
195C22×D61—41no—
196C22×D61—41no—
197C22×D61—41no—
198C22×D61—41no—
199C22×D61C2×SD1614yes—
200Q8×C61—22no—
201Q8×C61—22no—
202Q8×C61—41no—
203Q8×C61—41no—
204Q8×C61—48no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—24no—
208C2×C4×C61—22no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61#12012yes—
217C2×C4×C61#12012yes—
218C2×C4×C61C2×SD1612yes—
219C2×C4×C61AGL(1,ℤ/8)12yes—
220C2×C4×C61C22×D411yescharacteristic
221C2×C4×C61C22×D411yescharacteristic
222C2×C4×C61C2×#4111yescharacteristic
223C2×C4×C61C2×#4111yescharacteristic
224C2×C4×C61H(5,𝔽2)11yescharacteristic
225C2×C4×C612−1+411yescharacteristic
226C2×Dic61—22no—
227C2×Dic61—22no—
228C2×Dic61—22no—
229C2×Dic61—22no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61—42no—
233C2×Dic61—42no—
234C2×Dic61—42no—
235D244—22no—
236D244—22no—
237D244—28no—
238D244—416no—
239D244—41no—
240D244—41no—
241C4×D61—416no—
242C4×D61—42no—
243C4×D61—42no—
244C4×D61—42no—
245C4×D61—42no—
246C4×D61—48no—
247C8×C61—42no—
248#92418—22no—
249#92418—41no—
250#24120—611nomaximal, Sylow
251C22×C61C2×#12011yescharacteristic
252C22×C61#54611yescharacteristic
253C22×C61#54711yescharacteristic
254C22×C61C22×SD1611yescharacteristic
255C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
256C22×C61#57711yescharacteristic
257C22×C61#57811yescharacteristic
258C2×D61—24no—
259C2×D61—24no—
260C2×D61—24no—
261C2×D61—42no—
262C2×D61—42no—
263C2×D61—42no—
264C2×D61—42no—
265C2×D61—42no—
266C2×D61—42no—
267C2×D61—42no—
268C2×D61—81no—
269C2×D61—81no—
270C2×D61—81no—
271C2×D61—81no—
272C4×C61—22no—
273C4×C61—42no—
274C4×C61—44no—
275C4×C61—81no—
276C4×C61#45014yes—
277C4×C61#52511yescharacteristic
278C4×C61#53711yescharacteristic
279C4×C61#54811yescharacteristic
280C4×C61#55011yescharacteristic
281Dic61—42no—
282Dic61—42no—
283Dic61—82no—
284C2×#547—611no—
285C2×#475D6111yescharacteristic, complemented
286C2×C4×SD16—611no—
287#2538—611no—
288#2034—612no—
289#2000—612no—
290#1990—611no—
291#1954—611no—
292#1872—612no—
293#1853—612no—
294#1802—611no—
295C2×C61#240711yescharacteristic
296C2×C61C2×#45011yescharacteristic
297C2×C61#301311yescharacteristic
298C2×C61#329811yescharacteristic
299C2×C61#333211yescharacteristic
300C2×C61#333511yescharacteristic
301C2×C61#335311yescharacteristic
302D61—44no—
303D61—82no—
304C22×#41—1221no—
305C22×SD16—611no—
306#547—1224no—
307#547—1224no—
308C2×C4⋊Q8C2×D6111yescharacteristic
309C2×#123—1222no—
310C2×C22⋊Q8—612no—
311C2×C22⋊Q8—1221no—
312C2×#121—612no—
313C2×C4×Q8C2×D6111yescharacteristic
314C2×C4×D4—611no—
315C2×C4×D4—611no—
316C22×C4⋊C4—612no—
317#475—24no—
318C4×SD16—1224no—
319C2×#106—611no—
320C2×#106C2×D6112yes—
321C2×C4⋊C8C2×D6111yescharacteristic
322C2×#103—611no—
323C2×#103C2×D6112yes—
324C2×#102—611no—
325C2×#102—612no—
326C2×C22⋊C8—612no—
327C2×C22⋊C8—612no—
328C2×C4×C8—611no—
329#396—1222no—
330#390—1221no—
331#389—1221no—
332#384—611no—
333#380—612no—
334#340—612no—
335#340—612no—
336#340—612no—
337C61#2412011yescharacteristic, complemented, Sylow
338C2×#41—1228no—
339C2×#41—2444no—
340C2×#41—2444no—
341C22×Q8—21no—
342C22×Q8C22×D6111yescharacteristic
343C22×D4—611no—
344C22×D4—611no—
345C22×D4—1222no—
346C23×C4—612no—
347C23×C4—1222no—
348C23×C4—1221no—
349C2×SD16—1222no—
350C2×SD16—1222no—
351C2×SD16—1222no—
352C22×C8—611no—
353C22×C8—611no—
354C22×C8C22×D6112yes—
355C4⋊Q8—22no—
356C4⋊Q8C22⋊D6114yes—
357#123—12216no—
358C22⋊Q8—1228no—
359C22⋊Q8—1224no—
360C22⋊Q8—2442no—
361#121—1224no—
362#121—1224no—
363C4×Q8—22no—
364C4×Q8—22no—
365C4×D4—614no—
366C4×D4—1222no—
367C4×D4—1222no—
368C4×D4—1222no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—611no—
372C2×C4⋊C4—611no—
373C2×C4⋊C4—612no—
374C2×C4⋊C4—1224no—
375C2×C4⋊C4—1222no—
376C2×C4⋊C4—1224no—
377C2×C4⋊C4—1221no—
378C2×C4⋊C4—1222no—
379C2×C4⋊C4—1221no—
380C2×C4⋊C4C22×D6111yescharacteristic
381C2×C4⋊C4C22×D6112yes—
382C2×C22⋊C4—612no—
383C2×C22⋊C4—1222no—
384C2×C22⋊C4—1221no—
385C2×C22⋊C4—1222no—
386C2×C22⋊C4—1222no—
387C2×C42—21no—
388C2×C42C22×D6111yescharacteristic
389#106—24no—
390#106—1222no—
391C4⋊C8—22no—
392#103—24no—
393#103—1222no—
394#102—1222no—
395#102—2442no—
396C22⋊C8—1224no—
397C22⋊C8—2442no—
398C4×C8—1222no—
399#95—1222no—
400C24—612no—
401C24—1221no—
402#41—24432no—
403C2×Q8—21no—
404C2×Q8—24no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—42no—
410C2×Q8—42no—
411C2×Q8#1175822612yescomplemented
412C2×D4—612no—
413C2×D4—614no—
414C2×D4—1228no—
415C2×D4—1221no—
416C2×D4—1221no—
417C2×D4—1221no—
418C2×D4—1221no—
419C2×D4—1221no—
420C2×D4—1222no—
421C2×D4—1222no—
422C2×D4—2444no—
423C2×D4—2444no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—612no—
428C22×C4—1228no—
429C22×C4—1221no—
430C22×C4—1222no—
431C22×C4—1222no—
432C22×C4—1222no—
433C22×C4—1222no—
434C22×C4—1222no—
435C22×C4—1222no—
436C22×C4—1224no—
437C22×C4—1222no—
438C22×C4—2444no—
439C22×C4—2444no—
440C22×C4—2442no—
441C22×C4—2442no—
442C22×C4D4×D6111yescharacteristic
443C22×C4D4×D6111yescharacteristic
444C22×C4C23×D6111yescharacteristic
445SD16—1228no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—612no—
450C2×C8—1221no—
451C2×C8—1221no—
452C2×C8—1221no—
453C2×C8—1221no—
454C2×C8—1221no—
455C4⋊C4—22no—
456C4⋊C4—24no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—42no—
460C4⋊C4—612no—
461C4⋊C4—1228no—
462C4⋊C4—1221no—
463C4⋊C4—1222no—
464C4⋊C4—1222no—
465C4⋊C4—1222no—
466C4⋊C4—4881no—
467C4⋊C4—2444no—
468C4⋊C4—2442no—
469C4⋊C4C2×C22⋊D6114yes—
470C22⋊C4—1228no—
471C22⋊C4—1224no—
472C22⋊C4—1224no—
473C22⋊C4—4882no—
474C22⋊C4—2444no—
475C42—42no—
476C42D4×D6112yes—
477C42C2×C22⋊D6112yes—
478C23—614no—
479C23—1222no—
480C23—1222no—
481C23—1222no—
482C23—1222no—
483C23—1222no—
484C23—1222no—
485C23—1222no—
486C23—2441no—
487C23—2441no—
488C23—2441no—
489C23—2441no—
490C23—2441no—
491C23—2441no—
492C23C2×D4×D6111yescharacteristic
493Q8—22no—
494Q8—22no—
495Q8—41no—
496Q8—48no—
497Q8—41no—
498D4—1222no—
499D4—1222no—
500D4—1228no—
501D4—24416no—
502D4—2441no—
503D4—2441no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—1222no—
516C2×C4—1222no—
517C2×C4—1222no—
518C2×C4—1222no—
519C2×C4—24416no—
520C2×C4—2442no—
521C2×C4—2442no—
522C2×C4—2442no—
523C2×C4—2441no—
524C2×C4—2441no—
525C2×C4—2442no—
526C2×C4—2442no—
527C2×C4—2448no—
528C2×C4—2442no—
529C2×C4—2442no—
530C2×C4#1451974512yes—
531C2×C4#1451974512yes—
532C2×C4SD16×D6112yes—
533C2×C4#1451980412yes—
534C2×C4C2×D4×D6111yescharacteristic
535C2×C4C22×C22⋊D6111yescharacteristic
536C2×C4#41×D6111yescharacteristic
537C2×C4C2×#1175822611yescharacteristic
538C2×C4#1451982311yescharacteristic
539C2×C4#1451982511yescharacteristic
540C8—42no—
541C8—1222no—
542C8—2441no—
543C22—1224no—
544C22—1224no—
545C22—1224no—
546C22—4881no—
547C22—4881no—
548C22—4881no—
549C22—4881no—
550C22—2442no—
551C22—2442no—
552C22—2442no—
553C22—2442no—
554C22—2442no—
555C22—2442no—
556C22—2442no—
557C22C2×#1451974511yescharacteristic
558C22#1684040311yescharacteristic
559C22#1684041411yescharacteristic
560C22C2×SD16×D6111yescharacteristic
561C22C2×#1451980411yescharacteristic
562C22#1684057111yescharacteristic
563C22#1684057811yescharacteristic
564C4—22no—
565C4—44no—
566C4—42no—
567C4—81no—
568C4—4882no—
569C4—2442no—
570C4—2442no—
571C4#1683991114yes—
572C4#1684027511yescharacteristic
573C4#1684034211yescharacteristic
574C4#1684042311yescharacteristic
575C4#1684043111yescharacteristic
576C2—4882no—
577C2—2444no—
578C2#2144916611yescharacteristic
579C2C2×#1683991111yescharacteristic
580C2#2145513311yescharacteristic
581C2#2145768111yescharacteristic
582C2#2145814311yescharacteristic
583C2#2145817911yescharacteristic
584C2#2145840611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#475C2
socleC23×C61C22×D4
prime radicalC2×#16840415C2
hypercenterC2×#475D61
hypocenterC61#24120
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#475

Complements

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

NH ≅ G/Ntype
2. C2×C4×#11758201C2semidirect
3. #21450737C2semidirect
4. #21447010C2semidirect
5. #21446821C2semidirect
6. #21446781C2semidirect
7. #21446612C2semidirect
8. #21446123C2semidirect
9. #21446010C2semidirect
10. #21445820C2semidirect
11. C2×C61×#475C2semidirect
117. C2×Q8×C61#41semidirect
138. C2×D244SD16semidirect
184. C2×#92418#41semidirect
285. C2×#475D61semidirect
337. C61#24120semidirect
411. C2×Q8#11758226semidirect

Maximal subgroups

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

subgroupindex
C2×#168404152
C2×C4×#117582012
#214507372
#214470102
#214468212
#214467812
#214466122
#214461232
#214460102
#214458202
C2×C61×#4752
#2412061