Finite Group Databasebeta

← #29714905

Subgroups — #29714905

Order 15616.

1116 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840486C211yesmaximal, characteristic, complemented
2C2×#16839872C211yesmaximal, characteristic
3#21451173C211yesmaximal, characteristic, complemented
4#21447056C212yesmaximal, complemented
5#21446893C212yesmaximal
6#21446758C211yesmaximal, characteristic, complemented
7#21446640C211yesmaximal, characteristic, complemented
8#21446176C212yesmaximal, complemented
9#21446085C212yesmaximal, complemented
10#21445830C211yesmaximal, characteristic, complemented
11C2×C61×#461C211yesmaximal, characteristic, complemented
12C22×#11758199C2211yescharacteristic
13#16840486—24no—
14#16840486—24no—
15C2×#14519772—22no—
16C2×#14519766C2212yes—
17C2×#14519757—21no—
18C2×Q8×Dic61C2212yes—
19C2×C4×Dic122C2211yescharacteristic
20C2×C4×D244C2211yescharacteristic
21C2×#14519735—21no—
22#16839872—24no—
23C2×#14519708C2211yescharacteristic
24C2×#14519701C2212yes—
25C2×#14519701C2212yes—
26C2×#14519699C2212yes—
27C2×#14519698C2211yescharacteristic
28C2×#14519697C2211yescharacteristic
29C2×#14519688C2211yescharacteristic
30#16839698—21no—
31#16839674—21no—
32#16839665C2212yes—
33#16839658C2212yes—
34#16839635—22no—
35#16839628C2211yescharacteristic
36#16839525C2212yes—
37#16839516C2212yes—
38#16839516C2212yes—
39C22×SD16×C61C2212yes—
40C2×C4⋊Q8×C61C2211yescharacteristic
41C2×C4×D4×C61C2211yescharacteristic
42C61×#461—24no—
43C2×C4⋊C8×C61C2211yescharacteristic
44C2×#103×C61C2212yes—
45C22×Dic122D411yescharacteristic
46C22×D244C2311yescharacteristic
47C22×C4×D61—21no—
48C2×#11758199—22no—
49C2×#11758199—22no—
50C2×#11758199—22no—
51C22×#92418C2311yescharacteristic
52C22×#92418C2311yescharacteristic
53#14519772—216no—
54#14519766—24no—
55#14519766—24no—
56#14519757—28no—
57Q8×Dic61—24no—
58Q8×Dic61—24no—
59C4×Dic122—22no—
60C4×Dic122—22no—
61C4×D244—22no—
62C4×D244D414yes—
63#14519735—28no—
64C2×C4⋊Dic61—22no—
65C2×C4⋊Dic61D411yescharacteristic
66C2×C4⋊Dic61C2312yes—
67C2×C4⋊Dic61C2311yescharacteristic
68C2×#11758196—22no—
69C2×#11758196D412yes—
70C2×C22⋊Dic61—22no—
71C2×C22⋊Dic61—22no—
72C2×#11758198—21no—
73C2×#11758198—22no—
74C2×#11758198—22no—
75C2×C4×Dic61—22no—
76C2×C4×Dic61C2312yes—
77#14519708—22no—
78#14519701—24no—
79#14519701—42no—
80#14519699—42no—
81#14519698—22no—
82#14519697—22no—
83#14519688—22no—
84#14519684—21no—
85#14519684—21no—
86#14519684—22no—
87C22×Q8×C61C2312yes—
88C22×D4×C61C2311yescharacteristic
89C23×C4×C61—21no—
90C2×SD16×C61—24no—
91C2×SD16×C61—24no—
92C2×SD16×C61—42no—
93C22×C8×C61C2312yes—
94C4⋊Q8×C61—22no—
95C4⋊Q8×C61D414yes—
96C4×D4×C61—22no—
97C4×D4×C61—22no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61C2311yescharacteristic
101C2×C22⋊C4×C61—21no—
102C2×C42×C61C2311yescharacteristic
103C4⋊C8×C61—22no—
104#103×C61—24no—
105C24×C61—21no—
106C23×D61—21no—
107C2×Q8×C61—22no—
108C2×Q8×C61—22no—
109C2×Q8×C61—22no—
110C2×Q8×C61—22no—
111C2×Q8×C61—22no—
112C2×Q8×C61C2×D414yes—
113C2×D4×C61—21no—
114C2×D4×C61—21no—
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61#4112yes—
119C22×C4×C61—22no—
120C22×C4×C61—21no—
121C22×C4×C61—42no—
122C22×C4×C61—82no—
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2411yescharacteristic
126C22×Dic61—21no—
127C22×Dic61—22no—
128C22×Dic61C2×D412yes—
129C22×Dic61C2×D412yes—
130C2×D244—21no—
131C2×D244—21no—
132C2×D244—21no—
133C2×D244—21no—
134C2×D244—21no—
135C2×D244C2×D412yes—
136C2×C4×D61—24no—
137C2×C4×D61—44no—
138C2×Dic122—22no—
139C2×Dic122—42no—
140SD16×C61—48no—
141C2×C8×C61—22no—
142C2×C8×C61—22no—
143C2×C8×C61—22no—
144C4⋊C4×C61—24no—
145C4⋊C4×C61—24no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C22⋊C4×C61—42no—
149C42×C61C2×D412yes—
150C42×C61C2×D412yes—
151C22⋊Dic61—44no—
152C22⋊Dic61—82no—
153#11758199—28no—
154#11758198—28no—
155#11758198—28no—
156#11758198—24no—
157C4⋊Dic61—28no—
158C4⋊Dic61—22no—
159C4⋊Dic61—21no—
160C4⋊Dic61—22no—
161C4⋊Dic61—22no—
162C4⋊Dic61C2×D412yes—
163#11758196—24no—
164#11758196—82no—
165C4×Dic61—28no—
166C4×Dic61—22no—
167C4×Dic61—22no—
168C2×#92418—21no—
169C2×#92418—21no—
170C2×#92418—21no—
171C2×#92418—21no—
172C2×#92418—21no—
173C2×#92418#4112yes—
174C23×C61—22no—
175C23×C61—41no—
176C23×C61—41no—
177C23×C61—42no—
178C23×C61—42no—
179C23×C61C22×D411yescharacteristic
180C22×D61—22no—
181C22×D61—41no—
182C22×D61—41no—
183C22×D61—42no—
184C22×D61—42no—
185Q8×C61—24no—
186Q8×C61—24no—
187Q8×C61—42no—
188Q8×C61—42no—
189D4×C61—22no—
190D4×C61—22no—
191D4×C61—41no—
192D4×C61—41no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—24no—
196C2×C4×C61—42no—
197C2×C4×C61—42no—
198C2×C4×C61—41no—
199C2×C4×C61—42no—
200C2×C4×C61—84no—
201C2×C4×C61#12012yes—
202C2×C4×C61#12012yes—
203C2×C4×C61C22×D411yescharacteristic
204C2×C4×C61C22×D411yescharacteristic
205C2×C4×C61C2×#4111yescharacteristic
206C2×C4×C61C2×#4111yescharacteristic
207C2×C4×C61H(5,𝔽2)11yescharacteristic
208C2×C4×C612−1+411yescharacteristic
209C2×Dic61—22no—
210C2×Dic61—22no—
211C2×Dic61—22no—
212C2×Dic61—22no—
213C2×Dic61—22no—
214C2×Dic61—22no—
215C2×Dic61—22no—
216C2×Dic61—42no—
217C2×Dic61—42no—
218C2×Dic61—42no—
219C2×Dic61—42no—
220D244—22no—
221D244—22no—
222D244—41no—
223D244—41no—
224C4×D61—48no—
225Dic122—44no—
226C8×C61—42no—
227#92418—22no—
228#92418—41no—
229#24197—611nomaximal, Sylow
230C22×C61—44no—
231C22×C61—42no—
232C22×C61—82no—
233C22×C61—81no—
234C22×C61—81no—
235C22×C61C2×#12011yescharacteristic
236C22×C61#54811yescharacteristic
237C22×C61#55011yescharacteristic
238C22×C61#57512yes—
239C22×C61#57712yes—
240C2×D61—42no—
241C2×D61—42no—
242C2×D61—42no—
243C2×D61—82no—
244C2×D61—82no—
245C4×C61—22no—
246C4×C61—22no—
247C4×C61—44no—
248C4×C61—81no—
249C4×C61#52511yescharacteristic
250C4×C61#53711yescharacteristic
251C4×C61#54611yescharacteristic
252C4×C61#54711yescharacteristic
253Dic61—42no—
254Dic61—42no—
255Dic61—42no—
256Dic61—82no—
257C2×#556—611no—
258C2×#461D6111yescharacteristic, complemented
259C2×#442—611no—
260#2580—611no—
261#2042—612no—
262#2012—612no—
263#1985—611no—
264#1960—611no—
265#1880—612no—
266#1865—612no—
267#1804—611no—
268C2×C61—82no—
269C2×C61#240711yescharacteristic
270C2×C61#300511yescharacteristic
271C2×C61#301011yescharacteristic
272C2×C61#337612yes—
273C2×C61#340112yes—
274D61—82no—
275C22×SD16C2×D6112yes—
276C22×D8—611no—
277#556—1224no—
278#556—1224no—
279C2×C4⋊Q8C2×D6111yescharacteristic
280C2×#126—1222no—
281C2×#124—612no—
282C2×C22⋊Q8—1221no—
283C2×C4×Q8—611no—
284C2×C4×Q8—612no—
285C2×C4×D4—611no—
286C2×C4×D4C2×D6111yescharacteristic
287C2×#117—1221no—
288#461—24no—
289#442—1224no—
290C2×#106—611no—
291C2×C4⋊C8C2×D6111yescharacteristic
292C2×#103—612no—
293C2×#103—612no—
294C2×#103C2×D6112yes—
295C2×#102—611no—
296C2×#102—611no—
297C2×#102—612no—
298C2×#97—611no—
299#399—1221no—
300#393—1221no—
301#390—612no—
302#389—612no—
303#385—1222no—
304#382—611no—
305#340—612no—
306#336—612no—
307#336—612no—
308C61#2419711yescharacteristic, complemented, Sylow
309C22×Q8—611no—
310C22×Q8C22×D6112yes—
311C22×D4—611no—
312C22×D4C22×D6111yescharacteristic
313C23×C4—21no—
314C23×C4—1221no—
315C2×SD16—24no—
316C2×SD16—24no—
317C2×SD16—42no—
318C2×D8—1222no—
319C2×D8—1222no—
320C2×D8—1222no—
321C22×C8—611no—
322C22×C8—611no—
323C22×C8C22×D6112yes—
324C4⋊Q8—22no—
325C4⋊Q8C22⋊D6114yes—
326#126—12216no—
327#124—1224no—
328#124—1224no—
329C22⋊Q8—1228no—
330C4×Q8—1224no—
331C4×Q8—1224no—
332C4×Q8—1222no—
333C4×Q8—1222no—
334C4×D4—22no—
335C4×D4—22no—
336C4×D4—614no—
337C4×D4—1222no—
338#117—1228no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—611no—
342C2×C4⋊C4—611no—
343C2×C4⋊C4—612no—
344C2×C4⋊C4—612no—
345C2×C4⋊C4—1222no—
346C2×C4⋊C4—1222no—
347C2×C4⋊C4C22×D6111yescharacteristic
348C2×C22⋊C4—21no—
349C2×C22⋊C4—1222no—
350C2×C22⋊C4—1222no—
351C2×C22⋊C4—1221no—
352C2×C22⋊C4—1222no—
353C2×C22⋊C4—1222no—
354C2×C42—612no—
355C2×C42—1222no—
356C2×C42C22×D6111yescharacteristic
357#106—1222no—
358C4⋊C8—22no—
359#103—24no—
360#103—1224no—
361#103—2442no—
362#102—1222no—
363#102—1222no—
364#102—2442no—
365#97—1222no—
366#95—1222no—
367#95—1221no—
368#95—1221no—
369C24—21no—
370C24—1221no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—1222no—
377C2×Q8—2442no—
378C2×Q8C2×C22⋊D6114yes—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—612no—
385C2×D4—1221no—
386C2×D4—1221no—
387C2×D4—1221no—
388C2×D4—1221no—
389C2×D4—1221no—
390C2×D4#1175822412yes—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—42no—
394C22×C4—82no—
395C22×C4—612no—
396C22×C4—612no—
397C22×C4—1221no—
398C22×C4—1224no—
399C22×C4—1222no—
400C22×C4—2444no—
401C22×C4D4×D6111yescharacteristic
402C22×C4D4×D6111yescharacteristic
403C22×C4C23×D6111yescharacteristic
404SD16—48no—
405D8—1228no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—612no—
410C2×C8—1221no—
411C2×C8—1221no—
412C2×C8—1221no—
413C2×C8—1221no—
414C2×C8—1221no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—612no—
420C4⋊C4—1228no—
421C4⋊C4—1221no—
422C4⋊C4—1224no—
423C4⋊C4—1222no—
424C4⋊C4—1222no—
425C4⋊C4—1222no—
426C4⋊C4—4882no—
427C22⋊C4—42no—
428C22⋊C4—1228no—
429C22⋊C4—1228no—
430C22⋊C4—1224no—
431C22⋊C4—4882no—
432C22⋊C4—2444no—
433C42—1228no—
434C42—1222no—
435C42—1222no—
436C42D4×D6112yes—
437C42C2×C22⋊D6112yes—
438C23—22no—
439C23—41no—
440C23—42no—
441C23—42no—
442C23—41no—
443C23—1222no—
444C23—2442no—
445C23—2442no—
446C23—2441no—
447C23—2441no—
448C23C2×D4×D6111yescharacteristic
449Q8—24no—
450Q8—24no—
451Q8—42no—
452Q8—42no—
453Q8—2444no—
454D4—22no—
455D4—22no—
456D4—41no—
457D4—41no—
458D4—1222no—
459D4—1222no—
460D4—2441no—
461D4—2441no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—84no—
470C2×C4—1222no—
471C2×C4—1222no—
472C2×C4—1222no—
473C2×C4—1222no—
474C2×C4—1222no—
475C2×C4—1222no—
476C2×C4—1222no—
477C2×C4—2448no—
478C2×C4—2442no—
479C2×C4—2442no—
480C2×C4—2442no—
481C2×C4—2442no—
482C2×C4#1451974512yes—
483C2×C4#1451974512yes—
484C2×C4C2×D4×D6111yescharacteristic
485C2×C4C22×C22⋊D6111yescharacteristic
486C2×C4#41×D6111yescharacteristic
487C2×C4C2×#1175822411yescharacteristic
488C2×C4#1451982211yescharacteristic
489C2×C4#1451982411yescharacteristic
490C8—42no—
491C8—1222no—
492C8—2441no—
493C22—42no—
494C22—44no—
495C22—82no—
496C22—81no—
497C22—81no—
498C22—4882no—
499C22—4882no—
500C22—2442no—
501C22—2442no—
502C22—2442no—
503C22C2×#1451974511yescharacteristic
504C22#1684041811yescharacteristic
505C22#1684043011yescharacteristic
506C22#1684056112yes—
507C22#1684056812yes—
508C4—22no—
509C4—22no—
510C4—44no—
511C4—81no—
512C4—4882no—
513C4—2442no—
514C4—2442no—
515C4—2442no—
516C4#1684027211yescharacteristic
517C4#1684034111yescharacteristic
518C4#1684040611yescharacteristic
519C4#1684041111yescharacteristic
520C2—82no—
521C2—4882no—
522C2#2144916011yescharacteristic
523C2#2145506011yescharacteristic
524C2#2145510111yescharacteristic
525C2#2145863912yes—
526C2#2145889212yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#461C2
socleC23×C61C22×D4
prime radicalC22×#11758199C22
hypercenterC2×#461D61
hypocenterC61#24197
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#461

Complements

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

NH ≅ G/Ntype
1. C2×#16840486C2semidirect
3. #21451173C2semidirect
4. #21447056C2semidirect
6. #21446758C2semidirect
7. #21446640C2semidirect
8. #21446176C2semidirect
9. #21446085C2semidirect
10. #21445830C2semidirect
11. C2×C61×#461C2semidirect
258. C2×#461D61semidirect
308. C61#24197semidirect

Maximal subgroups

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

subgroupindex
C2×#168404862
C2×#168398722
#214511732
#214470562
#214468932
#214467582
#214466402
#214461762
#214460852
#214458302
C2×C61×#4612
#2419761