Finite Group Databasebeta

← #29714774

Subgroups — #29714774

Order 15616.

1214 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×#16840497C211yesmaximal, characteristic, complemented
2C2×#16839957C211yesmaximal, characteristic
3C2×C4×#11758200C211yesmaximal, characteristic, complemented
4#21447060C212yesmaximal, complemented
5#21446895C212yesmaximal, complemented
6#21446756C211yesmaximal, characteristic, complemented
7#21446624C211yesmaximal, characteristic, complemented
8#21446170C212yesmaximal, complemented
9#21446079C212yesmaximal, complemented
10#21445821C211yesmaximal, characteristic, complemented
11C61×#2538C211yesmaximal, characteristic, complemented
12C22×#11758200C2211yescharacteristic
13C22×#11758199C2212yes—
14#16840497—24no—
15#16840497—24no—
16C2×#14519774C2211yescharacteristic
17C2×#14519766—21no—
18C2×#14519765C2212yes—
19C2×#14519762—22no—
20C2×C4×Dic122C2211yescharacteristic
21C2×C4×D244C2212yes—
22C2×#14519739—21no—
23#16839957—24no—
24C4×#11758200—24no—
25C2×#14519708C2211yescharacteristic
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519700C2211yescharacteristic
28C2×#14519700C2212yes—
29C2×#14519698C2211yescharacteristic
30C2×#14519697C2212yes—
31C2×#14519697C2212yes—
32C2×#14519697C2212yes—
33C2×C4×#92418C2211yescharacteristic
34#16839524C2212yes—
35#16839517C2212yes—
36#16839517C2212yes—
37C2×C22⋊Q8×C61—21no—
38C2×C4×Q8×C61C2211yescharacteristic
39C2×C4×D4×C61C2211yescharacteristic
40C61×#396—22no—
41C61×#390C2212yes—
42C61×#389C2212yes—
43C61×#384C2211yescharacteristic
44C61×#380—21no—
45C22×Dic122C2311yescharacteristic
46C22×D244—21no—
47C22×D244—21no—
48C22×D244C2312yes—
49C22×C4×D61—22no—
50C2×#11758200—22no—
51C2×#11758200—22no—
52C2×#11758200—22no—
53C2×#11758199—24no—
54C2×#11758199—24no—
55C2×#11758199—42no—
56C22×#92418C2311yescharacteristic
57C22×#92418C2312yes—
58C22×#92418C2311yescharacteristic
59#14519774—22no—
60#14519774D414yes—
61#14519766—28no—
62#14519765—24no—
63#14519765—24no—
64#14519762—216no—
65C4×Dic122—22no—
66C4×Dic122D414yes—
67C4×D244—24no—
68C4×D244—24no—
69#14519739—28no—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×#11758196—21no—
72C2×#11758198—22no—
73C2×#11758198—22no—
74C2×#11758198—22no—
75C2×C4×Dic61—21no—
76#14519708—22no—
77C4⋊#92418—22no—
78#14519700—22no—
79#14519700—42no—
80#14519698—22no—
81#14519697—24no—
82#14519697—24no—
83#14519697—42no—
84C4×#92418—22no—
85C22×Q8×C61D411yescharacteristic
86C22×D4×C61C2311yescharacteristic
87C23×C4×C61—21no—
88C22⋊Q8×C61—44no—
89C4×Q8×C61—22no—
90C4×Q8×C61—22no—
91C4×D4×C61—22no—
92C4×D4×C61—22no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61—22no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61D411yescharacteristic
97C2×C4⋊C4×C61D412yes—
98C2×C4⋊C4×C61C2312yes—
99C2×C4⋊C4×C61C2311yescharacteristic
100C2×C22⋊C4×C61—22no—
101C2×C22⋊C4×C61—21no—
102C2×C22⋊C4×C61—22no—
103C2×C42×C61C2312yes—
104C2×C42×C61C2311yescharacteristic
105#95×C61—22no—
106C24×C61—21no—
107C23×D61—22no—
108C2×Q8×C61—22no—
109C2×Q8×C61—22no—
110C2×Q8×C61—22no—
111C2×D4×C61—21no—
112C2×D4×C61—21no—
113C2×D4×C61—21no—
114C2×D4×C61—21no—
115C2×D4×C61—21no—
116C2×D4×C61#4112yes—
117C22×C4×C61—21no—
118C22×C4×C61—22no—
119C22×C4×C61—42no—
120C22×C4×C61—42no—
121C22×C4×C61—42no—
122C22×C4×C61C2×D412yes—
123C22×C4×C61C2×D412yes—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2411yescharacteristic
127C22×Dic61—21no—
128C22×Dic61—21no—
129C2×D244—22no—
130C2×D244—22no—
131C2×D244—22no—
132C2×D244—22no—
133C2×D244—24no—
134C2×D244—22no—
135C2×D244—24no—
136C2×D244—42no—
137C2×D244—42no—
138C2×D244—81no—
139C2×D244—81no—
140C2×D244C2×D414yes—
141C2×C4×D61—28no—
142C2×C4×D61—82no—
143C2×C4×D61—82no—
144C2×Dic122—21no—
145C2×Dic122—21no—
146C2×Dic122—21no—
147C2×Dic122—21no—
148C2×Dic122—21no—
149C2×Dic122C2×D412yes—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—24no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—44no—
157C4⋊C4×C61—42no—
158C4⋊C4×C61—81no—
159C22⋊C4×C61—44no—
160C22⋊C4×C61—42no—
161C22⋊C4×C61—82no—
162C42×C61—22no—
163C42×C61—22no—
164C42×C61C2×D412yes—
165C42×C61C2×D412yes—
166#11758200—28no—
167#11758199—48no—
168#11758198—28no—
169#11758198—28no—
170#11758198—28no—
171C4⋊Dic61—21no—
172C4⋊Dic61C2×D412yes—
173#11758196—24no—
174C4×Dic61—24no—
175C2×#92418—21no—
176C2×#92418—21no—
177C2×#92418—21no—
178C2×#92418—22no—
179C2×#92418—22no—
180C2×#92418—21no—
181C2×#92418—22no—
182C2×#92418—21no—
183C2×#92418#4112yes—
184C23×C61—22no—
185C23×C61—41no—
186C23×C61—41no—
187C23×C61—41no—
188C23×C61—41no—
189C23×C61—41no—
190C23×C61—41no—
191C23×C61C22×D411yescharacteristic
192C22×D61—24no—
193C22×D61—42no—
194C22×D61—42no—
195C22×D61—42no—
196C22×D61—42no—
197C22×D61—42no—
198C22×D61—42no—
199Q8×C61—44no—
200D4×C61—22no—
201D4×C61—22no—
202D4×C61—41no—
203D4×C61—41no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—22no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—41no—
213C2×C4×C61—42no—
214C2×C4×C61—41no—
215C2×C4×C61—42no—
216C2×C4×C61—41no—
217C2×C4×C61—42no—
218C2×C4×C61—84no—
219C2×C4×C61#12012yes—
220C2×C4×C61#12012yes—
221C2×C4×C61#13512yes—
222C2×C4×C61AGL(1,ℤ/8)12yes—
223C2×C4×C61C22×D411yescharacteristic
224C2×C4×C61C22×D411yescharacteristic
225C2×C4×C61C2×#4111yescharacteristic
226C2×C4×C61C2×#4111yescharacteristic
227C2×C4×C61H(5,𝔽2)11yescharacteristic
228C2×C4×C612−1+411yescharacteristic
229C2×Dic61—22no—
230C2×Dic61—22no—
231C2×Dic61—41no—
232C2×Dic61—41no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235D244—24no—
236D244—24no—
237D244—42no—
238D244—42no—
239D244—48no—
240D244—84no—
241C4×D61—88no—
242Dic122—22no—
243Dic122—22no—
244Dic122—41no—
245Dic122—41no—
246#92418—22no—
247#92418—41no—
248#92418—42no—
249#24186—611nomaximal, Sylow
250C22×C61—42no—
251C22×C61—42no—
252C22×C61—42no—
253C22×C61—81no—
254C22×C61—81no—
255C22×C61—81no—
256C22×C61—81no—
257C22×C61C2×#12011yescharacteristic
258C22×C61#54811yescharacteristic
259C22×C61#55011yescharacteristic
260C22×C61C2×#13511yescharacteristic
261C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
262C22×C61#57611yescharacteristic
263C22×C61#57711yescharacteristic
264C2×D61—44no—
265C2×D61—44no—
266C2×D61—44no—
267C2×D61—82no—
268C2×D61—82no—
269C2×D61—82no—
270C2×D61—82no—
271C4×C61—22no—
272C4×C61—42no—
273C4×C61—42no—
274C4×C61—82no—
275C4×C61—81no—
276C4×C61#44914yes—
277C4×C61#52511yescharacteristic
278C4×C61#53711yescharacteristic
279C4×C61#54611yescharacteristic
280C4×C61#54711yescharacteristic
281Dic61—42no—
282Dic61—42no—
283C2×#559—611no—
284C2×#459—611no—
285C2×C4×SD16—611no—
286#2538D6111yescharacteristic, complemented
287#2042—612no—
288#2012—612no—
289#1984—611no—
290#1957—611no—
291#1879—612no—
292#1864—612no—
293#1803—611no—
294C2×C61—82no—
295C2×C61#240711yescharacteristic
296C2×C61C2×#44911yescharacteristic
297C2×C61#301211yescharacteristic
298C2×C61#336111yescharacteristic
299C2×C61#337311yescharacteristic
300C2×C61#339411yescharacteristic
301C2×C61#339911yescharacteristic
302D61—84no—
303C22×SD16—611no—
304C22×D8—612no—
305#559—1224no—
306#559—1224no—
307C2×#127—611no—
308C2×#124—612no—
309C2×#124—1221no—
310C2×#123—1222no—
311C2×C22⋊Q8—21no—
312C2×C4×Q8—611no—
313C2×C4×Q8C2×D6111yescharacteristic
314C2×C4×D4—612no—
315C2×C4×D4—1221no—
316C2×C4×D4C2×D6111yescharacteristic
317#459—1224no—
318C4×SD16—1224no—
319C2×#106—611no—
320C2×C4⋊C8—611no—
321C2×#103—611no—
322C2×#103—612no—
323C2×#102—611no—
324C2×#102—612no—
325C2×#102—612no—
326C2×#102—612no—
327C2×C4×C8—611no—
328#396—22no—
329#390C2×D6112yes—
330#389C2×D6112yes—
331#384C2×D6111yescharacteristic
332#380—21no—
333#340—612no—
334#336—612no—
335#336—612no—
336C61#2418611yescharacteristic, complemented, Sylow
337C22×Q8—611no—
338C22×Q8C22⋊D6111yescharacteristic
339C22×D4—612no—
340C22×D4—1221no—
341C22×D4—1221no—
342C22×D4C22×D6111yescharacteristic
343C23×C4—21no—
344C23×C4—1222no—
345C2×SD16—1222no—
346C2×SD16—1222no—
347C2×SD16—1222no—
348C2×D8—1224no—
349C2×D8—1224no—
350C2×D8—2442no—
351C22×C8—611no—
352C22×C8—612no—
353C22×C8—611no—
354#127—614no—
355#127—1222no—
356#124—1224no—
357#124—1224no—
358#124—1228no—
359#123—12216no—
360C22⋊Q8—44no—
361C4×Q8—22no—
362C4×Q8—22no—
363C4×Q8—614no—
364C4×Q8—1222no—
365C4×D4—22no—
366C4×D4—22no—
367C4×D4—1224no—
368C4×D4—1224no—
369C4×D4—1228no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—611no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4C22⋊D6111yescharacteristic
376C2×C4⋊C4C22⋊D6112yes—
377C2×C4⋊C4C22×D6111yescharacteristic
378C2×C4⋊C4C22×D6112yes—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—1222no—
383C2×C22⋊C4—1222no—
384C2×C22⋊C4—1222no—
385C2×C42—1221no—
386C2×C42C22×D6112yes—
387C2×C42C22×D6111yescharacteristic
388#106—1222no—
389C4⋊C8—1222no—
390#103—1222no—
391#103—2442no—
392#102—1224no—
393#102—1224no—
394#102—1222no—
395#102—2442no—
396C4×C8—1222no—
397#95—22no—
398C24—21no—
399C24—1222no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—612no—
404C2×Q8—1221no—
405C2×Q8—1221no—
406C2×Q8—1221no—
407C2×Q8—1221no—
408C2×Q8—1221no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—614no—
415C2×D4—1222no—
416C2×D4—1222no—
417C2×D4—1222no—
418C2×D4—1222no—
419C2×D4—1222no—
420C2×D4—1224no—
421C2×D4—1224no—
422C2×D4—4881no—
423C2×D4—4881no—
424C2×D4—2442no—
425C2×D4—2442no—
426C2×D4#1175822612yes—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—1228no—
433C22×C4—1221no—
434C22×C4—1221no—
435C22×C4—4882no—
436C22×C4—4882no—
437C22×C4C2×C22⋊D6112yes—
438C22×C4C2×C22⋊D6112yes—
439C22×C4C2×C22⋊D6111yescharacteristic
440C22×C4C2×C22⋊D6111yescharacteristic
441C22×C4C23×D6111yescharacteristic
442SD16—1228no—
443D8—2448no—
444C2×C8—612no—
445C2×C8—1221no—
446C2×C8—1221no—
447C2×C8—1221no—
448C2×C8—1222no—
449C2×C8—1222no—
450C2×C8—1221no—
451C2×C8—1222no—
452C2×C8—1221no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—44no—
460C4⋊C4—42no—
461C4⋊C4—81no—
462C4⋊C4—612no—
463C4⋊C4—1221no—
464C4⋊C4—1224no—
465C22⋊C4—44no—
466C22⋊C4—42no—
467C22⋊C4—82no—
468C22⋊C4—1228no—
469C22⋊C4—1228no—
470C22⋊C4—1228no—
471C42—22no—
472C42—22no—
473C42—1224no—
474C42D4×D6112yes—
475C42D4×D6112yes—
476C23—22no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—1224no—
484C23—2442no—
485C23—2442no—
486C23—2442no—
487C23—2442no—
488C23—2442no—
489C23—2442no—
490C23C22×C22⋊D6111yescharacteristic
491Q8—44no—
492Q8—1222no—
493Q8—1222no—
494Q8—2441no—
495Q8—2441no—
496D4—22no—
497D4—22no—
498D4—41no—
499D4—41no—
500D4—1224no—
501D4—1224no—
502D4—4884no—
503D4—2442no—
504D4—2442no—
505D4—2448no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—84no—
521C2×C4—1222no—
522C2×C4—1222no—
523C2×C4—4888no—
524C2×C4—2441no—
525C2×C4—2441no—
526C2×C4—2441no—
527C2×C4—2441no—
528C2×C4#1451974512yes—
529C2×C4#1451974512yes—
530C2×C4#1451979712yes—
531C2×C4#1451980312yes—
532C2×C4C2×D4×D6111yescharacteristic
533C2×C4C2×D4×D6111yescharacteristic
534C2×C4#41×D6111yescharacteristic
535C2×C4C2×#1175822611yescharacteristic
536C2×C4#1451982311yescharacteristic
537C2×C4#1451982511yescharacteristic
538C8—1222no—
539C8—2441no—
540C8—2442no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22—81no—
548C22—4882no—
549C22—4882no—
550C22—4882no—
551C22—4882no—
552C22—2444no—
553C22—2444no—
554C22—2444no—
555C22C2×#1451974511yescharacteristic
556C22#1684042311yescharacteristic
557C22#1684043111yescharacteristic
558C22C2×#1451979711yescharacteristic
559C22C2×#1451980311yescharacteristic
560C22#1684056511yescharacteristic
561C22#1684057011yescharacteristic
562C4—22no—
563C4—42no—
564C4—42no—
565C4—81no—
566C4—82no—
567C4—2442no—
568C4—2442no—
569C4#1683990514yes—
570C4#1684027511yescharacteristic
571C4#1684034211yescharacteristic
572C4#1684040311yescharacteristic
573C4#1684041411yescharacteristic
574C2—82no—
575C2—4884no—
576C2#2144916611yescharacteristic
577C2C2×#1683990511yescharacteristic
578C2#2145512011yescharacteristic
579C2#2145847911yescharacteristic
580C2#2145861111yescharacteristic
581C2#2145881511yescharacteristic
582C2#2145886111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2538C2
socleC23×C61C22×D4
prime radicalC2×#16839957C2
hypercenter#2538D61
hypocenterC61#24186
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2538

Complements

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

NH ≅ G/Ntype
1. C2×#16840497C2semidirect
3. C2×C4×#11758200C2semidirect
4. #21447060C2semidirect
5. #21446895C2semidirect
6. #21446756C2semidirect
7. #21446624C2semidirect
8. #21446170C2semidirect
9. #21446079C2semidirect
10. #21445821C2semidirect
11. C61×#2538C2semidirect
286. #2538D61semidirect
336. C61#24186semidirect

Maximal subgroups

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

subgroupindex
C2×#168404972
C2×#168399572
C2×C4×#117582002
#214470602
#214468952
#214467562
#214466242
#214461702
#214460792
#214458212
C61×#25382
#2418661