Finite Group Databasebeta

← #29647723

Subgroups — #29647723

Order 15616.

1070 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459620C211yesmaximal, characteristic
2#21454842C211yesmaximal, characteristic
3#21448730C211yesmaximal, characteristic
4#21444389C214yesmaximal, complemented
5#21444384C211yesmaximal, characteristic, complemented
6#21444383C211yesmaximal, characteristic, complemented
7#21444317C211yesmaximal, characteristic, complemented
8#328×D61C214yesmaximal, complemented
9C61×#1503C211yesmaximal, characteristic, complemented
10#16840473C412yescomplemented
11#16840472—22no—
12#16840472C412yescomplemented
13#16840468—22no—
14Q8×D244C412yescomplemented
15#16840427C412yescomplemented
16#16840390—21no—
17#16840387—22no—
18#16840319C2211yescharacteristic
19C22⋊Q8×D61C2214yes—
20#16840236—21no—
21C2×C4×D244—21no—
22C2×C4×D244C2211yescharacteristic
23C8×D244—22no—
24#16839827—22no—
25#16839815C2211yescharacteristic
26C22⋊C8×D61C2214yes—
27#16839660—21no—
28#16839633—22no—
29#16839628C2211yescharacteristic
30#16839621C2212yes—
31#95×D61C2212yes—
32#16839500C2214yes—
33#16839499C2214yes—
34#16839498C2212yes—
35#16839498C2212yes—
36C61×#533C2211yescharacteristic
37C61×#431C2211yescharacteristic
38C4×C4⋊C4×C61C2211yescharacteristic
39C61×#328C2212yes—
40C61×#328C2212yes—
41C2×Q8×D61C2×C414yescomplemented
42C22×D244D411yescharacteristic
43C22×C4×D61—22no—
44C22×C4×D61C2312yes—
45C2×C8×D61—24no—
46#14519777—22no—
47#14519777C2×C411yescharacteristic, complemented
48#14519777C2×C411yescharacteristic, complemented
49#14519776—22no—
50#14519769—21no—
51#14519769—22no—
52#14519768—22no—
53#14519757—24no—
54#14519757C2×C414yescomplemented
55#14519756—21no—
56#14519756C2312yes—
57#14519755—24no—
58#14519755—24no—
59#14519755—24no—
60#14519754—24no—
61#14519753—24no—
62#14519753C2×C414yescomplemented
63C4×Dic122—21no—
64C4×Dic122C2×C412yescomplemented
65C4×D244—22no—
66C4×D244—22no—
67C4×D244—44no—
68C4×D244C2×C412yes—
69C4×D244C2×C412yes—
70C4×D244D412yes—
71C4×D244D412yes—
72#14519739—22no—
73#14519739—22no—
74#14519736—21no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61D411yescharacteristic
77C2×#11758196—22no—
78C4⋊C4×D61—24no—
79C4⋊C4×D61C2×C414yescomplemented
80C2×#11758198—22no—
81C2×#11758198D412yes—
82C22⋊C4×D61—24no—
83C4⋊#92418—21no—
84C22⋊#92418C2312yes—
85#14519689—24no—
86C4×#92418—21no—
87#14519684—21no—
88#14519684—22no—
89#14519684C2312yes—
90C4⋊Q8×C61C2×C411yescharacteristic, complemented
91C4⋊Q8×C61C2×C411yescharacteristic, complemented
92#125×C61—21no—
93C22⋊Q8×C61—21no—
94C22⋊Q8×C61C2312yes—
95C4×Q8×C61—21no—
96C4×Q8×C61C2×C412yescomplemented
97#117×C61—21no—
98C2×C4⋊C4×C61—21no—
99C2×C42×C61—21no—
100C2×C42×C61C2311yescharacteristic
101C4⋊C8×C61—21no—
102C22⋊C8×C61C2312yes—
103C4×C8×C61—21no—
104#95×C61C2311yescharacteristic
105#95×C61C2311yescharacteristic
106C23×D61C2×D412yes—
107C2×Q8×C61—21no—
108C2×Q8×C61C22×C412yes—
109C22×C4×C61—21no—
110C22×C4×C61—21no—
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2411yescharacteristic
114C22×Dic61—22no—
115C22×Dic61C2×D412yes—
116Q8×D61—84no—
117C2×D244C22⋊C412yes—
118C2×D244C22⋊C412yes—
119C2×D244SD1614yes—
120C2×D244Q1614yes—
121C2×C4×D61—24no—
122C2×C4×D61—24no—
123C2×C4×D61—44no—
124C2×C4×D61—44no—
125C2×C4×D61—82no—
126C2×C4×D61—82no—
127C2×C4×D61C22×C414yes—
128C2×C4×D61C2×D414yes—
129C2×Dic122—21no—
130C2×Dic122—24no—
131C2×Dic122C22×C412yes—
132C2×C8×C61—22no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—42no—
139C4⋊C4×C61C22×C412yes—
140C22⋊C4×C61—22no—
141C42×C61—22no—
142C42×C61—42no—
143C42×C61C22×C411yescharacteristic
144C42×C61C22×C411yescharacteristic
145C42×C61C2×D411yescharacteristic
146C42×C61C2×D411yescharacteristic
147C22⋊Dic61—22no—
148#11758198—24no—
149#11758198—24no—
150#11758198—44no—
151#11758198C22⋊C414yes—
152#11758198C22⋊C414yes—
153C4⋊Dic61—21no—
154C4⋊Dic61—21no—
155C4⋊Dic61—21no—
156C4⋊Dic61—24no—
157C4⋊Dic61—42no—
158C4⋊Dic61C22⋊C412yes—
159C4⋊Dic61C22⋊C412yes—
160C4⋊Dic61C22×C412yes—
161#11758196—22no—
162#11758196—22no—
163#11758196—24no—
164#11758196—24no—
165#11758196—82no—
166C4×Dic61—22no—
167C2×#92418—22no—
168C8×D61—84no—
169C23×C61C22×D411yescharacteristic
170C22×D61—42no—
171C22×D61—42no—
172C22×D61C2×C22⋊C414yes—
173C22×D61C2×SD1614yes—
174C22×D61C2×Q1614yes—
175Q8×C61—42no—
176Q8×C61—81no—
177C2×C4×C61—22no—
178C2×C4×C61—22no—
179C2×C4×C61—22no—
180C2×C4×C61—41no—
181C2×C4×C61—41no—
182C2×C4×C61—41no—
183C2×C4×C61—41no—
184C2×C4×C61—41no—
185C2×C4×C61—41no—
186C2×C4×C61—41no—
187C2×C4×C61C2×C22⋊C411yescharacteristic
188C2×C4×C61C2×C22⋊C411yescharacteristic
189C2×C4×C61C2×C22⋊C411yescharacteristic
190C2×C4×C61C2×C22⋊C411yescharacteristic
191C2×C4×C61C2×SD1612yes—
192C2×C4×C61C2×Q1612yes—
193C2×C4×C61C23×C411yescharacteristic
194C2×C4×C61C22×D411yescharacteristic
195C2×Dic61—22no—
196C2×Dic61—22no—
197C2×Dic61—42no—
198C2×Dic61—42no—
199C2×Dic61—42no—
200C2×Dic61—42no—
201C2×Dic61C2×C22⋊C414yes—
202D244#10318yes—
203D244#10318yes—
204C4×D61—44no—
205C4×D61—44no—
206C4×D61—84no—
207C4×D61—84no—
208C4×D61—88no—
209Dic122—42no—
210Dic122—84no—
211Dic122—81no—
212C8×C61—42no—
213#92418—42no—
214#17614—611nomaximal, Sylow
215C22×C61—21no—
216C22×C61—21no—
217C22×C61C22×C22⋊C411yescharacteristic
218C22×C61C22×SD1611yescharacteristic
219C22×C61C22×Q1611yescharacteristic
220C2×D61—24no—
221C2×D61—24no—
222C2×D61—44no—
223C2×D61—44no—
224C2×D61C2×#10318yes—
225C4×C61—21no—
226C4×C61—21no—
227C4×C61—21no—
228C4×C61—21no—
229C4×C61—42no—
230C4×C61—42no—
231C4×C61—42no—
232C4×C61—81no—
233C4×C61C2×#10312yes—
234C4×C61C2×#10312yes—
235Dic61—24no—
236Dic61—42no—
237Dic61—42no—
238Dic61—82no—
239#3501—611no—
240#2972—611no—
241#2349—611no—
242#1520—614no—
243#1518—611no—
244#1518—611no—
245#1503—611no—
246#1503D6111yescharacteristic, complemented
247C2×#328—614no—
248C2×C61—21no—
249C2×C61—21no—
250C2×C61#288311yescharacteristic
251C2×C61C22×#10311yescharacteristic
252C2×C61#290111yescharacteristic
253D61—28no—
254#554—612no—
255#554—612no—
256#554—1222no—
257#554—1222no—
258D4×Q8—612no—
259D4×Q8—612no—
260#544—1221no—
261#543—1222no—
262#533—611no—
263#533C2×D6111yescharacteristic
264C2×C22⋊Q8—614no—
265#518—1221no—
266C2×C4×D4—611no—
267C2×C4×D4—1221no—
268C8×D4—1222no—
269C8×D4—1222no—
270#431—611no—
271#431C2×D6111yescharacteristic
272C2×C22⋊C8—614no—
273#389—1221no—
274#385—1222no—
275#382—611no—
276#380—612no—
277C4×C4⋊C4C2×D6111yescharacteristic
278C2×#95—612no—
279#328—614no—
280#328—614no—
281#328—612no—
282#328—612no—
283#328C2×D6112yes—
284#328C2×D6112yes—
285C61#1761411yescharacteristic, complemented, Sylow
286C22×Q8—614no—
287C22×D4—611no—
288C23×C4—612no—
289C23×C4—1222no—
290C22×C8—1224no—
291C4⋊Q8—611no—
292C4⋊Q8—611no—
293C4⋊Q8—1222no—
294C4⋊Q8—1222no—
295C4⋊Q8C4×D6111yescharacteristic, complemented
296C4⋊Q8C4×D6111yescharacteristic, complemented
297#125—21no—
298#125—1222no—
299#125—1222no—
300#125—1221no—
301C22⋊Q8—21no—
302C22⋊Q8—614no—
303C22⋊Q8—612no—
304C22⋊Q8—614no—
305C22⋊Q8—1224no—
306C22⋊Q8—1224no—
307C22⋊Q8—1224no—
308C22⋊Q8—1224no—
309C22⋊Q8—1224no—
310C22⋊Q8—1224no—
311C22⋊Q8—1221no—
312C22⋊Q8C22×D6112yes—
313C4×Q8—21no—
314C4×Q8—612no—
315C4×Q8—1221no—
316C4×Q8C4×D6112yescomplemented
317C4×D4—612no—
318C4×D4—612no—
319C4×D4—612no—
320C4×D4—612no—
321C4×D4—1222no—
322C4×D4—1222no—
323C4×D4—1222no—
324C4×D4—1222no—
325C4×D4—2444no—
326#117—21no—
327#117—1221no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—611no—
330C2×C4⋊C4—614no—
331C2×C4⋊C4—1224no—
332C2×C4⋊C4—1222no—
333C2×C4⋊C4—1221no—
334C2×C22⋊C4—612no—
335C2×C22⋊C4—1224no—
336C2×C22⋊C4—1222no—
337C2×C42—21no—
338C2×C42C22×D6111yescharacteristic
339C4⋊C8—21no—
340C4⋊C8—1221no—
341C22⋊C8—612no—
342C22⋊C8—1224no—
343C22⋊C8C22×D6112yes—
344C4×C8—21no—
345C4×C8—1221no—
346#95—612no—
347#95—1222no—
348#95—1221no—
349#95C22×D6111yescharacteristic
350#95C22×D6111yescharacteristic
351C24—612no—
352C2×Q8—21no—
353C2×Q8—612no—
354C2×Q8—1224no—
355C2×Q8—1221no—
356C2×Q8—4884no—
357C2×Q8C2×C4×D6112yes—
358C2×D4—612no—
359C2×D4—612no—
360C2×D4—614no—
361C2×D4—614no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—612no—
365C22×C4—614no—
366C22×C4—614no—
367C22×C4—1224no—
368C22×C4—1224no—
369C22×C4—1222no—
370C22×C4—4882no—
371C22×C4—4882no—
372C22×C4—2444no—
373C22×C4—2444no—
374C22×C4D4×D6111yescharacteristic
375C22×C4D4×D6111yescharacteristic
376C22×C4C23×D6111yescharacteristic
377C2×C8—22no—
378C2×C8—1222no—
379C2×C8—4884no—
380C4⋊C4—22no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—22no—
385C4⋊C4—42no—
386C4⋊C4—612no—
387C4⋊C4—612no—
388C4⋊C4—612no—
389C4⋊C4—1224no—
390C4⋊C4—1224no—
391C4⋊C4—1224no—
392C4⋊C4—1222no—
393C4⋊C4—1222no—
394C4⋊C4—1221no—
395C4⋊C4—1221no—
396C4⋊C4—1221no—
397C4⋊C4—4882no—
398C4⋊C4—2442no—
399C4⋊C4C2×C4×D6112yes—
400C22⋊C4—22no—
401C22⋊C4—614no—
402C22⋊C4—614no—
403C22⋊C4—1224no—
404C22⋊C4—1224no—
405C22⋊C4—1222no—
406C22⋊C4—2444no—
407C42—22no—
408C42—42no—
409C42—1222no—
410C42C2×C4×D6111yescharacteristic
411C42C2×C4×D6111yescharacteristic
412C42D4×D6111yescharacteristic
413C42D4×D6111yescharacteristic
414C23—614no—
415C23—614no—
416C23—614no—
417C23—2442no—
418C23—2442no—
419C23C2×D4×D6111yescharacteristic
420Q8—42no—
421Q8—81no—
422Q8—4881no—
423Q8—4884no—
424Q8—2442no—
425D4—618no—
426D4—618no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—614no—
438C2×C4—1222no—
439C2×C4—1222no—
440C2×C4—4888no—
441C2×C4—4884no—
442C2×C4—4884no—
443C2×C4—2444no—
444C2×C4—2444no—
445C2×C4—2442no—
446C2×C4—2442no—
447C2×C4—2442no—
448C2×C4—2442no—
449C2×C4C22⋊C4×D6111yescharacteristic
450C2×C4C22⋊C4×D6111yescharacteristic
451C2×C4C22⋊C4×D6111yescharacteristic
452C2×C4C22⋊C4×D6111yescharacteristic
453C2×C4SD16×D6112yes—
454C2×C4Q16×D6112yes—
455C2×C4C22×C4×D6111yescharacteristic
456C2×C4C2×D4×D6111yescharacteristic
457C8—42no—
458C8—2442no—
459C22—21no—
460C22—21no—
461C22—618no—
462C22—1224no—
463C22—1224no—
464C22—2444no—
465C22—2444no—
466C22C2×C22⋊C4×D6111yescharacteristic
467C22C2×SD16×D6111yescharacteristic
468C22C2×Q16×D6111yescharacteristic
469C4—21no—
470C4—21no—
471C4—21no—
472C4—21no—
473C4—42no—
474C4—42no—
475C4—42no—
476C4—81no—
477C4—1224no—
478C4—4882no—
479C4—2442no—
480C4—2442no—
481C4#103×D6112yes—
482C4#103×D6112yes—
483C2—21no—
484C2—21no—
485C2—1228no—
486C2#2145427811yescharacteristic
487C2C2×#103×D6111yescharacteristic
488C2#2145436311yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1503C2
socleC22×C61C22×C22⋊C4
prime radicalC22×D244D4
cosocleC22×C4×C61C24
hypercenter#1503D61
hypocenterC61#17614
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1503

Complements

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

NH ≅ G/Ntype
4. #21444389C2semidirect
5. #21444384C2semidirect
6. #21444383C2semidirect
7. #21444317C2semidirect
8. #328×D61C2semidirect
9. C61×#1503C2semidirect
10. #16840473C4semidirect
12. #16840472C4semidirect
14. Q8×D244C4semidirect
15. #16840427C4semidirect
41. C2×Q8×D61C2×C4semidirect
47. #14519777C2×C4semidirect
48. #14519777C2×C4semidirect
54. #14519757C2×C4semidirect
62. #14519753C2×C4semidirect
64. C4×Dic122C2×C4semidirect
79. C4⋊C4×D61C2×C4semidirect
90. C4⋊Q8×C61C2×C4semidirect
91. C4⋊Q8×C61C2×C4semidirect
96. C4×Q8×C61C2×C4semidirect
246. #1503D61semidirect
285. C61#17614semidirect
295. C4⋊Q8C4×D61semidirect
296. C4⋊Q8C4×D61semidirect
316. C4×Q8C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214596202
#214548422
#214487302
#214443892
#214443842
#214443832
#214443172
#328×D612
C61×#15032
#1761461